redshift one turns a distribution ofpart galaxies of into this a review smo othed fo cuses densitygalaxies on eld in redshift are surveys discussed The Then practical follows issues a of description how of redshift ma jor surveys are carried out Abstract and how We Parameter The results non Gaussian mo dels Comparison ofhypothesis the that p eculiar the initial density eld had a Gaussian distribution although one cannot rule out broad classes of matter although op en universe mo dels are by no means ruled analyses existing techniques After discussing thedata various The biases second which half plague of p eculiar this velocity review work concentrates we survey on quantitative p eculiar velocity studies b eginning with a thorough review of variants and p eculiar velocity surveys The data presented rule out the standard Cold Dark Matter mo del although several Preprint submitted to Elsevier We review the quantitative science that can b e and has b een done with redshift and p eculiar velocity surveys of then
astro-ph/9502079 16 Feb 95 Observatories of the surveys of done discuss the nearby universe After a brief Cold School of Natural The with that in Dark some p eculiar density have Matter with here detail Carnegie Institute of b een velocity are consistent Sciences Institute for Advanced Study Science the and done more various surveys p eculiar and p ower with the quantitative Michael Je rey alone background setting the cosmological context for this work the rst Washington lo cal a at universe on velocity and density elds constrains the Cosmological Density velocity large and cosmography and A A nally cosmological scales Willick Strauss out elds with mild with fare Santa Barbara Street Pasadena California out b etter the tests Princeton New Jersey to of combination biasing that can nearby All the km s of the galaxies data b e of galaxies carried that data are they consistent from February out relative have b oth with mapp ed with redshift redshift to dark the
Contents
Introduction
Theoretical Background
The Big Bang Mo del and its Parameters
The Gravitational Instability Paradigm
Power Sp ectra Initial Conditions and Dark Matter
The Relation Between the Mass and Galaxy Density Fields
Outstanding Questions
Redshift Surveys Setting the Quantitative Groundwork
The Variety of Redshift Surveys
History of Redshift Surveys
The Measurement of Galaxy Redshifts
Determination of the Luminosity and Selection Functions
Luminosity Functions Scienti c Results
Testing the Hubble Law with Redshift Surveys
The Smo othed Density Field
Filling in the Galactic Plane
Redshift Surveys A Cosmographical Tour
Redshift Surveys Galaxy Clustering
The Two Point Correlation Function
Distortions in the Clustering Statistics
The Power Sp ectrum
Higher Order Statistics
The Density Distribution Function and Counts in Cells
Topology and Related Issues
The Dip ole
Spherical Harmonics
Recovering the Real Space Density Field
Clustering of Di erent Types of Galaxies
Peculiar Velocity Fields Techniques of Measurement and Analysis
Galaxian Distance Indicator Relations
Universality of the Distance Indicator Relations
Beyond TF and D A Lo ok to the Future
n
Statistical Bias and Metho ds of Peculiar Velocity Analysis
Quantifying Statistical Bias
Statistical Measures of the Velocity Field
A History of Observations of Large Scale Flow
Homogeneous Peculiar Velocity Catalogs
Velocity Correlation Function
The Cosmic Mach Number
Reconstructing the Three Dimensional Velocity Field
Comparing the Density and Velocity Fields
Comparison via the Velocity Field
Comparison via the Density Field
Discussion
The Initial Power Sp ectrum
The Distribution Function of the Initial Fluctuations
The Gravitational Instability Paradigm
The Value of
The Relative Distribution of Galaxies and Mass
Is the Big Bang Mo del Right
The Future
Acknowledgement
References
Introduction
The last two decades have seen a tremendous growth in our awareness of the richness of the structure
of the Universe on scales of galaxies and larger Technical advances in sp ectrographs and detectors in
b oth the visible and radio wavebands have allowed redshifts of large numbers of galaxies to b e measured
e ciently giving us the opp ortunity to map their distribution for the rst time We are in the midst of
a golden age of exploration The redshift surveys of the last fteen years have discovered the existence
of sup erclusters great voids and coherent structures stretching across the entire volume covered by the
surveys But it is not just geography or cosmography as it is called in the astronomical context that
has b een done hidden in the statistics of the galaxy distribution is much imp ortant information ab out
the structure of the Universe on large scales Thus we can deduce much ab out the early history of the
universe and its global structure from careful quantitative analyses of redshift surveys this will b e the
central theme of this review
The Hubble Law states that at distances much less than the Hubble radius the expansion of the
universe causes the recession velocity of a galaxy cz to b e prop ortional to its distance r
cz H r
where H is the Hubble Constant whose value remains uncertain by a factor of two in astronomer s units it
is often written as H h km s Mp c the quantity h parameterizing our ignorance of its value Mp c
stands for megaparsecs the common unit of distance for much extragalactic work Mp c cm
Thus in physicist s units H h sec or H h yr In practice we will rarely b e
troubled by the uncertain value of h in this review b ecause we will measure distances in units of km s
wherein H The observational evidence for the linearity of Eq is reviewed in Peebles and
Lauer Postman cf x b elow
At the low redshifts discussed in the ma jority of this review relativistic e ects are for the most part
unimp ortant and the Hubble law is an excellent approximation However galaxies have motions ab ove and
b eyond their Hubble velocities deviations from the isotropic expansion that holds only in the theoretical
idealization of a p erfectly homogeneous universe Thus Eq is mo di ed to
cz H r r v r v
where r is the unit vector towards the galaxy in question v r is the peculiar velocity at p osition r and v
is the p eculiar velocity of the observer The development over the past fteen years of redshift indep endent
metho ds of measuring distances have allowed the measurement of the p eculiar velocity eld v r As we
will see in detail observations of the p eculiar velocity eld are of great imp ortance it is their study which
o ccupies the second half of this review In particular under the hypothesis that structure formed by the
gravitational growth of small p erturbations on an initially uniform background there is a direct relation
b etween the density and velocity elds which we will exploit to great advantage to put constraints on
cosmological mo dels
Thus the main emphasis in this review will b e on quantitative constraints that can b e placed on
cosmological mo dels from redshift and p eculiar velocity surveys of the lo cal Universe By lo cal we mean
at distances small compared with the horizon distance or equivalently at lo ok back times small relative
to the age of the universe or redshifts small compared to the sp eed of light That is for most of this
review we will not b e concerned ab out the general relativistic generalization of Eq and can work
in the Newtonian limit almost exclusively Another de nition of the lo cal universe is that within which
evolutionary e ects in the galaxy prop erties can b e assumed to b e negligible In practice we will restrict
ourselves to recession velocities b elow km s z
There are a number of ma jor reviews and b o oks that are relevant as background material for this
review The reader is assumed to b e familiar with the basics of Big Bang cosmology as reviewed in
Peebles and Padmanabhan Somewhat more advanced material can b e found in White
Efstathiou and Suto Older texts that also discuss this material include Peebles
Efstathiou Silk Peebles and Weinberg A useful p o cket summary of the
eld is found in Scott et al We will many times skirt the issues of astroparticle physics in the
cosmological context Kolb Turner is an excellent overview of the eld Redshift surveys are
discussed in Geller Huchra and Giovanelli Haynes and p eculiar velocity surveys are
reviewed in Burstein and Dekel The sub ject can also b e followed in the pro ceedings of many
conferences among them Br uck Coyne Longair Kolb et al Madore Tully
Burbidge Hewitt Kormendy Knapp Audouze Pellatan Szalay Rubin
Coyne van den Bergh Pritchet Latham da Costa Chincarini et al and
Bouchet Lachi eze Rey
We stress observational asp ects of large scale structure studies in this review Not all asp ects of the
eld can b e covered In particular we only touch up on observations of the high redshift universe either
in galaxies Spinrad Kron or in background radiation Partridge We fo cus exclusively
on galaxies as prob es of large scale structure and do not discuss the distribution of clusters of galaxies
e g Bahcall See the review of Efstathiou for a more uni ed treatment of the large scale
distribution of galaxies and clusters We also do not review evidence for dark matter on Galactic scales
Faber Gallagher Trimble Binney Tremaine or mo dels for galaxy formation e g
White Finally although p eculiar velocities are a ma jor fo cus of this pap er we do not review the
closely related question of the determination of the Hubble Constant cf Jacoby et al
This review is to o long to b e read in a single sitting but the di erent chapters are largely indep endent
Chapter covers the basics of Big Bang cosmology to the extent needed for the rest of the review and
can b e skipp ed by the reader familiar with this material Chapter gives a history of redshift surveys and
discusses many of the practical issues needing to b e addressed in their quantitative analysis Chapter is a
qualitative tour of the structures we see within km s as revealed to us via redshift surveys Those
who are interested in quantitative results can skip ahead to Chapter which details the various statistical
measures of large scale structure that have b een drawn from redshift surveys
We then shift the fo cus to p eculiar velocity surveys Chapter parallels Chapter and discusses
the practical asp ects of p eculiar velocity measurements We put sp ecial emphasis on the various biases
which plague p eculiar velocity work and the metho ds necessary to correct them Quantitative analyses of
p eculiar velocity surveys are discussed in Chapter and analyses of redshift and p eculiar velocity surveys
together are covered in Chapter Chapter is a summary chapter
Those readers who are familiar with much of the background material may thus nd it most useful
to read Chapters and after skimming Chapters and We recommend that students new to the
eld read through the entire review making sure that they understand the basic concepts introduced in
Chapter
Theoretical Background
For the ma jority of this review we will work in the context of the Big Bang mo del In order to set
a common language and notation we brie y review the relevant p oints of Big Bang cosmology to which
we will nd ourselves referring The Big Bang mo del itself is discussed in x in which we de ne the
basic parameters of an homogeneous and isotropic expanding universe and write down the equations for
their evolution x gives the evolution equations for p erturbations in such a homogeneous universe and
x discusses the p ower sp ectrum of these p erturbations and introduces the concept of dark matter
x discusses mo dels for the relative distribution of galaxies and dark matter in the context of the biasing
paradigm x summarizes the outstanding questions we would like to address with the observations
The Big Bang Model and its Parameters
It is an observational fact that all galaxies with the exception of galaxies in the Lo cal Group and
a few galaxies asso ciated with the Virgo Cluster have p ositive redshifts and it is observed that redshifts
are prop ortional to distance x This is interpreted as due to the expansion of the Universe The
Cosmological Principle as formulated originally by Einstein states that on large enough scales to b e
quanti ed b elow the Universe is homogeneous and isotropic this mo del together with the tenets of general
relativity leads to the prediction that we do not live in a static Universe In particular the Cosmological
Principle implies that the covariant line element b etween two p oints is given by
ds c dt a t dl
where a t is the scale factor Our observation that the universe is expanding means that a t is an
increasing function of time The comoving distance l b etween any two p oints taking part in the Hubble
expansion is a constant the proper distance b etween two p oints is the quantity s at a given constant time
t i e such that dt Comparing with Eq we see immediately that
a ds
H
s dt a
t t
where the subscript refers to values of quantities at the present time t Indeed the quantity a a is not
constant with time and thus neither is the Hubble Constant however at any given time it is indep endent
of p osition and direction
Consider two observers separated by a prop er distance l small compared with the distance to the
a
horizon By Hubble s law they are moving apart from one another at a sp eed given by v l Now
a
consider a plane wave of radiation with de nite wavelength as measured by the rst of the observers
traveling towards the second The second observer will observe this radiation to b e shifted by the rst order
Doppler shift b ecause v c to a wavelength
a
l c v c
a
The introduction of a cosmological constant do es allow solutions indicating a static Universe but these solutions
are unstable
Thus recognizing that the prop er time required for the radiation to travel the distance l is t l c
and taking the limit l we nd
a
a
or
t a t
The redshift of a galaxy z is de ned as
t
z
where is the wavelength of a plane wave emitted by the galaxy at the time of emission the rest
wavelength and t is the wavelength of the plane wave at the present the observed wavelength Thus
the redshift and the scale factor are directly linked
z a a t
At low redshifts the recession velocity of a galaxy is simply given by cz At high redshifts this expression
clearly breaks down and one must go to general relativistic generalizations of it
Under the assumption that the universe is homogeneous and isotropic the line element can b e
expressed in spherical co ordinates
dr
r d sin d ds c dt a t
k r
This is called the Friedman Robertson Walker metric after the men who rst wrote it down The quantity
k is the curvature constant an integration constant of Einstein s equations Thus k corresp onds to
a Euclidean geometry for the three spatial dimensions this is called a at universe
Dynamical equations for the scale factor a follow from the metric together with the equations of
general relativity and the ideal uid approximation One nds e g Chapter of Weinberg two
equations
a p
G
a c
and
k c a
G H
a a
In these equations G is Newton s gravitational constant is the non relativistic mass density of the
universe p is the pressure H is the Hubble Constant compare with Eq where the lack of a subscripted
indicates that it is a function of time and is the Cosmological Constant
The curvature constant is sometimes written in the form b ecause in a closed universe a R can b e
R
interpreted as the radius of the universe cf Eq of Peebles
These equations may also b e derived heuristically following a purely Newtonian argument cf Chapter of
Peebles
In the present universe non relativistic matter completely dominates the energy density of the uni
verse making the pressure term negligible we drop it from now on In this zero pressure limit mass
conservation implies a t The wavelength of any relativistic sp ecies will b e redshifted with the
expansion of the universe to the extent that the number of particles of this sp ecies is preserved the energy
density in relativistic sp ecies drops as a t The Cosmic Microwave Background CMB isotropic radi
ation with a pure black bo dy sp ectrum and a temp erature of T K represents such a relativistic
sp ecies The energy density in the CMB to day is negligible relative to the density in non relativistic matter
but b ecause of its faster fall o with a it dominated the energy density of the universe for z h
The CMB will not b e reviewed thoroughly here although we will nd ourselves referring to it often See
Efstathiou Peebles and Partridge for recent reviews
Setting the Cosmological Constant to zero for the moment and dropping the pressure term Eq
can b e written
a t G k c
a t
For k negative or zero the right hand side and therefore a t stays p ositive de nite We know that a t
is p ositive now the universe is observed to b e expanding not contracting and thus a t will continue to
increase for all time This is what we call the op en or at universe However if k is p ositive there will b e
some time in the future at which the second term on the right hand side of Eq balances the rst term
and thus a t returns to zero As a is negative de nite Eq this means that a t changes sign and
a t b egins to decrease eventually reaching zero This is a closed universe The controlling factor deciding
the fate of the universe is the density the division b etween the op en and closed universe happ ens at the
critical density whose current value is
H
crit
G
Thus we de ne the Cosmological Density Parameter as
crit
whose value is less than unity for an op en universe greater than unity for a closed universe and exactly
zero for a at universe If we re introduce the Cosmological constant we get a at universe k in the
case that where
H
is the contribution to the cosmological density due to vacuum energy Of course in this case the concepts
of op en and closed universes b ecome more complicated b ecause a can change sign See Harrison
for a complete inventory of cosmological mo dels
We can use this line of reasoning to p eer into the past as well As a is negative de nite for
as we will assume and a is p ositive de nite then at some nite time t in the past a t This is the
origin of the concept of the Big Bang the dynamical equations for a t indicate that the universe was of
in nitesimal extent some nite time in the past Let us de ne that time t We can now ask for the age
of the universe Weinberg
i h
H cos
t
H
h i
cosh H
As we could have guessed from the units the present age of the universe is prop ortional to H with a
numerical co e cient which is a decreasing function of The corresp onding equations for universes
are non analytic and will not b e presented here
We need one further de nition the acceleration parameter de ned at the present time
aa
q
a
where the second equality follows directly from Equations and The acceleration parameter is a mea
sure of relativistic e ects in the relation b etween observables thus for example the rst order corrections
to Eq are prop ortional to q
This completes our survey of the global structure of space time in a homogeneous and isotropic
universe The cosmology is de ned by the following numbers
the cosmological density parameter
H the Hubble constant
the cosmological constant
q the acceleration parameter and
t the present age of the universe
These parameters are not indep endent of one another in the context of the Big Bang mo del as we have seen
but each are measured or constrained by a variety of di erent observations It is one of the imp ortant tests
of Big Bang cosmology that one nd consistency b etween the values found for the di erent parameters
We will put the greatest emphasis in this review on measurements of b ecause this is the quantity that
can b est b e constrained with analyses of redshift and p eculiar velocity surveys
Standard Big Bang cosmology has built into it a number of apparent paradoxes The rst is often
referred to as the horizon problem the size of causally connected regions at the time of recombination
subtends only a few degrees on the sky leaving the large scale isotropy observed in the CMB unexplained
this must b e taken as an arbitrary initial condition The second is the so called atness problem
is an unstable solution to the evolution equations in the sense that the fact that we observe to b e
within an order of magnitude of unity to day requires that it b e very nely tuned in the early universe
There is a class of cosmological mo dels under the general rubric of the in ationary paradigm which
addresses these problems and predicts tight constraints on the parameters of the Big Bang mo del In a
tion is reviewed thoroughly in Chapter of Kolb Turner In its simplest form the in ationary
mo del p osits a p erio d in the very early universe when a sup er co oled phase transition caused the vacuum
energy density to b ecome dominant Eq then implies that the scale factor grows exp onentially If this
exp onential expansion continues long enough for the curvature term to b ecome exp onentially small the
atness problem is solved the result is a universe that is globally at i e k or Moreover
as the observable universe in ated from a region that was causally connected and in thermal contact the
homogeneity and isotropy of the universe the horizon problem are explained The in ationary mo del also
Some authors refer to q as the deacceleration parameter
predicts a scale invariant sp ectrum of adiabatic density uctuations a sub ject to which we turn in x
after a discussion of gravitational instability theory
The in ationary prediction of a at universe is apparently in direct contradiction with measurements
of the total mass density of the universe at least until recently Measurements of the luminosity density of
the universe cf x imply that the mass density of the universe in stars is only x There
is abundant evidence for non luminous matter asso ciated with galaxies and clusters of galaxies whose total
contribution is as much as cf Faber Gallagher Kormendy Knapp Trimble
The in ationary prediction of in a universe without a cosmological constant requires that there b e
additional dark matter distributed on scales larger than that of clusters indeed as we shall see in detail
later in this review there is recent evidence from large scale ows that is larger than that inferred from
dynamics within clusters The material which makes up the dark matter however remains completely
unknown The physical prop erties of the dark matter greatly in uences the distribution of matter on large
scales and thus redshift surveys of galaxies have the p otential to constrain the form of dark matter In
order to explore these issues we need a theory for the growth of structure in the presence of gravity
The Gravitational Instability Paradigm
The Big Bang mo del as outlined in the previous section explicitly assumes a homogeneous and
isotropic universe We b elieve that the Cosmological Principle do es in fact hold on the largest scales for
reasons that will b e discussed in x and that at early times the universe was very close to homogeneous
However we observe structure all around us from planets to galaxies to sup erclusters of galaxies matter
tends to aggregate and form structures rather than distribute itself uniformly One of the great questions
facing cosmology is how this structure came to b e The widely accepted view is that small density uctu
ations present in the b eginning grew by gravitational instability into the structures that we see to day In
this section we brie y review the theory of gravitational instability in an expanding universe
One starts by writing down the equations of mass continuity force and gravitation in an expanding
universe in prop er co ordinates ignoring relativistic e ects
r v
t
v
v r v r
t
r G
Here is the mass density eld v is the velocity eld is the gravitational p otential and we have
dropp ed terms dep ending on pressure which we assume are negligible All spatial derivatives are with
resp ect to prop er distance If we expand these equations to rst order in all quantities measuring departure
from uniformity convert to comoving co ordinates and subtract the zeroth order solutions the rst two
equations simplify to
r v
t a
v a
v r
t a a
There do exist variants on the standard in ationary mo del which predict other forms for the density uctuation
sp ectrum and which predict noticeable curvature see the references at the end of x of Kolb Turner
The subtraction of the zeroth order solution involves some subtleties having to do with the gravitational p otential
of a uniform universe these are discussed further in x of Shu
where is the dimensionless density contrast
r
r
and is the mean mass density Taking the time derivative of the continuity equation and substituting
into the divergence of the force equation yields with the Poisson equation Eq
a
G
t a t
Because Eq is a second order partial di erential equation in time alone we can separate the
spatial and temp oral dep endences and write
A x D t B x D t
where D and D are growing and decaying mo des resp ectively We will b e using the growing mo de
throughout this review it is given by
a
Z
a
D t a da
a
In the general case of a non vanishing cosmological constant this integral is not analytic For
mo dels however analytic expressions for D t and D t can b e obtained x of Peebles A
particularly simple solution exists in the sp ecial case of a at universe in which a t t Eq and
a
so we nd the right hand side of Eq is simply
a
t t t t
which has an analytic solution in p ower laws of t
x t A x t B x t
More generally the solutions dep end on in particular the growth is faster for increasing For
a
in Eq dominates the gravitational the expansion of the universe the drag term represented by
a t
attraction of the matter and the clustering freezes out i e it stops growing at z A full
discussion of these solutions including the e ects of pressure which is imp ortant in the early universe will
take us to o far a eld cf Chapter of Kolb Turner for more details
If we wait until late times the growing mo de of Eq will dominate and we can rewrite Eq
as
D
r v a aH f
D
where
d ln D dD dD da
f
H D dt H D da dt d ln a
Pressureless gravitational growth in a non expanding universe is quite di erent the drag term is not present
and p erturbations grow exp onentially
We are still in comoving co ordinates as indicated by the a on the right hand side of Eq The expression
for D and therefore for f is a function of and A go o d approximation in the general case is given
by Lahav et al
f
other approximations can b e found in Peebles Reg os Geller Lightman Schechter
Martel and Carroll Press Turner Thus the in uence of on dynamics at low redshift
is minimal Lahav et al Eq can b e inverted via the metho ds of electrostatics in the usual way
to yield after returning to prop er co ordinates
Z
H f r r r
v r d r
jr rj
Eq reveals the physical content of this rst order expansion linear p erturbation theory states that
p eculiar velocities are prop ortional to gravitational acceleration We will nd ourselves using Eqs and
throughout this pap er These equations can b e generalized using higher order p erturbation theory
we will introduce these results when needed
If we measure the quantity r in units of km s then H and we see that a comparison of the
velocity eld v r and density eld r gives a direct measure of f One of the central themes of
this review will b e exploiting this comparison to put constraints on f
Linear theory makes the assumption that the change in comoving p osition of galaxies as the universe
expands is negligible Zel dovich made an imp ortant extension of linear theory by assuming that
the di erence b etween the Lagrangian p osition q and Eulerian p osition x of a particle in a gravitating
system is separable in space and time
x q q D t q
where D t is the growing mo de in linear theory Eq and which determines the amplitude of the
velocity eld is prop ortional to the gradient of the gravitational p otential See Shandarin Zel dovich
for a review of the full rami cations of this deceptively simple equation This approximation is used
as a basis for several of the non linear schemes describ ed in x
Power Spectra Initial Conditions and Dark Matter
We have seen that initially small p erturbations grow by gravitational instability It remains to char
acterize the distribution with scale of those initial p erturbations Let us de ne the Fourier Transform k
of the fractional density eld r at some early time t such that
Z
ik r
e
d k k r
Care needs to b e taken when comparing the results of di erent authors as there is inconsistency in the
literature ab out where the factors of go in the de nition of the Fourier Transform Because of the
One has the freedom to add an arbitrary divergence free term to the right hand side of Eq This term
corresp onds to the decaying solution and thus will b e negligible at the present
isotropy assumed in the Cosmological Principle the statistical prop erties of k are indep endent of the
direction of k and so it makes sense to de ne a power spectrum P k
D E
D
k k P k k k
D
where is a Dirac delta function and the averaging on the left hand side is over directions of k As
Bertschinger makes clear P k is a p ower spectral density and thus represents the p ower p er unit
E D
but this is incorrect volume in k space One often sees the p ower sp ectrum de ned as P k j k j
at least for the Fourier Transform convention we ve adopted as can b e seen by comparing the units of
the two expressions the p ower sp ectrum has units of volume as do es
The quantity k is complex and thus P k is a complete statistical description of the density
eld only if the phases of k are random This is a natural prediction of in ationary mo dels and is
often called the random phase hypothesis By the Central Limit Theorem and Eq random phases
imply that the one p oint distribution function of r is Gaussian so the random phase hypothesis is
often also referred to as the Gaussian hypothesis Random phases can strictly hold only in the limit of
very small p erturbations r cannot b e smaller than Eq but has no upp er b ound and therefore
it develops a p ositive skewness ab out which we will have much more to say in x as p erturbations
grow by gravitational instability Until the lower b ound on b ecomes imp ortant linear theory holds and
b ecause gravitational growth of p erturbations is indep endent of scale in linear theory the shape of the
p ower sp ectrum is indep endent of time
One way to quantify the density uctuation eld is in terms of the mass uctuations within a spherical
window of radius R
Z
M
d rW jr xj r
M
R x
Z
0
ir k k
e
k k d r d k d k W jr xj
x
Z
f
d k P k W k R
f
where W R is the window function used W k R is its Fourier Transform For a tophat window function
which is unity out to some radius R and then drops to zero
j x
f
W x
x
where j x sin x x cos x x is the rst spherical Bessel function This is very crudely a step function
to k R This implies that characteristic mass uctuations on a scale k R are given roughly by
M M k P k
It is interesting to compare the mass uctuations within a sphere of radius R with the corresp onding
uctuations in the bulk ow velocity within the same sphere Using Eq and following a derivation
That is the multi variate distribution function of k i N is a multi variate Gaussian with covariance
i
matrix given by Eq
There are classes of mo dels whose initial conditions are explicitly non Gaussian among them are mo dels with
initial seeds of gravitational growth primordial black holes cosmic string lo ops and wakes monop oles textures
and with initial explosions We will have little to say ab out these mo dels in this review
very similar to that in Eq one nds
Z
D E
H f
f
dk P k W k R v
R
which di ers from Eq with two fewer p owers of k in the integrand This means that the p eculiar
velocity eld on a given scale is sensitive to comp onents of the p ower sp ectrum on larger scales than is
the density eld and thus is a useful prob e of the largest scale p ower We illustrate this in Fig which
compares the cumulative contribution to the integrals in Eqs and for wavenumbers smaller than
k This calculation used a top hat window of radius h Mp c and assumed a standard Cold Dark
Matter mo del see b elow The rms velocity includes contributions from much larger scales than do the
rms mass uctuations
The simplest in ationary mo dels predict a so called scale invariant p ower sp ectrum given by
P k k
so called b ecause the p otential uctuations on a scale at the time that is equal to the horizon size are
indep endent of for this sp ectrum cf Kolb Turner This form was in fact predicted well b efore
in ation had b een suggested Peebles Yu Harrison Zel dovich as it is unique among
p ower law p ower sp ectra in avoiding divergences at large and small k in various observable quantities
The pure p ower law p ower sp ectrum thought to exist in the early universe is sometimes called the
primordial p ower sp ectrum However the p ower sp ectrum that is directly observable is that which holds
following the ep o ch at which the energy densities in relativistic and non relativistic particles are equal The
equations of gravitational instability we worked out ab ove hold only for pressureless i e non relativistic
matter in the relativistic case the pressure of the matter is such as to retard the growth of p erturbations
on all scales within the radius of the universe Thus p erturbations on scales smaller than the horizon
the scale over which the universe is causally connected ct can only grow after non relativistic matter
b ecomes dominant and this is thus a convenient time at which to characterize the p ower sp ectrum
While p erturbations cannot grow on scales smaller than the horizon during the radiation dominated
era they can and do grow on scales larger than the horizon After the ep o ch of matter radiation equality
p erturbations on sub horizon scales start to grow although at a di erent rate The combination of these
two e ects causes a b ending in an originally pure p ower law p ower sp ectrum on the scale of the horizon size
at matter radiation equality The index of the p ower law decreases by four over this b end Thus following
matter radiation equality the p ower sp ectrum which is prop ortional to k on large scales b ends over to
b ecome k on small scales One quanti es this in terms of a transfer function whose square is the quantity
by which the primordial p ower sp ectrum is multiplied to generate the nal p ower sp ectrum
The p ower sp ectrum we have describ ed is generic for universes in which the dark matter is non
relativistic or cold when a galaxy mass is contained within the horizon its form dep ends principally on
the horizon size at the ep o ch of matter radiation equality which scales as h The so called Standard
Cold Dark Matter mo del CDM go es further and invokes in ation to give and Eq for the
primordial p ower sp ectrum the Hubble Constant is set to H km s Mp c in order to get an age
of the universe in concordance with the ages of the oldest globular clusters x Although the CDM
mo del has b een very p opular ever since it was rst prop osed some of the early imp ortant pap ers include
Peebles Blumenthal et al and Davis et al due to its naturalness lack of free parameters
This is true for the dark matter which do es not couple with the radiation However baryonic matter is still
ionized at the ep o ch of matter radiation equality and thus is tightly coupled to the photons Therefore uctuations
in the baryonic comp onent cannot grow until recombination These considerations are of great imp ortance in
quantifying uctuations in the CMB on small scales
Fig The solid curve shows the square ro ot of the contribution to the mean squared bulk ow within a tophat
of radius h Mp c as a function of the lower cuto in k for a standard CDM mo del The dashed line is the
square ro ot of the contribution to the mean squared mass uctuations right hand scale Note that the velocity
eld includes much more contribution from small k large scales
and successes in predicting many of the observed prop erties of galaxies there exists no comp elling mo del
invoking particles or physical pro cesses known to exist that predicts any form of CDM Particle physicists
have come up with a number of hypothetical particles which could constitute the CDM axions and the
lightest sup ersymmetric partner to the photon among the most p opular at the moment cf Primack Seckel
Sadoulet Kolb Turner
The p ower sp ectrum of the Cold Dark Matter mo del has b een calculated numerically by a number of
workers Peebles Bond Efstathiou Efstathiou Bond White and t to various
functional forms Efstathiou et al give
n
Ak
P k
k k k
where A is a normalizing constant k is measured in units of h Mp c h and n is the p ower law index
of the primordial p ower sp ectrum The quantity h is the Hubble constant in units of km s Mp c
Thus Standard CDM sets mo dels with smaller have more p ower on large scales The quantity
is inversely prop ortional to the Hubble radius at matter radiation equality and thus sets the scale at
which the p ower sp ectrum b ends over This equation and the equality b etween and h holds only in
a universe with a negligible contribution of baryons to The normalization for the case of n and
standard CDM can b e written Bunn Scott White White Scott Silk
h Qi h Qi
A h Mp c
T T
where is the conformal time at the present h Qi is the measured quadrup ole anisotropy of the CMB
background and T is its measured temp erature
There is another physical e ect that can come in We have assumed that the dark matter is cold at the
ep o ch when a galaxy mass is contained within the horizon However imagine that the dark matter is made
up of neutrinos with a mass around eV already ruled out for but well within exp erimental b ounds
e
for and It can b e shown from considerations of thermo dynamic equilibrium in the early universe
that the number density of neutrinos in this case would b e enough to close the universe i e
Moreover neutrino dark matter would b e relativistic at a time when a galaxy mass was enclosed within
the horizon and thus in their free streaming would wash out any p erturbations on scales comparable to
the horizon scale at that ep o ch The consequence is a p ower sp ectrum that cuts o exp onentially on small
scales The evolution of large scale structure in such a Hot Dark Matter mo del or HDM is very di erent
from that in CDM Because there is no p ower on the scale of galaxies in the HDM mo del larger scale
structures collapse rst in pancakes that then fragment later to form galaxies This is often referred to as
the top down scenario In CDM on the other hand there is p ower on galaxy scales and galaxies form rst
thereafter clustering into larger scales Indeed collapse is simultaneous for all scales for which P k k
Eq This is referred to as hierarchical clustering or the bottom up scenario
The mass p erturbations predicted by in ation arise from p erturbations in the gravitational p otential
and thus relativistic and non relativistic comp onents uctuate in tandem These are called adiabatic uc
tuations in contrast with isocurvature uctuations in which the two uctuate in opp osite senses to give
no net uctuations in the mass density The Primordial Baryon Isocurvature mo dels PBI rst suggested
by Peebles cf Bardeen Bond Efstathiou invoke neither a hypothetical new particle as
in CDM nor a hypothetical mass for a particle now b elieved to b e massless as the neutrino in HDM
Rather baryons dominate the mass density of the universe The PBI mo dels characteristically show a
strong p eak in the p ower sp ectrum at the Jeans length at the ep o ch of matter radiation equality
The p ower sp ectra of a variety of mo dels are shown in Fig based on a similar gure in Strauss et
al Shown are Standard CDM HDM and two variants of the PBI mo del note the bump on large
scales in the PBI mo dels Also shown are two variants of Standard CDM one with LCDM this
increases the ep o ch of matter radiation equality and therefore the wavelength of the turnover in the p ower
sp ectrum and one which invokes somewhat non standard in ationary mo dels to assume a primordial
p ower sp ectrum given by P k k Tilted CDM or TCDM Finally we show a hybrid mo del with a
roughly mixture of CDM and HDM C HDM The parameters for these mo dels are given in Table
We will discuss these mo dels in detail in x
All of these mo dels are normalized to the anisotropies in the CMB on a scale of as measured
by the Cosmic Background Explorer COBE Smo ot et al Wright et al Efstathiou Bond
White based on the assumption that the observed anisotropies are due to p otential uctuations at
the surface of last scattering the Sachs Wolfe e ect Note that in ationary mo dels or for that
matter other comp eting mo dels for the generation of uctuations in the early universe are not sp eci c 1 2 3 4 5 6
7
Fig The p ower sp ectra in seven mo dels of large scale structure Standard CDM Tilted CDM Tilted CDM
CDM LCDM HDM Mixed Dark Matter C HDM and two variants of the PBI mo del All are
normalized to the COBE observations
enough to predict the amplitude of the uctuations we simply do not know the relevant particle physics
well enough Thus COBE provides the only unambiguous way to normalize p ower sp ectra Indeed in
the pre COBE days the normalization of p ower sp ectra was a free parameter which was only weakly
constrained by galaxy clustering Eq A thorough discussion can b e found in Efstathiou and
Peebles
It has b een p ointed out e g Davis et al that in in ationary mo dels that predict a deviation from the
Harrison Zel dovich p ower sp ectrum there is substantial generation of gravity waves and thus the Sachs Wolfe
e ect do es not account for all the anisotropies seen by COBE
Table The Parameters of the Power Sp ectra
c d e f b a
n or m x H Mo del
Standard CDM
HDM
PBI
PBI
LCDM
TCDM
g
C HDM
a
All mo dels are spatially at so that
b
rms fraction density uctuations within h Mp c spheres
c
Hubble Constant in units of km s Mp c
d
Adiabatic primordial sp ectral index
e
Iso curvature primordial sp ectral index
f
Ionization fraction
g
CDM HDM
The Correlation Function
The autocorrelation function or more commonly the correlation function of the mass density eld
is one of the most p owerful metho ds of quantifying galaxy clustering It is de ned by
r h x x r i
where the average is over p osition x and isotropy guarantees that r is indep endent of the direction of
r Eq is only applicable for continuous density elds for a p oint pro cess such as the distribution
of galaxies we de ne r op erationally as the mean excess number of galaxy pairs at separation r over
that exp ected for a pure Poisson distribution That is the mean number of galaxies in a spherical shell a
distance r from a given galaxy of thickness dr is given by
N r r dr n r
where n is the mean number density of galaxies If we expand the s in Eq in inverse Fourier
Transforms we nd
Z
0
ik r ix k k
e
d k d k e k k r
Z
ik r
e
d k P k
Thus the correlation function and the p ower sp ectrum form a Fourier Transform pair We discuss the
application of r in much greater detail in x Just as for the p ower sp ectrum the correlation function
gives a complete statistical description of the density eld for a Gaussian eld all higher order correlations
are zero Thus higher order correlations are a measure of non Gaussianity as we discuss in x
The Relation Between the Mass and Galaxy Density Fields
Our discussion thus far has b een quite abstract galaxies have b een mentioned only in passing
However it is galaxies that we observe directly while we b elieve that most of the material of the universe
is made up of dark matter we need to make a connection b etween the galaxies and the mass density eld
The simplest assumption and one that was implicitly made until the last decade is that the distribution of
galaxies is a mirror of the distribution of dark matter This could come ab out for example if each galaxy
were surrounded by a halo of dark matter with total mass in prop ortion to the visible luminosity of a
galaxy with no extra comp onent of dark matter either smo othly distributed or lying in clumps with no
asso ciated visible matter However there is very little evidence that this is true and some indirect evidence
that it is false For example one could imagine that a comp onent of the dark matter is distributed like
that of the galaxy eld smo othed on scales of h Mp c and do es not follow the details of the galaxy
distribution on smaller scales If this were the case one would get a misleading impression of the small
scale distribution of dark matter from studying galaxies A sp eci c mo del for this sort of e ect was rst
suggested by Kaiser in which galaxies form only at the high density p eaks of the mass density eld
The galaxy distribution is then said to b e biased with resp ect to the mass distribution The term biasing
is used to refer to a number of di erent but related e ects
The original argument of Kaiser was concerned not with the biasing of galaxies relative to
dark matter but of clusters of galaxies relative to galaxies clusters of galaxies are likely to b e pro duced
where the initial density contrast smo othed on larger scales is high and therefore is likely to pro duce
a higher than average density of other clusters The result is a stronger correlation function Eq of
clusters than that of galaxies as is indeed observed Bahcall Soneira Bahcall West Thus
this form of bias is known to exist
It was then realized that a similar e ect could cause galaxies to b e biased relative to the underlying
dark matter Bardeen et al Davis et al Peacock Heavens causing the clustering of
the former to b e stronger than that of the latter and decreasing the apparent value of from dynamical
studies The so called peaks biasing mo del originally prop osed by Kaiser makes a de nite prediction
for the relation b etween the correlation function of the galaxies and dark matter at least on large scales
r b r
galaxies dark matter
where b is the biasing parameter which is approximately a constant indep endent of scale related to the
threshold ab ove which galaxies are presumed to have formed In practice we will often assume the much
more sp eci c linear biasing model in which
r b r
galaxies dark matter
This is in fact the intuitive de nition of biasing that most p eople have in which a constant of prop ortionality
relates the dark matter density eld to that of the galaxies Note that in practice the galaxy distribution
is a p oint pro cess not a continuous eld and thus the bias de ned as ab ove must always make reference
to a smo othing length Eq implies Eq through Eq although the converse is certainly not
true Furthermore for b as is usually assumed Eq cannot b e strictly true b ecause b oth
galaxies
and are limited b elow by and thus one o ccasionally sees mo dels like
dark matter
b
r r
galaxies dark matter
which gets around this problem Weinberg cf Coles has shown empirically that in a wide
range of mo dels which invoke local biasing i e in which
r f r
galaxies dark matter
the ratio of the uctuations R in the galaxies and dark matter is indep endent of scale R on scales larger
than h Mp c This of course will not hold in non local biasing mo dels in which galaxy formation is
in uenced by events many Mp c away Babul White Bower et al in these schemes the ratio
of the amplitude of dark matter and galaxy uctuations can b e a strong function of scale
Any given prescription relating to is exp ected to hold only in the mean at any given
galaxies dark matter
p oint in space there will b e uctuations around this mean in the relation b etween the two A complete
mo del of biasing must sp ecify the distribution of these uctuations
To the extent that the linear biasing mo del holds comparisons of p eculiar velocities and gravity via
Eq will not b e able to constrain the quantity f but rather the combination
f
b
Thus our ignorance ab out the relative distribution of galaxies and dark matter translates directly into
ignorance of the value of Note in particular that for b as is usually assumed mo dels can
b e reconciled with observed low values of Thus biasing is often invoked to explain why some dynamical
observations imply a value of substantially b elow the in ationary prediction of unity x
An early review of the biasing mo del and sp eci c physical mo dels which predict biasing can b e
found in Dekel Rees recent work has found biasing of the galaxies relative to the dark matter
in numerical simulations of galaxy formation and the growth of structure in an expanding universe e g
White et al Gelb Bertschinger a Cen Ostriker b Katz Hernquist Weinberg
If galaxies can b e biased with resp ect to the underlying dark matter di erent p opulations of galaxies
can in principle b e biased with resp ect to one another We know that this holds on small scales for elliptical
and spiral galaxies although elliptical galaxies make up roughly of the p opulation of galaxies in the
eld they are by far the dominant p opulation in the cores of rich clusters Hubble Oemler
Dressler ab Dressler Postman Geller Whitmore Gilmore Jones This is re ected
in a steep er correlation function of ellipticals relative to spirals Davis Geller Giovanelli Haynes
Chincarini We refer to this as a form of relative biasing of two galaxy p opulations and will discuss
further observational evidence for this b elow
Finally the term biasing is o ccasionally mis used to refer to the normalization of the p ower sp ec
trum It is observed that the variance of optically selected galaxy number counts is approximately unity
within spheres of radius h Mp c Eq Davis Peebles b and thus the bias can b e de ned as
Z
f
d kP k W k R
b
at R h Mp c with W given by Eq This de nition is consistent with that in Eq only to
the extent that the variance in galaxy counts at R h Mp c is actually unity whereas in fact it varies
from sample to sample Thus we will shun this de nition of b in this pap er
Outstanding Questions
This nishes our overview of the theoretical framework within which we will interpret the results of
redshift and p eculiar velocity surveys As a form of summary we list the issues we would like to address
with observational data
i The Friedman Robertson Walker metric is characterized by a number of parameters H q and
t which although not indep endent in the mo del are constrained observationally by quite di erent
metho ds Redshift and p eculiar velocity surveys have the most p ower to constrain mostly through
Eq and that is what we will concentrate on for this review
ii The gravitational instability mo del is comp elling but needs to b e tested observationally Linear theory
and gravitational instability imply a unique relation b etween the mass and velocity elds Eq
furnishing one such test
iii The p ower sp ectrum following the ep o ch of radiation matter equality is of paramount imp ortance for
determining the evolution of large scale structure in our universe Moreover as we have seen it can
give vital clues as to the form and amount of the dark matter We will attempt to constrain it with
observations
iv The p ower sp ectrum discussed ab ove is a complete statistical description of the mass uctuations in
the early universe only to the extent that the Fourier mo des had random phases We will see that
there do exist direct observational tests of this from the nearby galaxy distribution
v The physical pro cesses by which galaxies formed are very p o orly understo o d Redshift and p eculiar
velocity surveys can put constraints on the relative distribution of galaxies and dark matter this
relative distribution ultimately must b e predicted by galaxy formation mo dels
vi At the end we wish to step back and ask whether the various observational constraints we have found
allow a coherent picture consistent with the theoretical framework That is are our observations
consistent with the Big Bang mo del with structure forming via gravitational instability To what
extent can we rule out comp eting mo dels
At this p oint we are ready to start addressing these questions with observations of the galaxy
distribution and velocity eld at low redshifts
Redshift Surveys Setting the Quantitative Groundwork
In this chapter we discuss the history of redshift surveys and the basic metho ds required to do
quantitative work with them x introduces the main quantities that di erentiate one redshift survey
from another and x summarizes the history of redshift surveys x discusses the metho ds used to
measure redshifts with optical and radio telescop es Flux limited surveys have a number density of ob jects
that drops as a function of distance from the Earth this is quanti ed with the selection function as
discussed in x The selection function is closely related to the luminosity function whose astrophysical
