A Higher-Derivative Hubble Parameter Dark Energy Model: Cosmological Analysis and Scalar Field Correspondence

TL;DR

This work introduces a higher-derivative holographic dark energy model where the DE density depends on $H^2$ and the third derivative of the Hubble parameter via $\rho_D=3\left[\alpha\left(\dddot{H}/H^2\right)+\beta\left(\ddot{H}/H\right)+\gamma\dot{H}+\delta H^2\right]$, analyzed in a non-flat FLRW universe with a power-law scale factor $a(t)\propto t^n$. The authors derive a complete set of analytic cosmological quantities for both non-interacting and nine interacting dark-sector scenarios, including $\rho_m$, $\rho_D$, $\Omega_m$, $\Omega_D$, $H^2$, $q$, $\Omega'_D$, $p_D$, and $\omega_D$, and they explore limiting cases such as $n=2/3$ and $n(1-d^2)=2/3$. A central part of the study is the reconstruction of scalar-field correspondences—tachyon, k-essence, quintessence, Yang–Mills, and NLED—mapping the DE dynamics to familiar field theories and providing explicit expressions for the field kinetic terms and potentials. The results offer a versatile framework to study cosmic acceleration in a holographic setting and demonstrate how different interaction forms between dark matter and dark energy affect the evolution and effective equation of state, with potential implications for observational tests and model discrimination.

Abstract

In this work, we study a Dark Energy (DE) energy density model which depends on the Hubble parameter squared $H^2$ and on its first, second and third time derivatives $\dot{H}$, $\ddot{H}$ and $\dddot{H}$. Considering a scale factor $a$ with a power-law dependence on the time (with $n$ indicating the power-law index), we obtain some important cosmological quantities as function of the , like the energy densities of Matter $ρ_m$ and of DE $ρ_D$, the fractional energy densities of DM $Ω_m$ and of DE $Ω_D$, the Hubble parameter squared $H^2$, the deceleration parameter $q$, the evolutionary form of the fractional energy density of DE $Ω'_D$, the pressure of DE $p_D$ and the Equation of State (EoS) parameter of DE $ω_D$, for both non interacting and interacting cases. For the interacting case, we consider 9 different interacting term $Q$, all functions of the Hubble parameter $H$ and/or of $ρ_m$ and $ρ_D$. Finally, we establish a correspondence between the DE model we study and some scalar field theories, including tachyon, k-essence, quintessence, Yang-Mills (YM) and Nonlinear Electrodynamics (NLED) fields.