From Microwave Anisotropies to
Cosmology
Douglas Scott Joseph Silk Martin White
Center for Particle Astrophysics
and Departments of Astronomy and Physics
University of California Berkeley CA
Fluctuations in the temp erature of the cosmic microwave background have now b een
detected over a wide range of angular scales and a consistent picture seems to b e
emerging This article describ es some of the implications for cosmology Analysis of
all the published detections suggests the existence of a p eak on degree scales of height
to CL relative to the amplitude of the p ower sp ectrum at large angular
scales This result con rms an early prediction implies that the universe did in
fact recombine and limits theories of structure formation Illustrative examples are
provided of how the comparison of microwave background and large scale structure
data will b e a p otentially p owerful means of answering fundamental questions ab out
the universe
All indication s are that the large scale structure in the Uni of the CMB anisotropy sp ectrum on large angular scales
verse has developed by the pro cess of gravitational insta On degree scales we study CMB temp erature uctuations
bility from small primordial uctuations in energy density generated at the ep o ch of last scattering of the radiation
generated in the early universe Slightly overdense regions The large scale structure LSS of galaxies in the Universe
collapsed under their own gravity to b ecome more and more provides an indep endent measure of density uctuations of
overdense the density contrast increasing with time In the similar physical size but at the present ep o ch By probing
standard mo del the uctuations were laid down during an the uctuations at two di erent times the comparison of
in ationary phase when quantum uctuations inescapably CMB and LSS measurements constrains the growth of uc
present in any theory were b o osted in scale The outcome of tuations which in turn dep ends on the total matter content
the growth of these uctuations is the formation of galaxies of the Universe together with the value of the cosmological
and galaxy clusters by the present ep o ch constant
There are three distinct approaches to studying the pri In the standard Cold Dark Matter CDM and its vari
mordial uctuations from which large scale structure orig ants mo del one assumes that primordial gravitational p o
inated Numerical simulations of non linear growth and tential uctuations are generated in the in ationary era and
collapse have provided realistic descriptions of galaxies and are visible on the last scattering surface as photons propa
galaxy clusters and constrain the uctuation sp ectrum over gate out of the p otential wells that are destined to eventually
scales of a few Mp c in the recent past On large scales red form clusters and sup erclusters of galaxies The p otential
shift surveys measure the density uctuations directly out uctuations drive photon density and velocity uctuations
to ab out Mp c On still larger scales the COsmic Back which lead to anisotropies in the observed temp erature of
ground Explorer COBE satellite and a series of subsequent the microwave sky A schematic list of the sources of uctu
exp eriments have detected temp erature uctuations in the ations in rough order of imp ortance with decreasing angular
cosmic microwave background These temp erature uctu scale is given in Table
ations are the high redshift precursors emanating from a The largest scale anisotropy is the dip ole generated by
redshift of ab out of the uctuations that generated the motion of the Sun and Earth through the microwave
the structures we see to day background The other e ects arise as the photons interact
Thus cosmic microwave background CMB uctuations with p erturbations in the matter For example the Doppler
provide a key to understanding the origin of large scale shift arises when the photons gain energy by scattering o
structure in the Universe The ripples or anisotropies moving electrons For further discussion of all of these e ects
in the background radiation represent not only the initial see In this article we shall fo cus on the gravitational p o
seeds from which structure rst emerged but also contain tential and adiabatic contributions which are of primary im
0
co ded measures of various cosmological parameters Simul p ortance on large and intermediate angular scales in
taneously we are probing the thermal history of the early cosmological mo dels Here is the total density of
Universe One consequence of the analysis describ ed b elow
the universe in units of the critical density H G
c
is that very early reionization p opular in several cosmo
where H is the Hubble constant to day The Doppler shifts
logical mo dels wherein structure forms early cannot have
are of imp ortance only in scenarios where the universe was
o ccurred
reionized by energy injection at late times which we shall
We are now on the verge of a measurement of the total
argue are now disfavored
density of the Universe and there is a p ossibili ty of learn
A frenzy of exp erimental activity followed the COBE
ing ab out any ep o ch of in ation from the detailed shap e
DMR announcement in of the detection of uctuations
There have subsequently b een no fewer than claimed conventionally de nes C ja j where T T
m
P
detections in di erent regions of the sky by some inde
a Y the angled brackets representing an av
m m
m
p endent exp eriments ballo on and ground based Most
erage over the ensemble of p ossible uctuations and where
