Comparing fully general relativistic and Newtonian calculations of structure formation

In the standard approach to studying cosmological structure formation, the overall expansion of the Universe is assumed to be homogeneous, with the gravitational effect of inhomogeneities encoded entirely in a Newtonian potential. A topic of ongoing debate is to what degree this fully captures the dynamics dictated by general relativity, especially in the era of precision cosmology. To quantitatively assess this, we directly compare standard N-body Newtonian calculations to full numerical solutions of the Einstein equations, for cold matter with various magnitude initial inhomogeneities on scales comparable to the Hubble horizon. We analyze the differences in the evolution of density, luminosity distance, and other quantities defined with respect to fiducial observers. This is carried out by reconstructing the effective spacetime and matter fields dictated by the Newtonian quantities, and by taking care to distinguish effects of numerical resolution. We find that the fully general relativistic and Newtonian calculations show excellent agreement, even well into the nonlinear regime. They only notably differ in regions where the weak gravity assumption breaks down, which arise when considering extreme cases with perturbations exceeding standard values.

Figure 1

The density contrast as measured by an observer comoving with matter at the point of maximum overdensity (black and red curves) or underdensity (blue and green curves) for two cases with $\overline{\delta }={10}^{-3}$ in Eq. (9), and an initial velocity profile that is either zero or given by the Zel’dovich approximation. The general-relativistic (solid lines) and Newtonian (dotted lines) calculations closely track each other.

Figure 2

The fractional difference between the density contrast ${\delta }_{\mathrm{obs}}$ at an underdensity (top) or overdensity (bottom) in the GR calculation compared to Newtonian simulations (solid lines) or linear perturbation theory (dotted lines) for various magnitudes of the initial inhomogeneities.

Figure 3

Two-dimensional slice of ${\delta }_N$ from the Newtonian simulations at the initial ($a=1$; top) and final ($a=60$; bottom) times from the $\overline{\delta }={10}^{-2}$ case.

Figure 4

The fractional difference between the density contrast ${\delta }_{\mathrm{obs}}$ measured by an observer comoving with the fluid initially at $(x,y,z)=(\pi /k)\times (-1/2,1/2,1/2)$ (black lines) and $(\pi /k)\times (-1/2,-1/2,1/2)$ (blue lines) in the GR calculation compared to Newtonian simulations (solid lines) or linear perturbation theory (dotted lines) for $\overline{\delta }={10}^{-2}$.

Figure 5

The luminosity distance to the global maximum (OD) or minimum (UD) of the density field in the model with $\overline{\delta }={10}^{-3}$. Distances are computed for an ensemble of observers located along the line joining the two critical points. Approximate values of the scale factor at subsequent moments of light emission are provided in the bottom right corner. The upper panel shows deviation of the GR/Newtonian distances from the corresponding observable based on the FRW metric, and the bottom panel shows the fractional difference between the GR and Newtonian calculations. The shaded band indicates a linear scale of the vertical axis. The general-relativistic (thick solid lines) and Newtonian (dashed lines) simulations return consistent distances with compatible deviations from the FRW model and the predictions of linear Newtonian evolution (thin black lines, shown only for the latest emission time).

Figure 6

The same as Fig. 5 but for the case with $\overline{\delta }={10}^{-2}$. The green line shows the maximum super-Hubble flow in empty voids, i.e. ${\delta }_N\to -1$, based on Newtonian calculations.

Figure 7

Two-dimensional slice of ${\delta }_N$ from the Newtonian simulations at the initial ($a=1$; top) and final ($a=100$; bottom) times from the ${\overline{\delta }}_{\mathrm{PS}}={10}^{-2}$ case. The red and green points indicate the positions of fiducial observers initially at points of maximum overdensity and underdensity, respectively.

Figure 8

The density contrast as measured by an observer comoving with matter at the points of maximum overdensity (black and red curves) and underdensity (blue and green curves) shown in Fig. 7 for two cases with ${\overline{\delta }}_{\mathrm{PS}}={10}^{-3}$ and ${\overline{\delta }}_{\mathrm{PS}}={10}^{-2}$. The general-relativistic (solid lines) and Newtonian (dashed lines) calculations closely track each other except at the overdensity in the largest amplitude case where a noisy multistream region forms. In such regions the N-body density contrast is sensitive to the particular density estimator used, which we illustrate by also including the cloud-in-cell density estimate, as well as the one obtained from the potential through the Poisson equation (dotted lines), for one case.

Figure 9

The luminosity distance to the global minimum (left panels), the node point (middle panels), and the global maximum (right panels) of the initial density field in the cosmological model with ${\overline{\delta }}_{\mathrm{PS}}={10}^{-3}$. Distances are computed for observers located along photon rays emitted at the scale factor $a_{\mathrm{em}}=1$ in the $x$, $y$, and $z$ directions. The upper panels show deviations between the GR and Newtonian distances from the corresponding observables based on the unperturbed FRW metric, and the bottom panels show the fractional difference between the GR and Newtonian results. Both the GR and Newtonian simulations return consistent distances with nearly the same deviations from the FRW model.

Figure 10

The same as Fig. 9 but for the case with ${\overline{\delta }}_{\mathrm{PS}}={10}^{-2}$. The fractional differences are computed at the same value of affine parameters and plotted as a function of redshift from the GR simulations. The corresponding fractional differences in redshift are smaller than ${10}^{-2}$ at all times.

Figure 11

The volume average of the norm of the constraint violation ${\mathcal{C}}_a≔\square x_a-H_a$ as a function time (parametrized by a volume-averaged measure of the effective scale factor), for simulations with $\overline{\delta }={10}^{-2}$ (top) and ${\overline{\delta }}_{\mathrm{PS}}={10}^{-3}$ (bottom) at three different resolutions. The decrease in constraint violation with increasing resolution is consistent with roughly second-order convergence.

Figure 12

The relative difference in density contrast ${\delta }_{\mathrm{obs}}$ between the GR and Newtonian simulations, at three different sets of resolutions. The left and middle panels show the points of maximum overdensity and underdensity for $\overline{\delta }={10}^{-3}$ and $\overline{\delta }={10}^{-2}$, respectively (similar to Fig. 2), while the right panel shows the same intermediate points from the $\overline{\delta }={10}^{-2}$ case as in Fig. 4.

Figure 13

The relative difference in the luminosity versus redshift between the GR and Newtonian simulations at different resolutions for light rays emitted at the point of maximum overdensity that propagate to the point of maximum underdensity for the $\overline{\delta }={10}^{-3}$ (top) and $\overline{\delta }={10}^{-2}$ (bottom) cases, similar to Figs. 5 and 6.

Figure 14

The luminosity distance (normalized by the FRW value) versus redshift for a representative set of null rays in the ${\overline{\delta }}_{\mathrm{PS}}={10}^{-3}$ (top) and ${\overline{\delta }}_{\mathrm{PS}}={10}^{-2}$ (bottom) cases. We show results from both the highest resolution GR (solid black lines) and Newtonian (dashed red lines) simulations, as well as lower resolution results from both cases (grey solid and magenta dashed lines, for GR and Newtonian, respectively) to indicate the magnitude of the numerical truncation error.