Annu. Rev. Astron. Astrophys. 1994. 32: 319-70 Copyright © 1994 by Annual Reviews. All rights reserved

1.1. Recombination

The photons we observe from the microwave background have traveled freely since the matter was highly ionized and they suffered their last Thomson scatterings. If there has been no significant early heat input from galaxy formation, then this happened when the Universe became cool enough for the protons to capture electrons (the recombination epoch). If the Universe was reionized early enough, then the photons will have been scattered more recently, the effects of which we discuss in Section 5.1. To understand the CMB fluctuations we observe, it is crucial to have a good picture of the recombination process.

The process of recombination would proceed via the Saha equation (see e.g. Lang 1980), except that recombinations to the ground state are inhibited by the recombination process itself (Novikov & Zel'dovich 1967). Thus recombination is controlled by the population of the first excited state, and the physical processes which either populate or depopulate it in the expanding Universe, This problem was first worked out in detail by Peebles (1968) and at about the same time by Zel'dovich et al (1968).

Any modification in our understanding of recombination would be crucial for microwave background anisotropies, but in fact little has changed since the seminal work of the late 1960s; the only significant improvement was made by Jones & Wyse (1985), who refined some of the earlier assumptions and included the possibility of non-baryonic matter. Matsuda, Sato & Takeda (1971) considered the effects of collisional processes, which are negligible, and Krolik (1989, 1990) showed that two previously unconsidered scattering effects in the Ly line almost completely cancel one another. Sasaki & Takahara (1993) showed that an accurate treatment of the stimulated rate lowers the ionization "freeze-out" but has no real effect at the recombination epoch.

Solving the coupled equations for the ionized fraction and matter temperature gives the evolution of the ionized fraction x_e(z) n_e / n_B and the visibility function g(z) e^- d / dz, for Thomson scattering optical depth . This function measures the probability that the radiation was last scattered in a redshift interval dz. It is reasonably well approximated by a Gaussian with mean z_rec 1100 and width z 80, largely independent of ₀, _B, and H₀, as shown in Table 1 (see also Scott 1991). Thus the epoch and thickness of the last scattering surface can be assumed to be independent of the cosmological model, although the amount of scattering will depend on _B h², the angular scales will depend on ₀, etc. Useful approximations to x_e(z) and g(z) are given by Sunyaev & Zeldovich (1970), Zabotin & Nasel'skii (1982), Jones & Wyse (1985), Grachev & Dubrovich (1991), and Fink (1993). Note that z_rec = 1100 corresponds to T_rec = 0.26 eV, and t_rec = 5.6 × 10¹²(₀ h²)^-1/2 sec. The thickness of the last scattering surface z = 80 at this epoch corresponds to a comoving scale of 6.6₀^-1/2 h^-1 Mpc and an angular scale of 3.'8₀^1/2.

Table 1. Parameters for the redshift of recombination for a range of cosmologies ^a

0	0.1					0.2			1
b	0.1		0.5		0.1	0.05		0.01	0.1		0.05		0.01
h	0.5	1	0.5	1	1	0.5	1	1	0.5	1	0.5	1	1
zrec	1060	1110	1070	1060	1100	1080	1060	1100	1080	1070	1100	1080	1170
z	81.8	85.1	85.8	82.5	98.4	89.5	84.4	106	91.5	85.5	104	92.1	135
a The location and width are obtained by fitting a Gaussian to exp(- ) d / dz.

It is also worth pointing out that in the expanding Universe the fractional ionization approaches a constant, which is significantly different from zero: x_e(residual) (₀ h²)^1/2(_B h²)^-1. For models with significant reionization, radiation drag may also be important (see e.g. Peebles 1965, Rees 1977, Hogan 1979, Peebles 1993). The fluctuations cannot grow until the photons release their hold on the matter which happens at 1 + z_drag 120(₀ h²)^1/5 x_e^-2/5.

There is a prediction that there must be broad lines in the CMB spectrum due to the photons produced during H (and He) recombination at z_rec. These distortions maybe large in the Wien region (Peebles 1968, Zel'dovich et al 1968, Lyubarski & Sunyaev 1983, Fahr & Loch 1991), but there are few photons out there, so this effect will be swamped by the background at ~ 100µm. In the Rayleigh-Jeans region, where there are more photons, the distortions are small (Dubrovich 1975, Bernstein et al 1977), although they may be enhanced if there is some extra energy injection during recombination (Lyubarski & Sunyaev 1983). If such lines could ever be detected they would be a direct probe of z and the physics of recombination. Recombination can also lead to trace amounts of the primordial molecules H₂, HD, LiH, etc (Lepp & Shull 1984, Puy et al 1993). Resonance scattering by intergalactic LiH molecules at z 400 may possibly result in smoothing of CMB fluctuations up to degree scales (Dubrovich 1993, Melchiorri 1993, Maoli et al 1993) at long wavelengths, as the resonance line is redshifted.