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 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 due to the differential gravitational redshift and due to an accompanying contraction of the wavelength (see §3.3).
Potential decay due to residual radiation was introduced in §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 in the integral in Equation (23) could be set to and removed from the integral. The final effect then is proportional to 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 and line of sight instead of the full spherical shell at radius , only the ring with participate. Thus, the anisotropy induced is suppressed by a factor of (or in angular space). Mathematically, this arises in the line-of-sight integral of Equation (23) from the integral over the oscillatory Bessel function (see also Plate 3).
The ISW effect thus generically shows up only at the lowest 's in the power spectrum ([Kofman & Starobinskii, 1985]). This spectrum is shown in Plate 5a. Secondary anisotropy predictions in this figure are for a model with , , , , , and inflationary energy scale 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 §4.2.4).
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 despite the intervening growth of structure.