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
Combining these gives a damped, driven acoustic wave equation for :
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, ) (k, )exp(ik . x). The wave equation becomes
where we have defined the comoving Jeans wavenumber,
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:
The plasma frequency is p = (4 ne e2 / me)1/2 while the Jeans frequency is J = kJ cs = (4 G )1/2. Whereas electromagnetic waves with 2 < p2 do not propagate (k2 < 0 implies they are evanescent, e.g., they reflect off the Earth's ionosphere), gravitational modes with k < kJ 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 / cs is longer than the gravitational dynamical time (G)-1/2 for a perturbation of proper wavelength = 2a/k.
Including the Hubble damping term slows the growth of the Jeans instability from exponential to a power of time for k << kJ. In general there is one growing and one decaying solution for (k, ); these are denoted ±(k, ). For cs2 = 0 and an Einstein-de Sitter (flat, matter-dominated) background with a() 2, + 2 and - -3. For k >> kJ, 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 Ga2 a-1 for nonrelativistic matter (with cs2 << c2):
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 / a2 = - (1/2)K, yielding
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
where j2 and y2 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 kcs >> 1 and power-law behavior for kcs << 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), cs2 = c0s2 a-1 where c0s is a constant. In this case the solutions are powers of :
The solutions oscillate for kcs0 > 5/2 and they damp for kcs0 > 0.
In both of our solutions, and indeed for any reasonable equation of state in an Einstein-de Sitter universe, long-wavelength (kcs << 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 a2 a-1 from the Poisson equation. If K < 0 or k2cs2 > 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 cs, not the causal horizon distance c or the Hubble distance c/H. Setting cs to the acoustic speed of the coupled photon-baryon fluid at matter-radiation equality gives the physical scale at which the bend occurs, cs eq, in the power spectrum of the standard cold dark matter and other models.