Physical Foundations of Cosmology

6.3 Instability in an expanding universe 277

We have skipped in (6.64) the term proportional toc^2 sbecause it is determined by
the pressure of the cold matter alone and hence is negligible. The general solution
of (6.64) for an arbitraryw=const is given by a linear combination of hyperge-
ometric functions. However, at least in two important cases, they reduce to simple
elementary functions.

Cosmological constant(w=− 1 )One can easily verify that in this case,

δ 1 (x)=

√

1 +x−^3 (6.65)

satisfies (6.64). The other solution can be obtained using the properties of the
Wronskian.

Problem 6.5Verify that ifδ 1 (x) is a solution of (6.64), then the second independent
solution is given by

δ 2 (x)=δ 1 (x)

∫x

dy

y^3 /^2

(

1 +y−^3 w

) 1 / 2

δ^21 (y)

. (6.66)

Forw=−1, the general solution of (6.64) is thus

δ(x)=C 1

√

1 +x−^3 +C 2

√

1 +x−^3

∫x

0

(

y

1 +y^3

) 3 / 2

dy, (6.67)

whereC 1 andC 2 are constants of integration. At early times when the cold matter
dominates (x1), the perturbation grows as

δ(x)=C 1 x−^3 /^2 +^25 C 2 x+O

(

x^3 /^2

)

, (6.68)

in complete agreement with our previous result. Subsequently the cosmological
constant becomes dominant and in the limitx1wehave

δ(x)=(C 1 +IC 2 )−^12 C 2 x−^2 +O

(

x−^3

)

, (6.69)

where

I=

∫∞

0

(

y

1 +y^3

) 3 / 2

dy 0. 57. (6.70)

Thus, when the cosmological constant overtakes the matter density the growth
ceases and the amplitude of the perturbation is frozen. According to (6.45), the
induced gravitational potential decays in inverse proportion to the scale factor
sinceεd∝a−^3. The results obtained are not surprising because the cosmological
constant acts as “antigravity” and tries to prevent the growth of the perturbations.