11.3. Fluctuations in the CMBR

The Sunyaev-Zel'dovich effect not only causes changes in the integrated spectrum of the CMBR, but also induces fluctuations in its brightness which appear superimposed on the fluctuations arising from the formation of structure in the early Universe (Sec. 1.3). The angular scale of these new structures in the CMBR will depend on their origin, and on the large-scale structure of the Universe. Constraints on both the manner in which clusters evolve and ₀ have been obtained by the limits to the fluctuation power from arcminute-scale experiments.

The present review concentrates on the fluctuations induced in the CMBR by clusters and superclusters of galaxies, but a diffuse ionized intergalactic medium with density and velocity irregularities, such as those created as large-scale structure develops, will also produce significant CMBR fluctuations. The best-known of these is the Vishniac effect (Vishniac 1987), which is due to the kinematic Sunyaev-Zel'dovich effect of a perturbation in the electron density in the (re-ionized) diffuse intergalactic medium. Discussions of this, and other, structures that are superimposed on the primordial spectrum by inhomogeneities in the re-ionized intergalactic medium are given by Dodelson & Jubas (1995) for ₀ = 1, by Persi (1995) for open Universes, and in the review of White et al. (1994).

Cluster Sunyaev-Zel'dovich effects can have a strong influence on the CMBR because, unlike ``normal'' astrophysical sources, the surface brightness of the Sunyaev-Zel'dovich effect from a cluster is independent of redshift, and does not suffer (1 + z)^-4 fading. This is because the effect is a fractional change in the brightness of the CMBR, and the CMBR's energy density itself increases with redshift as (1 + z)⁴, cancelling out the dimming effect of cosmology. The integrated flux density of a cluster at observed frequency ,

(129)

in the Kompaneets approximation, where j (x) is the Kompaneets spectral function (defined by n = y j (x) in equation 59), x = h / k_B T_rad is the usual dimensionless frequency, and the first integral is over the solid angle of the cluster. Equation (129) can be written as an integral over the cluster volume

(130)

which for a constant electron temperature over the cluster can be written simply in terms of the total number of electrons in the cluster, N_e, and the angular diameter and luminosity distances, D_A and D_L, as

(131)

This indicates that the cluster's apparent luminosity increases as (1 + z)⁴ - or, alternatively, that its flux density is a function of intrinsic properties and angular diameter distance only.

As a result, a population of clusters with the same N_e and T_e, observed at different redshifts, will exhibit a minimum flux density at the redshift of maximum angular diameter distance in that cosmology (Korolyov et al. 1986). Although this might provide a cosmological test for ₀, in practice clusters exhibit a wide range of properties and change significantly with redshift so it might be difficult to distinguish the effects of cosmology, cluster populations, and cluster evolution. The realizable cluster source counts (the histogram of sky brightnesses observed by a particular telescope of given properties) will then depend on a complicated mix of observational characteristics of the telescope used, the cosmological parameters, and the evolution of the cluster atmospheres. Nevertheless, Markevitch et al. (1994) suggest that a study of source counts at the µJy level (at cm-wavelengths) or at the mJy level (at mm-wavelengths) can constrain the spectrum of cluster masses (which determines the value of y), the cosmological parameter ₀, and the redshift of cluster formation. Using the results of arcminute-scale measurements of the anisotropy in the CMBR (the OVRO RING experiment; Myers 1990, Myers et al. 1993), Markevitch et al. were able to rule out slowly-evolving (n = 1 in eq. 128) models in an open Universe with ₀ < 0.3.

More detailed treatments of the effects of foreground clusters on the CMBR express their results in the formalism of Sec. 1.3 that is used to describe primordial fluctuations. A number of different assumptions about the cosmology and evolution of large scale structure have been used to calculate the amplitude and angular pattern of the foreground fluctuations (Rephaeli 1981; Cavaliere et al. 1986; Cole & Kaiser 1988; Schaeffer & Silk 1988; Thomas & Carlberg 1989; Markevitch et al. 1992; Makino & Suto 1993; Bartlett & Silk 1994a, 1994b; Ceballos & Barcons 1994; Colafrancesco et al. 1994; see also the review by Rephaeli 1995b). A uniform result of the calculations is that the distribution of sky brightness fluctuations that result is strongly non-Gaussian and asymmetrical since it is composed of negative or positive sources (depending on the frequency of observation, and the sign of j (x)) with varying numbers of sources on any line of sight or contained in a particular telescope beam (e.g., Markevitch et al. 1992). However, the amplitude and angular scale of the cluster-generated fluctuations depend strongly on the pattern of cluster evolution and the cosmology assumed. If the negative evolution of cluster atmospheres is strong (negative n in the self-similar model used by Markevitch et al.; eq. 128), then the distribution will be dominated by low-redshift clusters and the value of ₀ will not be important. For slow evolution or no evolution, the value of ₀ becomes important, since the variation of D_A with redshift dictates the appearance of the microwave background sky.

