310 Gravitational instability in General Relativity
universe with perfect fluidδT
i
j(T)=^0 ,(7.43) simplifies to
(
h′′ij+ 2 Hh′ij−hij
)
= 0. (7.95)
Introducing the rescaled variablevvia
hij=v
a
eij, (7.96)whereeijis a time-independent polarization tensor, and considering a plane wave
perturbation with the wavenumberk,(7.95) becomes
v′′+(
k^2 −a′′
a)
v= 0. (7.97)In aradiation-dominateduniversea∝η,hencea′′=0 andv∝exp(±ikη).In
this case the exact solution of (7.95) is
hij=1
η(C 1 sin(kη)+C 2 cos(kη))eij. (7.98)The nondecaying mode of the gravitational wave with wavelength larger than the
Hubble scale(kη 1 )is constant. After the wavelength becomes smaller than
the Hubble radius, the amplitude decays in inverse proportion to the scale factor.
This is a general result valid for any equation of state.In fact, for long-wavelength
perturbations withkη 1 ,we can neglect thek^2 term in (7.97) and its solution
becomes
vC 1 a+C 2 a∫
dη
a^2. (7.99)
Hence,
hij=(
C 1 +C 2
∫
dη
a^2)
eij. (7.100)Forp<εthe second term describes the decaying mode.
For short-wavelength perturbations(kη 1 ),we havek^2 a′′/a andh∝
exp(±ikη)/a.
Problem 7.15Find the exact solution of (7.95) for an arbitrary constant equa-
tion of statep=wεand analyze the behavior of the short- and long-wavelengh
gravitational waves. Consider separately the casesp=±ε.
7.4 Baryon–radiation plasma and cold dark matter
Understanding the perturbations in a multi-component medium consisting of a mix-
ture of a baryon–radiation plasma and cold dark matter is important both to analyze