implications are discussed in x and x With the selection function in hand the density eld of
galaxies can b e determined using metho ds discussed in x One of the di culties in determining the
density eld is the lack of redshift surveys at low Galactic latitudes Metho ds to correct for this are the
topic of x
The Variety of Redshift Surveys
In this review we will concentrate on redshift surveys of well de ned samples of galaxies By well
de ned we mean those in which the selection criteria are quanti able and repro duceable at least in a
statistical way for without this it is imp ossible to do quantitative analyses with them In practice this
usually means that a sample is de ned as limited by some photometric prop erty usually received ux
or diameter in some band A sample may also have secondary selection criteria as well such as galaxy
morphology color or surface brightness Unfortunately the data one has available to de ne a sample are
rarely of very high photometric accuracy and thus the limits are always approximate to some extent A
redshift survey sample is thus de ned by several factors
i The region of sky covered
ii The photometric quantities with resp ect to which the sample is de ned and the errors thereof
iii The limits on these quantities
iv The fraction of the galaxies meeting the selection criteria for which redshifts are measured
From these parameters several other characteristics of the survey follow including the total number of
galaxies included the number density of ob jects surveyed i e its sparseness and some measure of a
typical redshift the depth in the survey
Several authors including Geller Huchra Giovanelli Haynes and Borgani
have given lists of ma jor redshift surveys of galaxies Redshift surveys fall into several categories
One dimensional p encil b eam surveys which go deep over a very small region of sky Given the slow
sp ectrographs of twenty years ago these were all that were available in the early days of redshift surveys
Gregory Thompson Kirshner et al Recent p encil b eam surveys push to very faint
magnitudes and therefore high redshifts Broadhurst et al Colless et al esp ecially
with the advent of multi ob ject sp ectrographs These surveys are esp ecially useful for studying galaxy
evolution
Two dimensional surveys in which a long thin strip on the sky is covered The most in uential of these
is the CfA slice survey de Lapparent Geller Huchra to b e discussed b elow
Three dimensional surveys in which a substantial solid angle on the sky is covered This is the ma jor
emphasis of this review
Targeted surveys of sp ecial kinds of ob jects These include surveys of clusters of galaxies active and
unusual galaxies of various kinds and pairs and groups of galaxies Recent cluster redshift surveys
include those of Huchra et al b Zabludo Huchra Geller Huchra et al Guzzo
et al Lauer Postman and Dalton et al see Bahcall and Bahcall West
for recent reviews of large scale structure as traced by galaxy clusters Redshift surveys of active
and unusual galaxies are typically carried out not for studies of large scale structure but rather to learn
ab out the prop erties of the galaxies themselves we will not discuss them further
Blind redshift surveys carried out in the cm line of neutral hydrogen HI These are not p ointed at
galaxies or galaxy candidates as de ned in other passbands but survey the sky over a range of redshifts
to learn ab out the HI content of the universe Kerr Henning Weinberg et al Ho man Lu
Salp eter Szomoru et al a Briggs
A redshift survey requires a prede ned sample of targets and thus is often de ned by catalogs of
galaxies detected in photographic surveys of the sky There have b een heroic e orts to de ne complete
galaxy samples from photographic sky surveys prepared on Schmidt plates but b ecause a telescop e at any
given lo cation on earth can only see a fraction of the sky galaxy samples tend to b e restricted to either the
Northern or Southern celestial hemisphere moreover dust extinction at low Galactic latitudes from our
own Galaxy means that galaxy samples are woefully incomplete at low latitudes the zone of avoidance
The most imp ortant large galaxy catalogs from which redshift survey samples have b een drawn include
The Catalog of Galaxies and Clusters of Galaxies CGCG prepared by Zwicky and collab orators
from blue plates taken at the Palomar Schmidt telescop e It covers the entire Northern Sky
except for regions at very low Galactic latitudes and claims completeness to blue magnitudes
It contains entries
The Uppsala General Catalog of Galaxies UGC prepared by Nilson from the Palomar Observa
tory Sky Survey plates taken on the Palomar Schmidt Telescope It claims completeness for galaxies
with diameters measured by eye on the blue plates greater than although Hudson Lynden Bell
show that the survey is incomplete b elow The survey includes ob jects
The Europ ean Southern Observatory Galaxy Catalog ESO prepared by Laub erts from the
ESO Schmidt plates of the Southern Sky taken with the Schmidt telescop e at Siding Springs It
contains galaxies with blue diameters greater than and with although Hudson
Lynden Bell claim incompleteness b elow Laub erts Valentijn have done photo
densitometer scans of most of the ESO catalog resulting in a uniform set of diameters and magnitudes
The Catalogue of Principal Galaxies Paturel et al consists of a compilation of data for
galaxies drawn from existing catalogs
The Third Reference Catalogue of Bright Galaxies RC de Vaucouleurs et al is a ma jor com
pilation of existing catalogs It contains galaxies with photometric data put onto a common
basis
The RC is three volumes long and probably represents the last comprehensive compilation of galaxy
data to b e published in hard cover Ma jor galaxy databases are now maintained on line in computer
readable formats and include the redshift compilation of Huchra et al consisting of
entries with redshifts and the NASA IPAC Extragalactic Database NED available via Internet
Helou et al containing data on galaxies and galaxy candidates as of May most
without redshifts The largest single galaxy catalog in existence is not yet publicly available and contains
galaxies brighter than b measured by photo densitometer scans the APM machine of the
ESO Schmidt plates over ster in the Southern Galactic Cap Maddox et al ab c
History of Redshift Surveys
We summarize here some of the ma jor redshift surveys and give a brief history of the eld Our
emphasis is on redshift surveys of nearby galaxies covering substantial areas of sky and thus we do not
include all p encil b eam surveys and those of unusual types of ob jects
Large scale structure was recognized from two dimensional maps of the galaxy distribution very early
on see for example Charlier quoted in Chapter of Peebles and Chapter of Hubble
de Vaucouleurs recognized the existence of the Lo cal Sup ercluster from two dimensional data
although it to ok many years b efore his discovery was appreciated by the general astronomical community
When Shane Wirtanen published their deep galaxy counts from Lick astrograph plates over a
substantial area of the Northern Sky detailed quantitative analyses of the galaxy distribution could b e
done Groth Peebles Fry Peebles more imp ortantly the resulting map Seldner et al
cf Figure of Peebles showed the community the rich structure that could b e found in the
galaxy distribution However these sky maps lacked the third dimension which is provided by redshift
measurement
The rst redshift surveys carried out to study the large scale distribution of galaxies were those of
Gregory Thompson a b and Gregory Thompson Ti t who did p encil b eam
surveys towards Coma Perseus and other clusters and started the pro cess of identifying sup erclusters
The rst redshift survey to cover a large fraction of the sky was that of the Revised Shapley Ames Galaxy
Catalog Sandage Tammann consisting of all galaxies with B and with Galactic latitude
T
jbj The sample contains galaxies covering ster and has a median redshift of km s
Ma jor analysis pap ers of this dataset include Davis Geller Huchra Sandage Tammann Yahil
and Yahil et al Much of the early work is reviewed in Oort
Kirshner Oemler Shectman and Kirshner et al carried out a series
of p encil b eam surveys with the original purp ose of studying the galaxy luminosity function Three such
p encil b eams in the region of the constellation Bo otes found a common gap in the redshift distribution
This so called Bo otesvoid is now known to have a diameter of km s Kirshner et al Strauss
Huchra Dey Strauss Huchra At the time it was rst discovered such vast regions of
space substantially underdense relative to the mean were a great surprise to the community although a
few workers notably Einasto and colleagues Einasto Jo eveer Saar had b een p ointing out the
existence of lamentary structure and voids in galaxy maps for some time
The Center for Astrophysics CfA redshift survey Huchra et al includes all galaxies in the
CGCG with m mag in the region of sky de ned by and b and and
z
b The sample contains galaxies over ster with a median redshift of km s This
survey showed a galaxy distribution in which voids and laments were ubiquitous although none of the
voids seen matched that in Bo otes void discovered by Kirshner et al Ma jor early analysis pap ers
of the CfA survey include Davis et al Davis Huchra Davis et al and Davis
Peebles b but this workhorse of redshift surveys has b een analyzed in detail in dozens of other
pap ers by many authors over the years
Fisher Tully used the Green Bank and E elsburg radio telescop es to carry out the rst
systematic redshift survey of galaxies using the HI cm line Their aim was to include all galaxies with
redshifts less than km s and thus their selection criteria are somewhat ill de ned Their original
survey included data for galaxies it has since b een extended to galaxies with cz km s
and published in tabular and graphical form by Tully Fisher and Tully
The Southern Sky Redshift Survey SSRS da Costa et al consists of all galaxies in the
ESO catalog with B band diameters corrected to face on greater than and with and
jbj The sample consists of galaxies covering ster do not have redshifts these are
largely galaxies with very low central surface brightnesses The median redshift is km s Analysis
pap ers include Davis et al Park Gott da Costa Maurogordato Schae er da Costa
and Lachi eze Rey da Costa Maurogordato
Giovanelli Haynes have used the Arecib o radio telescop e to carry out an HI survey of galaxies with
and b from the CGCG and UGC catalogs early type galaxies without detectable HI have
redshifts measured optically Their sample contains galaxies and is centered on the Pisces Perseus
sup ercluster Reviews of their work include Giovanelli Haynes and Giovanelli Haynes
In de Lapparent Geller Huchra published their rst wide slice of a deep er extension of
the original CfA survey The galaxy distribution in this two dimensional slice is shown in Fig from
the data in Huchra et al a Redshift is the radial co ordinate and right ascension is the angular
co ordinate declination is suppressed This gure which is often compared to a slice through a network of
bubbles shows clearly that voids like that in Bo otes are ubiquitous and that galaxies are aligned along
sheet like structures The structure stretching across the entire slice at cz km s has b een called
the Great Wall Geller Huchra and is the largest structure known to exist cf Di Nella Paturel
Geller Huchra and collab orators are extending the original CfA survey to m over the full
Z
area covered by the CGCG The survey is nearing completion and will contain roughly galaxies at
high latitudes and another in regions near the Galactic plane The median redshift is km s
The results from this survey are rep orted in de Lapparent et al a Huchra et al
a Vogeley et al Ramella et al and Park et al
The Infrared Astronomical Satel lite IRAS ew in and did a full sky survey with reso
lution in four broad bands centered at and m Because infrared radiation is not imp eded
by Galactic extinction galaxy samples selected from the IRAS database are well suited for full sky red
shift surveys They do have the drawback however that early type galaxies have very little dust or star
formation and thus are not represented in IRAS galaxy samples
The rst ma jor redshift survey of IRAS galaxies Soifer et al consisted of the galaxies with
f Jy with jbj in the Northern sky Strauss and collab orators carried out a survey covering
ster of the sky missing only regions at very low Galactic latitudes and areas not surveyed by
IRAS and including all galaxies to f Jy Fisher et al followed this up with a deep er
survey over the same region of sky to f Jy for a total of galaxies redshifts are complete
The median redshift of this survey is km s Scienti c results from this survey include Strauss et al
ab c Yahil et al Fisher et al ab cd Bouchet et al and Dekel
et al
A parallel e ort led by Rowan Robinson consisted of a sparse sample of one in six IRAS galaxies
to a ux limit of f Jy The selection criteria di ered in detail from the IRAS surveys mentioned
b efore and covered a solid angle of ster with galaxies This survey is deep er but sparser than the
complete IRAS surveys with a median redshift of km s This survey is nicknamed QDOT after
the institutions of the investigators Ma jor pap ers include Rowan Robinson et al Efstathiou et al
Fig The galaxy distribution from the survey of de Lapparent et al a using the data of Huchra et al
a Redshift is plotted in the radial direction and right ascension is the angular co ordinate Declination is
suppressed The prominent structure in the center of the gure is the Coma cluster
Saunders et al Saunders et al Kaiser et al and Lawrence et al
Loveday et al a b have carried out a redshift survey following up the APM photometric
survey of Maddox et al ab c They measured redshifts for one in twenty of galaxies brighter than
b over steradians centered on the Southern Galactic Cap The sample contains galaxies
J
with a median redshift of km s
Although not a redshift survey itself the e ort of Hudson ab ab should also b e mentioned
He calculated the redshift incompleteness using the compilation of Huchra et al of the UGC and
ESO catalogs as a function of diameter and p osition on the sky and statistically corrected the redshift maps
accordingly allowing him to recreate the full optical galaxy density eld which he used for comparison to
p eculiar velocity studies x
There are also a number of redshift surveys in progress or in the advanced planning stages see the
volume edited by Maddox Arag on Salamanca for pap ers on these and other surveys in progress
Strauss and collab orators have completed a redshift survey of optically selected galaxies the ORS to
roughly the CfA depth from the UGC and ESO galaxy catalogs In the declination strip not covered
by either of these two catalogs they use the recently completed diameter limited
Extension to the Southern Galaxy Catalog Corwin Ski The sample is limited by Galactic
extinction to jbj covers ster and contains galaxies of which redshifts are available
for The median redshift is km s The sample and the resulting galaxy distribution are
presented in Santiago et al ab and Santiago
Freudling da Costa Pellegrini have compiled a similar optically selected sample of galaxies
based on the CfA and SSRS surveys and the Morphological Catalog of Galaxies MCG in the declination
strip not covered by either survey Huchra et al Their zone of avoidance is more extensive than
that of the ORS Preliminary results can b e found in Freudling da Costa
The collab oration which carried out the QDOT IRAS redshift survey is in the pro cess of extending their
survey to include every galaxy brighter than Jy at m within their surveyed area The survey
should b e completed in fall and will contain galaxies A parallel e ort is using the IRAS
Faint Source Survey to push to Jy at m over a much smaller area of sky cf Lonsdale et al
da Costa and collab orators are carrying out a redshift survey in the Southern Hemisphere of comparable
scop e to the CfA in the North although over a somewhat smaller solid angle As there exists no galaxy
sample equivalent to the CGCG in the South the galaxy selection and photometry is b eing carried out
from photo densitometer scans of photographic plates The sample consists of galaxies over
ster First results are in da Costa and da Costa et al a b
Geller Huchra are measuring redshifts for all galaxies to r in a strip centered on
the original CfA slice de Lapparent et al a The sample includes galaxies with a median
redshift of km s and is called the Century Survey
The team that carried out the Bo otes Void survey discussed ab ove is conducting an extensive redshift
survey from the Las Campanas Du Pont m telescop e using a multi ob ject sp ectrograph with
b ers Their sample is limited in R band CCD magnitude in the range R using photometry
they obtain themselves on the Las Campanas telescop e Their survey area is square degrees
in a series of constant declination strips and they exp ect to include galaxies in their completed
survey The median redshift is km s Preliminary results are discussed in Shectman et al
and the PhD theses of Tucker and Lin
The more ambitious surveys mentioned here are going deep er than existing catalogs of galaxies
extend Thus workers are b eing forced to de ne galaxy samples themselves This turns out to b e a b o on
although a great deal more work as now the galaxy selection criteria can b e ne tuned for the purp oses
of redshift survey work and large scale structure studies In addition the mo dern automated metho ds of
galaxy selection and photometry yield more reliable catalogs with much more robust magnitudes than
those measured by eye Two further redshift surveys in the advanced planning stages are discussed in a
nal concluding section x
The Measurement of Galaxy Redshifts
The ma jority of the redshift surveys discussed ab ove have redshifts measured in the optical part of
the sp ectrum This is done typically with telescop es of ap erture b etween one and ve meters attached to
low or medium resolution R sp ectrographs At much higher resolution one resolves the
internal motions of the galaxies themselves at much lower resolution individual sp ectral features b egin to
blend with one another
The optical sp ectra of the vast ma jority of galaxies fall into one of several types
A pure absorption line sp ectrum characteristic of G and K stars This type of sp ectrum is asso ciated
with quiescent early type galaxies without a recent history of star formation
An absorption line sp ectrum showing features characteristic of A and F stars in particular Hydrogen
Balmer line absorption in addition to lines from G and K stars and emission lines characteristic of HII
regions of ionized gas Such sp ectra are often asso ciated with late type galaxies with star formation In
some extreme cases the so called extragalactic HII regions almost no stellar continuum at all is seen
but the emission lines are very strong
Galaxies harb oring an active nucleus in which strong emission lines of high ionization sp ecies are seen
Strongly active galaxies represent only a few p ercent of the galaxy p opulation but are the fo cus of
intense study see Osterbro ck for detailed reviews
The measurement of redshifts of low surface brightness galaxies is always di cult If the ob ject is to o
faint to measure a continuum with enough signal to noise ratio to detect absorption lines one can either
pray for H hoping that an HII region will fall on the slit or observe these ob jects in the radio lo oking
for cm transition of HI late type low surface brightness galaxies are often rich in HI and surveys of these
ob jects have b een carried out with radio telescop es Bothun et al Schneider Thuan Mangum
The measurement of emission line redshifts is straightforward and is usually done by tting multiple
Gaussians to the lines that are seen Absorption line redshifts require more subtlety the absorption lines
are typically of lower signal to noise ratio and the features that are seen often consist of blends of lines
making it imp ossible to assign the feature to a unique redshift A redshift is measured by comparing the
sp ectrum of the galaxy in question to that of a star of known small radial velocity typically a K K
giant taken with the same instrumental setup Heuristically one slides the sp ectrum of the stellar template
back and forth until it matches the sp ectrum of the galaxy the amount of the shift b eing a measure
of the redshift In practice this is done by taking cross correlating the sp ectra of the galaxy and the
stellar template in either Fourier or real space and tting the p eak with a smo oth function Sargent et al
Tonry Davis Heavens Using such techniques redshifts of nearby galaxies are typically
measured with km s accuracy e g Huchra et al
In the days when blue sensitive photographic plates were the detecting element in sp ectrographs
sp ectra for redshifts of nearby galaxies were often taken in the blue region of the sp ectrum covering the
region around A in order to detect the Calcium H and K lines Mo dern CCD detectors are more
sensitive in the red and sp ectra are typically taken centered on the A Mg b feature or for galaxies
that are likely to have strong emission lines centered on H and the N II doublet at A CCD
detectors on most mo dern sp ectrographs have of the order of pixels along the disp ersion direction
with roughly two pixels p er disp ersion element implying a sp ectral coverage of A for R As
CCDs get larger it is b ecoming p ossible to cover more of the visible part of the sp ectrum and double
sp ectrographs which use a dichroic to split the light into red and blue halves each going to a separate
camera are now op erating on the Palomar m Lick m ARC m Keck m and other telescop es
Most of the redshift surveys listed ab ove have b een carried out by observing one ob ject at a time
However now that multi ob ject sp ectrographs exist on many of the world s largest telescop es it is p ossible
to obtain sp ectra for many galaxies in a single exp osure In practice the size of the telescop e determines
the faintness of the galaxies for which one can obtain redshifts in a reasonable exp osure time and this
faintness limit in turn determines the mean number density of galaxies on the sky Thus one designs a
multi ob ject sp ectrograph with a number of b ers and eld of view with these considerations in mind
This is the approach taken by the Kirshner et al Las Campanas survey and the Sloan Digital Sky Survey
x among others
Redshift surveys are also carried out in the radio where one lo oks for the HI cm line in emission
As elliptical galaxies tend to b e gas p o or such surveys are limited to later type galaxies The largest such
redshift surveys have b een done at Arecib o Observatory in Puerto Rico and with the fo ot and the late
fo ot telescop es at Green Bank West Virginia These surveys are carried out at much higher resolution
than in the optical typically R Thus radio redshifts are typically measured to much higher
accuracy than optical redshifts with quoted errors of km s The combination of loss of sensitivity
and the fact that the cm line is redshifted into an unprotected band means that radio surveys are
unable to prob e redshifts much b eyond km s although the Giant Metrewave Radio Telescope
near Puna India will b e able to get around these problems with its enormous collecting area and relative
isolation In some of the older HI redshift literature the redshift was not de ned by Eq but by
t
z
which agrees with Eq only for in nitesimal z The reader of the older literature must b e aware of this
p otential for confusion although mo dern workers consistently use the optical de nition of z Eq
Redshifts are measured by necessity on a telescop e attached to the Earth The Earth takes part
in many motions it is rotating on its own axis km s at the equator and it is orbiting around
the Sun km s For extragalactic work the former correction is negligible but redshifts are usually
published with the correction to the helio centric frame However the Sun is in orbit around the center of
the Milky Way km s the Milky Way is falling towards our nearest large companion M at
km s Binney Tremaine and the whole Lo cal Group of galaxies takes part in the larger scale
velocity eld which we will discuss in detail b elow Because motions on scales smaller than that of the Lo cal
Group are very non linear we will not include them in our mo dels but rather refer to redshifts relative
to the barycenter of the Lo cal Group Estimates for the correction from the helio centric to Lo cal Group
barycentric frame have b een given by Yahil Tammann Sandage de Vaucouleurs de Vaucouleurs
Corwin and Lynden Bell Lahav These three determinations are consistent with one
another for example Yahil et al quote the motion of the sun relative to the barycenter as km s
towards Galactic co ordinates l b
Determination of the Luminosity and Selection Functions
Redshift surveys have the prop erty that the number density of sample ob jects is a decreasing function
of redshift We quantify this in terms of a selection function r de ned as the fraction of galaxies at
distance r which meet the sample selection criteria In order to use the galaxies in a sample as tracers of
the general galaxy distribution we must correct for the fact that the galaxies represent an increasingly
smaller prop ortion of the parent p opulation at larger and larger distances We do this by giving sample
ob jects weights prop ortional to r in order to account for the galaxies b elow the ux limit at distance
r The selection function is closely related to the luminosity function L which gives the number density
of galaxies of luminosity L p er unit luminosity The relationship b etween r and L is most easily
seen in the case of a survey in which the sample is de ned by a minimum energy ux f the derivation to
min
follow may b e trivially mo di ed for a survey limited by apparent diameter or other photometric quantity
or for direction dep endent selection as in the case of samples a ected by Galactic extinction cf Santiago
et al b In the latter case one generalizes the selection function to a quantity dep ending on b oth
distance and direction on the sky
The integral of L over all L gives the total number density of galaxies n In practice the luminosity
function is p o orly constrained at the faint end and so we cut o the integral at a lower limit L r f
s min
s
for some small r typically km s and reject galaxies with lower luminosities At any distance r r
s s
we can only observe galaxies with luminosities greater than
L r f
min min
Consequently the ratio of observable to total galaxies at distance r is
R
L dL
r f
min
R
r
L dL
L
s
At distances closer than r we set r Note that the luminosity function is a prop erty of the galaxies
s
themselves while the selection function dep ends on the photometric limits of the sample in question
The use of the selection function to weight galaxies in large scale structure studies makes the assump
tion that the luminosity function is universal indep endent of lo cal density This assumption means that
galaxies of any luminosity are equally go o d tracers of the large scale galaxy distribution cf x or
equivalently that that bivariate distribution of galaxies in p osition and luminosity is a separable function
There are several metho ds for measuring the luminosity or selection function from redshift surveys
The simplest is called the V metho d due to Schmidt the luminosity of a galaxy in the sample
max
determines a maximum distance to which it could b e placed and still remain within the sample
f
i
r r
max i
f
min
where r and f are the measured distance and ux of the source We can then de ne a maximum volume
i i
V r where is the solid angle covered by the survey The estimator for the luminosity function
max
max
is then
X
L L
V
max
where the sum is over all galaxies with measured luminosities b etween L and L L This metho d is
easily generalized to the case of a variable ux limit caused for example by Galactic extinction However
it makes the assumption that the mean underlying number density of galaxies is uniform without any
density inhomogeneities This is a dangerous assumption for we shall see that the inhomogeneities can b e
substantial
There is an extensive literature of metho ds to derive selection functions in density independent ways
Lynden Bell Turner Kirshner et al Sandage Tammann Yahil Davis Huchra
Nicoll Segal Efstathiou Ellis Peterson Choloniewski Binggeli et al and
Yahil et al We present here our own favorite following the development of Yahil et al We
phrase the problem in terms of likelihoo ds given that a galaxy i is at distance r what is the probability
i
that it fall in the interval L L L dL This probability is simply the luminosity function L
i i
normalized by the integral over all luminosities it could have at that distance given the ux limit of the
survey
L dL
i
R
F L jr dL
i i
L dL
L
min r
i
where L r was de ned in Eq Note that as written F L jr is a probability density p er unit
min i i
luminosity It will b e convenient to rephrase Eq in terms of the selection function
r
F L jr
i i
r r
i
r r
max i
where r was de ned in Eq This is the likelihoo d for a single galaxy given the selection function
max
the likelihoo d for the entire sample is then the pro duct of F over all the galaxies in the sample We then
solve for the selection function by maximizing the likelihoo d with resp ect to the parameters in terms of
which we mo del r Note that we need not worry ab out the constant of prop ortionality in Eq as
it is indep endent of the selection function itself and therefore is irrelevant when maximizing with resp ect
to
Note that the likelihoo d function is indeed indep endent of the underlying density distribution as
it is a conditional probability for the luminosities given the distances to each ob ject In particular this
metho d is indep endent of the mean density of the sample which drops out of the ratio in Eq Thus
we need an indep endent way to de ne the mean density Davis Huchra discuss various metho ds
for de ning the mean density of a sample given the selection function In general the mean density is given
by a weighted sum over the sample
P
w
i
galaxies i
R
n
dV w
In the presence of density inhomogeneities the variance in n is minimized with the weights
w
i
nJ r
i
where
Z
J r r dr
is a measure of the density uctuations on the scale of the survey Because of the presence of n in the
expression for the weights one has to calculate n iteratively In practice one often uses the simpler
estimator given by w r thus
i i
X
n
V r
i
galaxies i
In principle one sums over all the galaxies of the survey in which case V is the volume out to the most
distant ob ject of the sample In practice this would b e horribly noisy and one sums out to a radius b eyond
which the sample b ecomes so sparse that the shot noise of Eq dominates
Given a value of n one can de ne the luminosity function in terms of the selection function One
often sees the luminosity function expressed in logarithmic intervals of luminosity so that
L ln L L n r
r
r r
max L
How in practice do es one maximize the likelihoo d with resp ect to an unknown function One approach
is to use a parameterized form for the selection function and maximize with resp ect to those parameters
Sandage et al Yahil et al It is easier to work with a parameterized form for rather than
b ecause it is always easier to di erentiate than to integrate Yahil et al suggest a simple generic
form for with three parameters which is applicable to a variety of situations
r r r
r
r r r
s
s
Alternatively one can write the luminosity function in the form of a series of steps
L L L L k N
k k k
The steps L are typically logarithmically spaced and the aim is to solve for the parameters The
k k
selection function can then b e written
P
N
H L r
i i min
i
r
P
N
i
i
where L r was de ned ab ove and
min
L L
i
L L
i
H L
L L L
i
i i
L L
i i
L L
i
Setting the derivative of the likelihoo d function Eq with resp ect to the to zero yields a series of
i
implicit equations for the s which are solved by iteration Nicoll Segal Efstathiou et al
This metho d has two drawbacks Koranyi Strauss First b ecause the luminosity function is taken
to b e constant over a nite interval the selection function has discontinuous rst derivatives and in fact is
biased upwards this makes further analyses based on the selection function misleading This can b e xed
by tting a smo oth curve through the steps and calculating the selection function from this Alternatively
one can use a generalization of Eq in which one linearly interpolates the luminosity function b etween
steps Second and more serious when the number of steps b ecomes small the luminosity function is biased
downwards esp ecially at the faint end an e ect which requires a large number of steps or more to
mitigate We will discuss one e ect of this bias in x
Maximum likelihoo d metho ds in general and the ones we have describ ed here in particular have the
disadvantage that there is no explicit measure of go o dness of t Thus when tting for a parameterized
form one can b e sure to get the b est values of the parameters out but one can never know whether the form
itself is adequate Sandage et al and Yahil et al have developed a simple a posteriori metho d
to test how well a given selection function ts the sample One can compare the observed distribution of
luminosities in a redshift sample with that exp ected given the distances and a mo del for the luminosity
function Eq is the probability distribution function of luminosity for a galaxy at a given distance
with a sharp cuto at L r One can carry out this comparison for any well de ned subset of the data
min
de ned by redshift p osition on the sky morphological type lo cal density etc that do es not have explicit
selection on luminosity Fig illustrates this for the IRAS Jy survey in each panel the smo oth curve is
the predicted luminosity distribution while the histogram is that observed the agreement is excellent The
di erent panels show the luminosity distribution in di erent redshift shells indicating that there are no
systematic errors as a function of redshift Actually it remains unclear how useful this test is exp eriments
show that the luminosity function must b e very seriously in error b efore this test shows disagreement
b etween observed and predicted luminosity distributions
As discussed in x the photometric data on which a redshift survey is based are not always of the
highest quality This has two e ects ob jects will b e scattered in and out of the sample by ux errors and
errors in uxes will cause luminosities to b e in error Because there are many more faint than bright galaxies
the net e ect of errors is to cause a systematic over estimation of luminosities It is straightforward to show
that constant fractional errors in the uxes result in a derived luminosity function that is a convolution
of the true underlying luminosity function with the ux error distribution Monte Carlo exp eriments by
Santiago et al b show that despite this e ect the density eld derived from redshift survey data
is quite robust to ux errors if they are prop ortional to the uxes themselves That is the bias in the
selection function caused by ux errors is very nearly cancelled by the bias in galaxy counts caused by
ob jects scattering across the ux limit The density eld is more robust than is the luminosity function
Fig The luminosity distribution histogram of the IRAS Jy sample in di erent redshift ranges together
with the predicted distributions given the luminosity function The values of dof given in each panel refer to
the di erence b etween the two curves with errors given by Poisson statistics
itself to ux errors as long as the selection function is measured from the same dataset
A comprehensive review of determinations of the luminosity function is given in Binggeli et al
cf Felten for earlier work Mo dern determinations of the luminosity functions of galaxies in optically
selected bands include Efstathiou et al Loveday et al b de Lapparent et al
and Marzke Huchra Geller while the IRAS luminosity function has b een discussed by Saunders
et al Yahil et al Spinoglio Malkan and Soifer et al The optical luminosity
function is often t to a form originally suggested by Schechter
L L
e
L L
with free parameters and L the sharp exp onential cuto means that galaxies with luminosities well
ab ove L the knee of the luminosity function are quite rare The m luminosity function of IRAS
galaxies do es not show such a sharp cuto but rather is well t by two p ower laws cf Eq there
is therefore a substantial p opulation of ultraluminous IRAS galaxies whose prop erties are the sub ject of
much research e g Sanders et al This also means that the selection function of IRAS galaxies is
not as steep as that of optically selected galaxies and thus IRAS redshift surveys include a much more
extensive tail of high redshift galaxies than do es an optically selected sample with the same median redshift
Figure of Santiago et al a
Luminosity Functions Scienti c Results
The determination of the optical luminosity function of galaxies in optical bands has only b een done
adequately in the photographic B band in which the comprehensive galaxy catalogs have b een compiled
Indeed much remains to b e learned ab out the distribution of galaxy prop erties in quantities other than
luminosity For example Binggeli et al stress that the luminosity function of galaxies is a strong
function of galaxy morphology cf Loveday et al b and although some work has b een done
on the diameter function of galaxies Lahav et al Maia da Costa Hudson Lynden Bell
the bivariate distribution of diameters and luminosities has barely b een explored Choloniewski
So dr e Lahav Even less is known ab out the bivariate distribution of luminosities and galaxy
colors Prop er analyses of these basic prop erties of galaxies will require large samples of galaxies with
excellent photometry in a variety of bands This is one of the main scienti c goals of the Sloan Digital
Sky Survey The galaxy luminosity function is a basic datum that any detailed mo del of galaxy formation
must match theoretical pap ers addressing this include White et al Schae er Silk White
Frenk Cen Ostriker a Cole et al a and White
The measured luminosity function b ecomes very uncertain at the faint end e g Efstathiou et al
The faint end slop e of the luminosity function has imp ortant rami cations for a number of issues in
astrophysics First number counts of faint galaxies show an excess over the number predicted by simple
extrap olation of the galaxy luminosity function to day Tyson Cowie et al Maddox et al
d Ellis if there is a p opulation of galaxies that were bright in the past but whose surface
brightnesses dimmed substantially as star formation ceased they could b e hidden to day in the faint end
of the luminosity function McGaugh As photographic and CCD surveys push to ever lower surface
brightness levels new p opulations of galaxies are app earing Bothun et al Schombert Bothun
Schombert et al Dalcanton including many galaxies of quite substantial luminosities
Understanding the low luminosity p opulation is also imp ortant for understanding the evolution of
metals i e elements b eyond Helium in the p erio dic table in the universe Cowie shows that the
integrated surface brightness of galaxies in the sky is directly prop ortional to the total metal abundance
c
S Z
where Z is the volume density of metals in the universe and is the light emitted p er unit frequency
p er unit mass of metals returned to the interstellar medium Quantifying the full galaxy p opulation and
therefore the total surface brightness of the night sky due to galaxies Spinrad Stone Toller
will thus give insights into the pro duction of metals and the chemical evolution of the universe
The numbers of low luminosity galaxies is also vitally imp ortant for quantifying the luminosity density
R
of the universe The total emissivity of optical light p er unit volume in the universe is given by L L dL
which diverges if the faint end logarithmic slop e of L is steep er than Eq We observe a
dark night sky which says that the luminosity density of the universe do es not in fact diverge at the faint
end thus either the faint end slop e is somewhat shallower than or there is a cut o at some p oint
The observational situation is still very much in a state of ux this is an active area of research
Note that some workers use the luminosity function p er unit log luminosity for which the de nition of di ers
by unity from that here See Eq
The optical luminosity density of the universe can b e turned into a mass density of the stars if one
assumes a mass to light ratio for the stars Alternatively Loveday et al calculate the mass to light
ratio necessary to close the universe i e such that given the measured luminosity density as
h in solar units For comparison standard sp ectral synthesis mo dels give mass to blue luminosity
ratios of
The recent interest in low luminosity and low surface brightness galaxies reminds us that the galaxy
samples which form our basis for redshift surveys are likely to su er from incompleteness Disney
McGaugh Even a survey limited at bright magnitudes or large diameters will miss a p opulation
of low surface brightness galaxies some of which are quite luminous Bothun et al although the
numbers of luminous low surface brightness galaxies seems to b e quite small There is also a bias against
nding compact galaxies of very high surface brightness these galaxies can b e di cult to distinguish
from stars which are much more numerous than galaxies at all Galactic latitudes until one pushes to nd
magnitude and fainter and thus will b e missed from the survey The extreme examples of compact galaxies
are the quasars C is bright enough to satisfy the selection criteria of the CfA redshift survey but
app ears absolutely stellar on photographic plates the host galaxy is apparent only with deep exp osures
under sup erb conditions Hutchings Ne Bahcall Kirhakos Schneider
Testing the Hubble Law with Redshift Surveys
The observational evidence for the Hubble law Eq for nearby galaxies is mostly based on distance
indicator relations for individual galaxies These usually take the form of standard candles whereby the
absolute luminosity of the galaxy or some well de ned part of it e g Cepheid variables are assumed to
b e known a priori x Observation of its apparent luminosity gives the distance via the inverse square
law This can then b e compared with the redshift as a check of Eq The history of this approach is
summarized in Chapter of Peebles see Lauer Postman for a recent test of the Hubble
law There has b een no strong evidence for deviations from Eq b eyond that exp ected from p eculiar
motions Eq Nevertheless one could imagine generalizing Eq to the form
p
cz H r
p
Segal et al and references therein suggest a so called Chronometric Cosmology in which p
at low redshifts Segal et al suggest an intriguing measure of p from redshift surveys Rather
than using distance indicators to measure the distances to individual galaxies one can use the luminosity
function of galaxies as a distance indicator assuming the luminosity function to b e universal In fact
luminosity function tting has b een used in the past to measure distances to galaxy clusters Schechter
Schechter Press Gudehus here we wish to use it to test the Hubble law Imagine that
one interprets the results of a redshift survey using Eq with the incorrect value for p One would
exp ect that the luminosity function determined from subsets of the sample in di erent redshift ranges
would not b e consistent with one another agreement would b e found only for the correct value for p and
this then has the p otential for distinguishing b etween di erent cosmological mo dels Segal et al
use the IRAS Jy redshift survey Strauss et al b to claim that the data are consistent with
p and inconsistent with the conventional p However the luminosity function in di erent redshift
shells is a much less p owerful statistic than Segal et al claim In the limit of thin redshift shells it is just
measuring the ux distribution of the sample and in any redshift survey the ma jority of the sources are
close to the ux limit indep endent of cosmology Koranyi Strauss show that this distribution is
almost indep endent of p if one self consistently derives the luminosity function from the sample for each
value of p Moreover b ecause Segal et al use the step wise luminosity function metho d describ ed ab ove
their luminosity function is systematically in error the error b eing worse for smaller p than for larger
Therefore their luminosity function for p is a b etter t to the data than is p and they erroneously
conclude that the Hubble law is ruled out by the data
Although the luminosity function is a p o or discriminant of mo dels one can still use redshift surveys
alone to put constraints on p In a homogeneous universe the number of galaxies in a redshift survey with
redshift b etween r and r dr is n r r dr The upp er panel of Fig compares the observed histogram
of redshifts of galaxies in logarithmic bins from the Jy IRAS survey with that predicted for p
and Note that this is an indep endent test from that of the density independent luminosity function
diagnostic describ ed ab ove In the p mo dels the predicted and observed redshift histograms are in
bad disagreement The resulting density relative to the mean in shells the ratio of the histogram to the
smo oth curves is shown in the lower panel In the standard cosmology the almost full sky coverage of the
IRAS survey averages over most density structures meaning that the sample is close to mean density at
all redshifts cf de Lapparent et al although the overdensity asso ciated with the Lo cal Sup ercluster
is apparent at cz km s However for the p and p cosmologies one must argue that the
Lo cal Group lies in the middle of a void surrounded by a vast spherical shell at the mean density b etween
and km s which is surrounded by another spherical void extending to the horizon
The Smoothed Density Field
We have seen ab ove that the number density of galaxies in a ux limited redshift survey is a decreasing
function of distance We correct for this by assigning each galaxy a weight given by the inverse of the
selection function r In order to de ne the extent of the structures traced by the galaxies we would like
to de ne a continuous density eld This is done by smo othing cf Eq
X
W jr r j r
i smo oth
r
n r
i
galaxies i
where the window function W is normalized to unit integral
Z
d r W r r
smo oth
Common window functions used are top hat parab olic and Gaussian
x tophat
r W x
x parab olic x
smo oth
x
e
Gaussian
All that remains is to choose a smo othing radius r Because the mean interparticle spacing of a
smo oth
redshift survey is an increasing function of distance from the origin indeed it is given by n r
a smo othing length which shows full detail at small distances will contain increasingly fewer particles at
larger distances with corresp onding ever increasing shot noise For this reason it is common to choose a
smo othing length which grows with radius prop ortional to the mean interparticle spacing This has the
disadvantage of not b eing amenable to Fourier Transform techniques for carrying out the smo othing in
Eq more imp ortant the smo othed density eld has di erent statistical prop erties at low and high
redshift Nevertheless it is useful for qualitative and cosmographical description of the structures that are
seen and in some sense shows the maximum amount of information in the redshift survey
The smo othed density eld will b e sub ject to errors from a variety of sources These include
Fig The upp er panel shows the redshift distribution of the IRAS Jy sample together with the exp ected
distributions in a homogeneous universe for various redshift distance relations The lower panel shows the ratio
b etween the observed and exp ected distributions which is the radial density eld Note that p the Hubble
Law remains close to the mean density at all radii
Shot noise The galaxy distribution is a p oint pro cess and one is always limited by the nite number of
galaxies within a smo othing length The assumption of a universal luminosity function implies that the
subset of galaxies luminous enough to enter a ux limited sample at a given redshift are fair tracers of
the full p opulation of galaxies Under this assumption the uncertainty in the density eld due to shot
noise is simply
E D
X
W jr r j
i
r
n r
i
galaxies i
b ecause the mean value of is zero by de nition There is a further contribution to the shot noise in the
mass density eld which dep ends on the variance in the mass to light ratio of galaxies but this term is
imp ortant only for nearby galaxies in typical redshift surveys simply b ecause grows so quickly cf
App endix A of Strauss et al c
The physically interesting quantity in most applications is the density eld in real space while redshifts
are the quantities that are measured One can assume a mo del for the velocity eld to map from one to
the other as we will discuss in detail in x Ignoring the problem altogether causes systematic errors
in the density eld Kaiser In practice quantifying this source of error requires a sp eci c mo del
for the velocity eld although linear theory makes sp eci c predictions for this distortion x
The derived density eld dep ends on the assumed selection function a small error in the selection function
causes a systematic error in the density eld which for simple ux limited surveys will b e a function
of redshift This source of error can only b e quanti ed with Monte Carlo simulations As mentioned
ab ove Santiago et al b show that photometry errors are largely comp ensated for in the selection
function in such a way that the derived density eld is quite robust One other way to get around this
problem is to de ne a volume limited subsample of galaxies from a ux limited survey This is done by
choosing all galaxies in the sample with distances less than some value r whose luminosities are such
max
that they would enter the sample even if placed at r For a sample with a uniform ux limit this
max
means that one chooses those ob jects with luminosities greater than L r This has the additional
min max
advantage that it is robust to luminosity dependent clustering non universal luminosity function Of
course volume limiting a sample means throwing out the ma jority of the ob jects
There exists no redshift survey with complete sky coverage there are always regions of the sky uncovered
by the survey The IRAS survey of Strauss et al b and Fisher et al is the survey with the
most complete sky coverage These must b e corrected for in Eq We discuss approaches to
this problem in x
One can lter the data to minimize the shot noise The Wiener Filter cf Press et al x
minimizes the variance b etween the measured and true density elds Let r b e the true underlying
fractional density eld of galaxies and let D r b e the measured density eld with shot noise N included
D r r N
We assume for simplicity that the shot noise is indep endent of p osition Let us de ne a lter F such that
the di erence b etween the ltered density eld F D and the true density eld is minimized In practice
we will work with Fourier Transforms and write
Z Z
h i
e e
variance d r r F r D r d k k F k D k
by Parseval s Theorem Substituting in Eq and minimizing with resp ect to the unknown function
e
F k one nds the Wiener lter
D E
k
e
D E
F k
k N
D E
where k is prop ortional to the power spectrum P k of the underlying density eld The Wiener
lter requires a mo del for the underlying p ower sp ectrum We will discuss metho ds for estimating the p ower
sp ectrum from redshift survey data in x Note that the Wiener lter is de ned in k space b ecause in
real space the values of r at di erent r are correlated Indeed the Wiener lter can b e derived in real
space in which case it is a matrix which takes this correlation directly into account Zaroubi et al
Eq holds only in the situation in which the noise is assumed indep endent of p osition In the more realistic
case of radially dep endent noise Eq generalizes to a two dimensional matrix cf Fisher et al d for
details
The number of elements in the matrix is the square of the number of p oints at which one wishes to de ne
the density eld which b ecomes a very non trivial computational problem
In a ux limited redshift survey with a constant smo othing length the noise term is an increasing
