Annu. Rev. Astron. Astrophys. 2002. 40: 171-216 Copyright © 2002 by Annual Reviews. All rights reserved

2. OBSERVABLES

2.1. Standard Cosmological Paradigm

While a review of the standard cosmological paradigm is not our intention (see [Narkilar & Padmanabhan, 2001] for a critical appraisal), we briefly introduce the observables necessary to parameterize it.

The expansion of the Universe is described by the scale factor a(t), set to unity today, and by the current expansion rate, the Hubble constant H₀ = 100h km sec^-1 Mpc^-1, with h 0.7 [Freedman et al, 2001]. The Universe is flat (no spatial curvature) if the total density is equal to the critical density, _c = 1.88h² × 10^-29 g cm^-3; it is open (negative curvature) if the density is less than this and closed (positive curvature) if greater. The mean densities of different components of the Universe control a(t) and are typically expressed today in units of the critical density _i, with an evolution with a specified by equations of state w_i = p_i / _i, where p_i is the pressure of the ith component. Density fluctuations are determined by these parameters through the gravitational instability of an initial spectrum of fluctuations.

The working cosmological model contains photons, neutrinos, baryons, cold dark matter and dark energy with densities proscribed within a relatively tight range. For the radiation, _r = 4.17 × 10^-5h^-2 (w_r = 1/3). The photon contribution to the radiation is determined to high precision by the measured CMB temperature, T = 2.728 ± 0.004K [Fixsen et al, 1996]. The neutrino contribution follows from the assumption of 3 neutrino species, a standard thermal history, and a negligible mass m << 1eV. Massive neutrinos have an equation of state w = 1/3 0 as the particles become non-relativistic. For m ~ 1eV this occurs at a ~ 10^-3 and can leave a small but potentially measurable effect on the CMB anisotropies [Ma & Bertschinger, 1995, Dodelson et al, 1996].

For the ordinary matter or baryons, _b 0.02 h^-2 (w_b 0) with statistical uncertainties at about the ten percent level determined through studies of the light element abundances (for reviews, see [Boesgaard & Steigman, 1985, Schramm & Turner, 1998, Tytler et al, 2000]). This value is in strikingly good agreement with that implied by the CMB anisotropies themselves as we shall see. There is very strong evidence that there is also substantial non-baryonic dark matter. This dark matter must be close to cold (w_m = 0) for the gravitational instability paradigm to work [Peebles, 1982] and when added to the baryons gives a total in non-relativistic matter of _m 1/3. Since the Universe appears to be flat, the total _tot must be equal to one. Thus, there is a missing component to the inventory, dubbed dark energy, with 2/3. The cosmological constant (w = - 1) is only one of several possible candidates but we will generally assume this form unless otherwise specified. Measurements of an accelerated expansion from distant supernovae [Riess et al, 1998, Perlmutter et al, 1999] provide entirely independent evidence for dark energy in this amount.

The initial spectrum of density perturbations is assumed to be a power law with a power law index or tilt of n 1 corresponding to a scale-invariant spectrum. Likewise the initial spectrum of gravitational waves is assumed to be scale-invariant, with an amplitude parameterized by the energy scale of inflation E_i, and constrained to be small compared with the initial density spectrum. Finally the formation of structure will eventually reionize the Universe at some redshift 7 z_ri 20.

Many of the features of the anisotropies will be produced even if these parameters fall outside the expected range or even if the standard paradigm is incorrect. Where appropriate, we will try to point these out.

2.2. CMB Temperature Field

The basic observable of the CMB is its intensity as a function of frequency and direction on the sky . Since the CMB spectrum is an extremely good blackbody [Fixsen et al, 1996] with a nearly constant temperature across the sky T, we generally describe this observable in terms of a temperature fluctuation () = T / T.

If these fluctuations are Gaussian, then the multipole moments of the temperature field

(1)

are fully characterized by their power spectrum

(2)

whose values as a function of are independent in a given realization. For this reason predictions and analyses are typically performed in harmonic space. On small sections of the sky where its curvature can be neglected, the spherical harmonic analysis becomes ordinary Fourier analysis in two dimensions. In this limit becomes the Fourier wavenumber. Since the angular wavelength = 2 / , large multipole moments corresponds to small angular scales with ~ 10² representing degree scale separations. Likewise, since in this limit the variance of the field is d² C / (2)², the power spectrum is usually displayed as

(3)

the power per logarithmic interval in wavenumber for >> 1.

Plate 1 (top) shows observations of _T along with the prediction of the working cosmological model, complete with the acoustic peaks mentioned in Section 1 and discussed extensively in Section 3. While COBE first detected anisotropy on the largest scales (inset), observations in the last decade have pushed the frontier to smaller and smaller scales (left to right in the figure). The MAP satellite, launched in June 2001, will go out to ~ 1000, while the European satellite, Planck, scheduled for launch in 2007, will go a factor of two higher (see Plate 1 bottom).

