# Daniel J. Eisenstein, Wayne Hu, and Max Tegmark

## Transcript Of Daniel J. Eisenstein, Wayne Hu, and Max Tegmark

The Astrophysical Journal, 504:L57–L60, 1998 September 10

᭧ 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.

COSMIC COMPLEMENTARITY: H0 AND Qm FROM COMBINING COSMIC MICROWAVE BACKGROUND EXPERIMENTS AND REDSHIFT SURVEYS

Daniel J. Eisenstein, Wayne Hu, and Max Tegmark1

Institute for Advanced Study, Princeton, NJ 08540 Received 1998 May 19; accepted 1998 July 9; published 1998 July 31

ABSTRACT

We show that the detection of acoustic oscillations in both upcoming cosmic microwave background (CMB) satellite experiments and large-redshift surveys can yield 5% determinations of H0 and Qm, an order-of-magnitude improvement over CMB data alone. CMB anisotropies provide the sound horizon at recombination as a standard ruler. For reasonable baryon fractions, this scale is imprinted on the galaxy power spectrum as a series of spectral features. Measuring these features in redshift space determines the Hubble constant, which in turn yields Qm once combined with CMB data. Since the oscillations in both power spectra are frozen-in at recombination, this test is insensitive to low-redshift cosmology.

Subject headings: cosmic microwave background — cosmology: theory — dark matter — large-scale structure of universe

1. INTRODUCTION

In the usual cosmological paradigm, the cosmic microwave background (CMB) contains a vast amount of information about cosmological parameters (Hu, Sugiyama, & Silk 1997). With upcoming experiments, most notably the two satellite missions Microwave Anisotropy Probe (MAP)2 and Planck,3 detailed measurements of the angular power spectra of its anisotropy and polarization may accurately determine many cosmological parameters (Jungman et al. 1996; Bond, Efstathiou, & Tegmark 1997; Zaldarriaga, Spergel, & Seljak 1997). However, certain changes in the cosmological parameters can conspire to leave the CMB power spectra unchanged, resulting in degenerate directions in the parameter space (Bond et al. 1994, 1997; Zaldarriaga et al. 1997; Huey et al. 1998). For example, since the Hubble constant H0 and the matter density Qm can be varied while keeping the angular diameter distance and the matter-radiation ratio ﬁxed, their values remain uncertain but highly correlated. Such degeneracies must be broken with cosmological information from other sources.

Upcoming redshift surveys for the study of large-scale structure hold the potential for resolving this issue. In particular, the 2dF survey4 and the Sloan Digital Sky Survey (SDSS)5 should measure the galaxy power spectrum on large enough scales to allow detailed comparisons with the mass power spectra predicted by cosmological theories. In this Letter and a companion paper (Eisenstein, Hu, & Tegmark 1998b, hereafter EHT), we explore the potential of combining redshift surveys and CMB anisotropy data for the purpose of parameter estimation. Here we focus on the dramatic improvement possible in the measurement of H0 and Qm. Neither data set yields tight limits by itself, yet together they could yield errors better than 5% on H0 and 10% on Qm.

The key to this improvement is the presence of features in

1 Hubble Fellow. 2 See http://map.gsfc.nasa.gov, maintained at NASA by D. N. Spergel and G. Hinshaw. 3 See http://astro.estec.esa.nl/SA-general/Projects/Planck, maintained at ESA by J. Tauber. 4 See http://meteor.anu.edu.au/˜colless/2dF, maintained at MSSSO by M. Colless. 5 See http://www.astro.princeton.edu/BBOOK/, maintained at Princeton University by R. Lupton.

the matter power spectrum on scales exceeding 60 hϪ1 Mpc. With a nonnegligible baryon fraction, the acoustic oscillations that exist before recombination are imprinted not only on CMB anisotropies but also on the linear power spectrum (Holtzman 1989; Hu & Sugiyama 1996; Eisenstein & Hu 1998a). CMB anisotropies accurately calibrate their characteristic length scale; measurement of this standard ruler in the redshift survey power spectrum yields H0. With this added information, the CMB returns a signiﬁcantly more precise measure of Qm.

2. METHODOLOGY

We seek to quantify the potential sensitivity of these data sets to various cosmological parameters. For this, we use the Fisher matrix formalism (see Tegmark, Taylor, & Heavens 1997 for a review), which yields a lower limit on the statistical errors on cosmological parameters achievable by a set of experiments. This formalism operates within the context of a parameterized cosmological model. For this, we use a 12-variable parameterization of the adiabatic cold dark matter (CDM) model, described in detail in EHT. It includes CDM, baryons, massive neutrinos, a cosmological constant QL, curvature QK ({1 Ϫ QL Ϫ Qm), the Hubble constant H0 { 100 h km sϪ1 MpcϪ1, a reionization optical depth, and a primordial helium fraction. It assumes an initial scalar power spectrum Pi(k) ∝ k nSϩa log (k/kp) (k p { 0.025 MpcϪ1) with tilt nS, a logarithmic running of the tilt a, and an unknown amplitude as well as scale-invariant tensor contributions with an unconstrained amplitude. Finally, it allows an unknown linear bias to adjust the galaxy power spectrum relative to the mass (Pgal ϭ b2Pmass). All of the above parameters are determined simultaneously from the data.

