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CMB Anisotropies and the Determination of Cosmological Parameters

I review the basic theory of the cosmic microwave background anisotropies in adiabatic cold dark matter cosmologies. The latest observational results on the CMB power spectrum are consistent with the simplest inflationary models and indicate that the Universe is close to spatially flat with a nearly scale invariant fluctuation spectrum. We are also beginning to see interesting constraints on the density of CDM, with a best fit value of omega_c = Omega_c h^2 of about 0.1. The CMB constraints, when combined with observations of distant Type Ia supernovae, are converging on a Lambda-dominated Universe with Omega_m approx 0.3 and Omega_Lambda approx 0.7.


Introduction
The discovery of temperature anisotropies in the CMB by the COBE team (Smoot et al. 1992) heralded a new era in cosmology. For the first time COBE provided a clear detection of the primordial fluctuations responsible for the formation of structure in the Universe at a time when they were still in the linear growth regime. Since then, a large number of ground based and balloon borne experiments have been performed which have succeeded in defining the shape of the power spectrum of temperature anisotropies C ℓ 2 up to multipoles of ℓ ∼ 300 clearly defining the first acoustic peak in the spectrum. Figure 1 shows a compilation of band power anisotropy

measurements
∆T ℓ T = 1 2π ℓ(ℓ + 1)C ℓ (1) that is almost up to date at the time of writing. The horizontal error bars show the multipole range probed by each experiment. The recent results from the VIPER experiment (Peterson et al. 1999) and the Boomerang test flight (Mauskopf et al. 1999) are not plotted because the exact window functions are not yet publically available. Neither are the published results from the Python V experiment (Coble et al. 1999) which seem to be discrepant with the other experiments particularly in the multipole range ℓ ∼ < 100. The points plotted in figure 1 are generally consistent with each other and provide strong evidence for a peak in the power spectrum at ℓ ∼ 200. In this introductory article, I will review briefly the theory of CMB anisotropies in adiabatic models of structure formation and then discuss the implications of Figure 1 for values of cosmological parameters. The literature on the CMB anisotropies has grown enormously over the last few years and it is impossible to do the subject justice in a short article. General reviews of the CMB anisotropies are given by Bond (1996) and Kamionkowski and Kosowsky (1999). A recent review on constraining cosmological parameters from the CMB is given by Rocha (1999).

Basic Theory
Most of the key features of figure 1 can be understood using a simplified set of equations. The background universe is assumed to be spatially flat together with small perturbations h ij so that the metric is We adopt the synchronous gauge, h 00 = h 0i = 0, and ignore the anisotropy of Thomson scattering and perturbations in the relativistic neutrino component. With these assumptions, the equations governing the evolution of scalar plane wave perturbations of wavenumber k arė Here, ∆ is the perturbation to the photon radiation brightness and ∆ 0 and ∆ 1 are its zeroth and first angular moments, δ b and δ c are the density perturbations in the baryonic and CDM components, v b is the baryon velocity and h = Tr(h ij ). Dots denote differentiation with respect to the conformal time variable τ . It is instructive to look at the solutions to these equations in the limits of large (kτ R ≪ 1) and small (kτ R ≫ 1) perturbations, where τ R is the conformal time at recombination:

LARGE ANGLE ANISOTROPIES
In the limit kτ R ≪ 1, Thomson scattering is unimportant and so the term in square brackets in the Boltzmann equation for the photons can be ignored.
In the matter dominated era h 33 = h ∝ τ 2 and so equation (3a) becomeṡ with approximate solution This solution is the Sachs-Wolfe (1967) effect. Any deviation from the evolutionḧ = constant, caused for example by a non-zero cosmological constant, will lead to additional terms in equation (5) increasing the large-angle anisotropies (sometimes referred to as the late-time Sachs-Wolfe effect, see e.g. Bond 1996). The CMB power spectrum is given by where the perturbation ∆ has been expanded in Legendre polynomials, Inserting the solution of equation (5) into equation (6) gives and so for a power-law spectrum of scalar perturbations, |h| 2 ∝ k ns , the CMB power spectrum is giving the characteristic power-law like form, C ℓ ∝ ℓ (ns−3) at low multipoles (ℓ ∼ < 30).