exp eriments have degree scale resolution although two de
Y are the spherical harmonics and and are angular
m
tections almost overlap with the COBE scale of degrees
co ordinates on the sky Assuming the uctuations have a
We present these detections in Fig and Table where
gaussian probabili ty distribution the mo dels are uniquely
Q is a measure of T to b e precisely de ned later
at
sp eci ed by giving their C s which in any mo del are sim
Now that the existence of CMB uctuations over a wide
ply a function of the cosmological parameters
range of angular scales has b een established empha
Given an input primordial uctuation sp ectrum one can
sis is shifting toward studies that try to x the parame
follow the distribution of photons baryons neutrinos and
ters of theoretical mo dels There have b een several pap ers
dark matter as the universe evolves The output is the
that combine the data from two exp eriments usually the
sp ectrum of anisotropies observable to day What is usu
COBE DMR results on the largest scales plus a sp eci c
ally plotted is C vs which is the p ower p er
smaller angular scal e exp eriment to place constraints on
logarithmic interval in or a D p ower sp ectrum on the
some cosmological parameters or mo dels We b elieve that it
sphere As an example the solid line in Fig shows the
is now feasible to combine the available data from al l of the
anisotropy sp ectrum for the standard CDM mo del
high quality generally multi frequency exp erimental mea
and h where the Hubble constant is
B
surements of microwave background anisotropies to go one
H h km s Mp c
step b eyond simply constraining the normalizatio n How
ever rather than trying to rule out sp eci c cosmological
mo dels we adopt a more phenomenologi cal approach
Firstly we set up a toy model for the radiation p ower
sp ectrum which is at on large angular scales and has a
p eak in p ower around multipole or scales of
see Fig Secondly we take the data from the di erent
exp eriments and convert each of them into one measure of
p ower so that they can all b e plotted together for compari
son and so that they can b e combined to place constraints
Finally we calculate the b est tting height for the p eak in
our phenomenologi cal p ower sp ectrum
Despite the apparent scatter of rep orted T T values
we nd that a distinct pattern is emerging In particular a
totally at scale invariant p ower sp ectrum is ruled out by
the data at the con dence level which instead prefer
some sort of p eak with height relative to the large
angle part of the radiation p ower sp ectrum This result is
remarkably close to what theorists had b een anticipating
and has interesting implicatio ns for cosmology
Of course we are assuming that the error estimates for
individu al exp eriments allow for p ossible foreground con
tamination this we b elieve to b e the case for most of the
exp eriments utilized in our analysis While there is still
Fig The solid line shows the sp ectrum of temp erature uc
some concern that not all of the measurements see only
tuations for a standard Cold Dark Matter mo del The quantity
CMB uctuations for several exp eriments the case against
C is p ower p er logarithmic interval in multipole number
foreground contamination is quite comp elling We will pro
Note that the curve is fairly at at small large angular
ceed under the assumption that the data can b e taken at
scales and has considerable structure at larger small angular
face value and investigate what they app ear to b e telling
scales Our simple phenomenological t is shown by the dashed
us We caution that the true cosmic uctuation signal may
line Note that the y axis here is a measure of p ower which is
b e overestimated if contaminated by foreground and un
prop ortional to Q see Eq
flat
derestimated if to o much foreground has b een subtracted
either of these e ects may b e present in some of the ex
p eriments Particularly worrying are exp eriments with very
In the sp ectrum the large angle small plateau is due
limited frequency coverage or data sets with obvious con
to gravitational p otential and large scale adiabatic den
tamination However with many data p oints contributing
sity p erturbations which are sensitive primarily to the dark
to our analysis a wrong exp eriment should not skew our
matter uctuations A generic prediction of in ationary
conclusions unduly Clearly as the data improve and issues
mo dels is a primordial sp ectrum of adiabatic density uc
n
related to foregrounds are further understo o d our conclu
tuations with p ower sp ectrum P k k and n
matter
sions can b e re ned Nevertheless we are con dent that our
For an n or scale invariant sp ectrum the temp erature
analysis should give a avor of the kind of information al
anisotropies are indep endent of angle at large angular scales
ready available from CMB studies and an indication of what
i e they have a at angular p ower sp ectrum at small On
will so on b e p ossible
the surface of last scattering sound waves in the coupled
baryon photon uid enhance over the large angle value
The Radiation Power Sp ectrum
the radiation p ower sp ectrum on scales around the horizon
size at that ep o ch In the standard mo del with reioniza It is standard practice in CMB studies to work in terms of
tion at redshift this e ect sets in on scales b elow the multipole moments of the temp erature anisotropy One
or On the smallest scales the uctuations are unstable to their own gravity while on smaller scales p er
