The scalar particle production through a scalar field non-minimally coupled with geometry is investigated in the context of a spatially homogeneous and isotropic universe. In this paper, in order to study the evolution of particle production over time in the case of analytical solutions, we focus on a simple Horndeski theory. We first suppose that the universe is dominated by a scalar field and derive the energy conservation condition. Then from the thermodynamic point of view, the macroscopic non-conservation of the scalar field energy-momentum tensor can be explained as an irreversible production of the scalar particles. Based on the explanation, we obtain a scalar particle production rate and the corresponding entropy. Finally, since the universe, in general, could be regarded as a closed system satisfying the laws of thermodynamics, we naturally impose some thermodynamic constraints on it. The thermodynamic properties of the universe can provide additional constraints on the simple Horndeski theory.
We review and discuss some recent progress in Lovelock and Horndeski theories modifying Einstein's General Relativity. Using as our guide the uniqueness properties of these modified gravity theories we then discuss how Kaluza-Klein reduction of Lovelock theory can lead to effective scalar-tensor actions including several important terms of Horndeski theory. We show how this can be put to practical use by mapping analytic black hole solutions of one theory to the other. We then elaborate on the subset of Horndeski theory that has self-tuning properties and review a generic method giving scalar-…
The vector-tensor Horndeski theory is supplemented by a real, massless scalar field non-minimally coupled to the Horndeski interaction term. The generic dyonic Reissner-Nordstrom solutions characterized by electric and magnetic charges, acquire an instability and bifurcate into electromagnetically charged solutions presenting an horizon and a cloud of scalar hairs at specific values of the non-minimal coupling constant. These hairy black holes are studied in terms of the physical parameters of the model and the critical phenomena limiting their domain of existence are discussed.
We study the nature of constraints and the Hamiltonian structure in a scalar-tensor theory of gravity recently proposed by Gleyzes, Langlois, Piazza and Vernizzi (GLPV). For the simple case with A_5 = 0, namely when the canonical momenta conjugate to the spatial metric are linear in the extrinsic curvature, we prove that the number of physical degrees of freedom is three at fully nonlinear level, as claimed by GLPV. Therefore, while this theory extends Horndeski's scalar-tensor gravity theory, it is protected against additional degrees of freedom.
We constrain the viable models of Horndeski gravity, written in its equivalent Generalised Galileon version, by resorting to the Witten positive energy theorem. We find that the free function $G_3(ϕ, X)$ in the Lagrangian is constrained to be a function solely of the scalar field, $G_3(ϕ)$, and relations among the free functions are found. Other criterion for stability are also analysed, such as the attractiveness of gravity, the Dolgov-Kawasacki instability and the energy conditions. Some applications for Cosmology are discussed
Recently, a no inner (Cauchy) horizon theorem for static black holes with non-trivial scalar hairs has been proved in Einstein-Maxwell-scalar theories. In this paper, we extend the theorem to the static black holes in Einstein-Maxwell-Horndeski theories. We study the black hole interior geometry for some exact solutions and find that the spacetime has a (space-like) curvature singularity where the black hole mass gets an extremum and the Hawking temperature vanishes. We discuss further extensions of the theorem, including general Horndeski theories from disformal transformations.
We consider disformal transformations in a subclass of Horndeski theory in which a scalar field is kinetically coupled to the Einstein tensor. We apply a disformal transformation on a seed hairy black hole solution of this theory and we show that there is a transition of a black hole to a wormhole. We also show that the null energy condition is violated in the wormhole configuration and we study the stability of the wormhole solution by calculating the time evolution of scalar perturbations in this geometry.
We have discovered a new type of scalarized charged black holes in a surprisingly simple system: an Einstein-Maxwell-Klein-Gordon field theory where the three fields couple non-minimally through a Horndeski vector-tensor term. In addition to hairy charged black holes, this system exhibits Horndeski-Reissner-Nordstrom solutions and ordinary Reissner-Nordstrom ones which bifurcate to the scalarized solutions. There exist also vector-tensor black hole solutions which possess central curvature singularities although their metric components are finite there, and also solutions with naked singularit…
We study a "classical" bouncing scenario in beyond Horndeski theory. We give an example of spatially flat bouncing solution that is non-singular and stable throughout the whole evolution. The model is arranged in such a way that the scalar field driving the cosmological evolution initially behaves like full-fledged beyond Horndeski, whereas at late times it becomes a massless scalar field minimally coupled to gravity.
