Thermodynamics of sign-switching dark energy models

We perform a comprehensive thermodynamic analysis of three sign-switching dark energy models in a flat FLRW cosmology: graduated dark energy (gDE), sign-switching cosmological constant ($Λ_s$), and smoothed sign-switching cosmological constant ($Λ_t$). We systematically derive key cosmological thermodynamic quantities – horizon temperature, horizon entropy, internal entropy, total entropy, and their first and second derivatives – using the Generalised Second Law (GSL) as the fundamental evaluation criterion. We first confirm the compliance of the $Λ$CDM model with the GSL, establishing a baseline for comparison. We find that despite their unconventional negative-to-positive energy density transitions, both $Λ_s$ and $Λ_t$ remain thermodynamically consistent. In contrast, gDE exhibits significant issues: divergences in its equation-of-state lead to infinite horizon temperature and entropy derivatives; and asymptotically, the horizon temperature diverges while entropy approaches zero, causing entropy reduction and violating the GSL. We highlight a key insight: models with divergences in the product of the dark energy equation-of-state parameter and its energy density ($w_x Ω_x$) inevitably produce thermodynamic inconsistencies in standard cosmology. This thermodynamic approach provides a complementary criterion alongside observational constraints for evaluating the physical viability of cosmological models.

💡 Research Summary

The paper conducts a thorough thermodynamic assessment of three sign‑switching dark energy (DE) scenarios—graduated dark energy (gDE), an abrupt sign‑switching cosmological constant (Λₛ), and a smoothed version using a hyperbolic tangent (Λₜ)—within a spatially flat Friedmann‑Lemaître‑Robertson‑Walker (FLRW) universe. The authors adopt the apparent horizon as the thermodynamic boundary, employing the standard relations for horizon temperature (Tₕ = |H|/(2π)(1+q)) and entropy (Sₕ = π/(G H²)). The entropy of the cosmic fluid inside the horizon is taken as Sᵢ = (ρᵢ + pᵢ) V/Tₕ, and the total entropy S_tot = Sₕ + ΣSᵢ must satisfy the Generalised Second Law (GSL): dS_tot/dt ≥ 0 for all times and d²S_tot/dt² < 0 in the far future, indicating a move toward thermodynamic equilibrium.

First, the ΛCDM model is used as a benchmark. With a constant DE density (ρₓ = const) and wₓ = –1, the product wₓ Ωₓ remains zero, guaranteeing that both the horizon and fluid entropies evolve smoothly and that the GSL holds throughout cosmic history.

Next, the three sign‑switching models are examined. Λₛ is defined by ρₓ = ρₛ₀ sgn(z* – z) with a fixed equation‑of‑state wₓ = –1. Although the energy density flips sign at the critical redshift z*, the product wₓ Ωₓ never diverges, so temperature, entropy, and their derivatives stay finite. Numerical evaluation shows a modest kink in S_tot at the transition but an overall monotonic increase, satisfying the GSL.

Λₜ replaces the discontinuous sign function with a continuous tanh profile: ρₓ = ρₛ₀ tanh