Lagrangian theory of structure formation in relativistic cosmology. VI. Comparison with Szekeres exact solutions

TL;DR

This paper explores the relationship between Szekeres exact solutions and relativistic Lagrangian perturbation schemes, demonstrating how certain solutions fit within the Relativistic Zel'dovich Approximation and discussing implications for cosmological backreaction and structure formation.

Contribution

It establishes a precise connection between Szekeres models and the RZA, and introduces conditions for backreaction effects, advancing the understanding of relativistic structure formation.

Findings

Second class Szekeres solutions are contained within RZA for irrotational dust.

Conditions for vanishing backreaction are derived and expressed as integral constraints.

Domains with no backreaction can be modeled as a lattice, recovering homogeneous background as an average.

Abstract

We examine the relation between the Szekeres models and relativistic Lagrangian perturbation schemes, in particular the Relativistic Zel'dovich Approximation (RZA). We show that the second class of the Szekeres solutions is exactly contained within the RZA when the latter is restricted to an irrotational dust source with a flow-orthogonal foliation of spacetime. In such a case, the solution is governed by the first principal scalar invariant of the deformation field, proving a direct connection with a class of Newtonian three-dimensional solutions without symmetry. For the second class, a necessary and sufficient condition for the vanishing of cosmological backreaction on a scale of homogeneity is expressed through integral constraints. Domains with no backreaction can be smoothly matched, forming a lattice model, where exact deviations average out at a given scale of homogeneity, and the homogeneous and isotropic background is recovered as an average property of the model. Although the connection with the first class of Szekeres solutions is not straightforward, this class allows for the interpretation in terms of a spatial superposition of nonintersecting fluid lines, where each world line evolves independently and under the RZA model equations, but with different associated `local backgrounds'. This points to the possibility of generalizing the Lagrangian perturbation schemes to structure formation models on evolving backgrounds, including global cosmological backreaction.