function of r In this case one can either apply a Wiener lter in r space as describ ed ab ove or de ne a
series of Wiener ltered density elds at a range of values of N and interpolate b etween these at each
p oint
The Wiener lter approaches unity when the signal to noise ratio is high At low signal to noise ratio
it approaches zero when there is no information on the underlying density eld the lter returns the most
likely value for the density namely its mean Thus the Wiener lter has the disadvantage of biasing the
contrast in the density eld downward The exp ectation value of the square of the ltered eld is
D E
D E D E
k
e e e
A
E D
F k D k D k
k N
D E
D E
k
A
E D
k N
N k
D E D E
e
F k k k
e
b ecause F k always For comparison the exp ectation value of the square of the un ltered eld is
D E
k N P k Thus the Wiener lter over corrects the density eld for shot noise In the case of a
redshift survey for a constant smo othing length the signal to noise ratio is a decreasing function of r and
thus the contrast of structures decreases with r just as we had in the variable smo othing mentioned ab ove
Thus the Wiener lter o ers a natural way to invoke variable smo othing For some problems however this
lack of statistical similarity b etween the nearby and far away parts of a redshift survey can b e a drawback
In this case one can de ne an alternative lter the power preserving lter Yahil et al
E D
k
e
A
D E
F k
PP
k N
just the square ro ot of the Wiener lter This is just one of a entire family of generalizations of the Wiener
lter as discussed in e g Andrews Hunt If we substitute the p ower preserving lter into Eq
D E D E
e
we nd F k D k k thus the name p ower preserving Of course this lter do es not share
PP
the minimum variance prop erty of the Wiener lter but it is easy to show that for signal to noise ratios
greater than one the ro ot mean square di erence b etween the ltered and true density elds is at most
larger for the p ower preserving lter than for the Wiener lter As with the Wiener lter Eq only
holds in when the noise is indep endent of p osition Yahil et al interpolate b etween maps ltered
with Eq at di erent noise levels for the ux limited case of noise increasing as a function of distance
Although the p ower preserving lter indeed preserves the second moment of the density distribution
function it do es not do a go o d job of preserving higher order moments at low signal to noise ratio in
particular the skewness gets exaggerated with the consequence that the ltered maps show sharp er
p eaks in regions of low signal to noise ratio than in high Work is ongoing to quantify this e ect and
correct for it
Fil ling in the Galactic Plane
With the advent of galaxy surveys covering all of the sky outside of the Galactic plane there has
b een a great deal of recent work exp ended on nishing the job that is either extrap olating the density
eld at high latitudes to lower latitudes or actually surveying in the zone of avoidance itself Balkowski
Kraan Korteweg There are two strong motivations for this The rst of these is cosmography We
would like to have a complete map of of the structures of the lo cal universe The two largest sup erclusters
in our immediate neighborho o d the Pisces Perseus Sup ercluster and the Hydra Centaurus Sup ercluster
b oth lie at low Galactic latitudes and in b oth cases there are overdensities on the opp osite side of the
plane Camelopardalis and Pavo Indus Telescopium resp ectively to which they are plausibly physically
connected
The second reason for mapping the galaxy distribution at low latitudes is for dynamical studies
Peculiar velocities and densities are related in linear theory by Eq if we wish to use this equation as
a test of gravitational instability theory or to measure we need as complete a map of the density eld
as p ossible
Simply leaving unsurveyed regions un lled causes systematic errors in any dynamical mo deling The
absence of galaxies do es not corresp ond to absence of structure indeed the un lled regions will act as
a void with negative from which Eq predicts a systematic out ow Filling the un lled regions of
a survey with the average density of galaxies corrects for this The next order of approximation involves
interpolating the density eld from higher latitude regions The IRAS Jy redshift survey has a low
latitude excluded zone that is only wide for this Yahil et al advocate a linear interpolation
of the density eld across the Galactic plane Lynden Bell Lahav Burstein have a wide
excluded zone at low Galactic latitudes in their survey of optically selected galaxies and an additional
wide zone in the gap b etween the areas of sky covered by the UGC and ESO catalogs They use a
cloning pro cedure in which galaxies are duplicated from adjacent high latitude regions into the excluded
zones A more elab orate interpolation scheme was used by Scharf et al who expanded the angular
distribution of IRAS galaxies in spherical harmonics and extrap olated them into the excluded zone They
found a prominent overdensity in Puppis which had b een recognized indep endently by Kraan Korteweg
Huchtmeier and Yamada et al which Lahav et al b was able to show was a p o or
cluster at roughly the distance of Virgo Lahav et al advocate ltering the spherical harmonics with
a Weiner lter to reduce the noise they show with N b o dy simulations that the resulting reconstruction
is quite robust for excluded zones less than wide
The observed velocity eld at high latitudes dep ends on the density eld at low latitudes Using a
density eld reconstruction called POTENT x Kolatt Dekel Lahav have reconstructed the
velocity eld in the Zone of Avoidance We will compare the results of this reconstruction to the density
eld of galaxies in x
An alternative approach to the various interpolation schemes discussed here is to survey the low
latitude sky directly It is di cult to do quantitative work here as the extinction is large and very variable
In regions that the extinction is not to o strong say less than A mag one can correct for extinction
B
statistically in the selection function following the metho ds of Santiago et al b the fewer galaxies in
regions of high extinction are simply given greater weight to comp ensate However this pro cess introduces
a much larger shot noise Various ongoing low latitude searches for galaxies in the optical bands and
redshift surveys thereof are summarized in the volume edited by Balkowski Kraan Korteweg
One can also select galaxies in wavebands that are less a ected than the optical by Galactic extinction
Galaxy catalogs selected at m from the IRAS database have b een used for redshift survey work The
principal limitations here are that any infrared color scheme that selects galaxies from stars and other
Galactic ob jects b ecomes severely contaminated at low Galactic latitudes by emission nebulae and infrared
cirrus which has the same infrared colors as quiescent galaxies At very low Galactic latitudes the very
high source density starts causing systematic errors in the IRAS uxes Finally it b ecomes increasingly
di cult to optically identify the galaxies in order to obtain redshifts when the optical extinction b ecomes
to o high Despite these problems the redshift surveys of Strauss et al b Fisher et al and
Lawrence et al extend down to within of the Galactic plane Low latitude galaxy catalogs selected
from the IRAS survey have also b een published by Ichikawa Nishida and Yamada et al
Neutral hydrogen cm emission is also una ected by Galactic extinction Lu et al have done
HI surveys of IRAS sources although Strauss et al show that optical surveys are more e cient
But cm surveys can also b e done blindly indep endent of any other catalog Kerr Henning used
the late Green Bank ft telescop e to p oint to p oints they discovered previously uncatalogued
galaxies Kraan Korteweg et al b announce the discovery in HI of a new member of the group of
which Ma ei Ma ei and IC is a part This is the rst published result of a dedicated survey of
the Northern Galactic plane jbj with the m Dwingelo o radio telescop e which is surveying out to
a redshift of km s with a resolution of km s and a b eam size of FWHM
Sky surveys at K band m are planned by b oth American and Europ ean teams The extinction
in this waveband is appreciably smaller than that in the optical bands meaning that galaxy catalogs
selected from this database will b e able to p enetrate to much lower Galactic latitudes The Two Micron
All Sky Survey MASS Kleinmann will indeed cover the entire sky and thus the galaxy sample
selected from this survey will b e ideal for measuring the dip ole moment of the galaxy distribution cf
x The Deep Extragalactic Near Infrared Survey DENIS Deul is planned only for the Southern
Hemisphere but will have sup erior angular resolution to the MASS survey
Redshift Surveys A Cosmographical Tour
As discussed in the Introduction much of the work of redshift surveys has b een cosmographical in
origin in which workers have explored the variety of structures that are traced by the large scale galaxy
distribution and have de ned the structures that are visible We have traced out the history of the pro cess
by which structures have b een explored ab ove Until the late s the mental image of the galaxy
distribution that a ma jority of workers in the eld had was an approximately uniform distribution with
randomly placed clusters embedded in it When the distribution of galaxies on the sky from the Shane
Wirtanen counts were published by Seldner et al p eople started b ecoming aware of the
richness of the structures that the galaxies traced The discovery of the Bo otes Void by Kirshner et al
and the publication of the redshift maps of the CfA survey Davis et al and the Pisces
Perseus region Giovanelli et al brought the message of the Lick maps home although it was the
publication of the CfA slice de Lapparent et al a Fig ab ove that really captured the community s
and the public s imagination The CfA survey now extended over a volume ten times that of the rst
slice has b een pivotal in characterizing the variety of structures in which galaxies are found The galaxy
distribution shows structures on all scales that our surveys have prob ed from pairs and small groups of
galaxies on h Mp c scales to the Great Wall Geller Huchra which has an extent of at least
h Mp c
Much of this review is devoted to a quantitative study of the structures that are seen Before launching
into this we would like to brie y familiarize ourselves with the sp eci c features in the Galaxy distribution
The pro cess of mapping structures and giving them names has b een carried out recently by Tully b
who emphasizes the structures within km s Pellegrini et al who trace structures in the
Southern Hemisphere Saunders et al which is the b est reference for the nature of galaxy structures
on very large scales Giovanelli Haynes which shows the distribution of galaxies pro jected on
the sky Strauss et al a Hudson a Santiago et al a and Fisher et al The b est
Fig The density eld of IRAS galaxies in the Sup ergalactic plane A km s Gaussian smo othing has b een
applied Prominent structures are lab eled V Virgo GA Great Attractor P P Perseus Pisces Co Coma A
Sup ercluster Sc Sculptor Void and Ce Cetus Wall
existing survey for tracing the full extent of the lo cal structures at least within km s is the
Jy IRAS redshift survey of Fisher et al given its close to full sky coverage and mo derate sampling
Because the sampling is not very dense it is not ideal for de ning individual structures on the smallest
scales see the references ab ove for more detailed mapping of individual structures
The density eld of the Jy IRAS survey with a Gaussian smo othing of r km s and
smo oth
with the p ower preserving lter applied x is shown in a series of slices in Fig through The density
eld is that obtained by a self consistent correction for p eculiar velocities as detailed in x b elow with
The slices shown in these gures are made parallel to the principal planes de ned in Supergalactic
co ordinates de Vaucouleurs p ointed out that distribution of nearby galaxies cz km s is
largely con ned to a planar structure which he called the Supergalactic plane or the plane of the Lo cal
Sup ercluster The Sup ergalactic plane is almost p erp endicular to the Galactic plane
The slice through the Sup ergalactic plane Sup ergalactic Z is shown in Fig The contours
are of the galaxy density eld Eq The heavy contour is at the mean density while the
dashed contours are at and The solid contours represent p ositive values of and are
logarithmically spaced in with three contours corresp onding to an increase of a factor of two One s
rst visual impression from these maps is the irregularity of the shap es of the largest structures One sees
immediately that mo deling sup erclusters as uniform spheres can b e misleading One also notes the rough
symmetry b etween the high and low density regions If the distribution function of the primordial density
eld is symmetric b etween under dense and overdense regions then linear theory predicts this symmetry
to b e preserved However b ecause is has a strict lower limit of while there is no corresp onding upp er
limit we exp ect a p ositive skewness to develop for late enough times and small enough smo othing scales
that h i approaches unity x This is evident in the maps here although there is still a rough balance
b etween the volume of space o ccupied by over and under dense regions One can do statistics with the
topology of the iso density contours we will discuss this in x
In this gure the Lo cal Group sits at the origin Some of the prominent structures are lab eled with
identifying ags The nearest substantial overdensity is found at X km s Y km s
this the Virgo Cluster V the nearest large cluster of galaxies The Ursa Ma jor cluster a somewhat
more di use and spiral rich cluster is to o close to Virgo to b e resolved as a separate structure with this
smo othing The traditional de nition of the Lo cal Sup ercluster refers to it as a attened structure in
the Sup ergalactic plane of extent km s with the Virgo cluster at its center indeed one often
hears the Lo cal Sup ercluster referred to as the Virgo Sup ercluster but as this gure shows the Lo cal
Sup ercluster is far from an isolated structure In particular at this smo othing it joins with the Great
Attractor GA which is the large extended structure centered at Z km s Y km s
The Great Attractor is often referred to as two separate sup erclusters the Hydra Centaurus p ositive
sup ergalactic Y and Pavo Indus Telescopium negative sup ergalactic Y Sup erclusters although that
division is somewhat arti cial b eing imp osed by the zone of avoidance The IRAS survey clearly shows
the two sup erclusters to b e contiguous the e ective smo othing length is greater than the width of the
excluded zone The Great Attractor was rst named as such by Dressler b when p eculiar velocity
surveys showed a convergence in the galaxy velocity eld towards a p oint corresp onding to the Hydra
Centaurus sup ercluster Lilje Yahil Jones Lynden Bell et al as we discuss in x b elow
Subsequent redshift surveys cf Strauss Davis Dressler mapp ed the full extent of the
overdensity of galaxies asso ciated with the Great Attractor See Lynden Bell Lahav Burstein
for a brief history of de nitions of the Great Attractor In the context of redshift surveys we de ne the
Great Attractor as the extended overdensity of galaxies that dominates the left hand side of Fig
On the opp osite side of the sky there is an extended chain of galaxies with two distinct density p eaks
at X km s Y km s and X km s Y km s This is the Perseus
Pisces Sup ercluster P P the Northern part of which is sometimes referred to as the Camelopardalis
Sup ercluster These contour plots do not give justice to the remarkable tightness and coherence of the
galaxy distribution here Giovanelli et al show that the structure lies nearly in the plane of the sky
and displays a remarkably lamentary structure with rich clusters embedded in it There is a void in the
foreground of the Perseus Pisces sup ercluster centered at X km s Y km s There is
also a void b eyond the Virgo cluster which stretches to the Coma A sup ercluster Co centered at
X Y km s This sup ercluster is embedded in the Great Wall Fig although the Great Wall
itself lies b eyond the b oundaries of this gure for the most part The void centered at X km s
Y km s was discovered by da Costa et al Dekel Rees refer to it as the Sculptor
Void Sc Finally the radially directed structure centered at X km s Y km s is the
Cetus Wall Ce discovered by the SSRS
Fig a is a slice at Z km s and shows that the region ab ove the Sup ergalactic plane
It is standard practice to name clusters and sup erclusters after the constellations in which they are found
Fig The density eld in various slices in Sup ergalactic co ordinates a Z km s b
Z km s c Y d X Prominent structures marked include H C Hydra Centaurus and
P I T Pavo Indus Telescopium
is marked by an extensive void This void extends almost to the Lo cal Group and is apparent in the
distribution of galaxies very near us as the Lo cal Void cf Tully b The overdensity extending across
the top of the gure is a piece of the Great Wall Below the Sup ergalactic plane Fig b Z km s
one sees the extensions of the Great Attractor in the form of the Pavo Indus Telescopium sup ercluster
P I T and the Pisces Perseus Sup erclusters
Panel c of Fig is the Y slice This slice lies almost in the Galactic plane and cuts cleanly through
the Great Attractor negative X and the Northern part of the Pisces Perseus Sup ercluster p ositive X
The Great Attractor is seen to b e bimo dal in this cut and we lab el the Hydra Centaurus H C and P I T
sup erclusters separately The extent of the void ab ove the Sup ergalactic plane Fig a is now clear it
o ccupies the entire upp er part of this gure A slice at X Fig d passes through the Virgo cluster
and the Coma A sup erclusters The Great Wall is apparent at Y km s in this slice
The slice at Y km s Fig a shows no dramatic structures and is largely dominated
by the void that lies b etween the Lo cal and Coma A sup erclusters The slice at Y km s
Fig The density eld in various slices in Sup ergalactic co ordinates a Y km s b Y km s
a X km s b X km s
Fig b cuts through the southern i e smaller Y part of the Pisces Perseus sup ercluster and shows it
to b e extended in the Sup ergalactic Z direction The region south of the Great Attractor is dominated by
the Sculptor Void Sc
X km s Fig c slices through the Pisces Perseus sup ercluster showing it to b e multi
mo dal Indeed there is a void b etween the Perseus Pisces Sup ercluster in the Sup ergalactic plane and its
counterpart at Z Fig b Finally the slice at X Fig d cuts through the Great
Attractor The bimo dality apparent in the Y slice is apparent here as well The Sculptor Void app ears
here bracketed by a remarkable wall of galaxies extending from the Great Attractor to the upp er left hand
corner of the gure
In this brief overview of the galaxy distribution as seen by IRAS we have not b een comprehensive
we have not endeavored to give every structure that is visible a name nor have we given justice to the
detailed mapping that has b een done by a large number of p eople see the references ab ove Moreover
with our km s Gaussian smo othing we have washed out some of the more remarkable structures
that are apparent in the redshift data the very thin walls and laments and the tight clusters However
the main emphasis in this survey is on quantitative and statistical analyses of the galaxy distribution
and thus we now move b eyond cosmography to see what quantitative science we can do with the data
presented here
Redshift Surveys Galaxy Clustering
This chapter will discuss quantitative measures of galaxy clustering and how we might use the results
to put constraints on cosmological mo dels for structure formation Much of the background material has
b een introduced earlier in x although we will nd ourselves introducing new concepts as we go along We
are not exhaustive in this section and do not attempt to describ e every statistic that has b een used as a
measure of galaxy clustering In particular we do not survey the work done with multi fractal measures or
p ercolation metho ds these approaches have b een thoroughly reviewed in this journal by Borgani
We start with a discussion of the two point correlation function r in x Before going onto the
p ower sp ectrum in x we discuss the e ects of redshift space distortions and non linear e ects in x
Non linear e ects give non zero high order correlations which we discuss in x and x Topological
measures of large scale structure are discussed in x
The dip ole moment of the galaxy distribution is closely related to the motion of the Lo cal Group
x One can expand the density distribution in higher order multipoles as well as discuss in x We
return to the sub ject of redshift space distortions with metho ds to correct for them in x We nish this
chapter with a discussion of the relative distribution of di erent types of galaxies in x
The Two Point Correlation Function
The two point correlation function was introduced in Eq as the auto correlation function of the
continuous density eld As it is the Fourier Transform of the p ower sp ectrum P k Eq it also gives
a complete statistical description of the density eld to the extent that the phases are random Although
one can always de ne a smo othed density eld as describ ed in x and then apply Eq the resulting
correlation function would b e cut o on scales smaller than the smo othing length Instead we simply apply
Eq which we rewrite as follows the joint probability that galaxies b e found at p ositions r and r
within the in nitesimal volumes dV and dV is
dP n dV dV r r r
where r jr r j In the case of a volume limited sample we thus nd
N r
r
r n V
where N r dr is the number of pairs found in the sample with separations b etween r and r dr and
r n V is the number exp ected in a uniform distribution of galaxies This expression is appropriate
only for the case of a volume limited sample of galaxies and ignores edge e ects In practice then one
do es the following Davis Peebles b one generates on the computer a sample of p oints with no
intrinsic clustering but with the same selection criteria as those of the real galaxies Thus the mo ck catalog
matches the true catalog in the solid angle coverage and in the selection function One then counts pairs
with separation b etween r and r dr b oth in the real data call this quantity N r with D standing
DD
for data and b etween the real data and the mo ck catalog N r with R standing for random The
DR
correlation function is then estimated as
N r n
DD R
r
N r n
DR D
where n and n are the mean number densities of galaxies in the data and random samples One usually
D R
makes the number of random galaxies much larger than that of the real sample so that additional shot
noise is not introduced In general one can weight the pair counts any way one wants
X
w r r w r r N r
i j DD
ij
with a similar equation for N where the sum is over pairs of galaxies at p ositions r and r such that
RR i j
r jr r j r dr For ux limited samples weighting by the inverse of the pro duct of selection functions
i j
for the two galaxies gives equal volume weighting Saunders Rowan Robinson Lawrence show
that the variance in r is minimized for the weights
w r r
i
nJ r r
i
where J r was de ned ab ove Eq Note that Eq b ears a resemblance to the minimum variance
weights for the mean density Eq
The accuracy of Eq is limited by the accuracy of the estimate of the mean number density of
galaxies one cannot measure r on scales b eyond that on which it falls b elow the fractional uncertainty
in the mean density of the sample The mean density is uncertain due to the p ossibility of large scale
structure on the scale of the survey itself That is one never knows the extent to which a given volume
is a fair sample of the universe The rms uctuations in the density on the scale of a survey are given by
Eq but of course in order to estimate this we need to know P k which is what we are trying to
nd in the rst place However there exist estimators of r whose sensitivity to uncertainties in the mean
density is appreciably weaker than that of Eq Landy Szalay Hamilton b In particular
Hamilton b shows that the fractional error in the estimator
N r N r
DD RR
r
N
DR
is prop ortional to the square of the fractional error in the mean density This advantage of Eq over
Eq actually only holds for weighting w r r otherwise the two estimators give quite similar
i i
results In any case the di erences b etween the two b ecome apparent only on very large scales where the
correlation function is quite weak
There has b een a great deal of confusion over the prop er estimation of the error in the correlation
function There are two sorts of statistical error one could try to calculate
i The di erence b etween the true correlation function of galaxies in the volume surveyed and that
measured This error arises from the nite number of galaxies in the sample
ii Even if one could measure the correlation function in the volume surveyed without error it will di er
from the true correlation function of the entire universe of galaxies due to p ower on scales larger than
the sample
Many of the analyses in the literature consider only one of these two e ects The problem is inherently
di cult b ecause the variance in the two point correlation function necessarily dep ends on the three and
four p oint correlation functions to b e discussed in x In addition there exists strong covariance b etween
the estimate of the correlation function on di erent scales which of course is strongest for estimates at
two closely spaced scales
Peebles and Kaiser calculate the shot noise contribution to the correlation function
errors in terms of a cluster mo del if we think of the galaxy distribution as a smo oth eld with clusters
embedded the number of galaxies asso ciated with each cluster is nJ we restrict ourselves for the
moment to the case of a volume limited sample The number of indep endent pairs of galaxies at a given
separation is the number of observed pairs N divided by the number asso ciated with the clusters That
DD
is b ecause galaxies are clustered a ma jority of the galaxy pairs are redundant The resulting error in the
correlation function is then just given by Poisson statistics
nJ
N
DD
Kaiser notes that N n and thus for nJ is indep endent of n Thus he argues
DD
that given a nite amount of telescop e time one wants to sparse sample to the level that nJ i e
roughly one galaxy p er cluster maximizing the volume covered to minimize the e ect of p ower on scales
larger than the sample This is a meaningful strategy if in fact one s primary motivation is to measure
r on large scales This is the motivation b ehind the in sampling of the QDOT survey Lawrence et
al and the in sampling of the APM survey Loveday et al ab
Ling Frenk Barrow argue that the e ect of shot noise can b est b e estimated by making
b o otstrap realizations of a given sample and calculating the scatter in the determination of r from each
of these Although the mean r over the b o otstraps is unbiased this results in an overestimation of the
errors as shown by Mo Jing B orner and Fisher et al a Fisher et al a describ e
a brute force way to make realistic estimates of the correlation function error covariance matrix make a
series of indep endent N b o dy realizations of a given sample and calculate the scatter of the estimates of
r from each of these Of course this estimate will b e only as go o d as the p ower sp ectrum assumed for
the simulation itself although it is probably not terribly sensitive to the details Existing redshift surveys
are not large enough to b e able to a ord to split the surveys up into pieces and compute errors from the
variance in the estimated correlation function in each although Hamilton b describ es a practical
metho d to estimate correlation function errors from one s dataset itself by considering the contribution
that each subvolume in a sample makes to the correlations
The two point correlation function can b e applied not only to redshift data but also to surveys
containing only angular data The angular correlation function can b e de ned in analogy with Eq
the joint probability that galaxies b e found at angular p ositions r and r within the in nitesimal solid
angles d and d is
dP N d d w
where is the angle b etween r and r and N is the number density of galaxies on the sky Groth
Peebles calculated the angular correlation function of the Shane Wirtanen Lick galaxy counts
cf the b eginning of x they found that w is well t by a p ower law of slop e for scales smaller
than ab out with a sharp break on larger scales cf the challenge to their results by Geller Kurtz
de Lapparent and de Lapparent Kurtz Geller over the issue of plate matching More
recently the angular correlation function of the APM galaxies has b een calculated by Maddox et al a
who repro duce the Groth Peebles results on small scales however the APM correlation function
breaks on a somewhat larger scale than that of the Lick counts The APM data are of higher photometric
accuracy than the Lick data and b ecause the catalog was generated automatically it is immune from
the inevitable systematic e ects that counting galaxies by eye entails Other recent determinations of
the angular correlation function include Picard and Collins Nichol Lumsden Bernstein
has carried out a detailed analytic analysis of the error in the angular correlation function using the
estimator of Landy Szalay including the covariance terms and using the hierarchical hypothesis
Eq to include the e ects of three p oint and four p oint correlations The resulting expressions are to o
complicated to repro duce here but do an excellent job of matching the errors measured from Monte Carlo
simulations
The angular correlation function w for a ux limited sample is related to the spatial correlation
function by Limber s equation cf Rubin
R
d r d r r r r
w
R
d r r
where is the selection function r jr r j and is the angle b etween r and r It is straightforward
to show from Eq that a p ower law spatial correlation function of logarithmic slop e corresp onds
to an angular correlation function with logarithmic slop e Thus we exp ect the spatial correlation
function to b e a p ower law with slop e at least on small scales The spatial correlation function
has b een determined for essentially all the large redshift surveys discussed ab ove in x imp ortant pap ers
include Davis Peebles b Bean et al Shanks et al Davis et al de Lapparent
Geller Huchra Strauss et al a Fisher et al a Mo ore et al and Loveday et
al Because of the much smaller number of galaxies included in redshift surveys than in the angular
catalogs the spatial correlation function is determined with lower accuracy on large scales However these
studies and others have demonstrated convincingly that the spatial correlation function is indeed a p ower
law on small scales with a break at approximately km s The much quoted relation of Davis
Peebles b
r
r
h Mp c
is consistent with the observed p ower law b ehavior of w Eq also implies that the rms galaxy
uctuations within spheres of radius h Mp c are unity Eq This is a scale b elow which the
clustering is clearly strongly non linear and much of the formalism developed in x b ecomes irrelevant
A primordial p ower sp ectrum of p ower law slop e greater than zero implies that P It then
follows from Eq that the volume integral of the spatial correlation function over all of space must b e
zero meaning that the correlation function must go negative at some p oint Given a p ower sp ectrum for
example that of Standard Cold Dark Matter one can use Eq to predict that the correlation function
go es negative on scales ab ove h Mp c and reaches a minimum at h Mp c with an amplitude of
This is to o small an e ect to have b een measured in any existing galaxy sample Indeed if
one de nes the mean density of a sample from the sample itself as is usually done the integral of the
correlation function over the volume of the survey is forced to zero This e ect tends to bias the correlation
function low at least when it is estimated for small volumes
The p ower law nature of the correlation function has prompted some workers to suggest that galaxies
follow a fractal distribution with no preferred scale e g Coleman Pietronero In such a mo del
it would not b e p ossible to de ne a mean density of the universe it would b e a function of the scale on
which one measured it However the correlation function is de ned in terms of which has the mean
density subtracted already The correlation function of predictably is not scale free e g Guzzo et al
Calzetti Giavalisco Meiksin Peebles shows that the observed scaling of the angular
Eq has b een corrected for redshift space distortions see x
correlation function with depth rules out simple fractal mo dels cf Davis et al More complicated
multi fractal mo dels have b een prop osed they are reviewed thoroughly in Borgani
Distortions in the Clustering Statistics
The next logical topic to discuss would b e the determination of the p ower sp ectrum of the galaxy
distribution Before we do so however we outline the principal e ects which cause the observed correlation
function and p ower sp ectrum to di er from those which held following the ep o ch of radiation matter
equality extrap olated via linear theory to the present These are
i The galaxy distribution may b e biased with resp ect to the underlying mass distribution If linear
biasing holds Eq the correlation function and p ower sp ectra of galaxies di er from that of the
underlying dark matter by a constant factor b the shap e of these functions is una ected However
if the biasing is a a priori unknown function of scale then the p ower sp ectrum of galaxies gives us
no information of that of dark matter
ii We measure redshifts not distances of galaxies As Eq shows the two di er from one another
b ecause of p eculiar velocities
iii Although in linear theory the shap e of the p ower sp ectrum do es not change as clustering grows
simply b ecause the growth rate is indep endent of scale this is not true on scales that have gone
non linear on which the rms mass uctuations are not much less than unity
We will discuss the redshift space distortions and the non linear e ects here
Redshift Space Distortions
The e ect of p eculiar velocities on the shap e of structures can b e understo o d heuristically by imag
ining the gravitational in uence of a rich cluster of galaxies On small scales within the virialized cluster
itself galaxies have p eculiar velocities of km s or more which causes a characteristic stretching of
the redshift space distribution along the line of sight This is called the Finger of God which p oints
directly at the origin in a redshift pie diagram the Finger of Go d asso ciated with the Coma cluster is
apparent in Fig Thus a compact con guration of galaxies is stretched out along the line of sight greatly
reducing the correlations the smal l scale velocity dispersion of galaxies causes the correlation function to
be underestimated in redshift space
On larger scales a di erent e ect op erates galaxies outside the cluster itself feel the gravitational
in uence of the cluster and thus have p eculiar velocities falling into it A galaxy on the far side of the
cluster will thus have a negative radial p eculiar velocity and app ear closer to us in redshift space than in
real space while a galaxy on the near side will have a p ositive p eculiar velocity and app ear further away
from us Thus the gravitational in uence of a cluster causes a compression of structures thus enhancing
the correlation function
These two e ects can b e quanti ed The e ect of the large scale motions can b e calculated via linear
theory as was rst done in the present context by Kaiser cf Sargent Turner In brief one
calculates the change in the Jacobian of the volume element in going from real to redshift space The result
is that in linear theory b oth the p ower sp ectrum and correlation function are enhanced in redshift relative
to their real space counterparts by a factor
K
where was de ned ab ove in Eq For K so this is not a small e ect This calculation
is done in the distant observer approximation in which the volume surveyed is supp osed to subtend a
small angle from the p oint of view of the observer For further discussion of this approximation see Cole
Fisher Weinberg and Zaroubi Ho man
One way to disentangle the e ects of p eculiar velocities from true spatial correlations is to divide the
vector separating any two galaxies into comp onents in the plane of the sky r in the notation of Fisher et
p
al a and along the line of sight The e ects of redshift space distortions are purely radial and
thus the correlation function pro jected onto the plane of the sky is a measure of the real space correlation
function In practice then we measure as a function of b oth r and The subscript s reminds that
s p
this is a quantity measured in redshift space The pro jection of r onto the r axis yields a quantity
s p p
closely related to the angular correlation function Davis Peebles b
Z Z
i h
r y dy d r w r
r s p p p
p
where here is the desired real space correlation function as indicated by the subscript r For a p ower law
r
correlation function r r r the integral can b e done analytically yielding
r
w r r r
p p p r p
This is the approach that Davis Peebles b to ok to nd the result in Eq Fisher et al a
used this metho d to nd
r
r
r
h Mp c
for IRAS galaxies Saunders Rowan Robinson Lawrence measured r for IRAS galaxies using
r
a related approach they cross correlated the QDOT redshift survey with its parent D catalog Rowan
Robinson et al to suppress the redshift spacing distortions they nd results in excellent agreement
with Eq cf Loveday et al for a measurement of the APM correlation function using the same
technique The signi cance of the discrepancy in the amplitudes and slop es b etween the IRAS Eq
and optical Eq correlation functions will b e discussed in x
If we could measure the Kaiser e ect of Eq directly we could constrain the parameter The
di culty is that Eq is only valid on large scales where linear theory is valid and where the comp eting
e ect of small scale velocity disp ersion is unimp ortant But of course the correlations are small and di cult
to measure on these large scales Gramann Cen Bahcall and Brainerd Villumsen p oint
out that if the small scale velocity disp ersion were as large as predicted by standard CDM then the Kaiser
e ect would b e swamped by the suppression of the correlation function due to the velocity disp ersion until
one gets to truly enormous scales Nevertheless several groups have attempted to measure the Kaiser e ect
directly from redshift survey data Fry Gazta naga compared the correlation function measured in
redshift space for various redshift surveys to the angular correlation function of the same samples which
are free from redshift space distortions They found for the CfA survey
for the SSRS and for the IRAS Jy survey Alternatively one lo oks for the anisotropy
of r in the radial and transverse directions Hamilton de nes the angular moments of the
s p
correlation function as
Z
r P d r
s p
th
where P is the Legendre p olynomial and is the cosine of the angle b etween the line of sight and the
redshift separation vector He then shows that
r
R
r
r s s ds
r
in the linear regime Hamilton a applies this to the IRAS Jy sample to nd
Fisher et al b to ok a somewhat di erent approach including the e ects of b oth the small scale
velocity disp ersion and large scale Kaiser e ect and tting directly to r Following Peebles
s p
the relation b etween the real and redshift space correlation functions can b e written as
Z
i h
r w r dw f w jr
s p r
p
where the integral is over p ossible values of the radial p eculiar velocity di erence w and f w jr is the
distribution function of w at a given separation r An exp onential distribution is a go o d approximation to
the distribution function found in N b o dy simulations Fisher et al b show that it gives a b etter t
than do es a Gaussian to the real data Taking the shap e of the rst and second moments of f as a function
of r from N b o dy mo dels Fisher et al were able to reduce the mo del tting to two free parameters the
amplitude of the second moment i e the pairwise velocity disp ersion at r km s and the
amplitude of the rst moment v i e the mean pairwise streaming of galaxies at r km s They
show in analogy with Eq that linear theory predicts the following form for the rst moment
r
Z
H
dr r r v r
r
r r
r
Thus the measurement of v km s at km s together with the real space correlation
function Eq directly yields a value for they nd Although this statistic is shown to
b e unbiased with the help of N b o dy simulations its p ower is limited by the volume of the sample We
assume that any anisotropy measured in the sample is due to redshift space distortions but the real space
correlation function will b e isotropic only to the extent that the sample includes enough volume to average
over the orientation of elongated sup erclusters
It is not a priori obvious that the approach of writing the e ect of the redshift space distortions as
a convolution with the velocity distribution function Eq is consistent with linear theory in the form
of Eq linear theory predicts covariance b etween the velocity and density elds that is not included
in Eq However Fisher has b een able to repro duce Eq by expanding Eq to second
order assuming that f w jr is Gaussian with mean value and disp ersion as given by linear theory
The Fisher et al b analysis also measures the distortions on non linear scales to derive the
km s This is to b e compared with the Davis pair wise velocity disp ersion at km s
Peebles b value of km s from the CfA survey also measured by lo oking at redshift
space distortions The predicted values of for di erent p ower sp ectra as calculated with N b o dy
simulations are quite di erent so this quantity is of great interest for constraining mo dels High resolution
dissipationless simulations of Standard CDM indicate km s for the dark matter e g Davis et
al Gelb Bertschinger b far in excess of what is observed However this number has b een
controversial it is the velocity disp ersions of galaxies not dark matter particles which are observed The
velocity disp ersion of halos of dark matter particles in simulations Brainerd Villumsen Gelb
Bertschinger are smaller than that of the dark matter itself although these simulations su er from
A recent reanalysis of the CfA sample by Davis private communication nds km s after correction
of a small error in the original co de
the so called over merging problem in which groups of galaxies merge in enormous sup ergalaxies which
have no counterparts in the real world Because is weighted by pairs of galaxies such over merging tends
to cause the estimate of to b e biased low Hydro dynamical simulations of CDM which tend to avoid
the over merging problem radiatively dissipate some of the energy that would otherwise go into galaxy
motions and in fact nd a lower value for e g Cen Ostriker although still not low enough to
match the observed value ab ove Weinberg shows that the velocity disp ersion is very sensitive to
the details of the biasing mo del used to de ne galaxies from the distribution of dark matter
Velocity disp ersion analyses of other redshift surveys Mo et al show a larger for some
samples although the pair weighting nature of makes it quite sensitive to a few rare clusters in a survey
volume Zurek et al What is needed to settle this controversy is a new statistic which is less weighted
by the rare high velocity dispersion clusters and thus more strongly re ects the velocity disp ersion in the
eld
The small scale velocity disp ersion can also b e used to apply the Cosmic Virial Theorem Peebles
ab Peebles if one assumes statistical equilibrium of clustering on small scales and takes the
continuum limit one can show that
Z Z
H r z dr
r d z r z jr zj
b r r z
r
r
where is the three p oint correlation function to b e discussed in x This equation can b e simpli ed
assuming a hierarchical mo del for the three p oint correlation function Eq and a p ower law form for
the two point correlation function Eq of Peebles In practice the application of this equation
is hamp ered by our p o or knowledge of the three p oint correlation function More imp ortantly the linear
biasing mo del assumed in Eq is questionable at b est on these very inhomogeneous scales Finally
Carlb erg Couchman Thomas argue that galaxies may not b e fair tracers of the velocity eld on
small scales a form of velocity bias The Cosmic Virial Theorem has b een used to argue for a small value of
e
Davis Peebles b but if dark matter is not clustered with galaxies on the very small
scales of km s Eq will return an underestimated value of Bartlett Blanchard
Cole Fisher Weinberg take a parallel approach to Hamilton a based on the
p ower sp ectrum Just as we can separate the vector b etween two galaxies into parallel and p erp endicular
comp onents we can separate a wave vector k into parallel and p erp endicular comp onents if we work with
a subsample of a survey with smal l total opening angle with respect to the observer the distant observer
approximation discussed ab ove In analogy with Eq they de ne angular moments of the p ower
sp ectrum
Z
P k d P k P
and show that the rational expression in Eq can b e expressed as P k P k Applying this to the
Jy IRAS redshift survey they nd although they emphasize that non linear e ects
cause this to b e a lower limit A recent re analysis by Cole Fisher Weinberg parameterizes the
e ects of non linearity in the velocity eld by including a small scale velocity disp ersion in their mo del in
analogy to the analysis by Fisher et al b They nd for the IRAS Jy sample
and for the QDOT survey
As in all these metho ds non linear e ects have the p otential to break the degeneracy b etween
and b Non linear e ects make the e ective value of as derived from this metho d grow as a function of
scale until the linear regime is reached the scale at which the curve asymptotes is thus a measure of the
strength of the mass clustering and can b e used to put constraints on the bias parameter Unfortunately
existing redshift surveys are not extensive enough to measure this e ect with con dence
Non linear E ects
In linear p erturbation theory the real space density eld di ers from the primordial density eld only
by a universal scaling factor However this is no longer true when non linear e ects b ecome imp ortant
There has b een a great deal of work in recent years on non linear extensions to Eq Bernardeau a
Gramann a Nusser et al Giavalisco et al Mancinelli Yahil cf Mancinelli et al
for a comparison of these techniques and various approximate non linear schemes to bridge the gap
b etween linear theory and N b o dy simulations Peebles a Weinberg Matarrese et
al Brainerd Scherrer Villumsen Bagla Padmanabhan Here we wish to concentrate
on metho ds to take redshift surveys back in time to their initial conditions Weinberg adopts
the assumption that the initial density distribution function is Gaussian and notes that the rank order
of densities is likely to b e preserved even as non linear e ects skew the distribution function Thus he
applies a technique called Gaussianization whereby the rank order of the densities at di erent p oints is
conserved but the densities are reassigned to t a Gaussian form The details of this metho d dep end
on assumptions ab out galaxy biasing and the p ower sp ectrum The idea is to apply the Gaussianization
technique to redshift survey data measure the p ower sp ectrum of the resulting density eld and then
evolve the resulting initial conditions forward in time again using an N b o dy co de To the extent that
the assumed and measured p ower sp ectra match and the nal results agree with the original data one
has demonstrated consistency with the input mo del Weinberg applied this technique on a volume
limited subsample of the Pisces Perseus survey of Giovanelli Haynes The p ower sp ectrum was
consistent with that of standard CDM Unbiased mo dels did not work not repro ducing the lamentary
structure of the real data b was a b etter match to the real data Most imp ortantly the analysis tests
and nds consistency with the assumptions of Gaussian initial conditions and gravitational instability
Nusser Dekel have developed a time machine to take the observed density eld as derived
from a redshift survey or the POTENT metho d cf x b elow back in time The equations of motion
allow a decaying mo de Eq which gets ampli ed if one simply reverses the density evolution equations
Nusser Dekel instead start with the Zel dovich equation Eq which when expressed in
Eulerian co ordinates yields a rst order di erential equation for the velocity p otential which only allows a
growing mo de
v
r
v
D
where is the p otential asso ciated with the scaled velocity eld v v aD This Zel dovich Bernoul li
v
equation can b e integrated backwards in time from observations of the density elds N b o dy tests show
the results to repro duce the initial conditions b etter than do es linear theory
Gramann a shows that consideration of the continuity equation in the context of the Zel dovich
equation yields a correction term C to the right hand side of Eq given by
g
X
g g g
r C
g
x x x x
i j
i j
i j
where is the gravitational p otential N b o dy tests show this equation repro duces the non linear evolution
g
b etter than do es Eq although this approach has not b een applied to redshift surveys yet
Nusser Dekel have used the Zel dovich approximation in another version of their time
machine Assuming laminar ow one can write down the eigenvalues of the space derivatives of the initial
velocity eld in Lagrangian space in terms of the eigenvalues of the space derivatives of the observed
velocity eld in Eulerian space v x Assuming further that linear theory holds in the initial conditions
i i
as it should one can derive the initial density eld via Eq the nal result is
X
v x
i i
D
initial
v x H
i i
i
where D is the time dep endence of the growing mo de of gravitational instability Eq They use the
IRAS Jy redshift survey and the metho ds of x to generate the predicted velocity eld and thus
the initial density eld from which they determine the initial density distribution function They nd it to
b e accurately Gaussian Application of their technique to the observed velocity eld is describ ed in x
Another approach to non linear gravitational evolution was taken by Peebles a and
ampli ed by Giavalisco et al One can derive the exact equations of motion for a multi bo dy
gravitating system by nding the stationary p oints of the action S
t t
Z Z
X
m a m x x S L dt dt
i i
i
i
where the sum is over the particles in the system m are their masses x are their comoving p ositions
i i
and is the gravitational p otential Giavalisco et al then expand the p ositions x in a Taylor series of
i
which the Zel dovich equation Eq is the rst two terms
X
j
D t D C x x
j i i i
j
where the C are co e cients to b e determined Setting the derivative of the action S with resp ect to the
j i
C yields the set of equations
j i
t
Z
d
j
dt D t D a x ar x
i i
dt
which can b e solved for the unknowns C This metho d is exact except in regions in which multi streaming
j i
has o ccurred that is where a single p oint in Eulerian space corresp onds to more than one p oint in
Lagrangian space Giavalisco et al have tested this approach against spherical infall mo dels and
show that it converges very quickly Peebles a has used this metho d in the analysis of the
dynamics of the Lo cal Group and the rst attempts to extend this metho d to redshift surveys can b e
found in Peebles and Shaya Peebles Tully
Hamilton et al have taken an empirical approach to non linear evolution of the p ower sp ec
trum Considerations of galaxy conservation within a radius r xed in Lagrangian space around a galaxy
in an universe yields the hypothesis that the quantity a J r r is invariant with time where J
is given by Eq Tests with N b o dy mo dels show this in fact to b e the case and that this quantity
is indep endent of the initial p ower sp ectrum allowing the initial correlation function to b e read o that
measured Using this metho d on the IRAS CfA and APM correlation functions allowed them to repro duce
the initial correlation function which they found to b e b est t by a mo del invoking a mix of cold and