For CMB anisotropies, we use the experimental speciﬁcations of the MAP and Planck satellites for temperature and polarization (EHT). We assume that foregrounds and systematics can be eliminated with negligible loss of cosmological information. For large-scale structure, we use the projected speciﬁcations of the bright red galaxy (BRG) sample of the SDSS to determine its sensitivity to the linear power spectrum (Tegmark 1997). On small scales, the observed power spectrum reﬂects nonlinear effects and galaxy formation issues. We therefore employ only wavenumbers less than kmax ϭ 0.1 h MpcϪ1, under the assumption that the linear power spectrum on these scales can be reconstructed (up to the unknown linear bias)

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TABLE 1 Errors on H0 and Qm for LCDM

Experiment

DH0 General Flat

DQm General Flat

MAP (no polarization) . . . . . . . . . . 135

15

1.4

0.23

With SDSS . . . . . . . . . . . . . . . . . . . . . .

3.0 2.5 0.042 0.037

MAP (with polarization) . . . . . . . . 23

6.7 0.25 0.10

With SDSS . . . . . . . . . . . . . . . . . . . . . .

2.9 2.4 0.037 0.036

Planck (no polarization) . . . . . . . . 113

5.3 1.2

0.079

With SDSS . . . . . . . . . . . . . . . . . . . . . .

2.5 2.3 0.035 0.035

Planck (with polarization) . . . . . . 13

1.6 0.14 0.024

With SDSS . . . . . . . . . . . . . . . . . . . . . .

2.2 1.4 0.027 0.020

Note.—The ﬁducial model has Qm ϭ 0.35, H0 ϭ 65 km sϪ1 MpcϪ1, and QB ϭ 0.05. We use kmax ϭ 0.1 h MpcϪ1 for SDSS. Errors are 1 j; DH0 errors are in units of km sϪ1 MpcϪ1. General: QK estimated from data. Flat: QK ϭ 0 by ﬁat.

despite the effects of extinction, evolution, and redshift-space distortions (Hamilton 1998; Tegmark et al. 1998b). We vary kmax in § 4.2.

3. CMB RESULTS

Parameter degeneracies occur when changes in the model parameters leave the power spectra essentially unchanged relative to the size of the experimental uncertainties. In particular, since cosmic variance is substantial at large angular scales, changes that affect large angles while leaving the acoustic peaks unchanged will be difﬁcult to detect.

The angular diameter distance dA to the last scattering surface contains the most important degeneracy for the present discussion. The CMB acoustic peaks are a high-redshift pattern viewed at distance dA. The pattern may be held ﬁxed by keeping Qmh2 and the baryon density QB h2 constant. However, dA depends on the low-redshift effects of a cosmological constant or curvature. Changing QL and QK so as to keep the angular diameter distance constant leaves the acoustic peaks unchanged. Only large-angle (ഞ Շ 50) gravitational redshift effects or small-angle (ഞ տ 1000) gravitational lensing effects can resolve this ambiguity.

In short, the CMB data sets will yield precision information on the physical properties at high redshift, notably Qmh2, QB h2, and dA(Qmh2, QL, QK), but not on H0 and Qm individually. A similar situation occurs in quintessence models with tradeoffs between QQ and the equation of state of the Q-ﬁeld (Huey et al. 1998).

In Table 1, we present the error bars on H0 and Qm attainable by upcoming CMB satellite experiments within our 12dimensional parameter space. One sees that when varying both QL and QK, the constraints on H0 and Qm are poor, although the polarization information does provide considerable help. Even if one assumes a ﬂat cosmological model (QK ϭ 0), MAP, with its partial coverage of the acoustic peaks, will not yield tiny errors on H0 and Qm.

There are several caveats. The Fisher matrix expansion of the likelihood function is not accurate for large steps in parameter space, which means that large error bars accurately detect a degenerate direction but may inaccurately reﬂect its magnitude (Zaldarriaga et al. 1997). One artifact of this is that the ellipses in Figure 1 follow straight lines rather than curves (e.g., constant Qmh2 in the CMB case). Moreover, the errors are slightly overestimated in the case of Planck because we have not included gravitational lensing (Seljak 1996; Metcalf & Silk 1997), by which the differences in growth factor between otherwise degenerate models will alter the small-angle

Fig. 1.—The 68% allowed regions for MAP (with polarization) alone, SDSS

(kmax ϭ 0.1 h MpcϪ1) alone, and the two combined. The lines in the directions

of

constant

Qmh2,

Qmh,

and

Q hϪ0.33 m

are

shown.

power spectra. Nevertheless, the point remains that the CMB alone will not constrain H0 and Qm to a level capable of strong consistency checks against other cosmological tests.

4. RESULTS WITH REDSHIFT SURVEYS

4.1. Linear Analysis

When we include the Fisher information matrix from SDSS, the error bars on H0 and Qm drop by an order of magnitude. In Table 1, we see that for a ﬁducial LCDM model, the errors on H0 are below 3 km sϪ1 MpcϪ1 while those on Qm are around 0.035. EHT discuss improvements on other parameters of the model; however, none are nearly as dramatic. Figure 1 displays the situation. Note that the results are roughly independent of whether polarization information is available or not.

The reason for this dramatic improvement lies with the baryons. Table 2 shows the error bars on H0 for a sequence of ﬁducial models with increasing baryon fraction; those on Qm behave similarly. Increasing the baryon fraction from ∼1% to ∼15% results in a dramatic increase in the information provided by SDSS. A baryon fraction exceeding 10% is strongly indicated by cluster gas fractions (White et al. 1993; David, Jones, & Forman 1995; White & Fabian 1995; Evrard 1997).

