An essential building block for cosmological zoom-in perturbation theory

TL;DR

The paper identifies a matter horizon, defined by the vanishing of the expansion scalar $\Theta$, as a pre-caustic boundary in GR cosmology, and shows that this horizon precedes shell crossing for overdense regions on an expanding FLRW background. It introduces a geodesic-surgery framework that cuts spacetime at the horizon and glues to a mirrored sheet, effectively creating a two-sheeted, hierarchical spacetime that captures fast inner dynamics while preserving global expansion. The authors connect this relativistic construction to cosmological zoom-in N-body simulations, demonstrating an equivalence in the dynamical equations and deriving robust boundary conditions for high-resolution regions. This framework bridges linear perturbation theory and nonlinear structure formation, offering a principled foundation for boundary handling and potential backreaction studies in small-scale cosmology.

Abstract

The evolution of large-scale structure within the standard model of cosmology is well posed only up to the onset of shell crossing, where particle trajectories appear to intersect. Beyond this point, the evolution equations become non-predictive and perturbative approaches break down. We show that in General Relativity, a matter horizon forms before caustics develop for a well-defined initial over-density on an expanding FLRW spacetime. The matter horizon was first identified by Ellis and Stoeger in 2010 as a dynamical causal boundary that encloses a sub-region of spacetime where structure formation actually takes place. We construct a multi-scale hierarchical framework for the propagation of geodesic congruences that avoids the shell-crossing singularity by cutting the spacetime at the matter horizon and glueing to another spacetime with opposite orientation. We identify a relationship between the multi-scale hierarchical framework and the cosmological zoom-in N-body simulation approach, and relate the local sub-region that decouples from the Hubble flow to the region of interest in cosmological zoom-in N-body simulations. Most importantly, the multi-scale hierarchical framework provides a more robust way of implementing boundary conditions, which could benefit cosmological zoom-in N-body simulation approaches.

Figures (7)

• Figure 1: A sketch of the tractable resolution as a function of scale for zoom-in simulation. A large volume is simulated at resolution ${\rm{Res}}_{\rm{cos}}$ determined by the mass element. The dynamics of the sub-structure is probed by dividing the grid into sub-grids and sampling at smaller mass elements. The procedure is sequenced.
• Figure 2: We show the divergence of the initial relative velocity vector field(i.e equation \ref{['eq:velocity field']}) $\Theta_{\rm{ini L}} ={\partial}_i v^{i} _{\rm{ini}}$.. Note that $\Theta_{\rm{ini L}}$ has both negative and positive values, and the amplitude is much less than one at the initial hypersurface. The total expansion is positive, and it is dominated by the background component. We evaluated the initial data at a redshift of $z =10$.
• Figure 3: A schematic illustration of the propagation of two families (blue and red) of geodesics orthogonal to a deformed initial hypersurface $h^{ab}_{\rm{ini}}(x)$. The blue family of geodesics emerges from an initially underdense region and expands into the future ($\Theta_{L} >0$.), while the red family of geodesics emerges from an initially overdense region and converges in the future($\Theta_{L} <0$).
• Figure 4: The figure shows the plot of the expansion scalar as a function of proper time for various values of the density contrast. The decoupling timescale from the Hubble flow, $\Theta_{H}$, is very sensitive to the initial density contrast.
• Figure 7: This is a plot of the scale factor ($a(t)$) versus time $(t)$ with $H_0 =1$. Values of time are chosen to illustrate that the scale factor remains positive irrespective of the direction of flow of coordinate time.