. . HYSICAL .REVIEW

VOLUME 73 25 JULY 1994 NUMBER 4

Imprint of 0 on the Cosmic Microwave Background

Naoshi Sugiyama' * and Joseph Silk'~Departments of Astronomy and Physics and Center for Particle Astrophysics, University of California, Berkeley, Californt'a 94720

2Department of Physics, Faculty of Science, University of Tokyo, Tokyo I13, Japan(Received 16 March 1994; revised manuscript received 12 May 1994)

We investigate the dependence of large-angular-scale cosmic microwave background anisotropieson various initial conditions, including both adiabatic and isocurvature perturbations and the initialpower-law index n, in a variety of low-0 cosmological models. A-dominated flat models, inflationary

open adiabatic models, with n,«& 1, and open isocurvature models, with n, « = 2, are significantlyconstrained.

PACS numbers: 98.70.Vc, 98.80.Es

The discovery of cosmic microwave background(CMB) anisotropies by the Cosmic Background Explorer(COBE) satellite [1] has provided important informationabout initial conditions in terms of the spectrum ofprimordial density perturbations. In particular, theseobservations provide a powerful probe of the slope ofthe initial power spectrum. The first-year DifferentialMicrowave Radiometers (DMR) data suggests an initialpower-law index n = 1.1 ~ 0.5, [also in agreementwith one recent analysis of second-year combined data(n = 1.10 ~ 0.32) [2]], which is consistent with thescale-invariant prediction n = 1 of inflationary cosmol-ogy. An independent analysis of two-year data [3],however, may result in a steepening to n = 1.59 o55, therecent detection of CMB anisotropies by the Tenerifeexperiment [4] supports the higher power-law slope. Avalue of n significantly greater than unity is difficult toreconcile with inflation. In this paper, we consider a setof alternative models that allow a wide range of n.

Previous discussions of large-angular-scale CMBanisotropies almost invariably adopt the flat 0 = 1

universe with initially adiabatic perturbations. In an openuniverse, the shape of the CMB anisotropies inducedby adiabatic perturbations is affected by backgroundcurvature on large scales [5—7]. It is generically difficultto disentagle curvature effects from initial conditions.However, we have found that primordial isocurvatureperturbations in an open, baryon-dominated (BDM)

universe [8] result in a spectral shape that is almost scaleinvariant on large scales. The interpretation of smallerangular scale observations are confused by the subhorizonmicrophysics of Doppler peaks and rescattering; hence wefocus here on large angular scales (~5 degrees). In this

Letter, we present the results of numerical calculationsof large-scale CMB anisotropies for both adiabatic andisocurvature initial conditions, in order to see whetherthe effects of geometry can be distinguished from thedependence on fluctuation mode, 0, initial power-lawindex n, and thermal history of the universe.

Here for generality, we consider open universe models.The temperature anisotropy is usually expanded into mul-

tipole components 8 in Fourier space. Detailed treatmentsare given elsewhere [5,9]. In order to directly comparethe spectrum with specific observations, we introduce thecoefficients Ce [10] of the CMB anisotropies in f, space asCe/4' —= ((ae(go) ( ), where ae(71o) is the coefficient of the8th multipole component of the temperature anisotropy atpresent and go is the present conformal time. The ex-pected temperature anisotropy for each experiment is ex-pressed by using Cz and the specific window function W~.

(BT/T)„t= Ze z We(2& + 1)Ce/4'

There are several physical contributions to CMBanisotropies for generic density perturbations [11], i.e.,the Sachs-Wolfe (SW) effect, the integrated Sachs-Wolfe(ISW) effect, primordial entropy perturbations, inducedDoppler effect, and primordial adiabatic perturbations.

