Figure 1

The plane wave example: the $x$ axis always refers to the nontrivial spatial direction and is given in comoving units. The three columns show snapshots at three different redshifts, $z=100$ (perturbations are still linear), $z=3$ (the density perturbation is becoming nonlinear), and $z=0$ (today). The first row depicts the particle phase space of the simulation, it is easy to see how shell crossing leads to a spiral-like structure in $(x,v)$. In the second row we plot the density. As the central region around $x=50\mathrm{Mpc}/\mathrm{h}$ is overdense, this region undergoes gravitational collapse, resulting eventually in shell crossing and the associated formation of caustics visible as sharp spikes at $z=0$. The bottom row shows the behavior of the gravitational potentials $\Phi$ (dark blue dashed line) and $\Psi$ (light blue dot-dashed line). We see that in this example the potentials always remain small, of the order of the initial value of ${10}^{-4}$. To display the difference between the potentials more clearly, we plot it multiplied by a factor of 10 in the bottom-most panel (using otherwise the same scale as for the potentials). This difference mainly consists of a homogeneous mode that can be interpreted as a correction to the background scale factor due to nonlinear effects. The simulation was carried out with a resolution of 1024 grid points and 16384 particles (a representative subset is shown in the phase space diagram).Reuse & Permissions

Figure 2

A $\Lambda \mathrm{CDM}$ toy setup with planar symmetry: ${\Omega }_{\Lambda }\simeq 2/3$, and the initial power spectrum for $\Phi$ is flat for $k<0.075h/\mathrm{Mpc}$ and scales like $k^{-4}$ for higher wave numbers. As in Fig. 1 we show the phase space (top row), the density (middle row), and the gravitational potentials (bottom row), for three different redshifts. The last panel shows $\Phi -\Psi$ on the same scale as the potentials, but multiplied by a factor of 100 (as indicated in the panel). The difference consists again mostly of a homogeneous mode as in the plane wave example. The simulation was carried out with a resolution of 16384 grid points and contained 131072 particles.Reuse & Permissions

Figure 3

Power spectra averaged over 50 realizations of the $\Lambda \mathrm{CDM}$ toy setup for three different redshifts. The upper panels show the power spectra of the density (solid line) and of the velocity perturbations (dashed line). In the lower panels we plot the power spectra of the gravitational potentials $\Phi$ (dark blue dashed line) and $\Psi$ (light blue dot-dashed line) as well as their difference (orange solid line) compared to the second-order estimate (green dotted line) based on $\xi$ from 26. The red curves on the bottom of the plot show the truncation error (see text for definition) of the gravitational potentials and indicate the expected numerical accuracy, and hence the level to which we can trust $\Phi -\Psi$. Each simulation of the ensemble was carried out with a resolution of 4096 grid points and contained 32768 particles.Reuse & Permissions

Figure 4

The homogeneous mode of $\Phi$ for the plane wave setup (green, dashed) and the $\Lambda \mathrm{CDM}$ setup (blue, dot-dashed). This homogeneous mode shows the size of the backreaction effect for the plane-symmetric case, i.e. the amount by which the average evolution of the perturbed universe differs from the exact FLRW solution. More precisely, it can be absorbed into the background by a redefinition $a\to a(1+\int \Phi dx/\int dx)$.Reuse & Permissions

Figure 5

Difference of the phase space coordinates for $N$-body particles in the relativistic simulation of Fig. 2 and a purely Newtonian one, initialized on the same linear solution. The three colors yellow, orange, and red correspond to redshifts $z=100$, $z=3$, and $z=0$, respectively. As long as the solution is in the linear regime (yellow, $z=100$), the difference in particle positions is essentially due to gauge dependence. At later times, however, particle positions and velocities become affected by relativistic corrections. Nevertheless, the phase space distribution of particles for the two different simulation techniques remains highly correlated, with velocities in agreement to well within a percent, and particle positions within a few $\mathrm{kpc}/h$.Reuse & Permissions