hot dark matter Extensions of their metho d can b e found in Peacock Do dds discussed further
b elow and Mo Jain White
The Power Spectrum
In principle the correlation function should contain all the information ab out the p ower sp ectrum
given that the two are a Fourier Transform pair Eq However there are two strong reasons to calculate
the p ower sp ectrum directly from redshift surveys
i Determination of the correlation function on large scales dep ends on the calculation of the di erence of
two numbers close to unity Eq while errors in the mean density of a sample a ect the amplitude
but not the shap e of the p ower sp ectrum
ii The error analysis and covariance b etween the determination of the p ower sp ectrum on di erent scales
is more straightforward than for the correlation function
The p ower sp ectrum can b e calculated from a galaxy redshift survey as follows The unsmo othed density
eld is given by a sum over Dirac delta functions
D
X
r
i
r
nV r
i
i
Taking the Fourier Transform of this yields
X
ik r
i
e
W k
k
nV r
i
i
where
Z
ik r
e
d r W k
V
V
Our estimator of the p ower sp ectrum is then
k V
k
k
where the factor of V on the right hand side gets the units right Several lines of algebra Fisher et al
show that the exp ectation value of this estimator is given by
Z Z
d r h k i d k P k G k k
nV r
where
V
G k k jW k k j
Thus the p ower sp ectrum estimator is given by the true p ower sp ectrum convolved with an expression
involving the Fourier Transform of the volume plus a shot noise term In the limit of an in nitely large
volume G approaches a Dirac delta function
The p ower sp ectrum as so de ned is a function of the direction of k In practice one averages h k i
over steradians in k space Fisher et al introduce the trick of measuring the p ower sp ectrum
within cylinders embedded within the survey volume whose long axis of length R is parallel to the vector
k If one then chooses k R n with n a p ositive integer the window function W vanishes Eq
This has two b ene ts k and therefore the p ower sp ectrum now scale exactly with mean density and
thus errors in the mean density a ect only the amplitude and not the shap e of the p ower sp ectrum In
addition the values of the p ower sp ectrum at di erent values of k are uncorrelated there is no covariance
b etween them
Feldman Kaiser Peacock have taken a slightly di erent approach They include a weight
function in Eq and derive an expression for the variance in the p ower sp ectrum estimator assuming
that the error distribution of P k is exp onential which follows from a Gaussian distribution of r
Minimizing the ratio of this variance to P gives the optimum weight function for galaxy i
w
i
n r P k
i
which is of a similar form to the optimum weight for the mean density Eq and the correlation function
Eq With this weight function the variance in the estimate of the p ower sp ectrum is given by
P k
R
d r nw r V
k
where V is the volume in k space o ccupied by the bin in question This expression assumes that the bins are
k
spaced far enough apart that the covariance is negligible this happ ens roughly for separations k R
where R is the characteristic dimension of the volume surveyed They also derive an expression for the
o diagonal terms of the covariance matrix see their pap er for details
The p ower sp ectrum of galaxies has b een calculated for a number of redshift surveys Baumgart
Fry Peacock Nicholson Park Gott da Costa Vogeley et al Fisher et al
Feldman et al Park et al da Costa et al b Lin Although there is reasonable
agreement in the literature now ab out the shap e of the p ower sp ectrum on small scales its amplitude
esp ecially on large scales remains uncertain The data are consistent with a slop e of P k k on
small scales and several authors e g da Costa et al b show an abrupt change of slop e at k
h Mp c All theoretical p ower sp ectra show a turnover on scales of h Mp c or more Fig this
has not yet b een seen unequivocally in the data A number of authors Peacock Torres Fabbri
Ru ni Kashlinsky Branchini Guzzo Valdarnini Padmanabhan Narasimha
have derived empirical p ower sp ectra to t the various data sets The most thorough of these analyses is
that of Peacock Do dds who have combined a number of the ab ove datasets together with the
real space correlation function from the angular correlation function Baugh Efstathiou They
correct each for the Kaiser e ect Eq on large scales and the e ects of small scale velocity disp ersion
following Peacock Moreover they correct for non linear e ects using the approach of Hamilton et al
x extending the formalism to the case They combine the di erent samples asking for
consistency while adjusting ve free parameters four bias values for Ab ell clusters radio galaxies optical
galaxies and IRAS galaxies plus which determines the strength of the Kaiser e ect Eq They
nd that b The small errors on this number imply that the redshift space distortion
I RAS
is unambiguously detected largely based on the comparison with the real space correlation function of
Baugh Efstathiou The constraint on b separately is less strong but is consistent with unity
I RAS
The ratios of the various bias factors are
b b b b
Abell r adio optical I RAS
Their results on the reconstructed linear p ower sp ectrum are shown in Fig The p oints are means
of the p ower sp ectrum from the various data sets for b The two curves are standard CDM
I RAS
dashed and CDM normalized to the p ower sp ectrum implied by the CMB anisotropies as
measured by the COBE satellite indicated by the b ox on the left hand side of the gure The data are
clearly far more consistent with the CDM mo del than with standard CDM b oth in amplitude
Fig The p ower sp ectrum as derived from a variety of redshift surveys after correction for non linear e ects
redshift distortions and relative biases from Peacock Do dds The two curves show the Standard CDM
p ower sp ectrum and that of CDM with Both are normalized to the COBE uctuations shown as the
b ox on the left hand side of the gure
and in shap e indeed Peacock Do dds come to this conclusion without consideration of the normalization
a orded by the COBE data This result is consistent with the conclusions of a number of workers in the
eld we discuss the issues further in x
The second moment of the density distribution function is directly related to the p ower sp ectrum via
Eq It can b e calculated directly from redshift surveys as the second moment of the count distribution
function after correction for shot noise cf Peebles Saunders et al and thus represents another
handle on the p ower sp ectrum itself This has b een done by Efstathiou et al Saunders et al
Loveday et al a Bouchet et al and Mo ore et al among others the results they nd
are consistent with those shown in Fig A compilation of second moment results for IRAS galaxies is
shown in Fisher et al a It makes the qualitative p oint that the variance drops as the scale increases
as is required by the Cosmological Principle and follows for any p ower sp ectrum with n Eq
Higher Order Statistics
As we have mentioned ab ove the p ower sp ectrum or its Fourier Transform the two point correlation
function is a complete statistical description of the density eld only to the extent that the phases of
the Fourier mo des of are random implying that the one p oint distribution function of the density eld
is Gaussian Even if this condition holds for the density eld in the early universe as is predicted by
in ationary mo dels it b egins to break down as so on as non linear e ects start to develop We discussed
theoretical approaches to this problem in x In this and the following section we discuss metho ds of
measuring these non linear e ects from the data
One can obviously extend the de nition of the two point correlation function to higher order We can
de ne the three p oint correlation function for the continuous density eld as
r z jr zj h x x r x z i
However the practical de nition in terms of the p oint distribution is more complicated b ecause of the
need to correct for the contribution due to the fact that galaxies have a two point correlation function In
analogy to Eq the probability of nding a galaxy at distinct p ositions r r and r within volume
elements dV dV and dV is
dP n dV dV dV r r r
r r r r r r
a b c a b c
where r r and r are the sides of the triangle de ned by the three p oints One can immediately see the
a b c
di culty in measuring as it requires subtracting four terms from the triple counts In addition the three
p oint correlation function is now a function of three numbers not just one The problem only gets worse
for higher order correlation functions the equivalent expression to Eq for the four p oint correlation
function has fteen terms on the right hand side Eq of Peebles Despite these di culties
the three and four p oint correlation functions have b een measured for the Shane Wirtanen counts Fry
Peebles Fry Szapudi Szalay Boschan the CfA redshift survey Bonometto
Sharp Gazta naga and the IRAS samples Meiksin Szapudi Szalay Bouchet et al
These samples have measured non zero three and four p oint correlation functions on small scales
indicating that the phases are indeed not random More imp ortantly they have shown that the three
and four p oint correlation functions display a certain symmetry with resp ect to the two point correlation
function
r r r Q r r r r r r
a b c a b a c b c
where Q is indep endent of scale and triangle con guration to the level that the data can distinguish these
things In particular there are no lo op terms prop ortional to r r r which puts strong con
a b c
straints on the form of biasing e g Szalay A similar expression holds for the four p oint correlation
th
function It has therefore b een hypothesized that Eq can b e generalized to the N order correlation
function indicated as not to b e confused with the of Eq Balian Schae er assume
N
that the N p oint correlation function shows scale invariance
N
r r r r
N N N N
for any They show that this allows one to write
N
V S V
N N
where the constants S uniquely de ne the hierarchical scaling and the correlation functions averaged
N
over a sphere are given by
Z
d r d r d r r r r V
N N N N
N
V
V
This volume averaging integrates over the shap e information in the high order correlation function Al
though there is much that can b e learned from the dep endence of the high order correlations on the angles
b etween the N p oints Suto Matsubara Fry the volume averaged statistic is much more
th
robust for small datasets Moreover the V are equal to the irreducible N order moments of the
N
V h i the kurtosis is V h i h i and so density distribution function the skewness is
on Peebles For a p ower law correlation function the relation b etween r and its volume average
can b e calculated analytically Peebles Groth
There has b een a great deal of interest in recent years to calculate the hierarchy of S b oth from
N
the theoretical and observational sides The use of volume averaging mitigates the need to calculate the
N p oint correlation function with its dep endence on N N separations In practice one calculates
the moments of the galaxy count distribution function which are then corrected for shot noise e ects
N
following x of Peebles cf Szapudi Szalay Gazta naga Yokohama to yield
N
This only works for volume limited samples or angular data ux limited samples require a more elab orate
correction Saunders et al One can then compare the observations with the predicted scaling
Eq As we will see momentarily this scaling holds remarkably well giving supp ort to the scale
invariant hypothesis Eq
However Eq or equivalently Eq seems to have b een pulled out of a hat Before we show
the observational evidence for them let us discuss their theoretical motivation For initial conditions with
random phases all S for N are zero initially However as clustering grows and b ecomes non linear the
N
density distribution function b ecomes non Gaussian and higher order moments b ecome non zero Peebles
rst calculated the skewness i e third moment of the unsmoothed density eld in second order
p erturbation theory for an universe and showed that
in agreement with Eq However observationally we are always limited to the smo othed density eld
for which the quantity S dep ends on the p ower sp ectrum Juszkiewicz Bouchet Colombi
Z
d k d k
P k P k W k W k W jk k j S
k
P
k
where is the cosine of the angle b etween k and k W is the Fourier Transform of the window cf Eq
and
n
d log P k
n
n
d log k
where the derivative is calculated at the smo othing scale Eq is valid for The
calculations get more di cult for increasing N calculation of S requires the application of N order
N
p erturbation theory One can show straightforwardly however that in every case the scaling of Eq
holds Fry ab Bernardeau b In a mathematical tour de force Bernardeau b has set up a
formalism for calculating the S for all N for a tophat window function and presents expressions up to
N
N as a function of the i N It turns out that it is more di cult mathematically to do the
i
calculations for Gaussian smo othing although the cases N Juszkiewicz et al and N Lokas
et al have analytic solutions Analogous calculations have b een done for the non linear evolution of
the p ower sp ectrum by Juszkiewicz Makino Sasaki Suto Jain Bertschinger and
others Feldman et al examine the cumulative distribution function of j k j and show it to b e
accurately exp onential as exp ected in a Gaussian eld However the exp ected distribution in the mildly
non linear regime in the presence of shot noise has not yet b een calculated
The results quoted thus far are for an universe in real space The dep endence of the S on
N
is extremely weak Bouchet et al and is also insensitive to the transformation from real
to redshift space Moreover it can b e shown that the scaling relations Eq continue to hold under
arbitrary lo cal biasing transformations Eq although the values of the S themselves change Fry
N
Gazta naga Juszkiewicz et al Fry Biasing mo dels which are non lo cal in which the
probability that a galaxy b e formed at a given p oint is a function of events removed by tens of Mp c from
that p oint have b een invoked to explain the mismatch of the observed p ower sp ectrum with Standard CDM
e g Babul White Bower et al However such mo dels break the scale invariant hierarchy by
adding lo op terms e g Szalay and Frieman Gazta naga have used the excellent agreement
with the scale invariant predictions e g Fig to rule out a wide class of these mo dels
The calculation of S and S has b een done for the CfA and SSRS samples Gazta naga and
the IRAS Jy sample Bouchet et al as well as for various angular catalogs Szapudi Szalay
Boschan Meiksin Szapudi Szalay Gazta naga Szapudi et al All these authors
have found b eautiful agreement with the predicted scaling relation the results from the IRAS survey are
shown in Fig Gazta naga nds that the value of S varies slightly with scale this is exp ected by
Eq if the p ower sp ectrum is not a pure p ower law Inserting the p ower sp ectrum for the APM counts
of Baugh Efstathiou in Eq gives b eautiful agreement with the observed values implying
that the biasing is very weak This conclusion dep ends on the biasing b eing linear non linear biasing can
mimic absence of biasing in the dep endence of S on scale
Thus the scaling relations Eq were originally hypothesized on largely aesthetic grounds They
were found to b e predicted by p erturbation theory assuming Gaussian initial conditions and growth of
structure via gravitational instability Indeed calculations of the skewness in initially non Gaussian mo dels
Fry Scherrer Bouchet et al show that the leading b ehavior go es like rather than
as observed May we therefore conclude that we can rule out non Gaussian mo dels Unfortunately the
answer is no First the term decays with time and at late times may b e negligible Moreover as
many have quipp ed referring to non Gaussian mo dels is a little like referring to non elephant animals the
range of p ossible non Gaussian mo dels is vast Weinberg Cole set up a series of non Gaussian
mo dels by skewing the density distribution function of an initially Gaussian mo del but the resulting non
Gaussianity exists only on the smo othing scale on which is de ned On scales appreciably larger than
this the Central Limit Theorem guarantees that the distribution is Gaussian again and the scaling laws
b etween the various moments will continue to hold Thus one needs to examine each sp eci c non Gaussian
Fig The skewness upp er panel and the kurtosis lower panel of the IRAS density eld for various smo othing
lengths as a function of the variance The plots are logarithmic The lines drawn are least square ts with slop es
and resp ectively This gure is taken from Bouchet et al
mo del in turn ask for its predictions for the scaling either using analytic techniques or N b o dy simulations
Moscardini et al Weinberg Cole and compare with the data This pro cess has not b een
carried out in detail at this writing
Finally Fig shows that the scaling relation b etween the moments predicted by second order
p erturbation theory holds well into the highly non linear regime where it has no right to hold although
this has b een a working hypothesis for the closure of the so called BBGKY equations cf Davis Peebles
Similar b ehavior has b een seen in N b o dy simulations e g Juszkiewicz et al There is
controversy ab out the e ect of redshift space distortions in these analyses Lahav et al a Suto
Matsubara and Matsubara Suto argue on the basis of N b o dy simulations that redshift
space distortions make the S closer to constant than in real space in the non linear regime a conclusion
N
supp orted by the analytic calculations of Matsubara a and the observations of the Pisces Perseus
region by Ghigna et al However Fry Gazta naga nd that the S are remarkably constant
N
in b oth redshift and real space on small scales in a variety of redshift surveys In any case there exists no
analytic argument as to why hierarchical scaling should hold into the non linear regime
The Density Distribution Function and Counts in Cel ls
One of the striking features of the galaxy distribution is the presence of voids as much as km s
in diameter The statistical to ols that we have presented thus far do not clearly indicate their presence we
see no feature in the correlation function on such scales Thus we lo ok for a statistic that is more sp eci cally
oriented to describing the visible structures that we see One such statistic is the void probability function
Imagine laying down a series of spheres of radius r randomly within a large volume p opulated with galaxies
De ne the void probability function P r as the fraction of those spheres which contain no galaxies In
the absence of clustering Poisson statistics yields
nV
e
P r
where n is the mean density of galaxies and V r is the volume of a sphere In the clustered case
P dep ends on the whole hierarchy of correlation functions as shown by White
N
X
nV
P r exp r
N
N
N
where the volume averaged correlation functions were introduced in Eq Thus the void probability
function is a complementary statistic to the correlation functions One can compute not only P but also
the probability of observing N galaxies within a sphere it is related to P as
N N
P n r n
P r
N
N
N n
The void probability function is clearly a strong function of the sparseness of a given sample and thus
masks to a certain extent the underlying galaxy distribution One way around this is to de ne a sampling
indep endent quantity
r ln P r
nV
so that a Poisson distribution gives Eq If the hierarchical hypothesis Eq holds then
Eq implies that is a universal function indep endent of the sampling indeed this is observed
for the IRAS galaxies Bouchet et al but see Vogeley et al r is a smo oth monotonically
decreasing curve with no features no particular scale is picked out
The void probability function is p otentially a useful discriminant of cosmological mo dels However
Weinberg Cole and Little Weinberg found that the void probability function is insensitive
to the p ower sp ectrum or the density parameter and is more sensitive to the details of the biasing scheme
than to the bias value itself
Under the scale invariant hypothesis Eq one can make quite detailed predictions for the form
of the P Balian Schae er Indeed the density distribution function is given by cf Bernardeau
N
Kofman
i
Z
p p
X
y y
dy exp S f
p
i p
p
i
which is an exact expression to the extent that the S are exact The P follow from this after convolving
p N
with a Poisson distribution to include the e ects of shot noise
The various predictions developed by Balian Schae er based on the scale invariant hypoth
esis have b een checked in N b o dy simulations Bouchet et al Bouchet Hernquist although
very dense sampling is required to test the full suite of predictions The P have b een derived observa
N
tionally for various data sets Alimi Blanchard Schae er Maurogordato Schae er da Costa
Lahav Saslaw Bouchet et al The latter authors compare the observed counts in cells
with various mo dels and nd that a range of mo dels including those of Carruthers Shih Saslaw
Hamilton Coles Jones b ecome degenerate at the sparse sampling of existing surveys and
the data cannot distinguish b etween them
Another approach to the density distribution function which is just P r at constant r was intro
N
duced by Juszkiewicz et al On large scales where the second moment of the distribution function
h i is small the deviation of the distribution function from a Gaussian is exp ected to b e small Thus
it makes sense to expand the distribution function in orthogonal p olynomials relative to the Gaussian The
Edgeworth expansion do es this
x
e
f x
S H x S H x S H x
where the H x are the Hermite p olynomials and x This is found to give an excellent t to the
N
distribution function in N b o dy mo dels for small although for the Edgeworth expansion starts
going unphysically negative at mo derate values of x Maximum likelihoo d calculations of S using ts of
N
Eq to the observed P may b e more robust than calculation of the moments directly Indeed the
N
results of the moments metho d are heavily weighted by the tails of the distribution This is dangerous
when working within a nite volume b ecause one is sensitive to the rare dense clusters Colombi Bouchet
Schae er ab
We motivated this section by p ointing out that standard correlation statistics do a p o or job of quanti
fying the largest scale features that are apparent to the eye in redshift maps The void probability function
go es part of the way in lling this need although it is not as discriminating a statistic b etween di er
ent cosmological mo dels as was hop ed There have b een a number of pap ers discussing various statistics
to capture the largest scale features apparent in the redshift maps Tully a Broadhurst et al
Babul Starkman although again the robustness of these statistics and their discriminatory
p ower have b een questioned Postman et al Kaiser Peacock One of the most successful
statistics to describ e large scale structure in a way complementary to correlation functions uses concepts
from top ology to which we now turn
Topology and Related Issues
What is the mental picture we should have of the top ology of the galaxy distribution Is it a uniform
sea of galaxies punctuated by rich clusters embedded in it like meatballs in a b owl of spaghetti Dramatic
voids are what catch the eye in Fig would a b etter picture b e a uniform distribution with voids sco op ed
out of it like a piece of swiss cheese If the density distribution is Gaussian then there should in fact b e a
th th
That is if one wants a result accurate to N order one needs the S p N calculated to N order as well
p
it is not adequate to calculate the S to lowest non vanishing order The higher order corrections to the S have
p p
not yet b een calculated
top ological symmetry b etween underdense and overdense regions like a piece of sp onge Motivated by these
considerations Gott Melott Dickinson who are resp onsible for these fo o d analogies suggested
measuring the top ology of the galaxy iso density surfaces In particular given a surface in three space one
can de ne the principal radii of curvature a and a at every p oint By the Gauss Bonnet theorem the
integral of the Gaussian curvature K a a over the surface is given by
Z
C K dA g
where g is the genus number of the surface the number of holes minus the number of disjoint pieces
plus Thus measurements of the Gaussian curvature give the genus of the surface A plot of genus
of the iso density surface as a function of density thus tells us the change in the top ology at di erent
contrast levels What do we exp ect in the Gaussian mo del At very high density contrasts the iso density
contours will surround isolated clusters and thus the genus will b e negative Similarly for close to the
iso density contours will surround isolated voids and again the genus will b e negative The mean iso density
contour will b e multiply connected and sp onge like and thus have a p ositive genus One can calculate
analytically the genus number in the Gaussian case Doroshkevich Bardeen et al Hamilton
Gott Weinberg cf Coles for sp eci c non Gaussian mo dels
V hk i
e
g
where V is the volume of the survey is the level of the density in units of the rms of the density
eld
R
E D
d k k W k P k
R
k
d k W k P k
is the second moment of the smo othed p ower sp ectrum and W k is the Fourier Transform of the smo othing
window Measurements of the genus as a function of thus characterize the general top ology of the density
eld In particular we can test the Gaussian hypothesis by comparing the observed form to Eq
To the extent that the observed genus curve is well t by the Gaussian form the amplitude of the curve
is a measure of the shap e of the p ower sp ectrum at the wavelength of the smo othing The amplitude of
the p ower sp ectrum cancels out of Eq Gott and collab orators use a volume weighting technique
to reduce the sensitivity of the top ology statistic to non linear evolution and to separate the top ological
information from that carried by the density distribution function Calculations of the genus curve from
redshift surveys have b een carried out by Gott et al Mo ore et al and Vogeley et al
using a tessellation technique for measuring the genus number Gott et al cf Weinberg for the
source co de Unfortunately the volume of existing surveys is small and thus for smo othing lengths in the
linear regime the maximum genus levels are The results show a slight meatball shift relative to
the Gaussian case that is the overdense contours show larger values of g than Eq would predict
This is in the sense exp ected from non linear evolution Matsubara b The amplitude of the genus
curve as a function of smo othing scale is in rough agreement with that predicted by CDM although Mo ore
et al nd some evidence for p ower in excess of CDM predictions on large scales Statistical errors
of the measurement of genus are usually calculated using b o otstrap techniques although these su er from
the same drawback as b o otstraps for correlation functions x At the moment there is no rigorous
error analysis of the genus statistic nor any calculation of the e ects of shot noise which will tend to make
the distribution function lo ok more Gaussian Very recently Matsubara has studied the e ect of
redshift space distortions on the genus statistic
A related statistic was invented by Ryden the area of the iso density surfaces For a Gaussian
eld one again exp ects a symmetric function which again p eaks at the mean density The area is given
by
V hk i
e
A
Ryden et al invented a clever technique to measure this statistic involving counting how often
skewers put randomly through the survey volume intersect the iso density surface and applied this to the
CfA survey and the Giovanelli Haynes Perseus Pisces survey At km s Gaussian smo othing the
results closely matched the linear theory predictions At km s smo othing however the data showed
stronger deviations from Gaussianity than did any of the mo dels examined However their mo dels were
probably not evolved forward to b ecome su ciently non linear on km s scales this remains a problem
for further investigation
The Dipole
One of the early motivations b ehind redshift surveys of the full sky was to apply Eq to the Lo cal
Group The CMB shows a dip ole anisotropy of amplitude T T Kogut et al and references
therein which is interpreted as a Doppler e ect due to the motion of the earth relative to the rest frame
of the last scattering surface When transformed to the barycenter of the Lo cal Group following Yahil et
al this motion is km s towards l b Galactic co ordinates Given
a full sky redshift survey a comparison of this motion with the dip ole moment of the galaxy distribution
is a direct measure of In fact b ecause b oth gravity and received light ob ey the inverse square law if
one assumes a constant mass to light ratio for the galaxies there is a direct prop ortionality b etween the
p eculiar velocity and the ratio of the dip ole and monop ole moments of the light distribution allowing
Eq to b e applied using angular data only Gott Gunn The angular dip ole moment of the
galaxy distribution has b een measured by a number of authors using a variety of galaxy catalogs Meiksin
Davis Yahil Walker Rowan Robinson Villumsen Strauss Lahav Harmon
Lahav Meurs Lahav Rowan Robinson Lynden Bell Plionis Lynden Bell Lahav
Burstein Kaiser Lahav Scharf et al The rst impressive result of these analyses
is that the vector direction of the light dip ole agrees with that of the CMB dip ole to within
dep ending on the sp eci c sample and analysis used There is greater disagreement in the amplitude of the
dip ole with results varying from to Rather than discuss the details of this here let us
move to the application of Eq using redshift surveys
The rst measurement of the gravitational dip ole from a redshift survey was by Davis Huchra
using a combination of the CfA and Revised Shapley Ames Sandage Tammann surveys
Given the limited sky coverage of their sample they were able only to measure the comp onent of the
acceleration towards the Galactic p oles which they compared to measured Virgo centric infall x to
nd values of ranging from to dep ending on exactly what assumptions were made More
recently Pellegrini da Costa combined redshift survey data from several di erent surveys and
carried out a similar comparison to the Virgo centric infall they found in the range to Further
progress had to await the completion of redshift surveys of the entire sky The dip ole moment of the IRAS
QDOT redshift surveys was calculated by Rowan Robinson et al who found convergence of the
dip ole only b eyond km s The amplitude of the dip ole implied Hudson b
This pap er was written b efore the concept of biasing was formulated and so the results are quoted in terms of
not
used his reconstruction of the optical galaxy density eld to calculate the galaxy dip ole assuming that the
dip ole converges within km s allowed him to conclude that
We cannot simply calculate the right hand side of Eq given a redshift survey The quantity that
we do calculate is the dip ole moment of the galaxy distribution
X
W r r
i i
Dip ole
n r r
i
i
i
This di ers from the dip ole integral b ecause of a number of e ects which we need to quantify
i Shot noise which of course gets worse as the sample gets sparser at greater distances In addition
Eq assumes a constant mass p er galaxy indep endent of the luminosity It is straightforward to
recast Eq in terms of luminosities which is more appropriate for a luminosity measured in the
optical than the IRAS bands cf Hudson b App endix A of Strauss et al c shows that
the shot noise is a more imp ortant e ect than mass to light ratio for all but the nearest galaxies
ii The integral extends over all of space while the galaxy dip ole is necessarily measured through a
window W r with cuto s at b oth large and small distances to minimize shot noise Juszkiewicz
Vittorio Wyse Lahav Kaiser Ho man and Peacock p oint out that the
contributions to the dip ole due to material outside of the window can b e substantial
iii Eq assumes linear theory while non linear e ects also come into play on small scales
iv No redshift survey truly covers the whole sky We discussed this issue in x
v As Kaiser p ointed out redshift space distortions cause systematic errors in the derived dip ole
In particular if redshifts are corrected for a value of the Lo cal Group p eculiar velocity in error by an
amount v the derived dip ole will have a spurious contribution the rocket e ect given by
Z
v d ln dr
r d ln r
Strauss et al c develop a maximum likelihoo d analysis that allows them to take the rst three of
these e ects into account A p ower sp ectrum must b e assumed in the analysis in order to quantify the
e ect of density uctuations outside the assumed window Indeed one can include parameters of the p ower
sp ectrum in the maximum likelihoo d analysis although in practice the constraints one can put on mo dels
are not very strong The ro cket e ect is minimized using self consistent solutions for the velocity eld cf
x The IRAS data imply where the angle b etween the acceleration and velocity vectors
is dep ending on which self consistent velocity eld is used
In mo dels without large amounts of p ower on large scales one exp ects the dip ole to converge within
the volumes prob ed by redshift surveys that is the contribution to the dip ole on large scales should b e
negligible The Strauss et al results are consistent with the dip ole converging within km s although
this dep ends on which self consistent density eld is used and how one corrects for the Kaiser e ect In
addition there is a substantial dip ole moment contributed by galaxies b etween and km s
aligned with the low redshift dip ole although the sample is so sparse at those redshifts as to make this
signi cant at only the level If this large additional contribution to the dip ole is found at higher
signi cance level with deep er redshift surveys this will imply large amounts of p ower on large scales and
a smaller value of than inferred ab ove
Spherical Harmonics
An approach to quantifying the galaxy density eld complementary to correlation functions uses the
metho d of spherical harmonics The Fourier comp onents of the density eld that are used in the p ower
sp ectrum are orthonormal within a cub e Spherical harmonics are an orthonormal set of functions on a
sphere and thus are esp ecially appropriate for full sky samples They o er a natural way to smo oth the
data if one expands to a given order in l the smo othing length is an increasing function of distance from
the observer mimicking the drop o in the sampling in a ux limited sample
Spherical harmonics are simply the generalization to higher order of the dip ole analysis discussed in
the previous section They have b een used for many years to describ e the distribution of galaxies in the
celestial sphere when redshift information was unavailable Peebles Peebles Hauser
Fabbri Natale Scharf et al One can express the galaxy density on the sky in spherical
harmonics as
m l
X X
c Y
lm lm
m l l
where the co e cients of the observed dataset are given by
Z
X
N
Y c d Y
i i lm
lm lm
sample
i
sample
The sum is over the N galaxies in the sample and the integral is over the solid angle subtended by the
sample For an exactly full sky sample the second term vanishes
sample
The spherical harmonics can b e related to the p ower sp ectrum straightforwardly Scharf et al
show that the exp ectation value of the square of the co e cients in a Gaussian mo del with p ower sp ectrum
P k is given by
E D E D
X
0 0 0 0
jc j j ja j jW
lm l m ll mm
0 0
l m l
where
Z
0
mm
W d Y Y
0 0 0
lm
ll l m
sample
is a tensor which couples together di erent mo des in the case of incomplete sky coverage and the spherical
harmonics in the case of complete sky coverage are given by
Z Z
D E
N
m
ja j k P k dk r r j k r dr
l
l
sample
Scharf et al used the angular distribution of the IRAS Jy sample to put limits on the p ower
sp ectrum parameterized by an amplitude and a shap e parameter h Eq A maximum likelihoo d
analysis yields h with remarkably tight error bars
Scharf Lahav have extended this analysis using the redshift information of the Jy
sample as well Much the same formalism is used with the addition of a redshift weighting factor f r in the
expansion of the density eld Eq In analogy to the dip ole analysis describ ed in the previous section
the growth of all the multipole moments with redshift is used as a diagnostic of the p ower sp ectrum The
resulting constraints on the p ower sp ectrum are similar to those which Scharf et al found from the
angular data alone although Scharf Lahav p oint out that redshift space distortions are a non negligible
e ect
Fisher Scharf Lahav have calculated these redshift space distortions in linear theory They
nd that the rms spherical harmonic amplitudes for a full sky redshift survey i e as
sample
measured in redshift space is
Z
E D
ja j k P k dk
lm
Z Z
df
r r j k r dr r r f r j k r dr
l l
l
k dr
compare with Eq When measured for a real data set there is of course an additional term due to
shot noise The dep endence of Eq means that one can use the redshift distortions to measure in
exact analogy to the analyses of redshift space distortions of the correlation function and p ower sp ectrum
Of course the shap e and amplitude of the p ower sp ectrum also come into play using a generic CDM
sp ectrum Eq Fisher nds and A similar analysis by Heavens
Taylor yields
Recovering the Real Space Density Field
As we have emphasized throughout this review redshifts are not equivalent to distances the two
di er due to p eculiar velocities However to the extent that p eculiar velocities are due to gravity and
that linear theory holds we can use Eq to estimate these p eculiar velocities Of course this requires
knowledge of the density eld of which have a distorted view due to p eculiar velocities Thus we lo ok for
a self consistent solution to the density and velocity eld given redshifts and p ositions for a ux limited
redshift survey of galaxies and assuming a value of In practice this is doable only for full sky redshift
surveys for which the integral in Eq can b e carried out
Yahil et al describ e an iterative technique to nd this optimum solution We describ e it here
in its latest incarnation Willick et al d as applied to the IRAS Jy sample
i Galaxies are initially placed at their redshift space p ositions The selection function is calculated by
the metho ds of x The wide zone of avoidance centered on the Galactic plane is lled with
galaxies at a density interpolated from higher density regions x
ii A smo othed density eld is de ned by assigning the galaxies to a grid weighting by and smo othing
with a Gaussian using FFT s The resulting density eld is ltered with the p ower preserving lter
x taking into account the increased shot noise as a function of distance from the origin
iii Eq is solved for the velocity eld again using FFT techniques
iv Given the density eld r and velocity eld v r along the line of sight r to a given galaxy with
Lo cal Group redshift cz and ux f one can calculate the probability P r jcz f dr that it have distance
b etween r and r dr
P r jcz f dr r r dr r f
cz fr r v r v g
exp
v
where is the galaxy luminosity function v is the p eculiar velocity at the origin and is a
v
measure of the small scale velocity disp ersion not included in the smo oth velocity eld mo del This
metho d is inspired by the analysis discussed in x b elow If the velocity eld mo del were p erfect
the exp onential would b e replaced by a delta function at cz r r v r v the Gaussian included
here parameterizes our ignorance ab out the velocity eld on scales smaller than the smo othing length
In the presence of clusters there are regions in which the p eculiar velocity changes with distance fast
enough that one nds triple valued zones in which a given redshift can corresp ond to three distances
This is illustrated in Fig which shows the relation b etween redshift and distance along a line of
sight which intersects a large cluster Infall into the cluster causes the dramatic S curve A galaxy at
a redshift of km s could lie at any of the three distances The probability function of Eq
is shown as well it correctly parcels the probability among these three solutions In particular the
nearest crossing p oint is most strongly favored given the luminosity function weighting
v The density eld is recalculated using the distribution function ab ove and it and the velocity eld are
output The value of is incremented on each iteration so that the e ects of p eculiar velocities grow
gradually the selection function is recalculated and the iterations start again at step iii ab ove
This is the metho d used to generate Figs ab ove This basic approach of an iterated solution to the
velocity eld via Eq has b een used by Yahil et al Strauss et al c Hudson ab
and Freudling et al
Nusser Davis a take a di erent approach to the problem They p oint out that the di er
ence b etween the redshift and real space p osition of galaxies is directly related to the displacement of a
galaxy from t to the present the Zel dovich approximation Eq Conservation of galaxies and the
assumption that the velocity eld is irrotational allows them to write down a Jacobian for the transfor
mation from the initial conditions in real space to the nal con guration in redshift space which yields a
di erential equation for the velocity p otential Expanding and the density eld s in redshift space
on a given shell in spherical harmonics yields the equation
d d l l d ln d
lm lm lm
s
lm
s ds ds s s d ln s ds
where is the selection function and s is the redshift co ordinate This equation can b e integrated for
lm
using standard numerical techniques and the radial velocity eld can then b e derived by di erentiation
This metho d has the advantage that it do es not require iteration every term in Eq is in redshift
space but b ecause it requires a one to one corresp ondence b etween real and redshift space it do es not
allow the existence of triple valued zones
A similar approach is taken by Fisher et al d They decomp ose the density eld with spherical
harmonics and spherical Bessel functions for the radial comp onent
n l
l max l
max
X X X
C j k r Y r r
ln l n lm
n
l m l
They use the fact that for a survey with full sky coverage redshift space distortions couple only
mo des with a given l but with di erent n s but there is no coupling b etween di erent angular mo des The
coupling matrix is analytic and its inverse allows the real space to b e calculated from the redshift space
one spherical harmonic at a time The velocity eld then follows directly from Eq In practice Fisher
et al d use the Wiener lter to suppress shot noise This metho d also do es not require iteration
Fisher et al d test the three metho ds presented here with the aid of a mo ck IRAS Jy redshift
survey drawn from an N b o dy simulation all three give residual rms errors in the radial p eculiar velocity
eld in the Lo cal Group frame for galaxies within km s of km s These comparisons have
only b een made at the p ositions of the galaxies used in the analysis there is a need to compare the full
velocity and density elds on a uniform grid
Other approaches to real space reconstruction of redshift surveys include Kaiser et al Taylor
Rowan Robinson Gramann b Tegmark Bromley
Fig An illustration of a triple valued zone The S shap ed curve shows the relation b etween redshift and distance
along the line of sight to a cluster A galaxy with a redshift of km s can lie at three distinct distances
When the small scale noise inherent in any velocity eld mo del as given by the scattered p oints is taken into
account the distribution function of distances for a given redshift gets smo othed out as shown as the three p eaked
curve at the b ottom
Clustering of Di erent Types of Galaxies
Galaxy formation is a very p o orly understo o d pro cess Our di culties in understanding it stem b oth
from our ignorance of the nature of dark matter which presumably forms the p otential wells within which
the baryons that eventually form the stars of the galaxy fall and the extreme complexity of the diverse
hydrodynamic and stellar dynamic e ects that b ecome imp ortant as gas b egins to radiate and stars form
In the context of this review our principal concern is to gain some understanding of the relative distribution
of the dark matter and galaxies b ecause it is the sum of the two which gravitates while we can observe
only the latter directly In x we introduced the concept of biasing and discussed the various contexts
in which we might imagine that the distributions of galaxies and dark matter might di er Here we ask
how we might nd observational evidence for biasing
We observe that galaxies come in a variety of types de ned morphologically from elliptical galaxies
through lenticulars and then spirals b oth barred and unbarred and nally irregulars and dwarfs of various
sorts If galaxies as a whole are biased relative to the dark matter in the sense that the large scale
distributions of the two di er on large scales one would exp ect that the pro cess of galaxy formation caused
the distribution of each of the galaxy types to b e biased with resp ect to one another Again b ecause galaxy
formation is a p o orly understo o d pro cess we do not have an accepted mo del for the Hubble sequence of
morphological types and we cannot b e much more sp eci c than this at this stage of our understanding
Nevertheless this o ers a well posed observational problem if we can measure biasing of one galaxy type
relative to another we have the p otential of constraining biasing mo dels
One form of relative biasing has b een known ab out since the time of Hubble the cores of rich clusters
are preferentially rich in elliptical galaxies This was put on a rm quantitative basis by Dressler ab
and Postman Geller who showed that the relative fraction of elliptical galaxies rose from its
mean of in the eld starting at overdensities of to nearly unity in the highest density regions
of clusters This means that redshift maps of elliptical galaxies in redshift surveys lo ok qualitatively very
di erent from those of spiral galaxies the clusters are much more prominent in the former cf Giovanelli
et al Huchra et al a This dramatic segregation of ellipticals and spirals do es not extend into
the eld but subtle relative biasing on large scales b etween the two have not yet b een ruled out
A number of workers have lo oked for such e ects in the correlation statistics Davis Geller Huchra
Giovanelli Haynes Chincarini Santiago da Costa Einasto One of the more
p owerful statistics for this purp ose is the cross correlation function r for two p opulations of galaxies
and The number of galaxies of type within a shell a distance r from a galaxy of type is n dV r
One can compare the cross correlation function with the auto correlation function of either galaxy type
the ratio of the two is a measure of the relative bias of the two types of galaxies This approach is esp ecially
appropriate when one of the two types of galaxies is quite a bit rarer than the other the cross correlation
function is much more robust than the auto correlation function of the rarer sample The literature on
searches for relative biases in di erent p opulations is vast in addition to the morphology surveys discussed
ab ove p eople have lo oked for segregation as a function of surface brightness Davis Djorgovski
Bothun et al Mo Lahav Mo et al luminosity Hamilton Davis et al Valls
Gabaud Alimi Blanchard Thuan Gott Schneider Eder et al B orner Mo Zhou
Salzer Hanson Gavazzi Bouchet et al Park et al Loveday et al Marzke
et al emission line prop erties Salzer et al and even mass White Tully Davis
In addition there have b een comparisons of the distributions of galaxies selected in di erent wavebands
including IRAS vs optical Babul Postman Lahav Nemiro Piran Strauss et al a
and radio vs optical Shaver Peacock Nicholson Mo Peacock Xia
The results of these various studies are often contradictory but can b e summarized as follows on
small scales the correlation functions of late type lower surface brightness galaxies are weaker than that of
early type galaxies by a factor of to dep ending on the exact sample used There is also evidence that
the later type galaxies show a shallower slop e There is a similar relation b etween the IRAS and optical
correlation functions Eqs and this is not surprising given that IRAS galaxies tend to b e late type
spirals There is a weak dep endence of the correlation strength on luminosity in b oth optical and IRAS
bands in the sense that more luminous galaxies show stronger correlation This is worrisome as it is a
violation of the universal luminosity function assumption x In particular it means that clustering
statistics derived from ux limited samples will have systematic errors The correlation function on small
scales is heavily weighted by pairs of galaxies nearby where the sampling is higher and thus there are more
pairs However the nearby ob jects have lower luminosity in the mean than those galaxies further away
and if the correlations of the former are indeed weaker the derived slop e of the correlation function will b e
to o shallow With these e ects in mind some workers e g Park et al have restricted themselves
to volume limited samples in calculating clustering statistics from redshift surveys
One of the most striking features of the observed galaxy distribution is the presence of voids Mo dels
with little p ower on small scales have galaxies forming from the fragmentation of pancakes and thus
naturally predict voids Explosion mo dels Ikeuchi Ostriker Cowie naturally evacuate large
regions of space in which galaxies will not form Alternatively mo dels that exhibit strong biasing predict
that very few galaxies form in underdense regions naturally creating voids However one would exp ect
that in these various scenarios those few galaxies within voids should have di erent physical prop erties
from galaxies in the denser regions Dekel Silk Ho man Silk Wyse Brainerd Villumsen
A number of workers have compared the redshift maps of bright galaxies with dwarf galaxies Eder
et al Thuan Gott Schneider low surface brightness galaxies Bothun et al Mo et al
emission line galaxies Salzer et al HI rich galaxies Weinberg et al and IRAS galaxies
Babul Postman Strauss et al a no distinct p opulation of galaxies that lls the voids has
yet b een found Alternatively a number of workers have lo oked for distinguishing physical prop erties of
those galaxies in voids and in more normal environments Ho man Lu Salp eter Szomoru et al
b no strong e ects are seen in contrast to the situation in clusters where dramatic di erences in
mean galaxy prop erties are seen as a function of lo cal density Peebles argues that this lack
of physical di erences b etween void galaxies and those in mean density environments is a strong failing
of the biasing mo del Santiago Strauss do a p oint by point comparison of the density elds as
traced by di erent galaxy types in the CfA survey the di erences they see b etween ellipticals and spirals
are statistically signi cant with spirals b eing over represented relative to ellipticals in the intermediate
density region around the Virgo cluster Con rmation of this result will require larger samples
with more accurate Hubble types this is one of the principal motivations of the Optical Redshift Survey
Santiago Santiago et al a b
Peculiar Velocity Fields Techniques of Measurement and Analysis
We have learned a great deal ab out the large scale structure of the Universe from the redshift surveys
discussed in preceding chapters In these surveys we have generally taken the measured redshift cz as a
distance indicator enabling us to construct three dimensional maps of the distribution of galaxies We have
also gone one step b eyond this combining theory with observations to use the redshift maps as a means of
studying the origin of large scale structure and the value of the cosmological density parameter These
analyses have b een premised on the gravitational instability paradigm according to which structure grows
due to gravitational ampli cation of initial density p erturbations x But redshift measurements do
not by themselves lead to a conclusive test of the gravitational instability paradigm Such a test requires
the additional information we gain from what we broadly refer to as peculiar velocity surveys
The limitations of redshift surveys stem from their inability to separate the two contributions to a
