Measuring Cosmological Parameters with Cosmic Microwave ...

Measuring Cosmological Parameters with Cosmic Microwave Background Experiments J. Richard Bond,1 Robert Crittenden,2 Richard L. Davis,2 George Efstathiou,3 Paul J. Steinhardt2

arXiv:astro-ph/9309041v2 6 Oct 1993

(1)

Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 (2)

Department of Physics, University of Pennsylvania, Philadelphia, PA 19104 (3)

Department of Physics, Oxford University, Oxford, England OX1 3RH

Abstract

The cosmic microwave background anisotropy is sensitive to the slope and amplitude of primordial energy density and gravitational wave fluctuations, the baryon density, the Hubble constant, the cosmological constant, the ionization history, etc. In this Letter, we examine the degree to which these factors can be separately resolved from combined small- and large-angular scale anisotropy observations. We isolate directions of degeneracy in this cosmic parameter space, but note that other cosmic observations can break the degeneracy. PACS NOs: 98.80.Cq, 98.80.Es, 98.70.Vc

Typeset using REVTEX < 1◦ ) experThe observation of large-angular scale (∼ large- and small-angular scale (∼ 10◦ ) fluctuations in the Cosmic Microwave iments [3–10] are anticipated. In this LetBackground(CMB) [1,2] marks the beginning ter, we explore the degree to which the CMB of a new age of precision measurement in cos- anisotropy observations can determine cosmology [3–10]. Dramatic improvements in mological parameters such as the slope of the 1

initial power spectrum, the age of the uni- matter, etc., divided by the critical denverse and the cosmological constant. We find sity. We use the CMB quadrupole moments (S,T,Is,...)

that CMB anisotropy measurements alone C2

to parameterize the overall am-

cannot fix the parameters individually; how- plitudes of energy density (scalar metric), ever, a non-trivial combination of them can gravitational wave (tensor metric), isocurbe determined. More concretely, for models vature scalar and other primordial fluctuabased on the generation of gaussian, adia- tions predicted by the model. We parambatic fluctuations by inflation, we have iden- eterize the shape of the initial (e.g., poste s , a function of the ba- inflation) fluctuation spectra in wavenumber tified a new variable n

sic parameters that can be fixed to great pre- k by power law indices ns,t,is,..., defined at g cision by CMB anisotropy observations. Dis- time ti by k 3 h|(δρ/ρ)(k, ti )|2 i ∝ k nS +3 and 2 nT e e s k 3 h|h tinct models with nearly the same value of n +,× (k, ti )| i ∝ k , where δρ/ρ and h+,×

cannot be discriminated by CMB data alone. are the amplitudes of the energy density and In a likelihood analysis, this leads to error gravitational wave metric fluctuations (for contours centered around a highly elongated two polarizations), respectively. es maximum-likelihood surface inside which n

In this Letter, we restrict ourselves to

is approximately constant. However, when subdomains of this large space, in particular combined with other, independent cosmolog- to parameters consistent with inflation mode s is els of fluctuation generation. Inflation proical observations, the determination of n

a powerful tool for testing models and mea- duces a flat universe, hence ΩCDM + ΩHDM + suring fundamental parameters.

ΩB + ΩΛ ≈ 1. We also take ΩHDM = 0, > 10′, the but note that, for angular scales ∼

We parameterize the space by (S,T,Is,...)

(C2

anisotropy for mixed dark matter models , ns,t,is,..., h, ΩB , ΩΛ , ΩCDM , ΩHDM , . . .) , with ΩCDM + ΩHDM ≈ 1 is quite similar to

where H0 = 100 h km sec−1 Mpc−1 is the Hub- the anisotropy if all of the dark matter is cold. ble parameter, and ΩB,Λ,CDM,HDM,... are the Given ΩB , we impose the nucleosynthesis esenergy densities associated with baryons, cos- timate [11], ΩB h2 = 0.0125, to determine h, mological constant (Λ), cold and hot dark but also satisfy the globular cluster and other 2

< 0.65 for ΩΛ = 0 and as a reminder that r is determined by Eq. (1) age bounds, [12] h ∼ < 0.88 for ΩΛ < 0.6. (Gravitational lens given ns ; we have also assumed nt = ns − 1. h ∼ ∼ < 0.6. A straightOur results are based on numerical instatistics [13] suggest ΩΛ ∼ forward match to galaxy clustering data gives tegration of the general relativistic Boltzmann, Einstein, and hydrodynamic equa-

ΩΛ ≤ 1 − (0.2 ± 0.1)h−1 if ns ≤ 1. [14])

Inflation produces adiabatic scalar [15] tions for both scalar [21] and tensor metand tensor [16] Gaussian fluctuations. (For ric fluctuations using methods reported elsesimplicity, we do not consider isocurvature where [20]. fluctuations [17].) (T )

fixes C2

Included in the dynamical

The COBE quadrupole evolution are all the relevant components:

