Holographic Dark Energy from a Polynomial Expansion in the Hubble Parameter

TL;DR

The paper develops a generalized holographic dark energy framework in a flat FLRW universe, where the DE density scales as $\rho_{\mathrm{de}}=3\,c^{2}(t)\,H^{2}$ with a polynomial $c^{2}(t)=\beta_{1}+\beta_{2}H^{2}+\beta_{3}H^{4}$, effectively incorporating $H^{2}$, $H^{4}$, and $H^{6}$ corrections. By identifying the Hubble horizon as the infrared cutoff, the authors derive modified Friedmann equations, connect the apparent-horizon thermodynamics to a $P$--$v$ equation of state, and demonstrate a van der Waals–type phase structure with a Maxwell equal-area transition and a critical point. They solve for the Hubble parameter via a cubic equation in $X(z)=E^{2}(z)$, calibrate parameters to reproduce $\Lambda$CDM near $z=0$, and show a dynamical dark-energy equation of state with a transient phantom phase. An extended Granda–Oliveros term is introduced, yielding a slightly higher $E(z)$ at low redshift and offering a potential avenue to alleviate the $H_{0}$ tension, while the distance modulus and generalized second law are analyzed to gauge observational viability. Overall, the work provides a thermodynamically rich, holographically motivated extension to late-time cosmology with testable implications for expansion history and horizon thermodynamics.

Abstract

This work investigates a generalized holographic dark energy (HDE) model defined by a polynomial expansion in the Hubble parameter, incorporating the first three leading terms proportional to $H^{2}$, $H^{4}$, and $H^{6}$ through a variable parameter in the expression for the energy density. The analysis is developed within the framework of a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) Universe composed of non-interacting matter and this HDE fluid. We derive the complete set of Friedmann equations to study the cosmic evolution and subsequently examine the system for the existence of thermodynamic $P-V$ type phase transitions. Finally, a comprehensive comparison with the predictions of the standard $Λ$CDM model is presented.

Figures (7)

• Figure 1: Isotherms of the holographic equation of state. The critical point $(v_{c},P_{c},T_{c})$ is indicated. For $T<T_{c}$ the non–monotonic region is replaced by a horizontal Maxwell line (dashed).
• Figure 2: Reduced Gibbs free energy as a function of pressure. Subcritical isotherm ($0.8T_c$) exhibit the characteristic swallowtail structure, indicating a first–order phase transition. For $T_A>T_c$ (shown here at $1.2T_c$) the swallowtail disappears and the Gibbs free energy becomes smooth and single-valued.
• Figure 3: $H(z)/H_{0}$ for the holographic model compared to flat $\Lambda$CDM with $\Omega_{m0}=0.315$. Also it's observed that the term $\frac{c'^{2}}{c^{2}}$ it remains bounded above by $H(z)$ over the entire range considered. In this case the prime denotes derivatives with respect to the cosmological redshift.
• Figure 4: Evolution of the dark energy EoS parameter $\omega_{\mathrm{de}}(z)$. The curve remains in the phantom regime for $z>0$ and approaches to $-1$ as $z\rightarrow{0}$. Indicating an asymptotic approach to a de Sitter state.
• Figure 5: $H(z)/H_{0}$ for the EGOHDE model compared to flat $\Lambda$CDM with $\Omega_{m0}=0.315$.