Annu. Rev. Astron. Astrophys. 2002. 40: 171-216
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Once the acoustic peaks in the temperature and polarization power spectra have been scaled, the days of splendid isolation of cosmic microwave background theory, analysis and experiment will have ended. Beyond and beneath the peaks lies a wealth of information about the evolution of structure in the Universe and its origin in the early universe. As CMB photons traverse the large scale structure of the Universe on their journey from the recombination epoch, they pick up secondary temperature and polarization anisotropies. These depend on the intervening dark matter, dark energy, baryonic gas density and temperature distributions, and even the existence of primordial gravity waves, so the potential payoff of their detection is enormous. The price for this extended reach is the loss of the ability both to make precise predictions, due to uncertain and/or non-linear physics, and to make precise measurements, due to the cosmic variance of the primary anisotropies and the relatively greater importance of galactic and extragalactic foregrounds.

We begin in Section 4.1 with a discussion of the matter power spectrum to set the framework for the discussion of secondary anisotropies. Secondaries can be divided into two classes: those due to gravitational effects and those induced by scattering off of electrons. The former are treated in Section 4.2 and the latter in Section 4.3. Secondary anisotropies are often non-Gaussian, so they show up not only in the power spectra of Section 2, but in higher point functions as well. We briefly discuss non-Gaussian statistics in Section 4.4. All of these topics are subjects of current research to which this review can only serve as introduction.

4.1. Matter Power Spectrum

The same balance between pressure and gravity that is responsible for acoustic oscillations determines the power spectrum of fluctuations in the non-relativistic matter. This relationship is often obscured by focussing on the density fluctuations in the pressureless cold dark matter itself and we so we will instead consider the matter power spectrum from the perspective of the Newtonian potential.

4.1.1 PHYSICAL DESCRIPTION     After recombination, without the pressure of the photons, the baryons simply fall into the Newtonian potential wells with the cold dark matter, an event usually referred to as the end of the Compton drag epoch. We claimed in Section 3.5 that above the horizon at matter-radiation equality the potentials are nearly constant. This follows from the dynamics: where pressure gradients are negligible, infall into some initial potential causes a potential flow of vtot ~ (k eta) Psii [see Equation (19)] and causes density enhancements by continuity of deltatot ~ - (k eta) vtot ~ - (k eta)2 Psii. The Poisson equation says that the potential at this later time Psi ~ - (k eta)-2 deltatot ~ Psii so that this rate of growth is exactly right to keep the potential constant. Formally, this Newtonian argument only applies in general relativity for a particular choice of coordinates [Bardeen, 1980], but the rule of thumb is that if what is driving the expansion (including spatial curvature) can also cluster unimpeded by pressure, the gravitational potential will remain constant.

Because the potential is constant in the matter dominated epoch, the large-scale observations of COBE set the overall amplitude of the potential power spectrum today. Translated into density, this is the well-known COBE normalization. It is usually expressed in terms of deltaH, the matter density perturbation at the Hubble scale today. Since the observed temperature fluctuation is approximately Psi / 3,

Equation 25 (25)

where the second equality follows from the Poisson equation in a fully matter dominated universe with Omegam = 1. The observed COBE fluctuation of DeltaT approx 28µK [Smoot et al, 1992] implies deltaH approx 2 × 10-5. For corrections for Omegam < 1 where the potential decays because the dominant driver of the expansion cannot cluster, see [Bunn & White, 1997].

