Feynman Path Integral Formulation

320 9 Scale Dependent Gravitational Couplingswould overlook the fact that the relationship between density ρ(t) and scale factora(t) is quite different from the classical case.The running of G(t) in the above equations follows directly from the basic resultof 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 smallk 2 . Indeed from Eq. (9.2), the infrared regulated version of the above expressionshould read insteadG(t) ≃ G⎡( ) 1⎣ t2 2ν1 + c ξt 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 applywhenever one considers times for which t ≪ ξ is not valid. But since ξ ∼ 1/ √ λ isof the order the size of the visible universe, the latter regime is largely of academicinterest.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 insome scalar-tensor theories of gravity, where the gravitational Lagrangian is postulatedto be some singular function of the scalar curvature (Capozziello et al, 2003;Carroll et al, 2004; Flanagan, 2004). Indeed in the Friedmann-Robertson-Walkercase one has, for the scalar curvature in terms of the scale factor,and for k = 0 and a(t) ∼ t α one hasR = 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 indistinguishablefrom 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 correspondthe to an effective gravitational actionI ef f ≃ 1 ∫16πG()dx √ g R +f ξ − 1 ν|R| − 2λ , (9.83)2ν 1 −1with f a numerical constant of order one, and λ ≃ 1/ξ 2 . But this superficial resemblanceis seen here more as an artifact, due to the particularly simple form of theRobertson-Walker metric, with the coincidence of several curvature invariants notexpected to be true in general. In particular in Eqs. (9.75) and (9.76) it would seemartificial and in fact inconsistent to take λ ∼ 1/ξ 2 to zero while keeping the ξ inG(t) finite.