$P-V$ criticality of higher dimensional charged topological dilaton de Sitter black holes

There are both the black hole horizon and the cosmological horizon for the charged topological dilaton de Sitter spacetime. The thermodynamic quantities on both horizons satisfy the first law of black hole thermodynamics. Because all of these thermodynamic quantities depend on the mass $M$, the electric charge $Q$, and the cosmological constant $l$, the two horizons are not independent. Considering the connection between the black hole horizon and the cosmological horizon, we derive the effective thermodynamic quantities of the ($n+1$)-dimensional charged topological dilaton de Sitter spacetime. We find that the charged topological dilaton black hole in de Sitter spacetime has a similar phase transition and critical behavior to that in anti–de Sitter spacetime.

Figure 1

$P_{\text{eff}}-V$ diagram of ($n+1$)-dimensional charge Dilaton dS black hole with ($n=4$, $\gamma =0.1$, $b=1$) for different $Q$. The critical temperatures in figures (a), (b) and (c) are $T_c=0.03619$, $T_c=0.02280$ and $T_c=0.01838$ for $Q=1$, $Q=3$, and $Q=5$, respectively. The five isotherms in every figure correspond to $T_c-0.001,T_c-0.0005,T_c,T_c+0.0005,T_c+0.001$ from bottom to top.

Figure 2

$P_{\text{eff}}-V$ diagram of ($n+1$)-dimensional charge Dilaton dS black hole with ($n=4$, $Q=3$, $b=1$) for different $\gamma$. The critical temperatures in figures (a), (b) and (c) are $T_c=0.01758$, $T_c=0.01800$, and $T_c=0.01765$ for $\gamma =0$, $\gamma =0.01$, and $\gamma =0.001$, respectively. The five isotherms in every figure correspond to $T_c-0.001,T_c-0.0005,T_c,T_c+0.0005,T_c+0.001$.

Figure 3

$S-T$ diagrams of ($n+1$)-dimensional charge Dilaton dS black hole with ($\gamma =0.1$, $b=1$) for different $n$ and $Q$ at their respective critical pressures. The critical pressures are $P_c=0.009324$ in (a) for $n=4$ and $Q=1$, $P_c=0.003697$ in (b) for $n=4$ and $Q=3$, $P_c=0.012660$ in (c) $n=5$ and $Q=3$, respectively.

Figure 4

Plots of $\beta -x,{\kappa }_T-x,C_p-x$ for $n=4$, $Q=3$, $\gamma =0.1$ and $b=1$, corresponding the critical thermodynamic quantities $P_c=0.003697$, $T_c=0.02280$. The plots (a)–(c) are plotted below the critical quantities. $P=P_c-0.0002$ in plots (a) and (c), and $T=T_c-0.0005$ in plot (b). The plots (d)–(e) are plotted at the critical values. $P=P_c$ in plots (d) and (f), and $T=T_c$ in plot (e).