5 Cosmological Models
In Section 5.1 we turn to the ‘concordance’ or Friedmann–Lemaitre–Robertson–
Walker (FLRW) model of cosmology, really only a paradigm based on Friedmann’s
and Lemaitre’s equations and the Robertson–Walker metric, which takes both energy
density and pressure to be functions of time in a Copernican universe. Among the
solutions are the Einstein universe and the Einstein–de Sitter universe, both now
known to be wrong, as we shall see in Section 5.4, and the currently accepted
Friedmann–Lemaitre universe, which includes a positive cosmological constant.
In Section 5.2 we describe the de Sitter model, which does not apply to the Universe
at large as we see it now, but which may have dominated the very early universe, and
which may be the correct description for the future.
In Section 5.3 we introduce tne Schwartzschild solution to The Einstein equation.
This takes us to black holes in Section 5.4.
In Section 5.5 we briefly present extensions of general relativity.
5.1 Friedmann–Lemaitre Cosmologies
Let us now turn to our main subject, a model describing our homogeneous and
isotropic Universe for which the Robertson–Walker metric in Equation (2.32) was
derived. Recall that it could be written as a 4×4 tensor with nonvanishing com-
ponents [Equation (2.33)] on the diagonal only, and that it contained the curvature
parameter푘.
Friedmann’s Equations. The stress–energy tensor푇휇휈entering on the right-hand
side of the Einstein Equations (3.29) was given by Equation (3.30) in its diagonal
form. For a comoving observer with velocity four-vector푣=(푐, 0 , 0 , 0 ), the time–time
Introduction to Cosmology, Fourth Edition. Matts Roos
© 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd.