![]() | Annu. Rev. Astron. Astrophys. 1994. 32:
319-70 Copyright © 1994 by Annual Reviews. All rights reserved |
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) 1,
and we can write the temperature fluctuation power spectrum as
![]() | (12) |
where A is the amplitude for scalar perturbations and
0
3t0 =
2H0-1 (for
0 = 1) is
the conformal time today with scale factor normalized to unity. By using
k
0 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
![]() | (13) (14) (15) |
(see e.g.
Peebles 1982c,
Bond & Efstathiou
1987).
For the special case
of n = 1, we have C2/A =
4/3 and
C
-1
(
+ 1). This is often
referred to as "flat" since potential fluctuations (and the amplitude
of
/
at horizon
crossing) are independent of scale, and it also makes
(
+ 1)
C
= constant.
In some older literature, the normalization of the power spectrum is
given in terms of
H, the
dimensionless amplitude of matter fluctuations at horizon crossing. For
a flat spectrum this quantity is simply
H2
= (4/
)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
![]() | (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)
![]() |
which implies
![]() | (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 1.5 - 0.5
(Bennett et al 1994,
Wright et al 1994b)
[by combining both COBE and Tenerife data, a stronger limit
n
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
1 the best value for the normalization is
(Smoot et al 1992)
![]() | (18) |
which probes a range of
centered around
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
![]() | (19) |
where
Tm(k) is a matter transfer function (see later
section) not to be confused with T(k), and
2(r)
is the variance of the density field
within spheres of radius r. The variance of galaxies, possibly biased
relative to the matter
(
gal =
b
), 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)
![]() | (20) |
with a =
6.40
h-2 Mpc, b =
3
0
h-2 Mpc, c =
1.7
0
h-2 Mpc, and
= 1.13. We will set
0 = 1 and
h = 1/2 unless otherwise noted. For n = 1 the COBE
best fit gives
8
1.2, i.e. an
essentially unbiased model. However, this depends on the adopted values of
0,
h, etc. It is possible to have a nonstandard (e.g.
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):
![]() | (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).