suppressed by photon di usion b elow arcmin turbations oscillate as sound waves driven by gravity with
utes the angle subtended by the width of the last a restoring force from uid pressure The scale which is
scattering surface Between the horizon and damping scales just large enough to collapse is known as the Jeans scale
several p eaks of successively smaller amplitude are gener which initiall y grows with time The di erent p eaks and
ated By measuring T T on various angular scales it is troughs corresp ond to photon density and velocity p erturba
p ossible to di erentiate b etween the various contributions tions which have had an integral number of half oscill atio ns
and to b egin the task of confronting the mo del predictions b efore the Jeans scale reaches their size with complications
with the data in detail caused by the dark matter p otential wells and the thickness
It has b ecome apparent that variations caused of the last scattering surface Higher corresp onds to
B
by di erent cosmological parameters are not orthogonal fewer photons p er baryon and thus less pressure This leads
in the sense that somewhat similar sets of C s can b e found to a smaller Jeans length at any ep o ch allowing p erturba
for di erent parameters Attempts to extract cosmological tions to grow more b efore the Jeans scale reaches their size
parameters from CMB data are further complicated by the and they start to oscillate The oscillations will therefore
fact that theories only predict the exp ectation values of the b e of greater amplitude for higher leading to higher
B
a s for an ensemble of skies and not the a on our sky adiabatic p eaks when the photons are last scattered The
m m
hence the cosmic variance These di culties could b e exact heights of the various bumps and wiggles come from
largely overcome with a high sensitivity high angular reso a combination of p otentials for the rst p eak and density
lution map of a large fraction of the sky which would come and velocity e ects for all the p eaks and so dep end on the
from another satellite mission Until then it will prove al sp eci cs of the cosmological mo del For example the height
most imp ossible to separate inital conditions e g predic of the rst p eak is fairly insensitive to h when
B
tions of in ation from the evolution induced dep endence on while the relative heights of subsidiary p eaks have quite a
cosmological parameters using CMB data alone While we strong h dep endence However exp eriments are sensitive
do not work in a vacuum and these problems may b e par to a wide range of which will somewhat wash out these
tially overcome by using other observations at present the variations
required observations are highly uncertain as we describ e The p osition of the rst adiabatic p eak dep ends essen
b elow tially only on the geometry of the Universe Spatial
However the current CMB uctuation data are already curvature in an op en universe causes light rays to diverge
capable of tackling other imp ortant issues In particular as they propagate from the last scattering surface to the
we will address constraints that may b e placed on the ther observer Thus a xed length scale subtends a smaller an
mal history of the universe Could the universe have b een gle in an op en universe Sp eci call y the size of the hori
that the primordial de ionized su ciently early at z zon at last scattering subtends an angle corresp onding to
gree scale CMB uctuations would have b een erased It is
with a small amount of Hubble constant de
p ossible to formulate this question in a form that the data
p endence So for an mo del our assumption the
may already b e able to answer by resorting to some theo
p osition of the rst adiabatic p eak is well determined An
retical prejudice and falling back on some assumptions that
extremely signi cant step for the near future will b e when
have b een common in previous studies We will assume that
the data are up to the task of testing these assumptions and
the p ower sp ectrum of radiation uctuations is at least phe
obtaining a rm constraint on
nomenological ly similar to that obtained from mo dels like
The damping scale of the C s is also a fairly robust
the standard Cold Dark Matter CDM mo del although
physical quantity determined by the thickness of the last
we do not need all the dark matter to b e cold Sp eci call y
scattering surface The damping comes from photon di u
we assume that the p ower sp ectrum is at corresp onding to
sion out of overdensities and into underdensities on scales
n on the largest scales that and that the tensor
equal to the mean free path of the photons times the dura
mo de contribution to the uctuations is small compared
tion of recombination If the universe recombines at redshift
T S
to that of the scalar mo des i e C C These assump
z and this scale is ab out arcminutes Reion
tions have the advantage of minimali ty however they have
ization at late times generates a new last scattering surface
also received some supp ort from analyses of the COBE two
at lower redshift moving the damping scale to lower If the
year data which prefer a p ower sp ectrum that is either at
universe reionized at su ciently high redshift z and
or weakly rising towards high This is marginally in
remained ionized until the present the damping is su cient
consistent b oth with an appreciable tensor mo de and the
to remove the p eaks on degree scales
asso ciated negative tilt n or a low value of in a
We nd that the radiation p ower sp ectrum can b e rea
spatially at background though current limits are
