Effects of structure formation on the expansion rate of the Universe: An estimate from numerical simulations

General relativistic corrections to the expansion rate of the Universe arise when the Einstein equations are averaged over a spatial volume in a locally inhomogeneous cosmology. It has been suggested that they may contribute to the observed cosmic acceleration. In this paper, we propose a new scheme that utilizes numerical simulations to make a realistic estimate of the magnitude of these corrections for general inhomogeneities in ($3+1$) spacetime. We then quantitatively calculate the volume averaged expansion rate using N-body large-scale structure simulations and compare it with the expansion rate in a standard FRW cosmology. We find that in the weak gravitational field limit, the converged corrections are slightly larger than the previous claimed ${10}^{-5}$ level, but not large enough nor even of the correct sign to drive the current cosmic acceleration. Nevertheless, the question of whether the cumulative effect can significantly change the expansion history of the Universe needs to be further investigated with strong-field relativity.

Figure 1

Matter power spectrum for three different cosmologies from both a linear theory and simulations.

Figure 2

The growth of the peculiar gravitational potential $\Phi$ from redshift $z=9$ to $z=0$ on a comoving spatial slice in the X-Y plane in a three dimensional simulation. The value of $\Phi /c^2$ is drawn along the Z axis as indicated at the bottom of the figure.

Figure 3

Correction to the energy density, $({\rho }_{\mathrm{eff}}-{\rho }_{\mathrm{FRW}})/{\rho }_{\mathrm{FRW}}$ as a function of the smoothing length $L_d$ for various box sizes and numbers of particles. Note the convergence for the largest box sizes.

Figure 4

Effective EOS parameter $w_{\mathrm{eff}}$ as a function of the smoothing length $L_d$ for various box sizes and numbers of particles. Note the convergence for the largest box sizes.

Figure 5

$({\rho }_{\mathrm{eff}}-{\rho }_{\mathrm{FRW}})/{\rho }_{\mathrm{FRW}}$ as a function of redshift $z$ for various box sizes and numbers of particles. Note the convergence for the largest box sizes.

Figure 6

$w_{\mathrm{eff}}$ as a function of redshift $z$ for various box sizes and numbers of particles. Note the convergence for the largest box sizes.