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

**5.2. Second-Order Anisotropies**

The idea that reionization would erase the primary anisotropies, while
generating smaller secondary Doppler anisotropies (since the new
scattering surface was so thick), was overturned by the realization
that the Vishniac effect played an important role at small angular
scales. The first-order Doppler effects mainly cancel when the
redshifts and blueshifts are integrated through the thick last
scattering surface. However there is a second-order term in the
Boltzmann equation (23) coming from the
*v* cross term in
_{T}
_{e}
(1 + )*v*. In
solutions to the Boltzmann equation at small scales, this term
is a convolution over velocity and density perturbations, which does
not suffer from the cancellation effect. This leads to sonic models
giving *larger* secondary anisotropies than the primary anisotropies
which the reionization was invoked to erase.

We can obtain a scaling relation for the Doppler fluctuations from
the new last scattering surface as follows. Notice that the last
scattering shell is at *n*_{*} with width
~ _{*}. The number of independent
regions of scale
lying across the thickness of the shell is *N* ~
_{*} /
. The optical depth
through each region
of comoving scale is
_{}
~ 1/*N* ~ /
_{*}. The fluctuations on scale
(
) are
about *N*^{1/2} times the
fluctuation due to a single lump. The fluctuation for each lump is
second-order because of the approximate cancellation of redshifts and
blueshifts through a lump, leading to an effect
~ _{}^{2}*v*
(see Kaiser 1984).
From the continuity equation, the peculiar velocity is
*v* ~
_{k}(_{*}) /
_{*} and overdensities evolve as
^{2}.
Therefore an order-of
magnitude estimate for the temperature fluctuations (i.e.
*k*^{2} ,
see also Kaiser 1984,
Vishniac 1987) is

(35) |

Hence earlier reionization generally leads to larger large-scale
(first-order Doppler) fluctuations from the last scattering surface.
Similarly for the second-order (e.g. Vishniac) fluctuations, the
effect for each lump is
~ _{}
*v*, giving
(*T* /
*T*)_{}^{2}
~ *k*^{2}*P*^{2}(*k*)(1 +
*z*_{*})^{-5/2}. Again, this
argument is crude, but shows that second-order
small-scale fluctuations from the secondary last scattering surface
would be expected to be *smaller* for earlier reionization. There are
several experimental limits at scales ~ 1', which provide good
constraints on these secondary fluctuations rather than the primary
ones (e.g.
Uson & Wilkinson
1984a,
b,
c;
Readhead et al 1989;
Fomalont et al 1993;
Subrahmanyan et al 1993).
The strong k dependence of the
Vishniac effect means that the fluctuations will be very spiky, so
that a double- or triple-beam experiment will be dominated by the
zero-lag autocorrelation. Note, however, that nonlinear effects may
also be important, e.g. if the universe is very clumpy, as may happen
in BDM or other models with a large amount of small-scale power
(Hu et al 1994),
or if most of the baryons are trapped in galaxies at an
early epoch (R. Juszkiewicz & P.J.E. Peebles, private communication).

In general, there are many second-order terms to consider once a
fully second-order Boltzmann equation is derived
(Hu et al 1994,
Dodelson & Jubas
1994).
For most of these terms, there will be a
cancellation of redshifts and blueshifts
(Kaiser 1984)
through the
last scattering surface, so that the major contribution to the
fluctuations comes from modes with *k*-vector almost perpendicular to
the lines of sight. The only significant term that remains is the
Vishniac term, which is a convolution of *v* and
, coupling
large-scale
velocity perturbations with small-scale density perturbations. In a
model with significant reionization the primordial anisotropies are
erased, a new Doppler peak is generated at smaller
and of lower
amplitude, and a Vishniac bump is generated at arc-minute scales. For
CDM-type models, the Vishniac effect is negligible, but for models
with a large amount of small-scale power and/or open models (e.g. BDM
models, which also *require* reionization), the constraints can be
restrictive. Generally the ionization history,
*x*_{e}(*z*) can be varied in
such models, since there is no clear picture for the process of early
reheating. The Vishniac anisotropies tend to become smaller for an
earlier last scattering surface, while the first-order amplitude
becomes bigger; so there are limits to how much the constraints can be
avoided by invoking a different ionization history (see
Hu & Sugiyama
1994b,
c).

Another kind of second-order anisotropy is that coming from
nonlinear effects in the growth of perturbations, which causes the
potential to change with time even in an
_{0} = 1
universe. This
change in potential in the light-crossing time across a fluctuation
(Zel'dovich 1965,
Rees & Sciama 1968,
Dyer 1976)
is known as the
Rees-Sciama of integrated Sachs-Wolfe effect. It is small in almost
all models (e.g.
Argüeso &
Martínez-González 1989;
Martínez-González
et al 1990,
1992,
1993,
1994;
Anninos et al 1991),
although large voids
(Nottale 1984,
Scaramella et al 1989,
Thompson & Vishniac
1987,
van Kampen &
Martínez-González 1991,
Arnau et al 1993),
or structures such as the Great Attractor
(Bertschinger et al 1990,
Martínez-González & Sanz 1990,
Hnatyk et al 1992,
Goicoechea &
Martin-Mirones 1992,
Saez et al 1993)
or the Great Wall
(Atrio-Barandela &
Kashlinsky 1992,
Chodorowski 1992)
etc might give observable signatures.