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

A knowledge of the variation of optical depth (or equivalently the
ionized fraction *x*_{e}) and of the peculiar velocities
associated with different scales are enough to calculate
*T* /
*T*(). Reionization
will erase the fluctuations generated at *z* ~ 1000, by making the
Universe optically thick at much lower redshifts, but in the process extra
fluctuations will be generated due to the motions of the new last
scatterers. Because reionization cad be used to erase the primordial
fluctuations in a model that would otherwise conflict with
experimental upper limits, it is important to be able to calculate the
secondary fluctuations so that the minimal level of anisotropy can be
estimated.

The dominant contribution to anisotropies generated during
reionization are Doppler shifts of the scatterers (Equation 23). The
first (analytic) considerations of the velocity-induced perturbations
in a reionized model were by
Sunyaev (1977,
1978),
Davis (1980), and
Silk (1982).
However Kaiser (1984)
pointed out that an important term
contributing to the fluctuations had been neglected in the previous
work. The correct expression for
*T* / *T*
could not be written quite as
straightforwardly as in Equation (2). The evolution of the radiation
fluctuations needs to be followed more accurately, and requires a
numerical solution of the collisional Boltzmann equation for the
photons (see Section 3.2).

Ostriker & Vishniac (1986) extended the discussion to second-order fluctuations (see Section 5.2); detailed calculations are presented by Vishniac (1987), Efstathiou & Bond (1987; also Efstathiou 1988) made further calculations of the effects of reionization, including details of the pattern of polarization and approximations valid for small angular scales. The important effects of reionization on anisotropies are at arc-minute scales, in particular, where primary anisotropies are expected to be erased, and approximate calculations can be used which are valid only for small angles and correspondingly for small wavelength perturbations. More recent calculations have been carried out for reionization in BDM models (e.g. Hu et al 1994), generic open universe models (Persi & Spergel 1993), CDM models (Hu et al 1994, Dodelson & Jubas 1994, Sugiyama et al 1993, Chiba et al 1993, Hu & Sugiyama 1994b, c), and for the case of decaying dark matter (Scott et al 1991).

An important epoch is the redshift at which the optical depth for
photons becomes unity. Since Thomson scattering is independent of
frequency, the evaluation of *z*_{*} (or equivalently
_{*}) is relatively simple:

(33) |

which, for an
_{0} = 1
universe with a constant ionized fraction, becomes

(34) |

where *Y*_{p} is the primordial fraction of the baryonic
mass in helium,
which is assumed to be all neutral. Hence, for a particular model of
the ionization history, the redshift for which
= 1 can be
calculated. For example, for constant ionization fraction
*z*_{*}
69(*h*/0.5)^{-2/3}
(_{B} /
0.1)^{-2/3} *x*_{e}^{-2/3}. For an
open universe *z*_{*}
_{0}^{1/3} very
approximately, so that the last scattering surface can be at
significantly lower redshift. If the universe is ionized as early as
*z* 100, then the
exact *z* of reionization is unimportant, since
then >> 1.

Reionization need not be complete, i.e. the optical depth may not
become very large back to the new scattering surface. Obviously if
0.1 say, then
reionization has negligible effect. This would be the
case for a standard CDM model fully ionized even up to *z* ~
20. However if
0.1
1, then a good
approximation to the effects of erasure of
the primary anisotropies (i.e. those generated at the standard
recombination scattering surface) is that
*T* / *T*
is reduced by the fraction of photons scattered at
*z* ~ 1000 rather than at the new scattering surface. If
(1 - *e*^{-i}) is the fraction of photons scattered
since the Universe was reionized at *z*_{i} (i.e. this is
the integral of the visibility function), then we simply have
(*T* /
*T*)_{obs}
(*T* /
*T*)_{prim}(*e*^{-i}). There will also be
secondary anisotropies generated on
the new last scattering surface, although these will generally be
smaller (see below). However, second-order fluctuations at much
smaller angular scales also need to be considered.

Early reionization is likely to have been inevitable for
*COBE*-normalized CDM
[Tegmark et al (1994),
Sasaki et al (1993), and
Fukugita & Kawasaki
(1994),
following the early studies by
Couchman (1985) and
Couchman & Rees
(1986)].
On scales of order the horizon at
*z*_{*} there will still be Sachs-Wolfe
fluctuations and the polarization would be expected to be higher.