Newtonian versus relativistic cosmology

We show how the relativistic matter and velocity power spectra behave in different gauges. We construct a new gauge where both spectra coincide with Newtonian theory on all scales. However, in this gauge there are geometric quantities present which do not exist in Newtonian theory, for example the local variation of the Hubble parameter. Comparing this quantity to second-order Newtonian quantities, we find that Newtonian theory is inaccurate on scales larger than 10 Mpc. This stresses the importance of relativistic corrections to Newtonian cosmological $N$-body simulations on these scales.

Figure 1

Matter power spectrum ($P_{\delta \delta }\equiv |\delta {|}^2$), matter-velocity power spectrum ($P_{\mathbf{v}\delta }\equiv |\mathbf{v}\delta |$) and velocity power spectrum ($P_{\mathbf{v}\mathbf{v}}\equiv |\mathbf{v}{|}^2$) according to relativistic perturbation theory in different gauges (defined in the text) and according to Newtonian theory (labelled by “$N$”), for $z=0$ (left panels) and $z=100$ (right panels).

Figure 2

Density contrast in first- and second-order Newtonian cosmological perturbation theory, compared to the quantity ${\delta }_H$ (defined in the text), which represents the local modification of the Hubble rate and is a purely relativistic effect. The prefactor is chosen to make the quantities dimensionless. The different panels show the behavior on different scales: on comoving length scales larger than 10 Mpc the relativistic modification is always larger than second-order Newtonian effects, while on scales below 1 Mpc the relativistic modifications are small compared to second-order Newtonian effects. These results hold for a spatially flat dust universe.

Figure 3

The local modification of the Hubble parameter, ${\delta }_H$, plotted against the comoving wavenumber $k$. From top to bottom, the lines denote growing redshifts of 0, 2, 5, 10, 20, and 50, respectively. Solid lines mark the regime of validity of linear perturbation theory. Our results do not strictly apply in the nonlinear regime (dotted lines).