**4.3.2. Small scale anisotropy**

On small angular scales (<< *ct*_{0}, the horizon
scale at present), the
residual radiation temperature fluctuations at present reduce to an
approximate expression of the form

(4.19a) |

where the amplitude of the Fourier components of the temperature fluctuations,

(4.19b) |

the dimensionless wavenumber
*K* = *kt*_{d}*c* is normalized to the horizon
scale at decoupling, *µ* is the cosine of the angle between
wavenumber **k** and the observation direction
and
*K*
3(1 +
*Z*_{d})^{1/2}*K* = 1 defines a
comoving wave vector equal to the present horizon. The first term in
*T*_{rms}(**k**) yields the residual
temperature fluctuation at the epoch *t*_{d}
when decoupling is effective, and the remaining terms are associated
with secondary fluctuations induced by Thomson scattering in density
fluctuations due to the residual ionization remaining after
decoupling. The constants *C*_{1} and *C*_{2}
are associated with the two
adiabatic modes; the *C*_{3} term is due to the isothermal
mode; The *C*_{1} term is due to gravitational potential
fluctuations [~ (*l* / *ct*_{d})^{2}
/
]
and the *C*_{2} term is associated with peculiar motions
~ [(*l* / *ct*_{d})
/
]
on the last scattering surface. The former term dominates on large
angular scales, where the temperature fluctuations scale as
^{(1-n)/2} for
/
*M*^{-1/2-n/6}
(Sachs and Wolfe, 1967).

The angular correlation function for the temperature fluctuations is

(4.20) |

where cos =
_{1}
^{.}
_{2}.
This simplifies to

(4.21) |

where *x* =
|_{1} -
_{2}|
for
<< 1. Comparison
with actual observations
requires that the correlation function be smeared on small angular
scales by convolution with the antenna beam pattern. For example, a
Gaussian beam of half-width
leads to a smeared
correlation function

(4.22) |

where *I*_{0} is a modified Bessel function and
<< 1 has been
assumed. This reduces to
*C*() if
= 0. The root mean square
temperature fluctuation for an antenna beam that is switched over an
angle is

(4.23) |

Predicted fluctuations arc shown in Figure 4.3 for a beam with
= 3.6'
and a recent observational upper limit
(*T* / *T*
2 ×
10^{-4} =
3.6' = 9')
is also shown. We also show the far infrared experiment with a
beam of 5° and an angular resolution of 6°, for which
*T* / *T*
3 ×
10^{-5}.

The small-scale anisotropy experiment at 6°, while probably
an upper limit,
is still especially significant. It clearly rules out the *n* = 1
adiabatic model with
= 0.1 (although it
is just compatible if
= 1). It is also on
the verge of constraining an isothermal model with
= 1 and *n* =
0. Experiments on finer angular scales are not as
useful. While the 9' experiment also constrains the adiabatic model,
it is relatively insensitive to n because of the smearing on the
surface of last scattering over angular scales of less than a degree.

The approximate scaling with
is such that the
predicted values of
*T* / *T*
^{-1}, large
amplitudes of the initial density fluctuation
spectrum being required in a low
universe. Adiabatic
fluctuations are in conflict on small angular scales with the
observational upper limits only if
< 0.1. Lowering
the upper limit by a factor ~ 4 would
permit a test of primordial adiabatic fluctuation spectra with
*n* 3
and isothermal fluctuation spectra with
*n* 0 for
cosmological models with
1. This assumes
the standard model, in which the
intergalactic medium does not become reionized at an epoch
*z* 10, and
in which substantial amounts of non-baryonic matter are not present.

In a galaxy formation theory based on adiabatic fluctuations, the
first objects capable of ionizing the universe form late, at a redshift
10. Since
fluctuations are damped on scales smaller than
*M*_{D}, the objects must collapse at a relatively late
epoch in order to
produce the low density large-scale structure of the universe that is
observed. Hence reionization probably does not occur if only
adiabatic fluctuations are present initially.