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

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.