Interacting Holographic Dark Energy in $f(Q)$ Gravity: Cosmological Evolution and Gravitational Wave Signatures

Abstract: In this paper, an interactive Holographic Dark Energy (HDE) model is studied in the framework of modified gravity (f(Q)). By adopting a power parameterization for the Hubble parameter, the field equations are reconstructed and the evolution of the universe at the background level and tensor perturbations are investigated. Then, using observational data (H(z)), the model parameters are constrained and the dynamical behavior of dark energy throughout the history of the universe is analyzed. Also, the study of the evolution of energy density, pressure and the Equation of State (EoS) parameter of dark energy shows that dark energy in the late universe naturally tends to a region close to the cosmological constant behavior, while in the past it followed a distinct dynamical evolution. Stability analysis based on the speed of sound also indicates that the model has good classical stability around the present era. In addition, the compatibility of the current values of the relative density parameters of matter and dark energy with the observational constraints confirms the ability of the model to reproduce the main features of the observed universe. Next, the propagation of gravitational waves in the cosmological context of the model is investigated. The results show that the corrections due to (f(Q)) gravity and the interaction between matter and dark energy can affect the evolution of tensor perturbations and produce signatures distinct from the standard scenario. Overall, the findings of this study indicate that the interactive HDE in the gravitational framework (f(Q)) can provide a consistent framework for describing the cosmic acceleration and studying the cosmological consequences of gravitational waves.

• The paper introduces interacting holographic dark energy within the f(Q) gravity framework using an analytic power-law ansatz to reconstruct the model and address the Hubble tension.
• It employs a holographic energy density with the Hubble horizon cutoff and a DM-DE interaction term, yielding parameter estimates consistent with ΛCDM at low redshifts.
• Gravitational wave propagation is modified by nonmetricity and interaction effects, resulting in distinctive damping signatures that can be probed with next-generation detectors.

Interacting Holographic Dark Energy in $f(Q)$ Gravity: Synthesis and Implications

Theoretical Framework: $f(Q)$ Gravity and HDE Synthesis

The paper presents a comprehensive study of interacting Holographic Dark Energy (HDE) within the context of $f(Q)$ gravity—a modification of the symmetric teleparallel formulation where gravity is encoded in the nonmetricity scalar $Q$ rather than curvature or torsion. The model employs an analytic power-law ansatz for the scale factor $a(t)$, which enables closed-form reconstruction of the $f(Q)$ action and facilitates direct engagement with cosmological observational constraints.

In the construction, the energy density of the HDE component is determined via the holographic principle with the Hubble horizon as the IR cutoff, i.e., $\rho_D = 3c^2 M_p^2 H^2$. The inclusion of an explicit interaction term between dark matter (DM) and dark energy (DE), modeled as $\mathcal{Q} = 3b^2 H \rho_m$, allows dynamic energy exchange in the dark sector and introduces additional phenomenological flexibility. The resulting $f(Q)$ action is generalized to $f(Q) = c^2 Q + c_1 Q^m$, where the first term recovers GR in the appropriate limit and the second term captures dynamical deviations relevant at different cosmic epochs.

Cosmological Dynamics and Observational Constraints

The parameter space of the model is directly constrained using a 46-point $f(Q)$0 dataset spanning $f(Q)$1, using a $f(Q)$2 minimization approach. The best-fit values are found to be $f(Q)$3 and $f(Q)$4, with the model demonstrating excellent consistency with both $f(Q)$5CDM and observational data for $f(Q)$6. At higher $f(Q)$7, the model's dynamics diverge from $f(Q)$8CDM due to nontrivial $f(Q)$9 corrections and interaction terms, reflecting the dominance of nonmetricity-induced dynamics at early times.

A particularly notable result is the moderation of the so-called "Hubble tension": the derived $f(Q)$0 value bridges the gap between Planck-$f(Q)$1CDM ($f(Q)$2) and direct distance-ladder measurements ($f(Q)$3), indicating that energy exchange in the dark sector allowed by the $f(Q)$4 construction can accommodate both local and global expansion-rate constraints without significant tension.

