22 Kinematics and dynamics of an expanding universe
Problem 1.12Show that the equations of motion for the scalar field,
φ;α;α+∂V
∂φ= 0 , (1.57)
follow fromTβα;α= 0.
Ifφ,γφ,γ>0, then the energy–momentum tensor for a scalar field can be rewrit-
ten in the form of a perfect fluid ( 1.55) by defining
ε≡^12 φ,γφ,γ+V(φ), p≡^12 φ,γφ,γ−V(φ), uα≡φ,α/√
φ,γφ,γ. (1.58)In particular, assuming that the field is homogeneous (∂φ/∂xi=0), we have
ε≡^12 φ ̇^2 +V(φ),p≡^12 φ ̇^2 −V(φ). (1.59)For a scalar field, the ratiow=p/εis, in general, time-dependent. Additionally,
wis bounded from below by−1 for any positive potentialVand the weak energy
dominance condition,ε+p≥0, is satisfied. However, the strong energy domi-
nance condition,ε+ 3 p≥0, can easily be violated by a scalar field. For example,
if a potentialV(φ) has a local minimum at some pointφ 0 , thenφ(t)=φ 0 is a
solution of the scalar field equations, for which
p=−ε=−V(φ 0 ). (1.60)As far as Einstein’s equations are concerned, the corresponding energy–momentum
tensor,
Tβα=V(φ 0 )δαβ, (1.61)imitates a cosmological term
= 8 πGV(φ 0 ). (1.62)The cosmological term can therefore always be interpreted as the contribution of
vacuum energy to the Einstein equations and from now on we include it in the
energy–momentum tensor of matter and set=0 in (1.48).
1.3.3 Friedmann equations
How are the Newtonian equations of cosmological evolution (1.12), (1.15) and
(1.18) modified when matter is relativistic? In principle, to answer this question we
must simply substitute the metric (1.47) and energy–momentum tensor (1.55) into
the Einstein equations (1.48). The resulting equations are the Friedmann equations
and they determine the two unknown functionsa(t) andε(t).However, rather than
starting with this formal derivation, it is instructive to explain how the nonrelativistic
equations (1.12) and (1.15) must be modified.