**3.3. Shape of the fluctuation spectrum**

A common assumption to be found in the literature is that the amplitude of a fluctuation when it comes within the horizon varies as some power of the mass contained within the horizon,

(3.20) |

with the power-law index ,
and the amplitude (fixed by
*M*_{*}) as free parameters.

This hypothesis is made purely for simplicity rather than from any
compelling physical reason (a hypothesis of maximal ignorance).
Zel'dovich (1972)
has suggested an adiabatic spectrum with
= 0, i.e.
(
/
)_{H} =
*K* where *K* is some constant. The motivation
behind this choice of spectrum (often referred to as constant curvature
fluctuations) is that it presents no preferred scales. Zel'dovich
argues that the constant *K* might be related to the total entropy per
baryon, the entropy being generated by the damping of short wavelength
perturbations (though see
Barrow and Matzner, 1977).
If we choose
< 0, it implies that
the Universe is non-Friedmannian with divergent
curvature perturbations on large scales. If
> 0, it implies that
the Universe was non-Friedmannian at early times. In the latter case it is
likely that order-unity fluctuations on scales of the horizon would
lead to a prolific production of black holes with masses
*M*_{*}
(Carr, 1975;
Barrow and Carr, 1978)
(and a cosmological density in black holes
_{black hole}
>> 1) and would disrupt the standard picture of
nucleosynthesis. This is unattractive, but in order to avoid these
problems, some physical process is required which cuts off the
spectrum (3.20) such that
/
<< 1 for
*M*
*M*_{*}
(Press and Vishniac,
1980b).

Gott and Rees (1975)
suggested an initial spectrum of isothermal
perturbations obeying the Zel'dovich constant curvature hypothesis. If
we let *t*_{equ} be the epoch at which baryon and radiation
densities are equal, _{B}(*t*_{equ}) =
_{R}(*t*_{equ}), and
*M*_{equ} is
the total mass in baryons with the horizon at *t*_{equ},
then from Eq. (3.20),

(3.21) |

where *M*_{B}(*t*_{H}) is the total mass in
baryons within the horizon at time
*t*_{H}. Gott and Rees' constant curvature hypothesis then
yields
_{B} /
_{B}
*M*_{B}^{-1/3}.
As in the case of adiabatic perturbations, any spectrum of
the form (3.20) with
> 0 leads to
non-Friedmannian behaviour on
small scales. In the case of isothermal perturbations this occurs
when perturbations are matter-dominated on horizon scales at early
times. The evolution of non-linear isothermal perturbations, and their
effect on primordial nucleosynthesis has been considered in some
detail by
Hogan (1978).

Finally we note that the observed entropy per baryon ratio may have
an explanation in terms of baryon non-conserving and CP violating
processes acting at very early times when
*T*
10^{28} K. Several authors
(Weinberg, 1982;
Press and Vishniac,
1980b)
have made the important
point that if the entropy per baryon is fixed in terms of purely
microscopic parameters, then any initial isothermal perturbations
would be erased, and only adiabatic fluctuations could survive. Ways
out of this possible difficulty are discussed in
Section 9.