Scale-Invariant Bounce Cosmology in Weyl f(Q) Gravity with Quintom Signature

We investigate a bouncing cosmological model within the Weyl-type $f(Q)$ gravity framework, employing a power-law form of the non-metricity scalar $Q$. The model successfully resolves the initial singularity problem by demonstrating a nonsingular bounce, where the universe transitions from a contracting phase $ \dot{a}(t)<0 $ to an expanding phase ($ \dot{a}(t)>0 $) at the bouncing point $t \approx 0.$ Key features include the violation of the null energy condition (NEC) near the bounce and the crossing of the phantom divide line ($ω=-1$) by the equation of state (EoS) parameter, indicating quintom-like behavior. The model exhibits accelerated expansion post-bounce, suggesting an inflationary phase. Stability analysis via the adiabatic index reveals instability near the bouncing point, while energy conditions highlight the dominance of dark energy. Additionally, the study explores scalar fields, showing that quintessence-like kinetic energy becomes negative and phantom-like kinetic energy peaks positively near the bounce, aligning with dark energy dynamics. The Hubble parameter, deceleration parameter, and Hubble radius further validate the bouncing scenario, with the latter displaying symmetric behaviour around the bounce. These results underscore the viability of Weyl-type $f(Q)$ gravity as a framework for nonsingular bouncing cosmologies, offering insights into early universe dynamics and dark energy behaviour.

💡 Research Summary

The paper presents a nonsingular bouncing cosmology within the framework of Weyl‑type f(Q) gravity, where the non‑metricity scalar Q is taken to follow a power‑law form f(Q)=α Q^ξ. By introducing a massive Weyl vector field w_μ together with a Lagrange multiplier that enforces the vanishing of the total curvature, the authors construct a gravitational action that differs fundamentally from both standard f(Q) and f(T) models. Working in a spatially flat FLRW background, they derive modified Friedmann‑like equations and adopt a scale‑factor ansatz a(t)=a₀(1+β t²)^n, which guarantees a smooth transition from contraction ( \dot a<0 ) to expansion ( \dot a>0 ) at t≈0. The Hubble parameter H(t) therefore satisfies H(0)=0 and changes sign, while its time derivative \dot H>0 at the bounce, indicating a violation of the null energy condition (NEC) as required for a bounce.

From the effective energy‑momentum tensor the authors compute the effective energy density ρ_eff and pressure p_eff, and the equation‑of‑state parameter ω_eff = p_eff/ρ_eff. They find that ω_eff crosses the phantom divide ω=−1 in the vicinity of the bounce: ω_eff>−1 (quintessence‑like) before the bounce and ω_eff<−1 (phantom‑like) immediately after, a hallmark of quintom behavior. This crossing is driven by the dynamics of the Weyl vector and the chosen power‑law exponent ξ.

Stability is examined through the adiabatic index Γ_ad = (ρ_eff + p_eff)/p_eff · dp_eff/dρ_eff. The analysis shows Γ_ad>0 in a narrow window around the bounce, signalling a temporary dynamical instability that triggers the bounce but quickly subsides as the universe expands.

The scalar‑field reconstruction proceeds by introducing two effective scalar fields, φ_q (quintessence) and φ_p (phantom), whose kinetic terms reproduce the effective ρ_eff and p_eff. Near the bounce the kinetic energy of φ_q becomes negative, while that of φ_p peaks positively, reproducing the NEC‑violating effective stress‑energy required for the bounce.

Cosmological diagnostics such as the Hubble radius R_H = 1/|H| and the deceleration parameter q = −1−\dot H/H² are plotted. R_H reaches a symmetric minimum at the bounce, and q drops below −1, indicating a short‑lived inflationary‑like accelerated phase after the bounce.

The authors emphasize that the presence of the massive Weyl vector and the Lagrange multiplier introduces a novel geometric mechanism for NEC violation, distinct from the scalar‑field potentials used in many f(R) or f(T) bounce models. The parameters ξ and the vector mass m control the energy scale and duration of the bounce, offering flexibility to fit observational constraints.

In the concluding section the paper verifies that all four standard bouncing criteria—(i) scale‑factor sign change, (ii) Hubble sign change, (iii) NEC violation, and (iv) ω crossing the phantom divide—are satisfied. The model also passes consistency checks via adiabatic stability, scalar‑field reconstruction, and cosmological parameter evolution. The authors suggest future work on perturbation analysis, comparison with CMB data, and extensions to anisotropic Bianchi‑I settings, positioning Weyl‑type f(Q) gravity as a promising arena for singularity‑free early‑universe cosmology and dark‑energy phenomenology.