Realizing the phantom-divide crossing with vector and scalar fields

In generalized Proca theories, characterized by a vector field with broken $U(1)$ gauge invariance, late-time cosmic acceleration can be realized with a dark energy equation of state in the regime $w_{\rm DE} < -1$. In such scenarios, however, a phantom-divide crossing, as recently suggested by DESI observations, is not achieved without encountering theoretical inconsistencies. We incorporate a canonical scalar field with a potential, in addition to the vector field, and show that the phantom-divide crossing from $w_{\rm DE} < -1$ to $w_{\rm DE} > -1$ can occur at low redshifts. We propose a minimal model that admits such a transition and identify the region of parameter space in which all dynamical degrees of freedom in the scalar, vector, and tensor sectors are free from ghost and Laplacian instabilities. We further investigate the evolution of linear cosmological perturbations by applying the quasi-static approximation to modes well inside the Hubble radius. The dimensionless quantities $μ$ and $Σ$, which characterize the growth of matter perturbations and the bending of light rays, respectively, depend on the sound speed $c_ψ$ of the longitudinal scalar perturbation associated with the vector field. Since $c_ψ$ is influenced by the transverse vector mode, the model exhibits sufficient flexibility to yield values of $μ$ and $Σ$ close to 1. Consequently, unlike theories such as scalar Galileons, the present model can be consistent with observations of redshift-space distortions and integrated Sachs-Wolfe-galaxy cross-correlations.

💡 Research Summary

In this paper the authors address the recent DESI indication that the dark‑energy equation‑of‑state parameter (w_{\rm DE}) may cross the phantom divide ((w=-1)) at low redshifts. Within generalized Proca (GP) theories, a single vector field with broken (U(1)) gauge symmetry can drive cosmic acceleration but, as shown by previous no‑go theorems, it cannot achieve a phantom‑divide crossing without pathological instabilities. The authors therefore extend the GP framework by adding a canonical scalar field (\phi) with a potential (V(\phi)).

The action consists of the Einstein–Hilbert term, the standard Maxwell‑like kinetic term for the vector field (A_\mu), two X‑dependent functions (G_2(X)=b_2 X^{p_2}) and (G_3(X)=b_3 X^{p_3}) (with (X\equiv -A_\mu A^\mu/2)), and a canonical scalar sector (-\frac12(\nabla\phi)^2-V(\phi)). On a flat FLRW background the vector field has only a temporal component (\chi(t)) while the scalar depends on time only. The vector equation of motion yields two branches: (i) (\chi=0) (pure quintessence) and (ii) a non‑trivial solution satisfying (G_{2,X}+3HG_{3,X}=0). The latter leads to a power‑law relation (\chi\propto H^{-1/p}) with (p\equiv1-2p_2+2p_3>0). Choosing (b_2<0) makes the vector energy density (\rho_\chi=-G_2>0) and thus a genuine dark‑energy component.

The scalar field is taken to have an exponential potential (V(\phi)=V_0 e^{-\lambda\phi/M_{\rm Pl}}), which renders the dimensionless slope (\lambda) constant. Introducing the usual dimensionless variables (\Omega_\chi, x\equiv \dot\phi\sqrt{6}M_{\rm Pl}/(3H), y\equiv \sqrt{V}/(\sqrt{3}M_{\rm Pl}H)) and (\Omega_r), the background dynamics reduce to an autonomous system (Eqs. 2.22–2.25). The effective dark‑energy equation of state becomes
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