(S)

+ C2 , but the tensor-to-scalar baryons, photons, dark matter, and mass(T )

(S)

quadrupole ratio r ≡ C2 /C2

is undeter- less neutrinos. The temperature anisotropy,

mined [18]. Inflation does not produce strict ∆T /T (θ, φ) =

P

ℓm

aℓm Yℓm (θ, φ), is com-

power-law spectra, in general, but ns and nt puted in terms of scalar and tensor multipole (S)

(T )

can be defined from power-law best-fits to the components, aℓm and aℓm , respectively. For theoretical prediction over the scales probed inflation, each multipole for the two modes by the CMB. For generic models of inflation, is predicted to be statistically independent including new, chaotic, and extended models, and Gaussian-distributed, fully specified by (S)

angular power spectra, Cℓ

inflation gives [18–20] nt ≈ ns − 1 and r ≡

(S) (T ) C2 /C2

(T )

≈ 7(1 − ns ) . Cℓ (1)

D

(T )

E

= |aℓm |2 .

D

(S)

E

= |aℓm |2 and

Our results are presented in a series of

two-panel figures (e.g. see Fig. 1). The upMeasuring r and ns to determine whether per plots show the spectrum Cℓ ’s normalthey respect Eq. (1) is a critical test for ized to COBE, and the lower bar charts show inflation.

With this set of assumptions, the predicted (∆T /T )rms for idealized exper-

we have reduced the parameter-space to iments spanning 10◦ to 2′ . The bar chart three-dimensions, (r|ns , h, ΩΛ ) (where ΩB = is constructed by computing h(∆T /T )2 i = 0.0125h−2 and ΩCDM = 1−ΩB −ΩΛ ). We ex-

1 4π

P

(2ℓ + 1)Cℓ Wℓ , where Wℓ is a filter func-

plicitly display both r and ns but with a “|” tion that quantifies experimental sensitivity. 3

FIG. 1. Top:

Power spectra as a func-

FIG. 2. Power spectra as a function of ℓ for

tion of multipole moment ℓ for (r=0|ns =1), scale-invariant models, with r = 0|ns = 1. The (r=0.7|ns =0.9)

and

(r=1.4|ns =0.8)

where middle curve shows h = 0.5 and ΩΛ = 0. In

h = 0.5 and ΩΛ = 0 for all models. The spec- the upper curve, ΩΛ is increased to 0.4 while tra in all figures are normalized by the COBE keeping h = 0.5. In the lower curve, ΩΛ = 0 but σT2 (10◦ ) ≡ (4π)−1

P

(2ℓ+1)Cl exp(−ℓ(ℓ+1)/158.4).h is increased from 0.5 to 0.65 (hence ΩB drops

(a Gaussian filter with 10◦ fwhm), observed by from 0.5 to 0.3). The spectra are insensitive to DMR to be ∼ 1.2 × 10−10 , with about a 30% changes in h for fixed ΩB . Increasing ΩΛ or ΩB error. Bottom: (∆T /T )rms levels with 1-sigma increases the power at ℓ ∼ 200. cosmic variance error bars for nine experiments

each other to be largely uncorrelated. For

assuming full-sky coverage.

large ND , the likelihood function falls by

[For ND = 50

patches and a unity signal-to-noise ratio, the

e−ν

variance is 20%; see Eq. (2)].



The gaussian

coherence angle is indicated below each exper-

2 /2

∆T T

from a maximum at (∆T /T )max when 2



∆T = T

2

max

±

s



∆T 2 ν[ ND T

2

max

2 + σD ].

(2)

iment; see Refs. 1-11 for acronyms.

−5 [20,22] Errors arise from experimental noise An experimental noise σD below 10 is stan−6 is soon and “cosmic variance”, the latter a theoret- dard now, and a few times 10

ical uncertainty due to observing the fluc- achievable, hence if systematic errors and untuation distribution from only one vantage wanted signals can be eliminated, the 1-sigma point. The errors bars represent cosmic vari- (ν = 1) relative uncertainty in ∆T /T will be √ 2ND , falling from cosmic-variance alone, 1/ ance alone assuming full-sky coverage, exemplifying the limiting resolution achievable below 10% for ND > 50. The optimal variwith CMB experiments. For more realistic ance limits shown in the figures roughly corerror bars, consider a detection obtained from respond to filling the sky with patches sepameasurements (∆T /T )i ± σD (where σD rep- rated by 2θf whm .

Figure 1 shows a sequence of spectra with

resents detector noise) at i = 1, . . . , ND ex-

perimental patches sufficiently isolated from varying r|ns . The characteristic feature is increasingly suppressed small-angular signal as 4

FIG. 3. Examples of different cosmologies

FIG. 4.

Power spectra for models with

with nearly identical spectra of multipole mo- standard recombination (SR), no recombination ments and (∆T /T )rms .

