11.2. Contributions to the CMBR spectrum

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

(125)

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

(126)

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

(127)

For re-ionization redshifts 30, and any ₀ < 1, the scattering optical depth is less than about 2.6 _IGM h₁₀₀, 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

(128)

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.