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

Abstract

We disclose remarkable features of the scalar-tensor theory with the derivative coupling of the scalar field to the curvature in the Palatini formalism. Using disformal transformations, we show that this theory is free from Otrogradski ghosts. For a special relation between two coupling constants, it is disformally dual to the Einstein gravity minimally coupled to the scalar, which opens the way to constructing several exact solutions. The disformal transformation degenerates at the boundary of the physical region, near which the desingularization properties are revealed, illustrated by exact solutions: non-singular accelerating cosmology and a static spherically symmetric geon. We also construct the exact pp-waves of this theory propagating at the speed of light.Introduction. Multimessenger gravitational wave astronomy, which commenced with the detection of the binary neutron star merger GW170817 by LIGO-VIRGO collaboration [1] and subsequent observation of its electromagnetic counterparts [2], showed that the velocity of gravitational waves is equal to the speed of light to within 10 −15 . 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]. One of the major problems is to avoid the Ostrogradski ghosts. While the Horndeski models are free from ghosts in the metric approach, they can have ghosts in the Palatini formalism. Recently this problem was investigated using Bekenstein's disformal transformations [29], which turned out to be extremely useful both in the context of the metric [28,[30][31][32] and Palatini [24,27] theories. These transformations, which depend on the derivatives of the scalar field, are not point-like. Nevertheless, if they are invertible, two disformally dual theories are classically equivalent [33][34][35][36][37][38][39].Here we revisit the two-parameter scalar-tensor theory with derivative couplings to the Ricci tensor and scalar [40,41], which in the metric version is able to provide an inflationary mechanism without the potential [42,43], in the case when the scalar couples to the Einstein tensor. Unfortunately, this model is now in question. We study here the derivatively coupled theory in the first order formalism, essentially using the disformal t...

A generalized Weyl structure with arbitrary non-metricity

“…A similar transformation is considered in [58] in the Palatini formalism, in that case, there are some situations in which the disformal vector ∂ µ φ is null-like [59], which implies that gravitational waves propagate at the speed of light, thus preserving the causal structure.…”

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

“…whose roots can be found explicitly (for more details see [44]), so with such precautions, we can say that the transformation between two metrics is reversible.…”

“…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.…”

“…Introduced by Bekenstein [52] on the basis of Finsler geometry as generalization of conformal transformations, they reappeared in derivatively coupled ST theories as relations between different frames [53][54][55][56][57]. They can be used to obtain new Lagrangians, or as the solution generation tools [44,57], they also naturally arise in Palatini versions of STs [42,44,48] as relating two canonical frames. Special class constitute invertible disformal transformations: these do not change the number of degrees of freedom [58][59][60][61][62][63][64][65], so two ST theories related by an invertible disformal transformation mathematically are equivalent.…”

Conformally invariant proper time with general non-metricity

“…While the consequences of torsion have been fairly analyzed up to date, those of non-metricity have not yet been fully explored. Several classes of theories which feature torsion and/or non-metricity that have been studied in the literature are gauge theories of gravity [7][8][9], Ricci-Based gravity theories (which encompass Palatini f (R) or Born-Infeld gravity for instance) [10][11][12][13][14][15][16][17][18], teleparallel and symmetric teleparallel [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35], hybrid gravity [36], Palatini scalar-tensor theories [37], infinite derivative theories, etc. [38].…”