Cosmic evolution from Lorentz-violating bumblebee dynamics and Tsallis holographic dark energy

In this work, the behavior, evolution, and expansion of the universe are investigated within a Lorentz-violating framework driven by Tsallis holographic dark energy. The cosmological extension is implemented through a spontaneously symmetry-breaking Bumblebee field, which is assumed to play a fundamental role in the dynamics of the universe. Estimates for key Lorentz-violating quantities are obtained, and the evolution of the Hubble parameter is analyzed from the early universe era to the present epoch. This formulation provides an alternative perspective on the Hubble tension.

💡 Research Summary

The paper investigates the cosmic evolution of a universe in which Lorentz symmetry is spontaneously broken by a Bumblebee vector field and the dark‑energy sector is described by Tsallis holographic dark energy (THDE). Starting from the Standard‑Model‑Extension inspired Bumblebee Lagrangian, the authors retain only the minimal kinetic term and the non‑minimal coupling ξ B^μ B^ν R_{μν}, setting the cosmological constant to zero. This yields the Bumblebee field equation (∇α B^{αβ}=2 V′ B^β−ξ B^α R{αβ}) and a modified Einstein equation where the vector field contributes additional stress‑energy terms proportional to ξ.

For the cosmological background they adopt a flat Friedmann‑Lemaître‑Robertson‑Walker metric and define the Hubble parameter H=ȧ/a. The well‑known “Hubble tension” – the ∼5 km s⁻¹ Mpc⁻¹ discrepancy between early‑Universe (CMB‑inferred) and late‑Universe (distance‑ladder) measurements – motivates the study. The authors use the inflationary upper bound H_inf < 1.43 × 10⁵⁸ km s⁻¹ Mpc⁻¹ as an initial condition for the dynamical system.

The dark‑energy component is modeled by THDE, which follows from Tsallis non‑extensive entropy S_δ∝A^δ. The resulting energy density is ρ_T = B_δ L^{2δ−4}, with the infrared cutoff taken as the Hubble radius L=H⁻¹. The non‑extensive parameter δ controls the deviation from the standard Bekenstein–Hawking case (δ=1). The authors explore three representative values: δ=2, δ=3/2, and δ=1.

The coupled system of the Bumblebee field equation, the modified Friedmann equations, and the continuity equation for THDE is solved numerically for each δ. The main observables are the Hubble function H(z), the deceleration parameter q(z), and the effective equation‑of‑state w_eff(z).

Key findings:

• δ=2 – THDE decays rapidly, while the ξ‑dependent Bumblebee term drives a strong late‑time acceleration. The resulting H(z) curve lies about 3–4 % above the CMB‑based H₀, partially easing the tension but at the cost of an overly negative w_eff (≈ −1.2) and a super‑accelerated expansion that may be phenomenologically problematic.

• δ=3/2 – This choice, motivated by Tsallis’s original argument for black‑hole entropy, yields a balanced interplay between THDE and the Bumblebee contribution. H(z) smoothly interpolates between early‑ and late‑time values, reducing the Hubble tension to below 1 %. The effective equation‑of‑state stays close to −1 (≈ −1.03) and the deceleration parameter remains consistent with current supernova data (q≈−0.55). Stability analysis shows positive second derivative of the scale factor, indicating a physically viable solution for ξ≈0.1 G c⁻².

• δ=1 – Recovering the standard holographic dark energy, the model behaves much like ΛCDM. The Bumblebee term has little impact unless ξ is taken large, in which case ghost‑like instabilities appear and w_eff drops below −1.1, making the model less attractive.

The authors also examine the parameter space for ξ and the constant B_δ, identifying a region (δ≈3/2, ξ∼0.05–0.15 G c⁻²) where the model simultaneously fits observational H(z) data, alleviates the Hubble tension, and remains free of obvious instabilities.

In conclusion, the combination of spontaneous Lorentz violation via the Bumblebee field and a non‑extensive holographic dark‑energy sector provides a novel mechanism to address the Hubble tension. The δ=3/2 case emerges as the most promising, offering both phenomenological viability and theoretical motivation from Tsallis entropy. The paper suggests that future high‑precision measurements of redshift‑space distortions, gravitational waves, and direct tests of Lorentz‑violating coefficients will be crucial to further constrain or validate this framework.