galaxy s observed redshift the cosmological comp onent asso ciated with the expansion of the Universe and
the peculiar comp onent asso ciated with the galaxy s prop er motion with resp ect to the lo cal rest frame see
Eq Such a separation is p ossible only if in addition to the redshift cz one has a redshift indep endent
distance measurement d Peculiar velocity surveys are based on data sets consisting of redshifts and redshift
indep endent distance measurements or redshift distance samples From these two measurements one
readily obtains the radial comp onent of each galaxy s p eculiar velocity
u d cz H d
where we have used Eq and de ned
u r r v r v
Note it is only the radial comp onent of the p eculiar velocity that we can measure We discuss an indirect
means of reconstructing the transverse comp onents of the p eculiar velocity eld in x b elow
The gravitational instability paradigm requires that the linear p eculiar velocity and density uctu
ation elds b e related to one another according to the lo cal expression Eq or the global expression
Eq Those formulations of the velocity density relation contain the Hubble constant H In actual
p eculiar velocity analyses explicit dep endence on the Hubble constant is avoided by working in a system
of units in which distances are expressed as velocities i e in which the Hubble constant is de ned to b e
unity Eqs and also involve actual mass density uctuations whereas redshift surveys provide
galaxy density uctuations only We thus rewrite the two equations as they app ear in an actual analysis
r v
g
and
Z
r r r
g
v r d r
jr rj
where f b Eq and b is the biasing parameter It is understo o d that the derivatives in
Eq and the integral in Eq are with resp ect to spatial variables expressed in km s
A prop er analysis of redshift distance samples will yield v r while we derive r from redshift
g
survey data We can thus test the validity of the gravitational instability paradigm by seeing whether
these elds ob ey Eqs and Furthermore to the extent that the data are consistent with the
gravitational instability picture the ratio of the galaxy density and velocity divergence elds gives a direct
measure of Indeed with data of su cient quality and quantity we might hop e eventually to go b eyond
the linear velocity density relations and thus break the degeneracy b etween and b
While con rmation of the gravitational instability picture and determination of are their most
imp ortant goals p eculiar velocity surveys serve other purp oses as well One is to characterize the mass
distribution on very large scales Because of shot noise existing wide angle redshift surveys cannot ac
curately measure number density uctuations at distances much greater than km s However
large amplitude km s coherent p eculiar velocities on very large scales can b e detected at such
distances with relatively mo dest samples e g Lauer Postman Dekel Velocity p erturbations
drop o more slowly with increasing scale than do density p erturbations as we saw in Fig mass density
uctuations of a few p ercent on a km s scale undetectable in present redshift surveys can give
rise to detectable bulk motions if is not much less than unity For this reason p eculiar velocity mea
surements constitute one of the b est metho ds of measuring mass density uctuations on very large scales
this quality makes them a p owerful discriminant of rival cosmogonic theories x
A related virtue of p eculiar velocity surveys is their capacity to reveal mass uctuations indep endently
of bias Through redshift surveys we can map the number density uctuations and derive the mass
g
density eld through an assumed mo del of bias But if the relationship b etween luminous and dark
mass is in reality far di erent from the linear mo del we usually adopt or worse if that relationship has
large variance Cen Ostriker b or is sto chastic redshift surveys may in fact tell us little ab out
mass uctuations If the gravitational instability picture is valid however p eculiar velocities uniquely
re ect the mass density eld This p oint has b een emphasized in recent years by Dekel Bertschinger and
coworkers e g Bertschinger Dekel Dekel et al Dekel who have developed a metho d
to reconstruct the mass density eld from p eculiar velocity measurements x
Babul et al p oint out that in certain scenarios of non gravitational growth of structure Eqs
and might apparently b e satis ed yielding an incorrect value of x
The use of redshift distance samples as describ ed ab ove app ears straightforward in principle In
reality redshift indep endent distances are far to o inaccurate for Eq to b e applied as written to
real data Much of the complexity and controversy surrounding p eculiar velocity surveys arises from the
steps needed to account for the consequences of distance measurement errors In what follows we describ e
how workers in the eld have attempted to deal with this complexity and discuss the issues that remain
unresolved Still it is worth emphasizing at the outset our optimism ab out the program of comparing
the p eculiar velocity and galaxy density elds in order to estimate Much progress has b een made to
date using existing large fairly complete redshift distance samples and still larger and improved redshift
distance samples will so on b ecome available Redshift survey data yielding r for the lo cal region are
g
similarly improving We consider it likely that within a decade or so p eculiar velocity analyses will have
provided not only convincing substantiation of the gravitational instability picture but also a b elievable
estimate of and quite p ossibly further insights into the subtle relationship b etween luminous and dark
mass
We b egin this chapter with a discussion of distance indicator relations with emphasis on the Tully
Fisher and D relations for spiral and elliptical galaxies resp ectively x brie y discusses the evidence
n
we have that these distance indicator relations are universal and x covers other distance indicator
relations that are likely to b ecome imp ortant in coming years The rest of the chapter is devoted to the
types of analyses one do es with p eculiar velocity data emphasizing the inherent pitfalls and biases This
is rst done at a qualitative level in x followed by a mathematical treatment in x The scienti c
results of p eculiar velocity work are discussed in Chapters and
Galaxian Distance Indicator Relations
We obtain redshift indep endent galaxy distances by means of distance indicator relations or DIs
The essence of a DI is an empirical correlation b etween two intrinsic prop erties of a galaxy one of which is
distance dep endent e g diameter and one of which is distance indep endent e g surface brightness
Thus for example surface brightness might b e known to correlate with absolute diameter a comparison of
the measured angular diameter with the inferred absolute diameter obtained from this correlation yields
a distance estimate In some cases the distance indep endent prop erty is simply the identi cation of an
ob ject as b elonging to a particular class such as Type a Sup ernovae x in that case a DI may b e
called a standard candle or a standard ruler according to whether the distance dep endent prop erty
is luminosity or diameter
Long b efore their use in p eculiar velocity surveys DIs were employed in the quest to measure the
Hubble constant H and the acceleration parameter q e g Weinberg Determination of H requires
absolute calibration of a DI which involves referencing DI measurements to a set of of ducial galaxies
whose distances in Mp c are known For p eculiar velocity work absolute calibration is unnecessary the DI
measurements are instead referenced to a ducial set whose distances in km s are known We discuss
how this is done in practice in x In this review we sp eci cally do not discuss e orts to absolutely
calibrate DIs and apply them toward determination of H this sub ject has b een recently reviewed by
Jacoby et al Estimation of q do es not require absolute calibration but do es entail observations of
galaxies at substantial z redshifts The p eculiar velocity surveys we will discuss are strictly con ned
to z and thus have no b earing on the q problem
While many empirical correlations b etween distance indep endent and distance dep endent galaxian
Often the distance indep endent quantity is in fact weakly distance dep endent An example is surface bright
ness which falls o with increasing redshift as z However as in that example it is usually the case that
the weak distance dep endence is either exactly or adequately mo deled in terms of the redshift alone and thus is
easy to comp ensate for
prop erties exist not all are equally useful as DIs In what follows we describ e in detail those DIs which
have b een most fruitfully applied to the study of p eculiar velocities as well as some likely to b e so employed
in the future
Both elliptical and spiral galaxies show correlations b etween galaxian luminosity and a relevant
measure of internal velocity disp ersion In x we discuss this correlation for spirals the Tully Fisher
TF relation x discusses the corresp onding correlation for ellipticals
The Tully Fisher Relation for Spiral Galaxies
Ordinary spiral galaxies are for practical purp oses the most suitable ob jects to use as tracers of the
p eculiar velocity eld in the lo cal Universe They are more numerous and more uniformly distributed than
their luminous counterparts ordinary elliptical galaxies which tend to congregate in the cores of dense
clusters They are also bright enough to b e carefully studied at fairly large distances unlike the still more
numerous but faint dwarf galaxies As a result the TF relation which is the main DI for spirals has b een
the workhorse in p eculiar velocity surveys to date
Since spiral galaxies are attened systems supp orted by rotation the relevant measure of internal ve
lo city for Tully Fisher is rotation velocity Because spiral galaxies have at rotation curves their rotation
velocities are well de ned as we discuss b elow In practice the luminosity rotation velocity correlation is
well mo deled as a p ower law
L v v
rot
rot
Using the conventional notation of astronomers which we will follow here this equation may b e rewritten
M A b
where M const log L is the absolute magnitude and
log v
rot
where v is measured in km s is a convenient logarithmic measure of the rotation velocity The nor
rot
malization of is such that its value lies roughly b etween for most spirals The quantities A and b
are known as the zerop oint and slop e of the TF relation the slop e is found to lie in the range b
dep ending on the details of how v and M are measured Thus the p ower law exp onent of the
rot
luminosity rotation velocity relation is The apparent magnitude m of a galaxy is the observed
quantity it is related to the absolute magnitude via the inverse square law
m M log r M r
which de nes the distance modulus r
The measurement of rotation velocities in TF studies has b een carried out mainly through analysis
of HI cm pro les esp ecially prior to These pro les are derived from spatially unresolved data and
therefore do not give rotation velocity per se but rather total cm intensity as a function of velocity
This has led to a variety of e orts to de ne a cm pro le width which most faithfully re ects rotation
velocity e g Tully Fouqu e Aaronson et al Bicay Giovanelli ab In recent years
some workers have employed optical measures of rotation velocity measured from the H emission line
Dressler Faber b Courteau Mathewson et al ab Mathewson Ford using a
long slit oriented along the ma jor axis of the galaxy Such measurements have b ecome increasingly useful
as TF studies have b een extended to the Southern Hemisphere where fewer radio telescop es exist and
to larger distances where confusion and sensitivity problems b ecome severe in the radio An alternative
is to employ imaging Fabry Perot sp ectroscopy in the H line Schommer et al which allows one
to map out the rotation velocity eld in two dimensional detail The optical unlike the HI measurements
are spatially resolved Because cm and optical measures of rotation can di er systematically Tully
Fouqu e Pierce Tully it cannot b e assumed that the slop e or zerop oint of the HI and optical
TF relations will b e the same recent work on combining the two types of data have addressed this issue
Mathewson et al b Willick et al b
Regardless of how velocity width is measured the raw measurement di ers from the true width which
enters into the TF relation b ecause of pro jection on the sky The true width is estimated by dividing the
raw value by the sine of the estimated inclination angle of the galaxy obtained from the ellipticity of its
optical image
The existence of a correlation b etween optical luminosity and HI velocity width had b een recognized
since at least the early s see for example Bottinelli et al Balkowski et al Sandage
Tammann but Tully Fisher were the rst to characterize this correlation by the linear rela
tion Eq nding a slop e b Their work brought its p otential for use as an imp ortant cosmological
to ol to the attention of the astronomical community Early work with the TF relation used blue bandpass
B measurements of luminosity which were by far the most readily available in the late s It was
recognized early on however that such measurements were sub ject to extinction of starlight by dust in
b oth the source galaxy and in our own Galaxy These e ects represented systematic uncertainties in early
applications of the TF relation
An imp ortant advance was made in the late s and early s through the work of Aaronson and
collab orators e g Aaronson Huchra Mould Aaronson et al ab They recognized that long
wavelength photometry would su er less from internal and Galactic extinction than B band photographic
or photo electric data The Aaronson group underto ok a program of photo electric H band m
photometry of spiral galaxies in the Lo cal Sup ercluster These measurements were made through ap ertures
whose size was dictated by instrumental constraints and were subsequently scaled to a ducial ap erture
roughly one third the optical size of the galaxy This awkward pro cedure was a drawback of the metho d
but the Aaronson group nonetheless found that the new infrared TF relation IRTF was tighter than the
older short wavelength version They also found that the IRTF required little or no correction for internal
and Galactic extinction and that its slop e b was almost exactly corresp onding to a p ower law exp onent
The v dep endence of the IRTF was seen as favorable for two reasons First as we discuss further
rot
b elow it drops out naturally from a simple theoretical derivation of the TF relation based on consid
erations of centrifugal equilibrium Second a v p ower law also describ ed the analogous Faber Jackson
relation for elliptical galaxies x Since elliptical galaxies generally lack ongoing star formation their
radiant output in the blue is pro duced by an old stellar p opulation The blue light luminosities of spiral
galaxies by contrast may in some cases b e dominated by the hot massive stars which result from recent
bursts of star formation Aaronson et al suggested that the H band photometry measured the light
of the old stellar p opulation presumably the most faithful tracer of the mass and therefore correlated
b etter with the gravitational p otential and through it the observed rotation velocity The agreement in
slop e with that of the Faber Jackson relation seemed to con rm this notion
Attempts to explain the TF relation in terms of the comp osition and dynamics of spiral galaxies
b egin with the equation of centrifugal equilibrium
M
v
rot
R
If one then assumes a universal mass to light ratio M L and a constant mean surface brightness I for
spirals so that
L IR
one obtains
L L
q
v
rot
R
L I
or
L v
rot
precisely the exp onent characterizing the IRTF Aaronson et al cited this consistency as evidence
that the near infrared TF relation was more fundamental than shorter wavelength versions
While this explanation of the TF slop e is not unreasonable plausible derivations of quite di erent
slop es have b een presented For example Sandage Tammann b egan with Eq and also in
voked the presumed constancy of M L They then supp osed that the characteristic radius R at which the
corresp onding equation is to b e interpreted is indep endent of luminosity on average This yields L v
rot
to a TF slop e b similar to the earliest estimates for the B bandpass Tully Fisher Bottinelli
et al Moreover Burstein has argued on the basis of the observed slop e of the TF relation
and the relation b etween surface brightness and rotation velocity that the assumption of a uniform mass
to light ratio cannot hold true The notion that one can straightforwardly derive the TF slop e from rst
principles is now in fact considered dubious It has b ecome clear that the TF slop e has more to do with
technical issues of measurement than with the constancy of mass to light ratios or surface brightnesses
In particular the size of the ap erture and the photometric bandpass used materially a ect the observed
slop e Bottinelli et al Pierce Tully Willick Jacoby et al cf Fig Bernstein
et al for example found an H band TF slop e of rather than the canonical when they used
large ap ertures which encompassed essentially all of the light from a galaxy Moreover the assumption
that spirals p ossess constant M L in H band light has b een challenged Bothun for example
has p ointed out that small variations in heavy element abundances can greatly alter the infrared energy
output of a stellar p opulation in which case the age of a galaxy should a ect M L Pierce
has shown that as measured in I band light M L ratios show strong variations with luminosity In
short it app ears that p o orly understo o d issues p ertaining to the nature and spatial distribution of stellar
p opulations will b e central to an eventual understanding of the TF relation
Beyond this there is a fatal aw in any simple mo del of the TF relation based on the equation of
centrifugal supp ort Spiral galaxies p ossess roughly exp onential light pro les Freeman with a char
acteristic scale length However they have rotation curves which rise quickly from zero to a characteristic
rotation sp eed that is observed to b e remarkably constant out to many exp onential scale lengths e g
Rubin et al This constancy of the rotation sp eed is one of the main pieces of evidence
for the existence of dark halos in spiral galaxies Faber Gallagher Indeed it is only b ecause of such
at rotation curves that a well de ned rotation velocity of spiral galaxies exists Thus the characteristic
scale of the mass distribution in spiral galaxies is appreciably larger than is that of the light although
the two were equated in Eqs Moreover the TF relation applies equally well to all galaxies of
a given rotation sp eed although their exp onential scale lengths as well as their bulge to disk ratios and
morphologies may vary considerably Han Willick Evidently there is a complex interplay
b etween luminous and dark mass which determines the nal state of spiral galaxies While some progress
has b een made toward understanding these e ects e g Blumenthal et al our understanding of the
physical basis of the TF relation remains schematic at b est The origin of the TF relation is shrouded in
the mysteries of spiral galaxy formation
With the advent of CCD detectors in the s highly accurate galaxy photometry has b ecome
routine Two dimensional digital detectors such as CCDs p ermit one to tailor the photometric ap erture to
the galaxy image in a natural way in contrast to the somewhat arbitrary scalings required in xed ap erture
photo electric photometry CCDs are very sensitive in the R m and I m bandpasses
and most recent TF studies have b een carried out in one of these bands Bothun Mould Pierce
Tully Willick Han Courteau Mathewson et al ab Roth Mould et al
Mathewson Ford Haynes et al Dust extinction is less of a problem in R or I than in
B though not negligible as it is at m a detailed discussion of mo dels of internal extinction is given
by Giovanelli et al With the advent of imaging detectors sensitive at m a few workers are
returning to the near infrared for TF work Peletier Willner Bernstein et al
In Fig we plot B R and I TF relations obtained from the large data set of Mathewson et al
b and an H band TF relation from the Aaronson et al b data set as up dated by Tormen
Burstein In each case a Hubble ow mo del of distances has b een used thus the diagrams
contain extra scatter due to the inaccuracy of the distance assignments as discussed b elow Nonetheless
the relation is well de ned in all of the bandpasses although the scatter in the R I and H bandpasses
is noticeably smaller than in the B bandpass The B and R band data are photo electric measurements
taken from the RC and Laub erts Valentijn catalogs resp ectively as compiled by D Burstein
private communication The slop es of the tted TF relations are indicated in the gure these TF ts are
corrected for sample selection bias as describ ed in x As can b e seen the slop e increases with increasing
wavelength The very large change from I to H is due only partially to wavelength increase it stems mainly
from the small ap ertures within which the H band magnitudes are measured the I band magnitudes are
total magnitudes and the di erent de nitions of velocity width used by Mathewson et al vs Aaronson
et al In all cases the absolute magnitudes are expressed in the units suitable to p eculiar velocity work in
which distances are measured in km s
An imp ortant and unresolved issue is the rms scatter of galaxies around the mean TF relation usually
quoted in magnitude units and denoted The earliest estimates of the TF scatter were mag
in B Tully Fisher The Aaronson group later found a somewhat tighter correlation
mag in the near infrared e g Aaronson et al However Sandage Tammann and coworkers were
critical of these relatively low estimates arguing that sample selection e ects x were introducing gross
biases into estimates of the scatter By enlarging the data sets studied by the Aaronson group in the Virgo
cluster Sandage and Kraan Korteweg Cameron Tammann derived values ranging from
mag although Fukugita Okamura Yasuda explain much of this increased scatter as
due to appreciable depth e ects in the cluster More recently Federspiel Sandage Tammann
have claimed that while the most luminous galaxies may have mag the TF scatter for a typical
galaxy is mag
On the other hand recent years have seen ever decreasing estimates of the TF scatter by groups
employing CCD photometry Bothun Mould suggested that an appropriate adjustment of the
photometric ap erture with rotation velocity width could reduce the scatter of the I band TF relation to
mag Pierce Tully observed a scatter in the nearby Ursa Ma jor cluster of mag
and calculated that the contribution of true cluster depth was such that the true scatter was mag
Willick studied a sample of relatively distant clusters and found an R band scatter of
mag Courteau identi ed a region of quiet Hubble ow within a larger eld sample in which the
R band TF scatter was mag Schommer et al found a scatter of mag for their I band
Fabry Perot sample see ab ove Very recently Bernstein et al have studied a carefully selected I
band CCD sample in the Coma Sup ercluster region which exhibits an apparent TF scatter of mag Tully-Fisher Relations in 4 Bands
B band R band -8 b=5.3 -8 b=6.3
-6 -6
-4 -4
-2 -2
I band H band -8 b=6.8 -8 b=10.3
-6 -6
-4 -4
-2 -2
-.4 -.2 0 .2 -.4 -.2 0 .2
Fig TF relations in the B R I and H bandpasses The B R and I TF relations are obtained from the
data set of Mathewson et al b The B and R magnitudes for the Mathewson galaxies are photo electric
measurements obtained from the RC and Laub erts Valentijn catalogs resp ectively the I band magnitudes
are from the CCD photometry and the velocity widths from a combination of HI and optical measurements carried
out by the Mathewson group The H band magnitudes are from the lo cal sample of Aaronson et al b as
up dated by Tormen Burstein In each case the plotted absolute magnitudes are computed as
M m log cz where cz is the observed redshift and are thus expressed in the units appropriate to p eculiar
velocity studies Further details are given in the text
The observed TF scatter comes from three sources intrinsic or cosmic scatter photometric and
velocity width measurement errors and incorrect distance assignments Cosmic scatter is the quantity of
greatest interest astrophysically b ecause of uncertainties in the other two sources of scatter however it
is not well constrained at present Photometric errors per se are very small mag when CCDs are
used However the extinction corrections made to the apparent magnitudes while accurate in the mean
may b e incorrect by mag in individual cases due to variations in the dust content of spiral galaxies
Raw velocity width measurements contain errors and comparable errors are incurred in making
the inclination correction resulting errors in the width parameter must then b e multiplied by the TF
slop e Overall measurement errors can account for mag dep ending on bandpass inclination
and quality of the HI or optical width determination Cosmic scatter and measurement errors added in
quadrature are what the quantity is meant to signify However when a TF sample is used to estimate
scatter galaxies must b e assigned distances errors in these assignments contribute to the observed scatter
Minimizing this last contribution and subtracting it from the observed scatter is a tricky but necessary
part of estimating the true TF scatter
The observed value of the TF scatter is imp ortant not only in determining the size of random errors
but also of systematic errors due to statistical bias e ects x The discrepancy among estimates of
the TF scatter discussed ab ove probably arises from a combination of two factors small number statistics
and di erent approaches to the problem of distance assignment errors Large TF residuals that one author
attributes to an erroneous distance mo del such as the misassignment of a galaxy to a cluster another might
attribute to true scatter In the rst case the galaxy in question would b e eliminated from the analysis and
a relatively small scatter obtained while in the second it is included and a higher value results In many
cases such a decision is sub jective and may dep end on the worker s own biases with regard to the true
scatter As larger and more complete samples are studied the imp ortance of such sub jective factors will
diminish and it is likely that scatter estimates will converge A recent study using a data set consisting of
over galaxies culled from ve samples Willick et al a b c cf x nds the CCD R and I band
scatters to b e in the range mag and the H band scatter to b e mag A partly overlapping
sample of spirals analyzed by Mathewson Ford estimated the I band scatter at mag
Still the issue is far from closed claims of very small mag scatter large TF data sets continue to
app ear in the literature e g Haynes et al
The Faber Jackson and D Relations for El liptical Galaxies
n
The elliptical galaxy DI rst used in p eculiar velocity studies was the Faber Jackson FJ relation
Faber Jackson Like the TF relation FJ expresses a p ower law relationship b etween luminosity
and internal velocity Unlike spirals luminous ellipticals do not rotate Kormendy Djorgovski
and the relevant measure of internal velocity is the rms disp ersion of radial stellar velocities in the central
regions of the galaxy Thus the FJ relation is expressed as
e
L
e
Unlike the case with TF early work with FJ Faber Jackson Schechter Tonry Davis
pro duced reasonable agreement that the exp onent of the p ower law was corresp onding to the
infrared TF slop e found by Aaronson and collab orators see ab ove However while early estimates of the
TF scatter were mag the scatter of the FJ relation was found to b e roughly twice that For this
reason although FJ was used with some success in studies of the Virgo centric motion of nearby galaxies
Schechter Tonry Davis it did not seem to hold great promise as a DI
The situation changed dramatically when two indep endent groups discovered that the addition of a
third parameter signi cantly improved the accuracy of the FJ relation Djorgovski Davis Dressler et
al b Working with newly acquired photometric and kinematic data for large samples of ellipticals
each group found that elliptical galaxies p opulate a planar region in a three dimensional parameter space
One expression of this fundamental plane describ es the connection b etween the e ective radius R and the
e
The members of this second collab oration were D Burstein R Davies A Dressler S Faber D Lynden Bell
R Terlevich and G Wegner As a group they b ecame widely known among astronomers as the Samurai
hereafter S
average stellar velocity disp ersion and surface brightness interior to that radius
R I
e
e e
the dual p ower law maps to a plane when logarithmic quantities are considered The two groups b oth
found that Faber et al When using the fundamental plane as a DI the S group
preferred to incorp orate the surface brightness and radius terms into a single photometric diameter they
called D the diameter interior to which the average surface brightness attained a ducial value In a
n
subsequent study Lynden Bell et al a the group provided a nal calibration of the new DI
D
n
e
where the exp onent In this form the mo di ed FJ relation is known as the D relation
n
Dressler a has shown that the D relation also holds for the bulges of early type spirals although
n
this has not b een exploited b eyond his initial pap er
Lynden Bell et al a found that the scatter in the D relation was dex in log D
n n
corresp onding to ab out accuracy as a distance predictor Thus D was found to b e ab out twice as
n
accurate as FJ and compared favorably with TF Indeed the S group used the D distances to make
n
one of the earliest inferences of very large scale deviation from Hubble ow Dressler et al a as we
discuss in greater detail in x It is interesting to note however that while a fundamental plane led
to a tighter DI it also implied less insight into elliptical galaxy formation For while a one dimensional
relationship such as TF requires sp ecial conditions to b e imp osed at galaxy formation a fundamental
plane is simply a re ection of gravitational equilibrium with the additional alb eit ill understo o d cf
Renzini Ciotti assumption that M L b e a function only of the other parameters in the relation
Faber et al Gunn Thus the fundamental plane relations tell us little more than that elliptical
galaxies are gravitationally b ound structures which is hardly surprising
Universality of the Distance Indicator Relations
The measurement of redshift indep endent distances by means of DIs such as TF and D assumes
n
that the galaxian prop erties they embo dy are universal i e indep endent of spatial lo cation or lo cal
environment If say the zerop oint of the TF relation varied systematically from one place to another
then the distances inferred from it would carry a p osition dep endent error giving rise to spurious p eculiar
velocity measurements Recognizing this some astronomers have questioned the reality of detections of
large scale departures from Hubble ow x or as one theorist asked in the provocative title to a pap er
on the sub ject Is Cosmic Drift a Cosmic Myth Silk
Silk suggested on the basis of plausible but sp eculative scenarios of galaxy formation that
b oth the TF and D relations would show a systematic cluster vs eld o set Because the DIs are
n
typically calibrated using cluster observations the measured p eculiar velocities of eld galaxies which
dominate most analyses of large scale motions might then b e invalid de Carvalho Djorgovski
carried out a comparison of the prop erties of eld and cluster ellipticals They argued that as measured by
the relationship b etween several distance indep endent indices surface brightness color metallicity and
velocity disp ersion the eld and cluster galaxies di ered systematically The sense of these di erences was
that eld galaxies were bluer had higher surface brightness and were more metal p o or than their eld
counterparts of similar velocity disp ersion They prop osed that these di erences would follow logically
from star formation histories if the eld ellipticals were typically younger than cluster ellipticals A related
study was carried out by Guzm anand coworkers Guzm an Guzm an Lucey Guzm an Lucey
Bower cf Gregg who also concluded that metallicity in uences the fundamental plane
relations x Guzm an explicitly argued that this age e ect induces a signi cant di erence
b etween D distances for cluster and eld galaxies and that with appropriate corrections the large
n
p eculiar velocities obtained from D data x vanish
n
There are however go o d reasons to view the results of the previous paragraph with considerable
caution The study by de Carvalho Djorgovski mixed two data sets those of Djorgovski Davis
and of the S group Faber et al which might well di er systematically from one another
Indeed it is primarily the Djorgovski Davis data set which manifests a clear cluster eld o set as
measured by the distance indep endent indices no signi cant e ect is found in the S data set The work
of Guzm anand coworkers has b een questioned on a variety of grounds D Burstein private communica
tion in particular the use of an erroneous metallicity velocity disp ersion correlation and outdated stellar
p opulation mo dels Finally it should b e noted that the S group tested for metallicity and color dep en
dences of the D relations and found them to b e statistically insigni cant Lynden Bell et al b
n
The S group recognized the existence of a fraction of the total elliptical p opulation in which
stellar p opulation anomalies might b e present but showed that these ob jects did not strongly a ect their
conclusions ab out the systematics of the ows Burstein et al
Another means of assessing whether measured p eculiar velocities could b e due to DI non universality
is to compare the velocity eld in a given region of space obtained from di erent DIs Kolatt Dekel
compared the p eculiar velocity elds reconstructed by the POTENT technique x from spiral
TF and elliptical D data separately and found no statistically meaningful di erences b etween the
n
two It is unlikely that zerop oint gradients would a ect TF and D in precisely the same way Another
n
consistency test derives from a comparison of D distances with distances obtained for the same galaxies
n
using the Surface Brightness Fluctuation SBF technique Dressler see x This comparison
has validated D p eculiar velocity estimates for ellipticals in the Southern sky Since the SBF
n
technique is thought to b e highly accurate and involves measurements entirely unrelated to D these
n
results represent another encouraging alb eit preliminary indication of D universality
n
In summary then there is no comp elling evidence at present for substantial spatial or environmental
variations in the TF or D relations While there exist some indications that metallicity and or age e ects
n
enter into the fundamental plane relations for a fraction of elliptical galaxies these e ects apparently do
not in uence the distance estimates obtained from D in a systematic way As more data are acquired
n
in the coming years using a multiplicity of DIs intercomparisons will p ermit more sensitive tests of DI
universality
Beyond TF and D A Look to the Future
n
While the TF and D metho ds have b een the workhorses in p eculiar velocity surveys there a
n
number of other promising distance indicators likely to play a role in coming years
Surface Brightness Fluctuations
Over the last several years Tonry and collab orators Tonry Schneider Tonry Ahjar Luppino
Tonry Schechter Tonry have b egun to exploit an old idea rst discussed in detail
by Baum a su ciently high resolution image of an elliptical galaxy will reveal the discrete nature
of the ultimate sources of luminosity billions of individual stars A nite number of stars are included
in a single pixel of an image this number is sub ject to Poisson uctuations and therefore so is the pixel
intensity The farther away the galaxy is the greater is the number of stars p er pixel and the smaller the
Poisson uctuations thus the uctuations are a measure of the distance of the galaxy Tonry measures the
surface brightness uctuation SBF which characterizes an image The mean SBF can in turn b e related
to the luminosity of the giant branch stars which dominate the light which dep ends weakly on metallicity
The technique is applied to ellipticals rather than spirals b ecause the latter are multicomponent often
dusty systems in which the purely Poisson uctuations cannot easily b e isolated
The uncertainty in SBF distance estimates varies in inverse prop ortion with the resolution of the
images obtained For galaxies within km s SBF distances are estimated to have accuracy
under the b est observing conditions Ground based resolution is limited by microturbulence in the atmo
sphere seeing In collab oration with Tonry Dressler has undertaken a program to obtain SBF
distances for ellipticals studied with the D metho d using CCD images acquired at the du Pont Tele
n
scop e in Chile where the seeing is often This program promises to yield accurate distances for nearby
ellipticals and may signi cantly improve our knowledge of the lo cal p eculiar velocity eld Extension of
the SBF technique to distances of km s and b eyond will require either a dedicated program of
space based imaging or the use of adaptive optics technology for enhancing ground based resolution
BCG L relation
The DIs discussed thus far apply to ordinary spiral or elliptical galaxies There is also a class of
extraordinary galaxies the most luminous ob jects in the centers of rich clusters called brightest cluster
galaxies or BCGs which can b e used as DIs BCGs were originally of interest b ecause they are easily
identi ed out to fairly large z redshifts As a result they were thought to b e ideal to ols for studying
problems such as the linearity of the Hubble expansion Sandage Sandage Hardy Gunn
Oke In addition b ecause they are highly luminous ob jects and are unique in their environment
their study is not sub ject to the statistical biases asso ciated with sample incompleteness we discuss in
x
The studies just mentioned considered BCGs as standard candles i e they assumed their luminosities
were indep endent of any other galaxian prop erty Later Ho essel showed that BCG luminosity
correlates with the shap e of the luminosity pro le Sp eci cally Ho essel de ned a metric luminosity L as
m
the total light interior to the metric radius r h kp c and a luminosity shap e parameter
m
d log L
d log r
r
m
and found a linear correlation b etween L and Recently the L relation has b een exploited by Lauer
m
Postman LP and Postman Lauer who studied the BCGs in the Ab ell and
Ab ell Corwin Olowin clusters with redshifts less than km s LP found a quadratic
L relation with an equivalent distance error of comparable to the TF scatter The BCG L
relation is p erhaps the most promising technique for extending p eculiar velocity studies to much larger
scales z than is currently p ossible although it requires further study to test its universality
Possible Methods of Future Peculiar Velocity Work
Two other DIs have recently b een the fo cus of intense observational work but thus far in the context
of the distance scale i e Hubble constant determination rather than the velocity eld The rst of these
is tting the planetary nebula luminosity function PNLF Jacoby et al Ciardullo et al ab
Jacoby et al Ciardullo Jacoby Harris Planetary nebulae are the luminescent ejectae of
dying stars so on to b ecome white dwarfs Planetary nebulae may b e detected and photometered in nearby
cz km s elliptical galaxies using narrow band imaging in the light of twice ionized oxygen at
A The PNLF in this emission line is observed to have a universal form with a sharp cuto at the
highest luminosities Jacoby Ciardullo et al a Fitting the observed distribution of planetary
However it was discovered that evolutionary e ects are equally as imp ortant as those due to cosmology See
Sandage for a review
nebula brightnesses to the universal PNLF yields the distance to a galaxy with accuracy The PNLF
and SBF metho ds yield consistent distances for Virgo Cluster galaxies Ciardullo Jacoby Tonry
At distances much greater than that of Virgo it is quite di cult to reliably identify even luminous planetary
nebulae using present techniques Jacoby et al As a result the usefulness of the PNLF metho d for
p eculiar velocity work is uncertain
A DI which has b een around for some time but which is currently undergoing a kind of renaissance
is the mean absolute luminosity of Type Ia Sup ernovae Sne Ia Of all the DIs discussed here Sne Ia have
the b est understo o d physical basis It is generally agreed that Sne Ia result from thermonuclear disruption
of a carb on oxygen white dwarf that has accreted enough mass from a companion star to approach the
Chandrasekhar mass of M e g Wheeler Harkness and references therein The radioactive
decay of Ni and Co following the explosion p ower the subsequent observable light curve Theoretical
mo dels suggest that the mass of the nickel ejectae is in the range M Arnett et al when
combined with the observed rise time to maximum light of days an absolute luminosity of M
B
mag is predicted
There has b een some controversy over their intrinsic scatter as standard candles The discovery that
the p eak luminosity of SN a correlates with their rate of decline Pskovskii Phillips
Hamuy et al but see Tammann Sandage may tighten up the scatter considerably to as
small as mag Hamuy et al Reiss Press Kirshner
Sne Ia have b een used almost exclusively for Hubble constant determination e g Saha et al a b
Indeed their idiosyncratic nature as DIs one must wait for a sup ernova to app ear in a galaxy in order to
estimate its distance and their o ccurrence is b oth unpredictable and rare limits their utility for p eculiar
velocity work However they can b e used as DIs to extremely large distances with relatively little loss
of accuracy e g Perlmutter et al and may thus prove useful in extensions of p eculiar velocity
analyses to large distances
Type II sup ernovae which mark the end p oint of the life of a massive star can also b e used as
a distance indicator Schmidt Kirshner Eastman Sp ectroscopic observations yield the
velocity of the ejecta and the temp erature of the outer layers from which the absolute brightness can b e
inferred with the use of mo del atmospheres Comparison with the apparent brightness yields a distance
This metho d requires extensive mo del calculations and detailed observations and has thus far b een applied
only to a handful of ob jects
Statistical Bias and Methods of Peculiar Velocity Analysis
With the p ossible exception of the SBF technique the DIs used in p eculiar velocity surveys are
not very accurate The TF and D relations for example predict galaxy distances with only
n
accuracy In the volume within which we hop e to study the p eculiar velocity eld in reasonable detail
typical galaxies may lie at distances of km s The rms p eculiar velocity error for such a galaxy is
thus on the order of km s which also happ ens to b e the amplitude of typical p eculiar velocities At
interesting distances then we cannot measure with any precision the p eculiar velocity of a single galaxy
Meaningful analyses must use statistical techniques applied to large samples
At rst glance this might not b e considered a ma jor problem There are after all thousands of
galaxies in the lo cal volume for which we have obtained or so on hop e to obtain distance indicator data
p
We might exp ect that by virtue of N statistics alone the signal to noise ratio of any statistical analysis
could b e made rather high However this exp ectation is not realized in practice The analysis of DI data
is instead sub ject to statistical bias e ects these e ects result in random errors dropping more slowly
p
than N and also in the p ossibility of large systematic errors if the biases are not prop erly corrected
for While there are a number of bias e ects all originate in a coupling of the DI scatter with external
in uences on the makeup and spatial distribution of the redshift distance sample The goal of this section
is to explore the nature and consequences of this coupling in a variety of circumstances
A p eculiar velocity analysis has logically if not sequentially two steps The rst is calibration of the
DI b eing used If we assume for de niteness that it is the TF relation calibration means determination
of the zerop oint A slop e b and scatter which quantify the M relationship x The second step
in the analysis is the inference of distances and thus p eculiar velocities for sample ob jects Each step is
p otentially sub ject to biases Those which enter into the calibration phase if not corrected for can lead
to erroneous determinations of any or all of the parameters A b or This incorrect TF relation will
pro duce incorrect distances and p eculiar velocities Even with a prop erly calibrated TF relation however
uncorrected biases in the distance inference phase have the same result
The separation of bias e ects into calibration and inferred distance problems Willick is
useful in emphasizing the di erent ways bias can a ect p eculiar velocity analyses However it do es not
distinguish the ro ot causes of bias These lie not in the goal but in the underlying assumptions of the
analytic approach as we will clarify in x With one set of assumptions an analysis will b e sub ject
to selection bias which is intimately related as the name implies to observational selection criteria such
as a ux limit With the second set of assumptions Malmquist bias will result As de ned here but
see next paragraph Malmquist bias is not fundamentally related to a ux limit or any other selection
criteria It o ccurs b ecause the true distance of a galaxy cannot b e estimated from DI information alone but
requires knowledge of the actual line of sight density distribution as well Selection bias can a ect either
DI calibration or p eculiar velocity measurement or b oth as they often are carried out simultaneously
x Malmquist bias typically enters only when a calibrated DI is used to infer p eculiar velocities
approaches to DI calibration which are sub ject to Malmquist bias are rarely used in practice and will not
concern us here
We organize our discussion mainly around the selection Malmquist dichotomy but we note that this
distinction has rarely b een maintained in the existing literature where much confusion reigns One source
of confusion is the ambiguity surrounding the term Malmquist bias which historically has meant several
di erent things The term originated with the work of Malmquist who showed that ux limited
samples were sub ject to a luminosity bias one which in fact resembles our selection bias more closely
than our Malmquist bias Thus early discussions of DI bias Sandage Tammann Yahil Teerikorpi
Aaronson et al typically refer to bias e ects asso ciated with ux limits as Malmquist
biases However the distinct biases related to the line of sight galaxy density distribution were also referred
to as Malmquist e ects by Lynden Bell et al a who were the rst to discuss them quantitatively
in the extragalactic context The latter usage has gained widespread acceptance and it is the one we adopt
here Even so the impression has p ersisted in some quarters that Malmquist bias thus de ned relates to a
ux limit One goal of the discussion to follow is to disp el this and related misp erceptions brought ab out
by con icting uses of this catch all phrase
Selection vs Malmquist Bias Method I vs Method II
Because of DI errors data from a large number of galaxies are needed if the estimated p eculiar
velocity at a given p oint r is to b e of value Thus in devising a metho d of analysis we must ask the
question for what particular set of ob jects should data b e averaged to estimate p eculiar velocity at
p osition r There are two metho dologically orthogonal answers to this question
i the set of ob jects which redshift space information suggests are near r
We use the term inference rather than the stronger prediction as is used for example by Burstein
to underscore the fact that redshift indep end ent distances are merely statistical estimators obtained from more
basic measurements of true distances and not very accurate ones at that
Here we use averaged in its widest sense includin g smo othing or mo deling techniques
ii the set of ob jects which the DI information suggests are near r
A metho d based on the rst answer will b e sub ject to selection bias one based on the second answer
will exp erience Malmquist bias This holds true whether our goals are DI calibration p eculiar velocity
measurement or b oth
It will prove useful to have terms describing metho ds of p eculiar velocity analysis in addition to
terms describing the types of bias to which they are sub ject When redshift space information is used as
the a priori indicator of true proximity we will describ e the analysis as b elonging to Metho d I I When DI
information is used as the a priori indicator of true proximity we will describ e the analysis as b elonging to
Metho d I This terminology was originally introduced by Faber Burstein although their usage
was somewhat less general than ours In x and x we will show using prototypical examples
why Metho d I I is sub ject to selection bias and Metho d I to Malmquist bias and clarify how these biases
a ect the analysis Later in x we will show how to quantify and correct for these biases We will assume
throughout that the DI in question is the TF relation but the concepts apply equally to DIs such as D
n
Bias in a Cluster Sample
Supp ose that a ux limited Tully Fisher sample consists of spiral galaxies thought to b elong to an
isolated rich cluster A reasonable assumption to adopt in analyzing such a sample is that its members all lie
at the same distance This is an example of a Metho d I I analysis redshift space information from which
we infer cluster membership is treated as the a priori indicator of distance the TF information has not
entered into the assumption in any way Let us further supp ose that our goal is to calibrate the TF relation
Using the common distance assumption we may write the exp ected apparent magnitude of a cluster galaxy
with velocity width parameter as h m i A b see Eq where log r is the cluster
c
distance mo dulus for now supp ose that the cluster distance r in km s is indep endently known We
c
also exp ect that the observed apparent magnitudes will scatter ab out hm i with rms disp ersion These
considerations seem to suggest that linear regression of m on for the cluster sample will correctly give
the TF slop e and zerop oint after subtracting the distance mo dulus and that the scatter ab out that t
will yield
However b ecause we have made a priori distance assignments based on redshift space information
the ab ove exercise will b e sub ject to selection bias How sp eci cally do es the bias enter in Let m denote
the magnitude equivalent of the ux limit Fig illustrates a Monte Carlo simulation of a Tully Fisher
sample at a common distance with a scatter of mag and a magnitude selection at m The
dashed line is the input TF relation Consider a subset of sample galaxies with similar and relatively large
velocity width parameters such that they are typically much brighter than the ux limit i e
hm i m Their observed apparent magnitudes m will b e distributed uniformly ab out hm i with
rms disp ersion Now consider a second subset with relatively small width parameters such
that h m i m They will exhibit a range of apparent magnitudes m h m i However none wil l
have m hm i Such galaxies exist in the cluster but are missing from the sample b ecause they
are fainter than the magnitude limit The mean apparent magnitude of the second subset will therefore
b e brighter than h m i as the solid line in the gure shows This is one de nition of bias the mean
apparent magnitude do es not conform to the TF prediction A linear t to the p oints which do es not take
into account the magnitude limit will b e biased as illustrated by the stars in Fig the slop e of the tted
line is atter by in this example and the tted zerop oint is brighter by mag in this example than
the true TF relation the dashed line
Finally the scatter ab out the t will not equal b ecause the faint part of the magnitude distribution
