VOLUME 68, N U M B E R 6 P H Y S I C A L R E V I E W L E T T E R S 10 FEBRUARY 1992

Cold Dark Matter Confronts the Cosmic Microwave Background: Large-Angular-Scale Anisotropies in ilo + X = 1 Models

Krzysztof M. Gorski(a)

Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, New Jersey 08544 and Institute for Advanced Study, School of Natural Science, Princeton, New Jersey 08540

and Laboratory for Astronomy and Solar Physics, NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771

Joseph Silk Departments of Astronomy and Physics and Center for Particle Astrophysics, University of California, Berkeley, California 94720

Nicola Vittorio Dipartimento di Fisica, Universita dell'Aquila, Aquila, Italy

(Received 23 September 1991)

We use a new technique to compute the correlation function for large-angle cosmic microwave back­ground anisotropies resulting from both the space and time variations in the gravitational potential in flat, vacuum-dominated, cold dark matter cosmological models. Such models with fio—0.2 fit the excess power, relative to the standard cold dark matter model, observed in the large-scale galaxy distribution and allow a high value for the Hubble constant. The low order multipoles and quadrupole anisotropy that are potentially observable by COBE and other ongoing experiments should definitively test these models.

PACS numbers: 98.70.Vc, 98.80.Cq

The large-angular-scale anisotropy of the cosmic mi­crowave background (CMB) provides a fundamental probe of the primordial fluctuation spectrum back to the epoch of inflation, since no causal process could have erased such imprints on angular scale ^ 1 0 ° . The flat, vacuum-dominated, cold dark matter (CDM) model with fto — 0.2, h « 0 . 7 5 , where fto is the density parameter and h is the Hubble constant Ho in units of 100 k m s ~ , M p c ~ I , provides an attractive option for fitting large-scale structure observations and time-scale con­straints [1,2],

Kofman and Starobinskii [3] have computed the large-angular-scale anisotropies by direct integration of the Sachs-Wolfe formula for gravitational-potential-driven fluctuations in flat vacuum-dominated CDM mod­els, and predict a quadrupole anisotropy of about 10~5 . In view of the critical significance of CMB anisotropy ob­servations for testing this class of cosmological models, we have developed a new formalism for studying large-angular-scale anisotropy in the CMB. We utilize an ex­plicit expression for the temperature correlation function that involves a double integral along the lines of sight to the last scattering surface over the second derivative of the growth rate factor for adiabatic density fluctuations in CDM combined with simple integral transforms of the power spectrum of inhomogeneity. The technique is not limited to low multipoles, but allows computation of the angular correlation on all physically relevant angular scales. We present results here for both the temperature correlation function and the low-order multipoles in flat models with a range of values of fto and h.

We briefly review the formal developments described in detail elsewhere [4]. Our goal is to evaluate the correla­tion function of cosmic microwave background radiation

anisotropy induced by the curvature perturbations super­posed on the globally flat, vacuum-dominated, cold dark matter universe. The present fractional density of cold dark matter is taken to be fto, and we assume that the cosmological constant A = 3 / / o ( l — fto)/*?2 contributes the remaining fraction X=Ac2/3Ho of the total mean density required to make the universe flat. We define the dimensionless conformal time as dr]=Hodt/ait), where ait) is the scale factor, normalized today so aito) = \. After these specifications, the Friedman equation takes the form

n°_i_i u dab]) — - + 1 — fto, where a=* — dt]

(1)

The density perturbation field is assumed to be of sufficiently small amplitude that linear theory applies and the usual ansatz is possible: lpiri,y)—pir])]/piT]) =s6(r;)5(y). Here 8iy) is the spatial homogeneous and isotropic random field of zero mean statistically described by a certain power spectrum P(k), and eir\) is the gauge invariant growing mode of the adiabatic density perturba­tions [5]. The temporal evolution is governed by the equation iea)'**(3/2) floe, whose solution is given by

tit]) ftc

ail]) + i - f t o

1/2 a(n) du

° w3(n 0 /K 3+l-no) 3 / 2

(2)

To calculate the CMB temperature anisotropy correla­tion function CiO) we integrate the photon energy varia­tions along two paths through the random field of curva­ture perturbations specified by unit vectors yi,y2, such that yry2— cos0. The null geodesies are parametrized

© 1992 The American Physical Society 733

V O L U M E 68, N U M B E R 6 P H Y S I C A L REVIEW L E T T E R S 10 FEBRUARY 1992

with affine parameters 0<5"i, $2< 7]Q — rie, so that pho­tons1 coordinates along trajectories are 1̂,2 — ^+^1,2, and (conformal distances) y \%2

=c(rio — T]e — 5-1,2), with r\e

conformal time at last scattering and rj0 conformal time today. Hereafter we use velocity units for distances so the Hubble parameter HQ does not enter the calculations [except in the formula for P(k)—the cold dark matter perturbation power spectrum, which does not explicitly appear in this paper].

