Thermodynamic implications of the gravitationally induced particle creation scenario

A rigorous thermodynamic analysis has been done at the apparent horizon of a spatially flat Friedmann-Lemaitre-Robertson-Walker universe for the gravitationally induced particle creation scenario with constant specific entropy and an arbitrary particle creation rate $\Gamma$. Assuming a perfect fluid equation of state $p=(\gamma -1)\rho$ with $\frac{2}{3} \leq \gamma \leq 2$, the first law, the generalized second law (GSL), and thermodynamic equilibrium have been studied and an expression for the total entropy (i.e., horizon entropy plus fluid entropy) has been obtained which does not contain $\Gamma$ explicitly. Moreover, a lower bound for the fluid temperature $T_f$ has also been found which is given by $T_f \geq 8\left(\frac{\frac{3\gamma}{2}-1}{\frac{2}{\gamma}-1}\right)H^2$. It has been shown that the GSL is satisfied for $\frac{\Gamma}{3H} \leq 1$. Further, when $\Gamma$ is constant, thermodynamic equilibrium is always possible for $\frac{1}{2}<\frac{\Gamma}{3H} < 1$, while for $\frac{\Gamma}{3H} \leq \text{min}\left\lbrace \frac{1}{2},\frac{2\gamma -2}{3\gamma -2} \right\rbrace$ and $\frac{\Gamma}{3H} \geq 1$, equilibrium can never be attained. Thermodynamic arguments also lead us to believe that during the radiation phase, $\Gamma \leq H$. When $\Gamma$ is not a constant, thermodynamic equilibrium holds if $\ddot{H} \geq \frac{27}{4}\gamma ^2 H^3 \left(1-\frac{\Gamma}{3H}\right)^2$, however, such a condition is by no means necessary for the attainment of equilibrium.