We consider a generic scalar-tensor theory involving a shift-symmetric scalar field and minimally coupled matter fields. We prove that the Noether current associated with shift-symmetry vanishes in regular, spherically symmetric and static spacetimes. We use this fact to prove the absence of scalar hair for spherically symmetric and static stars in Horndeski and beyond theories. We carefully detail the validity of this no-hair theorem.
We study the superradiance effect in a class of scalar-tensor Horndeski theory. We first study the dynamics of a massive charged scalar wave scattered off the horizon of a Reissner-Nordström black hole which except its canonical coupling to gravity it is also coupled kinetically to curvature. We find that a trapping potential is formed outside the horizon of a Reissner-Nordström black hole, due to this coupling, and as the strength of the new coupling is increased, the scattered wave is superradiantly amplified, resulting to the instability of the Reissner-Nordström spacetime. We then consider…
We study spherically symmetric solutions with a scalar field in the shift-symmetric subclass of the Horndeski theory. Constructing an effective energy-momentum tensor of the scalar field based on the two-fluid model, we decompose the scalar field into two components: dark matter and dark energy.We find the dark-matter fluid is pressure-less, and its distribution of energy density obeys the inverse-square law. We show the scalar field dark matter can explain the galaxy rotation curve and discuss the time evolution of the dark matter in the cosmic background.
The Einstein-Hilbert action with a cosmological constant is the most general local four-dimensional action leading to second-order derivative equations of motion that are symmetric and divergence free. In higher dimensions, additional terms can appear. We investigate a generalised metric decomposition involving a scalar degree of freedom to express the higher-dimensional action as an effective four-dimensional scalar-tensor theory. From the higher-dimensional Ricci scalar alone and a subclass of our metric ansatz, we recover the subset of Horndeski theories with luminal speed of gravitational …
We generate new spherical and time-dependent solutions of viable Horndeski gravity by disforming a solution of the Einstein equations with scalar field source and positive cosmological constant. They describe dynamical objects embedded in asymptotically FLRW spacetimes and contain apparent horizons and a finite radius singularity that evolve in time in peculiar ways apparently not encountered before in Einstein and "old" scalar-tensor gravity.
We present all scalar-tensor Lagrangians that are cubic in second derivatives of a scalar field, and that are degenerate, hence avoiding Ostrogradsky instabilities. Thanks to the existence of constraints, they propagate no more than three degrees of freedom, despite having higher order equations of motion. We also determine the viable combinations of previously identified quadratic degenerate Lagrangians and the newly established cubic ones. Finally, we study whether the new theories are connected to known scalar-tensor theories such as Horndeski and beyond Horndeski, through conformal or disf…
The Horndeski theories are extended into the Lovelock gravity theory. When the canonical scalar field is uniquely kinetically coupled to the Lovelock tensors, it is named after Lovelock scalar field. The Lovelock scalar field model is a subclass of the new Horndeski theories. A most attractive feature of the Lovelock scalar field is its equation of motion is second order. So it is free of ghosts. We study the cosmology of Lovelock scalar field in the background of $7$ dimensional spacetime and present a class of cosmic solutions. These solutions reveal the physics of the scalar field is rather…
We study holographic conformal anomalies and the corresponding $a$-theorem for Einstein gravity extended with Horndeski terms that involve up to and including linear curvature tensors. We focus on our discussion in $D=5$ bulk dimensions. For the generic Horndeski coupling, the $a$-charge is the same as that in Einstein gravity, but the inclusion of the Horndeski term violates the $a$-theorem. However, there exists a critical point of the Horndeski coupling, for which the theory admits nearly AdS spacetimes with non-vanishing Horndeski scalar. The full AdS isometry is broken down by the logarit…
In this paper, we analyze a simple class of Horndeski scalar-tensor theory. We investigate several cosmological solutions for inflation, bounce scenario and finite future-time singuarities. Perturbations respect to Friedmann-Robertson-Walker metric are studied and discussed in the context of inflation.
We investigate a cosmological scenario by finding solutions using first-order formalism in the Horndeski gravity that constrains the superpotential and implies that no free choice of scalar potential is allowed. Despite this we show that a de Sitter phase at late-time cosmology can be realized, where the dark energy sector can be identified. The scalar field equation of state tends to the cosmological scenario at present time and allows us to conclude that it can simulate the dark energy in the Horndeski gravity.
