Cosmology after Phantom Crossing by Horndeski Gravity

TL;DR

The paper investigates how phantom-crossing dark energy can arise within shift-symmetric cubic Horndeski gravity while preserving near-GR growth and no gravitational slip. By deriving the modified Friedmann equations and the scalar equation of motion, it links the expansion history to the braiding parameter $\alpha_B$ and the effective gravitational strength $G_{\rm eff}$, showing that $\alpha_B$ must remain small to fit structure formation and lensing data. It then analyzes several scenarios—natural DE domination, kinetic-structure deviations, field-evolution deviations, slow-roll limits, and the no-kinetic-term case—to map possible future evolutions of the equation of state $w$ after crossing $-1$, including convergence to $w=-1$, settling at $w<-1$, or approaching a matter-like or other nontrivial asymptote. The results highlight that both the canonical kinetic term $K$ and the cubic term $G_3$ are generally required, and that the theory predicts no gravitational slip with a slightly enhanced gravity $G_{\rm eff}\ge1$, offering concrete tests via large-scale structure growth and lensing measurements.

Abstract

One possible way to explain the observed effective dark energy equation of state crossing $w=-1$ (the phantom divide) is through modified gravity. A key point is to not view the expansion history in isolation but to take into account the other gravitational impacts on growth of large scale structure, lensing, etc. Within shift symmetric Horndeski gravity this implies three main paths for the late time cosmic expansion. All require unusual kinetic structure and we analyze their various implications for how $w$ should behave after phantom crossing.