astro-ph/9506028 5 Jun 95 National Science Foundation Young Investigator ryden mps ohio state edu large scale structure of universe Subject in the real universe redshift di erent a The maximizes the area of the empty ellipses and ensures that ellipses do not overlap a free source in voids central redshift will provide an As of in used Voronoi measurement of depth q the redshift the 0 Because To test this metho d of estimating noted of accuracy to will have a size and voids of p eculiar headings estimate the space z cells values uncertainty for by the space universe the Department of by Alco ck of b ecome increasingly The shap e of velocities transformation tting Distortion the in the will b e greater q cosmology redshift 0 galaxies and volume and estimated deceleration parameter of In Paczy nski ellipses W th the a elongated volume in redshift the in q Measuring 0 space Astronomy The are voids which values Barbara theory within values from distance of estimate of ABSTRACT then Ave Columbus OH the space Peculiar velocities will along distorted Voids In redshift space the of universe axis q mapp ed 0 the of the q S Ryden I create a two dimensional toy universe 0 galaxies galaxies I q the measured observed ratio estimate the 0 toy voids q q is in 0 q Ohio State University as line of requires 0 to from universe distinguishing limited by into of the Redshift are If 1 redshift using voids distances than q redshift redshift redshift in 0 sight lo cated is a N b o dy area an nearby voids greater survey as is the create In algorithm a nonlinear distortions space and z and near Space function intrinsic addition b ecomes greater simulations and which than redshifts the an the adopting additional axis which walls Accurate go es scatter maps in of b etween can distant ApJ their ratio voids to b e of in press
Introduction
A quarter century ago cosmology was characterized as the search for two numbers
H and q Sandage Although the scop e of cosmology has broadened those same
0 0
two numbers are still b eing assiduously searched for I don t intend at the moment to
plunge into the H fray However in this pap er I will describ e how q may b e determined
0 0
in principle by measuring the size and shap e of voids in redshift space
The classical tests for q as describ ed by Sandage involve measuring the ux of
0
standard candles or the angular size of standard rulers The standard candles used to test
for q are usually galaxies Accurately determining q therefore requires knowing how the
0 0
intrinsic luminosity and color of galaxies evolve with time Unfortunately for cosmologists
galaxy evolution is a complex pro cess Uncertainties in the evolutionary mo dels lead to a
large uncertainty in the derived values of q Among the ob jects used as standard rulers to
0
determine q are brightest cluster galaxies Sandage extended radio sources Miley
0
Kapahi Daly and clusters of galaxies Hickson Adams Bruzual
Spinrad Once again the di culty of determining the intrinsic evolution results in
a large uncertainty in the value of q which is determined
0
It is desirable therefore to measure cosmological parameters by using metho d which
are uncontaminated by evolutionary e ects One such metho d was prop osed by Alco ck
Paczy nski who prop osed measuring the width and depth in redshift space of
ob jects which are intrinsically spherical or nearly so in real space The distortion of the
spheres in redshift space would provide an estimate of q and the cosmological constant
0
At the time of publication Alco ck and Paczy nski had no data to which they could apply
their test so their pap er remained an undercited curiosity Recently however interest
in their geometrical metho d has revived Phillips has prop osed using quasar pairs
assumed to b e randomly oriented in space as a substitute for the intrinsically spherical
ob jects of Alco ck and Paczy nski Placing interesting limits on q using quasar pairs will
0
require observations of approximately pairs at redshifts z In this pap er I will
apply the metho d of Alco ck Paczy nski to galaxies at relatively small redshift z
I will describ e how the size as well as the shap e of voids in redshift maps can b e used
to estimate q In section I describ e how measurements in redshift space can b e used
0
in principle to measure q In section I explain a practical metho d for measuring the
0
shap es of voids and apply it to simple toy universes Section contains a discussion of
the practicality of applying the void distortion test to real redshift surveys
Theoretical Exp ectations
Consider the idealized case of a spherical void which is surrounded by a thin wall of
galaxies At a time t the void wall has a radius r t and the galaxies within the wall are
v
moving outward with a velocity u t At some later time t an observer lo cated outside
v 0
the void collects light that was emitted from the void walls at the time t plus or minus the
light travel time across the void The redshift of a galaxy on the near side of the void is
z z and the redshift of a galaxy on the far side is z z where z is the redshift of the
