Crossing the phantom divide in scalar-tensor and vector-tensor theories

TL;DR

DESI and complementary data indicate dynamical dark energy with a phantom-divide crossing at low redshift. The authors show that such a crossing is difficult to realize in shift-symmetric Horndeski and generalized Proca theories with luminal GW speed due to ghost- and strong-coupling constraints, and they demonstrate that breaking shift symmetry via a scalar potential can enable crossing without ghosts or Laplacian instabilities. They construct an explicit low-energy EFT-like model with Lagrangian $\mathcal L = \frac{M_{Pl}^2}{2}R + a_1 X + a_2 X^2 + 3 a_3 X \Box \phi - V(\phi)$ and analyze an exponential potential, showing crossing can occur for $0<z_c<1$ with stable perturbations and modest deviations from GR in growth. The work provides a concrete, testable mechanism for phantom-divide crossing that can be constrained by DESI and Euclid, highlighting the necessity of shift-symmetry breaking in scalar-tensor/vector-tensor dark-energy models.

Abstract

DESI observations of baryon acoustic oscillations (BAOs), combined with cosmic microwave background (CMB) and type-Ia supernova (SN Ia) data, suggest that the dark energy equation of state $w_{\rm DE}$ crosses the phantom divide from $w_{\rm DE} < -1$ to $w_{\rm DE} > -1$ at low redshifts. In shift-symmetric Horndeski and generalized Proca theories with luminal gravitational-wave speed and no direct couplings to dark matter, we show that such a phantom-divide crossing is generically difficult without theoretical pathologies. Breaking the shift symmetry in Horndeski theories allows this transition. We construct an explicit model with broken shift symmetry, in which the scalar field has a potential in addition to a Galileon self-interaction and a quadratic kinetic term. This model realizes the desired phantom-divide crossing at low redshifts without introducing ghosts and Laplacian instabilities.

Figures (3)

• Figure 1: Evolution of $w_{\rm DE}$ versus the redshift $z$ for $a_1=-1$, $\lambda=1$, and three different cases: (a) $x_1=0.442$, $x_2=0$, $x_3=0.519$, $x_4=0.356$, (b) $x_1=0.792$, $x_2=0.512$, $x_3=0.672$, $x_4=0.124$, and (c) $x_1=0.792$, $x_2=0.569$, $x_3=0.605$, $x_4=0.134$ at $z=0$, respectively, where $\Omega_{m}=0.32$ and $\Omega_r=9.0 \times 10^{-5}$ today. Case (a) corresponds to $a_2 = 0$, whereas $a_2 \neq 0$ in cases (b) and (c).
• Figure 2: Evolution of $\mu$ versus redshift $z$ for cases (a)-(c) in Fig. \ref{['fig1']}, and for the tracker solution of covariant Galileons ($x_2=0$, $x_4=0$).
• Figure 3: Evolution of ${\cal F}$ versus redshift $z$ for cases (a)-(c) in Fig. \ref{['fig1']}, together with that for covariant Galileons.