Gravitational backreaction in cosmological spacetimes

We develop a new formalism for the treatment of gravitational backreaction in the cosmological setting. The approach is inspired by projective techniques in nonequilibrium statistical mechanics. We employ group averaging with respect to the action of the isotropy group of homogeneous and isotropic spacetimes (rather than spatial averaging), in order to define effective Friedmann-Robertson-Walker variables for a generic spacetime. Using the Hamiltonian formalism for gravitating perfect fluids, we obtain a set of evolution equations for the evolution of the effective variables; these equations incorporate the effects of backreaction by the inhomogeneities. Specializing to dust-filled spacetimes, we find regimes that lead to a closed set of backreaction equations, which we solve for small inhomogeneities. We then study the case of large inhomogeneities in relation to the proposal that backreaction can lead to accelerated expansion. In particular, we identify regions of the gravitational state space that correspond to effective cosmic acceleration. Necessary conditions are (i) a strong expansion of the congruences corresponding to comoving observers, and (ii) a large negative value of a dissipation variable that appears in the effective equations (i.e., an effective “antidissipation”).

Figure 1

The normalized scale factor $\alpha /{\alpha }_0$ as a function of time $t$ in units of the Hubble time $H_0^{-1}$; $H_0=\frac{\dot{\alpha }}{\alpha }(t_0)$, for ${\theta }_0^2=0.01$, $u_0^2=0.0001$, ${\Gamma }_0=-0.1$. The dashed line shows the FRW solution.

Figure 2

Plots of the decelerations parameter $q_0$ as a function of ${\Omega }_m$, for $\eta =0.95$, ${\theta }_0=0.4$ and different values of $\delta {\gamma }_0$: from top to bottom: $\delta {\gamma }_0=0.8$, $\delta {\gamma }_0=1.0$, $\delta {\gamma }_0=1.2$.