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2.2. Linear instability 1: isentropic fluctuations and Jeans criterion

We begin with some nomenclature from thermodynamics. Isentropic means nablaS = 0: the same entropy everywhere. Adiabatic means dS/dtau = 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 nablaS = 0, the linearized fluid and gravitational field equations are

Equation 2.11 (2.11)

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

Equation 2.12 (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, delta(x, tau) rightarrow delta(k, tau)exp(ik . x). The wave equation becomes

Equation 2.13 (2.13)

where we have defined the comoving Jeans wavenumber,

Equation 2.14 (2.14)

Neglecting Hubble damping (by setting a = 1), the time dependence of the solution to eq. (2.13) would be delta propto exp(- i omega tau), 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:

Equation 2.15 (2.15)

The plasma frequency is omegap = (4pi ne e2 / me)1/2 while the Jeans frequency is omegaJ = kJ cs = (4pi G rhobar)1/2. Whereas electromagnetic waves with omega2 < omegap2 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 (omega2 < 0), as was first noted by Jeans (1902). In physical terms, pressure forces cannot prevent gravitational collapse when the sound-crossing time lambda / cs is longer than the gravitational dynamical time (Grho)-1/2 for a perturbation of proper wavelength lambda = 2pia/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 delta(k, tau); these are denoted delta±(k, tau). For cs2 = 0 and an Einstein-de Sitter (flat, matter-dominated) background with a(tau) propto tau2, delta+ propto tau2 and delta- propto tau-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 phi rather than delta using nabla2 phi = 4pi Ga2 rhobar delta propto a-1 delta for nonrelativistic matter (with cs2 << c2):

Equation 2.16 (2.16)

where we used the Friedmann equation (1.6); recall that K = (Omega - 1)(aH)2 is the spatial curvature constant. In a matter-dominated universe, differentiating the Friedmann equation gives  ddot a/a - (1/2)dot a2 / a2 = - (1/2)K, yielding

Equation 2.17 (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

Equation 2.18 (2.18)

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 kcstau >> 1 and power-law behavior for kcstau << 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 tau:

Equation 2.19 (2.19)

The solutions oscillate for kcstau0 > 5/2 and they damp for kcstau0 > 0.

In both of our solutions, and indeed for any reasonable equation of state in an Einstein-de Sitter universe, long-wavelength (kcstau << 1) growing density modes have corresponding potential phi+ = constant, while the decaying density modes have phi_ propto integ a-3 dtau. The density perturbation and potential differ by a factor of rhobar a2 propto a-1 from the Poisson equation. If K < 0 or k2cs2 > 0, then phi+ decays with time, although delta+ still grows. Note that the important physical length scale where the transfer function phi+(k, tau) / phi(k, 0) falls significantly below unity is the acoustic comoving horizon distance cstau, not the causal horizon distance ctau 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 taueq, in the power spectrum of the standard cold dark matter and other models.

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