The angular pattern of fluctuations that results generally shows significant power in the two-point correlation function (eq. 8) at the level 10^-6 < T / T > 10^-5 on sub-degree scales (e.g., Colafrancesco et al. 1994), but some models for the evolution of clusters (and cluster atmospheres) can be ruled out from the absence of large anisotropies in the OVRO data of Readhead et al. (1989), Myers et al. (1993), or other experiments, and some cosmological parameters can be excluded under particular models for the evolution of cluster atmospheres. Since different models can make quite different predictions for the angular pattern and the amplitude of fluctuations, there is a potential for studying the processes that lead to the accumulation of cluster atmospheres through a study of the microwave background radiation on the range of angular scales (arcminute to degree) on which the cluster signal should be significant.

If the evolution of clusters is to be studied in this way, then observations of the cluster-induced Sunyaev-Zel'dovich fluctuation pattern would need to be made over a wide range of angular scales in order to validate or falsify any one of the models unambiguously. This range of angular scales overlaps that occupied by the stronger ``Doppler peaks'' in the primordial spectrum of fluctuations, so that the cluster signal may be hard to detect (see, e.g., the review of Bond 1995). The cluster signal is also an important contaminant of the Doppler peaks, which are expected to be a useful cosmological indicator and whose characterization is an important aim of the coming generation of CMBR satellites (MAP and Planck). Fortunately, measurements of the anisotropy pattern at several frequencies can be used to separate Sunyaev-Zel'dovich effects imposed by clusters and the primordial fluctuation background (Rephaeli 1981): the sensitivity required to achieve clean separations is formidable, but achievable with the current baseline design of the satellites' detectors.

Figure 27. The zero-frequency power spectrum of primordial microwave background anisotropies (solid line; calculated using the CMBFAST code of Zaldarriaga et al. 1998), the thermal Sunyaev-Zel'dovich effect (dotted line), the kinematic Sunyaev-Zel'dovich effect (short dashed line), and the Rees-Sciama effect from moving clusters (long dashed line) predicted in the -CDM cosmology discussed by Bahcall & Fan (1998). Figure from Molnar (1998).

An illustration of these results is given in Fig. 27, which shows the relative strengths of the power spectra of primordial fluctuations, the thermal and kinematic Sunyaev-Zel'dovich effects, and the moving-cluster Rees-Sciama effect in a -CDM cosmology (involving a significant cosmological constant and cold dark matter) with an evolving cluster population (Molnar 1998). Although the details of the power spectra depend on the choice of cosmology and the physics of cluster evolution (compare, e.g., Aghanim et al. 1998), the general features are similar in all cases. For l 3000, the power spectrum is dominated by the signal from primordial structures. The kinematic Sunyaev-Zel'dovich and Rees-Sciama effects from the cluster population are a factor 10² less important than the thermal Sunyaev-Zel'dovich effect. Thus the evident detectability of the thermal Sunyaev-Zel'dovich effect (Sec. 9) is principally due to its strongly non-gaussian nature and its association with clusters known from optical or X-ray observations, and not to its intrinsic power. Future work, for example the all-sky surveys that MAP and Planck will perform, will have the spectral discrimination to detect the thermal Sunyaev-Zel'dovich effect on a statistical basis, and should measure the power spectra of the thermal Sunyaev-Zel'dovich effect on small angular scales (l 300; Aghanim et al. 1997; Molnar 1998).

Figure 28. The zero-frequency power spectrum of primordial microwave background anisotropies (solid lines; calculated using the CMBFAST code of Zaldarriaga et al. 1998), and the Sunyaev-Zel'dovich effects of an evolving population of clusters (dotted lines) predicted in three cosmological models consistent with the COBE anisotropies: the open CDM, -CDM, and ``standard'' CDM models discussed by Bahcall & Fan (1998). Figure from Molnar (1998).

The sensitivity of the Sunyaev-Zel'dovich effect power spectrum to cosmology is illustrated in Fig. 28 (Molnar 1998). Variations of a factor > 10 in the power of fluctuations induced by the thermal Sunyaev-Zel'dovich effect are evident at l > 300: although this might be used as a cosmological test, the locations and strengths of the Doppler peaks in the primordial anisotropy power spectrum are more powerful. However, the amplitude of the Sunyaev-Zel'dovich effect power spectrum depends on how clusters evolve, and measurements of this power spectrum over a wide range of l should provide an important test of models of the formation of structure in the Universe.

Superclusters of galaxies, and the gas pancakes from which superclusters may have formed, are expected to make only a minor contributions to the fluctuation spectrum (Rephaeli 1993; SubbaRao et al. 1994). Once again, the angular scales on which the supercluster signals appear are similar to those of the Doppler peaks, and both good frequency and angular coverage will be needed to distinguish the primordial and foreground signals.