sonably approximated by a constant p ower sp ectrum plus
not very strong Under these assumptions the most promi
a Lorentzian p eak lo cated at log of width log
nent feature of the theoretical p ower sp ectrum is the rise
Analyticall y we take
near and the series of p eaks at larger s see Fig
The bumps and wiggles at s of a few hundred in
A A
p eak p eak
C C
the radiation p ower sp ectrum known as adia
y y
batic p eaks are caused by sound waves propagating in the
baryon photon uid b efore the Universe recombined These
with
wiggles would also b e seen in the matter p ower sp ectrum
log
y if the Universe were dominated by baryons the absence of
oscillation s in a mo del like CDM is b ecause the dominant
where the amplitude of the Lorentzian at has b een di
comp onent of the matter is not coupled to the photons
vided out so that A is the height of the p eak ab ove the
As the universe evolves p erturbations on large scales are
p eak
low plateau This is plotted as the dashed line in Fig
The parameters for the center and width of the Lorentzian The Height of the Adiabatic Peak
were tted to accurate C s for a standard CDM mo del
Taking the data from Table and the toy model p ower sp ec
which is the solid line in the gure Our tting function has
trum of Eq we can employ a likeliho o d analysis to t the
the virtue of simplicity and although it will not b e a go o d t
two parameters using the data i e the overall normalization
in the complicated adiabatic p eaks region it is a extremely
and the height of the adiabatic p eak For each set of param
go o d on the rise to the main p eak where most of the ex
eters we use Eq and the window functions to predict
p erimental data p oints lie see for further discussion
Q for each exp eriment These are then compared with the
We also note that the detailed shape of the p ower sp ectrum
data in Table Contrary to common wisdom we nd that
rising into the p eak is an imp ortant theoretical prediction
the b est t p eak mo del is allowed at the CL showing
which can b e checked by future exp eriments
that there is no statistical reason to increase the error bars
on the p oints The fact that at least one mo del provides a
Di erent Exp erimental Results
reasonable t to the data in Table is a sign that the ex
p erimental situation on degree scales is more coherent now
In order to use the results from several exp eriments at once
than it was even last year and is indicative of the very rapid
we need to convert them into a consistent system The most
exp erimental progress in this eld
straightforward and robust datum from each exp eriment is
the total measured p ower A simple parameterization of this
p ower integrated across the window function bandpass of
the exp eriment is given by the amplitude of a at p ower
sp ectrum C constant Q T
at CMB
required to repro duce the measured p ower
1
X
Q
at
W Power
T
CMB
Here W is the window function of the exp eriment which
de nes the sensitivity of the exp eriment to any given scale
The constants in this expression have b een chosen so
that Q has the same meaning as the familiar ro ot mean
at
square p ower sp ectrum estimated quadrup ole Q for
rms PS
n see also
Our estimated values for di erent exp eriments are shown
in Fig Each p oint represents a t for the amplitude of
a at sp ectrum convolved with the sp eci c window function
of the exp eriment The vertical error bars are errors on
this amplitude while the horizontal lines show the widths
of the window functions at half p eak height and so should
Fig A Contours of for a t to the data of Table using
not b e regarded as error bars For the error bars on the
our Eq The cross marks the b est t Q K A
flat p eak
p ower we have taken them to b e symmetric in Q or
while the contours mark and CL regions for the t
at
the same quantity as a T T measurement
parameters B The marginal likelihoo d or likelihoo d integrated
We have chosen only to use quoted detections see Ta
over Q as a function of A for our tting form in Eq The
flat p eak
ble and to neglect exp eriments that have given upp er
likelihoo d has b een normalized to p eak at
limits Generally the error bars on these upp er limits are
A plot of the and allowed regions in the pa
large enough that they would not a ect our results e g
rameters Q and A is shown in Fig A The p ower
at p eak
On Fig we have shown three of the tightest constraints
sp ectrum normalization is well xed by large scale measure
at smaller angular scales The upp er limits are plotted as
ments To fo cus on the adiabatic p eaks we show Fig B
CL error bars for the White Dish OVRO and
the marginal likelihoo d or L Q A integrated over
at p eak
ATCA exp eriments see for more details These
Q assuming a uniform prior distribution for Q The
at at
upp er limits may p ose constraints for op en or at mo dels
b est t is A with mean and A
p eak p eak
with low but do not strongly constrain standard CDM
with con dence
mo dels
We note that by p erforming a t of the functional form of
Cosmology with CMB Fluctuations
Eq to the CMB data to get each Q we automatically
at
insure that b oth the cosmic and sample variance are
So what is all this telling us For almost thirty years there
fully included in the error analysis We also add the quoted
has b een the promise of learning the answers to some truly
calibration uncertainty in quadrature to all of the error bars
fundamental questions by studying the anisotropies on the
If the p ower sp ectrum was actual ly a pure n p ower law
