Feynman Path Integral Formulation

316 9 Scale Dependent Gravitational Couplings
[
]
−6 −2k ä(t) − 5ȧ 2 (t)ä(t)+a(t)ä 2 (t)+3a(t)ȧ(t)a (3) (t)+a 2 (t)a (4) (t) /a 3 (t) ,
(9.53)
and then ✷ 2 on R etc. Since the resulting expressions are of rapidly escalating complexity,
one sets a(t) =r 0 t α , in which case one has first for the scalar curvature
itself
[ ]
k α (−1 + 2α)
R = 6
r0 2 +
t2α t 2 . (9.54)
Acting with ✷ n on the above expression gives for k = 0 and arbitrary power n
with the coefficient c n given by
c n 6α (−1 + 2α)t −2−2n , (9.55)
3α−1
n
Γ (n + 1)Γ (
2
)
c n = 4
Γ ( 3α−1
2
− n)
Here use has been made of the relationship
( ) d α
(z − c) β =
dz
. (9.56)
Γ (β + 1)
Γ (β − α + 1) (z − c)β−α , (9.57)
to analytically continue the above expressions to negative fractional n (Samko et al,
1993; Zavada, 1998). For n = −1/2ν the correction on the scalar curvature term R
is therefore of the form
[
1 − a 0 c ν (t/ξ ) 1/ν] · 6α (−1 + 2α)t −2 (9.58)
with
c ν = 2 − 1 Γ (1 − 1 3α−1
ν 2ν
)Γ (
2
)
Γ ( 3α−1
2
+
2ν 1 ) . (9.59)
Putting everything together, one then obtains for the trace part of the effective field
equations
[
1 − a 0 c ν
( t
ξ
) 1/ν
+ O
(
(t/ξ ) 2/ν)] 6α (2α − 1)
t 2 = 8πGρ(t) (9.60)
The new term can now be moved back over to the matter side in accordance with
the structure of the original effective field equation of Eq. (9.47), and thus avoids
the problem of having to deal with the binomial expansion of 1/[1 + A(✷)]. One
then has
6α (2α − 1)
t 2
= 8πG
[
1 + a 0 c ν
( t
ξ
) 1/ν
+ O
(
(t/ξ ) 2/ν)] ρ(t) , (9.61)