void center and
u t
v
z z
c
The angular radius of the void as seen on the sky is
H r t z
0 v
c y z
where H is the Hubble constant at time t and y z is the dimensionless angular size
0 0
distance Peebles In a spatially at universe the angular size distance is
Z
z
dz
y z H
0
H z
0
If the void is plotted in redshift space the ratio of its depth to its width is
z u y z
v
e z
v
z H r z
0 v
In general the function e is not equal to one even for a void which is intrinsically spherical
v
and which is simply expanding along with the Hubble ow
If the galaxies in the void wall are expanding along with the Hubble ow then
u H r However voids tend to expand more rapidly than the Hubble ow Thus for a
v v
spherically expanding void
u t H t r t F t
v v
with the excess expansion factor F b eing greater than zero for voids expanding more
rapidly than the Hubble expansion In a spatially at matter dominated universe an
isolated spherical void of nite radius evolves into a self similar state with F if
the void is comp ensated that is if the net mass de cit is equal to zero and F if
the void is uncomp ensated Bertschinger If the initial underdensity of the void
has a p ower law form / r then F when Fillmore Goldreich
When the initial density pro le has then the void wall is a density wave and
matter moves outward with F Ryden If the universe is not at then F is
a function of the density parameter In an op en universe Reg os and Geller nd
0 6
that comp ensated voids have F
Thus an isolated void when plotted in redshift space will b e stretched in the radial
direction as a result of its p eculiar expansion velocity in addition to the distortion which
results from cosmological factors The observed distortion of a void will b e
H t y z
e t F t
v
H z
0
with the term in parentheses representing the cosmological distortion and the term in square
brackets representing the distortion due to the p eculiar velocity of the void s expansion
In order to determine cosmological parameters from the observed distortion of voids
in redshift space I must somehow distinguish b etween the cosmological distortion and the
p eculiar distortion Observed voids are not isolated structures The distribution of galaxies
in redshift space is bubbly galaxies are typically lo cated in relatively narrow walls b etween
voids which are nearly space lling The average void size when measured in comoving
units can only increase by the merger of two adjoining voids to form a single void or by
the expansion of one void at the exp ense of a neighboring void Consider two adjacent
voids with radii r and r If r r then the wall b etween the voids will have no p eculiar
1 2 1 2
velocity in the direction p erp endicular to its surface if r r the void wall will b e pushed
1 2
in the direction of the smaller void with an excess velocity Reg os Geller
0 6
F r r
2 1
compared to the Hubble expansion velocity H r In addition to the velocity p erp endicular
1
to the void wall the galaxies within the wall tend to ow outward within the plane of the
wall The pair of neighboring voids is thus converted into a single void as the intervening
wall breaks up The velocities tangential to the void wall have a maximum excess velocity
F when compared to the Hubble expansion velocity H r Dubinski et al
1
In a hierarchical scenario the universe at any given time is lled with voids of a
characteristic size R Dubinski et al Since all voids are roughly the same size the
v
p eculiar velocities in the direction p erp endicular to void walls as found from equation
should b e small even if is close to one The p eculiar velocities in the direction parallel
to the void walls may have values of F as large as p ercent However if the evolution of
the hierarchical universe is approximately self similar then the average value of F should
remain constant with redshift The average distortion of voids in redshift space due to
p eculiar velocities will b e constant with redshift the distortion of voids due to cosmological
e ects will increase with increasing redshift
Observationally there is no tendency for nearby voids to show distortions resulting
from p eculiar velocities The analysis by Slezak de Lapparent and Bijaoui of the
rst CfA redshift slice picks out voids whose centers are at a redshift z This
small sample of nearby voids shows no tendency to b e elongated along the line of sight in
a redshift plot indicating that the p eculiar velocities of the void walls cannot b e large
The largest void in the slice is lo cated at z with z Comparison of the
redshifts and Tully Fisher distances for the galaxies within the walls of this void shows
no detectable out ow the upp er limit on the p eculiar void expansion is F
Bothun et al However the sample of voids within the CfA slice is small and is at
low redshift For accurate measurement of the cosmological distortion a deep er survey is
required
Measuring Distortions
For simplicity I am going to measure the distortion of voids in a universe in which the
galaxies have no p eculiar velocities Thus I am assuming that the void walls are xed in