SMALL ANGLE ANISOTROPIES AND ACOUSTIC PEAKS
In the matter dominated era, equation (3a) becomeṡ and taking the zeroth and first angular moments giveṡ Prior to recombination, τ /τ c ≫ 1 where τ c = 1/(σ T n e a) is the mean collision time, and so the matter is tightly coupled to the radiation. In this limit ∆ 1 ≈ 4/3v b from equation (3c) and ∆ 2 in equation (11b) can be ignored. With these approximations, equation (11b) becomeṡ Neglecting the expansion of the universe, equations (11a) and (12) can be combined to give a forced oscillator equation with solution where ∆ 0 (0) and∆ 0 (0) are evaluated when the wave first crosses the Hubble radius, kτ ∼ 1. For adiabatic perturbations the first term dominates over the second because the perturbation breaks at kτ ∼ 1 with∆ 0 ≈ 0. It is useful to define (gauge-invariant) radiation perturbation variables then the solution of equation (10) is If kτ ≫ 1, the second term in the square brackets is smaller than the first by a factor of kτ , and the solution of equation (15) gives a power spectrum with a series of modulated acoustic peaks spaced at regular intervals of k m r s (a r ) = mπ, where r s is the sound horizon at recombination (Hu and Sugiyama 1995). Here a equ is the scale factor when matter and radiation have equal densities and a r is the scale factor at recombination. The multipole locations of the acoustic peaks in the angular power spectrum are given by where α is a number of order unity and d A is the angular diameter distance to last scattering where Ω k = 1−Ω Λ −Ω m and sin k ≡ sinh if Ω k > 0 and sin k = sin if Ω k < 0. The general dependence of the CMB power spectrum on cosmological parameters is therefore clear. The positions of the acoustic peaks depend on the geometry of the Universe via the angular diameter distance of equation (18) and on the value of the sound horizon r s . The relative amplitudes of the peaks depend on the physical densities of the various constituents ω b ≡ Ω b h 2 , ω c ≡ Ω c h 2 , ω ν ≡ Ω ν h 2 , etc. and on the scalar fluctuation spectrum (parameterized here by a constant spectral index n s ). Clearly, models with the same initial fluctuation spectra and identical physical matter densities ω i will have identical CMB power spectra at high multipoles if they have the same angular diameter distance to the last scattering surface. This leads to a strong geometrical degeneracy between Ω m and Ω Λ (e.g. Efstathiou and Bond 1999 and references therein). The power spectrum on large angular scales (equation 9) is sensitive to the spectral index and amplitude of the power spectrum, geometry of the Universe and, for extreme values of Ω k can break the geometrical degeneracy via the late-time Sachs-Wolfe effect. We will discuss briefly some of the constraints on cosmological parameters from the current CMB data in the next section. Before moving on to this topic, I mention some important points that cannot be covered in detail because of space limitations: • Inflationary models can give rise to tensor perturbations with a characteristic spectrum that declines sharply at ℓ ∼ > 100 (see e.g. Bond 1996 and references therein). In power-law like inflation, the tensor spectral index n t is closely linked to the scalar spectral index, n t ≈ n s − 1, and to the relative amplitude of the tensor and scalar perturbations.
• The anisotropy of Thomson scattering causes the CMB anisotropies to be linearly polarized at the level of a few percent (see Bond 1996, Hu and White 1997, and references therein). Measurements of the linear polarization can distinguish between tensor and scalar perturbations and can constrain the epoch of reionization of the intergalactic medium (Zaldarriaga, Spergel and Seljak 1997).
• The main effect of reionization is to depress the amplitude of the power spectrum at high multipoles by a factor of exp(−2τ opt ) where τ opt is the optical depth to Thomson scattering. In the 'best fit' CDM universe described in the next section (ω b = 0.019, h = 0.65, Ω m = Ω c + Ω b ≈ 0.3  and Ω Λ ≈ 0.7) and a reionization redshift of z reion ≈ 20 (a plausible value) τ opt ≈ 0.2 which is significant. There is a reasonable chance that we might learn something about the 'dark ages' of cosmic history from precision measurements of the CMB.