As the baryon fraction increases, signiﬁcant acoustic oscillations develop in the matter power spectrum (see Fig. 2, Hu & Sugiyama 1996, and Eisenstein & Hu 1998a). There is a characteristic scale of these oscillations that is known as the sound horizon; morphologically, the power spectrum acquires a sharp break near the sound horizon and a series of oscillations marking the harmonics of this scale. The size of the sound horizon can be calculated given knowledge of the physical conditions at high redshift, particularly Qmh2 and QB h2. Since

TABLE 2 Errors on H0 as Function of QB

QB

0.005 . . . . . . 0.02 . . . . . . . 0.05 . . . . . . . 0.10 . . . . . . .

QB/Qm (%)

1.4 5.7 14 29

MAP (with polarization)

36 29 23 24

With SDSS

12 9.2 2.9 1.3

Note.—Same parameters as in Table 1 save for the baryon fraction.

No. 2, 1998

COSMIC COMPLEMENTARITY

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Fig. 2.—Power spectra for three LCDM models, showing a progression of

QB (0.005, 0.05, and 0.1). The spectra are normalized on large scales. The thin curves show the power spectrum extended to second order (Jain & Bertschinger 1994), assuming j8 ϭ 1 for the linear power spectrum. The perturbation formalism uses Qm ϭ 1, but this is a good approximation (Bernardeau 1994).

these are exactly the quantities that are well constrained by the

relative heights of the CMB acoustic peaks, we can accurately

infer the scale of these features in real space. Its measurement

in the redshift-space power spectrum then yields an accurate

measure of H0. At low baryon fractions, the SDSS power spectrum still

reduces the error bars on H0 and Qm. This results from the one scale left in the matter power spectrum, that of the horizon at

matter-radiation equality. In redshift space, this yields a measure of G { Qmh, with considerable inaccuracies due to confusion with scalar tilt. However, the fact that the SDSS ellipse

in Figure 1 lies along a line very different from constant Qmh indicates that at moderate baryon fractions, this feature is not providing the primary leverage on H0. Note that the break and oscillation morphology of the baryonic features cannot be

mimicked by the effects of tilt or massive neutrinos.

In short, baryonic features yield a standard ruler, whose

length can be accurately inferred by the CMB and can be

measured in redshift space using large-redshift surveys. The comparison of lengths yields H0; combining this with Qmh2 gives Qm. This inference is independent of the angular diameter distance to last scattering (provided that the peaks are visible

at all!), so this method will function regardless of cosmological

TABLE 3 Errors on H0 for Differing SDSS Assumptions

kmax

MAP alone . . . . . . 0.025 . . . . . . . . . . . . 0.05 . . . . . . . . . . . . . 0.1 . . . . . . . . . . . . . . . 0.2 . . . . . . . . . . . . . . . 0.4 . . . . . . . . . . . . . . .

DH0

P(k) PS(k)

23 23 16 15 9.7 10.7 2.9 10.0 1.2 9.0 0.9 8.6

DQm

P(k) PS(k)

0.25 0.25 0.17 0.16 0.098 0.11 0.037 0.11 0.016 0.10 0.014 0.10

Note.—MAP with polarization has been taken in

each case. Limits with the actual linear power spectrum P(k ≤ kmax) are compared with those from a smoother analytic form PS(k ≤ kmax) (see text). Same model and notation as in Table 1.

constant, spatial curvature, or more exotic smooth components (see, e.g., Turner & White 1997). With Qm known, the location of the CMB peaks and the information from Type Ia supernovae (Perlmutter et al. 1998; Riess et al. 1998) may be focused on distinguishing these low-redshift effects (Tegmark et al. 1998a; Hu et al. 1998).

4.2. Nonlinearities

Baryonic oscillations are a feature of the linear power spectrum. Nonlinear evolution erases these signatures. Hence, we should test the extent to which the above results depend upon our use of linear theory.

Heretofore, we have assumed that we could use the linear power spectrum on scales longward of kmax ϭ 0.1 h MpcϪ1. This is close to the point at which nonlinearities will smooth the oscillations. We therefore display in Table 3 the effect of altering kmax, again simply ignoring information on all smaller scales. For several ﬁducial models, we ﬁnd that moving kmax from 0.1 to 0.2 h MpcϪ1 decreases the errors on H0 and Qm by about a factor of 2.5. However, these gains saturate as one extends kmax to 0.4 h MpcϪ1; the acoustic oscillations there are of such small amplitude that little information is gained.

We also consider an alternative formulation in which we model the matter power spectrum by a ﬁtting formula (Eisenstein & Hu 1998b) that includes the break at the sound horizon but not the oscillations. The resulting errors are shown in Table 3. For kmax տ 0.08 h MpcϪ1, the performance is signiﬁcantly worse than that achieved with the actual linear power spectrum. This is close to the location of the ﬁrst bump in this ﬁducial model. Note that including the featureless power spectrum on scales from 0.1 to 0.4 h MpcϪ1 adds very little additional information on H0 or Qm.