comoving co ordinates and am ignoring the presence of virialized clusters embedded within
the void walls Such clusters if lo cated on the near or far side of a void will alter the void
shap e by sticking ngers of go d into the void A future study Melott Ryden
will use n b o dy simulations to examine the e ects of virialization on the shap es of voids in
redshift space In this pap er for the sake of simplicity I will lo ok at a synthetic universe
which is expanding smo othly with the Hubble ow
In the absence of any p eculiar velocities the distortion of a spherical void in redshift
space its depth divided by its width will b e
H z y z
e z
v
H z
0
The cross sectional area of the void normalized to the area it would have at z is
H z z
a z
v
H y z
0
The volume of the void normalized to the volume it would have at z is
2
z H z
V z
v
2
H y z 0
All of these functions e a and V are in principle measurable The theoretical value
v v v
2
dep ends on the current density parameter G H where is the current
0 b b
0
mean density of the mass within the universe If the cosmological constant is not zero
2
the distortion will also dep end on the parameter H Figure shows the
0
theoretically exp ected values of e a and V for three sets of cosmological parameters
v v v
In the limit that q z
0
q
0
e z z a
v
q
0
a z z b
v
and
q
0
z c V z
v
In an exp onentially in ating de Sitter universe is negligibly small and q
0 0
Consequently an exp onentially expanding universe shows no cosmological distortions in
then signi cant cosmological distortions redshift space However if and
0
e should o ccur at redshifts near unity
v
To test algorithms for measuring the sizes and shap es of voids I created a synthetic
two dimensional universe in which the galaxies were lo cated near the cell walls of a Voronoi
foam Icke van de Weygaert This mo del creates a bubbly distribution for the
large scale structure in which voids are isolated convex structures with a characteristic
length scale Icke van de Weygaert The number density of nuclei for the
2
Voronoi cells in my toy universe is H c chosen so that the number density of
0 0
cells is comparable to the number density of voids in the rst CfA redshift slice Slezak et al
1 2
1
The characteristic length scale for the Voronoi cells is thus h Mp c
0
To make the cells more nearly uniform in size the nuclei of the cells are anticorrelated
nuclei were laid down at random on the plane with the constraint that no nucleus b e closer
than a distance from a previously p ositioned nucleus Once the nuclei of the Voronoi
cells were lo cated galaxies were p ositioned randomly with the constraint that they lie
within a distance of a wall dividing two cells The number of galaxies was chosen
to b e times the number of cells The galaxy distribution created by this pro cedure
is shown in the left panel of Figure The mean number density of galaxies is constant
out to z no observational selection e ects are used Each cell in the Voronoi foam
corresp onds to a void in the galaxy distribution each wall in the Voronoi foam corresp onds
to a wall of galaxies b etween voids
The left panel of Figure shows the undistorted distribution of galaxies as would b e
seen by an observer in a q universe The right panel shows the distorted pattern that 0
would b e seen by an observer at the origin in a q universe This value of q is much
0 0
larger than we exp ect in our own universe I adopt it for illustrative purp oses to make the
distortions clearly visible at a mo dest redshift of z The toy universe which I have
created is a particularly simple one the voids are clearly delineated structures which are
totally empty of galaxies The metho d which I use to measure the size and shap es of voids
must b e a robust one however capable of coping with the more complicated distribution
of voids in the real universe
One practical algorithm for detecting voids and measuring their size and shap e
is the VOIDSEARCH algorithm of Kau mann Fairall When applied to a
two dimensional galaxy distribution Kau mann Melott this algorithm divides
the plane into a number of square cells The largest square of empty cells is lo cated and
the sides of the square are p ermitted to deform outward into adjoining empty regions
Constraints are placed on the p ermitted deformation to ensure that the void remains
compact The VOIDSEARCH algorithm can nd the characteristic size of voids in
simulations where such a characteristic scale exists it also gives a go o d t to the shap e
of nearly circular voids Kau mann Melott Unfortunately VOIDSEARCH do es
not always accurately repro duce the shap e of elongated voids The initial square of empty
cells which is tted within an elongated void can expand outward along the axes of a
cartesian grid but not in the diagonal directions Whether the shap e of an elongated void
is accurately measured thus dep ends on whether or not it happ ens to b e aligned with the
sup erimp osed grid of cells
Since I want to measure the elongation of cells as accurately as p ossible I have adopted
an algorithm which is di erent from the VOIDSEARCH algorithm Instead of inscribing