In addition to the usual ST/T — vltic1) Sachs-Wolfe

[6] contribution in a flat, matter-dominated cosmological model, there is also a term in ST/T that arises from the integral over the time-changing part of the gravitational potential fluctuations 8<p that enters at redshift 1+z < [(1 — noVHol , /3, when the cosmological constant starts to dominate the dynamics of cosmic expansion, and the linear inhomogeneities stop growing. Gorski and Silk [4] have found that in terms of the growing mode (2), the correlation function of 8T/T induced by both the Sachs-Wolfe and 4V" terms can be compactly written as

c(o)-(f(yi)Y(y2))

- ; / . no ne I y \yi *>

ds]ds2€(rie + S\)€(rie + S2)\y/\\(y)cosO+ii/-(y)—— sin20

where ( • • • ) denotes the usual ensemble averaging, and the conformal distance y between two points on the paths specified by s\ and s2 is y2=y2+y2 — 2y\y2cosO. More­over,

Vn(y)- - 1 2nlc 2^2 J o Jo dkP(k)

dji(ky) dky

^ - ^ " T - T T JTdkP(k)j2(ky), (4)

where j \ and j 2 are the spherical Bessel functions. It may be noted that with our choice of units [P(k)] = (km/s)3, and both y functions are dimensionless.

The usual orthogonal expansions of the correlation function (with monopole and dipole terms excluded) and the expectation values of the multipole coefficients in the spherical harmonic expansion # r / r ( y ) ==2/,ma/mF/m(y) are given by

C(0)=-j- X (2/+lW/>,(cos0), 47T / - 2

a? =<kw |2> =2n f l d(cosO)C(9)Pi(cos6)

(5)

Although ai coefficients suffice to quantify the CMB an-isotropy, other definitions are popular in the literature. In the particular case of a quadrupole component of the anisotropy, the / = 2 component of C(0) is frequently used under various guises, e.g., as Q or as (AT/T)2 coef­ficients. They are related to a2 as follows: (?2 = (A77 T)l =C/- 2 (0) =5a2

2/4;r, or Q =0.631a2. We have applied the formalism given above to calcu­

late the predicted amplitudes of the 8T/T correlation function and the multipole coefficients in the flat models with a fraction fto of critical density in cold dark matter, and the remaining 1 — Ho contribution from the cosmo­logical constant. We assumed that the last scattering of CMB photons occurred around redshift z —1100. The power spectrum of adiabatic cold dark matter density perturbations was taken from Davis et ai [7]. This spec-

(3)

trum corresponds to a zero-baryon-content model; expli­cit inclusion of baryons only tends to increase the numeri­cal predictions. The spectrum was normalized in a stan­dard fashion to yield mass fluctuations of order unity in a sphere of radius 800 km s ~':

2

1 = g(qo) '

2K2 f°°dkk2P(k)

Jo

3y,(800A:)

800 A:

sin/2(0/2) 0 0.2 0.4 0.6 0.8

2x10"

—— as above, SW only

— • n 0 =1.0 f h = 0 . 5 _1._._L._J....J...

50

L 1...LU...UUL ..

100 180

0 ( d e g )

FIG. 1. rms ST/T predictions for single-subtraction experi­ment anisotropy measurements as functions of beam-throw 0. Heavy lines indicate the intrinsic anisotropy signals as coded in the graph. The long-dashed heavy line shows the contribution from the Sachs-Wolfe [6] effect only in the Ho =0 .2 model. Two triplets of thin lines correspond to smoothed signals. From top to bottom, they correspond to antennas with FWHM —1.5°, 3.8°, and 6°, respectively.

734

VOLUME 68, NUMBER 6 PHYSICAL REVIEW LETTERS 10 FEBRUARY 1992

Since the matter content of the models is only a fraction of the critical density, there is no need to invoke any bias in the matter distribution, especially in models with fto~0.2. The results of our calculations are presented in Figs. 1-3.