We therefore conclude that detection of at least the ﬁrst of the acoustic oscillations (k ≈ 0.07 h MpcϪ1 in this model) is critical to enabling a precision measure of H0 and Qm. Detecting additional peaks improves the possible error bars, but with diminishing returns, because the oscillations damp down in amplitude. In Figure 2, we present the results of second-order perturbation theory for the power spectrum (Jain & Bertschinger 1994) assuming a linear power spectrum normalization of j8 ϭ 1.0. For this value, linear theory indeed holds to kmax ≈ 0.1 h MpcϪ1, so that the ﬁrst baryonic peak survives while higher peaks are smeared out. The second-order correction scales as j82; note that a linear j8 ϭ 1.0 would yield an observed (nonlinear) value well in excess of unity. Cosmological simulations normalized to the cluster abundance reach similar conclusions (Meiksin, Peacock, & White 1998).

5. DISCUSSION

Detection of acoustic oscillations in the matter power spectrum would be a triumph for cosmology, since it would conﬁrm the standard thermal history and the gravitational instability paradigm. Moreover, because the matter power spectrum displays these oscillations in a different manner than does the CMB, we would gain new leverage on cosmological parameters. In particular, we have shown in this Letter that the combination of power spectrum measurements from a galaxy redshift survey with anisotropy measurements from CMB satellite experiments could yield a precision measurement of H0 and Qm.

The potential measurement of H0 and Qm depends critically on the ability of the redshift survey to detect the baryonic features in the linear power spectrum. The best possible error

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bars are a strong function of the baryon fraction but are surprisingly good even if the fraction is ∼10%, roughly the minimum implied by cluster observations. For such cases, the fractional limits achievable with the SDSS are 5% for H0 and 10% for Qm, if only the ﬁrst acoustic peak in P(k) is detected. Detecting the smaller scale peaks could allow an additional factor of 3 reﬁnement; the exact limits would depend on the scale at which nonlinear effects smooth out the power spectrum. The results depend only mildly on the details of the CMB experiment: we ﬁnd only slight gains as our presumed CMB data set improves from MAP without polarization to Planck with polarization. While we have quoted numbers for SDSS, it is possible that the 2dF survey will be able to make signiﬁcant progress on the detection of features in the power spectrum on very large scales. Unfortunately, the hints of excess power on 100 hϪ1 Mpc scales are not likely to be due to baryons (Eisenstein et al. 1998a).

We have treated the galaxy power spectrum by assuming linear bias on large scales. There is some theoretical motivation for this (Scherrer & Weinberg 1997); moreover, if bias tends toward unity as structure grows (Fry 1996; Tegmark & Peebles 1998), then scale dependences in the bias at the time of formation will be suppressed. Most importantly, this method of measuring H0 and Qm depends on extracting an oscillatory feature from the power spectrum. While one cannot prove that scale-dependent bias should be monotonic on the largest scales, this seems more likely than an oscillation! Finally, the assumption of linearity can be tested by constructing the power spectrum with different types of galaxies (see, e.g., Peacock 1997); future redshift surveys will allow this to be done on very large scales with good statistics.

The method proposed here yields H0 independent of local distance measurements and Qm without the complications inherent in dynamical methods; i.e., this method is free of many confusing astrophysical problems. On the other hand, it does depend on restricting oneself to a class of models with observable acoustic oscillations in both CMB anisotropies and the galaxy power spectrum. This assumption will be deﬁnitively tested from the data itself.

If the method described in this Letter can yield tight constraints on H0 and Qm, it will then be very important to compare these with other measurements of these quantities. In the coming decade, there will be a number of paths toward a precision measure of H0, such as the local distance ladder (see, e.g., Freedman et al. 1998), gravitational lensing (see, e.g., Blandford & Kundic 1997), and the Sunyaev-Zeldovich effect (see, e.g., Cooray et al. 1998). Similarly, good estimates of Qm may be possible from velocity ﬁelds (see, e.g., Dekel 1997), cluster evolution (Carlberg et al. 1997a; Bahcall, Fan, & Cen 1997), and M/L measurements (see, e.g., Carlberg, Yee, & Ellingson 1997b). If the results from these diverse sets of measurements are found to agree, we will have a secure foundation on which to base our cosmology.

Numerical power spectra were generated with CMBFAST (Seljak & Zaldarriaga 1996). We thank Martin White for useful discussions. D. J. E. is supported by a Frank and Peggy Taplin Membership; D. J. E. and W. H. by NSF-9513835; W. H. by the Keck Foundation and a Sloan Fellowship; M. T. by NASA through grant NAG5-6034 and a Hubble Fellowship HF01084.01-96A from STScI, operated by AURA, Inc., under NASA contract NAS4-26555.

