274 Gravitational instability in Newtonian theory
time-translational-invariant. For smallτ, the time-shifted solutionε 0 (t+τ)can be
considered as a perturbation of the backgroundε 0 (t)with amplitude
δd=ε 0 (t+τ)−ε 0 (t)
ε 0 (t)≈
ε ̇ 0 τ
ε 0∝H(t).Once we know one solution of the second order differential equation,δd, then the
other independent solutionδican easily be found with the help of the Wronskian,
W≡δ ̇dδi−δd ̇δi. (6.51)Taking the derivative of the Wronskian and using (6.47) to expressδ ̈in terms ofδ ̇
andδ, we find thatWsatisfies the equation
W ̇ =− 2 HW, (6.52)which has the obvious solution
W≡δ ̇dδi−δd ̇δi=C
a^2, (6.53)
whereCis a constant of integration. Substituting the ansatzδi=δdf(t)into (6.53),
we obtain an equation forfthat is readily integrated:
f=−C∫
dt
a^2 δd^2. (6.54)
Thus the most general longwave solution of (6.47) is
δ=C 1 H∫
dt
a^2 H^2+C 2 H. (6.55)
In a flat, matter-dominated universe,a∝t^2 /^3 andH∝t−^1 .In this case, we have
δ=C 1 t^2 /^3 +C 2 t−^1. (6.56)Hence, we see that in an expanding universe, gravitational instability ismuch less
efficientand the perturbation amplitude increases only as a power of time. In the im-
portant case of a flat, matter-dominated universe, the growing mode is proportional
to the scale factor. Therefore, if we want to obtain large inhomogeneities
(
δ 1)
today, we have to assume that at early times (for example, at redshiftsz=1000) the
inhomogeneities were already substantial (δ 10 −^3 ).This imposes rather strong
constraints on the initial spectrum of perturbations. We will see in Chapter 8 that
the required initial spectrum can be explained naturally in inflationary cosmology.
Problem 6.3Calculate the peculiar velocities and gravitational potential for the
long-wavelength perturbations. Analyze their behavior and give the physical inter-
pretation of the behavior of the gravitational potential for the growing mode.