The paradox of soft singularity crossing and its resolution by distributional cosmological quantitities

TL;DR

This paper addresses the paradox of a universe crossing a soft singularity by redefining the anti-Chaplygin gas in a distributional sense, allowing consistent expansion and contraction phases while obeying fundamental equations.

Contribution

It introduces a novel distributional approach to resolve the soft singularity crossing paradox in cosmology, ensuring the validity of key equations and junction conditions.

Findings

The universe can pass through the soft singularity with a jump in the Hubble parameter.

Friedmann, Raychaudhuri, and continuity equations hold at the singularity.

Israel junction conditions are satisfied across the singular hypersurface.

Abstract

A cosmological model of a flat Friedmann universe filled with a mixture of anti-Chaplygin gas and dust-like matter exhibits a future soft singularity, where the pressure of the anti-Chaplygin gas diverges (while its energy density is finite). Despite infinite tidal forces the geodesics pass through the singularity. Due to the dust component, the Hubble parameter has a non-zero value at the encounter with the singularity, therefore the dust implies further expansion. With continued expansion however, the energy density and the pressure of the anti-Chaplygin gas would become ill-defined, hence from the point of view of the anti-Chaplygin gas only a contraction is allowed. Paradoxically, the universe in this cosmological model would have to expand and contract simultaneously. This obviosly could not happen. We solve the paradox by redefining the anti-Chaplygin gas in a distributional sense. Then a contraction could follow the expansion phase at the singularity at the price of a jump in the Hubble parameter. Although such an abrupt change is not common in any cosmological evolution, we explicitly show that the set of Friedmann, Raychaudhuri and continuity equations are all obeyed both at the singularity and in its vicinity. We also prove that the Israel junction conditions are obeyed through the singular spatial hypersurface. In particular we enounce and prove a more general form of the Lanczos equation.