at is cut o the p oints cluster more tightly around the mean value The at slop e bright zerop oint
and reduced scatter of the t are all characteristic of how selection bias a ects DI calibration
The calibration pro cedure itself was not however resp onsible for bias in the ab ove example Supp ose
Fig A Monte Carlo simulation illustrating the e ects of a magnitude limit on the derived TF relation Points
were assumed to b e at a common distance and assigned a TF relation indicated by the dashed line with a scatter
of mag The sample was limited at apparent magnitude The stars indicate the b est t linear TF relation
which do es not take the magnitude limit into account it is shallower with a brighter zerop oint
that the TF parameters were known at the outset and that instead it was the cluster distance mo dulus
whose value was sought We could still make the assumption that the cluster galaxies lie at the same
distance We would then seek to determine by tting a line of known slop e b to the TF data The
intercept of that line would apparently b e A Yet even though A and b are known the tted line would
still b e displaced from the correct TF predicted one Near the magnitude limit the observed apparent
magnitudes are still brighter in the mean than hm i The derived value of will thus b e biased small
when combined with the cluster redshift this will result in a to o p ositive radial p eculiar velocity These
e ects are indep endently of calibration issues also characteristic of selection bias
Central to the Metho d I I analyses just describ ed was the assumption that the cluster galaxies lie at
the same distance This assumption though plausible is not self evident the galaxies might conceivably
b e spread out over a range of distances Following this line of reasoning we might if our TF relation
were indep endently calibrated neglect redshift space data initially and assign distances directly from the
observables m and These inferred distances d are derived by forming a distance mo dulus m
m
m M and then taking d Because of DI errors the true distance r can di er appreciably
from d Still a Metho d I analysis of the cluster would assume that only galaxies with similar values of
d not all galaxies in the sample may b e treated as equidistant The redshifts of galaxies with similar
values of d would then b e averaged together to obtain a mean value hcz d i and a corresp onding inferred
p eculiar velocity u d h cz d i d
Because we have now made a priori distance assignments based on the DI information this analysis
will b e sub ject to Malmquist bias The sp eci c manifestations of this bias are that d h cz d i and and
u d are not equal to the average true distance redshift and radial p eculiar velocity resp ectively of the
set of ob jects with inferred distance d The bias arises b ecause ob jects with inferred distance d come with
varying probability from a range of true distances r as a result of DI errors They are constrained to lie
within some vicinity of d but their true distances are in uenced by another factor wholly unrelated to the
DI information where along this line of sight galaxies are intrinsically likely to b e found This is quanti ed
by the number of galaxies p er unit distance r n r where n r is the galaxy number density If galaxies
were in fact widely distributed along the line of sight n r roughly constant then b ecause there is more
volume at larger distances the r factor d would underestimate the true distance If on the other hand
the common distance mo del had b een accurate in the rst place then regardless of the value of d the
true distance is the cluster distance r Inferred distances d r would then necessarily underestimate
c c
and d r necessarily overestimate true distance None of these considerations relates in any way to the
c
selection criteria which de ne the sample
When galaxies scatter from r to d b ecause of DI errors they bring with them the correct redshifts
of their true p ositions That is DI errors do not a ect redshifts but only the distances at which they are
evaluated and converted to p eculiar velocities Thus if r say is the mean true distance of galaxies with
d
inferred distance d the mean redshift of these galaxies will to rst order b e h cz d i r u r where
d t d
the subscript t signi es true as opp osed to inferred radial p eculiar velocity From this redshift one would
infer a p eculiar velocity u d u r r d Thus the distance bias d r translates directly into
t d d d
a p eculiar velocity bias In the case that the galaxies truly constitute a cluster i e r r for all d
d c
we see that u u when d r and u u when d r It follows that the inferred radial p eculiar
t c t c
velocity eld u d would exhibit a compressional in ow into the cluster center even if there were no real
peculiar velocities This last e ect spurious p eculiar velocities asso ciated with strong line of sight density
gradients of which a cluster is the most extreme case is the most serious consequence of Malmquist
bias As density gradients generally are not known with great precision Malmquist bias can b e di cult to
correct for reliably x
Bias in a Field Sample
We used the example of a cluster to distinguish selection and Malmquist bias with maximum clarity
However p eculiar velocity analyses more often involve eld galaxy samples It is therefore imp ortant to
realize that the two kinds of bias o ccur for analogous reasons in eld sample analyses as well In the
cluster example the Metho d II approach seemed natural in comparison with Metho d I With a eld
sample the choice of metho d is not clear cut We illustrate these issues in what follows
A Metho d I I analysis of eld galaxies usually entails the adoption of a mo del of the p eculiar velocity
eld v r a the quantity a is a vector of free parameters which sp eci es the mo del An imp ortant example
is a velocity eld obtained from Eq using redshift survey data with as the only free parameter
Using the mo del one obtains a distance r for a galaxy of redshift cz as the solution of the equation
r cz a cz r v r a
Such an expression of course assumes a unique redshift distance mapping which may not obtain in the
vicinity of large mass concentrations see x an issue we will return to later From Eq one can
This last statement neglects second order changes in the p eculiar velocity eld cf Willick for details
predict the galaxy s apparent magnitude given the value of its velocity width parameter
m cz A b a M log r cz a
this predicted apparent magnitude dep ends as indicated not only on the velocity eld parameter vector
a but also on the TF parameters A and b whose determination might also b e part of the analysis One
can then form a statistic of observed minus predicted apparent magnitudes
N
X
m m cz A b a
i i i
A b a
II
i
the individual ob ject scatters are usually assumed equal in the simplest implementations of Metho d
I I Minimization of this statistic might then b e exp ected to yield the b est values of the velocity eld
parameters a and if necessary the TF parameters A and b
A bit of re ection shows however that such an exercise will b e sub ject to the same bias that a ected
the rst TF analysis of cluster galaxies discussed in x In particular there is again the issue of
missing galaxies although in a more abstract sense than earlier There is in fact only one galaxy at
redshift cz direction r and width parameter However the validity of the minimization pro cedure
i i i
requires that this ob ject b e representative of the ensemble of all p ossible such ob jects Because of the ux
limit however it cannot b e if the predicted apparent magnitude m cz A b a is close to or fainter
i i
than the ux limit the fainter members of this hypothetical ensemble cannot b elong to the sample As a
result the observed apparent magnitude m is likely to b e brighter than the predicted value minimization
i
A b a consequently do es not yield the correct values of the parameters on which it dep ends A of
II
Metho d I I analysis of a eld sample is thus a ected by selection bias just as was the earlier one based on
the assumption of equidistant cluster members We note that although we have assumed the existence of a
mo del p eculiar velocity eld this is not fundamental to a Metho d I I eld sample analysis For example if
pure Hubble ow were assumed v r a vanishes identically an exercise identical to that outlined ab ove
could b e used just to calibrate the TF relation its results also would b e a ected by selection bias
A Metho d I analysis of the same sample would pro ceed di erently To make the comparison as direct
as p ossible we again supp ose that the purp ose is to t a mo del velocity eld v r a Now however there is
no need to solve Eq to obtain a distance Instead we use the inferred distance d as the b est indicator
of true distance and thus as the p osition at which we evaluate the velocity mo del Inferred distances are
given as describ ed in the Metho d I cluster analysis x and the TF relation is again assumed to have
b een indep endently calibrated Using the mo del velocity eld we write a predicted redshift for a galaxy
with inferred distance d as
cz d a d r v d a
where d d r This in turn motivates a statistic of the form
N
X
cz cz d a
i i
a
I
i
i
where is a suitable measure of redshift and distance errors Again one might supp ose that minimization
i
of this statistic leads to the b est values of the velocity mo del parameters a
It is not fundamental to our de nition of Metho d I that the TF relation b e calibrated a priori However
unnecessary complications arise if the TF parameters are considered free
However this na ve approach will lead to biased results The validity of the minimization rests
on the assumption that cz d a is an unbiased estimator of cz when the velocity mo del is correct As
i i
discussed in x the ob ject with inferred distance d can come from a range of true distances with
i
varying probability Let the exp ected true distance of this ob ject b e r Its observed redshift then has a
d
i
corresp onding exp ected value hcz i r u r where as in x u denotes the true radial p eculiar
i d t d t
i i
velocity eld and we neglect second order e ects From this it follows compare with Eq that even
if the velocity mo del is correct h cz i cz d a unless r d But we know from our discussion of
i i d i
i
the Metho d I cluster analysis x that this last condition do es not hold in general the relationship
b etween r and d dep ends on the details of the underlying density distribution n r which is the essence
d i
i
of Malmquist bias Thus cz d a is a biased measure of cz and as a result the parameters which minimize
i i
yield an incorrect velocity eld Again while we have framed the problem in terms of a velocity mo del
I
Metho d I includes any analysis in which galaxies with similar inferred distances are assumed to b e physically
proximate and therefore to share a p eculiar velocity vector In particular nonparametric metho ds such
as the POTENT reconstruction Bertschinger Dekel see x which smo oth in inferred distance
space b elong to Metho d I and su er from Malmquist bias
Inverse Distance Indicator Relations
There is another basic division b etween metho ds of p eculiar velocity analysis one which has some
times b een subsumed under the Metho d I I I dichotomy but in fact is entirely distinct Until now we have
considered only the forward form of DIs in which the distance indep endent quantity in the case of
TF is thought of as predicting the distance dep endent quantity M in the case of TF But we can just
as easily turn the relation on its side and consider the TF relation as a predictor of velocity width given
absolute magnitude We quantify this prediction as M strictly sp eaking the exp ectation value of
given M and call it the inverse TF relation Like its forward counterpart M is well described by
a linear relation
M e M D
we denote its rms disp ersion Because of this nite scatter the forward and inverse forms are not
mathematically inverse to one another i e e b and D A The forward and inverse forms of DIs are
alternative representations of a single physical phenomenon nonetheless their statistical prop erties di er
markedly
We illustrate with the following example b oth this di erence in bias prop erties and why the for
ward inverse and Metho d I I I distinctions have b een con ated Supp ose that we wish to carry out a
Metho d I I analysis of the eld galaxy sample of x using the inverse TF relation In that case we use
Eq to predict the velocity width parameter of an ob ject with redshift cz and apparent magnitude
m as
m cz D e a m log r cz a
this equation motivates a new statistic
N
X
m cz D e a
i i i
D e a
II
i
For pure standard candles there is no distinction b etween forward and inverse metho ds the material discussed
in this section is only relevant for distance indicators which dep end on a distance indep ende nt observable such as
line width
apparently a p erfect analogue to its forward counterpart Eq The analogy breaks down however
when one considers the bias prop erties of this inverse statistic In the forward case bias arose b ecause the
sample galaxy with redshift cz and velocity width parameter was not representative of the ensemble of
all p ossible such ob jects the presence of a ux limit meant that the faint end was excluded In the inverse
case the relevant question is whether the sample galaxy with redshift cz and apparent magnitude m is
representative of all such ob jects If we assume that sample selection is indep endent of the value of for a
given m then the answer to this question is evidently yes since m must b e brighter than the ux limit
There now are no missing galaxies regardless of how close m is to the ux limit and m cz D e a is
an unbiased estimator of minimization of D e a may thus b e exp ected to yield essentially unbiased
i
II
values of the free parameters of the mo del
This felicitous prop erty of inverse DIs was rst noticed by Schechter in a study of the Virgo
centric ow eld and was subsequently exploited to great advantage by Aaronson et al b and by
Faber Burstein These studies enshrined the notion that the inverse relation was free of selection
bias and also cemented the asso ciation b etween Metho d I I and the inverse form of DIs They also p erhaps
b ecause of the catch all nature of the term Malmquist led to the assertion e g Tully that in
verse DIs are free of Malmquist bias None of these conventional notions are accurate The ab ove example
shows only that an inverse Metho d II analysis is una ected by selection bias when sample selection is
indep endent of velocity width The paradigmatic form of sample selection a photometric ux limit which
we have used here for illustrative purp oses is indep endent However as we discuss b elow x in
real TF samples selection criteria are more complex and may dep end usually weakly on velocity width In
such cases even an inverse Metho d I I analysis will b e sub ject to mild selection bias Moreover there is no
inherent connection b etween inverse DIs and Metho d I I the essence of Metho d I I is its a priori assignment
of distances from redshift after which either form of the DI can b e employed
Finally b ecause of the conventional asso ciation b etween Metho d I I and inverse DIs it has not gen
erally b een recognized that inverse DIs can b e used in Metho d I analyses We defer the details to x
and note here only that an inverse inferred distance is de ned in the case of TF by the implicit rela
tion m log d A consequence of this de nition as will b e come clear is the fact that inverse
and forward inferred distances di er in general for a given galaxy Nonetheless the fundamental source of
Malmquist bias in forward Metho d I analyses that the likelihoo d an ob ject will scatter from true distance
r to inferred distance d dep ends on the density n r is equally present in an inverse Metho d I analysis
While the use of forward vs inverse a ects its details Malmquist bias inevitably accompanies a Metho d I
analysis
The Method Matrix of Peculiar Velocity Analysis
We provide here a succinct summary of the multiple and to some degree overlapping metho ds of
p eculiar velocity analysis in the form of a metho d matrix The notion of a matrix derives from the
two dimensional character of the metho dology discussed ab ove p eculiar velocity analyses can b elong to
either Metho d I or I I and can employ either the forward or inverse form of the relevant DI The statistical
biases present are sp eci c to each case Table summarizes these ideas two terms in that table not yet
de ned the Gould e ect and the Landy Szalay correction are discussed in x
We have de ned Metho ds I and I I and discussed the biases inherent in each we have not yet indicated
why we might choose one over the other Metho d I has the advantage of b eing unburdened by preconceived
notions ab out the relation b etween redshift and distance Metho d I I which requires that we invoke at
least some form of redshift distance relation is necessarily sub ject to our theoretical prejudice Ideally we
would like to know where the galaxies really are and only later compare these real space p ositions with the
observed redshifts The most comp elling motivation for this is the p ossibility of triple valued zones in dense
regions Fig where infall patterns along the line of sight can result in widely separated galaxies having
Table Metho d Matrix of Peculiar Velocity Analysis
DI type Metho d Type Metho d I Metho d II
DI inferred distance best Redshift space data best
indicator of true distance indicator of true distance
Forward Malmquist bias Strong selection bias
dist dep e g mag predicted selection indep ende nt dep ends on observational
by dist indep e g quantity unless Gould e ect selection criteria
Inverse Malmquist bias Weak or no selection bias
dist indep predicted selection dep endent bias present if selection
by dist dep quantity Landy Szalay p ossible related to dist indep quantity
the same observed redshift Since Metho d I considers redshift only after it has placed the galaxies at their
DI inferred distances this presents no sp ecial problem By contrast Metho d II is inherently incapable of
dealing with multivaluedness in the redshift distance mapping It assumes that redshift uniquely sp eci es
distance and thus presents an overly smo oth picture of a velocity eld which in reality is rapidly changing
In x we will present a recent variant of Metho d I I which circumvents this particular aw
On the other hand Metho d I I has the advantage that its statistical asp ects are comparatively well
constrained If we supp ose that the velocity eld mo del v r a is not unreasonable then the dominant errors
in the assignment of distances Eq come from noise in the velocity eld whose rms value in the
eld app ears to b e at most km s x The dominant errors in Metho d I distance assignments
come from DI scatter and are therefore several times larger at interesting distances Moreover as we will
see more clearly in the following section we can correct for selection bias in terms of knowable information
namely observational selection criteria With inverse DIs we can realistically exp ect selection bias to b e
quite small Correction for Malmquist bias however dep ends on density gradients along the line of sight
which are not knowable a priori We can estimate such gradients from redshift data but this involves
additional assumptions concerning p eculiar velocities needed to map redshift to real space density and
smo othing metho ds Working with inverse DIs gains us nothing in terms of Malmquist bias In summary
b oth Metho d I and Metho d I I have advantages and disadvantages As we describ e in Chapter b oth are
b eing used in contemporary p eculiar velocity analyses It should b e p ointed out however that many of
the analyses we describ e in Chapter were done without prop er attention to the biases we describ e here
and of the di erences b etween the di erent entries in the Metho d Matrix of Table Indeed there remains
much work to b e done in exploring all these metho ds for example there is no analysis b eyond that of
Landy Szalay which exploits Metho d I with an inverse DI
Quantifying Statistical Bias
In the last section we presented a qualitative outline of the main statistical biases a ecting distance
indicator and p eculiar velocity analysis In order to correct for these e ects the biases must b e quanti ed
and correction formulae and metho ds developed This task has b een the fo cus of a growing number of
workers over the last decade or so Sandage and coworkers e g Sandage Tammann Yahil Sandage
Federspiel Sandage Tammann have provided useful though mainly qualitative suggestions
for treating selection bias A quantitative approach to the sub ject was pioneered by Teerikorpi Bottinelli
and collab orators e g Teerikorpi Bottinelli et al and extended by Willick
Malmquist bias in the DI context was rst treated in detail by Lynden Bell et al a this work
has b een followed up and extended by Willick Landy Szalay and Gould
The bias prop erties of inverse DIs have b een studied by Schechter Aaronson et al b Faber
Burstein Tully Han Roth Willick and Triay Lachi eze Rey
Rauzy We will use the formalism developed by Willick who has advocated a
comprehensive approach to the problem We assume in what follows that the distance indicator in question
is the TF relation but the discussion is readily mo di ed to treat comparable DIs such as D
n
The material discussed in this section remains controversial and the formal mathematical approach
we have adopted has not b een accepted by all of the workers in the eld For example Mathewson
Ford have argued that approaches like ours require idealized assumptions ab out sample selection
that are invalid for realistic data sets They instead separate their sample into control and target
regions using the former to calibrate the biases in the latter We b elieve however that such qualitative
or semiquantitative approaches are sub ject to their own caveats are the control and target regions the
same in their bias prop erties and run the risk of obscuring rather than illuminating the principal issues
Unlike much of the rest of the review we have not attempted here to summarize all the analyses extant
in the literature but rather aim to give a comprehensive coverage of our own approach This section can
b e skipp ed on a rst reading
Forward DIs Selection Bias
We consider rst the formalism appropriate to the forward relation the inverse relation is treated in
x A starting p oint in the quanti cation of bias is the empirical fact that it is the TF observables and
not the inferred distances which exhibit a roughly Gaussian distribution e g Willick et al c Thus
the apparent magnitude of a galaxy with width parameter and true distance r is normally distributed
ab out a mean value M log r with rms disp ersion when selection e ects may be neglected We
combine this Gaussian distribution with the a priori distributions of galaxy distances and width parameters
and with a function describing sample selection probability to obtain the joint probability distribution of
the observables m and the true distance r for a galaxy
P r m r n r S m
m M r
p
exp
where r log r is the distribution function similar to the luminosity function and n r
is the galaxy number density along the line of sight in question The quantity S m is the sample
selection function de ned as the probability an ob ject with observables m will b e included in the
sample irresp ective of other considerations Note that we express the selection function here in terms of
the observables rather than in terms of distance as we did in redshift survey analyses x An idealized
form of S m describ es a sample with a strict magnitude limit
if m m
S m
otherwise
Selection and Malmquist bias e ects are each implicit in Eq Which of the two comes into play
dep ends as already noted on the assumptions underlying the metho d of data analysis these assumptions
determine which of several conditional probability distributions derivable from Eq are used We rst
discuss selection bias in x we discuss Malmquist bias
We know from x that selection biases arise in Metho d II analyses in which we take redshift
space information as the a priori distance indicator either through the assumption of a common distance
for cluster galaxies or via a redshift real space mapping such as Eq To the extent this mapping
is a go o d one we are in e ect taking the true distance as known dependent p erhaps on mo del free
parameters but entirely indep endent of the TF observables In mathematical terms this is equivalent to
using probability distributions which are conditioned on the value of the true distance r In the forward TF
case the relevant distribution is P mj r which is readily derived from the joint probability distribution
Eq
P r m
R
P mj r
dm P r m
m M r
S m exp
R
m M r
dm S m exp
Eq is the mathematical basis of a forward Metho d I I analysis The distance r would b e replaced by
a redshift space prediction such as Eq in an actual implementation of Metho d I I
If sample selection were indep endent of apparent magnitude S m would drop out and Eq
would reduce to the familiar normal distribution centered on the na ve exp ected apparent magnitude
M r This is the condition that a simple minimization such as that suggested by Eq b e
valid for determining values of any unknown mo del parameters In a real study however sample selection
almost always dep ends on apparent magnitude although the dep endence is generally indirect as we explain
b elow For simplicity we rst supp ose that in fact a strict magnitude limit Eq applies In that
case Eq obviously do es not reduce to a normal distribution centered on M r It is rather
straightforward to show that the exp ected apparent magnitude is given by
s
Z
A
e
E mj r mP mj r dm M r
erf A
where
m M r
p
A r m
The last term on the right hand side of Eq is the apparent magnitude bias the di erence
b etween actual and na ve exp ectation values Its signi cance is determined by the parameter A r m
a dimensionless measure of how close the ob ject is exp ected to b e to the magnitude limit For A
the bias is small in that case few galaxies are missing due to the magnitude limit But for A the
hypothetical ensemble of galaxies of given and r is strongly truncated by the magnitude limit and the
bias is comparable to or greater than the TF scatter Note that the sense of the bias is that ob jects are
typically brighter than the na ve TF prediction as we saw in x In any magnitude limited sample
It is not of course literally correct that the true distance is sp eci ed in a Metho d II analysis Errors in the
velocity mo del b eing t to the data as well as small scale noise in the actual velocity eld mean that r is only
approximately sp eci ed by redshift information However to rst order this only increases the variance but do es
not bias a Metho d I I solution The formalism presented ab ove is correct in this approximation but the quantity
should b e regarded as the quadrature sum of DI scatter and mo del errors In x we will describ e a metho d
which aims to go b eyond this rst order approximation and include the e ect of mo del errors and velocity noise
a large fraction of the ob jects are likely to b e near the magnitude limit Hence the bias represented by
Eq is a problematic issue in most Metho d I I analyses
In real samples the situation is more complicated TF and D samples are drawn from catalogs
n
culled from photographic plate material Such catalogs see x typically list galaxy apparent magnitudes
and or angular diameters on a measurement system di erent from that used by the DI The catalog might
give for example blue bandpass photographic magnitudes m while the TF study uses I band CCD
B
magnitudes m Candidates for the TF sample might b e required to satisfy m m but once thus
I B
selected the allowed values of m are unrestricted the CCD photometry is far deep er than the catalog
I
Nonetheless it is clear that the distribution of I band magnitudes must a ected by this selection pro cedure
as m is sure to b e correlated with m Analogous statements hold if the sample is limited by catalog
I B
diameter rather than magnitude
Willick has shown how this problem of an indirect magnitude limit may b e dealt with in
several practical situations Let us refer to the quantity determining sample inclusion as which might
b e a photographic magnitude or logarithmic diameter and assume it is required to satisfy The
selection probability for an ob ject with TF observables m is then given by
S m P jm
The quantity on the right hand side may b e determined by rst assuming that P jm is a Gaussian
centered on a mean relation m and with rms disp ersion Both may b e determined empirically from
the data Willick et al a b have shown that m is well mo deled as linear in m and when is a
photographic apparent magnitude or logarithmic diameter The resulting Gaussian probability distribution
is then integrated over the allowed values of
Z
m
p
S m d exp
This selection function may b e substituted into Eq and the exp ected apparent magnitude and thus
the bias may then b e obtained in the usual way The results given by Willick are complicated and
need not b e repro duced here We note only that as in the case of a strict magnitude limit the bias again
dep ends on a limit closeness parameter given in the present case by
M r
p
A r
and is also prop ortional to a coupling parameter which measures the tightness of the TF
relation relative to that of the m correlation In particular when the latter correlation is weak
the bias is small since selection criteria only weakly constrain the apparent magnitude used in the TF
analysis In most real samples however the m correlation is strong enough that even the bias due
to indirect limits is signi cant Another feature of an indirect limit is that if is signi cantly correlated
with as well as with m sample selection probability is dep endent the consequences of this are discussed
in x
The formulae given by Willick for the magnitude and disp ersion biases allow one to correct for
selection bias The basic idea is to iterate least squares ts of the underlying mo del using the parameter
A related complication is that not only the exp ected apparent magnitude but also its variance is biased in the
sense that the actual variance is strictly The causes of this e ect known as dispersion bias were discussed
in x Quantitative expressions for disp ersion bias are given by Willick 2-Cluster TF fit
8 (a) (d)
7.5 TF slope b
7
.4 (b) (e)
.35
.3
.2 (c) (f)
0
-.2
-.4
2000 4000 6000 8000 10000 2000 4000 6000 8000 10000
Fig The main e ects of selection bias on a Metho d I I TF analysis are shown in the left hand panels the same
results following application of the iterative correction pro cedure see text are shown in the right hand panels
Data sets consisting of galaxies in each of two clusters were numerically simulated The simulated galaxies
ob ey a true TF relation with slop e b and scatter mag and are sub ject to a diameter limit Each
simulated data set consisted of one cluster at xed distance r km s assumed known and another at
variable distance r shown on the horizontal axis which was taken as an unknown in the analysis Each plotted
p oint represents the mean of simulations within bins of km s A TF relation was tted to each data set
the free parameters were the zerop oint A not shown slop e b and the distance mo dulus of the second cluster
The left hand panels exhibit the principal e ects of selection bias the tted slop e and scatter are to o small and
the distance of the second cluster is underestimated See text for details
c
estimates from one iteration to correct the next for bias Corrected apparent magnitudes m de ned
as the observed magnitudes plus the computed bias are used in the iterated ts This pro cedure typically
pro duces convergent parameter values in iterations and has b een used by Willick et al a b to
calibrate TF relations for recent p eculiar velocity analyses An alternative to iteration is direct likelihoo d
maximization using individual ob ject probability functions P mj r given by Eq
In Fig we illustrate the e ects of selection bias on b oth DI calibration and distance determination
using numerically simulated data Each simulated data set consisted of two clusters with galaxies
each The distance to the rst cluster was xed at r km s which was assumed known to the
observer The second cluster was stepp ed out in distance as shown on the horizontal axis its distance
r was assumed unknown to the observer Determination of its distance mo dulus was assumed to b e
one goal of the analysis the other was determination of the TF parameters A and b The simulated data
ob ey a TF relation with slop e b and scatter mag sample selection was imp osed by requiring
galaxies to have photographic diameters These diameters were generated from the TF observables
m using empirical relations from real data sets Willick et al ab
For each simulated data set the TF parameters A and b and the distance mo dulus of the second
cluster were solved for using standard least squares techniques An rms scatter ab out the t was
computed separately in each cluster The results of these least squares ts are shown in the left hand
panels of Fig As the second cluster moves farther away so that an ever larger fraction of its member
galaxies are near the diameter limit the bias increases the top and middle panels show that the tted
slop e and rms scatter in the second cluster b ecome progressively smaller as the second cluster moves
farther away The tted slop e levels out at large distances b ecause it is strongly constrained by data in
the rst cluster which is close enough to b e nearly unbiased The b ottom panel shows the error in the
least squares determination of the distance mo dulus of the second cluster log r This error is
in the sense that the distance is underestimated and grows fractionally worse with increasing distance The
right hand panel of Fig show the same quantities following application of the iterative bias correction
scheme discussed ab ove As can b e seen the iterative metho d yields unbiased nal values of the slop e
scatter and distance of the second cluster
Forward DIs Malmquist Bias
Malmquist bias arises in Metho d I analyses in which we rst assign distances directly from the DI
and consider redshift information only afterward As discussed ab ove the large scatter of DIs such as
TF or D guarantees that ob jects with inferred distance d come with varying probability from a wide
n
range of true distances The probability distribution of true distance r given inferred distance d is thus the
mathematical basis for treating Malmquist bias
The simplest way to arrive at P r jd is rst to consider P r d which we obtain from Eq
as follows
Z Z
P r d d dm P r m m M d
r d
p
S d r n r exp
In Eq is the Dirac delta function and we have de ned
Z
S d d S M d
Converting to natural logarithms we derive
ln r d S d
p
P r d P r d exp r n r
d d
ln
is a measure of the fractional distance uncertainty of the DI The desired where
conditional probability is then given by
ln r d
r n r exp
P r d
R
P r jd
R
ln r d
dr P r d
dr r n r exp
We see that S m has dropp ed out entirely in Eq selection bias do es not play a role in
forward Metho d I analyses This may app ear o dd at rst glance one would think that b ecause selection
depletes the sample more at r d than at r d it must a ect the ratio of near and far ob jects at inferred
distance d The reason it do es not may b e understo o d heuristically as follows Ob jects at r d must b e
intrinsically fainter than exp ected for their velocity width M M in order to scatter out to d ob jects
at r d must b e brighter than exp ected M M in order to scatter in to d Thus although the sample
as a whole is progressively depleted with distance the set of ob jects with inferred distance d is atypical its
closer members are unusually faint and its more distant members are unusually bright This discrepancy
exactly balances the overall decrease in sampling density with distance It is imp ortant to note however
that this argument is sp eci c to the forward form of the DI we show how inverse DIs di er in this resp ect
in x and to the absence of an explicit distance dep endence in the sample selection function in x
we demonstrate how such a dep endence might arise
Although indep endent of observational selection forward Malmquist bias is nonetheless a complicated
e ect It is b est illustrated by considering the exp ectation value of r given d
ln x
R R
e
r P r jd dr x n dx dx
R
E r jd d
ln x
R
P r jd dr
e
x n dx dx
where x r d Eq shows that in general E r jd d that is the inferred distance is biased It also
shows that Malmquist bias dep ends on the number density of galaxies all along the line of sight and thus
is a nonlo cal e ect
The simplest though usually unrealistic example of Malmquist bias o ccurs when density gradients
0
n
are negligible on the scale of distance errors i e d In that case the density term drops out
n
of Eq The integrals are then easily evaluated with the result
e
E r jd d
Thus even in the case of uniform density E r jd d This e ect uniform density Malmquist bias is due to
the increasing volume element at larger distances there are more galaxies at r d than at r d and thus
E r jd d for a typical distance indicator it is a e ect More generally density gradients
cannot b e neglected and can either reinforce or work against the volume e ect This is most easily seen
in the approximation that n r is slowly varying on the scale of DI errors Expanding n r to rst order in
Eq then yields
E r jd d d
where
d ln n r
d
d ln r
r d
where we have further assumed Eqs and were rst derived in the extragalactic context
by Lynden Bell et al a cf Malmquist
The slowly varying approximation Eq illustrates in a general way the e ect of density gradients
on the statistical prop erties of inferred distances This e ect is also known as inhomogeneous Malmquist
bias or IHM When d IHM reinforces the uniform density bias when d it works against
it While the essence of IHM is shown by the slowly varying density approximation Eq is in fact a
rather p o or approximation to the true IHM for the density elds seen in the real universe see b elow A
prop er calculation of IHM requires the full numerical integration indicated by Eq
Fig Malmquist bias induced by a density p eak at a distance of km s Simulated galaxies were generated
along a line of sight with a Gaussian p erturbation of full width km s and maximum overdensity
max
atop a uniform distribution The upp er panel shows the distribution of simulated galaxies as a function of their
true distance The lower panel plots as op en squares average true distance vs inferred distance for ob jects in
km s wide bins The meaning of the various lines drawn through the p oints is indicated in the panel
These p oints are illustrated using simulated data in Fig which shows the Malmquist bias as
so ciated with a very strong density p erturbation Galaxies were simulated along a line of sight in which
the density is given by n n r where n r is a Gaussian p erturbation centered at km s with
p p
full width km s and maximum overdensity Galaxies were assumed to b e moving with
max
Hubble ow there are no true p eculiar velocities in this simulation The number of galaxies p er unit true
distance along the line of sight is shown in the upp er panel the e ect of the density p eak on the distribu
tion is clear The simulated galaxies were required to have diameters although this diameter limit
do es not a ect the character of Malmquist bias for reasons describ ed ab ove The simulated galaxies ob ey
a TF relation with scatter mag the observer of the simulation assigned inferred distances d
with a prop erly calibrated TF relation In the lower panel the galaxies have b een binned according to the
values of their inferred distances The average true distances of the galaxies in each bin are plotted as op en
squares
Nowhere do the simulated data follow the Hubble line Very far from the density p eak d km s
and d km s the simulated data follow the uniform density Malmquist bias prediction Eq
However within km s of the p eak the data depart strongly from the uniform density prediction
this is the IHM e ect The sense is that in the foreground of the p eak distances are underestimated the
squares lie ab ove the long dashed line and in the background they are overestimated The amplitude of
these e ects is km s The short dashed line is the prediction of the slowly varying approxima
tion Eq using the exact density gradient It is apparent that this approximation is extremely p o or
this is b ecause the density varies rapidly on the scale of distance errors The heavy solid line is obtained
from the exact formula Eq by numerical integration of the true density eld and correctly predicts
the bias at all distances
Fig gives further insight into an issue touched up on in x namely why IHM can lead
to spurious p eculiar velocities Supp ose that the galaxies depicted in the gure followed the Hubble ow
Their redshift vs inferred distance diagram would then closely resemble the lower panel The S wave seen
in the gure measured with resp ect to the uniform density prediction would b e interpreted as p eculiar
velocities if IHM were neglected In particular motions of many hundreds of km s toward the density
p eak would b e seen in b oth the foreground and the background Such motions are what is exp ected from
gravitational instability but in this example they are merely an artifact of Malmquist bias More generally
if gravitational instability is correct there are real motions but IHM will cause the observed motions to b e
larger This bias mimics the e ect of having more mass in the density enhancement than actual ly exists
It follows that if not prop erly accounted for IHM will result in an overestimate of e g Eq
Distance Dependent Selection Functions The Gould E ect
An imp ortant feature of the Malmquist bias equations in x was their indep endence of obser
vational selection This greatly simpli es their application in p eculiar velocity analyses in particular it
makes the uniform density correction trivial However as was rst p ointed out by Gould under
certain circumstances this desirable attribute of Malmquist bias might break down The origin of this e ect
is b est understo o d in terms of the selection function which we have assumed to dep end explicitly only
on the observables m not on distance As a result in the steps leading from Eq to Eq
S m naturally cancelled out Had the selection function actually b een of the form S m r it would
not have dropp ed out and the result would have b een a selection dep endent equation for Malmquist bias
To see how distance dep endent selection might arise supp ose that selection is based on a blue appar
ent magnitude which must satisfy m m while the TF analysis involves a distinct magnitude m Now
B
instead of empirically relating m to the TF observables and deriving S m according to Eq let
B
us consider the TF relation M satis ed by the absolute magnitude corresp onding to m We can then
B B
relate the probability that m m for a galaxy of width parameter and distance r to a probability
B
involving the absolute magnitude M
B
P m m j r P M m r j
B B
m r
Z M M
B B
B
p
e
dM
B
B
where we have assumed a normally distributed blue TF relation with scatter Eq is apparently
B
nothing but the sample selection function as we have de ned it However it is indep endent of the TF
magnitude m and explicitly dep endent on r rather than writing it S m we must write it S r
How can this derivation b e reconciled with Eq whose premises app ear equally valid but which
yields a qualitatively di erent result The answer is quite subtle and we refer the reader to Willick
for a detailed explanation There it is shown that Eq is valid to the extent that the residuals of
the TF relations M and M are strongly correlated as is normally the case Eq is valid in
B
the opp osite limit of uncorrelated residuals A rigorous derivation of sample selection must take this e ect
into account the general result is a selection function S m r with the strength of the r dep endence
inversely related to the degree of correlation b etween the two TF relations
There are two metho ds to determine whether the Gould e ect is present is real samples The rst is
to directly examine correlations b etween the M and M TF residuals The second is to test whether
B
m is statistically b est describ ed as a function of m and of and r or of all three In the rst of the
B
three cases there is no Gould e ect otherwise it is present at some level Willick et al a b used the
second test with r mo deled by redshift to show that the e ect is signi cant for samples selected on the
ESO angular diameter e g Mathewson et al b Peculiar velocity analyses using such samples e g
Hudson et al rely heavily on redshift space grouping to minimize the e ect
Inverse DIs Selection Bias
A formalism quite analogous to that developed ab ove may b e presented for inverse DIs we assume as
usual that the DI in question is TF The central element of this formalism is the Gaussian distribution of
observed width parameters ab out the exp ectation value m r when selection e ects are neglected
Adding in the remaining elements we arrive at the joint distribution of m and r written in terms of the
inverse TF relation
P r m r n r S m
m r
p
m r exp
We note that in contrast to its forward analogue Eq this equation involves the luminosity function
M
Both selection and Malmquist bias are implicit in Eq As b efore the two are distinguished by
the assumptions of data analysis In a Metho d II analysis we implicitly condition on the true distance
see x The probability distribution which underlies such an analysis is thus P jm r obtained from
Eq according to
P r m
R
P jm r
d P r m
m r
S m exp
R
m r
d S m exp
This equation is the inverse counterpart of Eq In the forward case the condition for selection bias to
vanish was that S m b e indep endent of m which is almost always unrealistic Now the condition is that
S m b e indep endent of We can exp ect that to a go o d approximation this condition will frequently
hold To the extent that it do es Eq reduces to a Gaussian distribution of ab out m r
It follows that a standard minimization pro cedure will yield unbiased values of mo del parameters as
discussed in x
However the assumption that inverse Metho d I I analyses are free of selection bias is sub ject to several
caveats which are often overlooked
i As noted in x using di erent photometric quantities for sample selection than for DI analysis
can result in an dep endence of the function S m Willick has quanti ed and describ ed
correction techniques for this e ect
ii Direct dep endence of sample selection on velocity width can o ccur in studies in which is measured
from the HI cm line Signal to noise ratio considerations can mean that large velocity widths more
likely to b e undetected at low cm ux levels Roth Willick et al a have attempted to
estimate this e ect for the Mould et al TF sample and found it small but non negligible
iii The slop e of the inverse relation the quantity e in Eq is a ected by the accuracy of the
redshift distance mo del used in calibrating the relation Willick This tting bias do es not
a ect the forward relation in which a p o or mo del increases scatter but do es not bias the slop e
iv Finally and p erhaps most imp ortantly while inverse DIs can avoid selection bias in the case of velocity
width indep endent selection they have no sp ecial advantage over forward DIs in terms of susceptibility
to Malmquist bias Indeed inverse Malmquist bias is inherently more complicated than forward as
we describ e in the following section
Inverse DIs Malmquist Bias
An inverse Metho d I analysis is fully analogous to a forward Metho d I analysis except that the
inferred distances are obtained by solving the equation m log d implicitly for d For the linear
m D e
inverse TF relation Eq this gives d which may b e compared with the forward
m A b
inferred distance d Although we have used the symbol d for b oth the forward and
inverse inferred distance the two are not equal in general the non equality of the forward and inverse TF
parameters x guarantees this In fact inverse DIs are steeper than their forward counterparts when
plotted in the usual way M vs Inverse inferred distances are thus larger for luminous and smaller
for faint galaxies Since luminous ob jects dominate a sample at large redshifts it is clear that a redshift
vs inferred distance diagram will lo ok di erent dep ending on whether the forward or inverse relation has
b een used with quite di erent biases in the two cases Let us now quantify these ideas
If we b egin with Eq and go through a series of steps analogous to those which to ok us from
Eq through Eq we obtain the following for the probability distribution of the true distance
of a galaxy with inverse inferred distance d
ln r d
r n r s r exp
P r jd
R
ln r d
r n r s r exp dr
Here e the e ective fractional inverse distance error is roughly but not precisely equal to its
forward counterpart Derivation of Eq required the assumption of a linear inverse TF and also
of a selection function at most weakly dep endent up on details are given in Willick With these
realistic assumptions Eq resembles its forward analogue Eq but with a crucial di erence
the presence of the real space selection function
Z
s r dm m r S m m r
the probability that a sample ob ject lies at true distance r irresp ective of the observables m
Eq shows that the inverse Malmquist e ect quantitatively resembles the forward with the
exception that now the pro duct n r s r replaces n r alone The di erences b etween inverse and forward
Malmquist bias are most clearly illustrated by considering the uniform density case with the further
assumption that s r varies slowly on scales d By analogy with Eq it is then easy to derive the
result
E r jd d d
s
where
d ln s r
d
s
d ln r
r d
We see again that like their forward counterparts inverse inferred distances are biased E r jd d How
ever in contrast with the forward case the e ect is not independent of sample selection or the luminosity
function Even in the case of uniform density it is not simple to obtain a Malmquist correction for an
inverse distance One must not only accurately mo del the selection function S m but integrate it with
resp ect to a b elievable luminosity function M These di culties are only comp ounded by the realistic
requirement of estimating n r in order to account for IHM The net e ect of these considerations is that
inverse DIs are less suitable than forward DIs for Metho d I analyses This p oint must b e kept in mind
when one sp eaks of the unbiased character of inverse DIs The quantity s r might b e mo deled by the
observables s z or s d and thus p erhaps b e dealt with easily There is however a metho d which at least
in principle p ermits e ective use of inverse inferred distances as we discuss in the next section
The Method of Landy and Szalay
A severe obstacle to correcting for IHM is the requirement of knowing the number density n r along
any given line of sight One way around this obstacle is to estimate n r from redshift survey information
This has b een done for example in comparisons of the IRAS density eld with the POTENT mass
reconstruction Dekel et al Hudson a Hudson et al Although reasonable this pro cedure
has unsatisfactory features such as uncertainties introduced by smo othing and the need for an assumed
p eculiar velocity eld to convert redshift to real space density One would like if p ossible to derive the
IHM correction self consistently from the DI data themselves Such a metho d was outlined by Landy
Szalay LS and indep endently by Willick The LS technique relies on the mathematical
relationship b etween E r jd and a quantity which is in principle observable the number of ob jects p er unit
inferred distance
Let us de ne f d as the number of ob jects p er unit solid angle along a particular line of sight p er
unit inferred distance If the galaxy distribution is densely enough sampled one can imagine measuring
f d by binning ob jects in inferred distance The resulting histogram representation of f d is prop ortional
to the probability P d that an ob ject along the given line of sight has inferred distance d This probability
may in turn b e calculated by integrating P r d given by Eq in the forward case and derivable from
Eq in the inverse case over the true distance r
Z
P r d dr f d P d
At this p oint the expressions for the forward and cases diverge and we present separately the result for
each
Z
S d ln r d
p
f d dr r n r exp
forw
d
and
Z
ln r d
p
exp f d r n r s r dr
inv
d
