Physical Foundations of Cosmology

7.3 Hydrodynamical perturbations 305

and it immediately follows from ( 7.73) that

(^) f =

(

1 +wf

1 +wi

)(

5 + 3 wi

5 + 3 wf

)

(^) i. (7.74)
For a matter–radiation universe,wi= 1 /3 andwf= 0 ,and we obtain the familiar
result (^) f=( 9 / 10 ) (^) i.
Problem 7.11Verify that for a mode with wavenumberk,equation (7.65) can be
rewritten in the following integral form:
uk(η)=C 1 θ+C 2 θ

∫

dη

θ^2

−k^2 θ

∫η⎛

⎝

∫η ̃

c^2 sθukdη ̄

⎞

⎠^1

θ^2 (η ̃)

dη. ̃ (7.75)

Using this equation, calculate the subleadingk^2 -corrections to the long-wavelength
solution (7.67) and determine the violation of the “conserved” quantityζ.

Short-wavelength perturbationsWhencskη 1 ,the last term in (7.65) can be
neglected. The resulting equation,

u′′+cs^2 k^2 u 0 , (7.76)

can easily be solved in the WKB approximation for a slowly varying speed of
sound. Its solution describes sound waves with the time-dependent amplitude.

Matching conditionsSometimes it is convenient to approximate the continuous
change of the equation of state by a sharp jump. In this case the pressurep(ε)is
discontinuous on the hypersurface of transition#, εT=const, and its derivatives
become singular.Therefore we cannot directly use the equation for the gravitational
potential and, instead, must derive matching conditions for and ′on#.These
conditions can be obtained if we recast (7.65) in the following form:
[
θ^2

(u

θ

)′]′

=cs^2 θ^2 

(u

θ

)

. (7.77)

Evidentlyu/θshould be continuous. Because the scale factoraand the energy
densityεare both continuous, the gravitational potential does not jump during
the transition, or equivalently, the 3-metric induced on#is continuous.
To determine the jump in the derivative ofu/θlet us integrate (7.77) within an
infinitesimally thin layer near#:

[

θ^2

(u

θ

)′]

±

=

#∫+ 0

#− 0

c^2 sθ^2 

(u

θ

)

dη, (7.78)