On scales below the horizon at matter-radiation equality, we have seen in Section 3.5 that pressure gradients from the acoustic oscillations themselves impede the clustering of the dominant component, i.e. the photons, and lead to decay in the potential. Dark matter density perturbations remain but grow only logarithmically from their value at horizon crossing, which (just as for large scales) is approximately the initial potential, deltam approx - Psii. The potential for modes that have entered the horizon already will therefore be suppressed by Psi propto - deltam / k2 ~ Psii / k2 at matter domination (neglecting the logarithmic growth) again according to the Poisson equation. The ratio of Psi at late times to its initial value is called the transfer function. On large scales, then, the transfer function is close to one, while it falls off as k-2 on small scales. If the baryons fraction rhob / rhom is substantial, baryons alter the transfer function in two ways. First their inability to cluster below the sound horizon causes further decay in the potential between matter-radiation equality and the end of the Compton drag epoch. Secondly the acoustic oscillations in the baryonic velocity field kinematically cause acoustic wiggles in the transfer function [Hu & Sugiyama, 1996]. These wiggles in the matter power spectrum are related to the acoustic peaks in the CMB spectrum like twins separated at birth and are actively being pursued by the largest galaxy surveys [Percival et al, 2001]. For fitting formulae for the transfer function that include these effects see [Eisenstein & Hu, 1998].

4.1.2 COSMOLOGICAL IMPLICATIONS     The combination of the COBE normalization, the matter transfer function and the near scale-invariant initial spectrum of fluctuations tells us that by the present fluctuations in the cold dark matter or baryon density fields will have gone non-linear for all scales k gtapprox 10-1 hMpc-1. It is a great triumph of the standard cosmological paradigm that there is just enough growth between z* approx 103 and z = 0 to explain structures in the Universe across a wide range of scales.

In particular, since this non-linear scale also corresponds to galaxy clusters and measurements of their abundance yields a robust measure of the power near this scale for a given matter density Omegam. The agreement between the COBE normalization and the cluster abundance at low Omegam ~ 0.3 - 0.4 and the observed Hubble constant h = 0.72 ± 0.08 [Freedman et al, 2001] was pointed out immediately following the COBE result (e.g. [White et al, 1993, Bartlett & Silk, 1993]) and is one of the strongest pieces of evidence for the parameters in the working cosmological model [Ostriker & Steinhardt, 1995, Krauss & Turner, 1995].

More generally, the comparison between large-scale structure and the CMB is important in that it breaks degeneracies between effects due to deviations from power law initial conditions and the dynamics of the matter and energy contents of the Universe. Any dynamical effect that reduces the amplitude of the matter power spectrum corresponds to a decay in the Newtonian potential that boosts the level of anisotropy (see Section 3.5 and Section 4.2.1). Massive neutrinos are a good example of physics that drives the matter power spectrum down and the CMB spectrum up.

The combination is even more fruitful in the relationship between the acoustic peaks and the baryon wiggles in the matter power spectrum. Our knowledge of the physical distance between adjacent wiggles provides the ultimate standard candle for cosmology [Eisenstein et al, 1998]. For example, at very low z, the radial distance out to a galaxy is cz / H0. The unit of distance is therefore h-1 Mpc, and a knowledge of the true physical distance corresponds to a determination of h. At higher redshifts, the radial distance depends sensitively on the background cosmology (especially the dark energy), so a future measurement of baryonic wiggles at z ~ 1 say would be a powerful test of dark energy models. To a lesser extent, the shape of the transfer function, which mainly depends on the matter-radiation scale in h Mpc-1, i.e. Omegam h, is another standard ruler (see e.g. [Tegmark et al, 2001] for a recent assessment), more heralded than the wiggles, but less robust due to degeneracy with other cosmological parameters.

For scales corresponding to k gtapprox 10-1 h Mpc-1, density fluctuations are non-linear by the present. Numerical N-body simulations show that the dark matter is bound up in a hierarchy of virialized structures or halos (see [Bertschinger, 1998] for a review). The statistical properties of the dark matter and the dark matter halos have been extensively studied in the working cosmological model. Less certain are the properties of the baryonic gas. We shall see that both enter into the consideration of secondary CMB anisotropies.

4.2. Gravitational Secondaries

Gravitational secondaries arise from two sources: the differential redshift from time-variable metric perturbations [Sachs & Wolfe, 1967] and gravitational lensing. There are many examples of the former, one of which we have already encountered in Section 3.8 in the context of potential decay in the radiation dominated era. Such gravitational potential effects are usually called the integrated Sachs-Wolfe (ISW) effect in linear perturbation theory (Section 4.2.1), the Rees-Sciama (Section 4.2.2) effect in the non-linear regime, and the gravitational wave effect for tensor perturbations (Section 4.2.3). Gravitational waves and lensing also produce B-modes in the polarization (see Section 3.7) by which they may be distinguished from acoustic polarization.

