Thermodynamics in the emergent Universe model

The emergent Universe scenario is obtained in flat Universe with equation of state (EoS) ( $$P =B\rho -A\rho ^{{1}/{2}}$$ ) (where A and B are constants) as per Mukherjee et al. The EoS of this emergent Universe (EU) model can describe the current accelerated expansion and the initial singularity of the Universe. Following the standard thermodynamical criteria, stability of the EU models has been discussed. It is noted from thermal stability and positivity of adiabatic sound speed that B satisfies the values $$B=\frac{1}{3}, 1$$ in the EU model. So, the emergent models with $$B=0, -\frac{1}{3}$$ are not supported with the stability issue. Further, the third law of thermodynamics is obeyed in this case for $$B <-1$$ (or with $$A=0$$ , but it is outside the EU), i.e., any of the four discrete value of B $$ (= 0, -\frac{1}{3}, \frac{1}{3}, 1)$$ does not support this third law. The specific heat at constant volume $$c_{v}$$ obeys the relation $$c_{v}\ge 0$$ for $$T\ge 0$$ in the EU models. Two characteristic volume scales, critical volume $$V_{c}$$ and flip volume $$V_{f}$$ are obtained from zero pressure and zero deceleration condition in this model. Physically, these should follow the relation $$V_{f}> V_{c}$$ , which are actually followed in the EU model for $$B=\frac{1}{3},1$$ .