We find spherically symmetric and static black holes in shift-symmetric Horndeski and beyond Horndeski theories. They are asymptotically flat and sourced by a non trivial static scalar field. The first class of solutions is constructed in such a way that the Noether current associated with shift symmetry vanishes, while the scalar field cannot be trivial. This in certain cases leads to hairy black hole solutions (for the quartic Horndeski Lagrangian), and in others to singular solutions (for a Gauss-Bonnet term). Additionally, we find the general spherically symmetric and static solutions for …
Shift-symmetric Horndeski theories admit an interesting class of Schwarzschild black hole solutions exhibiting time-dependent scalar hair. By making use of Lemaître coordinates, we analyze perturbations around these types of black holes, and demonstrate that scalar perturbations around black hole backgrounds inevitably have gradient instabilities. Taken together with previously established results, this newly-discovered instability rules out black holes with time-dependent scalar hair in Horndeski theories.
We study certain bi-scalar-tensor theories emanating from conformal symmetry requirements of Horndeski's four-dimensional action. The former scalar is a Galileon with shift symmetry whereas the latter scalar is adjusted to have a higher order conformal coupling. Employing technics from local Weyl geometry certain Galileon higher order terms are thus constructed to be conformally invariant. The combined shift and partial conformal symmetry of the action, allow us to construct exact black hole solutions. The black holes initially found are of planar horizon geometry embedded in anti de Sitter sp…
We present different Taub-NUT/Bolt-anti de Sitter (AdS) solutions in a shift-symmetric sector of Horndeski theory of gravity possessing nonminimal kinetic coupling of scalar fields to the Einstein tensor. In four dimensions, we find locally and asymptotically locally AdS solutions possessing nontrivial scalar field. In higher dimensions, analytical Taub-NUT/Bolt-AdS p-branes and solitons are obtained, supported by the existence of p Horndeski scalar fields with axionic profile. The thermodynamical properties are studied through Euclidean methods and it is found that the first law of thermodyna…
We consider the newly proposed Bahamonde-Dialektopoulos-Levi Said (BDLS) theory, that is the Horndeski analog in the teleparallel framework and thus contains a non-minimally coupled scalar field, including higher order derivatives, that leads however to second order field equations both for the tetrad and the scalar field. This theory was mostly constructed to revive those models that were severely constrained in the scalar-tensor version of the theory from the GW170817, but includes also much richer phenomenology because of the nature of the torsion tensor. For this theory we determine the pa…
The scalar-tensor theories have become popular recently in particular in connection with attempts to explain present accelerated expansion of the universe, but they have been considered as a natural extension of general relativity long time ago. The Horndeski scalar-tensor theory involving four invariantly defined Lagrangians is a natural choice since it implies field equations involving at most second derivatives. Following the formalisms of defining covariant global quantities and conservation laws for perturbations of spacetimes in standard general relativity we extend these methods to the …
Recently it was pointed out that in shift-symmetric scalar-tensor theories a black hole can have nontrivial scalar hair which depends linearly on time. We develop black hole perturbation theory for such solutions and compute the quadratic action of odd-parity perturbations. We show that around all the solutions known so far with such time-dependent scalar hair the perturbations trigger instabilities or are presumably strongly coupled.