4.2.1 ISW EFFECT     As we have seen in the previous section, the potential on a given scale decays whenever the expansion is dominated by a component whose effective density is smooth on that scale. This occurs at late times in an Omegam < 1 model at the end of matter domination and the onset dark energy (or spatial curvature) domination. If the potential decays between the time a photon falls into a potential well and when it climbs out it gets a boost in temperature of delta Psi due to the differential gravitational redshift and - delta Phi approx delta Psi due to an accompanying contraction of the wavelength (see Section 3.3).

Potential decay due to residual radiation was introduced in Section 3.8, but that due to dark energy or curvature at late times induces much different changes in the anisotropy spectrum. What makes the dark energy or curvature contributions different from those due to radiation is the longer length of time over which the potentials decay, on order the Hubble time today. Residual radiation produces its effect quickly, so the distance over which photons feel the effect is much smaller than the wavelength of the potential fluctuation. Recall that this meant that jl(kD) in the integral in Equation (23) could be set to jl(kD*) and removed from the integral. The final effect then is proportional to jl(kD*) and adds in phase with the monopole.

The ISW projection, indeed the projection of all secondaries, is much different (see Plate 3). Since the duration of the potential change is much longer, photons typically travel through many peaks and troughs of the perturbation. This cancellation implies that many modes have virtually no impact on the photon temperature. The only modes which do have an impact are those with wavevectors perpendicular to the line of sight, so that along the line of sight the photon does not pass through crests and troughs. What fraction of the modes contribute to the effect then? For a given wavenumber k and line of sight instead of the full spherical shell at radius 4pi k2 dk, only the ring 2pikdk with k perp n participate. Thus, the anisotropy induced is suppressed by a factor of k (or ell = kD in angular space). Mathematically, this arises in the line-of-sight integral of Equation (23) from the integral over the oscillatory Bessel function integ dxj ell(x) approx (pi / 2ell)1/2 (see also Plate 3).

The ISW effect thus generically shows up only at the lowest ell's in the power spectrum ([Kofman & Starobinskii, 1985]). This spectrum is shown in Plate 5. Secondary anisotropy predictions in this figure are for a model with Omegatot = 1, OmegaLambda = 2/3, Omegab h2 = 0.02, Omegam h2 = 0.16, n = 1, zri = 7 and inflationary energy scale Ei << 1016 GeV. The ISW effect is especially important in that it is extremely sensitive to the dark energy: its amount, equation of state and clustering properties [Coble et al, 1997, Caldwell et al, 1998, Hu, 1998]. Unfortunately, being confined to the low multipoles, the ISW effect suffers severely from the cosmic variance in Equation (4) in its detectability. Perhaps more promising is its correlation with other tracers of the gravitational potential (e.g. X-ray background [Boughn et al, 1998] and gravitational lensing, see Section 4.2.4).

Plate 5

Plate 5. Secondary anisotropies. (a) Gravitational secondaries: ISW, lensing and Rees-Sciama (moving halo) effects. (b) Scattering secondaries: Doppler, density (delta) and ionization (i) modulated Doppler, and the SZ effects. Curves and model are described in the text.

This type of cancellation behavior and corresponding suppression of small scale fluctuations is a common feature of secondary temperature and polarization anisotropies from large-scale structure and is quantified by the Limber equation [Limber, 1954] and its CMB generalization [Hu & White, 1996, Hu, 2000a]. It is the central reason why secondary anisotropies tend to be smaller than the primary ones from z* approx 103 despite the intervening growth of structure.