microwave sky While exp eriments were giving only upp er
then the p oints would scatter ab out a horizontal line on this
limits the emphasis was on predicting the level of uctua
plot The fact that there would app ear to b e a trend for the
tions from various cosmological theories Now that uctua
smaller angular scale exp eriments to lie ab ove such a line
tions have b een detected on a range of scales theorists have
will b e examined next
b een exploring in detail the predictions for the sp ectrum as
a whole
Our analysis suggests that there is an increase in the the CMB what has b ecome obvious is that it will b e di
p ower measured on degree scales over that measured on cult to disentangle the variation due to simultaneous changes
larger scales If we interpret this increase as due to an adia of di erent cosmological parameters until a high resolution
batic p eak by far the most natural and comp elling explana high sensitivity map of the sky can b e obtained This e ect
tion then there were oscillatin g baryon uctuations in the is sometimes known as cosmic confusion In par
early Universe and early reionization could not have played ticular it seems that proving in ation by simultaneously
T S
a signi cant role in erasing primordial anisotropies in the
extracting C C and n from the C s will b e very di
CMB The Thomson scattering optical depth since the Uni
cult particularly b ecause of cosmic variance at small
verse b ecame ionized must have b een small This is the rst
In the near future it seems more likely that the
de nitive evidence that the Universe did in fact recombine
questions b eing explored will b e concerned with more clas
c f ref
sical cosmology and h On the largest scales there
B
Seeded mo dels generally are exp ected to have early non
are already constraints on mo dels which give the sp ectrum
T S
linearity and early reionization In calculations to date early
a negative slop e i e a combination of C C n
reionization has b een assumed which suppresses the p eaks
and cosmological constant CDM or low in ationary
at However reionization is not inevitable
mo dels The situation here will improve particularly af
in such mo dels and without it we would exp ect adiabatic
ter the full four years of COBE data have b een analysed
p eaks although no explicit calculation has yet b een done
On smaller scales there is a wealth of cosmological informa
Certainly the microwave background would b e exp ected to
tion to b e gained from the shap e of the bumps and wiggles
b e non Gaussian on degree angular scales roughly the hori
However it will b e di cult to accurately separate the var
zon size at last scattering in such texture or monop ole mo d
ious e ects without a ma jor new presumably space based
els though not in string mo dels The similarity in uctu
exp erimental e ort
ations in the three indep endent dust free regions scanned
In order to break the degeneracy b etween the variations
by the MAX exp eriment may already b e evidence against
caused by di erent cosmological parameters we can use con
strongly non Gaussian mo dels
straints from other areas of astrophysics A particularly
The existence of the adiabatic p eaks on degree
fruitful area of study is the combination of CMB measure
scales would con rm a fundamental theoretical prediction
ments with measurements of Large Scale Structure LSS
that complements the large angular scale COBE
These two elds provide indep endent prob es of the p ower
DMR detection of presumably gravitational p otential uc
sp ectrum in complementary regions with some overlap at
tuations predicted by in ationary cosmology The lat
di erent cosmological ep o chs In Fig we show an example
ter are acausal re ections of the initial conditions at the end
of the sort of information now available in these two elds
of in ation the former provide a glimpse of the physics of
c f the matter p ower sp ectrum inferred from b oth
the dark ages of the Universe long b efore the most distant
CMB and LSS observations the latter from It is im
galaxies or quasars had formed via the p ossibili ty of probing
p ortant to realize that the conversion from the CMB to the
back to the surface of last scattering
matter p ower sp ectrum is very theory dep endent we have
The fact that our tting formula which has a plateau for
assumed a sp eci c mo del in order to convert the numbers
low b efore rising into the p eak manages to pass through
in Table into the b oxes shown on the plots We show in
much of the data could also b e taken as evidence against the
the top panel of Fig b oth a standard CDM mo del and a
Baryonic Dark Matter BDM also called primordial iso cur
mixed dark matter MDM mo del i e CDM with a com
vature baryons PBI mo del which rises rapidly into the
p onent of massive neutrinos The middle panel shows an
adiabatic p eaks and is not at at large scales This
in ationary op en universe CDM mo del chosen to satisfy Big
is just one example of the kind of information that accurate
Bang Nucleosynthesis typical recent determinations of H
mapping of the adiabatic p eaks will give Another intriguing
and the LSS shape constraint h The apparent di
p ossibili ty may lie just ahead So far the data cannot accu
vergence at small k is an outcome of a sp eci c in ationary
rately pin p oint the p osition of the main p eak but it is clear
mo del The b ottom panel shows a cosmological
that in the next few years this situation should improve If
constant dominated mo del with the same parameters as the
it turns out that rather than then this
p eak
