Comparing metric and Palatini approaches to vector Horndeski theory

Abstract

We compare cosmologic and spherically symmetric solutions to metric and Palatini versions of vector Horndeski theory. It appears that Palatini formulation of the theory admits more degrees of freedom. Specifically, homogeneous isotropic configuration is effectively bimetric, and static spherically symmetric configuration contains non-metric connection. In general, the exact solution in metric case coincides with the approximative solution in Palatini case. The Palatini version of the theory appears to be more complicated, but the resulting non-linearity may be useful: we demonstrate that it allows the specific cosmological solution to pass through singularity, which is not possible in metric approach.

Metric-affine gravity and inflation

“…However, most of these researches are done through the purely metric approach in which the geometry is Riemannian. There are extensions to a non-Riemannian case [44][45][46][47][48], however, the fully metric-affine formulated theory and, more importantly, their applications to cosmology are yet to be explored. This is the main purpose of this paper.…”

Conformal and kinetic couplings as two Jordan frames of the same theory

“…This led to Horndeski [33] class, rediscovered as generalized Galileons [34,35], beyond-Horndeski [36,37] and DHOST theories encompassing finally the whole set [4,38]. Initially they were proposed in the metric formalism, but later also considered in Palatini [39][40][41][42][43][44][45][46][47][48] and hybrid [45,49] versions. Generically, Palatini formulation of nonminimal theories leads to equations of motion different from their metric counterpart, however, the question may be subtle in some cases (see discussion of the f (R) theory in Ref.…”

Ghost-free Palatini derivative scalar–tensor theory: Desingularization and the speed test

“…This discovery, anticipated in [3], has already had a major impact [4-9] on modified theories of gravity [10][11][12][13][14] partially refuting the most popular Horndeski (covariant Galileon) [15][16][17] and beyond Horndeski models [18][19][20].At present, the restrictions imposed by the speed of gravitational waves are more stringent than traditional cosmological bounds. Therefore, one is urged to revise the list of existing models and look for new ones that could pass this test.The extended theories of gravity studied over the last decade [13,14] included the conventional second-order theories with the metric connection, the first order theories (Palatini) with an independent connection [21][22][23][24][25], and the hybrid models [26,27]. If in Einstein's and some modified theories both formalisms are equivalent, this is not so in the derivatively coupled scalar-tensor theories [27,28].…”

“…Now we derive the corresponding solution to DSTP theory. Since the metric is diagonal and the scalar field depends only on t, from (21), (23) and (25) we obtain a cubic algebraic equation for g tt :…”

“…The extended theories of gravity studied over the last decade [13,14] included the conventional second-order theories with the metric connection, the first order theories (Palatini) with an independent connection [21][22][23][24][25], and the hybrid models [26,27]. If in Einstein's and some modified theories both formalisms are equivalent, this is not so in the derivatively coupled scalar-tensor theories [27,28].…”