We show in Fig. 1 the predicted 5T/T(0) = {2[C(0) — C(0)]} , / 2 for single-subtraction experiments as a func­tion of beam-throw (switching angle) 0. The difference between the solid, heavy line (no =0.2, h =0.75) and the long-dashed heavy line, which corresponds to inclusion of only the Sachs-Wolfe [6] effect at the last scattering sur­face in the same model, demonstrates the contribution to 8T/T correlations from both (symbolically) / f(<ip<p) and jiw) terms. As already stated, the X models with fto^-0.2 were not biased, but one should bear in mind that the fto = 1, h =0.5, cold-dark-matter-dominated models require biasing as the means of making sensible predictions at small scales. This would shift the appropri­ate set of curves (four short-dashed ones) by about a fac­tor of 2 downwards, making the difference in signal pre­dictions between the models with or without X more pro­nounced. The triplets of thin lines show the smoothed signal predictions for experiments operating with the an­tennas of FWHM=1.5°, 3.8°, 6°, corresponding to the UCSB, MIT, and the COBE differential microwave ra­diometer (DMR) instruments, respectively. A final re­mark on Fig. 1 involves our omitting the Doppler contri­butions to anisotropy at angular scales ;S2° in our calcu­lations. Including these contributions would shift the in­trinsic signal curves by roughly a factor of 2 upwards, and would also somewhat increase the FWHM=1.5° smoothed signal predictions. However, there would be little effect seen for larger FWHM curves.

In Fig. 2 the rms multipole coefficients corresponding to correlations from Fig. 1 are displayed. The multipole of order / has (2/+ 1 ^independent components, each of

o

-5.5

-6.5

T T

a # O " o * o

A ! s y *

Q0=0.1F h = 0 . 5

00=0.2, h = 0 . 5

O 0 =0.1 , h = 1 . 0

n 0 = 0 . 2 , h = 0 . 7 5

n 0 = 1 . 0 , h = 0 . 5

' !!•- . ' * • Hi

• ^ ! 0 o o « o « > A A

i i B i »**

* 2 2 ? i i

10 15

I

FIG. 2. Multipole coefficients derived from correlations displayed in Fig. 1.

FIG. 3. Quadrupole coefficient ai as a function of fto in fto+k—1 models parametrized by Hubble constant h. Power spectrum of adiabatic perturbations corresponds to negligible-baryon-content models (cold dark matter only). Inclusion of baryons increases the amplitude of anisotropy.

them Gaussian distributed, as are the density fluctuations in these models. The squared magnitude of each mul­tipole has a x1 distribution with 2/+ 1 degrees of freedom over the ensemble of possible microwave skies [8,9]. Fig­ure 3 displays the entire range of quadrupole coefficient a2 predictions parametrized by fto and h.

The present limits on quadrupole anisotropy are too weak by a factor of a few to directly test the vacuum-dominated cold dark matter models discussed here. How­ever, the new set of maps anticipated in early 1992 from the COBE DMR, involving more than a year-long data­base should improve the Q limits by about a factor of 4. This would mean that any signal of amplitude in excess of Q~~ 1 x 10 ~5 would be detectable. Other experiments, both on balloons and at the South Pole, are underway that are expected to have a ST/T sensitivity of a few parts in 106 at angular scales of up to 10°, and are well-matched to our low-order multipole predictions at / ;S 20. The theoretical appeal of CDM can most simply be reconciled with large-scale-structure observations in a low-ft0 universe, provided that a cosmological constant is introduced to maintain a flat geometry. Hence we may reasonably hope that these ongoing experiments should be sufficient to definitively test not merely the class of cold dark matter, fto~0.2, models that incorporate a cosmo­logical constant, but indeed the entire CDM paradigm.

This work started when K.G. was at Theoretical Astro­physics, Los Alamos National Laboratory. K.G. ac­knowledges the hospitality and help of J. P. Ostriker, J. N. Bahcall, and J. Mather. N.V. acknowledges financial support from MURST. The research of J.S. is supported in part by a grant from the NSF, and was previously sup­ported in part by a grant from NASA. The Laboratory for Astronomy and Solar Physics is funded by USRA.

735

V O L U M E 68, N U M B E R 6 P H Y S I C A L R E V I E W L E T T E R S 10 FEBRUARY 1992

(a)On leave from Copernicus Astronomical Center, Poland. Ill G. Efstathiou, W. J. Sutherland, and S. J. Maddox, Na­

ture (London) 348, 705 (1990). [2] M. S. Turner, Phys. Scr. T36, 167 (1991). [3] L. A. Kofman and A. A. Starobinskii, Pis'ma Astron. Zh.

11, 10 (1985) [Sov. Astron. Lett. 11, 5 (1985)]. [4] K. M. Gorski and J. Silk (to be published). [5] J. Bardeen, Phys. Rev. D 22, 1882 (1980).

[6] R. K. Sachs and A. M. Wolfe, Astrophys. J. 147, 73 (1967).

[7] M. Davis, G. Efstathiou, C. S. Frenk, and S. D. M. White, Astrophys. J. 292, 371 (1985).

[8] L. F. Abbott and M. B. Wise, Astrophys. J. Lett. 282, L47 (1984).

[9] R. Scaramella and N. Vittorio, Astrophys. J. 353, 372 (1990).

736