6 Gravitational instability in Newtonian theory
Measurements of the cosmic microwave background tell us that the universe was
very homogeneous and isotropic at the time of recombination. Today, however, the
universe has a well developed nonlinear structure. This structure takes the form of
galaxies, clusters and superclusters of galaxies, and, on larger scales, of voids, sheets
and filaments of galaxies. Deep redshift surveys show, however, that when averaged
over a few hundred megaparsecs, the inhomogeneities in the density distribution
remain small.The simple explanation as to how nonlinear structure could develop
from small initial perturbations is based on the fact of gravitational instability.
Gravitational instability is a natural property of gravity. Matter is attracted to
high-density regions, thus amplifying already existing inhomogeneities. To ensure
that the small initial inhomogeneities present at recombination produce the nonlin-
ear structure observed today, we have to study how fast they grow in an expanding
universe. The complete general relativistic analysis of gravitational instability is
rather involved and the physical interpretation of the results is not always straight-
forward. For this reason we develop the theory of gravitational instability in several
steps.
In this chapter we consider gravitational instability in the Newtonian theory of
gravity. The results derived in this theory are applicable only to nonrelativistic
matter on scales not exceeding the Hubble horizon. First, we find out how small
inhomogeneities grow in a nonexpanding universe (Jeans theory). The main pur-
pose here is to determine which types of perturbations can exist in homogeneous,
isotropic media, and to introduce methods to analyze them. Although the formulae
describing the rate of instability in a nonexpanding universe are not very useful,
the results obtained help us gain a solid intuitive understanding of the behavior
of perturbations. Next, we consider linear perturbations in an expanding universe.
This is not only a useful exercise, but a realistic theory describing the growth of
inhomogeneities on subhorizon scales after recombination. We apply this theory
to study the rate of instability in a matter-dominated universe and then see how a
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