The present-day energy budget is recovered as $f(Q)$5 and $f(Q)$6, in strong agreement with CMB+BAO combined constraints. The model yields a cosmic age $f(Q)$7 Gyr, again well within current empirical estimates.

Dynamical Behavior of Dark Energy Sector

The evolution of DE energy density and pressure demonstrates quintessential thermodynamic characteristics: dark energy density is always positive and declining across cosmic time, while its pressure remains negative, thereby sustaining late-time acceleration. The reconstructed equation-of-state (EoS) parameter, $f(Q)$8, is dynamically rich: at high $f(Q)$9 ($Q$0), $Q$1 (quintessence-like), but transitions sharply toward $Q$2 (phantom regime) near $Q$3. This "quintom" behavior is unattainable in standard HDE with $Q$4 without interaction and is directly enabled by nontrivial $Q$5 and energy transfer terms.

The deceleration parameter is fixed by the power-law ansatz and is found to be negative ($Q$6), reflecting the accelerating epoch, though its constancy is a limitation of the reconstruction method rather than the theory itself.

Stability Analysis and Physical Consistency

Analysis of the squared sound speed $Q$7 in the DE sector reveals classical stability ($Q$8) in the late universe ($Q$9), but a Laplacian instability ($a(t)$0) at high redshifts. This feature—shared with various HDE and interacting DE models—prevents DE clustering in the matter-dominated era, supporting the correct sequence of structure formation. The cosmological critical density is appropriately generalized for the $a(t)$1 background, and the usual flatness constraint ($a(t)$2) holds identically.

Gravitational Wave Sector: Propagation and Observational Signatures

A central advancement of the paper is the detailed treatment of gravitational wave (GW) propagation in the $a(t)$3-HDE background, including the effects of both nonmetricity modifications and DM-DE interaction. The resulting tensor perturbation equation exhibits modified friction and source terms:

$a(t)$4

The analysis demonstrates that while GWs experience standard cosmological amplitude damping at early times (mimicking GR), late-time deviations arise from the $a(t)$5 and interaction terms. The relative GW amplitude $a(t)$6 displays scale-dependent, regular oscillatory damping from $a(t)$7 to $a(t)$8, with higher-frequency modes showing more rapid oscillations and more pronounced amplitude suppression.

These modifications manifest as changes to the friction term and GW amplitude evolution, providing specific observational signatures that could be probed by third-generation GW detectors (e.g., LISA, DECIGO). Such differences in GW propagation in the late universe could serve as discriminants between $a(t)$9-HDE models and $f(Q)$0CDM, especially if future GW standard siren measurements attain percent-level precision.

Conclusions and Outlook

This analysis substantiates that interacting HDE in the $f(Q)$1 gravity framework is both cosmologically viable and theoretically robust. The model accommodates current cosmological background data, moderates late-universe tensions (notably $f(Q)$2), and offers distinct GW phenomenology. The dynamical structure of the DE sector—especially phantom crossing enabled by interaction—represents a notable departure from non-interacting HDE and standard $f(Q)$3CDM scenarios.

On the theoretical front, $f(Q)$4 models retain second-order field equations (unlike $f(Q)$5), affording calculational tractability and avoiding higher-derivative instabilities. The formalism also generalizes the critical density and preserves background flatness, ensuring physical consistency.

Future work should focus on cosmological perturbation spectra, the nonlinear regime of structure formation, tighter joint constraints with GW and large-scale structure data, and systematic analysis of the stability and attractor structure of the interaction-modified phase space. The model's predictions for GW propagation render it testable with next-generation experiments, affording a pathway to empirically discriminate modified gravity scenarios with dynamical, interacting dark energy sectors.

Reference:

"Interacting Holographic Dark Energy in $f(Q)$6 Gravity: Cosmological Evolution and Gravitational Wave Signatures" (2607.02792)