4.2.2. REES-SCIAMA AND MOVING HALO EFFECTS     The ISW effect is linear in the perturbations. Cancellation of the ISW effect on small scales leaves second order and non-linear analogues in its wake [Rees & Sciama, 1968]. From a single isolated structure, the potential along the line of sight can change not only from evolution in the density profile but more importantly from its bulk motion across the line of sight. In the context of clusters of galaxies, this is called the moving cluster effect [Birkinshaw & Gull, 1983]. More generally, the bulk motion of dark matter halos of all masses contribute to this effect [Tuluie & Laguna, 1995, Seljak, 1996b], and their clustering gives rise to a low level of anisotropies on a range of scales but is never the leading source of secondary anisotropies on any scale (see Plate 5a).

4.2.3. GRAVITATIONAL WAVES     A time-variable tensor metric perturbation similarly leaves an imprint in the temperature anisotropy [Sachs & Wolfe, 1967]. A tensor metric perturbation can be viewed as a standing gravitational wave and produces a quadrupolar distortion in the spatial metric. If its amplitude changes, it leaves a quadrupolar distortion in the CMB temperature distribution [Polnarev, 1985]. Inflation predicts a nearly scale-invariant spectrum of gravitational waves. Their amplitude depends strongly on the energy scale of inflation, 3 (power propto Ei4 [Rubakov et al, 1982, Fabbri & Pollock, 1983]) and its relationship to the curvature fluctuations discriminates between particular models for inflation. Detection of gravitational waves in the CMB therefore provides our best hope to study the particle physics of inflation.

Gravitational waves, like scalar fields, obey the Klein-Gordon equation in a flat universe and their amplitudes begin oscillating and decaying once the perturbation crosses the horizon. While this process occurs even before recombination, rapid Thomson scattering destroys any quadrupole anisotropy that develops (see Section 3.6). This fact dicates the general structure of the contributions to the power spectrum (see Figure 4, left panel): they are enhanced at ell = 2 the present quadrupole and sharply suppressed at multipole larger than that of the first peak [Abbott & Wise, 1984, Starobinskii, 1985, Crittenden et al, 1993]. As is the case for the ISW effect, confinement to the low multipoles means that the isolation of gravitational waves is severely limited by cosmic variance.

Figure 4

Figure 4. Gravitational waves and the energy scale of inflation Ei. Left: temperature and polarization spectra from an initial scale invariant gravitational wave spectrum with power propto Ei4 = (4 × 1016 GeV)4. Right: 95% confidence upper limits statistically achievable on Ei and the scalar tilt n by the MAP and Planck satellites as well as an ideal experiment out to ell = 3000 in the presence of gravitational lensing B-modes.

The signature of gravitational waves in the polarization is more distinct. Because gravitational waves cause a quadrupole temperature anisotropy at the end of recombination, they also generate a polarization. The quadrupole generated by a gravitational wave has its main angular variation transverse to the wavevector itself [Hu & White, 1997a]. The resulting polarization that results has components directed both along or orthogonal to the wavevector and at 45° degree angles to it. Gravitational waves therefore generate a nearly equal amount of E and B mode polarization when viewed at a distance that is much greater than a wavelength of the fluctuation [Kamionkowski et al, 1997, Zaldarriaga & Seljak, 1997]. The B-component presents a promising means of measuring the gravitational waves from inflation and hence the energy scale of inflation (see Figure 4, right panel). Models of inflation correspond to points in the n, Ei plane [Dodelson et al, 1997]. Therefore, the anticipated constraints will discriminate among different models of inflation, probing fundamental physics at scales well beyond those accessible in accelerators.

4.2.4. GRAVITATIONAL LENSING     The gravitational potentials of large-scale structure also lens the CMB photons. Since lensing conserves surface brightness, it only affects anisotropies and hence is second order in perturbation theory [Blanchard & Schneider, 1987]. The photons are deflected according to the angular gradient of the potential projected along the line of sight with a weighting of 2(D* - D) / (D* D). Again the cancellation of parallel modes implies that it is mainly the large-scale potentials that are responsible for deflections. Specifically, the angular gradient of the projected potential peaks at a multipole ell ~ 60 corresponding to scales of a k ~ few 10-2 Mpc-1 [Hu, 2000b]. The deflections are therefore coherent below the degree scale. The coherence of the deflection should not be confused with its rms value which in the model of Plate 1 has a value of a few arcminutes.