Cosmological Parameters from the CMB
In this section, we review some of the constraints on cosmological parameters from the CMB data plotted in figure 1. The analysis is similar to that presented in Efstathiou et al. (1999, hereafter E99), in which we map the full likelihood function in 5 parameters Ω Λ , Ω m , ω c , n s and Q 10 (the amplitude of √ C ℓ at ℓ = 10 relative to that inferred from COBE). The baryon density is constrained to ω b = 0.019, as determined from primor-dial nucleosynthesis and deuterium abundances measurements from quasar spectra (Burles and Tytler 1998). The results presented below are insensitive to modest variations (∼ 25%) of ω b and illustrate the main features of cosmological parameter estimation from the CMB. Recently, Tegmark and Zaldarriaga (2000) have performed a heroic 10 parameter fit to the CMB data, including a tensor contribution and finite optical depth from reionization. I will discuss the effects of widening the parameter space briefly below but refer the reader to Tegmark and Zaldarriaga for a detailed analysis.
The best fit model in this five parameter space is plotted as the solid line in figure 1. It is encouraging that the best fitting model has perfectly reasonable parameters, a spatially flat universe with a nearly scale invariant fluctuation spectrum and a low CDM density ω c ∼ 0.1. Marginalised likelihood functions are plotted in various projections in the parameter space in figures 2, 3 and 5 (uniform priors are assumed in computing the marginalized likelihoods, as described in E99). Figure 2 shows constraints on the position of the first acoustic peak measured by the 'location' parameter i.e. the parameter γ D measures the location of the acoustic peak relative to that in a spatially flat model with zero cosmological constant. The geometrical degeneracy between Ω m and Ω Λ described in the previous section is expressed by γ D = constant. Figure 2 shows that the best fitting value is γ D = 1 with a 2σ range of about ±0.3. The position of the first acoustic peak in the CMB data thus provides powerful evidence that the Universe is close to spatially flat. Figure 3 shows the marginalized likelihoods in the Q 10 − n s and ω c − n s planes. The constraints on Q 10 and n s are not very different to those from the analysis of COBE alone (see e.g. Bond 1996). The experiments at higher multipoles are so degenerate with variations in other cosmological parameters that they do not help tighten the constraints on Q 10 and n s . The constraints on ω c and n s show an interesting result; if n s ≈ 1, then the best fit value of ω c is about 0.1 with a 2σ upper limit of about 0.3. This constraint on ω c comes from the height of the first acoustic peak, as shown in figure 4. In this diagram, the CMB data points have been averaged in 10 band-power estimates as described by Bond, Knox and Jaffe (1998). The solid curve shows the best-fit model as plotted in figure 1, which has ω c = 0.1. The dashed lines show models with ω c = 0.25 and ω c = 0.05 with the other parameters held fixed. Raising ω c lowers the height of the peak and vice-versa. This result is not very sensitive to variations of ω b in the neighbourhood of ω b ∼ 0.02. Reionization and the addition of a tensor component can lower the height of the first peak relative to the anisotropies at lower multipoles and so the upper limits on ω c are robust to the addition of these parameters. The CMB data have now reached the point where we have good constraints on the height of the first peak, as well as its location, and this is beginning to set interesting constraints on ω c . The best fit value of Ω m ≈ 0.3, derived from combining the CMB data with results from distant Type Ia supernovae (see figure 5) implies ω c ≈ 0.11 for a Hubble constant of h = 0.65, consistent with the low values of ω c favoured by the height of the first acoustic peak.
The left hand panel of figure 5 shows the marginalized likelihood for the CMB data in the Ω Λ -Ω m plane. The likelihood peaks along the line for spatially flat universes Ω k = 0 and it is interesting to compare with the equivalent figure in E99 to see how the new experimental results of the last year have caused the likelihood contours to narrow down around Ω k = 0. (See also Dodelson and Knox 1999 for a similar analysis using the latest CMB data). As is well known, the magnitude-redshift relation for distant Type Ia supernovae results in nearly orthogonal constraints in the Ω Λ -Ω m plane, so combining the supernovae and CMB data can break the geometrical degeneracy. The right hand panel in Figure 5   It is remarkable how the CMB data and the supernovae data are homing in on a consistent set of cosmological parameters that are compatible with the simplest inflationary models and also with parameters inferred from a number of other observations (e.g. galaxy clustering, baryon content of clusters and dynamical estimates of the mean mass density, see Bahcall et al. 1999 for a review). It is also remarkable that the 'best fit' model requires a non-zero cosmological constant, a result that few cosmologists would have thought likely a few years ago.
The next few years will see a revolutionary increase in the volume and quality of CMB data. The results of the Boomerang Antarctic flight are awaited with great interest and should be of sufficient quality to render all previous analyses of cosmological parameters from the CMB obsolete. The polarization of the CMB has not yet been discovered, but a number of ground based and balloon borne experiments designed to detect polarization are under construction (Staggs, Gundersen and Church 1999). The MAP satellite, scheduled for launch in late 2000, will have polarization sensitivity and should determine the power spectrum C ℓ accurately to about ℓ ∼ 800, defining the first three acoustic peaks. Further into the future, the Planck satellite, scheduled for launch in 2007, should determine the power spectrum to ℓ ∼ > 2500, provide sensitive polarization measurements and extremely accurate subtraction of foregrounds. Evidently, the era of precision cosmology is upon us and the next decade should see a dramatic improvement in our knowledge of fundamental cosmological parameters and in our understanding of the origin of fluctuations in the early Universe.