Exact solutions, finite time singularities and non-singular universe models from a variety of $Λ(t)$ cosmologies

Abstract: Cosmological models with time dependent $\Lambda$ (read as $\Lambda (t)$) have been investigated widely in the literature. Models that solve background dynamics analytically, are of special interest. Additionally, the allowance of past or future singularities at finite cosmic time in a specific model signals for a generic test on its viabilities with the current observations. Following these, in this work we consider a variety of $\Lambda (t)$ models focusing on their evolutions and singular behaviour. We found that a series of models in this class can be exactly solved when the background universe is described by a spatially flat Friedmann-Lema^{\i}tre-Robertson-Walker (FLRW) line element. The solutions in terms of the scale factor of the FLRW universe offer different universe models, such as power law expansion, oscillating, and the singularity free universe as well. However, we also noticed that a large number of models in this series permits past or future cosmological singularities at finite cosmic time. At last we close the work with a note that the avoidance of future singularities is possible for certain models under some specific restrictions.