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M. Livio, M. Donahue, & N. Panagia (Cambridge: Cambridge Univ. Press), in press (astro-ph/9611229) Bond, J. R., Crittenden, R., Davis, R. L., Efstathiou, G., & Steinhardt, P. J. 1994, Phys. Rev. Lett., 72, 13 Bond, J. R., Efstathiou, G., & Tegmark, M. 1997, MNRAS, 291, L33 Carlberg, R. G., Morris, S. L., Yee, H. K. C., & Ellingson, E. 1997a, ApJ, 479, L19 Carlberg, R. G., Yee, H. K. C., & Ellingson, E. 1997b, ApJ, 478, 462 Cooray, A. R., Carlstrom, J. E., Joy, M., Grego, L., Holzapfel, W. L., & Patel, S. K. 1998, in Proc. Dark Matter Workshop, ed. D. Cline, in press (astroph/9804149) David, L. P., Jones, C., & Forman, W. 1995, ApJ, 445, 578 Dekel, A. 1997, in Third ESO-VLT Workshop, Galaxy Scaling Relations: Origins, Evolution, and Applications, ed. L. da Costa & A. Renzini (Springer: Berlin), 245 Eisenstein, D. J., & Hu, W. 1998a, ApJ, 496, 605 ———. 1998b, ApJ, submitted Eisenstein, D. J., Hu, W., Silk, J., & Szalay, A. 1998a, ApJ, 494, L1 Eisenstein, D. J., Hu, W., & Tegmark, M. 1998b, ApJ, submitted (astro-ph/ 9807130) (EHT) Evrard, A. E. 1997, MNRAS, 292, 289 Freedman, W. L., Mould, J. R., Kennicut, R. C., & Madore, B. F. 1998, IAU Symp. 183, Cosmological Parameters and the Evolution of the Universe, ed. K. Sato (Dordrecht: Kluwer), in press (astro-ph/9801080) Fry, J. N. 1996, ApJ, 461, 65 Hamilton, A. J. S. 1998, in Ringberg Workshop on Large-Scale Structure, ed. D. Hamilton (Dordrecht: Kluwer), in press (astro-ph/9708102)

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᭧ 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.

COSMIC COMPLEMENTARITY: H0 AND Qm FROM COMBINING COSMIC MICROWAVE BACKGROUND EXPERIMENTS AND REDSHIFT SURVEYS

Daniel J. Eisenstein, Wayne Hu, and Max Tegmark1

Institute for Advanced Study, Princeton, NJ 08540 Received 1998 May 19; accepted 1998 July 9; published 1998 July 31

ABSTRACT

We show that the detection of acoustic oscillations in both upcoming cosmic microwave background (CMB) satellite experiments and large-redshift surveys can yield 5% determinations of H0 and Qm, an order-of-magnitude improvement over CMB data alone. CMB anisotropies provide the sound horizon at recombination as a standard ruler. For reasonable baryon fractions, this scale is imprinted on the galaxy power spectrum as a series of spectral features. Measuring these features in redshift space determines the Hubble constant, which in turn yields Qm once combined with CMB data. Since the oscillations in both power spectra are frozen-in at recombination, this test is insensitive to low-redshift cosmology.

Subject headings: cosmic microwave background — cosmology: theory — dark matter — large-scale structure of universe

1. INTRODUCTION

In the usual cosmological paradigm, the cosmic microwave background (CMB) contains a vast amount of information about cosmological parameters (Hu, Sugiyama, & Silk 1997). With upcoming experiments, most notably the two satellite missions Microwave Anisotropy Probe (MAP)2 and Planck,3 detailed measurements of the angular power spectra of its anisotropy and polarization may accurately determine many cosmological parameters (Jungman et al. 1996; Bond, Efstathiou, & Tegmark 1997; Zaldarriaga, Spergel, & Seljak 1997). However, certain changes in the cosmological parameters can conspire to leave the CMB power spectra unchanged, resulting in degenerate directions in the parameter space (Bond et al. 1994, 1997; Zaldarriaga et al. 1997; Huey et al. 1998). For example, since the Hubble constant H0 and the matter density Qm can be varied while keeping the angular diameter distance and the matter-radiation ratio ﬁxed, their values remain uncertain but highly correlated. Such degeneracies must be broken with cosmological information from other sources.

Upcoming redshift surveys for the study of large-scale structure hold the potential for resolving this issue. In particular, the 2dF survey4 and the Sloan Digital Sky Survey (SDSS)5 should measure the galaxy power spectrum on large enough scales to allow detailed comparisons with the mass power spectra predicted by cosmological theories. In this Letter and a companion paper (Eisenstein, Hu, & Tegmark 1998b, hereafter EHT), we explore the potential of combining redshift surveys and CMB anisotropy data for the purpose of parameter estimation. Here we focus on the dramatic improvement possible in the measurement of H0 and Qm. Neither data set yields tight limits by itself, yet together they could yield errors better than 5% on H0 and 10% on Qm.

The key to this improvement is the presence of features in

1 Hubble Fellow. 2 See http://map.gsfc.nasa.gov, maintained at NASA by D. N. Spergel and G. Hinshaw. 3 See http://astro.estec.esa.nl/SA-general/Projects/Planck, maintained at ESA by J. Tauber. 4 See http://meteor.anu.edu.au/˜colless/2dF, maintained at MSSSO by M. Colless. 5 See http://www.astro.princeton.edu/BBOOK/, maintained at Princeton University by R. Lupton.

the matter power spectrum on scales exceeding 60 hϪ1 Mpc. With a nonnegligible baryon fraction, the acoustic oscillations that exist before recombination are imprinted not only on CMB anisotropies but also on the linear power spectrum (Holtzman 1989; Hu & Sugiyama 1996; Eisenstein & Hu 1998a). CMB anisotropies accurately calibrate their characteristic length scale; measurement of this standard ruler in the redshift survey power spectrum yields H0. With this added information, the CMB returns a signiﬁcantly more precise measure of Qm.