We present new cosmological parameter constraints for general Horndeski scalar-tensor theories, using CMB, redshift space distortion, matter power spectrum and BAO measurements from the Planck, SDSS/BOSS and 6dF surveys. We focus on theories with cosmological gravitational waves propagating at the speed of light, $c_{\rm GW} = c$, implementing and discussing several previously unaccounted for aspects in the constraint derivation for such theories, that qualitatively affect the resulting constraints. In order to ensure our conclusions are robust, we compare results for three different parametri…
We study the cosmological evolution of a scalar field in Horndeski gravity, assuming that the scalar field is subdominant with respect to the cosmic fluid. We first analyse the most general shift-symmetric action that respects local Lorentz symmetry. We show that the bound on the speed of gravitational waves set by GW170817+GRB170817A imposes a constraint only on the linear coupling between the scalar and the Gauss-Bonnet invariant and this constraint is rather mild. Then, we consider some interesting examples of theories that break shift-symmetry, such as the Damour-Esposito-Farèse model of s…
We consider Einstein-Horndeski gravity with a negative bare constant as a holographic model to investigate whether a scale invariant quantum field theory can exist without the full conformal invariance. Einstein-Horndeski gravity can admit two different AdS vacua. One is conformal, and the holographic two-point functions of the boundary energy-momentum tensor are the same as the ones obtained in Einstein gravity. The other AdS vacuum, which arises at some critical point of the coupling constants, preserves the scale invariance but not the special conformal invariance due to the logarithmic rad…
Theories with higher order time derivatives generically suffer from ghost-like instabilities, known as Ostrogradski instabilities. This fate can be avoided by considering "degenerate" Lagrangians, whose kinetic matrix cannot be inverted, thus leading to constraints between canonical variables and a reduced number of physical degrees of freedom. In this work, we derive in a systematic way the degeneracy conditions for scalar-tensor theories that depend quadratically on second order derivatives of a scalar field. We thus obtain a classification of all degenerate theories within this class of sca…
We study the metric-affine versions of scalar-tensor theories whose connection enters the action only algebraically. We show that the connection can be integrated out in this case, resulting in an equivalent metric theory. Specifically, we consider the metric-affine generalisations of the subset of the Horndeski theory whose action is linear in second derivatives of the scalar field. We determine the connection and find that it can describe a scalar-tensor Weyl geometry without a Riemannian frame. Still, as this connection enters the action algebraically, the theory admits the dynamically equi…
We show that the generalized McVittie spacetime, which represents a black hole with time-dependent mass in an expanding universe, is an exact solution of a subclass of the Horndeski family of actions. The heat-flow term responsible for the energy transfer between the black hole and the cosmological background is generated by the higher-order kinetic gravity braiding term, which generalizes the cuscuton action that yields McVittie with constant mass as a solution. Finally, we show that this generalization can be understood in terms of a duality realized by a disformal transformation, connecting…
This article is intended to review the recent developments in the Horndeski theory and its generalization, which provide us with a systematic understanding of scalar-tensor theories of gravity as well as a powerful tool to explore astrophysics and cosmology beyond general relativity. This review covers the generalized Galileons, (the rediscovery of) the Horndeski theory, cosmological perturbations in the Horndeski theory, cosmology with a violation of the null energy condition, degenerate higher-order scalar-tensor theories and their status after GW170817, the Vainshtein mechanism in the Hornd…
A New Definition of Horndeski Theory and the Possibility of Multiple Scalar Field Extensions
Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK)（高能加速器研究机构粒子与核学研究所）；The Graduate University for Advanced Studies (SOKENDAI)（高级研究院）
We consider all degenerate scalar-tensor theories that depend quadratically on second order derivatives of a scalar field, which we have identified in a previous work. These theories, whose degeneracy in general ensures the absence of Ostrogradski instability, include the quartic Horndenski Lagrangian as well as its quartic extension beyond Horndeski, but also other families of Lagrangians. We study how all these theories transform under general conformal-disformal transformations and find that they can be separated into three main classes that are stable under these transformations. This lead…
We discuss two spherically symmetric solutions admitted by the Horndeski (or scalar tensor) theory in the context of solar system and astrophysical scenarios. One of these solutions is derived for Einstein-Gauss-Bonnet gravity, while the other originates from the coupling of the Gauss-Bonnet invariant with a scalar field. Specifically, we discuss the perihelion precession and the bending angle of light for these two different spherically symmetric spacetimes derived in references \cite{Maeda:2006hj} and \cite{Sotiriou:2014pfa} respectively. The later in particular, applies only to the black ho…
We study entanglement entropy in a particular tensor-scalar theory: Horndeski gravity. Our goal is two-fold: investigate the Lewkowycz-Maldacena proposal for entanglement entropy in the presence of a tensor-scalar coupling and address a puzzle existing in the literature regarding the thermal entropy of asymptotically AdS Horndeski black holes. Using the squashed cone method, i.e. turning on a conical singularity in the bulk, we derive the functional for entanglement entropy in Horndeski gravity. We analyze the divergence structure of the bulk equation of motion. Demanding that the leading dive…
