Integration of inhomogeneous cosmological spacetimes in the BSSN formalism

We present cosmological-scale numerical simulations of an evolving universe in full general relativity and introduce a new numerical tool, cosmograph, which employs the Baumgarte-Shapiro-Shibata-Nakamura formalism on a three-dimensional grid. Using cosmograph, we calculate the effect of an inhomogeneous matter distribution on the evolution of a spacetime. We also present the results of a set of standard stability tests to demonstrate the robustness of our simulations.

Figure 1

Initial matter power spectrum (solid) for an $N^3={128}^3$ grid, including lines (dashed) indicating $k$ and $k^{-3}$ scalings. The power spectrum is cut at $k=50\mathrm{\Delta }k$.

Figure 2

The top panel shows deviation from Minkowski space vs time for varying ${\rho }_{\text{AwA}}$ for one box crossing time. Blue and orange curves (topmost) correspond to ${\rho }_{\text{AwA}}=1$, green (middle) to ${\rho }_{\text{AwA}}=2$, and red (bottom) to ${\rho }_{\text{AwA}}=4$. Dashed lines include random fluctuations around $\tilde{D}=0$. The orange curves include damping with $c_H=1$. At later times, the growth of deviations from Minkowski can be seen when noise in the matter sector is included. The bottom panel shows the maximum constraint violation vs time, with colors as in the top figure. The matter coupling can be seen to introduce a growing mode; the fixed error present in the static $\tilde{D}$ field will source the metric continuously. The constraint damping term helps diminish constraint violation in the short term.

Figure 3

The top panel shows the analytic (blue, solid) and numerical (red, dashed) solutions for the linear wave test after 1000 box crossing times for ${\rho }_{\text{AwA}}=4$. The bottom panel shows the difference between the two curves in the top panel, scaled so differences are noticeable.

Figure 4

A cross section of the conformal field $\varphi$ after a period of relaxation, analytically (dashed green line) and numerically (blue dots) for a black hole.

Figure 5

Deviation of FRW quantities from analytic solutions for varying time steps, and resulting fractional constraint violation. The colors red, yellow, and green correspond to time steps $\mathrm{\Delta }t=\mathrm{\Delta }x/10$, $\mathrm{\Delta }t=\mathrm{\Delta }x/20$, and $\mathrm{\Delta }t=\mathrm{\Delta }x/40$ respectively. Here $\mathrm{\Delta }x=L/N=H_I^{-1}/2N$.

Figure 6

We show the satisfaction of the constraints, Hamiltonian (left panel) and momentum (right panel), for our fiducial model. The bold lines show the average value of the constraint, the shaded regions envelop 68% of the points of the grid and the dashed line shows the worst (most violation) point.

Figure 7

The level of inhomogeneity in the extrinsic curvature, ${\sigma }_K/\overline{K}$ as a function of the average conformal factor, $\overline{\varphi }$, for our fiducial model.

Figure 8

We vary the value of $c_H$ in Eq. (22). The top panel shows the effect of varying this parameter on the satisfaction of the Hamiltonian constraint, Eq. (19), over time. The bottom panel shows the level of Hamiltonian constraint violation as we vary the same parameter. In both panels we take $c_H$ to be 0 (green) and 40 (red). The bold lines show the average value of the constraint, the shaded regions envelop 68% of the points of the grid and the dashed line shows the worst (most violation) point.

Figure 9

We track the generation of inhomogeneities of the extrinsic curvature, ${\sigma }_K/\overline{K}$, (top panel) and the satisfaction of the Hamiltonian constraint, Eq. (19), as we vary the resolution of the simulation. In both plots the color scheme corresponds to resolutions ${64}^3$ (red), ${128}^3$ (fiducial, green), and ${256}^3$ (yellow). We see the behavior of ${\sigma }_K/\overline{K}$ is stable and the amount of constraint violation diminishes as resolution is increased. The bold lines show the average value of the constraint, the shaded regions envelop 68% of the points of the grid and the dashed line shows the worst (most violation) point.

Figure 10

Sixth-order convergence can be seen as resolution is varied. The ratio of errors $\parallel H{\parallel }_{\infty }$ at two resolutions $\mathrm{\Delta }x=L/N$ and $\mathrm{\Delta }{x}^{\prime }=L/(2N)$ for an $\mathcal{O}(\mathrm{\Delta }{x}^6)$ method should be $2^6=64$, plotted as a blue line. Curves for $N^3={64}^3$ and $N^3={128}^3$ are shown.

Figure 11

The generation of inhomogeneities in the extrinsic curvature, ${\sigma }_K/\overline{K}$ as we vary $k_{\text{cutoff}}$. The color scheme follows $k_{cut}=c\mathrm{\Delta }k$, for $c=2$, 4, 8, 10, 12, 14, 16, 20, 40, from blue to red (bottom to top). Note that as we initialize more modes, we are also increasing ${\sigma }_{\rho }/\overline{\rho }$.

Figure 12

Here we show how our fiducial model compares to the numerical FRW models.

Figure 13

The top panel shows variations in the extrinsic curvature, ${\sigma }_K/\overline{K}$, versus variations in ${\sigma }_{\rho }/\overline{\rho }$ over the course of many runs. The initial $\sigma \rho /\overline{\rho }$ for these runs were $\sigma \rho /\overline{\rho }=0.009$, 0.0133, 0.019, 0.027, 0.038, 0.053, 0.076, and 0.107 from bottom to top (blue to red). The bottom panel shows the parametric dependence of ${\sigma }_K/\overline{K}$ on ${\sigma }_{\rho }/\overline{\rho }$ at $\overline{\varphi }=0.5$.

Figure 14

The top panel shows the ratio of the proper length to the initial length of six sets of $2^7$ paths in our simulation. The color coding goes from paths of initial coordinate distance of $4\mathrm{\Delta }x$ (purple) to paths of initial coordinate distance $128\mathrm{\Delta }x$ (red) in spectral order. The bottom panel shows the same set of paths, but compares the ratio of proper length to initial proper length with the FLRW estimate.

Figure 15

Here we show the distribution (and growth) of $\mathrm{\Delta }$, the amount of violation of the Killing equation for the translational FRW Killing vectors, relative to the scale $K$ for initial conditions (top) and after half an $e$-fold (bottom).