2. METHODOLOGY

We seek to quantify the potential sensitivity of these data sets to various cosmological parameters. For this, we use the Fisher matrix formalism (see Tegmark, Taylor, & Heavens 1997 for a review), which yields a lower limit on the statistical errors on cosmological parameters achievable by a set of experiments. This formalism operates within the context of a parameterized cosmological model. For this, we use a 12-variable parameterization of the adiabatic cold dark matter (CDM) model, described in detail in EHT. It includes CDM, baryons, massive neutrinos, a cosmological constant QL, curvature QK ({1 Ϫ QL Ϫ Qm), the Hubble constant H0 { 100 h km sϪ1 MpcϪ1, a reionization optical depth, and a primordial helium fraction. It assumes an initial scalar power spectrum Pi(k) ∝ k nSϩa log (k/kp) (k p { 0.025 MpcϪ1) with tilt nS, a logarithmic running of the tilt a, and an unknown amplitude as well as scale-invariant tensor contributions with an unconstrained amplitude. Finally, it allows an unknown linear bias to adjust the galaxy power spectrum relative to the mass (Pgal ϭ b2Pmass). All of the above parameters are determined simultaneously from the data.

For CMB anisotropies, we use the experimental speciﬁcations of the MAP and Planck satellites for temperature and polarization (EHT). We assume that foregrounds and systematics can be eliminated with negligible loss of cosmological information. For large-scale structure, we use the projected speciﬁcations of the bright red galaxy (BRG) sample of the SDSS to determine its sensitivity to the linear power spectrum (Tegmark 1997). On small scales, the observed power spectrum reﬂects nonlinear effects and galaxy formation issues. We therefore employ only wavenumbers less than kmax ϭ 0.1 h MpcϪ1, under the assumption that the linear power spectrum on these scales can be reconstructed (up to the unknown linear bias)

L57

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TABLE 1 Errors on H0 and Qm for LCDM

Experiment

DH0 General Flat

DQm General Flat

MAP (no polarization) . . . . . . . . . . 135

15

1.4

0.23

With SDSS . . . . . . . . . . . . . . . . . . . . . .

3.0 2.5 0.042 0.037

MAP (with polarization) . . . . . . . . 23

6.7 0.25 0.10

With SDSS . . . . . . . . . . . . . . . . . . . . . .

2.9 2.4 0.037 0.036

Planck (no polarization) . . . . . . . . 113

5.3 1.2

0.079

With SDSS . . . . . . . . . . . . . . . . . . . . . .

2.5 2.3 0.035 0.035

Planck (with polarization) . . . . . . 13

1.6 0.14 0.024

With SDSS . . . . . . . . . . . . . . . . . . . . . .

2.2 1.4 0.027 0.020

Note.—The ﬁducial model has Qm ϭ 0.35, H0 ϭ 65 km sϪ1 MpcϪ1, and QB ϭ 0.05. We use kmax ϭ 0.1 h MpcϪ1 for SDSS. Errors are 1 j; DH0 errors are in units of km sϪ1 MpcϪ1. General: QK estimated from data. Flat: QK ϭ 0 by ﬁat.

despite the effects of extinction, evolution, and redshift-space distortions (Hamilton 1998; Tegmark et al. 1998b). We vary kmax in § 4.2.

3. CMB RESULTS

Parameter degeneracies occur when changes in the model parameters leave the power spectra essentially unchanged relative to the size of the experimental uncertainties. In particular, since cosmic variance is substantial at large angular scales, changes that affect large angles while leaving the acoustic peaks unchanged will be difﬁcult to detect.

The angular diameter distance dA to the last scattering surface contains the most important degeneracy for the present discussion. The CMB acoustic peaks are a high-redshift pattern viewed at distance dA. The pattern may be held ﬁxed by keeping Qmh2 and the baryon density QB h2 constant. However, dA depends on the low-redshift effects of a cosmological constant or curvature. Changing QL and QK so as to keep the angular diameter distance constant leaves the acoustic peaks unchanged. Only large-angle (ഞ Շ 50) gravitational redshift effects or small-angle (ഞ տ 1000) gravitational lensing effects can resolve this ambiguity.

In short, the CMB data sets will yield precision information on the physical properties at high redshift, notably Qmh2, QB h2, and dA(Qmh2, QL, QK), but not on H0 and Qm individually. A similar situation occurs in quintessence models with tradeoffs between QQ and the equation of state of the Q-ﬁeld (Huey et al. 1998).

In Table 1, we present the error bars on H0 and Qm attainable by upcoming CMB satellite experiments within our 12dimensional parameter space. One sees that when varying both QL and QK, the constraints on H0 and Qm are poor, although the polarization information does provide considerable help. Even if one assumes a ﬂat cosmological model (QK ϭ 0), MAP, with its partial coverage of the acoustic peaks, will not yield tiny errors on H0 and Qm.

There are several caveats. The Fisher matrix expansion of the likelihood function is not accurate for large steps in parameter space, which means that large error bars accurately detect a degenerate direction but may inaccurately reﬂect its magnitude (Zaldarriaga et al. 1997). One artifact of this is that the ellipses in Figure 1 follow straight lines rather than curves (e.g., constant Qmh2 in the CMB case). Moreover, the errors are slightly overestimated in the case of Planck because we have not included gravitational lensing (Seljak 1996; Metcalf & Silk 1997), by which the differences in growth factor between otherwise degenerate models will alter the small-angle

Fig. 1.—The 68% allowed regions for MAP (with polarization) alone, SDSS

(kmax ϭ 0.1 h MpcϪ1) alone, and the two combined. The lines in the directions

of

constant

Qmh2,

Qmh,

and

Q hϪ0.33 m

are

shown.

power spectra. Nevertheless, the point remains that the CMB alone will not constrain H0 and Qm to a level capable of strong consistency checks against other cosmological tests.

