Incorporating curved geometry in cosmological simulations

Abstract

Spatial curvature is one of the fundamental cosmological parameters that is routinely constrained from observations. The forward modelling of observations, in particular of large-scale structure, often relies on large cosmological simulations. While the so-called separate universe approach allows one to account for the effect of curvature on the expansion rate in small sub-volumes, the non-Euclidean geometry is harder to accommodate. It becomes important when observables are computed over large distances, e.g. when photons travel to us from high redshift. Here we present a fully relativistic framework to run cosmological simulations for curved spatial geometry. The issue of consistent boundary conditions is solved by embedding a spherical cap of the curved spacetime into a hole within a flat exterior, where it can undergo free expansion. The geometric nature of gravity is made explicit in our framework, allowing for a consistent forward modelling of observables inside the curved patch. Our methodology would also work with any Newtonian code to a good approximation, requiring changes only to the initial conditions and post-processing.

Figures (6)

• Figure 1: We embed a sphere of radius $r_{1}$ and initial overdensity $\delta_{1}$ inside a "hole" of radius $r_{2}$ carved from a flat FLRW exterior. This setup allows periodic boundary conditions in a cubic simulation box. Observer A at the center sees a curved FLRW spacetime on the full sky out to the distance $r_1$. Observer B close to the perimeter sees the same curved FLRW spacetime when looking inward. Observations in the curved region can reach higher redshift in this case, but only for a field of view smaller than a quarter of the full sky (example illustrated as hatched region).
• Figure 2: Illustration of the relation between the time slicing in matter-synchronous coordinates (horizontal lines correspond to constant synchronous time) and the one in Poisson gauge (blue lines), as a function of spacetime position. The x-axis indicates comoving distance from the origin. For distances larger than the outer matching radius, $r_2$, both time coordinates coincide exactly. Inside the curved patch, the matter-synchronous clocks run slower than the coordinate time of Poisson gauge. To compute the simulation time, one first chooses the proper look-back time to initial conditions for the observer, placed at the origin in this example. Then one adds a small correction due to the fact that the initial constant-time hypersurface in Poisson gauge is slightly tilted. Finally, one accounts for the time dilation between exterior coordinate clock and matter clock at the observer position, see text for details. We make some extreme parameter choices to exaggerate the effects; for realistic values used in simulations the differences in time slicing would not be visible to the eye.
• Figure 3: The distance--redshift relations inferred by an observer A at the center of the curved domain (left panel) and by an observer B close to the perimeter when looking at a field of view pointing into the curved domain (right panel). The simulations are consistent with the curved Einstein--de Sitter input model ($\tilde{\Omega}_K = -0.25$, $\tilde{\Omega}_\mathrm{m} = 1.25$, $\tilde{h} = 0.5$) to high accuracy for both observers. For the central observer A, all lines of sight are equivalent (up to numerical effects). For observer B, different lines of sight take qualitatively different paths through the curved domain. We therefore compare results for two cases: the line of sight going through the center of the domain, and the lines of sight that are 30 degrees away from that axis. We also indicate the distance--redshift relations in the flat embedding space (red dashed curves) and in the separate universe approximation that does not account for the non-Euclidean spatial geometry (blue dashed curves). The tick labels of the top axis are derived using the distance--redshift relation of the curved space.
• Figure 4: The distance--redshift relation inferred by an observer A at the center of the curved domain (left panel) and by an observer B close to the perimeter when looking at a field of view pointing into the curved domain (right panel). The simulations are consistent with the curved $\Lambda$CDM input model ($\tilde{\Omega}_K = -0.1$, $\tilde{\Omega}_\mathrm{m} = 0.4$, $\tilde{h} = 0.7$) to very high accuracy for both observers. The small residual is a nearly constant offset, corresponding to an error in the fiducial Hubble parameter of about $0.05\,\%$. For the central observer A, all lines of sight are equivalent (up to numerical effects). For observer B, different lines of sight take qualitatively different paths through the curved domain. We therefore compare results for two cases: the line of sight going through the center of the domain, and the lines of sight that are 30 degrees away from that axis. We also indicate the distance--redshift relations in the flat embedding space (red dashed curves) and in the separate universe approximation that does not account for the non-Euclidean spatial geometry (blue dashed curves). The tick labels of the top axis are derived using the distance--redshift relation of the curved space.
• Figure 5: Observed angular power spectra $C_\ell$ in the curved patch once matter perturbations are included (left panel). We show results for two redshift bins, each with a top-hat selection window of width $\Delta z_\mathrm{obs} = 0.05$, centered on $z_\mathrm{obs}=0.5$ and $z_\mathrm{obs}=1.0$. The lower redshift value can be observed from any position within the domain, and we show separate measurements for observer A at the center of the domain and observer B close to the perimeter. For a fair comparison, each observer uses the same survey footprint of about $4500$ square degrees, although observer A has more area at their disposal. To reduce sample variance, we average the measurements over four nearly independent surveys. For observer A this simply means using different patches on the sky, while for observer B we choose four different locations within the curved domain. Our results show good agreement with the linear prediction from CLASS (dashed lines), except for the highest multipoles where linear theory is expected to break down. The right panel shows a density slice of our simulation at redshift $z=0$ for illustration purpose.