squares within voids I inscrib e ellipses within voids with the constraint that one of the
principal axes of the ellipse must lie along the line of sight from the origin The axis ratio
of the ellipse then provides an estimate of the elongation e of the void
v
Let me now outline how my void nding algorithm works I start by sup erimp osing
a grid of p oints up on the galaxy distribution in redshift space To provide adequate
resolution of voids the distance b etween adjacent grid p oints must b e small compared
to the characteristic size of voids In the toy universe examined in this pap er the
characteristic void size is z for nearby voids The sup erimp osed grid which I used
has z For each grid p oint I nd the largest ellipse which is centered on the
p oint and which contains no galaxies one of the principal axes of the ellipse is constrained
to lie along the line of sight from the origin to the grid p oint I allow the axis ratios of
1 2 1 2
the ellipse to vary from to and determine the axis ratio which maximizes the
area of the vacant ellipse The constraint that the vacant ellipse b e aligned with the line of
sight is imp osed b ecause I am interested solely in the radial distortions in redshift space A
more general tting pro cedure if the radial distortions were not the main p oint in question
would allow the axes of the ellipse to b e aligned arbitrarily
For each grid p oint I thus nd the area and the axis ratio of the largest vacant ellipse
centered on that p oint I rank all of the vacant ellipses which I have found according to
their area The largest ellipse is designated the largest void in the sample I test the
second largest ellipse to see whether it overlaps with the largest ellipse If it overlaps it is
discarded if it do esn t overlap it is designated the second largest void in the sample This
pro cedure is rep eated in turn with all of the ellipses in the list in order of decreasing area
If the tested ellipse overlaps with any of the previously designated voids it is discarded If
it do esn t overlap it is added to the list of voids
The end result of this pro cedure is a list of non overlapping ellipses empty of
galaxies whose principal axes are aligned with the line of sight from the origin and whose
areas are maximized in a well de ned manner These ellipses are lab eled voids A
similar pro cedure for a three dimensional distribution of galaxies would de ne voids as
non overlapping ellipsoids whose axes of symmetry lie along the line of sight from the
origin Figure displays the largest elliptical voids found in the toy universe the left panel
assumes q and the right panel assumes q Only voids whose centers are at
0 0
z are shown ignoring the region where edge e ects are imp ortant due to the abrupt
cuto in galaxies at z
The ellipse tting algorithm is a simple minded one and takes no notice of the fact
that there is a characteristic void size in the toy universe Once it ts an elliptical void
to each of the Voronoi cells compare Figure and Figure it go es on and ts smaller
and smaller ellipses into the available vacant corners Figure shows only those elliptical
5
voids whose dimensionless area is greater than the smaller ellipses those which
are tucked into the corners of the Voronoi cells are not displayed A cursory glance at
Figure demonstrates that the area of voids and the elongation of voids b oth increase with
redshift in a q universe However if the value of q is in the more plausible range of
0 0
q the distortions will b e more subtle It will b e useful to know how deep a
0
redshift survey must go in order to distinguish b etween a spatially at universe in which
matter dominates q and a at universe dominated by a cosmological constant
0
q
0
For my two dimensional toy universe Figure plots the area of the largest
elliptical voids with centers at z The theoretical value of a z as given by
0 v
equation b is shown as the dashed line in each panel Most of the elliptical voids fall
b elow the dashed line This is to b e exp ected it merely says that the ellipses are smaller in
area than the p olygonal Voronoi cells in which they are inscrib ed The true elliptical voids
5
those with areas greater than do a fair job of tracking the increasing
0
area of the voids with redshift The solid lines in Figure shows the b est least squares t
of the form
Area z
0 1
using only those voids with areas larger than In the q universe the
0 0
4
least square estimators with standard errors are and
0
4
with a co e cient of correlation r In the q
1 0
4 4
universe and with r
0 1
From the slop e and intercept of the b est least squares t an estimate
1
q
a
0
can b e made for the deceleration parameter q If the errors in and are assumed to
0 0 1
b e Gaussian then the estimate with standard error is q in the q
a 0
universe and q in the q universe Thus in the q universe
a 0 0
the estimate of q is to o large at ab out the con dence level If the wall b etween two
0
adjoining voids is not detected for any reason for instance if the galaxies which formed
within the wall are all to o faint to b e included within a particular redshift survey then the