0031-9007/94/73 (4)/509 (5)$06.001994 The American Physical Society

509

VOLUME 73, NUMBER 4 PH YS ICAL REVIEW LETTERS 25 JU1 v 1994

On very large scales, the Doppler and adiabatic terms canbe neglected. Hence the 4th moment of the temperatureanisotropy is written in gauge-invariant variables [12] as

l a'acI. '9o) = (+Ivies) — &('gpss) &.(&'g)

k a

g0

——X,'(5 rl ),

where O', V, S, and gi ss are the gravitational potential,the velocity perturbation, the entropy perturbation, andconformal time on the last scattering surface, Ag —= go-rlLss and ' =—8/drl. X,(hri) is a radial eigenfunctionin the open universe with v = k/Q —K, where K is thecurvature constant and k = v k2 + K, with k being wavenumber. If flat universe models are assumed, X~ reducesto the usual spherical Bessel function jr(k(571)). Eachcomponent in this equation corresponds to SW, ISW, andentropy terms, respectively. In the case of Oat 0 = 1

models, the SW term is 4/3 and the ISW term vanishesfor growing modes in the matter-dominated universe. Foradiabatic initial conditions, the entropy term is negligible.On the other hand, for isocurvature initial conditions,the combination of entropy term and SW terms becomes2V [13]. For 0 = 1 models, one can easily calculatelarge-scale CMB fluctuations from the matter powerspectrum through the Poisson equation. The relationbetween C~ and the initial power-law index n of the matterspectrum is C~ ~ I ((4 + (n —1)/2)/I (8 —(n —5)/2);(n ( 3) [6]. For isocurvature models, n is replaced byn + 4 in the above equation if n is defined as the initialpower-law index of the entropy perturbations.

For flat cosmological constant (A)-dominated models,however, the contribution of the ISW term is dominant onlarge scales for small (), and this modifies the temperaturespectrum [14]. For open models, in addition to the ISWeffect, the cutoff in the k integration and the modificationof the Poisson equation near the curvature scale add furthercomplications. For open adiabatic models, semianalyticcalculations on large scales have been recently performedby Kamionkowski and Spergel [7], who, however, onlyincluded the SW and ISW terms and therefore were unableto accurately probe the Tenerife scale. For BDM models,analytic formulas for C~ were given by Gorski and Silk[15], who included the effects of geometry but neglectedthe contribution of the ISW effect. Moreover the favoredrange for n in BDM models (—1.5 to 0) [16] is largerthan the value expected by inflation (—3), and is out ofthe region of validity of the analytic formulation by theGamma function. Gorski, Silk, and Vittorio [17]examinedlarge angular scale CMB anisotropies in A-dominatedadiabatic models, again including only the SW and ISWterms. There has been no previous work on large angularscale anisotropies in A-dominated BDM models.

The interplay of the ISW term, which enhances large-scale power, and geometry, which suppresses it viawhat in effect is gravitational focusing, i» sufficientlyintricate that we have been motivated to perform afull range of numerical calculations with both adiabaticand isocurvature initial conditions. We investigate thebehavior of C& with varying $l, and include the effectsof varying the primordial index, the thermal history,and the vacuum contribution in spatially flat models.We use the gauge invariant method [12, 18] to treatperturbation variables. The perturbation equations aresolved numerically until the present epoch [19].

First, we present numerical results for CDM withadiabatic perturbations. Here we take (k~ = 0.03 andthe dimensionless Hubble constant normalized by 100km/s Mpc to h = 0.5. In Fig. 1, the C~'s for differentA are plotted (a) for open and (b) for flat A models.Each C~ distribution is normalized to the quadrupoleanisotropy. The initial conditions for these models aretaken as imp/pi ~ k, namely, a Harrison-Zeldovich (HZ)spectrum in an Einstein-de Sitter universe. For openuniverse models, this is not the only reasonable choice,the Cq's having a weak dependence on different scalingof the initial spectrum [7]. However, we also consider aninitial condition produced by a low () inflationary model

(a)Op

CD

(b)A, +CD

(c

00

I I I I 1 I I I 1 2 I I I

(d

I I I I I I II I I I I I I I I

.'0= 1.00=0.1 ———X=0.9 "- ""

10 100 10L

100

FIG. 1. CMB power spectrum 4(f + 1)C~ normalized at Z =2 as a function of 8 for adiabatic CDM. Open and (l + A = 1

models (A —= A/3H&) with Il = 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, and1.0 are shown in (a) and (b), respectively. Bold solid lines are0, = 0.1 and A = 1. Dashed line» in (a) are low 0, modelswith initial power spectrum of Ref. [20]. Il = O. l open andA models and 0 = 1 model with optical depth» r = 1 andr = 0 (no reionizationl are shown in (c). Same model» withn = 1 and n = 1.5 are shown in (d). The hatched region is theexpected power-law slope 1.4 ~ 0.6 from the combined first-and second-year COBE data [2, 3]