The solid curve is (NR), and ‘late’ reionization (LR) at z = 50. In

(r = 0|ns = 1, h = 0.5, ΩΛ = 0).

The all models, h = 0.5 and ΩΛ = 0. NR or reion-

other two curves explore degeneracies in the ization at z ≥ 150 results in substantial suppres(r = 0|ns = 1, h, ΩΛ ) and (r|ns , h = 0.5, ΩΛ ) sion at ℓ ≥ 100. Models with reionization at planes. In the dashed curve, increasing ΩΛ is al- 20 ≤ z ≤ 150 give moderate suppression that most exactly compensated by increasing h. In can mimic decreasing ns or increasing h; e.g., the dot-dashed curve, the effect of changing to compare the ns = 0.95 spectrum with SR (thin, r = 0.42|ns = 0.94 is nearly compensated by dot-dashed) to the ns = 1 spectrum with reionincreasing ΩΛ to 0.6.

ization at z = 50 (thick, dot-dashed).

r increases and ns decreases. [18,20] Although Λ-suppression of the growth of scalar fluctucosmic variance is significant for large-angle ations [24]. The bar chart shows that either experiments, [23] it can shrink to insignifi- r|ns , ΩΛ , or h can be resolved if the other two cant levels at smaller scales if large maps are parameters are known. made. It appears that r|ns would be experi-

A degree of “cosmic confusion” arises,

mentally resolvable if Λ, h and ionization his- though, if r|ns , ΩΛ and h vary simultanetory were known.

ously.

Figure 3 shows our baseline spec-

Figure 2 shows the effects of varying ΩΛ trum and spectra for models lying in a twoor H0 compared to our baseline (solid line) dimensional surface of (r|ns , h, ΩΛ ) which spectrum (r = 0|ns = 1, h = 0.5, ΩΛ = 0). produce nearly identical spectra. In one case, Increasing ΩΛ enhances small-angular scale r|ns is fixed, and increasing ΩΛ is nearly comanisotropy by reducing the red shift zeq at pensated by increasing h. In the second case, which radiation-matter equality occurs; in- h is fixed, but increasing ΩΛ is nearly comcreasing h increases zeq and so has the op- pensated by decreasing ns (with r given by posite effect.

Increasing ΩΛ also changes Eq. (1)). [25]

< 10 due to slightly the spectral slope for ℓ ∼

Further cosmic confusion arises if we also

5

consider ionization history. [26] We expand equivalently, (1 − ΩΛ )h2 ), and to the optical the parameter-space to include zR , the red depth at last scattering for late-reionization 3/2

shift at which we suppose sudden, total reion- models, ∼ zR . These observations are the ization of the intergalactic medium. Fig. 4 basis of an empirical formula (accurate to < 15%) compares spectra with standard recombina- ∼ tion (SR), no recombination (NR) and late

ℓ(ℓ + 1)Cℓ ≈ A eB n˜ s 2πσT2 (10◦ ) max

reionization (LR) at zR = 50, where h = 0.5 and ΩΛ = 0.

(3)

NR represents the behav- where A = 0.1, B = 3.56, and

ior if reionization occurs early (zR >> 200).

n ˜ s ≈ ns − 0.28 log(1 + 0.8r)

The spectrum is substantially suppressed for

1

3/2

−0.52[(1 − ΩΛ )h2 ] 2 − 0.00036 zR + .26 ,

> 200 compared to any SR models. Experiℓ∼

(4) ◦ < ments at ∼ 0.5 scale can clearly identify NR > 150 gives quali- where r and ns are related by Eq. (1) for or early reionization (zR ∼ < tatively similar results to NR). Reionization generic inflation models, and zR ∼ 150 is ns has < zR < 150 results in modest suppres- needed to have a local maximum. (˜ for 20 ∼ ∼ ˜ s = ns for r = 0, h = sion at ℓ ≈ 200, which can be confused with been defined such that n 0.5, ΩΛ = 0, and zR = 0.)

a decrease in ns (see figure).

Our central result is that CMB anisotropy

The results can be epitomized by some

˜ s , but variations simple rules-of-thumb: Over the 30′ − 2◦ experiments can determine n ˜s range, (∆T /T )2rms is roughly proportional to of parameters along the surface of constant n

the maximum of ℓ(ℓ + 1)Cℓ (the first Doppler produce indistinguishable CMB anisotropy. peak). Since the maximum (corresponding to Given present uncertainties in h, ΩΛ and zR , ∼ .5◦ scales) is normalized to COBE DMR it will be possible to determine the true spec-

(at ∼ 10◦ ), its value is exponentially sensi- tral index ns (or r) to within

10% accu-

tive to ns . Since scalar fluctuations account racy using the CMB anisotropy alone. Quanfor the maximum, the maximum decreases as titative improvement can be gained by inr increases. The maximum is also sensitive to voking constraints from large-scale structure, the red shift at matter-radiation equality (or, e.g., galaxy velocity and cluster distributions, 6

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