op en mo del but to make the universe at
will b e very strong evidence that the Universe is op en since
We see in the top panel that CDM normalized by the
the op en geometry makes the same physical scale subtend a
CMB predicts to o much p ower on small scales as is well
smaller angular scale Alternatively should the data show a
known An MDM mo del predicts less small scale
rapid rise near this would b e hard to reconcile with
p ower but p erhaps to o little to form galaxies early enough
op en mo dels
and is also not a p erfect t to the shap e Tilting the mo del
Already it seems that op en mo dels with
from scale invariance n adding a tensor comp onent
have a hard time tting b oth the large and degree scale
lowering the Hubble constant or introducing decaying
data if there has b een no reionization Early reionization
neutrinos are p ossibili ties for xing b oth of these prob
is at least as likely as in at mo dels however and due to
lems However the t for the CMB alone is fairly go o d as
the longer path length to a given redshift the redshift for
indicated by the scatter in the b oxes b oth ab ove and b elow
which the Universe b ecomes optically thick is slightly
the curves Note also that the fraction of hot dark matter
reduced in op en mo dels The shap e of the p ower sp ec
has negligibl e e ect for degree scale anisotropies
trum should b e distingui shab le from the case even
In the middle panel we see that the in ationary op en
with partial reionization It seems that op en mo dels with
mo del manages to agree with the large angle CMB and LSS
up to ab out will b e fully testable within only a few
data quite well but these mo dels predict a falling C
years
on COBE scales which may b e de nitively tested in future
Also lo oking at the b oxes we see that it predicts uctuations
The CMB and LSS Together
on degree scales which are somewhat small though this is
While some qualitative features can already b e derived from
more a re ection of the parameters chosen a low due sis we know that cannot b e arbitrarily large in fact a
B B
to the high h than a generic prediction of op en mo dels value of larger than seems unlikely Since the adia
B
In the b ottom panel the normalizatio n inferred from batic p eak height increases with a lower limit on the tilt
B
CMB and large scale structure observations has galaxies of such a mo del is a conservative lower limit for any mo del
anti biased i e they need to b e less clustered than the dark with a more reasonable value of We nd that the CMB
B
matter This p oint has b een realized b efore how data alone even with such an unusually large app ear to
B
ever the increased CMB normalization has strengthened it require n at the CL This limit is comp etitive with
Notice that for the normalization of the mo dels combinations of large angle CMB and LSS data We
predict to o much p ower on LSS scales despite the reduced should p oint out that for mo dels based on in ation the sp ec
growth rate due to lowering However one can lower the tral index is not exp ected to b e exactly Including some
small scale p ower in al l of these mo dels by tilting the p ower amount of tilt will b e an imp ortant complication for future
sp ectrum away from n and or attributing some of the work and will change the heights of the inferred p eaks and
large angle CMB anisotropy to gravitational waves the amplitude of uctuations on smaller scales
Additionall y there is still some freedom in the normaliza We can crudely convert our limit on n into a limit on
tion of the matter p ower sp ectrum through the biassing of the optical depth of the Universe now once more as
galaxies relative to dark matter suming that n Recall that degree scale anisotropies
Now turn to Fig Here we have chosen di erent pa
are reduced by e in a universe with signi cant reioniza
n
rameters for each of the classes of mo dels from Fig to
tion and by in a tilted mo del Thus our limit
further demonstrate the p ower of using a range of CMB
ab ove which compares the COBE scales to the adi
and LSS constraints together In each case we have chosen
abatic p eak scales translates into If we
somewhat non standard mo dels which are p erhaps more
assume and full ionization fraction x from
e
B
realistic e g by having some tilt n as well as provid
z until to day this corresp onds to a redshift of reionization
ion
ing generally b etter ts to the data The top panel shows a
i e the Universe must have b een neutral b etween z
ion
CDM mo del with n h and h as may
B
redshifts and
b e suggested by the most recent nucleosynthesis considera
Knowledge of other cosmological parameters would to
tions The increase of approximately counteracts
B
some extent a ect our ts For example if or
T S
the e ect of lowering n on the height of the rst CMB p eak
C C etc then the height of the adiabatic p eaks will
This mo del ts the data fairly well with a reasonable level
change relative to the COBE normalization For example
of p ower on cluster scales and a passable t to the shap e
T S
a combination of n and C C would lower the
around the turnover in the matter p ower sp ectrum It would
predicted p eak height relative to large scales and may b e
b e p ossible to further decrease the p ower on small scales by
preferred if the t to from Big Bang Nucleosynthesis is
B
T
allowing C but it b ecomes harder to accomo date an
more than a few p ercent Because of these p ossibil iti es it
T
appreciable p eak near as C is increased By al
is hard to constrain at present However very low
B B
