MODERN COSMOLOGY

The cosmological framework 313

11.2.1 Friedmann cosmological background

What is generally referred to as thestandard cosmological frameworkis the result
of the solution of the Einstein equations in the hypothesis that the universe is,
on very large scales,homogeneousandisotropic. There are several pieces of
observational evidence which support thiscosmological principle, such as the
distribution of galaxies and clusters of galaxies on large scales and the remarkable
isotropy of the cosmic microwave background (CMB).
The FRW models provide thebackgroundon which the formation and
evolution of the large-scale structure in the universe can be studied as the
evolution of small perturbations to an otherwise uniform FRW model. The
application of the cosmological principle leads to the following FRW spacetime
line element (see Landau and Lifshitz (1971) for an elegant and simple
derivation):

ds^2 =c^2 dt^2 −R^2 (t)

[

dr 12

1 −kr 12

+r 12 (dθ^2 +sin^2 θdφ^2 )

]

(11.1)

=c^2 dt^2 −R^2 (t)[dr^2 +Sk^2 (r)(dθ^2 +sin^2 θdφ^2 )] (11.2)

where two possible definitions of thecomoving coordinate, r ,have been used.
This is the coordinate measured by observers at rest with respect to the local
matter distribution. The first expression is commoonly used in the literature. In
the second form, following the notation by Peacock (1999), we have defined:

Sk(r)=

{sin(r) k=1 (close)

rk=0(flat)

sinh(r) k=−1 (open).

(11.3)

The casesk =− 1 , 0 ,1 represent, respectively, anopen universe(infinite,
hyperbolic space), aflat universe(infinite, flat space) and aclosed universe(finite,
spherical space).
The solution of the Einstein field equations (with cosmological constant)
leads to the following equation for the evolution of the scale factor,R(t):

( ̇

R

R

) 2

=

8 πG

3

ρM+

1

3

c^2 −

kc^2

R^2

. (11.4)

This shows three competing terms driving the universal expansion: a matter
term, a cosmological constant term and a curvature term. We are neglecting
here a radiation term, as appropriate when the universe is dominated by non-
relativistic matter (‘dust’) with densityρM, i.e. the directly observable universe.
The respective fractional contributions to the energy density in the universe at the
present epoch are commonly defined as

m≡

8 πG

3 H 02

ρM 0 ,
≡

c^2

3 H 02

,
k≡−

kc^2

H 02 R^20

(11.5)