Ultralocality and slow contraction

Abstract

We study the detailed process by which slow contraction smooths and flattens the universe using an improved numerical relativity code that accepts initial conditions with non-perturbative deviations from homogeneity and isotropy along two independent spatial directions. Contrary to common descriptions of the early universe, we find that the geometry first rapidly converges to an inhomogeneous, spatially-curved and anisotropic ultralocal state in which all spatial gradient contributions to the equations of motion decrease as an exponential in time to negligible values. This is followed by a second stage in which the geometry converges to a homogeneous, spatially flat and isotropic spacetime. In particular, the decay appears to follow the same history whether the entire spacetime or only parts of it are smoothed by the end of slow contraction.

Lessons for adaptive mesh refinement in numerical relativity

“…While the refinement in AMR is every bit as deterministic as it is in a box-in-a-box approach, the complexity of the underlying algorithm makes it practically impossible for a user to predict when, if and where refinement will take place. Consider for example the convergence analysis of a simulation using the truncation-error based tagging criterion of (47); in some regions of the spacetime a low resolution run may encounter a sufficiently large truncation error to trigger refinement whereas a higher-resolution run will not. To counteract this effect, one may adjust the tagging threshold in anticipation of the reduction in the truncation error, but some experimentation is often necessary because different ingredients of the code have different orders of accuracy.…”

“…We note that the error (47) clearly must be computed before we average the finer grid values onto the coarser grid. As Chombo uses a cell-centered scheme, in order to compare the values of f on the two levels, we interpolate f from the coarser level onto the finer level using fourth order interpolation.…”

“…We also show the efficacy of truncation error tagging, outlined in section 3.4. We choose the variables f as defined in (47) to be χ, K, φ and π, as these capture the information in the Hamiltonian constraint H. We use the tagging criterion (48), explicitly…”

“…In contrast to the moving boxes approach, this approach allows for highly flexible "many-boxes-in-many-boxes" mesh topologies, enabling the study of dynamical systems where the spacetime dynamics are not driven by localized compact systems e.g. in studying non spherical collapse scenarios [33,34], higher dimensional black holes/black string evolution [35][36][37][38][39][40], cosmic string evolution [41][42][43], and the behavior of strongly inhomogeneous cosmological spacetimes [44][45][46][47][48][49][50][51][52].…”

Numerical Relativity as a New Tool for Fundamental Cosmology

“…This stiffness problem has been successfully addressed by using a 'Hubble-normalized' formulation in which the Hubble radius does not enter explicitly (see e.g. [20,6]). The analogous problem pertaining to studying the robustness of inflationary spacetimes to cosmic initial conditions is yet unresolved, as we will discuss below in Sec.…”