Plate 1: Top: temperature anisotropy data with boxes representing 1- errors and approximate -bandwidth. Bottom: temperature and polarization spectra for tot = 1, = 2/3, b h2 = 0.02, m h2 = 0.16, n = 1, zri = 7, Ei = 2.2 × 1016 GeV. Dashed lines represent negative cross correlation and boxes represent the statistical errors of the Planck satellite.

The power spectra shown in Plate 1 all begin at = 2 and exhibit large errors at low multipoles. The reason is that the predicted power spectrum is the average power in the multipole moment an observer would see in an ensemble of universes. However a real observer is limited to one Universe and one sky with its one set of _m's, 2 + 1 numbers for each . This is particularly problematic for the monopole and dipole ( = 0, 1). If the monopole were larger in our vicinity than its average value, we would have no way of knowing it. Likewise for the dipole, we have no way of distinguishing a cosmological dipole from our own peculiar motion with respect to the CMB rest frame. Nonetheless, the monopole and dipole - which we will often call simply and v - are of the utmost significance in the early Universe. It is precisely the spatial and temporal variation of these quantities, especially the monopole, which determines the pattern of anisotropies we observe today. A distant observer sees spatial variations in the local temperature or monopole, at a distance given by the lookback time, as a fine-scale angular anisotropy. Similarly, local dipoles appear as a Doppler shifted temperature which is viewed analogously. In the jargon of the field, this simple projection is referred to as the freestreaming of power from the monopole and dipole to higher multipole moments.

How accurately can the spectra ultimately be measured? As alluded to above, the fundamental limitation is set by "cosmic variance" the fact that there are only 2 + 1 m-samples of the power in each multipole moment. This leads to an inevitable error of

(4)

Allowing for further averaging over in bands of , we see that the precision in the power spectrum determination scales as ^-1, i.e. ~ 1% at = 100 and ~ 0.1% at = 1000. It is the combination of precision predictions and prospects for precision measurements that gives CMB anisotropies their unique stature.

There are two general caveats to these scalings. The first is that any source of noise, instrumental or astrophysical, increases the errors. If the noise is also Gaussian and has a known power spectrum, one simply replaces the power spectrum on the rhs of Equation (4) with the sum of the signal and noise power spectra [Knox, 1995]. This is the reason that the errors for the Planck satellite increase near its resolution scale in Plate 1 (bottom). Because astrophysical foregrounds are typically non-Gaussian it is usually also necessary to remove heavily contaminated regions, e.g. the galaxy. If the fraction of sky covered is f_sky, then the errors increase by a factor of f_sky^-1/2 and the resulting variance is usually dubbed "sample variance" [Scott et al, 1994]. An f_sky = 0.65 was chosen for the Planck satellite.

2.3. CMB Polarization Field

While no polarization has yet been detected, general considerations of Thomson scattering suggest that up to 10% of the anisotropies at a given scale are polarized. Experimenters are currently hot on the trail, with upper limits approaching the expected level [Hedman et al, 2001, Keating et al, 2001]. Thus, we expect polarization to be an extremely exciting field of study in the coming decade.

The polarization field can be analyzed in a way very similar to the temperature field, save for one complication. In addition to its strength, polarization also has an orientation, depending on relative strength of two linear polarization states. While classical literature has tended to describe polarization locally in terms of the Stokes parameters Q and U ¹, recently cosmologists [Seljak, 1997, Kamionkowski et al, 1997, Zaldarriaga & Seljak, 1997] have found that the scalar E and pseudo-scalar B, linear but non-local combinations of Q and U, provide a more useful description. Postponing the precise definition of E and B until Section 3.7, we can, in complete analogy with Equation (1), decompose each of them in terms of multipole moments, and then, following Equation (2), consider the power spectra,

(5)

Parity invariance demands that the cross correlation between the pseudoscalar B and the scalars or E vanishes.

The polarization spectra shown in Plate 1 [bottom, plotted in µK following Equation (3)] have several notable features. First, the amplitude of the EE spectrum is indeed down from the temperature spectrum by a factor of ten. Second, the oscillatory structure of the EE spectrum is very similar to the temperature oscillations, only they are apparently out of phase but correlated with each other. Both of these features are a direct result of the simple physics of acoustic oscillations as will be shown in Section 3. The final feature of the polarization spectra is the comparative smallness of the BB signal. Indeed, density perturbations do not produce B modes to first order. A detection of substantial B polarization, therefore, would be momentous. While E polarization effectively doubles our cosmological information, supplementing that contained in C, B detection would push us qualitatively forward into new areas of physics.

¹ There is also the possibility in general of circular polarization, described by Stokes parameter V, but this is absent in cosmological settings. Back.