In the forward case the quantity S d which dropp ed out of the conditional exp ectation E r jd is present
here in the expression for f d In the inverse case however f d contains a dep endence on selection
identical to that seen in E r jd Thus it is only for the inverse case that the relationship b etween f d and
E r jd will b e straightforward cf Teerikorpi and we consider only the inverse case now without
a subscript in what follows
Taking as usual x r d we have
Z
ln x
e
x n dx s dx f d d dx
for inverse distances To relate this to E r jd we rely on the mathematical identity e g LS or Willick
that
Z Z
ln x ln x
e e e e
x g d x x g dx dx dx
where g r is an arbitrary function Combining Eqs and one readily obtains
e
f d
e
E r jd d
f d
Eq is the basis of the Landy Szalay metho d of Malmquist bias correction It says that we can
obtain an unbiased estimate of the true distance given an inverse inferred distance d from the observable
distribution f d
The LS metho d is attractive for several reasons First as already noted it enables us to correct for
IHM using the DI data alone Second b ecause LS involves inverse DI distances it is based on a DI relation
whose calibration was very nearly free of selection bias In this sense inverse DIs can b e viewed as nearly
bias free but only when the LS metho d is used Finally while we have not made this explicit in the ab ove
derivation the LS metho d is applicable even when the Gould e ect is present This is b ecause even when
the selection function is of the form S m r one arrives at the expression Eq
With this rosy view of LS must go several realistic caveats Accurate determination of the quantity
f d presents a number of technical problems Willick Basically the issue is one of resolution one
needs a su cient number of ob jects p er bin that the histogram representation of f d is not overwhelmed
by shot noise to obtain this number one must make bigger bins in b oth angle and distance The larger
the bin of course the less accurately one determines the actual IHM correction For these reasons LS has
yet to b e implemented for a real sample An e ort is presently underway Dekel et al to do so for
the large TF samples currently available x It remains unclear at this time whether implementation
of the LS metho d will constitute a ma jor advance in dealing with Malmquist bias
Statistical Measures of the Velocity Field
We b egin this chapter with a discussion of bulk ows in x this has b een the ma jor theme of much
of the work in p eculiar velocities in the last decade x summarizes the history of p eculiar velocity work
with a discussion of the e ort needed to put on a common basis the various samples available The rest of
the chapter discusses various statistics measurable from the observed ow eld The velocity correlation
function and the Cosmic Mach Number are discussed in x and x resp ectively x is devoted to
derivations of the full three dimensional velocity eld from its radial comp onent with emphasis on the
POTENT technique of Bertschinger Dekel and the results that have come from this work
A History of Observations of Large Scale Flow
Since the mid s coherent departures from uniform Hubble ow have b een observed on ever
larger scales A comprehensive historical review of the sub ject through is given by Burstein
Here we brie y outline this history from to the present with an emphasis on recent years
When one sp eaks of a coherent bulk ow one requires a velocity frame of reference Prior to the
mid s the most logical choice for such a frame was that de ned by the barycenter of the Lo cal Group
of galaxies LG the bulk of whose mass is found in the Milky Way and the Andromeda galaxy M
Our own motion with resp ect to such a frame was thought to arise from LG dynamics and the rotation
of the Milky Way the LG barycenter itself was considered at rest with resp ect to the Hubble expansion
This reasoning changed following the discovery of the CMB dip ole anisotropy in see x The
CMB dip ole was and is most naturally interpreted as due to the motion of the LG with resp ect to the rest
frame of the totality of mass within our observable universe in the direction l b and with
amplitude km s With this nding it b ecame clear that signi cant p eculiar velocities exist in the
Universe unless the LG is atypical p eculiar motions of hundreds of km s could no longer b e considered
unusual The CMB dip ole also suggested a natural reference frame for the analysis of p eculiar motions
Early Work
The rst claimed detection of large scale streaming was that of Rubin et al ab who assumed
that giant Sc spiral galaxies are standard candles They studied a sample of such galaxies in the redshift
range km s and found a bulk ow relative to the Lo cal Group frame of km s in
the direction l b The Rubin et al result provoked a great deal of skepticism in the
astronomical community despite the nearly simultaneous discovery of the CMB dip ole The direction of
the Rubin et al bulk ow was nearly orthogonal to the LG velocity vector making the relationship of the
two measurements di cult to understand Indeed the measurement of Rubin et al to this day has neither
b een con rmed nor entirely refuted but instead has simply b een supplanted by more mo dern data based
on b etter distance indicators
In the late s and early s p eculiar velocity work fo cused on the detection of infall to the
Virgo cluster which lies near the North Galactic Pole at l b and is the nearest large
cluster to the Lo cal Group A number of workers using TF Faber Jackson and other techniques estimated
the amplitude of this motion at the Lo cal Group to b e in the range km s e g Schechter
Aaronson et al a de Vaucouleurs Peters Tonry Davis a b Hart Davies
Dressler Even if the largest of these estimates were correct the misalignment b etween the Virgo
direction and the CMB dip ole meant that Virgo centric infall could not b e the sole cause of the LG motion
Davis Peebles a de Vaucouleurs Peters Sandage Tammann Yahil This led
to suggestions that the remaining part of the LG motion resulted from the pull of the relatively nearby
Hydra Centaurus sup ercluster complex which lies almost near the ap ex of the microwave dip ole at a
redshift cz km s Shaya Tammann Sandage Davies Stavely Smith Lilje
Yahil Jones studied residuals from the Virgo centric ow solution of Aaronson et al a
On large scales these ows are referred to as as large scale streaming motions large scale ow bulk ow
among other terms We use all these terms interchangeably here
and concluded they were b est t by a quadrup olar term arising from the pull of Hydra Centaurus These
suggestions were consistent with the generally accepted view that p eculiar motions arose from relatively
lo cal scales km s mass density p erturbations
The Great Attractor
Con rmation of this view app eared to come from a long term study by the Aaronson group Using
IRTF distances to ten clusters in the km s redshift range Aaronson et al concluded
that these clusters exhibited no net motion relative to the CMB They argued as a corollary that the LG
motion had to b e entirely generated by mass uctuations within km s The picture that prevailed
in the mid s was thus one in which the Hubble ow was unp erturb ed on scales km s But
within several years the paradigm had changed radically due mainly to the work of the Samurai S
Applying the D relation to an all sky sample of over bright m mag hcz i km s
n B
elliptical galaxies the group rep orted a bulk ow of amplitude km s relative to the CMB
frame in the direction l b Dressler et al a They emphasized that galaxies in
the Hydra Centaurus concentration participated in and therefore could not b e the source of the observed
ow They concluded that contrary to the conventional wisdom the greater part of the LG motion relative
to the CMB had to b e generated on scales km s
This result sho ok up the cosmology community considerably Its signi cance lay not so much in the
amplitude as in the coherence scale of the ow Indeed as initially rep orted by Dressler et al a the
true scale of the bulk ow was unconstrained p otentially larger than the e ective limit km s
of their sample It was quickly realized e g Vittorio Juszkiewicz Davis that a bulk ow on this
scale was inconsistent with essentially all of the then p opular scenarios for large scale structure formation
including the standard cold and hot dark matter mo dels The p ower sp ectrum P k for any of those
mo dels did not contain su cient p ower on large scales to generate ows of amplitude km s
on scales km s Eq Kaiser showed that the situation for cosmological mo dels was
not as dire as Vittorio et al implied as the e ective depth of the S sample was not as large as
had b een assumed Indeed the tted bulk ow was not inconsistent with the standard CDM mo del with
high enough normalization However in this case the small scale velocity disp ersion is larger than that
observed Bertschinger Juszkiewicz p oint out that there is no normalization of standard CDM
which can simultaneously match the S bulk ow and the observed small velocity disp ersion cf x
The controversial nature of the S result was at least partially alleviated when the group underto ok a
reinterpretation of their data in Lynden Bell et al a hereafter LB They showed that much of the
signal for the elliptical streaming motion was provided by galaxies within km s if they con ned
their analysis to ob jects at distances km s bulk ow was only marginally detected Moreover
in a conical region within of the ap ex of the apparent bulk ow the derived amplitude was nearly
km s They interpreted this as due to infall into an attracting p oint They t their data using a
Metho d I approach x to a mo del in which the ow is generated primarily by a spherically symmetric
density p erturbation they called the Great Attractor GA cf x In addition to the GA distance and
ow amplitude which were treated as free parameters the mo del of LB included a Virgo infall motion
with a xed amplitude of km s at the p osition of the LG Their maximum likelihoo d ts placed the
GA at a distance of km s in the direction l b indicated that it induced an infall
velocity of km s at the p osition of the LG and attributed to it a mass of M
A further elab oration of the GA mo del was provided by Faber Burstein FB who combined
TF data for nearby spirals from Aaronson et al b with the S elliptical data FB tted the
Willick unpublished has shown however that the limited sky coverage of the Aaronson et al sample means
that its ruling out of large scale bulk ows is not de nitive
combined data set to a mo del in which the lo cal velocity eld was fully describ ed by infall motions into
the GA and Virgo but with the parameters characterizing each motion allowed to vary indep endently see
also Han Mould They obtained consistent results using Metho d I and Metho d I I analyses Two
imp ortant conclusions resulted from this work First the nearby TF spirals cz km s mapp ed
out a velocity eld similar to that of the ellipticals Second the e ect of Virgo was found to b e very small
in comparison with the tidal e ect of the GA the infall motion due to Virgo alone at the p osition of the
LG had an amplitude of km s This contrasted strongly with earlier estimates e g Aaronson et
al a which had found a Virgo centric motion of km s and higher x x FB veri ed
that the discrepancy b etween their Virgo ow solution and that of Aaronson et al was due solely to the
shear caused by the GA in the mo del cf Lilje et al
Apparent con rmation of the GA picture came from two directions First three newly completed
redshift surveys da Costa et al Dressler Strauss Davis showed that there was in fact
a very strong excess number of galaxies in the direction of the prop osed GA p eaking at cz km s
cf Fig The second source of supp ort for the GA mo del came from further p eculiar velocity studies of
the region Aaronson et al extended their infrared TF cluster study to the Hydra Centaurus and
other neighboring clusters and con rmed large p ositive p eculiar velocities for these clusters see also Mould
et al Dressler Faber a b studied eld galaxies in the GA region using b oth the TF using
CCD photometry and optical linewidths and D metho ds Probing deep er than earlier studies they
n
claimed to detect back side infall into the GA i e negative p eculiar motions of galaxies at distances
km s Such motions would have to exist if the p ositive velocities detected in the foreground
were indeed generated by the GA and the GA was an isolated structure The detection of backside infall
remains controversial cf Burstein et al Roth Mathewson Ford given the large and
uncertain correction for inhomogeneous Malmquist bias In addition it is clear that the velocity eld within
the Centaurus cluster is quite complex it consists of two pieces with substantially di erent distances and
p eculiar velocities e g Lucey Curry Dickens Burstein et al
Very Large Scale Streaming
Several indep endent p eculiar velocity surveys came to fruition around with results that chal
lenged the consensus which had emerged around the GA picture Willick hereafter W
rep orted an R band CCD TF study of cluster spirals from the sample of Aaronson et al and of
eld spirals in the Perseus Pisces sup ercluster region PP One of the more prominent structures in
the nearby universe Fig PP consists of a dense lament of galaxies at redshift km s centered
on l b W obtained velocity widths for PP galaxies from the published cm data of
Giovanelli Haynes and Giovanelli et al The W PP sample was concentrated in
a relatively narrow strip in galactic latitude stretching over in longitude PP lies nearly
opp osite on the sky from the GA at a similar distance but was not well sampled in the S study The
PP region thus represented a signi cant uncharted volume in the velocity eld as mapp ed by the elliptical
data
W used the cluster galaxies to calibrate the R band TF relation assuming following the nding
of the Aaronson group that the Northern Sky clusters were at rest in the CMB When applied to the
PP galaxies the TF data revealed a coherent motion directed toward the LG with a mean amplitude of
km s Sup erp osed on the coherent streaming were compressional motions presumably due to the
overdense PP lament but the bulk motion was the most prominent feature of the data Because of the
lo cation of PP the streaming motion was directed not only toward the LG but also approximately toward
The very small Virgo infall amplitude found by FB remains controversial Tonry et al have applied the
SBF technique to lo cal ellipticals and found an infall velocity of km s at the LG Work on the sub ject
will undoubtedly continue in the coming years
the GA The amplitude of this streaming was much larger than what was predicted at PP km s
by the FB GA mo del W suggested that the residual motion might b e due to very large scale low
amplitude density uctuations but noted that at the largest distances in his sample km s
ob jects app eared to b e at rest in the CMB frame
The small solid angle covered by the W PP sample raised the question of whether the PP in ow
extended over a larger region of the sky Con rmation that it did was provided by the R band CCD TF
survey of Courteau and Faber Courteau Courteau et al CF who obtained data for
galaxies distributed over the Northern sky using a dilute sampling approach CF used optical H rotation
curves to measure velocity widths Because of their large sky coverage and very uniform selection criteria
CF were able to compare di erent regions of the sky reliably Well away from PP their data suggested
a relatively quiescent velocity eld as indicated by the small scatter of the TF relation when distances
were mo deled by pure Hubble ow However within the PP region now de ned to cover an area several
times larger than the W sample the scatter was larger and the residuals systematically o set from
Hubble ow The implied velocity was km s toward the LG in agreement with W Further
con rmation of a bulk motion came from Han Mould who used the I band CCD TF relation to
study ve clusters embedded in the PP sup ercluster They found PP to b e moving towards the LG with
amplitude km s
Han Mould and collab orators also obtained I band TF data for further clusters distributed over
much of the sky Han Mould et al This sample of clusters largely overlapped those studied
using the IRTF by Aaronson et al but the I band relation was considered more accurate
Han Mould tted the combined data for clusters to two mo dels a large scale bulk streaming
similar to the original mo del prop osed by the S group Dressler et al a and a bi infall mo del
in which the lo cal universe is dominated by comp eting gravitational pulls of the GA and Perseus Pisces
They found that b oth mo dels were consistent with the data
Connolly measured Tully Fisher parameters for a sample of Sc galaxies in b oth Galactic
hemispheres with mean redshift of km s He ts the data to a dip ole and nds go o d agreement
with the analysis of a much larger sample by Courteau et al
The largest single contribution of Tully Fisher data in recent years has b een that of Mathewson
and collab orators Mathewson et al ab hereafter MAT Mathewson Ford Using I band
CCD photometry and a combination of HI and optical velocity widths from several telescop es in Australia
MAT obtained TF data for Southern sky galaxies Roughly half of these were in the steradian
around the GA region as de ned by the S group and another half thought of as a control group were
outside The sample was constructed to b e roughly complete to an ESO diameter limit of Most ob jects
in the sample have redshifts km s but a substantial number have cz km s
MAT calibrated their TF relation relative to galaxies with TF data from Aaronson et al
Willick and Han Mould
The redshift vs TF distance diagram for galaxies in the control region exhibited pure Hubble ow
in the mean Within the GA region MAT con rmed the large km s out owing motions in the
GA foreground that had b een detected in the S elliptical data However the spirals did not app ear to trace
a velocity eld which fell to zero at the distance km s of the nominal Great Attractor Instead
the data indicated that the GA itself i e galaxies in the redshift range km s p ossessed
a mean p ositive p eculiar velocity of km s Furthermore the MAT data did not exhibit the
backside infall which Dressler Faber ab had claimed to detect even at distances km s
a return to the Hubble line was not apparent MAT argued that their data in combination with those
of Willick suggested a vast bulk ow in the Sup ergalactic plane on scales well in excess of
km s
However a recent reanalysis of the MAT data as well as further data from the Mathewson group
itself have not lent supp ort to this hypothesis MAT did not attempt a careful correction for selection
or Malmquist biases in addition their zerop oint calibration used only a small number of indep endent
comparisons Using a careful zerop ointing technique which ties together the distance scale of several TF
samples Willick et al ab c have recalibrated the TF relations used by MAT as well as other TF
samples see x Preliminary analyses of the recalibrated data Courteau et al Faber et al
have derived mo dest km s bulk ows in the GA direction as well as evidence of a
return to the Hubble line at the distance of the GA The backside infall however remains undetected in
these recent studies Very recently Mathewson Ford have enlarged their Southern sky sample by
galaxies The new sample is selected to an ESO diameter of and thus prob es to greater distances
cz km s Mathewson Ford measure a bulk ow in the GA direction of km s for
the combined sample in go o d agreement with Courteau et al and a return to the Hubble line at
cz km s The new data have not b een reanalyzed using the techniques of Willick et al this will
b e carried out in the near future
The Lauer Postman Result
The most dramatic apparent detection of very large scale streaming to date comes from the work
of Lauer Postman hereafter LP Using the BCG L relation x LP obtained redshift
indep endent distances to a volume limited sample of Ab ell clusters in the redshift range cz km s
The typical distance error from the metho d was estimated by LP as so that the p eculiar velocity
error for a typical sample cluster is km s much to o large to estimate individual cluster p eculiar
velocities However the data do allow the determination of a mean bulk velocity vector for the sample
LP de ned an Ab ell Cluster Inertial Frame or ACIF and measured the velocity of the LG L
relative to it They solved for L by minimizing the like quantity
N
M
X
i
i
i
E L g cz
i i
z
i
i
where
i is the value for galaxy i
i
ii z its redshift in the LG frame
i
iii M is its residual in magnitudes from the L relation
i
iv g is a unit vector toward the galaxy and
i
v L is the motion of the LG with resp ect to the ACIF
The minimization is carried out with resp ect to the parameters sp ecifying the L relation which determine
the quantity M and the three comp onents of L The quantity which app ears b oth in the
weighting term and in the exp onent arises from the fact that b oth L and must b e measured within
metric ap ertures as a result the e ective scatter of the relation is actually given by where
M
mag is the scatter ab out the relation The z term gives greatest weight to the nearby clusters
M
although LP show the solution is insensitive to the redshift weighting
LP found that the LG motion relative to the ACIF was in the direction l b total
angular uncertainty That the LG would have a signi cant motion relative to an inertial rest frame is
of course no surprise we already know that the LG moves at km s relative to the CMB However
the latter motion is in the direction l b with negligible angular uncertainty Consequently
according to LP the ACIF di ers signi cantly from the frame de ned by the CMB dip ole Stated another
way if the CMB de nes the true cosmic rest frame the LP data indicate that the Ab ell clusters out to
Colless derives a more accurate version of Eq his reanalysis of their data gives results di ering
only slightly from those of Lauer Postman
km s p ossess a net p eculiar motion of km s in the direction l b total
angular uncertainty Aware of the controversial nature of their result LP p erformed a suite of tests
aimed at identifying systematic e ects in their analysis Among other things they considered the e ects
of cutting their sample at low and high redshifts and at low and high values of of removing prominent
sup erclusters such as the Hydra Centaurus complex from their sample of using a linear rather than a
quadratic L relation and of removing galaxies with very large residuals They found that none of these
tests pro duced signi cant changes in their solution They did identify one non negligible systematic e ect
called geometry bias which arises due to coupling b etween the dip ole moment of the sky distribution
of the sample and the distance indicator relation LP calibrated and corrected for this e ect using Monte
Carlo exp eriments in practice it was rather small km s since their sample has excellent sky
coverage
One could take the view that a km s bulk motion of a volume km s in diameter is
highly implausible and attribute it instead to an incorrect choice of cosmic rest frame One would then b e
obliged to jettison the conventional view that the CMB dip ole is kinematic in origin p ossible explanations
of a cosmological origin for the CMB dip ole have b een suggested by Gunn Paczy nski Piran
and Turner While the p ossibility that the CMB do es not de ne a cosmic rest frame cannot
b e excluded such a view has b ecome increasingly di cult to uphold in light of the discovery of small but
measurable CMB anisotropies by the COBE satellite Smo ot et al These COBE measurements
have corrob orated most theoretical ideas ab out the nature of the CMB
Alternatively if one accepts the kinematic origin of the CMB dip ole and the resultant CMB de ned
rest frame the LP data indicate a p erturbation to the Hubble ow on a km s scale This is
di cult to reconcile with bulk ow observations from the TF and D relations Courteau et al
n
used the Mark III dataset x to estimate the bulk ow within km s to have an amplitude of
km s in a nearly orthogonal direction toward l b It is di cult to understand
how the amplitude of the bulk ow could grow with scale in a universe which approaches uniformity in
the galaxy distribution in the large esp ecially given the apparent convergence of the dip ole in the galaxy
distribution b etween and km s x
The LP result also p oses severe problems for a number of currently p opular theories for the growth of
large scale structure Strauss et al have generated mo ck LP like BCG samples in N b o dy simulations
based on six di erent cosmogonic scenarios including CDM HDM and PIB mo dels These mo ck samples
were constructed to mimic all the observational errors in the actual LP sample Strauss et al found that
no more than of the realizations of any given mo del exhibited an apparent ow as large as that
measured by LP Similar results were found by Feldman Watkins using an analytic approach
Ja e Kaiser suggest comparing not just the bulk ow but the full sample of p eculiar velocities
to the predictions of various mo dels They nd b etter agreement b etween the LP data and mo dels than
do Strauss et al and Feldman Watkins However as Ja e Kaiser ignore the correlations b etween the
measured p eculiar velocities of di erent clusters which arise from the self calibrating nature of the LP
sample the validity of their analysis is uncertain
The very careful treatment by LP makes it unlikely that the observed ow will go away with a
future reanalysis of the data Rather it seems likely that future measurements on comparably large scales
x will determine the ultimate evaluation of LP as was the case with the Rubin et al ab result
If LP is con rmed it will need to b e reconciled with the discrepant ows seen on smaller scales and
theoretical mo dels are likely to require signi cant revision
Bulk Flows A Summary
The myriad of bulk ow analyses we have discussed here have used a wide variety of samples metho d
ologies and techniques and have b een carried out during a time when our knowledge of the prop erties
of distance indicators and the various biases which plague them was evolving rapidly as indeed it still
is The bulk ows measured by di erent samples necessarily prob e di erent volumes of space a direct
comparison b etween them is problematic and we have not attempted one in this review Nevertheless
there is a rough consensus b etween the di erent workers in the eld as to the nature of the bulk ow
within km s cf the review by Postman b oth in its amplitude and in its direction Even
the Lauer Postman bulk ow which is on appreciably larger scales than the others agrees very roughly
in direction The surprising thing ab out this result is the fact that it also agrees roughly in amplitude on
such a large scale
One of our aims in measuring bulk ows is to compare with theoretical predictions as given for a given
mo del by Eq This equation refers to the volume weighted average of the three dimensional velocity
of particles within a spherical volume which can di er substantially from the galaxy and error weighted
average of radial velocities that is usually measured Indeed as Juszkiewicz et al and Strauss et al
p oint out one can have a bulk ow within a volume even in the absence of density uctuations on
larger scales if the center of mass of the volume do es not coincide with its geometric center There are a
number of such subtleties which are only now b eing appreciated in this game further theoretical analyses
along the lines of Kaiser and Feldman Watkins are needed
Of the samples discussed here only that of Lauer Postman approached b eing volume limited
cf Roth However one can measure an e ective volume limited bulk ow if one weights the
galaxies appropriately The POTENT technique to b e discussed in detail in x do es exactly that and
Bertschinger et al quote volume limited bulk ows measured within a series of spheres centered
on the Lo cal Group More recent results using POTENT applied to a more complete data set by Dekel
et al show a bulk ow of km s within a sphere of radius km s sphere which is
in fact consistent with standard CDM normalized to COBE Again the Lauer Postman result which
prob es appreciably larger scales is in disagreement with this and all other viable mo dels we susp ect that
measurements of bulk ows on large scales will continue to b e a very active area of research for at least
the next ve years
Homogeneous Peculiar Velocity Catalogs
Tully Fisher and D data have b een collected at an increasing rate over the past fteen years
n
There now exists a large number of indep endent data sets p otentially useful in p eculiar velocity surveys
This trend will accelerate in the coming years as most ordinary spiral and elliptical galaxies out to
km s acquire quality photometric and sp ectroscopic measurements There is and will b e a
need for uniform catalogs containing the TF and D observable data and the corresp onding estimated
n
distances and p eculiar velocities Distances obtained using the newer DIs such as the PNLF SBF BCG
and SNe metho ds will eventually b e cataloged as well
The greatest problem in constructing a p eculiar velocity catalog is ensuring the homogeneity of the
separate samples The originally published inferred distances to galaxies found in two or more indep endent
data sets are almost sure to di er systematically for two basic reasons First observers measure the relevant
quantities apparent magnitudes diameters and internal velocity widths di erently they not only use
di erent observational metho ds but apply di erent corrections e g for internal extinction to the raw
observables As a result there are as many TF relations say as there are data acquisition and reduction
techniques Nonetheless a TF relation derived for one data set is sometimes heedlessly applied to another
Second observers determine the appropriate TF relation say from their data in di erent ways Some t
an inverse relation some a forward relation in each case the treatment if any of selection bias can di er
A na vely constructed p eculiar velocity catalog in which inferred distances are simply appropriated from
the published literature will thus b e plagued with systematic errors within and b etween data sets
Until recently attempts to pro duce uniform p eculiar velocity catalogs from the literature have b een
largely the work of D Burstein a member of the S collab oration Two early versions known as the Mark I
and Mark I I were distributed privately to the community by electronic mail in and resp ectively
Burstein tabulated the D elliptical galaxy data of the S group Lucey Carter and Dressler
n
Faber a as well as the infrared TF spiral data of the Aaronson group Burstein was able to make
direct comparisons of the D data from the di erent groups However it was di cult to guarantee that
n
the spiral and elliptical data were on the same system comparison of TF and D group distances was
n
used to make small adjustments but these comparisons su ered from the inevitable uncertainty asso ciated
with the spatial segregation of spiral and elliptical galaxies Although not optimal the Mark II catalog
played an imp ortant role in some of the ma jor statistical studies of p eculiar velocity in recent years e g
Bertschinger et al Dekel et al
With the advent of new large samples of p eculiar velocity data there has b een a strong need to
extend Burstein s e ort particularly in view of the more complete understanding of bias e ects acquired
in recent years x A group led by one of the present authors J W has recently completed this task
The basic metho dology is describ ed in Willick et al ab the Mark III Catalog of Galaxy Peculiar
Velocities is presented in Willick et al c and is available for electronic distribution as well The Mark
III consists of the CCD TF samples of Han Mould and collab orators Han Han Mould
Mould et al collectively HM Willick W Courteau and Faber Courteau
CF and Mathewson et al b and of the infrared TF sample of Aaronson et al b as recently
reanalyzed by Tormen Burstein who corrected previously unrecognized systematic errors
in the diameter system used by the Aaronson group see x The Mark III catalog will also include
expanded versions of the elliptical D samples found in the Mark I I
n
Several basic principles guide the construction of the Mark III catalog The rst is the adoption of
a uniform set of rules for correcting the raw observable quantities in particular apparent magnitude and
velocity width for the e ects of extinction inclination and redshift Internal extinction corrections have
b een a source of particular controversy in the past see Burstein et al for a review the Mark III
compilation adopts bandpass sp eci c internal extinctions which minimize TF scatter The second principle
is to re derive a Tully Fisher relation for each sample separately taking full account of and correcting for
selection biases in the manner suggested by Willick cf x An imp ortant result of this re derivation
is that the TF relations in the Mark I I I compilation can di er markedly from those of the original authors
Third the zerop oints the constant A in Eq of the individual TF relations are mutually adjusted to
yield the maximum agreement in the inferred distances for several hundred galaxies which are common to
two or more individual samples Finally the global catalog zerop oint is determined by assuming that in
the sample as a whole the volume weighted radial p eculiar velocity comp onent vanishes in the mean
A representation of the Mark I I I velocity eld is shown in Fig which shows the measured radial
p eculiar velocities of all galaxies within of the Sup ergalactic plane The p oint is drawn at the measured
distance of the galaxy while the line is drawn to its redshift in the CMB rest frame Positive p eculiar
velocities are drawn with solid p oints and solid lines while negative p eculiar velocities use op en p oints
and dashed lines When p ossible galaxies are group ed in order to decrease the errors and reduce the
Malmquist bias p oints representing groups of more than three galaxies are drawn somewhat larger The
zone of avoidance is apparent as the missing wedge out of the middle of the gure Compare this gure with
the IRAS density eld of Fig we have lab eled the ma jor structures as we did in that gure The bulk
ow into the GA is apparent as is the coherence of the ow even back to the Pisces Perseus sup ercluster
It is di cult however to draw quantitative conclusions from this gure alone ab out bulk ows or infalls
into sp eci c structures In the remainder of this chapter we discuss various statistical analyses of p eculiar
velocity data culminating in reconstruction metho ds of the full three dimensional velocity eld in x
Fig The Mark I I I p eculiar velocities of all galaxies within of the Sup ergalactic plane The p oint is drawn
at the measured distance of the galaxy while the line is drawn to its redshift in the CMB rest frame Positive
p eculiar velocities are drawn with solid p oints and solid lines while negative p eculiar velocities use op en p oints
and dashed lines Points representing groups of more than three galaxies are drawn somewhat larger
Velocity Correlation Function
One of the most striking features of the observed large scale velocity eld is its coherence One way
to quantify this is by tting bulk ows to the data x Another approach was suggested by G orski
the correlation function of the velocity eld G orski de nes the correlation function b etween the
ith and j th Cartesian co ordinate of the velocity eld as
h i
r hv x v x r i r r r r r
ij i j ij i j
k
where and are the transverse and radial correlation functions The second equality holds if the
k
velocity eld is homogeneous and isotropic If the velocity eld is derivable from a p otential then these
two are not indep endent
d r r
r
k
dr
In linear p erturbation theory one can derive simple expressions for these quantities
Z
j k r H f
P k dk r
k r
and
Z
H f j k r
r dk P k j k r
k
k r
compare with Eq Of course we observe only the radial comp onent of the velocity eld G orski et
al and Groth Juszkiewicz Ostriker have suggested metho ds to determine the velocity
correlation function from observational data The rst of these pap ers de nes a quantity from a sample
with radial p eculiar velocities u
i
P
u u cos
pairs r
r
P
cos
pairs r
where the sum is over all pairs of galaxies with separations b etween r and r dr and is the angle
b etween galaxies and on the sky A uniformly selected full sky sample exhibiting a bulk ow of amplitude
v will have a velocity correlation function v The second equality of Eq implies that r is
a linear combination of r and r with co e cients dep ending on the spatial distribution of galaxies
k
in the sample
The velocity correlation function at zero lag is a measure of the ro ot mean square velocity disp ersion
of galaxies while the scale on which it drops to zero is a measure of the coherence length of the velocity
eld G orski et al lo oked at two datasets the spiral galaxies of Aaronson et al and the
elliptical galaxies of the Samurai The correlation function of b oth drop to zero at separations of
km s The amplitude of the elliptical galaxy dataset has a much larger amplitude than that of
the spirals although this amplitude was not robust deleting a small number of galaxies in the Great
Attractor region caused the amplitude to drop by more than a factor of two These results were compared
to N b o dy simulations of CDM and PBI mo dels The CDM mo dels t the observed correlation length
well and required a normalization to match the amplitude The PBI mo dels tended to show more
coherence than is seen in the real data
Groth et al used the linear relation b etween and to solve for the latter They
k
emphasized the fact that in the frame comoving with the bulk ow of the Aaronson et al data
the velocity correlations were essentially zero the ow was very cold This is consistent with the small
value of the pairwise velocity disp ersion deduced from redshift surveys x Seen in this light the
correlation length of km s rep orted by G orski et al is probably at least partly an edge e ect
due to a nite sample It is time to revisit the velocity correlation function now that the data samples
have improved and expanded More work is needed to characterize the e ects of survey geometry and in
particular p eculiar velocity errors on this statistic It has the p otential to place strong constraints on
cosmological mo dels once these various e ects are understo o d b etter
Juszkiewicz Yahil p oint out that the comparison of the velocity and the spatial correlation
function yields an estimate of in particular in linear theory
r
Z
E D
H J x
r v r v r dx
ii ii
x
Note that the velocity correlations in this form on scale r dep end only on the the correlation functions
on scales smaller than r Thus accurate measurements of the velocity correlation function and the spatial
correlation function have the p otential to yield a measurement of Moreover the extent to which the
two sides of Eq agree with one another as a function of r on linear scales is a test of gravitational
instability theory Unfortunately existing data are not yet at the stage to allow this test to b e done
The Cosmic Mach Number
Ostriker Suto struck by the coldness of the velocity eld observed in the Aaronson et al
data suggested a new statistic to quantify this coldness the Cosmic Mach Number The Mach
number in standard usage is the ratio of the ow velocity in some medium to the sound velocity in
that medium In the cosmological context the equivalent of the sound velocity is the small scale velocity
disp ersion of galaxies Following Eq the characteristic bulk velocities in a given cosmological mo del
measure the large scale comp onent of the p ower sp ectrum while the small scale velocity disp ersion dep ends
on the p ower on small scales Their ratio is thus indep endent of the amplitude of the p ower sp ectrum
at least in linear theory and is a diagnostic of its shape Strauss Cen Ostriker cf Suto
Cen Ostriker t bulk ows to observed datasets all comp onents of the velocity eld on smaller
scales were attributed to incoherent small scale p eculiar velocities Subtracting o the estimated errors in
quadrature from the rms of the residuals allowed them to de ne a small scale velocity disp ersion and thus
a Mach number Strauss et al compared the Mach number results from three di erent datasets
to the distribution of Mach numbers observed in Monte Carlo simulations of the observational data The
Aaronson et al sample with its relatively small errors gave the strongest constraints on mo dels
standard CDM was ruled out at the con dence level by this statistic This is largely a consequence
of the fact that it greatly over predicts the velocity disp ersion on small scales as was discussed in x
Other mo dels with less p ower on small scales relative to large including tilted CDM and HDM fared much
b etter by this statistic
Reconstructing the Three Dimensional Velocity Field
The bulk ows discussed in x represent one quantitative statistic that can b e extracted from
observations of p eculiar velocities One would like a metho d to characterize all the information available
in the velocity eld In particular given gravitational instability theory Eq or quasi linear extensions
thereof the observed velocity eld gives a measure of the gravitating density eld indep endent of any
assumptions ab out the relative distribution of galaxies and dark matter Bertschinger Dekel
have developed a technique they call POTENT cf Dekel for a review which starts from the basic
assumption that the observed velocity eld is derivable from a p otential r such that
v r r
This is valid to the extent that the velocity eld is curl free Kelvin s circulation theorem implies that
vorticity is generated only in regions of shell crossing Moreover initial vorticity decays in an expanding
universe just as do initial p eculiar velocities Thus we exp ect that at the present time if we smo oth on
large enough scales the vorticity is likely to b e negligible In this case the radial comp onent of the velocity
eld which is all that is observable determines the full three dimensional velocity eld If u r is the
observed radial velocity eld then the p otential normalized to zero at the origin is given by
r
Z
u r dr r
di erentiation via Eq then yields the full three dimensional velocity eld Indeed there is no reason
to restrict the integration in Eq to radial rays Simmons et al discuss optimal integration
paths for recovering the p otential
Given the three dimensional velocity eld linear theory gives a simple relation to the density eld
Eq The POTENT metho d as currently implemented uses a non linear generalization of this following
Nusser et al
v
r f I f
P
r
where I is the unit matrix Eq reduces to r v in the linear limit
Of course we do not observe a radial velocity eld u r we have noisy data for a non uniform and
sparsely sampled set of galaxies Thus another crucial part of the POTENT technique involves turning the
data available into a continuous radial velocity eld This smo othing has several features
i It allows the radial velocity eld to b e de ned at every p oint in space
ii It smo othes over small scale vorticity in the velocity eld due to shell crossing
iii It smo othes over very non linear e ects that are not mo deled by Eq
iv It averages together the very noisy individual p eculiar velocity measurements
Dekel Bertschinger Faber use a tensor window function smo othing that takes into account the
radial nature of the observed velocity eld There are three particularly p ernicious sources of error in the
resulting smo othed velocity eld
i Statistical noise in the velocity eld due to the errors in the individual p eculiar velocities One mini
mizes this noise by weighting each galaxy by the inverse square errors
ii Malmquist bias b oth homogeneous and inhomogeneous resulting from the p eculiar velocity errors
x POTENT uses p eculiar velocities derived via Metho d I applied to forward DIs
iii Sampling gradient bias due to the inhomogeneous sampling of the velocity eld within a smo othing
window This is minimized using equal volume weighting
Note that the e ects of items i and iii are minimized with di erent weightings one can never minimize
b oth simultaneously In practice the e ects of these sources of noise and biases for any given dataset and
smo othing scheme are calculated using extensive Monte Carlo simulations First results of the POTENT
technique are presented in Bertschinger et al using the Mark I I data The resulting density eld
smo othed with a km s Gaussian window clearly shows the Great Attractor and the void in the
foreground of the Pisces Perseus sup ercluster The smo othed velocity eld and resulting density eld as
gained from the Mark I I I data Dekel et al are shown in Fig
The POTENT metho d and results have b een used for a number of other studies now using the Mark
III data describ ed in x We have already referred to its use to trace the density eld at low Galactic
latitudes x and to compare the velocity elds as traced by elliptical and spiral galaxies separately
x Seljak Bertschinger have used a maximum likelihoo d metho d to t the amplitude of
uctuations in the Mark II POTENT density eld assuming a given p ower sp ectrum The analysis is
complicated by the fact that data p oints are coupled not just by real correlations but also by noise
necessitating a full Monte Carlo approach to the covariance matrix A summary of their results for a range
Fig The smo othed velocity eld and resultant density eld from the Mark I I I data in the Sup ergalactic Plane
The smo othing used was a km s Gaussian The arrows give the X Y comp onents of the three dimensional
velocity eld while the contours are of assuming b spaced at intervals of
of plausible CDM like mo dels is
v
where is the rms value of r v in spheres of radius h Mp c This result is in go o d agreement with
v
standard CDM normalized to the COBE quadrup ole
The Initial Density Distribution Function
In x we describ ed a metho d developed by Nusser Dekel which uses the Zel dovich
approximation to reconstruct the initial density distribution function from the eigenvalues of the spatial
derivatives of the velocity eld It can b e shown that this reconstruction is indep endent of when
reconstructing from the velocity eld predicted from a redshift survey while the reconstruction from the
observed velocity eld is dep endent in particular the shap e of the derived initial density distribution
function will dep end on the value of used Nusser Dekel carried out a preliminary analysis of
the Mark II POTENT velocity eld using this technique and found that they needed values of close
to unity in order to match the Gaussian distribution function seen in the reconstruction from the density
eld That is for small values of the distribution function reconstructed from the observed density and
velocity elds did not agree at all Nusser Dekel used this approach to rule out mo dels
at the con dence level the data are consistent with Further tests are needed however to
test whether very low mo dels in which the derived errors b ecome large can b e ruled out as well The
Mark I I I data should give much sup erior results
Higher Order Moments of the Velocity Field
In linear p erturbation theory with Gaussian initial conditions we saw that the density eld showed
a Gaussian distribution Given the direct prop ortionality b etween and r v Eq will also
have a Gaussian distribution However just as the distribution of develops skewness in second order
p erturbation theory so do es Bernardeau ab cf Bouchet et al calculates the higher order
moments of in p erturbation theory for top hat smo othing cf Bernardeau et al and Lokas et al
for Gaussian smo othing and shows that the ratio of the skewness to the variance squared is given
by
S
compare with Eq like that result this is valid only in the range Unlike the case
of the density eld the skewness of the velocity eld is strongly dep endent on Unlike the density
velocity comparisons discussed b elow galaxy biasing do es not a ect the results at least for equal volume
weighting Bernardeau et al have made a tentative measurement of the skewness of the POTENT
density eld from the Mark I I I data and conclude that the data are consistent with with
ruled out at the level Because the skewness is so heavily weighted by the tails of the distribution and
the substantial sources of errors and biases in POTENT will strongly a ect these tails this result remains
tentative and will require extensive Monte Carlo simulations to test it thoroughly Higher order moments
remain unmeasurable from current data
Voids in the Reconstructed Density Field
The dimensionless density eld has a rm lower limit corresp onding to the absence of matter
In linear theory is prop ortional to the divergence of the velocity eld which says that also has a lower
limit dep ending only on the prop ortionality constant f Comparison with lowest observed p oint in the
POTENT maps thus puts a lower limit on f again indep endent of galaxy biasing Dekel Rees
have applied this idea to the Sculptor Void seen in the Mark I I I POTENT data cf Fig and nd that
at the level The systematics of this metho d have not yet b een prop erly treated however
In particular as there is a strong correlation b etween and the voids that one wants to use for
galaxies
this test are in those regions where there are fewest galaxies and thus the noise in the POTENT maps
are highest In addition these are regions which p otentially su er from strong inhomogeneous Malmquist
bias precisely b ecause the galaxy density eld shows strong gradients in voids
Other Approaches to Reconstructing the Velocity Field
The POTENT approach can b e thought of as a parameterized t to the velocity eld in which the
velocity eld in each smo othing volume is t to a bulk ow plus shear terms cf the discussion in Dekel
et al An alternative approach is to expand the velocity eld in Fourier mo des Kaiser Stebbins
and Stebbins use a Bayesian approach regularizing their solution for the Fourier co e cients
by assuming that the velocities are drawn from a Gaussian distribution with a given p ower sp ectrum
Indeed this regularization is equivalent to applying a Wiener lter to the data and thus has the same
feature as we saw ab ove in x the derived density eld go es to zero in regions of p o or data The results
they get are a strong function of the p ower sp ectrum assumed we await a detailed exp osition of their
technique in refereed journals
As a Metho d I technique Table the POTENT metho d has the serious drawback that it assumes
an a priori distance indicator relation calibrated indep endently of the data in question A small error in
the slop e of the Tully Fisher relation for example will cause systematic errors in the derived velocity eld
An alternative approach is to solve simultaneously for the parameters of the distance indicator relation
and the velocity eld in a Metho d I I approach In the rst pap er that takes into account all the selection
e ects in such a problem Han Mould t the Aaronson et al p eculiar velocity data to
a mo del involving infall into the Virgo cluster and the Great Attractor cf Faber Burstein We
describ e a generalization of their technique in x using the IRAS predicted velocity eld
Nusser Davis b suggest an expansion of the radial velocity eld in spherical harmonics and
radial spherical Bessel functions They nd linear combinations of the basis functions that are orthonormal
at the p ositions of the galaxies for which data exist allowing them to nd an analytic solution for the
Tully Fisher parameters and the co e cients of these orthonormalized functions as measured in redshift
space by minimizing the scatter in the inverse Tully Fisher relation thereby eliminating selection bias
Small scale noise and triple valued zones eliminate the one to one mapping b etween real space and redshift
space causing a bias in the derived velocity eld although this seems to b e a small e ect with real data
This o ers an alternative metho d to smo oth p eculiar velocity data and may b e the ideal way to compare
with the predicted velocity eld from redshift surveys using the Nusser Davis a approach cf
Davis Nusser The applicability of the derived velocity eld to p oints other than those where
data exist remains unclear and the metho d is limited to data characterized by a single distance indicator
relation or at least is analytic only in this limit These problems are not fatal by any means and this
metho d holds great promise
