The osgood criterion and finite‐time cosmological singularities

Abstract

In this paper, we apply Osgood's criterion from the theory of ordinary differential equations to detect finite-time singularities in a spatially flat FLRW universe in the context of a perfect fluid, a perfect fluid with bulk viscosity, and a Chaplygin and anti-Chaplygin gas. In particular, we applied Osgood's criterion to demonstrate singularity behaviour for Type 0/big crunch singularities as well as Type II/sudden singularities. We show that in each case the choice of initial conditions is important as a certain number of initial conditions leads to finitetime, Type 0 singularities, while other precise choices of initial conditions which depend on the cosmological matter parameters and the cosmological constant can avoid such a finite-time singularity. Osgood's criterion provides a powerful and yet simple way of deducing the existence of these singularities, and also interestingly enough, provides clues of how to eliminate singularities from certain cosmological models.

Singularities in particle-like description of FRW cosmology

“…It was proposed to constrain the position of singularities based directly on the ansatz of an approximation for the scale factor near the singularity [12,20,[23][24][25]. It is a model independent approach as it is based only on the mathematics of singularity analysis.…”

“…Our methodology of searching for singularities of the finite scale factor is similar to the method of detection of singularities by Odintsov et al [2,12,20,[23][24][25], by postulating the non-analytical part in a contribution to the Hubble function. In our approach we assume that singularities are related with the lack of analyticity in the potential itself or its derivatives.…”

Some Remarks on Non-Singular Spherically Symmetric Space-Times

“…When F(ρ) is a positive or negative function, in general, the Big Bang singularity is present. This may be understood as a consequence of the Osgood Criterion (OC) (see, for example, [104]): given y ≡ y(t), if f (y) is never vanishing (always positive or negative) and one has the initial value problem ẏ = f (y(t)) y(t 0 ) = y 0 , (…”