MODERN COSMOLOGY

50 An introduction to the physics of cosmology

be specified in terms of the sound speed

c^2 s≡

∂p

∂ρ

Now think of a plane-wave disturbanceδ ∝ e−ik·r,wherekis a comoving
wavevector; in other words, suppose that the wavelength of a single Fourier mode
stretches with the universe. All time dependence is carried by the amplitude of
the wave, and so the spatial dependence can be factored out of time derivatives in
these equations (which would not be true with a constant comoving wavenumber
k/a). An equation for the amplitude ofδcan then be obtained by eliminatingu:

δ ̈+ 2 a ̇

a

δ ̇=δ( 4 πGρ 0 −c^2 sk^2 /a^2 ).

This equation is the one that governs the gravitational amplification of density
perturbations.
There is a critical proper wavelength, known as theJeans length,atwhich
we switch from the possibility of exponential growth for long-wavelength modes
to standing sound waves at short wavelengths. This critical length is

λJ=cs

√

π

Gρ

,

and clearly delineates the scale at which sound waves can cross an
object in about the time needed for gravitational free-fall collapse. When
considering perturbations in an expanding background, things are more complex.
Qualitatively, we expect to have no growth when the ‘driving term’ on the right-
hand side is negative. However, owing to the expansion,λJwill change with time,
and so a given perturbation may switch between periods of growth and stasis.

2.6.2 Dynamical effects of radiation

At early enough times, the universe was radiation dominated (cs=c/

√

3. and
   the analysis so far does not apply. It is common to resort to general relativity
   perturbation theory at this point. However, the fields are still weak, and so it is
   possible to generate the results we need by using special relativity fluid mechanics
   and Newtonian gravity with a relativistic source term. For simplicity, assume
   that accelerations due to pressure gradients are negligible in comparison with
   gravitational accelerations (i.e. restrict the analysis toλλJfrom the start).
   The basic equations are then a simplified Euler equation and the full energy and
   gravitational equations:

Euler:

Dv

Dt

=−∇

energy:

D

Dt

(ρ+p/c^2 )=

∂

∂t

(p/c^2 )−(ρ+p/c^2 )∇·v

Poisson: ∇^2 = 4 πG(ρ+ 3 p/c^2 ).