This large coherence and small amplitude ensures that linear theory in the potential is sufficient to describe the main effects of lensing. Since lensing is a one-to-one mapping of the source and image planes it simply distorts the images formed from the acoustic oscillations in accord with the deflection angle. This warping naturally also distorts the mapping of physical scales in the acoustic peaks to angular scales Section 3.8 and hence smooths features in the temperature and polarization [Seljak, 1996a]. The smoothing scale is the coherence scale of the deflection angle Delta ell approx 60 and is sufficiently wide to alter the acoustic peaks with Delta ell ~ 300. The contributions, shown in Plate 5a are therefore negative (dashed) on scales corresponding to the peaks.

For the polarization, the remapping not only smooths the acoustic power spectrum but actually generates B-mode polarization (see Plate 1 and [Zaldarriaga & Seljak, 1998]). Remapping by the lenses preserves the orientation of the polarization but warps its spatial distribution in a Gaussian random fashion and hence does not preserve the symmetry of the original E-mode. The B-modes from lensing sets a detection threshold for gravitational waves for a finite patch of sky [Hu, 2001b].

Gravitational lensing also generates a small amount of power in the anisotropies on its own but this is only noticable beyond the damping tail where diffusion has destroyed the primary anisotropies (see Plate 5). On these small scales, the anisotropy of the CMB is approximately a pure gradient on the sky and the inhomogeneous distribution of lenses introduces ripples in the gradient on the scale of the lenses [Seljak & Zaldarriaga, 2000]. In fact the moving halo effect of Section 4.2.2 can be described as the gravitational lensing of the dipole anisotropy due to the peculiar motion of the halo [Birkinshaw & Gull, 1983].

Because the lensed CMB distribution is not linear in the fluctuations, it is not completely described by changes in the power spectrum. Much of the recent work in the literature has been devoted to utilizing the non-Gaussianity to isolate lensing effects [Bernardeau, 1997, Bernardeau, 1998, Zaldarriaga & Seljak, 1999, Zaldarriaga, 2000] and their cross-correlation with the ISW effect [Goldberg & Spergel, 1999, Seljak & Zaldarriaga, 1999]. In particular, there is a quadratic combination of the anisotropy data that optimally reconstructs the projected dark matter potentials for use in this cross-correlation [Hu, 2001c]. The cross correlation is especially important in that in a flat universe it is a direct indication of dark energy and can be used to study the properties of the dark energy beyond a simple equation of state [Hu, 2001b].

4.3. Scattering Secondaries

From the observations both of the lack of of a Gunn-Peterson trough [Gunn & Peterson, 1965] in quasar spectra and its preliminary detection [Becker et al, 2001], we know that hydrogen was reionized at zri gtapprox 6. This is thought to occur through the ionizing radiation of the first generation of massive stars (see e.g. [Loeb & Barkana, 2001] for a review). The consequent recoupling of CMB photons to the baryons causes a few percent of them to be rescattered. Linearly, rescattering induces three changes to the photon distribution: suppression of primordial anisotropy, generation of large angle polarization, and a large angle Doppler effect. The latter two are suppressed on small scales by the cancellation highlighted in Section 4.2.1. Non-linear effects can counter this suppression; these are the subject of active research and are outlined in Section 4.3.4.

4.3.1. PEAK SUPPRESSION     Like scattering before recombination, scattering at late times suppresses anisotropies in the distribution that have already formed. Reionization therefore suppresses the amplitude of the acoustic peaks by the fraction of photons rescattered, approximately the optical depth ~ tauri (see Plate 5b, dotted line and negative, dashed line, contributions corresponding to |delta DeltaT2|1/2 between the zri = 7 and zri = 0 models). Unlike the plasma before recombination, the medium is optically thin and so the mean free path and diffusion length of the photons is of order the horizon itself. New acoustic oscillations cannot form. On scales approaching the horizon at reionization, inhomogeneities have yet to be converted into anisotropies (see Section 3.8) and so large angle fluctuations are not suppressed. While these effects are relatively large compared with the expected precision of future experiments, they mimic a change in the overall normalization of fluctuations except at the lowest, cosmic variance limited, multipoles.