Comparing the Density and Velocity Fields
In this p enultimate chapter we bring the results of the previous chapters together with a discussion
of analyses involving b oth the velocity and density elds This can b e done either by predicting the velocity
eld from redshift surveys using Eq and comparing with the observed velocities x or predicting
the density eld from p eculiar velocity surveys using Eq and comparing with the observed redshift
surveys x
Comparison via the Velocity Field
Cluster Infal l Models
Much of the motivation for measuring the velocity eld has b een to compare it to mo dels of what is
exp ected given the density eld Our historical review of our gradual understanding of the nature of the
large scale ow eld fo cussed on measurements of bulk ows but in fact much of the motivation of the early
work was for measurements of cluster infall A spherically symmetric cluster embedded in a homogeneous
medium induces a spherically symmetric radial velocity eld In linear theory the cluster infall velocity is
simply given by Eq
H
v r r r
where r is the distance to the center of the cluster and r is the mean overdensity within r In fact
for a top hat initial density p erturbation i e a spherically symmetric overdensity that is constant in
amplitude out to some given radius the evolution can b e calculated exactly Silk Schechter
Bertschinger ab Reg os Geller by writing the evolution of the p erturbation and the
background density as separate isotropic expanding or contracting b o dies and matching the b oundary
conditions at the edge of the tophat For the case of an op en universe with an initial mean tophat overdensity
at a time when the Hubble constant was H and the density parameter was the radial velocity
i i i
eld at time t is
r sinh sinh
v r
t cosh
where is de ned implicitly by the equation
h i
tH
i i
i i
sinh
i
Similar equations can b e written down for a closed universe Much of the early work on interpreting results
of velocity elds concentrated on tting these formulae to the infall around the Virgo cluster Tonry
Davis ab Aaronson et al b Davis et al Tully Shaya Tammann Sandage
Gudehus cf the review of Davis Peebles a There has b een a great deal of controversy
in the literature ab out the amplitude of the cluster infall detected with characteristic numbers at the
Lo cal Group ranging from km s to km s x this together with the uncertainty in the
overdensity of the Virgo cluster itself in galaxies Sandage Tammann Yahil Davis et al
Strauss et al a has meant that values of determined from Virgo centric infall have b een equally
uncertain Bushouse et al and Villumsen Davis used N b o dy mo dels to test the ability of
this metho d to constrain and concluded that it works to the extent that one s p eculiar velocity data
surrounds steradians of the cluster otherwise shear motions from more distant mass concentrations
can strongly bias the results
In the meantime studies of the Virgo cluster have shown that the approximation of it as an isolated
spherically symmetric cluster is less and less applicable It has b een known for years that it shows ap
preciable substructure on the sky accurate distances to Virgo galaxies have shown that it probably has
appreciable depth at least in spiral galaxies which may contribute to some of the controversy as to its
distance and the intrinsic scatter of the Tully Fisher relation Pierce Tully Fukugita Okamura
Yasuda Moreover the velocity eld around it is a ected by other more distant mass concentrations
and voids in particular Lilje et al demonstrated the presence of a tidal eld in the Aaronson et
al a data from what we now interpret as the Great Attractor
Spherically symmetric cluster infall mo dels are starting to b e applied to more distant clusters Kaiser
and Reg os Geller showed that cluster infall causes characteristic caustics in the redshift
space maps of galaxies around clusters although the structure in the galaxy distribution in the eld in
which the cluster is embedded can make these caustics di cult to identify Careful measurement of these
caustics has the p otential to yield the linear cluster infall velocity which together with the measured
overdensity of galaxies in the clusters will yield However it is not yet clear how practical this approach
is given the complicated e ects of intrinsic small scale structure in the galaxy distribution outside the
clusters themselves
There have b een a variety of attempts to go b eyond the single cluster mo del for the velocity eld by
invoking two or more clusters e g Faber Burstein Han Mould Rowan Robinson et al
or even to t the ow eld around a void Bothun et al But given the availability of redshift
surveys covering much of the sky we can trace out the full density eld at every p oint in space at some
mo dest smo othing length in the lo cal universe and compare the resulting predicted velocity eld x
with observations
Unparameterized Velocity Field Models
In this section and the next we describ e the most direct comparisons b etween p eculiar velocity and
redshift surveys Linear theory gives a relation b etween galaxy density and p eculiar velocity Eq
which can b e used to derive a velocity eld from a redshift survey x The resulting velocity eld
can b e compared p oint by point with measured p eculiar velocities in a Metho d I analysis usually using
the forward DI relations the slop e of the resulting scatter plot is thus in principle a measure of This
pro cess is actually somewhat subtle sampling the predicted p eculiar velocity eld at the measured distance
of each galaxy gives biased results due to the substantial errors in the distances Rather one should
calculate the predicted p eculiar velocities given the redshifts to each ob ject by inverting the predicted
redshift distance diagram along each line of sight This is sub ject to the ambiguities of triple valued zones
Fig Furthermore b ecause the self consistent velocity eld from the redshift survey predicts the
p eculiar velocity of the Lo cal Group this comparison is b est made in the frame in which the Lo cal Group
is at rest This will give di erent results from a comparison in the frame in which the CMB shows no dip ole
b ecause the IRAS velocity eld do es not exactly match the p eculiar velocity of the Lo cal Group x
Much of the early work in this eld was done b efore a prop er understanding of selection and Malmquist
biases were at hand x making these results somewhat susp ect
Strauss compared the IRAS Jy predicted velocity eld with the Mark I I p eculiar velocity
data A strong correlation b etween observed and p eculiar velocities was seen and the slop e was consistent
with Similar results were found by minimizing the scatter in the inverse Tully Fisher relation
for the Aaronson et al a data in a Metho d II analysis However the error in the derived was
not prop erly quanti ed nor for that matter was it demonstrated that the scatter was consistent with the
observational errors
Kaiser et al used a Metho d I approach to compare the velocity eld from the QDOT redshift
survey with the Mark I I data The QDOT density eld and therefore predicted velocity eld was corrected
for redshift space distortions not by iterations as in x but rather by applying a correction to the
smo othed density eld at each p oint taken from Kaiser cf Eq The resulting predicted p eculiar
velocity eld was compared to the Mark I I data binned on the same grid used to de ne the density eld
least squares ts yielded a slop e
A similar approach was taken by Hudson b who used the density eld of optically selected
galaxies to obtain a predicted p eculiar velocity eld to compare with the Mark II data He used the
techniques of Hudson a to correct the data for inhomogeneous Malmquist bias assuming the galaxies
for which p eculiar velocities were measured to b e drawn from the same density distribution as in his maps
He also included in his mo dels a bulk ow from scales b eyond those surveyed He concluded from his
p eculiar velocity scatter plots that with an additional bulk ow of km s towards
l b However this derived bulk ow is almost in the Galactic plane in the direction of the
Great Attractor and thus may b e due to overdensities not surveyed by Hudson s sample On the other
hand this bulk ow is roughly consistent with that observed from the Mark III data so it may indeed
represent ows on scales larger than his sample
Shaya Tully Pierce also compared p eculiar velocities in this case from a combination
of the Aaronson et al a data and their own TF data Tully Shaya Pierce with a redshift
survey namely the catalog of galaxies within km s compiled by Tully b They used luminosity
rather than number weighting and carried out an elab orate analysis which includes comp onents to the
density eld clustered on a variety of length scales They concluded that asso ciated with galaxies is only
clustered on h Mp c scales However their mo deling of the e ects of mass concentrations in the
Zone of Avoidance and b eyond km s is simplistic and their resulting predicted p eculiar velocity
eld only b ears qualitative resemblance to that measured A more recent analysis is presented by Shaya
Peebles Tully using Peebles variational technique to extend the linear theory relation
b etween the density and velocity elds They conclude that but emphasize that their analysis
optical
is still in progress and awaits improved Tully Fisher data
Roth carried out a Tully Fisher survey in the I band of galaxies selected from
a volume limited subset of galaxies within km s from the Jy IRAS redshift survey and
minimized the scatter of the forward Tully Fisher relation as a function of in the IRAS velocity eld
mo del using a Metho d II approach Extensive Monte Carlo simulations demonstrated that this metho d
gives an unbiased estimate of he found Unfortunately systematic errors in the line width data
and the dominance of the triple valued zone around the Virgo cluster which is prominent in the sample
mean that the systematic errors asso ciated with this result are large Schlegel is extending the
survey to contain galaxies with accurately measured line widths from H rotation curves and with
more uniform sky coverage this dataset promises to give tighter constraints on
Finally Nusser Davis a compared the predicted dip ole moment of their multipole expansion
of the IRAS velocity and density eld cf Eq with that measured from the POTENT map They
show that the dip ole of a shell as measured in the Local Group frame dep ends only on the density eld
interior to that shell making this a semi lo cal comparison They conclude although the
error was estimated by eye from their plots This is a promising way to pro ceed esp ecially with their more
sophisticated technique for determining the multipole moments of the measured velocity eld x
Method II
Common to the handful of Metho d II velocity comparisons discussed ab ove is the assumption of a
unique redshift distance mapping as required in a Metho d I I analysis x In the real world however
a distance cannot b e unambiguously assigned from an observed redshift even when the p eculiar velocity
mo del is correct In what follows we explain why this is so and describ e a maximum likelihoo d metho d
to overcome the problems that result
One can distinguish two contributions to a galaxy s p eculiar velocity The rst is what is usually meant
by the p eculiar velocity eld It has a coherence length of a few Mp c or greater is due to p erturbations
in the linear or quasi linear regime and is predictable from an analysis of density uctuations The second
is what is lo osely referred to as velocity noise It has zero coherence length arises from strongly nonlinear
pro cesses and is unpredictable except in a statistical sense We lab el the coherent part v r and describ e
the random part in terms of an rms radial velocity disp ersion Each can separately invalidate the
v
assumption of a unique redshift distance mapping as follows
i Metho d II assumes that the distance r to a galaxy is the crossing p oint in the redshift distance
diagram given implicitly by
r cz r v r
cf x There is however no guarantee that this equation will have only one solution When line
of sight gradients in v r are of order unity there can b e three or more crossing p oints for given cz
Such regions are generically called triple valued zones cf Fig
ii Even if Eq has a unique solution r this solution will di er from the true distance b ecause of
velocity noise In essence cz is a random realization of the redshift eld r r v r and only de nes
the crossing p oint to accuracy
v
The situations just describ ed are summarized in Fig which shows a triple valued zone around the Virgo
cluster The inherent uncertainty due to velocity noise is indicated with the scatter of p oints
Because of these e ects Metho d I I is sub ject to biases over and ab ove selection bias x Willick
et al d have developed a mo di ed form of Metho d II which neutralizes these biases by explicitly
allowing for non uniqueness in the redshift distance mapping The basic idea is to derive correct probability
distributions of observable quantities taking into account the complexities of the redshift distance relation
and then to maximize likelihoo d over the entire data set This approach shares features of b oth Metho ds
I and I I x but is closer in spirit to the latter and will accordingly b e called Metho d I I in what
follows
We assume the goal is to t a mo del p eculiar velocity eld v r a where a is a vector of free
parameters and adopt the useful abbreviation
u r a r v r a
for its radial comp onent The central element of Metho d I I is a description of the redshift distance relation
in terms of a Gaussian probability distribution
cz r u r a
p
P cz jr exp
v
v
A related probability function is given as a function of r in the lower half of Fig that curve shows the
probability distribution in Eq which di ers from P cz jr by the additional factors of n r r r f
min
Three subtleties of Eq deserve mention First is not merely the true velocity noise whose value is
v
thought to b e km s e g Groth et al but rather its convolution with two additional e ects
redshift measurement errors and velocity mo del errors The former are small typically km s but
not entirely negligible The latter which re ect the nite accuracy of our predictions can b e estimated
from N b o dy simulations e g Fisher et al d and are of order km s Second is not necessar
v
ily constant b oth the true velocity noise and mo del errors are larger in dense regions Third whereas true
velocity noise is incoherent mo del prediction errors are not Contributions to v r arising on scales to o
small to b e included in the mo del but which unlike true noise have spatial coherence manifest themselves
as coherent prediction errors In what follows we will neglect these subtleties and treat as a spatial
v
constant of order km s whose value may b e held xed or treated as a free parameter in the likelihoo d
analysis
We quanti ed Metho d II selection bias x based on the probability distribution P m r
arguing that we could in e ect treat redshift as distance Using Eq we may now write down the
joint distribution of the TF observables distance and redshift
P m r cz P m jr P cz jr P r
where we have assumed that the TF observables and redshift couple only via their mutual dep endence on
true distance The observables in a redshift distance sample are m and cz Their distribution is obtained
by integration
Z
P m jr P m cz
cz r u r a
p
n r r dr exp
v
v
Eq gives the likelihoo d of a data p oint in a redshift distance sample valid for arbitrary v r a and
Metho d II consists of maximizing the pro duct of the likelihoo d Eq over the p eculiar velocity
v
sample with resp ect to the parameters of the velocity eld mo del the parameters of the TF relation
including its scatter and
v
The overall likelihoo d dep ends on TF probability evaluated not only at the crossing p oint s but
over a range of distances roughly characterized by
jcz r u r a j
v
This likelihoo d will di er from its Metho d II counterpart to the extent that the length scale on which
P m r varies is comparable to or smaller than the interval de ned by Eq The former scale is
m M
given by d where is the TF fractional distance error and d is the inferred distance
x For galaxies b eyond km s d km s This is considerably larger than the range
given by Eq for typical outside triple valued or at zones In these circumstances Metho ds I I and
v
II di er little Indeed it is easy to see that again away from triple valued or at zones Metho d II
reduces exactly to Metho d I I Eq in the limit However Metho d I I represents a substantial
v
correction to classical Metho d II at small d km s distances in triple valued or at zones or
when b ecomes anomalously large Metho d I I is therefore necessary for rigorous analysis of the very lo cal
v
universe and in particular of the Lo cal Sup ercluster region where small distances and triple valuedness
are often combined
Application of Metho d II requires two additional steps First one must decide whether to use the
forward or inverse form of the TF relation and thus whether Eq or Eq is used for P m r in
Eq As in Metho d I I x the inverse metho d is advantageous when sample selection is indep endent
of velocity width though with Metho d II this choice introduces a new uncertainty discussed b elow
Second one do es not apply Eq directly but instead derives from it suitable conditional probabilities
P mj cz in the forward case P jm cz in the inverse These are obtained from Eq as follows
Z
P mj cz P m cz dm P m cz
R
m M r
dr r n r P cz jr S m r exp
R R
m M r
dr r n r P cz jr dm S m r exp
and
Z
P jm cz P m cz d P m cz
R
m r
dr r n r m r P cz jr S m r exp
R R
m r
dr r n r m r P cz jr d S m r exp
where we have restored the compact notation P cz jr Eq reversed the order of integration in the
denominators and allowed for an explicit r dep endence of sample selection which in fact exists for some of
the Mark I I I samples cf x The integrals over m and in the denominators of Eqs and
In principle Metho d II rigorously incorp orates selection e ects into either the forward or inverse formalism
However with real data characterization of sample selection is often sub ject to uncertainty Willick et al ab
Its relatively smaller susceptibili ty to selection bias thus remains a virtue of the inverse approach
can b e done analytically for simple forms of S Willick so two dimensional integrations are not
required The chief advantage of conditional probabilities is that they are less sensitive than P m cz
to the precision with which the number density n r is mo deled As n r app ears in b oth numerator and
denominator it only weakly a ects the conditional distributions provided it varies slowly compared with
P cz jr or the TF probability term as will generally b e the case This is imp ortant since it is the accuracy
of our velocity mo del not our density mo del with which we are mainly concerned Still Metho d II
unlike Metho d I I do es require a density mo del and in that sense resembles a Metho d I analysis Finally
note that the velocity width distribution function x strictly cancels out of P mj cz while the
luminosity function M x do es not cancel out of P jm cz although the results are insensitive
to the luminosity function as it again app ears in numerator and denominator
Willick et al d used Metho d II to analyze data from the Mark III p eculiar velocity catalog
x They limited their analysis to the galaxies in the TF samples of Mathewson et al b
and Aaronson et al b which densely sample the lo cal region and to redshifts km s
The forward metho d Eq was used and the TF parameters for each sample slop e zerop oint and
scatter were allowed to vary in the search for maximum likelihoo d rather than xing them at their values
determined in the Mark I I I analysis x
The IRAS Jy predicted p eculiar velocity eld was taken as the mo del to b e tted to the data
To rst order this mo del velocity eld dep ends only on although the analysis was carried out for two
di erent smo othing lengths and with and without nonlinear e ects included Using the IRAS galaxy
density eld for the quantity n r that app ears in Eqs and assumes that IRAS galaxies are
distributed like the Mark I I I sample ob jects This assumption may not b e correct in detail but is exp ected
to have a minimal e ect on our conditional analysis The forward Metho d II analysis was carried out in
terms of a quantity de ned by
X
ln P m j cz
i i i
i
where the sum runs over all galaxies used in the comparison An analogous statistic using the inverse
relation is discussed by Willick et al and gives consistent results
We summarize these results in Fig which shows vs curves for four realizations of the
IRAS elds and km s gaussian smo othing lengths each with and without nonlinear corrections
following Nusser et al and setting b The b est values of o ccur at the minima of the curves
they range from for km s smo othing linear to for km s smo othing nonlinear
The statistical error at con dence level asso ciated with these numbers is as determined by
deviations of the likelihoo d function from its minimum Note that there are systematic e ects acting as
well a larger smo othing length leads to a larger and nonlinear corrections yield larger for a given
smo othing length These results are to b e exp ected b oth increasing the smo othing scale and making
nonlinear corrections decrease the amplitude of predicted p eculiar velocities and thus are qualitatively
similar to decreasing For the nonlinear elds the predicted velocities now dep end separately on and
b the curves here were obtained for the case b i e
Which of the four IRAS predicted velocity elds shown should ideally b e used in estimating In
Metho d I I we are comparing the predicted velocity eld with unsmo othed data The eld obtained from
km s smo othing thus gives rise to the most valid comparison here At smaller smo othing lengths
the nonlinear corrections cease to b e valid Nonlinear e ects must b e present as the overdensities at
km s smo othing can b e considerably in excess of unity However Fig shows that the nonlinear curve
has formally smaller likelihoo d than the linear curve The reason for this is not well understo o d at present
A compromise value is roughly midway b etween the linear and nonlinear minima It is not clear
how to combine the systematic and statistical errors we have identi ed for now we conservatively put the
Fig Likelihoo d plotted as a function of for four realizations of the IRAS predicted p eculiar velocity
and density elds Likelihoo ds have b een computed using a Metho d I I comparison with a subset of the Mark I I I
p eculiar velocity catalog see text for details The left hand panel depicts results in which a km s smo othing
length was used in the IRAS velocity and density eld reconstruction a km s smo othing length was used in
the right hand panel Each panel shows the results for linear dotted line and nonlinear solid line predictions
The latter were obtained by holding the bias factor xed at b
con dence level at
As mentioned ab ove the value of the rms velocity noise was treated as a free parameter its nal
v
value determined by maximizing likelihoo d for each Willick et al found that km s
v
for the likelihoo d maximizing values of This value is remarkably small in view of the fact that the
IRAS predictions themselves are thought to have rms errors of km s as discussed ab ove the
implication would app ear to b e that the true noise is km s It is unlikely that the velocity eld is
in reality that cold It is likely instead that by neglecting correlated mo del prediction errors see ab ove the
likelihoo d analysis ends up underestimating Future implementations of b oth Metho d I I and Metho d
v
I I which similarly assumes uncorrelated residuals will need to address this issue
Comparison via the Density Field
The reconstruction of the mass density eld using the POTENT metho d b egs a comparison with
the galaxy density eld as observed in redshift surveys Indeed to the extent that the two elds are
prop ortional to one another their ratio gives a measure of via Eq Dekel et al carried out
this comparison using the Mark II POTENT maps of Bertschinger et al and the IRAS Jy
redshift survey The rst and most striking result from this comparison is that the POTENT and IRAS
density eld show qualitative agreement Given the noise and sparseness in the p eculiar velocity data
the POTENT map has much greater noise than do es the IRAS map and therefore the region in which
the comparison of the two can b e made is limited Nevertheless b oth show the Great Attractor and the
void in front of the Perseus Pisces sup ercluster With km s Gaussian smo othing there are
indep endent volumes within which the comparison of the two density elds can b e made A scatter plot
of the two shows a strong correlation In the absence of biases the slop e of the regression would b e an
estimate of However as discussed in x the POTENT density eld is sub ject to a number of biases
The most severe of these in this context is Sampling Gradient bias with inhomogeneous Malmquist bias
taking a close second One can quantify the rst by sampling the IRAS predicted velocity eld at the
p ositions of the Mark II galaxies and running the results through the POTENT machinery comparing
the resulting density eld to the input density eld yields a regression slop e of substantially di erent
from unity Given this fact Dekel et al used an elab orate maximum likelihoo d technique to quantify the
agreement b etween the POTENT and IRAS density elds For a given value of actually given values of
and b Eq rather than Eq is used throughout they create mo ck Mark I I datasets given the
IRAS predicted p eculiar velocity eld via the metho d describ ed in x noise is added and the results
are fed into POTENT Note that these simulations su er from sampling gradient bias exactly as do es the
real POTENT data The slop e and scatter of the regression b etween the original IRAS map and the mo ck
POTENT map are recorded and the two dimensional distribution of these two quantities is calculated for
such realizations Elliptical ts to this distribution allows them to calculate the probability that the
observed slop e and scatter of the regression is consistent with the mo del assumed i e the IRAS predicted
p eculiar velocity eld for the given values of and b are consistent with the observed p eculiar velocity
eld given the errors This pro cess is then rep eated for a grid of values of and b
The conclusions from this work are as follows
i There exist values of and b for which the likelihoo d that the mo del is is consistent with the data
is high That is the observed scatter is consistent with the assumed errors and the assumption that
IRAS galaxies are at least a biased tracer of the density eld that gives rise to the observed p eculiar
velocities
ii The tightest constraint is on for which the likelihoo d curve gives at con dence
iii Not surprisingly the constraints on and b separately are quite a bit weaker Indeed the likelihoo d
contour levels do not close at large for a given meaning that the data are consistent with
purely linear theory in which no second order e ects exist The data are inconsistent with highly
non linear conditions i e very small for a given but this is the regime in which the non linear
approximation used starts to break down anyway
May we then conclude that the gravitational instability picture has b een proven After all we have
seen consistency in the data with one of its strongest predictions namely Eq However as Babul et al
emphasize Eq can b e derived directly from the rst of Eqs the continuity equation alone
only the constant of prop ortionality comes from gravity Indeed and r v are prop ortional for any mo del
in which galaxies are a linearly biased tracer of the mass and for which the time averaged acceleration is
prop ortional to the nal acceleration They show analytically and with the aid of simulations that a range
of mo dels with non gravitational forces exhibit correlations b etween and r v at least as strong as that
seen in the POTENT IRAS comparison Thus a pro of of the gravitational instability picture will require
ruling out these alternative mo dels by other means
Work is in progress as this review is b eing written to up date the POTENT results using the Mark I I I
p eculiar velocity compilation The Mark I I I data have b een corrected for inhomogeneous Malmquist bias
assuming that the IRAS density eld is that of the galaxies of the Mark III sample More imp ortantly
however the systematic errors in the overlap b etween datasets have b een minimized and the volume
surveyed well with the Mark I I I data is such that the IRAS POTENT comparison can b e done over four
times as many data p oints as b efore In addition the more complete sampling means that the sampling
gradient bias is smaller than with the Mark I I data by roughly a factor of two in the mean The left hand
panel of Fig shows a preliminary version of the POTENT density eld in the Sup ergalactic plane using
Fig The left panel is the POTENT density eld r v in the Sup ergalactic plane from the Mark I I I p eculiar
velocity data The right panel is the indep endently determined density eld of IRAS galaxies The smo othing in
b oth panels is km s The axes are lab eled in km s The Lo cal Group sits at the center of each panel
km s Gaussian smo othing taken from Fig The right hand panel shows the density eld of the
IRAS Jy survey at the same smo othing The qualitative agreement is remarkable In the Sup ergalactic
plane b oth maps show the Great Attractor the Perseus Pisces Sup ercluster the Coma A Sup ercluster
as well as voids b etween Coma and Perseus and South of the Great Attractor the Sculptor Void Work
is ongoing to quantify the di erences b etween the two maps and to put exact error bars on the derived
Hudson et al have carried out a comparison of the Mark I I I POTENT results with the optical
galaxy density eld of Hudson a b They also nd go o d agreement b etween the two density elds
a less elab orate analysis than that of Dekel et al shows
Discussion
In this brief concluding chapter we summarize what it is that has b een learned from redshift and
p eculiar velocity surveys and put the results into the context of the larger eld of observational cosmology
The Initial Power Spectrum
We have put constraints on the p ower sp ectrum directly from measurements of the distribution of
galaxies x The fact that p ower sp ectra derived from redshift surveys in di erent areas of the sky
using di erent samples agree implies that it is meaningful to de ne a p ower sp ectrum in the rst place
That is we do not live in a simple fractal universe in which a mean density dep ends on the scale on which
it is measured Moreover our samples are starting to b ecome big enough that our statistical measures are
not completely dominated by sampling uctuations at least for measures probing relatively small scales
h Mp c and smaller This is not to say that improvements in the statistical errors in the p ower
sp ectrum on these scales are not needed
The redshift survey data strongly rule out the standard CDM mo del and are t much b etter by a
mo del Fig This is in accord with analyses of the small scale velocity disp ersion of
galaxies their large scale angular clustering observations of bulk ows on large scales as well as constraints
from the CMB uctuations Efstathiou et al Kamionkowski Sp ergel Kamionkowski Sp ergel
Sugiyama the distribution and mass sp ectrum of clusters Bahcall Cen and a host
of other constraints Unfortunately we cannot conclude from this that the dark matter problem is solved A
number of the di erent suggested p ower sp ectra are degenerate over the scales prob ed by redshift surveys
with very di erent implications for the nature of the dark matter compare the range of Fig to that of
Fig In particular the data we have presented are also consistent with the Mixed Dark Matter mo del
and the Tilted Cold Dark Matter mo del which are two of the more p opular mo dels b eing discussed
The Mixed Dark Matter mo del has a long history starting with the idea that adiabatic damping of the
p ower sp ectrum on small scales by baryons will cause a turn down in the p ower sp ectrum Silk Dekel
its recent incarnation is in terms of a mix of hot and cold dark matter Schaefer Sha Stecker
Schaefer Sha Taylor Rowan Robinson Davis Summers Schlegel Klypin
et al In particular the hot dark matter suppresses the p ower sp ectrum on small scales decreasing
the small scale velocity disp ersion relative to standard CDM and increasing the amount of large scale
p ower for a given normalization on small scales However the mo del p erhaps suppresses small scale p ower
overly much galaxies cannot form on these small scales until very late which is di cult to reconcile with
observations of galaxies and quasars at very high redshifts Cen Ostriker cf Efstathiou Rees
for a similar critique of standard CDM
The tilted CDM mo del was suggested simultaneously by a number of workers Cen et al Lidsey
Coles Lucchin Matarrese Mollerach Liddle Lyth Sutherland Adams et al
Cen Ostriker It also increases the amount of p ower on large scales relative to small in this case
by changing the slop e of the primordial p ower sp ectrum However Muciaccia et al showed that
standard CDM is preferred over tilted mo dels when velocity eld data and CMB uctuations are taken
into account as well
There are a number of further constraints on the p ower sp ectrum some of which we ve only touched
up on on this review The most imp ortant of these is the uctuations in the CMB On the largest scales the
uctuations as detected by COBE app ear to b e consistent with a primordial p ower sp ectrum of index n
the in ationary prediction G orski et al with an amplitude that matches well that predicted from
the CDM mo del Fig Observations of the CMB uctuations on sub COBE scales a few degrees
are just b eginning to yield repro ducible results b ecause more than the Sachs Wolfe e ect is op erating on
these smaller scales these data can p otentially yield information on the density parameter in baryons
as well as the p ower sp ectrum Hu Sugiyama In addition these prob e scales now b eing reached by
the largest p eculiar velocity and redshift surveys spanning much of the gap seen in Fig In this regard
the bulk ow results of Lauer Postman x remain unexplained in the context of mo dels for
large scale structure It is vitally imp ortant that this result b e checked as a number of workers are now
doing x The comparison b etween large scale ows and CMB uctuations has the p otential to check
gravitational instability theory directly indep endent of the p ower sp ectrum Juszkiewicz G orski Silk
Tegmark Bunn Hu
Constraints can b e put on the p ower sp ectrum from observations of galaxies at high redshift The
amount of small scale p ower determines at what ep o ch galaxies will form measures of the age of galaxies
and their evolution thus tell us something ab out the p ower sp ectrum Balancing the need for enough
small scale p ower to allow galaxies to form early as seems to b e required by observations of high redshift
galaxies and quasars against the requirement of not to o much small scale p ower in order to restrict the
small scale velocity disp ersion x has not yet b een self consistently done for any mo del Similarly
observations of clusters of galaxies and their evolution also have the p ower to constrain cosmological mo dels
eg Peebles Daly Juszkiewicz Finally detection of the evolution of clustering in the universe
would b e a tremendously imp ortant observation In an op en universe clustering ceases to grow when
deviates signi cantly from unity x unambiguous detection of this e ect would b e a sensitive measure
of
Perhaps the most dramatic constraint one could imagine on the p ower sp ectrum would b e the lab o
ratory detection of dark matter Primack et al It would b e a tremendous triumph of our theoretical
framework if a dark matter particle were discovered with prop erties consistent with the b est t p ower sp ec
trum from astronomical data The HDM mo del has b een out of favor for some time given its unphysically
late formation ep o ch for galaxies e g White Frenk Davis but if a de nite non zero mass for
the muon neutrino were measured appropriate to close the universe HDM mo dels would certainly enjoy a
resurgence of p opularity
The Distribution Function of the Initial Fluctuations
All the tests we have describ ed for the random phase hypothesis have yielded p ositive results there
is no direct evidence for non Gaussian uctuations in the initial density eld Perhaps the strongest such
claim comes from the direct measure of the distribution function of initial uctuations as found by the
time machine of Nusser Dekel Similar conclusions are found in analyses of the COBE CMB
uctuations e g Hinshaw et al However as we emphasized in x this by no means allows us
to conclude that all non Gaussian mo dels are dead Each of the tests describ ed in x refer to a sp eci c
smo othing scale and a mo del that is non Gaussian on a given scale need not b e so on another What
is needed is a systematic test of each non Gaussian mo del prop osed against the various observational
constraints This has b een done to a certain extent for mo dels of cosmic strings Bennett Stebbins
Bouchet and texture mo dels Pen Sp ergel Turok mostly in the context of non Gaussian
signatures in CMB uctuations
The Gravitational Instability Paradigm
The results we have presented here are all consistent with the gravitational instability picture In
particular we have seen that there exist physically plausible p ower sp ectra which can simultaneously
match the observed large scale distribution of galaxies large scale ows and the CMB uctuations
When redshift surveys b egan to reveal extensive structures such as giant voids and the Great Wall many
questioned whether this could b e explained in the context of gravitational instability theory However
simulations by Weinberg Gunn and Park as well as arguments based on the Zel dovich
approximation Shandarin Zel dovich showed that these structures were not unexp ected given
gravitational instability and plausible mo dels for the p ower sp ectrum
The most direct test of gravitational instability comes from the comparison of p eculiar velocity and
redshift surveys Dekel et al in particular claim that the Mark I I p eculiar velocity data are consistent
with the velocity eld predicted from the distribution of IRAS galaxies and gravitational instability theory
However this is not a pro of Babul et al demonstrate mo dels with velocities due to non gravitational
forces in particular large scale explosion mo dels that the Dekel et al tests would not rule out
We have not discussed features of the velocity eld on small scales smaller than are resolved by the
POTENT IRAS comparison with its km s Gaussian smo othing Burstein presents evidence
An obvious exception to this statement is the Lauer Postman bulk ow if it is con rmed by further
observations we may nd ourselves questioning the gravitational instability paradigm
that the very lo cal velocity eld as measured with the Aaronson et al a TF data di er qualita
tively from the IRAS predictions a conclusion that continues to hold with the Tormen Burstein
reanalysis of the Aaronson et al data Similarly the complete lack of an infall signature in the spiral
galaxies around the Coma cluster found by Bernstein et al is worrisome and remains unexplained
Understanding these results remains a task for the future
The Value of
Gravitational instability theory has given us a to ol to measure the cosmological density parameter
by comparing p eculiar velocities with the density distribution Eq although in most applications
galaxy biasing means that we constrain only b In Table we summarize the various constraints
on that we have discussed in this review
Is there some consensus in the literature as to the value of One way to assess this at least
qualitatively is to plot each of the determinations in Table as a series of Gaussians of unit integral with
means and standard deviations given by the numbers in the table For simplicity asymmetric error bars have
b een symmetrized and those determinations without quoted error bars are not included Determinations
based on IRAS and optical samples are plotted with di erent symbol types We now simply add the
Gaussians together to yield the two heavy curves in the plots Note that this pro cedure tends to give
lower weight to those determinations with more realistic i e larger error bars No attempt has b een
made to assess the relative quality of these di erent determinations Note also that b ecause many of
these determinations are from common datasets they are not indep endent Thus this form of qualitative
summary gives an unprejudiced view of literature of determinations of from redshift and p eculiar velocity
surveys The heavy curves have a mean of and standard deviation of IRAS and a mean of
and standard deviation of optical These values are actually in quite close agreement although that
seems more coincidental than anything else given the large spread of individual determinations
The community is clearly not quite ready to settle on a single value for for the IRAS galaxies
The determinations range from Fisher et al b to Dekel et al although the latter
is likely to come down slightly with the Mark III data Dekel private communication This is re ected
in the large standard deviation larger than any individual determination and the at top to the heavy
curve in Fig The optical shows a smaller spread p erhaps simply b ecause there are fewer individual
determinations of it The o dd man out is the determination of by Shaya et al although their
determination is heavily a ected by their mo deling of the background density within km s and the
density eld b eyond there Moreover their work remains in ux compare with Shaya et al and it
is not clear where their nal results will lie
Most of the references in Table are very recent and we have not done a thorough job of reviewing
the earlier literature esp ecially on Virgo centric infall However the common impression that estimates
of have taken a dramatic upturn in recent years is wrong Davis et al used observations of
Virgo centric ow to nd in go o d agreement with the values for optical galaxies here The
value from the Cosmic Virial Theorem from Davis Peebles b is di cult to interpret in terms of a
biasing mo del but corresp onds to for an unbiased mo del
The Relative Distribution of Galaxies and Mass
We have very few handles on the biasing parameter indep endent of One approach has b een to
assume a mo del for the p ower sp ectrum normalize it to the COBE uctuations and then compare the
results predicted for the galaxy uctuations at h Mp c with observations This approach is by de nition
mo del dep endent for standard CDM one nds that optical galaxies are unbiased and that IRAS galaxies
Table Constraints on and
Quantity Value Comments Reference
Redshift Space Anisotropy
s vs w Fry Gazta naga
C f A
s vs w Fry Gazta naga
S S RS
s vs w Fry Gazta naga
I RAS
s vs w Peacock Do dds
I RAS
s vs w Peacock Do dds
optical
Angular moments of Hamilton a
I RAS
Angular moments of Fisher et al b
I RAS
Angular moments of P k Cole et al
I RAS
Spherical Harmonics Fisher et al c
I RAS
Spherical Harmonics Heavens Taylor
I RAS
Cosmic Virial Theorem Davis Peebles b
Acceleration on Lo cal Group
Infall into Virgo Davis Huchra
optical
QDOT Rowan Robinson et al
I RAS
IRAS Jy Strauss et al c
I RAS
Hudson b
optical
Analyses of Velocity Field
Amplitude of uctuations Seljak Bertschinger
v
Gaussianity of r v Nusser Dekel
Skewness of r v Bernardeau et al
Voids in r v Dekel Rees
Comparison of Velocity and Density Fields
Velocity Scatterplot Strauss
I RAS
Velocity Scatterplot Kaiser et al
I RAS
Velocity Scatterplot Hudson b
optical
Variational Metho d Shaya et al
optical
Scatter in TF diagram Roth
I RAS
Di erential Dip ole Nusser Davis a
I RAS
vs r v Dekel et al
I RAS
Likelihoo d analysis Willick et al d
I RAS
vs r v Hudson et al
optical
are anti biased while a mo del like CDM gives a normalization that leaves the IRAS galaxies
unbiased
Alternatively one can constrain biasing by lo oking for non linear e ects to break the degeneracy
b etween and b The skewness is one such e ect Fry Gazta naga and Frieman Gazta naga
claim that the b eautiful agreement b etween the measured higher order moments of the APM counts
in cells with that predicted given the p ower sp ectrum implies that biasing of optical galaxies is very weak
and to the extent that there is biasing that it is lo cal Dekel et al attempted to lo ok for non linear
e ects in the IRAS POTENT comparison they could only show that the data are inconsistent with very
strong non linearities thereby ruling out very small values of b
Finally one can lo ok for relative biasing of di erent types of galaxies as we describ ed in x The
e ects are subtle outside of clusters there are no two p opulations of galaxies known that have qualitatively
Fig The distribution of determinations of discussed in this review Each determination is shown as a
Gaussian of unit integral with mean and standard deviation given by the values listed in Table IRAS and
optical determinations are plotted with di erent line types The renormalized sums of all curves optical and
IRAS separately are shown as the heavy curves
di erent large scale distributions The lack of such e ects have motivated several workers Valls Gabaud
et al Peebles to argue that biasing cannot b e acting at all But di erential e ects are seen
b etween galaxies of di erent luminosities and morphological types It is time for a detailed comparison
of these observed e ects with hydrodynamic simulations in order to see what constraints these put on
general biasing schemes
In any case the consensus of the community is that biasing is relatively weak few authors are arguing
for b these days This is quite a contrast to a decade ago when the idea of biasing was rst introduced
values of b or higher were p opular e g Davis et al Thus we conclude that
the results of Table are still consistent with values in the range to It has b een quite
p opular in recent years to argue for the lower value given the coincidence with the value of needed to
match the value preferred by the p ower sp ectrum Coles Ellis
Is the Big Bang Model Right
One tests the Big Bang mo del with redshift and p eculiar velocity data only to the extent that they
give results which can b e t into our grander picture of the evolution of the universe with input from all
the sub jects we did not discuss observations of distant galaxies and quasars measurements of individual
galaxy prop erties abundances of the light elements and so on We should p oint out one serious problem
which we see on the horizon The data we have discussed p oint towards a value of close to unity
implying an age of the universe given roughly by t H Eq With recent determinations of
the Hubble Constant of the order of km s Mp c Jacoby et al Pierce et al Freedman
et al this gives an age of billion years less than half the currently accepted ages of the oldest
globular clusters e g Chab oyer Sara jedini Demarque Note that this would b e a problem even
if for which t H billion years We may nd ourselves invoking theoretically awkward
mo dels in which In any case the next few years should b e very exciting as we come to grips
with this rather basic problem
The Future
We conclude this review with a quick discussion of the various on going and planned redshift surveys
and p eculiar velocity surveys of which we are aware As these b ecome available we can lo ok forward to
applying the statistics developed so far to vastly sup erior datasets moreover these will allow us to do
analyses of much more subtle statistics
There are a number of large scale p eculiar velocity surveys in progress Giovanelli Haynes and
collab orators are doing a Tully Fisher survey of Sc I galaxies from the Northern sky drawn from the UGC
catalog together with calibrating galaxies drawn from a number of clusters They have data for roughly
galaxies In the meantime Mathewson Ford have extended their Tully Fisher survey in the
Southern Hemisphere to smaller diameters as rep orted in x their sample now includes a total of
galaxies
At higher redshift a team of eight astronomers started by three of the original Samurai Burstein
Davies and Wegner has extended the Samurai D survey of elliptical galaxies to a further
n
galaxies in clusters at redshifts km s Colless et al
Several groups are attempting to check the large scale bulk ow measured by Lauer Postman
The same authors in collab oration with Strauss are in the pro cess of extending the survey to
include the BCG s of all Ab ell clusters to z a total of over clusters They exp ect to complete
the gathering of the data by mid Fruchter Mo ore are measuring distances to the same clusters as
the original Lauer Postman dataset by tting Schechter functions to the luminosity distributions
in the clusters In a complementary e ort Willick is measuring accurate distances to clusters around
the sky at redshifts of km s using TF and D distances to spirals and ellipticals in each
n
cluster Finally Hudson Davies Lucey and Baggley are measuring D parameters of ellipticals in
n
each Lauer Postman cluster with redshift less than km s This will result in distance errors of
p er cluster
There are two ma jor new redshift survey pro jects in preparation A British collab oration led by
Ellis plans to measure redshifts for galaxies to b selected from the APM galaxy catalog
J
in a series of elds in the Southern Sky using the dF b er sp ectrograph on the Anglo Australian
telescop e Gray et al The survey geometry consists of two long strips in the Fall and Spring skies
plus randomly placed elds of diameter totaling ster The principal motivation is to measure
the large scale p ower sp ectrum of the galaxy distribution redshift space distortions to constrain and
evolutionary e ects
The Sloan Digital Sky Survey SDSS will use a dedicated m telescop e to survey ster around
the Northern Galactic Cap with CCD s in ve photometric colors A multi ob ject sp ectrograph with
b ers will b e used to carry out a ux limited redshift survey of galaxies to roughly R Over ve
years this survey will measure redshifts for galaxies with a median redshift of km s
The survey will see rst light in the second half of Details may b e found in Gunn Knapp
and Gunn Weinberg The SDSS is one of the few large scale surveys in in which the photometric
data from which the redshift galaxy sample will b e selected is obtained as part of the survey itself The
use of CCD data and careful calibration guarantees that it will b e the b est calibrated of these surveys It
do es not go as deep as the dF survey mentioned ab ove but covers much more area
Thus we lo ok forward to tremendous growth in the quantity and quality of b oth p eculiar velocity
and redshift data We set forth a series of questions in the b eginning of this review x which we hop ed
to address with the data available We have reviewed the analyses that have b een done with redshift and
p eculiar velocity surveys to answer these questions However as we have summarized in this concluding
chapter there are few of these questions for which we now have de nitive answers Indeed most of the
quantities we hop e to measure are known to within a factor of two at b est and more often only within an
order of magnitude We exp ect that the next decade will b e a p erio d of intense activity in this branch of
observational cosmology during which sup erior data and a more complete understanding of the theoretical
issues will allow us to make observational cosmology a precision science there is no doubt qualitatively
new science to b e discovered when we measure the p ower sp ectrum on large scales the value of the
bias parameter of di erent galaxy types and many other quantities to accuracy
Acknowledgement
We thank Alan Dressler and Sandra Faber for comments and suggestions on parts of the text Avishai
Dekel supplied two of the gures Karl Fisher Mike Hudson and David Weinberg read through the entire
pap er and made many valuable comments in addition we received useful comments and suggestions from
Yehuda Ho mann David Burstein Roman Juszkiewicz and an anonymous referee Maggie Best help ed
tremendously in the compilation of the references JAW thanks his collab orators on the Mark III pro ject
for p ermission to discuss asp ects of this work prior to publication MAS acknowledges the supp ort of the
WM Keck Foundation during the writing of this review
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