4. RESULTS WITH REDSHIFT SURVEYS

4.1. Linear Analysis

When we include the Fisher information matrix from SDSS, the error bars on H0 and Qm drop by an order of magnitude. In Table 1, we see that for a ﬁducial LCDM model, the errors on H0 are below 3 km sϪ1 MpcϪ1 while those on Qm are around 0.035. EHT discuss improvements on other parameters of the model; however, none are nearly as dramatic. Figure 1 displays the situation. Note that the results are roughly independent of whether polarization information is available or not.

The reason for this dramatic improvement lies with the baryons. Table 2 shows the error bars on H0 for a sequence of ﬁducial models with increasing baryon fraction; those on Qm behave similarly. Increasing the baryon fraction from ∼1% to ∼15% results in a dramatic increase in the information provided by SDSS. A baryon fraction exceeding 10% is strongly indicated by cluster gas fractions (White et al. 1993; David, Jones, & Forman 1995; White & Fabian 1995; Evrard 1997).

As the baryon fraction increases, signiﬁcant acoustic oscillations develop in the matter power spectrum (see Fig. 2, Hu & Sugiyama 1996, and Eisenstein & Hu 1998a). There is a characteristic scale of these oscillations that is known as the sound horizon; morphologically, the power spectrum acquires a sharp break near the sound horizon and a series of oscillations marking the harmonics of this scale. The size of the sound horizon can be calculated given knowledge of the physical conditions at high redshift, particularly Qmh2 and QB h2. Since

TABLE 2 Errors on H0 as Function of QB

QB

0.005 . . . . . . 0.02 . . . . . . . 0.05 . . . . . . . 0.10 . . . . . . .

QB/Qm (%)

1.4 5.7 14 29

MAP (with polarization)

36 29 23 24

With SDSS

12 9.2 2.9 1.3

Note.—Same parameters as in Table 1 save for the baryon fraction.

No. 2, 1998

COSMIC COMPLEMENTARITY

L59

Fig. 2.—Power spectra for three LCDM models, showing a progression of

QB (0.005, 0.05, and 0.1). The spectra are normalized on large scales. The thin curves show the power spectrum extended to second order (Jain & Bertschinger 1994), assuming j8 ϭ 1 for the linear power spectrum. The perturbation formalism uses Qm ϭ 1, but this is a good approximation (Bernardeau 1994).

these are exactly the quantities that are well constrained by the

relative heights of the CMB acoustic peaks, we can accurately

infer the scale of these features in real space. Its measurement

in the redshift-space power spectrum then yields an accurate

measure of H0. At low baryon fractions, the SDSS power spectrum still

reduces the error bars on H0 and Qm. This results from the one scale left in the matter power spectrum, that of the horizon at

matter-radiation equality. In redshift space, this yields a measure of G { Qmh, with considerable inaccuracies due to confusion with scalar tilt. However, the fact that the SDSS ellipse

in Figure 1 lies along a line very different from constant Qmh indicates that at moderate baryon fractions, this feature is not providing the primary leverage on H0. Note that the break and oscillation morphology of the baryonic features cannot be

mimicked by the effects of tilt or massive neutrinos.

In short, baryonic features yield a standard ruler, whose

length can be accurately inferred by the CMB and can be

measured in redshift space using large-redshift surveys. The comparison of lengths yields H0; combining this with Qmh2 gives Qm. This inference is independent of the angular diameter distance to last scattering (provided that the peaks are visible

at all!), so this method will function regardless of cosmological

TABLE 3 Errors on H0 for Differing SDSS Assumptions

kmax

MAP alone . . . . . . 0.025 . . . . . . . . . . . . 0.05 . . . . . . . . . . . . . 0.1 . . . . . . . . . . . . . . . 0.2 . . . . . . . . . . . . . . . 0.4 . . . . . . . . . . . . . . .

DH0

P(k) PS(k)

23 23 16 15 9.7 10.7 2.9 10.0 1.2 9.0 0.9 8.6

DQm

P(k) PS(k)

0.25 0.25 0.17 0.16 0.098 0.11 0.037 0.11 0.016 0.10 0.014 0.10

Note.—MAP with polarization has been taken in

each case. Limits with the actual linear power spectrum P(k ≤ kmax) are compared with those from a smoother analytic form PS(k ≤ kmax) (see text). Same model and notation as in Table 1.

constant, spatial curvature, or more exotic smooth components (see, e.g., Turner & White 1997). With Qm known, the location of the CMB peaks and the information from Type Ia supernovae (Perlmutter et al. 1998; Riess et al. 1998) may be focused on distinguishing these low-redshift effects (Tegmark et al. 1998a; Hu et al. 1998).

4.2. Nonlinearities

Baryonic oscillations are a feature of the linear power spectrum. Nonlinear evolution erases these signatures. Hence, we should test the extent to which the above results depend upon our use of linear theory.

Heretofore, we have assumed that we could use the linear power spectrum on scales longward of kmax ϭ 0.1 h MpcϪ1. This is close to the point at which nonlinearities will smooth the oscillations. We therefore display in Table 3 the effect of altering kmax, again simply ignoring information on all smaller scales. For several ﬁducial models, we ﬁnd that moving kmax from 0.1 to 0.2 h MpcϪ1 decreases the errors on H0 and Qm by about a factor of 2.5. However, these gains saturate as one extends kmax to 0.4 h MpcϪ1; the acoustic oscillations there are of such small amplitude that little information is gained.

