The cosmological effects of the Sunyaev-Zel'dovich effect fall into two categories: the integrated effect on the spectrum (discussed in this section) and the angular fluctuation pattern that is created (Sec. 11.3). Both the gas in clusters of galaxies and the distributed hot intergalactic medium between clusters will contribute to these effects: indeed, at a general level we can consider the cluster gas to be merely a strongly clumped fraction of the hot intergalactic medium. The cosmological Sunyaev-Zel'dovich effects then measure the projected electron pressure distribution since recombination.

It is convenient in discussing the effect of the intergalactic medium (IGM) on the CMBR to work in terms of the fraction of the critical density that this gas comprises. This is described by the quantity

where the critical density, _{crit}
(equation 2) just closes the Universe. Limits to the
contribution of *neutral* gas to _{IGM} are already
stringent, because of the absence of neutral hydrogen absorption
features in the spectra of high-redshift quasars (the Gunn-Peterson
test;
Gunn & Peterson 1965),
with a recent
limit on the optical depth _{GP} < 0.07 at redshifts near 4.3
based on a spectrum of a quasar at *z* = 4.7
(Giallongo *et
al.* 1994).
Further limits on the contribution of hot
gas to _{IGM} can be set
based on the X-ray background, most of which can be accounted for by
the integrated emission of active galaxies and quasars
(Comastri *et
al.* 1995).
At low energies it has been suggested
that the bremsstrahlung of hot gas in clusters and groups of galaxies may make
a significant contribution to the X-ray background, or even
over-produce the background under some models for cluster evolution
(Burg *et al.*
1993),
while the possibility that a
diffuse intergalactic medium is responsible for much of the X-ray
background was suggested by
Field & Perrenod (1977).

If some significant contribution to the X-ray background does come
from distributed gas, then the assumption that the gas is fully
ionized out to some redshift *z*_{ri} (at time
*t*_{ri}) leads to an optical depth for inverse-Compton
scatterings between ourselves and the epoch of recombination of

where *n*_{e0} is the electron density today and I have assumed a
Friedmann-Robertson-Walker cosmology with zero cosmological
constant. If the thermal history of this intergalactic plasma is
parameterized by a redshift-dependent electron temperature,
*T*_{e} *(z)*, then the Comptonization parameter is

For re-ionization redshifts 30, and any _{0} < 1, the
scattering optical depth is less than about 2.6 _{IGM}
*h*_{100}, and when the integral in (127) is
performed for plausible thermal histories of the intergalactic medium
(e.g.,
Taylor & Wright 1989;
Wright *et al.*
1996),
then the recent COBE FIRAS limit *y* < 15
x 10^{-6}
(Fixsen *et al.*
1996)
leads to a limit
on the electron scattering optical depth (averaged over the sky) of
less than 3 x 10^{-4}
(Wright *et al.*
1994).
This
corresponds to an electron density that is 100 times less
than the density needed to produce a significant fraction of the X-ray
background by thermal bremsstrahlung, which in turn suggests that a
uniform, hot, IGM produces less than 10^{-4} of the X-ray
background, and that a significant fraction of the X-ray background
can only arise from thermal bremsstrahlung if the gas has a filling
factor < 10^{-4} on the sky.

Direct calculations of the effects of clusters of galaxies on the
spectrum of the CMBR have been made by
Markevitch *et
al.* (1991)
and Cavaliere et
al. (1991).
An integration like that in
(127) must now be performed over an evolving
population of clusters of galaxies, with varying space density, size,
gas properties, etc. Markevitch *et al.* used self-similar models for
the variations of cluster properties with redshift. These models
are characterized by a power-law index *n*, which defines the
relationship between the redshift and density, size, mass, and
comoving number density scales of a population of
clusters. Specifically, the mass scale of the population is

if _{0} = 1 and a more
complicated expression for other values
of _{0}
(White & Rees 1978;
Kaiser 1986).
For the physical range -3 < *n* < 1, slower
evolution of *M** is obtained for larger values of *n*.

Markevitch *et
al.* (1991)
normalized the properties of
a population of clusters using present-day observed density,
temperature, and structure based on X-ray data, and integrated over
this population as it evolved to calculate the mean Comptonization
parameter that would result. The important parameters of this
calculation are *n*, _{0}, and *z*_{max}, the maximum
redshift for which clusters can be said to follow the evolution model
(128). Using the most recent limits on the
Comptonization parameter from the analysis of the COBE FIRAS data
(Fixsen *et al.*
1996),
the numerical results obtained by
Markevitch *et al.* can be interpreted as implying that
*z*_{max}
10 for a non-evolving cluster population, and
that _{0} 0.1
if the cluster population evolves with
-1 *n* 1. Similar conclusions can be drawn
from the results given by
Cavaliere *et
al.* (1991).

The closeness of the COBE FIRAS limit to the Comptonization parameter
to the prediction from these models for the change of cluster
properties with redshift indicates the power of the FIRAS data in
constraining models for the evolution of clusters, and
perhaps the value of _{0}
(Markevitch *et
al.* 1991;
Wright *et al.*
1994).
It should now be possible to
take into account all the constraints on the population of clusters
containing dense atmospheres, including the controversial ``negative
evolution'' of the population of X-ray clusters
(Edge *et al.*
1990;
Gioia *et al.*
1990a),
to place strong restrictions on the range of acceptable models of cluster
evolution.