Realizing the phantom-divide crossing with vector and scalar fields

TL;DR

This work addresses realizing a phantom-divide crossing in dark energy without instabilities by embedding a canonical scalar with a shift-symmetry-breaking potential into a scalar-vector-tensor (generalized Proca) framework. The model uses a vector field to drive $w_{DE}< -1$ and a scalar potential to trigger a late-time transition to $w_{DE}>-1$, while keeping the speed of gravitational waves equal to light and avoiding ghosts. Through a detailed perturbative analysis (tensor, vector, scalar) and a quasi-static treatment of linear perturbations, the authors show ghost- and Laplacian-free regions in parameter space and derive expressions for the growth and lensing modifications, notably $\mu=\Sigma=1+\dfrac{s\Omega_{\chi}}{3(1+s\Omega_{\chi})c_\psi^2}$, which can be close to unity with appropriate choices of the transverse-vector influence $c_\psi^2$ and $\nu_v$. The model can yield observationally viable growth and ISW-galaxy cross-correlations, offering a flexible alternative to scalar Galileon scenarios and providing a framework for accommodating DESI hints of dynamical dark energy with a late-time phantom-divide crossing.

Abstract

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.

Figures (3)

• Figure 1: Plots of $w_{\rm DE}$ as a function of the redshift $z=1/a-1$ in the ranges $10^{-4} \le 1+z \le 10^{4}$ (left) and $0 \le z \le 3$ (right). The scale factor is normalized to $a=1$ today. Each case corresponds to the model parameters (a) $s=0.2$, $\lambda=1.5$, (b) $s=0.2$, $\lambda=3$, and (c) $s=0.3$, $\lambda=3$. The present-day values of $\Omega_{\chi}$, $x$, and $y$ are (a) $\Omega_{\chi}^{(0)}=0.330$, $x^{(0)}=0.178$, $y^{(0)}=0.564$, (b) $\Omega_{\chi}^{(0)}=0.512$, $x^{(0)}=0.180$, $y^{(0)}=0.368$, and (c) $\Omega_{\chi}^{(0)}=0.508$, $x^{(0)}=0.187$, $y^{(0)}=0.371$, with $\Omega_r^{(0)}=9.00 \times 10^{-5}$ in all cases. The dashed line corresponds to the DE equation of state in the $\Lambda$CDM model.
• Figure 2: Left panel: Evolution of $\Omega_\chi$, $x^2$, $y^2$, $\Omega_r$, and $\Omega_m$ as functions of $1+z$ in case (b) of Fig. \ref{['fig1']}. Right panel: Evolution of $q_s/m^2$, $c_\psi^2$, and $u$ as functions of $1+z$ in case (b) of Fig. \ref{['fig1']}, with $p=2$ and $\nu_v=\nu^{2/[p(1+s)]}=0.1$.
• Figure 3: Left panel: Evolution of the quantity $\mu$ as a function of $1+z$ for case (b) of Fig. \ref{['fig1']}, with $p=2$. Each line corresponds to (i) $\nu_v=0.01$, (ii) $\nu_v=0.1$, (iii) $\nu_v=1$, and (iv) $\nu_v=10$. Right panel: Evolution of the quantity ${\cal F}$ as a function of $1+z$ for case (c) of Fig. \ref{['fig1']}, with $p=2$. Each line corresponds to (i) $\nu_v=0.01$, (ii) $\nu_v=0.1$, (iii) $\nu_v=1$, and (iv) $\nu_v=10$.