We also consider an alternative formulation in which we model the matter power spectrum by a ﬁtting formula (Eisenstein & Hu 1998b) that includes the break at the sound horizon but not the oscillations. The resulting errors are shown in Table 3. For kmax տ 0.08 h MpcϪ1, the performance is signiﬁcantly worse than that achieved with the actual linear power spectrum. This is close to the location of the ﬁrst bump in this ﬁducial model. Note that including the featureless power spectrum on scales from 0.1 to 0.4 h MpcϪ1 adds very little additional information on H0 or Qm.

We therefore conclude that detection of at least the ﬁrst of the acoustic oscillations (k ≈ 0.07 h MpcϪ1 in this model) is critical to enabling a precision measure of H0 and Qm. Detecting additional peaks improves the possible error bars, but with diminishing returns, because the oscillations damp down in amplitude. In Figure 2, we present the results of second-order perturbation theory for the power spectrum (Jain & Bertschinger 1994) assuming a linear power spectrum normalization of j8 ϭ 1.0. For this value, linear theory indeed holds to kmax ≈ 0.1 h MpcϪ1, so that the ﬁrst baryonic peak survives while higher peaks are smeared out. The second-order correction scales as j82; note that a linear j8 ϭ 1.0 would yield an observed (nonlinear) value well in excess of unity. Cosmological simulations normalized to the cluster abundance reach similar conclusions (Meiksin, Peacock, & White 1998).

5. DISCUSSION

Detection of acoustic oscillations in the matter power spectrum would be a triumph for cosmology, since it would conﬁrm the standard thermal history and the gravitational instability paradigm. Moreover, because the matter power spectrum displays these oscillations in a different manner than does the CMB, we would gain new leverage on cosmological parameters. In particular, we have shown in this Letter that the combination of power spectrum measurements from a galaxy redshift survey with anisotropy measurements from CMB satellite experiments could yield a precision measurement of H0 and Qm.

The potential measurement of H0 and Qm depends critically on the ability of the redshift survey to detect the baryonic features in the linear power spectrum. The best possible error

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EISENSTEIN, HU, & TEGMARK

Vol. 504

bars are a strong function of the baryon fraction but are surprisingly good even if the fraction is ∼10%, roughly the minimum implied by cluster observations. For such cases, the fractional limits achievable with the SDSS are 5% for H0 and 10% for Qm, if only the ﬁrst acoustic peak in P(k) is detected. Detecting the smaller scale peaks could allow an additional factor of 3 reﬁnement; the exact limits would depend on the scale at which nonlinear effects smooth out the power spectrum. The results depend only mildly on the details of the CMB experiment: we ﬁnd only slight gains as our presumed CMB data set improves from MAP without polarization to Planck with polarization. While we have quoted numbers for SDSS, it is possible that the 2dF survey will be able to make signiﬁcant progress on the detection of features in the power spectrum on very large scales. Unfortunately, the hints of excess power on 100 hϪ1 Mpc scales are not likely to be due to baryons (Eisenstein et al. 1998a).

We have treated the galaxy power spectrum by assuming linear bias on large scales. There is some theoretical motivation for this (Scherrer & Weinberg 1997); moreover, if bias tends toward unity as structure grows (Fry 1996; Tegmark & Peebles 1998), then scale dependences in the bias at the time of formation will be suppressed. Most importantly, this method of measuring H0 and Qm depends on extracting an oscillatory feature from the power spectrum. While one cannot prove that scale-dependent bias should be monotonic on the largest scales, this seems more likely than an oscillation! Finally, the assumption of linearity can be tested by constructing the power spectrum with different types of galaxies (see, e.g., Peacock 1997); future redshift surveys will allow this to be done on very large scales with good statistics.

The method proposed here yields H0 independent of local distance measurements and Qm without the complications inherent in dynamical methods; i.e., this method is free of many confusing astrophysical problems. On the other hand, it does depend on restricting oneself to a class of models with observable acoustic oscillations in both CMB anisotropies and the galaxy power spectrum. This assumption will be deﬁnitively tested from the data itself.

If the method described in this Letter can yield tight constraints on H0 and Qm, it will then be very important to compare these with other measurements of these quantities. In the coming decade, there will be a number of paths toward a precision measure of H0, such as the local distance ladder (see, e.g., Freedman et al. 1998), gravitational lensing (see, e.g., Blandford & Kundic 1997), and the Sunyaev-Zeldovich effect (see, e.g., Cooray et al. 1998). Similarly, good estimates of Qm may be possible from velocity ﬁelds (see, e.g., Dekel 1997), cluster evolution (Carlberg et al. 1997a; Bahcall, Fan, & Cen 1997), and M/L measurements (see, e.g., Carlberg, Yee, & Ellingson 1997b). If the results from these diverse sets of measurements are found to agree, we will have a secure foundation on which to base our cosmology.

Numerical power spectra were generated with CMBFAST (Seljak & Zaldarriaga 1996). We thank Martin White for useful discussions. D. J. E. is supported by a Frank and Peggy Taplin Membership; D. J. E. and W. H. by NSF-9513835; W. H. by the Keck Foundation and a Sloan Fellowship; M. T. by NASA through grant NAG5-6034 and a Hubble Fellowship HF01084.01-96A from STScI, operated by AURA, Inc., under NASA contract NAS4-26555.

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