The Horndeski action is the most general one involving a metric and a scalar field that leads to second order field equations in four dimensions. Being the natural extension of the well known Scalar-Tensor theories, its structure and properties are worth analysing along the experience accumulated in the latter context. Here we argue that disformal transformations play, for the Horndeski theory, a similar role to that of conformal transformations for Scalar-Tensor theories a l`a Brans-Dicke. We identify the most general transformation preserving second order field equations and discuss the issu…
Observations of neutron star mergers in the late Universe have given significant restrictions to the class of viable scalar-tensor theories. In this paper we construct black holes within the "self-tuning" class of this restricted set, whereby the bare cosmological constant is absorbed by the dynamics of the scalar, giving a lower effective cosmological constant. We use analytic expansions at the singularity, black hole and cosmological horizon, and asymptotic region, coupled with numerical solutions, to find well-behaved black holes that asymptote to the self-tuned de Sitter geometry. The geom…
In this essay I show that there exists a new way to obtain scalar-tensor field theories by combining a special scalar field on the tangent bundle of a four-dimensional manifold with a scalar field on that manifold. These two scalar fields act as a generating function for the metric tensor. When using these two scalar fields in the Horndeski Lagrangians, we discover, while seeking Friedmann-Lemaitre-Robertson-Walker type cosmological solutions, that hidden in the Horndeski Lagrangians are non-degenerate second-order scalar Lagrangians. In accordance with Ostrogradsky's work, these hidden scalar…
Invertible disformal transformations are a useful tool to investigate ghost-free scalar-tensor theories. By performing a higher-derivative generalization of the invertible disformal transformation on Horndeski theories, we construct a novel class of ghost-free scalar-tensor theories, which we dub generalized disformal Horndeski theories. Specifically, these theories lie beyond the quadratic/cubic DHOST class. We explore cosmological perturbations to identify a subclass where gravitational waves propagate at the speed of light and clarify the conditions for the absence of ghost/gradient instabi…
We construct spinning black holes (BHs) in shift-symmetric Horndeski theory. This is an Einstein-scalar-Gauss-Bonnet model wherein the (real) scalar field couples linearly to the Gauss-Bonnet curvature squared combination. The BH solutions constructed are stationary, axially symmetric and asymptotically flat. They possess a non-trivial scalar field outside their regular event horizon; thus they have scalar hair. The scalar "charge" is not, however, an independent macroscopic degree of freedom. It is proportional to the Hawking temperature, as in the static limit, wherein the BHs reduce to the …
The Horndeski scalar-tensor theory and its recent extensions allow nonlinear derivative interactions of the scalar degree of freedom. We study the matter bispectrum of large scale structure as a probe of these modified gravity theories, focusing in particular on the effect of the terms that newly appear in the so-called "beyond Horndeski" theories. We derive the second-order solution for the matter density perturbations and find that the interactions beyond Horndeski lead to a new time-dependent coefficient in the second-order kernel which differs in general from the standard value of general …
We reconsider the issue of whether scalar-tensor theories can admit stable wormhole configurations supported by a non-trivial radial profile for the scalar field. Using a recently proposed effective theory for perturbations around static, spherically symmetric backgrounds, we show that scalar-tensor theories of "beyond Horndeski" type can have wormhole solutions that are free of ghost and gradient instabilities. Such solutions are instead forbidden within the more restrictive "Horndeski" class of theories.
We construct a large class of explicit, asymptotically flat and regular wormhole solutions in higher order scalar tensor theories. The solutions are vacuum solutions of scalar tensor theory and no matter (exotic or regular) is introduced in order to support them. They are constructed via a general disformal transformation of a seed black hole solution. The seed solutions belong to a particular Horndeski theory which requires the presence of all extended Galileons and has a higher dimensional Lovelock origin. As a result, the resulting wormholes are always solutions of general beyond Horndeski …
We have recently proposed a new class of gravitational scalar-tensor theories free from Ostrogradski instabilities, in arXiv:1404.6495. As they generalize Horndeski theories, or "generalized" galileons, we call them G$^3$. These theories possess a simple formulation when the time hypersurfaces are chosen to coincide with the uniform scalar field hypersurfaces. We confirm that they contain only three propagating degrees of freedom by presenting the details of the Hamiltonian formulation. We examine the coupling between these theories and matter. Moreover, we investigate how they transform under…
A general framework for effective theories propagating two tensor and one scalar degrees of freedom is investigated. Geometrically, it describes dynamical foliation of spacelike hypersurfaces coupled to a general background, in which the scalar mode encodes the fluctuation of the hypersurfaces. Within this framework, various models in the literature---including $k$-essence, Horndeski theory, the effective field theory of inflation, ghost condensate as well as the Hořava gravity---get unified. Our framework generalizes the Horndeski theory in the sense that, it propagates the correct number of …
