Annu. Rev. Astron. Astrophys. 1994. 32:
319-70
Copyright © 1994 by . 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
*T*^{2}(*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}
3*t*_{0} =
2*H*_{0}^{-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 10^{n -
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
*P*_{mat}(*k*) = *Bk*^{n} on large
scales, which means that the dimensions of *B* will depend on n (and
will be length^{4} 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 *C*_{2}/*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
_{H}^{2}
= (4/)*A*.

The normalization convention used by the *COBE* group,
*Q*_{rms - 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
*C*_{2}, 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
*Q*_{rms - 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
*Q*_{rms - 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
*Q*_{rms - PS} assumed a flat spectrum, one
should not use (17) to normalize other spectra, although
<*Q*_{rms}^{2}>^{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 *Q*_{rms - 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 *Q*_{rms - PS} uses the full correlation function.

Another normalization sometimes used is the *bias*, defined through

(19) |

where
*T*_{m}(*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 *j*_{1} and the
final result is
dependent on the transfer function assumed. For CDM, we will take
(Efstathiou 1990)

(20) |

with *a* =
6.4_{0}
*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 *v*_{rms} 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).