4.3.2. LARGE-ANGLE POLARIZATION     The rescattered radiation becomes polarized since, as discussed in Section 3.8 temperature inhomogeneities, become anisotropies by projection, passing through quadrupole anisotropies when the perturbations are on the horizon scale at any given time. The result is a bump in the power spectrum of the E-polarization on angular scales corresponding to the horizon at reionization (see Plate 1). Because of the low optical depth of reionization and the finite range of scales that contribute to the quadrupole, the polarization contributions are on the order of tenths of µK on scales of ell ~ few. In a perfect, foreground free world, this is not beyond the reach of experiments and can be used to isolate the reionization epoch [Hogan et al, 1982, Zaldarriaga et al, 1997]. As in the ISW effect, cancellation of contributions along the line of sight guarantees a sharp suppression of contributions at higher multipoles in linear theory. Spatial modulation of the optical depth due to density and ionization (see Section 4.3.4) does produce higher order polarization but at an entirely negligible level in most models [Hu, 2000a].

4.3.3. DOPPLER EFFECT     Naively, velocity fields of order v ~ 10-3 (see e.g. [Strauss & Willick, 1995] for a review) and optical depths of a few percent would imply a Doppler effect that rivals the acoustic peaks themselves. That this is not the case is the joint consequence of the cancellation described in Section 4.2.1 and the fact that the acoustic peaks are not "Doppler peaks" (see Section 3.8). Since the Doppler effect comes from the peculiar velocity along the line of sight, it retains no contributions from linear modes with wavevectors perpendicular to the line of sight. But as we have seen, these are the only modes that survive cancellation (see Plate 3 and [Kaiser, 1984]). Consequently, the Doppler effect from reionization is strongly suppressed and is entirely negligible below ell ~ 102 unless the optical depth in the reionization epoch approaches unity (see Plate 5b).

4.3.4. MODULATED DOPPLER EFFECTS    The Doppler effect can survive cancellation if the optical depth has modulations in a direction orthogonal to the bulk velocity. This modulation can be the result of either density or ionization fluctuations in the gas. Examples of the former include the effect in clusters, and linear as well as non-linear large-scale structures.

CLUSTER MODULATION:     The strongly non-linear modulation provided by the presence of a galaxy cluster and its associated gas leads to the kinetic Sunyaev-Zel'dovich effect. Cluster optical depths on order 10-2 and peculiar velocities of 10-3 imply signals in the 10-5 regime in individual arcminute-scale clusters, which are of course rare objects. While this signal is reasonably large, it is generally dwarfed by the thermal Sunyaev-Zel'dovich effect (see Section 4.3.5) and has yet to be detected with high significance (see [Carlstrom et al, 2001] and references therein). The kinetic Sunyaev-Zel'dovich effect has negligible impact on the power spectrum of anisotropies due to the rarity of clusters and can be included as part of the more general density modulation.

LINEAR MODULATION:     At the opposite extreme, linear density fluctuations modulate the optical depth and give rise to a Doppler effect as pointed out by [Ostriker & Vishniac, 1986] and calculated by [Vishniac, 1987] (see also [Efstathiou & Bond, 1987]). The result is a signal at the µK level peaking at ell ~ few × 103 that increases roughly logarithmically with the reionization redshift (see Plate 5b).

