Polarization modes of gravitational waves in Palatini-Horndeski theory

Quintessence in the Weyl-Gauss-Bonnet model

“…These 'counter terms' are precisely those which cause c GW ̸ = 1 in the metric formalism [40]. Similar attempts were made to address this question for more complicated Horndeski models [54,55], but the connection field equations turn out to be differential rather than algebraic, so the connection Γ α µν becomes a propagating degree of freedom of the theory. This entails relevant new aspects in the physical content of the theory [51].…”

“…Contrary to this intuition however, it was found that the constraint is easily violated at present time. 2 Additionally, there has been an intensive research on Horndeski models in the framework of the Palatini formalism [51][52][53][54][55]. 3 A striking result of that investigation has been the realisation that Horndeski models in this formalism, with a conformal coupling depending on the kinetic term X ≡ − 1 2 g µν ∂ µ ϕ∂ ν ϕ (that is, G(ϕ, X)R), do allow GWs propagating at the speed of light [53].…”

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

“…Besides a handful of special assumptions to avoid the instabilities in Horndeski theory such as "asymptotically strong gravity" or very specific models [11,[15][16][17], the alternative of Horndeski theory on spacetimes with torsion has been recently analyzed in [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. In particular, it has been shown that in Horndeski-Cartan gravity (considering torsion in the second order, metric formalism), a similar No-Go theorem also holds (in up to the quartic case ) [19]: namely, the alltime sub/luminality, stability and nonsingularity of an spatially flat FLRW cosmology are mutually inconsistent, up to a few special cases.…”

“…This amounts to introduce torsion in the second order formalism, because the equations for the metric remain of second order. Other approaches in the context of Horndeski have been recently analyzed in[20][21][22][23][24][25][26][27][28][29][30][31][32][33] …”

Scalar fields with derivative coupling to curvature in the Palatini and the metric formulation

“…Some works have taken the metric formulation Horndeski action and simply replaced the Levi-Civita connection with a connection treated as an independent variable. In general, the resulting theories do not have second order equations of motion and are not stable[64][65][66][67].…”