The elusive physical nature of Horndeski gravity is elucidated in a new approach depicting this class of theories as a dissipative effective fluid. Requiring the constitutive equations of the latter to be those of a Newtonian fluid restricts the theory to only two disconnected subclasses of “viable” Horndeski gravity. Therefore, a stress-energy tensor of the Horndeski effective fluid, linear in the first derivatives of the fluid’s 4-velocity, is a sufficient condition for gravitational waves to propagate at light speed. All other Horndeski theories correspond to exotic non-Newtonian effective fluids. The two linear Horndeski classes are studied in the framework of first-order thermodynamics of viscous fluids, which further constrains the functional form of the theory.

We apply to tensor-multi-scalar gravity the effective fluid analysis based on the representation of the gravitational scalar field as a dissipative effective fluid. This generalization poses new challenges as the effective fluid is now a complicated mixture of individual fluids mutually coupled to each other, and many reference frames are possible for its description. They are all legitimate, although not all convenient for specific problems, and they give rise to different physical interpretations. Two of these frames are highlighted, and implications for cosmology are pointed out.

We delve into the first-order thermodynamics of Horndeski gravity, focusing on spatially flat, homogeneous, and isotropic cosmologies. Our exploration begins with a comprehensive review of the effective fluid representation within viable Horndeski gravity. Notably, we uncover a surprising alignment between the constitutive relations governing the “Horndeski fluid” and those of Eckart’s thermodynamics. Narrowing our focus, we specialize our discussion to spatially flat Friedmann-Lemaître-Robertson-Walker spacetimes. Within this specific cosmological framework, we systematically analyze two classes of theories: shift symmetric and asymptotically shift symmetric. These theories are characterized by a nonvanishing braiding parameter, adding a nuanced dimension to our investigation. On the one hand, unlike the case of the “traditional” scalar-tensor gravity, these peculiar subclasses of viable Horndeski gravity never relax to General Relativity (seen within this formalism as an equilibrium state at zero temperature), but give rise to additional equilibrium states with nonvanishing viscosity. On the other hand, this analysis further confirms previous findings according to which curvature singularities are “hot” and exhibit a diverging temperature, which suggests that deviations of scalar-tensor theories from General Relativity become extreme at spacetime singularities. Furthermore, we provide a novel exact cosmological solution for an asymptotically shift-symmetric theory as a toy model for our thermodynamic analysis.

We investigate McVittie and generalized McVittie solutions for Horndeski gravity with a spatially homogeneous gravitational scalar field, which is stealth at small scales near the central object but, at large scales, sources the Friedmann–Lemaître–Robertson–Walker universe in which the central inhomogeneity is embedded. Unlike previous studies, we include matter and obtain generalized McVittie solutions in the extended cuscuton model. The possible configurations are classified according to the time dependence of the gravitational coupling, the radial energy flow, the accretion rate onto the central object, and the Hubble rate.

We explore alternative formulations of the analogy between viable Horndeski gravity and Eckart’s first-order thermodynamics. We single out a class of identifications for the effective stress-energy tensor of the scalar field fluid that, upon performing the imperfect fluid decomposition, yields constitutive relations that can be mapped onto Eckart’s theory. We then investigate how different couplings to Einstein’s gravity, at the level of the field equations, can affect the thermodynamic formalism overall. Last, we specialize the discussion to the case of “traditional” scalar-tensor theories and identify a specific choice of the coupling function that leads to a significant simplification of the formalism.

Loop quantum cosmology (LQC) is a theory which renders the Big Bang initial singularity into a quantum bounce, by means of short-range repulsive quantum effects at the Planck scale. In this work, we are interested in reproducing the effective Friedmann equation of LQC, by considering a generic f(R, P, Q) theory of gravity, where $$R=g^{\mu \nu }R_{\mu \nu }$$R = g μ ν R μ ν is the Ricci scalar, $$P=R_{\mu \nu }R^{\mu \nu }$$P = R μ ν R μ ν , and $$Q=R_{\alpha \beta \mu \nu }R^{\alpha \beta \mu \nu }$$Q = R α β μ ν R α β μ ν is the Kretschmann scalar. An order reduction technique allows us to work in f(R, P, Q) theories which are perturbatively close to General Relativity, and to deduce a modified Friedmann equation in the reduced theory. Requiring that the modified Friedmann equation mimics the effective Friedmann equation of LQC, we are able to derive several functional forms of f(R, P, Q). We discuss the necessary conditions to obtain viable bouncing cosmologies for the proposed effective actions of f(R, P, Q) theory of gravity.