GENERAL DENSITY MODULATION:     Both the cluster and linear modulations are limiting cases of the more general effect of density modulation by the large scale structure of the Universe. For the low reionization redshifts currently expected (zri approx 6 - 7) most of the effect comes neither from clusters nor the linear regime but intermediate scale dark matter halos. An upper limit to the total effect can be obtained by assuming the gas traces the dark matter [Hu, 2000a] and implies signals on the order of DeltaT ~ few µK at ell > 103 (see Plate 5b). Based on simulations, this assumption should hold in the outer profiles of halos [Pearce et al, 2001, Lewis et al, 2000] but gas pressure will tend to smooth out the distribution in the cores of halos and reduce small scale contributions. In the absence of substantial cooling and star formation, these net effects can be modeled under the assumption of hydrostatic equilibrium [Komatsu & Seljak, 2001] in the halos and included in a halo approach to the gas distribution [Cooray, 2001].

IONIZATION MODULATION:     Finally, optical depth modulation can also come from variations in the ionization fraction [Aghanim et al, 1996, Gruzinov & Hu, 1998, Knox et al, 1998]. Predictions for this effect are the most uncertain as it involves both the formation of the first ionizing objects and the subsequent radiative transfer of the ionizing radiation [Bruscoli et al, 2000, Benson et al, 2001]. It is however unlikely to dominate the density modulated effect except perhaps at very high multipoles ell ~ 104 (crudely estimated, following [Gruzinov & Hu, 1998], in Plate 5b).

4.3.5. SUNYAEV-ZEL'DOVICH EFFECT     Internal motion of the gas in dark matter halos also give rise to Doppler shifts in the CMB photons. As in the linear Doppler effect, shifts that are first order in the velocity are canceled as photons scatter off of electrons moving in different directions. At second order in the velocity, there is a residual effect. For clusters of galaxies where the temperature of the gas can reach Te ~ 10 keV, the thermal motions are a substantial fraction of the speed of light vrms = (3Te / me)1/2 ~ 0.2. The second order effect represents a net transfer of energy between the hot electron gas and the cooler CMB and leaves a spectral distortion in the CMB where photons on the Rayleigh-Jeans side are transferred to the Wien tail. This effect is called the thermal Sunyaev-Zel'dovich (SZ) effect [Sunyaev & Zel'dovich, 1972]. Because the net effect is of order taucluster Te / me propto ne Te, it is a probe of the gas pressure. Like all CMB effects, once imprinted, distortions relative to the redshifting background temperature remain unaffected by cosmological dimming, so one might hope to find clusters at high redshift using the SZ effect. However, the main effect comes from the most massive clusters because of the strong temperature weighting and these have formed only recently in the standard cosmological model.

Great strides have recently been made in observing the SZ effect in individual clusters, following pioneering attempts that spanned two decades [Birkinshaw, 1999]. The theoretical basis has remained largely unchanged save for small relativistic corrections as Te / me approches unity. Both developements are comprehensively reviewed in [Carlstrom et al, 2001]. Here we instead consider its implications as a source of secondary anisotropies.

The SZ effect from clusters provides the most substantial contribution to temperature anisotropies beyond the damping tail. On scales much larger than an arcminute where clusters are unresolved, contributions to the power spectrum appear as uncorrelated shot noise (Cell = const. or DeltaT propto ell). The additional contribution due to the spatial correlation of clusters turns out to be almost negligible in comparison due to the rarity of clusters [Komatsu & Kitayama, 1999]. Below this scale, contributions turn over as the clusters become resolved. Though there has been much recent progress in simulations [Refregier et al, 2000, Seljak et al, 2001, Springel et al, 2001] dynamic range still presents a serious limitation.

Much recent work has been devoted to semi-analytic modeling following the technique of [Cole & Kaiser, 1988], where the SZ correlations are described in terms of the pressure profiles of clusters, their abundance and their spatial correlations [now commonly referred to an application of the "halo model" see [Komatsu & Kitayama, 1999, Atrio-Barandela & Mücket, 1999, Cooray, 2001, Komatsu & Seljak, 2001]]. We show the predictions of a simplified version in Plate 5b, where the pressure profile is approximated by the dark matter haloprofile and the virial temperature of halo. While this treatment is comparatively crude, the inaccuracies that result are dwarfed by "missing physics" in both the simulations and more sophisticated modelling, e.g. the non-gravitational sources and sinks of energy that change the temperature and density profile of the cluster, often modeled as a uniform "preheating" of the intercluster medium [Holder & Carlstrom, 2001].