lowing similar parameter freedom one can also nd MDM
as inferred from recent primordial deuterium measurements
mo dels which t the data as well or b etter than the one in
would b e in con ict with an appreciable p eak height
Fig but which are tilted and or have gravity waves
The indications are that none of these complicating ef
The op en mo del in Fig has h again with
fects are so imp ortant as to invalidate our general result
B
h and n Raising and provides the adiabatic p eak is p oking up ab ove the noise More am
B
a b etter t to the degree scale CMB data while lowering bitious analyses could clearly b e done but we feel the data
h keeps the age shap e and small scale p ower roughly con do not yet warrant multi parameter ts In particular we
stant The dominated mo del with h and have avoided the temptation to derive any sp eci c cosmo
logical quantity instead of our phenomenologica l amplitude
now h has b een tilted to n with a con
B
T S
A However if we were to adopt a particular mo del like
p eak
tribution of gravity waves C C This is enough to
standard CDM with h say then there obviously is a
stop the galaxies from b eing anti biased as in Fig These
b est tting baryon fraction Recalling that our tting form
two gures show that as the data improve it will b e p ossible
somewhat over estimates p ower for mo dels near the p eak
to considerably narrow the range of viable mo dels
our result A would corresp ond to for
p eak B
Other Parameters
this sp eci c mo del At the moment this result is almost
meaningless as a measurement of since it dep ends sen
B
With the ongoing explosion in the amount of useful data it
sitively on what is assumed for the other parameters But as
will so on b e p ossible to simultaneously set some constraints
the CMB data and other astrophysical constraints improve
on a number of di erent cosmological parameters For the
this technique is likely to complement the conventional BBN
moment we have concentrated on a simpler question ob
metho d providing a measurement of that has a very
B
taining two constraints from the data rather than just one
di erent h dep endence Perhaps one day a combination of
Already the information we have gleaned has proved of in
BBN and re ned CMB measurements will constrain h
terest for cosmology In this section we lo ok at some param
eters that we could choose to constrain instead of A
p eak
Conclusions
In the context of an in ationary dark matter based the
ory we can ask for information on the primordial sp ectral In summary we b elieve that the new intermediate angular
slop e n Determining this parameter accurately is well b e scale CMB anisotropy data provide supp ort for the existence
yond the scop e of this work requiring a multi parameter t of an adiabatic p eak This already has dramatic implicatio ns
However the p eak in the data at in combination for the early Universe recombination o ccurred on schedule
with the COBE measurement allows us to set a lower limit at a redshift of and the Universe remained neutral
on n Such a lower limit is most conservative if we ignore the until a redshift less than The primordial p ower sp ec
p ossibili ty of gravity waves From Big Bang Nucleosynthe trum is not to o far from scale invariant and the increase
in p ower on degree scales is consistent in p osition and am Bunn E White M Srednicki M Scott D Astrophys J
plitude with the adiabatic p eak predicted by dark matter
Tucker G S Gri n G S Nguyen H T Peterson J B
dominated mo dels at the critical density The former was al
Astrophys J L WD
ready hinted at by the COBE DMR exp eriment in and
Readhead A C S Lawrence C R Myers S T Sargent
subsequent exp eriments have provided some evidence for the W L W Hardeb eck H E Mo et A T Astrophys J
OVRO
latter This development represents an imp ortant advance in
Subrahmanyan R Ekers R D Sinclair M Silk J Mon
our mo delling of the deviations from uniformity in the early
Not R astr Soc ATCA
Universe that complements and p otentially surpasses our
White M Scott D in CMB Anisotropies Two Years After
COBE ed L Krauss World Scienti c Sihgap ore emerging understanding of the very early Universe origin of
the primordial density uctuations The new exp eriments
Scott D Srednicki M White M Astrophys J L
prob e the physics of last scattering and are signi cantly nar
rowing the class of viable mo dels A new technique for mea
Since A is de ned in terms of the p ower sp ectrum Q
p eak flat
squared the corresp onding enhancement at on Fig suring is b eing developed utilizin g the p ower spectral
B
is A times the amplitude at low
signature of the temp erature uctuations that already rep
p eak
Bartlett J G Stebbins A Astrophys J
resents a triumph for dark matter dominated cosmological
Coulson D Ferreira P Graham P Turok N Nature
mo dels with late reionization The joint use of information
from z i e CMB and from z i e LSS will
Silk J Nature
Sachs R K Wolfe A M Astrophys J
b e a p owerful to ol for cosmology The determination of the
Sugiyama N Silk J Phys Rev Lett
sp eci c cosmological mo del that describ es our Universe is an
Hu W Sugiyama N Bunn E Astrophys J in press
exciting challenge that still lies ahead of us but the eld is
already providing answers to some fundamental questions
Tegmark M Silk J Astrophys J
Knox L Turner M Phys Rev Lett
Acknowledgements
White M Krauss L Silk J Astrophys J
We would like to thank John Peacock for providing us with
Peacock J A Do dds S J Mon Not R astr Soc
the LSS data and Naoshi Sugiyama Wayne Hu and Ted
Lyth D H Stewart E D Phys Lett B