510

VOLUME 73, NUMBER 4 PH YS ICAL REVIEW LETTERS 25 JUL+ 1994

[20]. This initial spectrum is proportional to k ' on scaleslarger than the curvature scale, while it coincides with the

HZ spectrum on small scales. The hatched region is the

expected power-law slope 1.4 ~ 0.6 from the combinedfirst- and second-year COBE data [2, 3]. The width of thewindow functions up to the half-power points are I ~ 11and 13 ~ t ~ 30 for the COBE and Tenerife experiments,respectively. Note that all A models have effective n

smaller than unity on the COBE scales due to the ISWterms. Because of the curvature effect, n, ff & 1 for the

open HZ models at low 4. On the other hand, n, ff & 1 forthe model of Ref. [20] because of the strong enhancementof large scale fluctuations. Even for this model, thecurvature effect is apparent for 0 = 0.1.

The effect of reionization of the universe is shownin (c). Our adopted reionization model is the late-time

fully ionized universe with electron-scattering opticaldepth unity. We compare reionized models with standardrecombination models. Fluctuations on COBE scales arenot affected by reionization. However, this is not the caseon the Tenerife scale. There is a notable difference in

particular for A models. Because of the curvature effecton the geodesics, the horizon scale of the last scatteringsurface of an open model corresponds to a much smaller

angle, i.e., larger 8 than that of a A model. For larger0, the last scattering surface is earlier. This suggeststhat the A model with small 0, has the greatest effect viareionization on large-scale CMB anisotropies for modelswith similar optical depths. In Fig. 1(d), the dependenceon the initial power-law index n is shown for the 0, = 0.1

open and A models, and the 0 = 1.0 model. In order tomore directly compare with observations, we show the

(a

10 -OI'- BD

I I I I I I I li

(b)

- BDM— n= —1

expected temperature fluctuations for quadrupole, FIRS(far infrared survey) [11],and Tenerife scales normalizedto the COBE 10 scale in Table I. The values of n, ff onthe COBE scale for each model are also shown. Using therelation between Cq and n for 0 = 1 models, we definethis n,«by taking the ratio of Cz at 4 = 10 and 4 = 2.

The Cq's for BDM with primordial isocurvature pertur-bations are shown for different II in Fig. 2. Figures 2(a)and 2(b) are fully ionized models with n = —1 for openand A models, respectively. Here the initial power-lawindex n is defined as ~S~ ~ k". In Fig. 2(c), models with

no reionization for 0, = 0.1, n = —1 open and A models,and the 0 = 1.0 model, are plotted together with the cor-responding fully ionized models. The dependence on n

is shown in (d) for fully ionized models. For both openand A models, the dependence of Cq on n is weak. Thethermal history of the universe is also weakly affected in

open models as shown in (c). However the shape of the

Cq distribution for A and A = 1.0 models is sensitive tothe thermal history. In BDM models, the last scatteringsurface is much closer to the present for larger A. To-gether with the geodesic effect, the open low-0 modelis least affected by reionization on large scales. The ex-pected fluctuations of the quadrupole, FIRS, and Tenerifeanisotropies and the effective n are shown in Table II.

In this Letter, we have investigated the shape of CMB

TABLE I. Expected AT(pK) for quadrup. ole (Q), FIRS and

Tenerife of CDM adiabatic models. Observed values and n, ff

are also shown. *'s are models with reionization (r = 1) andf's are models with initial power spectrum of Ref. [20].

(cI I I I I I II

0.1

0.1*

0.1'

0.20.2'0.3031.01.0*

0.1

0.1*

0.20.3

Obs.

