Signature flip in deceleration parameter: A thermodynamic phase transition?

Using the Hayward-Kodama temperature for the apparent horizon, it is found that matter content in the Universe is not thermodynamically stable, and the entry to the late accelerated expansion is actually a second order phase transition. The cosmological model used for the purpose is one that imitates the $\Lambda$CDM model, the favoured model for the present Universe.Comment: Title has been change

The Raychaudhuri equation for a quantized timelike geodesic congruence

A recent attempt to arrive at a quantum version of Raychaudhuri's equation is looked at critically. It is shown that the method, and even the idea, has some inherent problems. The issues are pointed out here. We have also shown that it is possible to salvage the method in some limited domain of applicability. Although no generality can be claimed, a quantum version of the equation should be useful in the context of ascertaining the existence of a singularity in the quantum regime. The equation presented in the present work holds for arbitrary $n + 1$ dimensions. An important feature of the Hamiltonian in the operator form is that it admits a self-adjoint extension quite generally. Thus, the conservation of probability is ensured.Comment: 14 pages, major changes, version accepted for publication in EPJ