**2.3. Linear instability 2: entropy fluctuations and isocurvature
mode**

Entropy gradients act as a source term for density perturbation growth.
Using eq. (2.10) and repeating the derivation of the linear
acoustic equation, we obtain (for
*c*_{s}^{2} << *c*^{2})

(2.20) |

For *adiabatic* evolution,
= 0, so what counts
is the initial
entropy gradient. Entropy gradients may be produced in the early universe
by first-order phase transitions resulting in spatial variations in
the photon/baryon ratio or other abundance ratios. If there were no
entropy gradients present before such a phase transition, then the
entropy variations can only have been produced by nonadiabatic
processes. (This may explain the "adiabatic vs. isocurvature"
nomenclature used by some cosmologists.) In practice, these entropy
fluctuations are taken as initial conditions for subsequent adiabatic
evolution.

Equation (2.20) is not applicable to the early universe because it assumes the matter is a one-component nonrelativistic gas. However, the behavior of its solutions are qualitatively similar to those for a relativistic multi-component gas and so its analysis is instructive.

The **isocurvature mode** is given by the particular solution of
density perturbation growth having
=
= 0 but
^{2}*S*
0 at some early initial
time _{i}. The
initial conditions may
be regarded as a perturbation in the equation of state in an otherwise
unperturbed Robertson-Walker (constant spatial curvature) spacetime,
accounting for the name "isocurvature." Variations in entropy at
constant density correspond to variations in pressure, which lead through
adiabatic expansion to changes in the density. Therefore, initial entropy
fluctuations seed density fluctuations.

The solution to eq. (2.20) is obtained easily in Fourier
space using the source-free (isentropic) solutions
_{±}(*k*,
):

(2.21) |

where primes are used to indicate that the variables are evaluated
at =
'. We see that both growing
and decaying density perturbations are induced. After the source
(*aT*_)
becomes small, the density fluctuations evolve the same way as isentropic
fluctuations - e.g., they oscillate as acoustic waves if
*kc*_{s}
>> 1.
To reinforce the point about nomenclature made earlier, I note that
in our approximation, *both* isocurvature and "adiabatic" (i.e.,
isentropic) modes are *adiabatic* in the sense of thermodynamics:
= 0 after the initial
moment. For a realistic multi-component
gas the evolution is not truly adiabatic, but that is a complication
we shall not consider further. In the literature, modes are described
as being adiabatic or isocurvature depending only on whether the
initial density is perturbed with negligible initial entropy perturbation,
or vice versa.