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.

💡 Research Summary

The paper addresses a recent observational hint from DESI baryon acoustic oscillation data, combined with CMB and Type‑Ia supernova measurements, that the dark‑energy equation‑of‑state parameter (w_{\rm DE}) may cross the phantom divide ((w=-1)) at low redshift, moving from a phantom regime ((w<-1)) to a quintessence‑like regime ((w>-1)). The authors first examine whether such a crossing can be realized within shift‑symmetric (SS) Horndeski scalar‑tensor theories and their vector‑tensor counterparts, generalized Proca (GP) models, under the constraint that the speed of gravitational waves equals the speed of light (as required by GW170817) and that there is no direct coupling to dark matter. Using the effective field theory (EFT) of dark energy, they express the background dynamics in terms of two key functions, (\alpha_K) (the kinetic coefficient) and (\alpha_B) (the braiding term). In SS Horndeski/GP models (\alpha_K>0) guarantees (w_{\rm DE}<-1). To cross (w=-1) one must drive (\alpha_K) through zero. However, when (\alpha_K\to0) the combination (\rho_{\rm DE}+p_{\rm DE}) diverges unless (\alpha_B) simultaneously vanishes, and the scalar perturbation kinetic term (Q_s=\alpha_K+6\alpha_B^2) also goes to zero, signalling a strong‑coupling pathology. Thus, within the shift‑symmetric sector the phantom‑divide crossing is generically accompanied by ghosts, Laplacian instabilities or loss of predictivity.

To evade this obstacle the authors break the shift symmetry by adding a scalar potential (V(\phi)). They propose the Lagrangian
\