12.1

1 1.615.413.816.714.616.814.714.8

16.717.016.015.4

6~3

FIRS

Open

39.439.835.537.735.337.736.038.837.3

38.1

35.238.338.6

45 + 13

Tenerife

25.326.418.122. 1

17.722.319.224.321.8

22.517.723.023.6

42 ~ 9

neff

1.51.60.841.10.631.00.681.11.0

0.720.640.840.94

1.4 ~ 0.6

10 -B

10 100 10 100

FIG. 2. Same as Fig. 1 for isocurvature BDM. Norecombination open and 0 + A = 1 models withII = 0.1, 0.2, 0.3, 0.4, 0.6, 0.8 and 1.0 are shown in (a)and (b), respectively. In (a), the 0 = 1 adiabatic CDM modelwith n = 1 is also plotted for comparison. Bold solid lines are0 = 0.1 and 0, = l. 0 = 0.1 open and A models and 0, = 1

model with no recombination and standard recombination (noreionization) are shown in (c). No recombination II = 0.1

open and A models and 0 = 1 model with initial power lawindex n = —1.5, —1 and —0.5 are shown in (d).

511

VOLUME 73, NUMBER 4 PHYSICAL REVIEW LETTERS 25 JUL+ 1994

TABLE 11. Same as Table I of fully ionized BDM isocurvature models. *'s are models with standard recombination (noreionization).

—1.0 —0.5FIRS—1.0 —0.5

Tenerife—1.0 —0.5 —1.5

&et'—1.0 —0.5

0. 1

0. 1*

0.20.30.41.01.0*

10.210.1

10.310.510.712.37 ' 8

9.69.79.69.59.5

10.26.2

9.1

9.1

9.1

9.08.98.45.3

44.749.843.442.842.441.255.9

47.852.847.247. 1

47.248. 1

64.6

Open

50.755.650.550.951.757.672. 1

33.638.931.730.529.726.846. 1

37.441.736.435.935.735.254.3

40.544.340.040. 1

40.544.660.5

1.91.91.81.81.71.42.3

2.01.92.02.02.01.82.7

2. 1

2.02. 1

2. 1

2. 1

2.22.9

0. 1

0. 1*

0.20.3

Obs.

12.89.8

12.612.4

1 1.98.8

1 1.51 1. 1

6 3

1 1.68.4

1 1.210.7

37.548.538.439.1

40.255.1

41.743 ~ 1

45 ~ 13

42.761.644.947. 1

21.938.023. 1

24.0

26. 1

45. 1

27.929.6

42~9

29.550.831.834.1

1.31.91.41.4

1.52. 1

1.61.7

1.4 ~ 0.6

1.62.21.71.8

anisotropies on very large scales for models with adiabaticand isocurvature initial conditions. On such large scales,cosmic variance cannot be negligible. Before summariz-

ing our results, we discuss the effects of cosmic variance.We can assume that the 8th moment of the expected tem-perature fluctuations obeys g2 statistics with 2f + 1

degrees of freedom. The 90% confidence region of theexpected temperature fluctuations on scale 4 is expressedin terms of the rms temperature fluctuations as u &AT(Z)//JT, (f) & P. Here [AT, (Z)/T] —= (2Z + 1) X

Ct/47r. n and p are functions of Z. If 8 is largeenough, n = [1 —1.96/2/(2Z + I)]'t2 and p = [1 +1.96/2/(2Z + 1) ]'t2 because of the Gaussian nature of theg~ distribution with many degrees of freedom. For small

(8, n, P) = (2, 0.48, 1.49), (5, 0.65, 1.34), (10,0.74, 1.25),(20, 0.82, 1.18). In Tables I and II, we show the rmstemperature fluctuations. Even though the quadrupoleanisotropy contains a large cosmic variance, models withrms quadrupole anisotropy larger than 12 pK wouldbe ruled out by the COBE detection, if this result isconfirmed. Our normalization of fluctuations to theCOBE 10 degree scale involves 30% cosmic variance.On the Tenerife scale, the effect of cosmic variance is lessthan 20%.