void tting algorithm will treat the two adjacent voids as a single large void and nd one
oversized ellipse within the area
Failure to detect an authentic void wall will result in the identi cation of an arti cial
void which is larger than the characteristic void size at that redshift assuming that there is
such a characteristic size Thus using the size of voids to estimate q is no longer a valid
0
pro cedure at redshifts where the number of detected galaxies is to o small to accurately
de ne the void walls To illustrate this p oint I created a two dimensional Voronoi universe
which has only galaxies p er Voronoi cell let s call it the sparsely sampled universe since
it contains so few galaxies around each void The sparsely sampled universe was created by
starting with the universe shown in Figure which has galaxies p er cell and randomly
removing seven eighths of the galaxies In a Voronoi universe with only galaxies for every
cell the mean number of galaxies p er cell wall is Poisson statistics will therefore result
in many void walls which contain no galaxies at all
Applying the void tting algorithm to the sparsely sampled universe yields the results
shown in Figure Note that the characteristic void size is no longer easily detected There
is no longer a sharp distinction b etween the true elliptical voids which nearly ll a Voronoi
cell and the smaller ellipses that are tucked into the cell corners The lesson to b e learned
from comparing Figures and is that the characteristic void area is easier to detect and
hence q is easier to estimate when the large scale structure is well de ned with several
0
galaxies within each void wall and no stray galaxies within voids
An estimate of the deceleration parameter q can also b e made by measuring the axis
0
ratio e of each tted ellipse instead of its area Figure is a plot of the natural logarithm
v
of e as a function of the redshift z only the elliptical voids with area greater than
v 0
are included The dashed line in each panel is the exp ected relation
q
0
ln e z
v
which is exp ected for circular voids when q z Since the voids are not intrinsically
0
spherical there is a signi cant scatter in e for a given redshift In the q universe
v 0
the standard deviation in ln e is approximately
v
Despite the scatter in the intrinsic axis ratios the trend for increasing elongation at
greater redshifts can b e picked out of the data For b oth the q universe and the
0
q universe I made a least squares t to the data using the functional form
0
ln e z
v 0 1
For the q universe the b est t with standard errors had and
0 0
with correlation co e cient r For the q universe
1 0
the b est t gave and with r The b est ts
0 1
to equation are shown as the solid lines in Figure nearly indistinguishable on this
scale from the theoretically exp ected dashed lines
An estimate of the cosmological deceleration parameter comparing equations and
is
q
e 1
In the q toy universe the estimated value of q derived from the axis ratio of the
0 0
tted elliptical voids is q For the q universe q
e 0 e
Thus measurements of the axis ratio are capable in a toy universe with no p eculiar
velocities of distinguishing b etween a at universe which is matter dominated and one
which is dominated by a cosmological constant However the scatter in the intrinsic axis
ratios of voids leads to a fairly large uncertainty in the estimate of q derived by this
0
metho d
How deep must a redshift survey go in order to yield an accurate estimate of q A
0
survey with a depth of z will contain only a small number of voids and will display
only a small cosmological distortion A deep er survey is necessary to distinguish b etween
a universe with q and one with q To give some indication of how deep the
0 0
survey must go I computed the estimated value of q in the toy universe as a function of
0
survey depth that is I estimated q using only those voids with redshifts less than some
0
limiting value z The two dimensional toy universe corresp onds to a thin slice in the
d
real universe
Figure is a plot of the estimated value of q as a function of the depth z in redshift
0 d
space On the left the estimates q drawn from the areas of the tted elliptical voids are
a
shown On the right the estimates q drawn from the axis ratios are shown In addition
e
to the b est estimate shown as the solid line each panel shows the standard errors around
the b est estimate the dotted lines ab ove and b elow the solid line The errors decrease
in size with increasing redshift At a depth of z which contains voids in the
d
q universe and voids in the q universe there are simply to o few voids to
0 0
accurately determine q
0
Discussion
In an idealized universe where voids are clearly delineated by galaxies along their walls
and where the galaxies have no p eculiar velocities it is p ossible to estimate the value of
q from the distortion of voids in redshift space Even in an idealized universe a redshift
0
in order to distinguish b etween a q and slice would have to have a depth z
0
