Friedmann–Lemaˆıtre models 131the Raychaudhuri equation (3.34):
3
S ̈
S
=−^12 (μ+ 3 p)+λ (3.83)and the Friedmann equation (3.40), where^3 R= 6 k/S^2 ,
3
S ̇^2
S^2
−κμ−λ=−3 k
S^2. (3.84)
The Friedmann equation is a first integral of the other two whenS ̇ =0. The
solutions, of course, depend on the equation of state; for the currently favoured
universe models, going backward in time there will be
(1) a cosmological-constant-dominated phase,
(2) a matter-dominated phase,
(3) a radiation-dominated phase,
(4) a scalar-field-dominated inflationary phase and
(5) a pre-inflationary phase where the physics is speculative (see the last section
of this chapter). The normalized density parameter is≡κμ/ 3 H^2 ,where
as usualH=S ̇/S.3.6.1 Phase planes and evolutionary paths
From these equations, one can obtain phase planes
(i) for the density parameteragainst the deceleration parameterq,see [115];
(ii) for the density parameteragainst the Hubble parameterH, see [128] for
the caseλ=0; and
(iii) for the density parameteragainst the scale parameterS, see [94], showing
howchanges in inflationary and non-inflationary universes.It is a consequence of the equations that the spatial curvature parameterk
is a constant of the motion. In particular, flatness cannot change as the universe
evolves: eitherk =0 or not, depending on the initial conditions, and this is
independent of any inflation that may take place. Thus while inflation can drive
the spatial curvatureK=k/S^2 very close indeed to zero, it cannot setK= 0.
If one has a scalar field matter sourceφwith potentialV(φ), one can obtain
essentially arbitrary functional forms for the scale functionS(t)by using the
arbitrariness in the functionV(φ)and running the field equations backwards,
see [46].
3.6.2 Spatial topology
The Einstein field equations determine the time evolution of the metric and its
spatial curvature, but they do not determine its spatial topology. Spatially closed