Annu. Rev. Astron. Astrophys. 1994. 32: 319-70
Copyright © 1994 by . All rights reserved

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3. POWER SPECTRUM

3.1. Power Spectrum on Large Scales

Let us take the power spectrum of primordial fluctuations to be a power law in comoving wavenumber k. In the "processed" radiation power spectrum, this simple power law is multiplied by a transfer function T2(k). On COBE scales T(k) approx 1, and we can write the temperature fluctuation power spectrum as

Equation 12 (12)

where A is the amplitude for scalar perturbations and eta0 appeq 3t0 = 2H0-1 (for Omega0 = 1) is the conformal time today with scale factor normalized to unity. By using keta0 as our fundamental variable, we have A as a dimensionless number multiplied by something of order 10n - 1 on COBE scales. The connection between A and the normalization of the matter power spectrum is discussed in Section 3.3. [Another common convention is to define the matter power spectrum as Pmat(k) = Bkn on large scales, which means that the dimensions of B will depend on n (and will be length4 for n = 1); see Equation (24).]

We can write the average over universes of the moments of the temperature anisotropy as

Equation 13 (13)

(14)

(15)

(see e.g. Peebles 1982c, Bond & Efstathiou 1987). For the special case of n = 1, we have C2/A = 4pi/3 and Cell-1 propto ell(ell + 1). This is often referred to as "flat" since potential fluctuations (and the amplitude of delta rho / rho at horizon crossing) are independent of scale, and it also makes ell(ell + 1) Cell = constant.

In some older literature, the normalization of the power spectrum is given in terms of epsilonH, the dimensionless amplitude of matter fluctuations at horizon crossing. For a flat spectrum this quantity is simply epsilonH2 = (4/pi)A.

The normalization convention used by the COBE group, Qrms - PS is obtained by a best fit to the correlation function assuming a flat spectrum of fluctuations and allowing the normalization to vary. In terms of C2, this corresponds to

Equation 16 (16)

[For n = 1 the factor in parenthesis is (5/3)A, which allows a simple conversion from quadrupole normalization to our normalization in terms of A.] We would like to stress that Qrms - PS is the COBE group's best estimate, measured from our sky, of the power spectrum normalization. It is not the quadrupole measured by the COBE team from their maps. The value quoted for Qrms - PS, including the effects of systematic error, is (Smoot et al 1992, Wright et al 1994a, Bennett et al 1994)

Equation 17

which implies

Equation 17 (17)

Since the analysis for Qrms - PS assumed a flat spectrum, one should not use (17) to normalize other spectra, although <Qrms2>0.5 is still a valid way of quoting the normalization of the power spectrum.

For the first year data, a fit to the correlation function gives n = 1.1 ± 0.5 (Smoot et al 1992). Including the second year data gives n approx 1.5 - 0.5 (Bennett et al 1994, Wright et al 1994b) [by combining both COBE and Tenerife data, a stronger limit n gtapprox 0.9 has been obtained (Hancock et al 1994)] and the inferred value of Qrms - PS is quite correlated with a (Seljak & Bertschinger 1993, Watson & Gutierrez de la Cruz 1993). For n neq 1 the best value for the normalization is (Smoot et al 1992)

Equation 18 (18)

which probes a range of ell centered around ell approx 4 (Wright et al 1994a). Note that for n = 1, these two normalizations differ by ~ 10%, since the fit to Qrms - PS uses the full correlation function.

Another normalization sometimes used is the bias, defined through

Equation 19 (19)

where Tm(k) is a matter transfer function (see later section) not to be confused with T(k), and sigma2(r) is the variance of the density field within spheres of radius r. The variance of galaxies, possibly biased relative to the matter (deltagal = b deltarho), is roughly unity on a scale of 8 h-1 Mpc (Davis & Peebles 1983). Equation (19) is nontrivial to evaluate numerically because of the "ringing" of the j1 and the final result is dependent on the transfer function assumed. For CDM, we will take (Efstathiou 1990)

Equation 20 (20)

with a = 6.4Omega0 h-2 Mpc, b = 3Omega0 h-2 Mpc, c = 1.7Omega0 h-2 Mpc, and nu = 1.13. We will set Omega0 = 1 and h = 1/2 unless otherwise noted. For n = 1 the COBE best fit gives sigma8 appeq 1.2, i.e. an essentially unbiased model. However, this depends on the adopted values of Omega0, h, etc. It is possible to have a nonstandard (e.g. OmegaLambda appeq 0.8) CDM model with the galaxies significantly biased on small scales as seems to be required (e.g. Davis et al 1985, Bardeen et al 1986, Frenk et al 1990, Carlberg 1991).

Large-scale flows also provide a measure of the power spectrum (Peebles 1993):

Equation 21 (21)

where vrms is the 3-D velocity dispersion smoothed with a Gaussian filter of width r. This tends to probe scales similar to the degree-scale CMB experiments. Whether there is agreement between the two measures for a particular theory is still a matter of debate (see e.g. Vittorio & Silk 1985; Juszkiewicz et al 1987; Suto et al 1988; Atrio-Barandela et al 1991; Kashlinsky 1991, 1992, 1993a; Górski 1991, 1992).

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