Feynman Path Integral Formulation

9.5 Cosmological Solutions 319
One can find a closed form expression for the coefficients c tt and c rr = c tt /3 as functions
of ν which are not particularly illuminating, except for providing an explicit
proof that they exist.
As a result, in the simplest case, namely for a universe filled with non-relativistic
matter (p=0), the effective Friedmann equations then have the following appearance
k
a 2 (t) + ȧ2 (t)
for the tt field equation, and
a 2 (t) = 8πG(t)
3
= 8πG
3
k
a 2 (t) + ȧ2 (t)
a 2 (t) + 2ä(t)
a(t)
ρ(t)+ 1
3ξ 2
[
1 + c ξ (t/ξ ) 1/ν + ...
= − 8πG
3
]
ρ(t)+ 1 3 λ , (9.75)
[
]
c ξ (t/ξ ) 1/ν + ... ρ(t)+λ , (9.76)
for the rr field equation. The running of G appropriate for the Robertson-Walker
metric, and appearing explicitly in the first equation, is given by
[
]
G(t) =G
1 + c ξ
( t
ξ
) 1/ν
+ ...
, (9.77)
with c ξ of the same order as a 0 of Eq. (9.1). Note that the running of G(t) induces
as well an effective pressure term in the second (rr) equation. 1 One has therefore an
effective density given by
and an effective pressure
ρ ef f (t) = G(t) ρ(t) , (9.78)
G
p ef f (t) = 1 3
( G(t)
G − 1 )
ρ(t) , (9.79)
with p ef f (t)/ρ ef f (t)= 1 3
(G(t) − G)/G(t). Strictly speaking, the above results can
only be proven if one assumes that the pressure’s time dependence is given by a
power law. In the more general case, the solution of the above equations for various
choices of ξ and a 0 has to be done numerically. Within the FRW framework, the
gravitational vacuum polarization term behaves therefore in some ways (but not all)
like a positive pressure term, with p(t) =ωρ(t) and ω = 1/3, which is therefore
characteristic of radiation. One could therefore visualize the gravitational vacuum
polarization contribution as behaving like ordinary radiation, in the form of a dilute
virtual graviton gas: a radiative fluid with an equation of state p = 1 3ρ. But this
1 We wish to emphasize that we are not talking here about models with a time-dependent value
of G. Thus, for example, the value of G ≃ G c at laboratory scales should be taken to be constant
throughout most of the evolution of the universe.