Feynman Path Integral Formulation

320 9 Scale Dependent Gravitational Couplings
would overlook the fact that the relationship between density ρ(t) and scale factor
a(t) is quite different from the classical case.
The running of G(t) in the above equations follows directly from the basic result
of Eq. (9.1), following the more or less unambiguously defined sequence G(k 2 ) →
G(✷) → G(t). For large times t ≫ ξ the form of Eq. (9.1), and therefore Eq. (9.77),
is no longer appropriate, due to the spurious infrared divergence of Eq. (9.1) at small
k 2 . Indeed from Eq. (9.2), the infrared regulated version of the above expression
should read instead
G(t) ≃ G
⎡
( ) 1
⎣ t
2 2ν
1 + c ξ
t 2 + ξ 2
+ ...
⎤
⎦ . (9.80)
For very large times t ≫ ξ the gravitational coupling then approaches a constant,
finite value G ∞ =(1 + a 0 + ...)G c . The modification of Eq. (9.80) should apply
whenever one considers times for which t ≪ ξ is not valid. But since ξ ∼ 1/ √ λ is
of the order the size of the visible universe, the latter regime is largely of academic
interest.
It should also be noted that the effective Friedmann equations of Eqs. (9.75) and
(9.76) also bear a superficial degree of resemblance to what might be obtained in
some scalar-tensor theories of gravity, where the gravitational Lagrangian is postulated
to be some singular function of the scalar curvature (Capozziello et al, 2003;
Carroll et al, 2004; Flanagan, 2004). Indeed in the Friedmann-Robertson-Walker
case one has, for the scalar curvature in terms of the scale factor,
and for k = 0 and a(t) ∼ t α one has
R = 6 ( k + ȧ 2 (t)+a(t)ä(t) ) /a 2 (t) , (9.81)
R =
6α(2α − 1)
t 2 , (9.82)
which suggests that the quantum correction in Eq. (9.75) is, at this level, nearly indistinguishable
from an inverse curvature term of the type (ξ 2 R) −1/2ν ,or1/(1 +
ξ 2 R) 1/2ν if one uses the infrared regulated version. The former would then correspond
the to an effective gravitational action
I ef f ≃ 1 ∫
16πG
(
)
dx √ g R +
f ξ − 1 ν
|R| − 2λ , (9.83)
2ν 1 −1
with f a numerical constant of order one, and λ ≃ 1/ξ 2 . But this superficial resemblance
is seen here more as an artifact, due to the particularly simple form of the
Robertson-Walker metric, with the coincidence of several curvature invariants not
expected to be true in general. In particular in Eqs. (9.75) and (9.76) it would seem
artificial and in fact inconsistent to take λ ∼ 1/ξ 2 to zero while keeping the ξ in
G(t) finite.