For adiabatic fluctuations, the difference between openmodels and A models is significant. Flat A models, whichare favorable for large-scale structure formation, appear tobe unable to account for the new COBE results and theTenerife detection. We caution that a precise comparisonmust be made using our non-power-law power spectrumbefore any definitive conclusions can be drawn. Theinflationary open models [20] provide rather similar C&'s

to ~Y-dominated models. These models will be discussedin detail in a forthcoming paper. Open HZ modelsare well fitted to COBE results, but have difficulty in

producing large enough fluctuations on the Tenerife scale(Table!). The standard 0 = 1 model has a slightly tiltedslope n, ff

——1.1 even on the COBE scale.Isocurvature perturbations reveal intrinsically dif-

ferent shapes for CMB anisotropies on large scales.The value of n, « ——2 required by viable BDM models(—1.5 ~ n ~ —0.5) is only marginally consistent with theobservational data. Open and A-dominated models havedifferent n, ff. For A models, the dependence on thermalhistory is important. Other parameter dependences, andin particular the initial n dependences, are weak on theCOBE scales for both open and A-dominated models.Geometry dominates over ISW in the absence of intrinsiccurvature fluctuations. BDM models may be distinguish-able from models with adiabatic perturbations via thelarge-scale CMB anisotropies. The third- and fourth-yearCOBE results and new large-scale experiments shouldprovide a definitive probe of curvature in a BDM universethat may be written on the sky.

The authors would like to thank W. Hu,M. Kamionkowski, D. H. Lyth, B. Ratra, and M. Whitefor valuable discussions. This research has been sup-ported at Berkeley in part by grants from NASA andNSF. N. S. acknowledges financial support from a JSPSPostdoctoral Fellowship for Research Abroad.

*Electronic address: sugiyama@astron. berkeley. eduElectronic address: silkpac2. berkeley. edu

[1] G. F. Srnoot et al. , Astrophys. J. Lett. 396, Ll (1992).[2] K. M. Gorski et al. , Astrophys. J. Lett. (to be published).[3] C. L. Bennett et al. , Astrophys. J. (to be published).

[4] S. Hancock et a/. , Nature (London) 367, 333 (1994).[5] M. L. Wilson, Astrophys. J. 273, 2 (1983).

VOLUME 73, NUMBER 4 PHYSICAL REVIEW LETTERS 25 JvLv 1994

[6] L. Abbot and R. K. Scheafer, Astrophys. J. 308, 546(1986).

[7] M. Kamionkowski and D. N. Spergel, Astrophys. J. (to bepublished).

[8] P. J.E. Peebles, Astrophys. J. Lett. 315, L73 (1987).[9] N. Gouda, N. Sugiyama, and M. Sasaki, Prog. Theor.

Phys. 85, 1023 (1991).[10] J.R. Bond, G. Efstathiou, P. M. Lubin, and P. R. Mein-

hold, Phys. Rev. Lett. 66, 2179 (1989).[11] For references, see e.g. , M. White, D. Scott, and

J. Silk, Ann. Rev. Astro. Astrophys. (to be published).Here we distinguish between the Sachs-Wolfe effectand the integrated Sachs-Wolfe effect as a matter ofconvenience although both effects were discussed in R. K.Sachs and A. M. Wolfe, Astrophys. J. 147, 73 (1967).

[12] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 7$,1 (1984).

[13] H. Kodama and M. Sasaki, Internat. J. Mod. Phys. A 1,

265 (1986).[14] L. A. Kofman and A. A. Starobinskii, Sov. Astron. Lett.

11, 271 (1985).[15) K. M. Gorski and J. Silk, Astrophys. J. Lett. 346, Ll

(1989).[16] Formation of large scale structure was studied by

T. Suginohara and Y. Suto, Astrophys. J. 387, 431(1992); R. Cen, J. Ostriker, and P. J.E. Peebles,Astrophys. J. 415, 423 (1993). Constraints fromCMB anisotropies were investigated by W. Hu andN. Sugiyama, Astrophys. J. (to be published).

[17] K. M. Gorski, J. Silk, and N. Vittorio, Phys. Rev. Lett. 68,733 (1992).

[18] J.M. Bardeen, Phys. Rev. D 22, 1882 (1980).[19] For detailed treatment, see, N. Sugiyama and N. Gouda,

Prog. Theor. Phys. 88, 803 (1992).[20] D. H. Lyth and E.D. Stewart, Phys. Lett. B 252, 336

(1990); B. Ratra and P. J.E. Peebles, (unpublished).

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