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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 finite‐time, 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.

The osgood criterion and finite‐time cosmological singularities