Integrability of irrotational silent cosmological models

TL;DR

This paper investigates the mathematical conditions required for the integrability of irrotational silent cosmological models, revealing a linearisation instability and suggesting the rarity of such models with certain curvature properties.

Contribution

It formulates integrability conditions in covariant and orthonormal frame formalisms, demonstrating their failure in the non-linear case and proposing a conjecture about the scarcity of certain silent models.

Findings

Integrability conditions are satisfied in linearized models but not in the full non-linear case.

There is a linearisation instability in silent cosmological models.

It is conjectured that no inhomogeneous solutions with Petrov type I Weyl curvature exist.

Abstract

We revisit the issue of integrability conditions for the irrotational silent cosmological models. We formulate the problem both in 1+3 covariant and 1+3 orthonormal frame notation, and show there exists a series of constraint equations that need to be satisfied. These conditions hold identically for FLRW-linearised silent models, but not in the general exact non-linear case. Thus there is a linearisation instability, and it is highly unlikely that there is a large class of silent models. We conjecture that there are no spatially inhomogeneous solutions with Weyl curvature of Petrov type I, and indicate further issues that await clarification.