New HDE models with higher derivatives of the Hubble parameter $H$

TL;DR

This work investigates two higher-derivative holographic dark energy models in a non-flat FRW universe under a power-law expansion $a(t)=b_0 t^n$. Analytic expressions for the DE density, pressure, EoS, deceleration parameter, and the evolution of the DE fraction are derived for both non-interacting and interacting dark sectors with a coupling $Q=3 d^{2} H \rho_m$, and for a second model an extra parameter $\zeta$ is introduced. The EoS satisfies $\omega_D=-1+2/(3n)$ in the non-interacting case, approaching a cosmological-constant-like behavior for large $n$, while the DE fraction is constant in these setups; the interacting case modifies the evolution of $\rho_m$, $\omega_D$ and $H(z)$ but preserves late-time acceleration. Numerical estimates of the present-age $t_0$ show values close to $13.8$ Gyr, with ages increasing with $n$ and with inter-sector coupling, highlighting both the phenomenological viability and limitations of higher-derivative DE models. Overall, the study demonstrates the impact of higher derivatives of the Hubble parameter on cosmic evolution and sets the stage for data-driven parameter constraints and exploration of alternative scale factors.

Abstract

In this work, we investigate two Dark Energy (DE) models characterized by higher-order derivatives of the Hubble parameter $H$, which generalize previously proposed DE scenarios. Assuming a power-law form of the scale factor $a(t)$ given by $a(t)=b_0t^n$, we derive analytical expressions for the DE energy density, pressure, the Equation of State (EoS) parameter, the deceleration parameter and the evolutionary form of the fractional DE density. Both non-interacting and interacting dark sector frameworks are examined, with the interaction modeled through a coupling term proportional to the Dark Matter (DM) energy density. For specific parameter sets corresponding to power-law indices $n=2$, $n=3$, and $n=4$, we compute the present age of the Universe. The values obtained slightly deviate from the observationally inferred age of $\approx 13.8$ Gyr; moreover, a systematic trend is identified, with larger $n$ leading to higher ages. Furthermore, interacting scenarios consistently predict larger ages compared to their non-interacting counterparts. These results highlight the phenomenological viability and limitations of higher-derivative DE models in describing the cosmic evolution.

Figures (2)

• Figure 1: Plot of the expression of $\omega_{D,1}$ for $n \in \left[0.1-10\right]$.
• Figure 2: Plot of the expression of $\omega_{D,1,I}$ for $n \in \left[0.1-10\right]$.