Cosmological backreaction and its dependence on spacetime foliation

Abstract

The subject of cosmological backreaction in General Relativity is often approached by coordinate-dependent and metric-based analyses. We present in this letter an averaging formalism for the scalar parts of Einstein's equations that is coordinate-independent and only functionally depends on a metric. This formalism is applicable to general 3 + 1 foliations of spacetime for an arbitrary fluid with tilted flow. We clarify the dependence on spacetime foliation and argue that this dependence is weak in cosmological settings. We also introduce a new set of averaged equations that feature a global cosmological time despite the generality of the setting, and we put the statistical nature of effective cosmologies into perspective.

On average properties of inhomogeneous fluids in general relativity III: general fluid cosmologies

“…One question is, how much a change of foliation quantitatively affects the result on the magnitudes of the backreaction terms. This question is considered in an accompanying Letter [30] and will be addressed more explicitly in a forthcoming work [58]. A second question is, whether a foliation choice can make individual backreaction terms disappear completely.…”

Does spatial flatness forbid the turnaround epoch of collapsing structures?

“…In the almost-FLRW numerical QZA modelling in which curvature is allowed to vary above and below zero on any given spatial hypersurface and averages in Lagrangian domains are studied, gauge dependence is again not an issue (provided, again, that no restrictions are imposed by a gauge transformation), but foliation dependence could, in principle, be significant. Buchert, Mourier & Roy [64] argue that volume would differ by a factor of the order of the mean Lorentz factor γ relating the fluid rest frame to an alternative reference frame. If the latter is that of a best-fit FLRW model to observational data, then fluid velocities of the order of 200 km/s would yield changes in volume or volume-based functionals such as Ω D R by about γ − 1 < 10 −6 , that is, the tilt between vectors normal to the different foliations is negligible in the present context.…”

Towards statistically homogeneous and isotropic perfect fluid universes with cosmic backreaction

“…This term, the kinematical backreaction, is defined as Q := 2 3 Θ 2 − Θ 2 − 2 σ 2 , where Θ is the local expansion rate of the fluid and σ 2 := 1 2 σ µν σ µν is its shear scalar. Besides the kinematical backreaction term, the averaged Hamiltonian constraint deviates from the Friedmann equation by permitting (3) R to evolve differently than proportional to a −2 D . In fact, the evolution of (3) R is linked to the evolution of Q by the integrability condition, (3) R = 0, which must be fulfilled in order for equation (1) to be the integral of equation (2).…”

“…Besides the kinematical backreaction term, the averaged Hamiltonian constraint deviates from the Friedmann equation by permitting (3) R to evolve differently than proportional to a −2 D . In fact, the evolution of (3) R is linked to the evolution of Q by the integrability condition, (3) R = 0, which must be fulfilled in order for equation (1) to be the integral of equation (2). This equation shows that (3) R ∝ a −2 D when Q = 0.…”

“…In fact, the evolution of (3) R is linked to the evolution of Q by the integrability condition, (3) R = 0, which must be fulfilled in order for equation (1) to be the integral of equation (2). This equation shows that (3) R ∝ a −2 D when Q = 0. In this case, the resulting average evolution is that of an FLRW model.…”