Physical Foundations of Cosmology

6.3 Instability in an expanding universe 273

6.3.1 Adiabatic perturbations

Taking the Fourier transform of (6.46) with respect to the comoving coordinatesq,
we obtain the ordinary differential equation

δ ̈k+ 2 Hδ ̇k+

(

c^2 sk^2

a^2

− 4 πGε 0

)

δk= 0 (6.47)

for every Fourier modeδ=δk(t)exp(ikq).The behavior of each perturbation
depends crucially on its spatial size; the critical lengthscale is the Jeans length

λphJ =

2 πa

kJ

=cs

√

π

Gε 0

. (6.48)

Hereλphis the physical wavelength (measured, for example, in centimeters), related
to the comoving wavelengthλ= 2 π/kviaλph=a·λ.In a flat, matter-dominat-
ed universeε 0 =

(

6 πGt^2

)− 1

and hence

λJph∼cst, (6.49)

that is, the Jeans length is of order the sound horizon. Sometimes instead of the
Jeans length, one uses the Jeans mass, defined asMJ≡ε 0 (λphJ )^3.
Perturbations on scales much smaller than the Jeans length (λλJ) are sound
waves. Ifcschanges adiabatically, then the solution of (6.47) is

δk∝

1

√

csa

exp

(

±k

∫

csdt

a

)

. (6.50)

Problem 6.2Derive the solution in (6.50). Explain why the amplitude of sound
waves decays with time. (HintUsing conformal timeη≡

∫

dt/ainstead of the
physical timet, derive the equation for the rescaled amplitude

√

aδkand solve it in
the WKB approximation.)

On scales much larger than the Jeans scale (λλJ), gravity dominates and we
can neglect thek-dependent term in (6.47). Then one of the solutions is simply
proportional to the Hubble constantH(t). In fact, substitutingδd=H(t)in (6.47),
where we setc^2 sk^2 =0, one finds that the resulting equation coincides with the time
derivative of the Friedmann equation (6.34). Note thatδd=H(t)is the decaying
solution of the perturbation equation (Hdecreases with time) in a matter-dominated
universe witharbitrarycurvature.
Actually, one could have guessed this solution using the following simple ar-
gument. Both the background energy densityε 0 (t)and the time-shifted energy
densityε 0 (t+τ), whereτ=const, satisfy (6.33), (6.34). Indeed, using (6.33)
to expressHin terms ofε 0 and substituting this into (6.34), we obtain an equa-
tion forε 0 (t)in which the time does not explicitly appear. Hence its solution is