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

**3.3. The Observed Power Spectrum**

In this section, we consider the current status of measurements of the matter power spectrum (see Peacock 1991, Peacock & Dodds 1994), and the radiation power spectrum (see Bond 1993) in the context of inflation (see Liddle & Lyth 1993). In Figure 3 we compare the radiation power spectrum with the matter power spectrum measured by IRAS-selected galaxies (Fisher et al 1993, Feldman et al 1994), the CfA redshift survey (Vogeley et al 1992), and the APM galaxy survey (Baugh & Efstathiou 1993a, b).

Since one cannot make a theory-independent extrapolation from the radiation power spectrum (probed by CMB measurements) to the matter power spectrum (probed by large-scale structure work) in the following, we will assume CDM and write the matter power spectrum as

(24) |

where *A* is as in (12) and *T*_{m} is a matter
transfer function (see
Equation 20), not to be confused with *T*(*k*). The change in
units and the index of the power law from *k*^{0} to
*k*^{1} is a matter of convention. We
follow the usual convention in large-scale structure (LSS) work that a
"flat" spectrum has *P*_{mat}(*k*)
*k* (see
Liddle & Lyth 1993
for more
discussion on the various definitions of power spectra) and work in
units of *h*^{-1} Mpc. To convert from our normalization to
that of the
bias or _{8}
conventionally used in LSS work, see Equation (19); note
that this conversion is senstive to any variation in the theory. The
large-scale structure and CMB data are shown in
Figure 3.

For the CMB anisotropy measurements, we have chosen some recent
experiments for which we could estimate the best-fit normalization.
Specifically, we show results from *COBE*
(Smoot et al 1991,
1992;
Wright et al 1994a),
FIRS/MIT
(Page et al 1990;
Ganga et al 1993;
Bond 1993,
1994), Tenerife
(Davies et al 1992,
Watson et al 1992,
Hancock et al 1994),
Python (Dragovan et al
1994), ARGO
(de Bernardis et al 1993,
1994),
SP91/ACME (13-point)
(Schuster et al 1993),
Saskatoon
(Wollack et al 1993),
MAX [MuP
(Meinhold et al 1993)
and GUM
(Devlin et al 1993,
Gundersen et al 1993)],
and MSAM
(Cheng et al 1994).
We have concentrated here on those experiments that quote a detection,
leaving out those that give only upper limits, e.g. Relikt
(Klypin et al 1992),
19.2 GHz
(Boughn et al 1992),
SP89
(Meinhold & Lubin
1991),
SP91-9pt (Gaier et al
1992),
ULISSE
(de Bernardis et al 1992),
White Dish
(Tucker et al 1993),
OVRO
(Readhead et al 1989,
Myers et al 1993),
and VLA
(Fomalont et al 1993).
Note we have also avoided complicated
issues involving redshift space corrections (e.g.
Kaiser 1987)
to the large-scale structure data.

The *P*_{mat}(*k*) inferred from CMB anisotropies depends
on the assumed
theory (_{0}
for example shifts the correspondence between
and *k*, and
shifts the amplitude for a given
*T* / *T*).
Consequently, the boxes in
Figure 3 would have to be redrawn for each theory.

In addition to the survey data shown in Figure 3,
there is information on large-scale flows (e.g.
Kashlinsky & Jones
1991).
Bertschinger et al (1990a)
estimated the 3-D velocity
dispersion of galaxies within spheres of radius 40 *h*^{-1}
Mpc and 60 *h*^{-1}
Mpc. After smoothing with a Gaussian filter on 12 *h*^{-1}
Mpc scales they found
_{v}(60) = 327
± 82 km s^{-1} and
_{v}(40) = 388
± 67 km s^{-1}. The unbiased
CDM estimate for these quantities are 224 km s^{-1} and 287 km
s^{-1}
respectively. Translating this information into the power on scales of
40 h^{-1} Mpc and 60 *h*^{-1} Mpc gives roughly
_{8}
1.15
(Efstathiou et al 1992).

One can also consider the CMB data independently of theories of
structure formation. In Figure 4, we show the
current situation with
regard to experiments that have quoted detections on degree scales or
larger. We plot the normalization which, for an *n* = 1 power spectrum,
would reproduce the quoted
*T* /
*T*_{rms} for each experiment. If there is a
Doppler peak in the power spectrum on degree scales, then this would
show up as a higher required normalization for a flat spectrum to fit
the data. All of these points should be interpreted as approximations
to the results of a full analysis.