Dynamical constraints on variable vacuum energy in Brans-Dicke theory

In this research work, we investigate the late-time accelerated expansion of the universe within the framework of Brans-Dicke theory by considering dynamical vacuum energy models with a time-varying cosmological constant. Two vacuum energy models are studied, namely the hybrid vacuum law $Λ(t)=αH^{2}+β\dot{H}$ and the power vacuum law $Λ(H)=α_{1}H^{n}$, where $α$, $β$, $α_{1}$ and $n$ are free parameters. We derive analytical solutions for the Hubble parameter and other relevant cosmological quantities. The evolution of the deceleration parameter, the effective equation of state parameter, the cosmographic parameters, the behaviour of Om($\mathit{z}$) diagnostics and the present age of the universe are examined. Furthermore, the analysis of the $ω_{\rm eff}-ω’_{\rm eff}$ plane shows that the model evolves in the freezing region and the thermodynamic analysis confirms that the generalized second law of thermodynamics is satisfied within the power vacuum law model.

💡 Research Summary

In this work the authors explore late‑time cosmic acceleration within the scalar‑tensor framework of Brans‑Dicke (BD) theory by allowing the vacuum energy density (equivalently the cosmological constant Λ) to evolve with cosmic time. Two phenomenological prescriptions for Λ are considered: (i) a hybrid law Λ(t)=α H²+β Ĥ that depends on both the Hubble parameter H and its time derivative, and (ii) a power‑law form Λ(H)=α₁ Hⁿ where n is a free exponent. The BD scalar field φ is assumed to follow a simple power‑law dependence on the scale factor, φ∝a^ε, with ε≪1 so that variations of the effective Newton constant remain compatible with Solar‑System bounds (large ω_BD). Substituting these ansätze into the BD field equations yields a single evolution equation for H:  Ĥ+(3+ε)² H² = 3 Λ (6+6ε−ω_BD ε²).

For the hybrid model the substitution leads to a first‑order linear differential equation in H, which integrates to a power‑law solution  H(z)=H₀(1+z)^{