$P$--$V$ criticality, Joule--Thomson expansion, and holographic heat engine of charged Hayward-AdS black holes with a cloud of strings and perfect fluid dark matter

Abstract

We construct the charged Hayward-anti-de Sitter (AdS) black hole (BH) with a cloud of strings (CS) and perfect fluid dark matter (PFDM), and analyze its extended thermodynamic phase structure. The Hayward parameter $g$ replaces the central singularity with a de Sitter (dS) core, while the CS parameter $a$ and the PFDM parameter $β$ encode astrophysically motivated matter content. Treating the cosmological constant as pressure, we derive the thermodynamic quantities, verify the Smarr relation, and establish $P$--$V$ criticality with a van der Waals (vdW)-like small-large BH phase transition and mean-field critical exponents. The Gibbs free energy (GFE) exhibits the characteristic swallowtail below the critical pressure. The Joule-Thomson (JT) expansion yields $T_i^{\rm min}/T_c \approx 0.247$, roughly half the Reissner--Nordström-AdS value. The parameters $g$ and $Q$ contract the cooling region, $β$ expands it, and $a$ reshapes it non-monotonically. A holographic heat engine with a rectangular cycle gives efficiencies $η= 0.362$--$0.396$ and Carnot benchmarking ratios $η/η_C = 0.625$--$0.791$ across six configurations. The CS parameter improves the engine efficiency by reducing the enthalpy at fixed thermodynamic volume, while the PFDM parameter degrades it by adding gravitational enthalpy without contributing to the mechanical work.

Figures (15)

• Figure 1: Metric function $f(r)$ versus $r/M$ for the charged Hayward-AdS BH with CS and PFDM at $M=1$ and $\Lambda=-0.03$. The dotted horizontal line marks $f=0$. Navy, red, and green solid curves correspond to NE BH configurations with two horizons: $(g,Q)=(0.3,0.3)$, $(g,Q,a,\beta)=(0.2,0.4,0.1,0.5)$, and $(g,Q,a,\beta)=(0.25,0.5,0.1,0.8)$, respectively. The magenta curve is the Schwarzschild-AdS limit ($g=Q=a=\beta=0$), the dashed orange curve is the Letelier-AdS case ($a=0.1$), and the cyan dot-dashed curve is a NS configuration ($g=0.8$, $Q=1.2$, $\beta=0.3$) where $f(r)>0$ for all $r>0$.
• Figure 2: Three-dimensional embedding of the metric function $f(r)$ for the charged Hayward-AdS BH with CS and PFDM at $M=1$ and $\Lambda=-0.03$. The turquoise surface represents $f(r)$ over the $(x,y)$-plane, and the red ring marks the outer EH at $r = r_+$ where $f(r_+)=0$. From (a) to (d), the progressive narrowing of the throat reflects the combined effect of the Hayward regularization, electric charge, CS tension, and PFDM logarithmic potential in driving the BH toward extremality.
• Figure 3: Hawking temperature $T$ versus outer horizon radius $r_+/M$ for the charged Hayward-AdS BH with CS and PFDM at $g = 0.3$, $Q = 0.3$, $a = 0.1$, $\beta = 0.5$. Different curves correspond to $P = 0.0010$ (navy), $P = 0.0030$ (red), $P = 0.0060$ (green), and $P = 0.0100$ (orange). All curves share a local maximum near $r_+ \approx 0.6M$, while the large-$r_+$ behavior is controlled by the AdS pressure term $\sim 2Pr_+$.
• Figure 4: Effect of individual parameters on the Hawking temperature $T(r_+)$ at fixed $P = 0.003$. (a) Increasing $g$ suppresses the local maximum and extends the negative-temperature region; for $g = 0.6$, $T < 0$ persists up to $r_+ \approx 1.5M$. (b) Larger $Q$ deepens the negative-temperature region; for $Q = 0$ the temperature diverges at small $r_+$, while for $Q = 0.7$ it remains negative up to $r_+ \approx 1.2M$. (c) The CS parameter $a$ uniformly suppresses $T$ through the factor $(1-a)$ without altering the shape of the curve. (d) For $\beta \geq 0.6$, the logarithmic PFDM potential dominates at small $r_+$, driving $T$ to large positive values and qualitatively changing the near-horizon thermal behavior.
• Figure 5: Specific heat at constant pressure $C_P$ versus $r_+/M$ for the charged Hayward-AdS BH with CS and PFDM at $g = 0.3$, $Q = 0.3$, $a = 0.1$, $\beta = 0.5$. Curves correspond to $P = 0.0010$ (navy), $P = 0.0030$ (red), $P = 0.0060$ (green), and $P = 0.0100$ (orange). The divergences separate the locally stable small and large BH branches ($C_P > 0$) from the locally unstable intermediate branch ($C_P < 0$). Higher pressure shifts the divergences to smaller $r_+$, stabilizing the large BH phase at progressively smaller radii.