Physical and Geometrical Interpretation of the epsilon <= 0 Szekeres Models

TL;DR

This paper explores the properties and interpretations of epsilon <= 0 Szekeres models, focusing on their geometric and physical characteristics, regularity conditions, and implications for inhomogeneous cosmological structures.

Contribution

It provides a detailed analysis of quasi-pseudospherical and quasi-planar Szekeres models, clarifying their geometric properties, regularity conditions, and physical interpretations, especially regarding mass and radius functions.

Findings

Pseudospherical models can have spatial maxima and minima without origins.

Mass and radius functions can vary independently without shell crossings.

Planar models lack extrema and origins, and cannot be more inhomogeneous than Ellis models.

Abstract

We study the properties and behaviour of the quasi-pseudospherical and quasi-planar Szekeres models, obtain the regularity conditions, and analyse their consequences. The quantities associated with "radius" and "mass" in the quasi-spherical case must be understood in a different way for these cases. The models with pseudospherical foliation can have spatial maxima and minima, but no origins. The "mass" and "radius" functions may be one increasing and one decreasing without causing shell crossings. This case most naturally describes a snake-like, variable density void in a more gently varying inhomogeneous background, although regions that develop an overdensity are also possible. The Szekeres models with plane foliation can have neither spatial extrema nor origins, cannot be spatially flat, and they cannot have more inhomogeneity than the corresponding Ellis model, but a planar surface can be the boundary between regions of spherical and pseudospherical foliation.