a q universe Distinguishing b etween q and q would require an even
0 0 0
deep er survey
Alco ck and Paczy nski prop osed using the ratio of depth in redshift space z
to width in redshift space z as a means of measuring q One can as easily use areas in
0
redshift space depth times width or in a three dimensional survey volumes in redshift
space depth times the square of the width If voids were all intrinsically spherical but
with di erent intrinsic volumes then the axis ratio in redshift space e would give the b est
v
estimate of q Conversely if the spheres were intrinsically ellipsoidal but with identical
0
intrinsic volumes then the volume in redshift space V would give the b est estimate of q
v 0
Since real voids have a scatter b oth in size and in shap e all estimators for the value of q
0
e a and V will contain uncertainties
v v v
A great advantage of using the metho ds outlined in this pap er is that unlike the
standard candle metho ds they do not strongly dep end on the luminosity evolution of
galaxies As long as the galaxies are bright enough to b e included in the redshift survey
their precise luminosities and colors are irrelevant The survey must simply contain enough
galaxies to accurately delineate the void walls In addition for the purp oses of this pap er
the origin of the voids is largely irrelevant Whether they form by gravitational forces alone
or were triggered by explosions they have stamp ed a convenient pattern up on the universe
whose distortions can b e measured
The origin of the voids is only relevant in the extent to which p eculiar velocities are
imparted to the galaxies For instance numerical simulations of an universe show
0
that large voids have excess expansion velocity of F in a CDM universe but
only F in an HDM universe van de Weygaert van Kamp en If voids
expand more rapidly than the Hubble ow they will p ossess an additional elongation in
redshift space b eyond the cosmological elongation Studies of voids at z Bothun et
al Slezak et al indicate that the distortion of voids due to p eculiar velocities is
small An additional e ect of p eculiar velocities is to convert virialized clusters into ngers
of go d in redshift space These ngers by p oking into the near side and far side of voids
decrease the volume of the largest ellipsoid which can b e t into a void If the ratio of the
average nger length to the average void radius is roughly constant with time then the
e ect of virialized clusters can b e more easily comp ensated for In any case the p eculiar
velocities asso ciated with clusters will prove the greatest stumbling blo ck to measuring the
distortion of real voids in redshift space and thereby measuring the value of q
0
I thank Tod Lauer for introducing me to the work of Alco ck Paczy nski and for
encouraging me to undertake this study Cheongho Han provided invaluable calculations
and discussions
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A
This preprint was prepared with the AAS L T X macros v E
Fig The top panel shows the line of sight distortion e in redshift space the short
v
dashed line indicates a universe with and the solid line indicates a universe
0
with and and the long dashed line indicates a universe with and
0 0
The middle panel shows the cross sectional void area in redshift space and the
b ottom panel shows the void volume V in redshift space the meaning of line types is the
v
same as in the upp er panel
Fig The distribution of galaxies in a toy universe in which the voids are the cells of a
Voronoi foam The left panel shows the redshift map for an observer lo cated at the origin
in a universe with q The right panel shows a similar map for a universe with q
0 0
Fig The elliptical voids which are tted to the toy universes shown in Figure Voids
whose centers are at z and whose dimensionless area is a are shown In the
0
q universe left panel there are such voids in the q universe right panel
0 0
there are
Fig The dimensionless area of the largest elliptical voids whose centers are at a
redshift z The top panel assumes q the b ottom panel assumes q
0 0
In each panel the dashed lines shows the theoretically exp ected value for a z the solid
0 v
line is the b est least squares t for those voids with area greater than
0
Fig The same results as the previous Figure but for a universe with only galaxies
p er void instead of galaxies p er void Because of the much larger scatter in areas a
least squares t was not made
Fig The natural logarithm of the void elongation only voids with area greater than
are included The top panel assumes q and the b ottom panel q The
0 0 0
dashed line shows the theoretically exp ected value the solid line shows the b est least squares
t
Fig The estimated value of q as a function of survey depth in redshift space The
0
left hand panels show q the value estimated from the void areas the right hand panels
a
show q the value estimated from the axis ratios of the voids The heavy solid line is the
e
estimated value the dotted lines are the estimated value plus or minus the standard error
The horizontal line in each panel shows the true value of q 0