One-loop quantum gravitational backreaction on the local Hubble rate

Abstract

We determine corrections to the Hubble rate due to graviton loops in a cosmological background spacetime of constant deceleration parameter. The corrections are gauge-invariant, based on a recent proposal for all-order gauge-invariant observables in perturbative quantum gravity. We find explicit expressions for the cases of matter-and radiation-dominated eras and slow-roll inflation with vanishing second slow-roll parameter. Interestingly, in the latter case the corrections can be described by a quantum-corrected first slow-roll parameter, which brings the spacetime closer to de Sitter space.

Graviton backreaction on the local cosmological expansion in slow-roll inflation

“…Our aim in this paper is to make use of the proposal of references [40][41][42] to compute the quantum-gravitational one-loop correction to a gauge-invariant observable measuring the space-time local expansion rate in slow-roll inflation, with both slow-roll parameters finite. Similar calculations were recently performed in de Sitter space-time [26] and in single-field inflation with constant deceleration parameter [27]. The calculation of Miao et al [26]h a s confirmed earlier expectations [18,19] that there is no quantum-gravitational backreaction on the expansion rate in de Sitter at one-loop order, although they leave the door open for effects at two-loop order.…”

“…The condition on the scalar field perturbation also implies thatX (0) (2) = 0, as can be seen from equation (14), and this gauge choice simplifies equation (27)to 1) = H (1) , H (2) = H (2) .…”

“…Since we are interested in the causal evolution of the observables expectation value, the natural choice in that case is to impose the initial conditionsX (i) (η 0 , x) = x i and ∂ tX (i) (η 0 , x) = 0 and use the retarded Green's function. For further discussions on this point, see references [27,42].…”

“…We remark that equation ( 18) is invariant under diffeormorphisms that preserve the background fields by construction [40]. Nevertheless, it is possible to explicitly check this for equation (21) by using the gauge transformation of the inflaton and the metric perturbation and the definitions of the coordinatesX (α) , and we refer the reader to reference [27] for further details.…”

“…The calculation of Miao et al [26]h a s confirmed earlier expectations [18,19] that there is no quantum-gravitational backreaction on the expansion rate in de Sitter at one-loop order, although they leave the door open for effects at two-loop order. Shortly after Fröb [27] performed the calculation of the one-loop correction to the gauge-invariant local expansion rate as defined in reference [42] in single-field inflation with constant deceleration parameter, and found a secular backreaction effect that accelerates the expansion.…”