Thermodynamical properties of dark energy

We have investigated the thermodynamical properties of dark energy. Assuming that the dark energy temperature $T\sim a^{-n}$ and considering that the volume of the Universe enveloped by the apparent horizon relates to the temperature, we have derived the dark energy entropy. For dark energy with constant equation of state $w>-1$ and the generalized Chaplygin gas, the derived entropy can be positive and satisfy the entropy bound. The total entropy, including those of dark energy, the thermal radiation, and the apparent horizon, satisfies the generalized second law of thermodynamics. However, for the phantom with constant equation of state, the positivity of entropy, the entropy bound, and the generalized second law cannot be satisfied simultaneously.

Figure 1

The evolution of $S_w$ for $w=-0.9$ and $S_r$. The dotted line is for $(S_w/S_{r0})\times (T_{w0}\times {10}^{-16}/T_{r0})$ with $n=-5.0$, the dash-dotted line is for $(S_w/S_{r0})\times (T_{w0}\times {10}^{-8}/T_{r0})$ with $n=-3.5$, the dashed line is for $S_r/S_{r0}$, and the solid line is for the apparent horizon entropy $S_A/S_{r0}$.

Figure 2

The evolutions of $S_c$, $S_r$, and $S_A$ with $w_{c0}=-0.88$ and $\alpha =1.57$. The dotted lines, for $(S_c/S_{A0})\times (T_{c0}/T_{r0})$, from top to down are $n=-2$, $n=-1$, $n=1$, and $n=2$. (Note that $S_c$ is negative after MD when $n=2$.) The dashed line is for $S_r/S_{A0}$, and the solid line is for $S_A/S_{A0}$.