Quartic Horndeski-Cartan theories: a review on the stability of nonsingular cosmologies

Abstract

We briefly review recent progress in Horndeski theory (generalized Galileons) on a spacetime with torsion and the implications for the stability of the cosmological background against the graviton and the scalar mode. We highlight a critical modification of a well known NO-GO theorem that holds for nonsingular cosmological solutions in the torsionless theory. We show that these results trivially extend to a broader family of Lagrangians containing the contraction of the torsionful higher derivative terms -typical of Horndeski -with the fully antisymmetric tensor.

Healthy Horndeski cosmologies with torsion

“…(C.1). Thus, the branch (C.2) is a natural continuation to previous results in up to quartic Horndeski-Cartan theory in [27,43,44]. With (C.2) the remaining equations can be written with…”

“…be massless, but its speed is modified due to its coupling c 2 , c 3 to one of the torsion perturbations, T (1) ij . The essential aspect is that in this case T (2) ij fully decouples, because d 1 = d 2 = d 3 = 0 (namely, its equation gives c 1 T (2) ij ≡ 0), and as a consequence the modification to the graviton of the up to quartic Horndeski-Cartan theory is not as marked as in (3.1), as reported in [27,43,44].…”

“…Thus, there are more ways to build Lorentz invariant combinations of the type ( ∇µ ∇ν ϕ) p and still keep the second order equations of motion. These multiple choices of Horndeski Lagrangians on a spacetime with torsion may be named by a handful of free parameters, 5 which for different choices may lead to fundamentally different dynamics within the family of Horndeski-Cartan theories [43,44].…”

“…(2.4) 5 Namely, for Horndeski with torsion with up to L4 there are two free parameters [44]. Including L5 it is clear that many more free parameters are possible, but it has not been classified.…”

“…6 Namely, the theory that besides a graviton also propagates a scalar mode with a relativistic dispersion relation. This corresponds to a particular choice of parameters of a two-parameter family of (up to quartic G4) Horndeski-Cartan theories [43,44]. 7 In particular, note that the Einstein tensor Gµν is not symmetric due to torsion.…”