**2.2. Linear instability 1: isentropic fluctuations and Jeans
criterion**

We begin with some nomenclature from thermodynamics. **Isentropic**
means *S* = 0: the same
entropy everywhere. **Adiabatic** means
*dS*/*d* = 0: the
entropy of a given fluid element does not change.
The two concepts are distinct. It is common in cosmology to say "adiabatic"
when one means "isentropic." This usage is confusing and I shall adopt
instead the standard terminology from thermodynamics.

Isentropic fluctuations are the natural outcome of quantum fluctuations
during inflation followed by reheating: rapid particle interactions in
thermal equilibrium eliminate entropy gradients. If
*S* = 0,
the linearized fluid and gravitational field equations are

(2.11) |

Combining these gives a damped, driven acoustic wave equation for :

(2.12) |

Aside from the Hubble damping and gravitational source terms, this equation is identical to what one would get for linear acoustic waves in a static medium.

To eliminate the spatial Laplacian we Fourier transform the wave
equation. For one plane wave,
(** x**,
)
(

(2.13) |

where we have defined the comoving Jeans wavenumber,

(2.14) |

Neglecting Hubble damping (by setting *a* = 1), the time dependence
of the solution to eq. (2.13) would be
exp(- *i*
), yielding a dispersion
relation very similar to that
for high-frequency waves in a plasma, but with an important sign difference
because gravity is attractive:

(2.15) |

The plasma frequency is
_{p} =
(4 *n*_{e}
*e*^{2} / *m*_{e})^{1/2} while the
Jeans frequency is
_{J} =
*k*_{J} *c*_{s} =
(4 *G*
)^{1/2}. Whereas electromagnetic waves with
^{2} <
_{p}^{2}
do not propagate
(*k*^{2} < 0 implies they are evanescent, e.g., they
reflect off the Earth's ionosphere), gravitational modes with
*k* < *k*_{J} are *unstable*
(^{2} < 0), as
was first noted by
Jeans (1902).
In physical terms,
pressure forces cannot prevent gravitational collapse when the
sound-crossing time
/
*c*_{s} is longer than the gravitational dynamical time
(*G*)^{-1/2} for a perturbation of proper wavelength
=
2*a*/*k*.

Including the Hubble damping term slows the growth of the Jeans instability
from exponential to a power of time for
*k* << *k*_{J}. In general there
is one growing and one decaying solution for
(*k*,
); these are denoted
_{±}(*k*,
). For
*c*_{s}^{2} = 0 and an Einstein-de Sitter
(flat, matter-dominated) background with
*a*()
_{2},
_{+}
_{2} and
_{-}
^{-3}. For
*k* >> *k*_{J}, we
obtain acoustic oscillations. In a static universe the acoustic
amplitude for an adiabatic plane wave remains constant; in the expanding
case it damps in general. An important exception is oscillations in the
photon-baryon fluid in the radiation-dominated era; the amplitude of
these oscillations is constant. (Showing this requires generalizing
the fluid equations to a relativistic gas, a good exercise for the
student.) In any case, acoustic oscillations suppress the growth
relative to the long-wavelength limit.

It is interesting to write the linear wave equation in terms of
rather than using
^{2}
=
4 *Ga*^{2}
*a*^{-1}
for nonrelativistic matter (with
*c*_{s}^{2} << *c*^{2}):

(2.16) |

where we used the Friedmann equation (1.6); recall that
*K* = ( -
1)(*aH*)^{2} is the spatial curvature constant. In a
matter-dominated universe, differentiating the Friedmann equation gives
/*a* -
(1/2)^{2}
/ *a*^{2} = - (1/2)*K*, yielding

(2.17) |

When written in terms of the gravitational potential rather than the density, the wave equation loses its gravitational source term.

The solutions to eq. (2.17) depend on the time-dependence of the sound speed as well as on the background cosmology. To get a rough idea of the behavior, consider the evolution of the potential in an Einstein-de Sitter universe filled with an ideal gas. For a constant sound speed, the solutions are

(2.18) |

where *j*_{2} and *y*_{2} are the spherical
Bessel functions of the first
and second kinds of order 2. Although simple, this is not a realistic
solution even before recombination (in that case, the photons and baryons
behave as a single tightly-coupled relativistic gas, and relativistic
corrections to the fluid equations must be added), except insofar as
it illustrates the generic behavior of the two solutions: (damped)
oscillations for
*kc*_{s}
>> 1 and power-law behavior for
*kc*_{s}
<< 1.

An alternative approximation, valid after recombination, is to assume
that the baryon temperature roughly equals the photon temperature (this
is a reasonable approximation because the small residual ionization
thermally couples the two fluids for a long time even though there is
negligible momentum transfer),
*c*_{s}^{2} = *c*_{0s}^{2}
*a*^{-1} where
*c*_{0s} is a constant. In this case the solutions are
powers of :

(2.19) |

The solutions oscillate for
*kc*_{s}_{0} > 5/2 and they damp for
*kc*_{s}_{0} > 0.

In both of our solutions, and indeed for any reasonable equation of
state in an Einstein-de Sitter universe, long-wavelength
(*kc*_{s}
<< 1) growing density modes have corresponding potential
_{+} = *constant*,
while the decaying density modes have
_
*a*^{-3}
*d*. The
density perturbation and potential differ by a factor of
*a*^{2}
*a*^{-1} from the Poisson equation. If *K* < 0 or
*k*^{2}*c*_{s}^{2} > 0,
then _{+} decays
with time, although
_{+} still
grows. Note that the important physical length scale where the
**transfer function**
_{+}(*k*,
) /
(*k*, 0)
falls significantly below unity is the
*acoustic* comoving horizon distance
*c*_{s}, not
the causal horizon distance
*c* or the Hubble
distance *c*/*H*. Setting
*c*_{s} to the acoustic speed of the coupled photon-baryon
fluid at matter-radiation equality gives the physical scale at which the
bend occurs, *c*_{s}
_{eq}, in the power
spectrum of the standard cold dark matter and other models.