Distinguishing modified gravity with just two tensorial degrees of freedom from general relativity: Black holes, cosmology, and matter coupling

Abstract

We consider spatially covariant modified gravity in which the would-be scalar degree of freedom is made non-dynamical and hence there are just two tensorial degrees of freedom, i.e., the same number of dynamical degrees of freedom as in general relativity. Focusing on a class of such modified gravity theories characterized by three functions of time, we discuss how modified gravity with two tensorial degrees of freedom can be distinguished observationally or phenomenologically from general relativity. It is checked that the theory gives the same predictions as general relativity for weak gravitational fields and the propagation speed of gravitational waves. We also find that there is no modification to asymptotically flat black hole solutions. Due to a large degree of freedom to choose the time-dependent functions in the theory, the homogeneous and isotropic cosmological dynamics can be made close to or even identical to that of the ΛCDM model. We investigate the behavior of cosmological perturbations in the long and short wavelength limits and show that in both limits the effects of modified gravity appear only through the modification of the background evolution. Finally, it is remarked that in the presence of a galileon field in the matter sector, the scalar degree of freedom is revived, ruining the essential feature of the theory.

PPN meets EFT of dark energy: post-Newtonian approximation in higher-order scalar-tensor theories

“…[31] (see also [88]). The action was originally given in terms of the Arnowitt-Deser-Misner variables in the unitary gauge [31], and later it was written in a fully covariant form as a limiting case of the U-DHOST theory [33]. The functions in the action of the shift-symmetric TTDOF theory we consider are given by…”

“…The spatially covariant gravity theory includes the ghost condensation [20], Hořava-Lifsitz gravity [21,22], and its infrared limit, the khronometric theory [23], as well as the effective field theory (EFT) of inflation/dark energy [24][25][26][27] as concrete examples. Moreover, there exists an interesting family of the spatially covariant gravity theory having only two tensorial degrees of freedom (TTDOF) and no scalar propagating mode, which has drawn much attention recently [28][29][30][31][32][33].…”

Spatially covariant gravity with nonmetricity

“…In [13,14], a class of spatially covariant gravity (SCG) theories that propagate a single scalar degree of freedom was constructed, which was further extended by introducing the velocity of lapse function [15,16]. A large class of SCG Lagrangians without any scalar degree of freedom has been investigated in [17,18] and applied in the study of cosmology and black holes [19][20][21][22]. 1 Additionally, constraints from gravitational waves on SCG have been explored in [28][29][30][31], from which one can see that SCG provides a broad framework to study various modified gravity theories in a unifying manner.…”

Gravitational wave constraints on spatial covariant gravities

“…One natural question now is whether the new terms beyond GR introduced in the spatial covariant gravities can lead to any observational effects in the current and/or forthcoming experiments and observations, so the spatial covariant gravities can be tested or constrained directly by observations. Such considerations have attracted a great deal of attention lately and several phenomenological implications of the spatial covariant gravities have been already investigated [26][27][28][29][30][31][32]. In particular, the effects of the spatial covariant gravities on the propagation of GWs in the cosmological background has been preciously explored in [26].…”