Although the SZ effect is expected to dominate the power spectrum of secondary anisotropies, it does not necessarily make the other secondaries unmeasurable or contaminate the acoustic peaks. Its distinct frequency signature can be used to isolate it from other secondaries (see e.g. [Cooray et al, 2000]). Additionally, it mainly comes from massive clusters which are intrinsically rare. Hence contributions to the power spectrum are non-Gaussian and concentrated in rare, spatially localized regions. Removal of regions identified as clusters through X-rays and optical surveys or ultimately high resolution CMB maps themselves can greatly reduce contributions at large angular scales where they are unresolved [Persi et al, 1995, Komatsu & Kitayama, 1999].

4.4. Non-Gaussianity

As we have seen, most of the secondary anisotropies are not linear in nature and hence produce non-Gaussian signatures. Non-Gaussianity in the lensing and SZ signals will be important for their isolation. The same is true for contaminants such as galactic and extragalactic foregrounds. Finally the lack of an initial non-Gaussianity in the fluctuations is a testable prediction of the simplest inflationary models [Guth & Pi, 1985, Bardeen et al, 1983]. Consequently, non-Gaussianity in the CMB is currently a very active field of research. The primary challenge in studies of non-Gaussianity is in choosing the statistic that quantifies it. Non-Gaussianity says what the distribution is not, not what it is. The secondary challenge is to optimize the statistic against the Gaussian "noise" of the primary anisotropies and instrumental or astrophysical systematics.

Early theoretical work on the bispectrum, the harmonic analogue of the three point function addressed its detectability in the presence of the cosmic variance of the Gaussian fluctuations [Luo, 1994] and showed that the inflationary contribution is not expected to be detectable in most models ([Allen et al, 1987, Falk et al, 1993]). The bispectrum is defined by a triplet of multipoles, or configuration, that defines a triangle in harmonic space. The large cosmic variance in an individual configuration is largely offset by the great number of possible triplets. Interest was spurred by reports of significant signals in specific bispectrum configurations in the COBE maps [Ferreira et al, 1998] that turned out to be due to systematic errors [Banday et al, 2000]. Recent investigations have focussed on the signatures of secondary anisotropies [Goldberg & Spergel, 1999, Cooray & Hu, 2000]. These turn out to be detectable with experiments that have both high resolution and angular dynamic range but require the measurement of a wide range of configurations of the bispectrum. Data analysis challenges for measuring the full bispectrum largely remain to be addressed (c.f. [Heavens, 1998, Spergel & Goldberg, 1999, Phillips & Kogut, 2001]).

The trispectrum, the harmonic analogue of the four point function, also has advantages for the study of secondary anisotropies. Its great number of configurations are specified by a quintuplet of multipoles that correspond to the sides and diagonal of a quadrilateral in harmonic space [Hu, 2001a]. The trispectrum is important in that it quantifies the covariance of the power spectrum across multipoles that is often very strong in non-linear effects, e.g. the SZ effect [Cooray, 2001]. It is also intimately related to the power spectra of quadratic combinations of the temperature field and has been applied to study gravitational lensing effects [Bernardeau, 1997, Zaldarriaga, 2000, Hu, 2001a].

The bispectrum and trispectrum quantify non-Gaussianity in harmonic space, and have clear applications for secondary anisotropies. Tests for non-Gaussianity localized in angular space include the Minkowski functionals (including the genus) [Winitzki & Kosowsky, 1997], the statistics of temperature extrema [Kogut et al, 1996], and wavelet coefficients [Aghanim & Forni, 1999]. These may be more useful for examining foreground contamination and trace amounts of topological defects.

3 Ei4 ident V(phi), the potential energy density associated with the scalar field(s) driving inflation. Back.

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