Bunn for many useful conversations This work was sup
Ratra B Peebles P J E Astrophys J L
p orted in part by grants from the NSF
In an op en geometry there is ambiguity in de ning wavenum
b er Here k is the eigenvalue of the Laplacian The upturn is
due to the choice of k as the x ordinate the total p ower is of
References and Notes
course nite
Acronyms in square brackets indicate exp erimental pap ers from
Bartlett J G Blanchard A Silk J Turner M S Science
which data have b een taken
White M Gelmini G Silk J Phys Rev D
White M Scott D Silk J Ann Rev Astron Astro
phys
Efstathiou G Bond J R White S D M Mon Not R
Smo ot G et al Astrophys J L
astr Soc p
Silk J Astrophys J
We note in passing that there is a correction to the
Doroshkevich A G Zel dovich Ya B Sunyaev R A Sov
scaling used in due to the integrated Sachs Wolfe e ect Astron
since the p otentials are not constant at late times in a J rgensen H E Kotok E Nasel ski P D Novikov I
dominated mo del D Astr Astrophys
Kofman L Starobinski A A Sov Astron Lett
Hu W Sugiyama N Phys Rev D
Bond J R Crittenden R Davis R L Efstathiou G
Krauss L M Kernan P Phys Lett B in press Steinhardt P J Phys Rev Lett
White M Bunn E F Astrophys J in press Seljak U Astrophys J L
Kofman L Gnedin N Y Bahcall N A Astrophys J As well as the usual scalar density p erturbations in ation
can generate tensor ripples in the background space time If
Lyth D H Liddle A R Astrophys Lett Commun in present these gravity waves cause additional uctuations in
press the microwave background In in ationary mo dels one gener
T S
Hu W Scott D Silk J Phys Rev D ally exp ects C C if n is close to
Ganga K Page L Cheng E Meyer S Astrophys J
G orski K M et al Astrophys J L COBE
L FIRS
Bunn E F Sugiyama N Astrophys J in press
Hanco ck S et al Nature Ten
Bunn E F Scott D White M Astrophys J L
Gundersen J O et al Astrophys J L SP
Netter eld C B Jarosik N C Page L A Wilkinson D
Peebles P J E Yu J T Astrophys J
Wollack E Astrophys J in press SK
Wilson M L Silk J Astrophys J
Dragovan M et al Astrophys J L Pyth
Bond J R Efstathiou G Mon Not R astr Soc
de Bernardis P et al Astrophys J L ARGO
Piccirillo L Calisse P Astrophys J
Sugiyama N Gouda N Prog Theor Phys
IAB
Kamionkowski M Sp ergel D N Sugiyama N Astrophys
Alsop D C et al Astrophys J MAX
J L
Gundersen J O et al Astrophys J L MAX
Scott D White M in CMB Anisotropies Two Years After
COBE ed L Krauss World Scienti c Sihgap ore
Devlin M et al Astrophys J L MAX
Meinhold P R et al Astrophys J L MAX
White M Srednicki M Astrophys J
Clapp A C et al Astrophys J L MAX
Bond J R in Proceedings of the IUCAA Dedication Cere
Cheng E S et al Astrophys J L MSAM
monies ed T Padmanabhan New York John Wiley Sons
in press
Figure Captions
Fig The amplitude of T uctuations in each exp eriment as a function of scale multipole Q is
at
the b est tting amplitude of a at p ower sp ectrum quoted at the quadrup ole see Eq The vertical error bars are
while the horizontal lines represent the half p ower ranges of the window functions The data p oints are listed
in Table where the references to the exp eriments are also given We have plotted the data for of the MAX
results as one p oint with the discrepant Peg p oint plotted separately The MSAM exp eriment has two indep endent
mo des There are also three smaller scale upp er limits plotted at the CL The general rise in the area around
' can b e interpreted as evidence for an adiabatic p eak in the radiation p ower sp ectrum
Fig The matter p ower sp ectrum P k on a range of scales as inferred from Large Scale Structure and Cosmic
Microwave Background data The b oxes are values of P k inferred from CMB measurements on large and
intermediate scales assuming top CDM h or MDM midd le an op en universe
B
in ationary mo del h h bottom CDM h h The
B B
horizontal width of each b ox represents the range of scales to which the exp eriment is most sensitive The LSS data
are a compilation taken from
Fig The matter p ower sp ectrum P k for more realistic sp eci c choices of the mo del parameters The data
p oints are as in Fig The mo dels are top CDM h h tilted to n midd le
B
op en h h n bottom CDM h h n with a
B B
gravity wave comp onent In general these mo dels provide a b etter t than the more standard mo dels of Fig
illustrating the p otential p ower in using CMB and LSS data together Note that the overall amplitude of the LSS
data is uncertain to p erhaps
Tables
Table The primary sources of temp erature uctuations roughly in order of increasing imp ortance with decreasing
angular scale
T T V c dip ole anisotropy where V is our motion relative
to the radiation
T T gravitational p otential or Sachs Wolfe uctuations
T T density p erturbations if the p erturbations are adiabatic
T T S entropy p erturbations if the p erturbations are iso curvature
T T v c Doppler shifts when the photons were last scattered
Adiabatic initial conditions naturally generated during in ation have so that the
rad mat
entropy is constant Iso curvature initial conditions have so that there is no p erturbation to the total
rad mat
energy density
Table Summary of angular scales and measured uctuations for current exp eriments The parameters and
are the p eak and the lower and upp er half p eak p oints of the window function resp ectively Q is the b est t
at
amplitude for a at sp ectrum through the window function quoted at the quadrup ole scale The error bars are
Exp eriment Q K
at
COBE
FIRS
Ten
SP
SK
Pyth
ARGO
IAB
MAX UMi
MAX UMi
MAX UMi
MAX Peg
MAX Her
MAX Dra
MSAM
MSAM
Fig 4 Fig 5