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 issue of viable frames for this kind of theories, in particular the possibility to cast the action in the so called Einstein frame. Finally, we investigate the physical equivalence of such frames and their reciprocal relationship.
In this paper, we have studied bi-gravity theory in a very specific limit where we focused only on one degree of freedom generated by the massive graviton. We have analyzed the model in the context of cosmology and demonstrated that the model can give rise to late time cosmic acceleration as an attractor of the dynamical system. However, the observational constraints due to tensor perturbations are stringent giving rise to large fine tuning.
In this note we collect, systemise and generalise the existing results for relations between general Horndeski theories and beyond Horndeski theories via disformal transformations of metric. We derive additional disformal transformation rules relating Lagrangian functions of beyond Horndeski theory and corresponding Horndeski theory and demonstrate that some of them become singular at some moments(s) once one constructs a non-singular cosmological solution in beyond Horndeski theory that is free from ghost, gradient instabilities and strong gravity regime during the entire evolution of the sys…
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.
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 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…
It is known that Horndeski theories can be transformed to a sub-class of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories under the disformal transformation of the metric $g_{μν} \to Ω^2(ϕ)g_{μν}+Γ(ϕ,X) \nabla_μ ϕ\nabla_ν ϕ$, where $Ω$ is a function of a scalar field $ϕ$ and $Γ$ is another function depending on both $ϕ$ and $X=g^{μν}\nabla_μ ϕ\nabla_ν ϕ$. We show that, with the choice of unitary gauge, both curvature and tensor perturbations on the flat isotropic cosmological background are generally invariant under the disformal transformation. By means of the effective field theories encompa…
Matter coupling in modified gravity theories is a nontrivial issue when the gravitational Lagrangian possesses a degeneracy structure to avoid the problem of the Ostrogradsky ghost. Recently, this issue was addressed for bosonic matter fields in the generalized disformal Horndeski class, which is so far the most general class of ghost-free scalar-tensor theories obtained by performing a higher-derivative generalization of invertible disformal transformations on Horndeski theories. In this paper, we clarify the consistency of fermionic matter coupling in the generalized disformal Horndeski theo…
The invariance of physical observables under disformal transformations is considered. It is known that conformal transformations leave physical observables invariant. However, whether it is true for disformal transformations is still an open question. In this paper, it is shown that a pure disformal transformation without any conformal factor is equivalent to rescaling the time coordinate. Since this rescaling applies equally to all the physical quantities, physics must be invariant under a disformal transformation, that is, neither causal structure, propagation speed nor any other property of…
We show that under a general disformal transformation the linear comoving curvature perturbation is not identically invariant, but is invariant on superhorizon scales for any theory that is disformally related to Horndeski's theory. The difference between disformally related curvature perturbations is found to be given in terms of the comoving density perturbation associated with a single canonical scalar field. In General Relativity it is well-known that this quantity vanishes on superhorizon scales through the Poisson equation that is obtained on combining the Hamiltonian and momentum constr…
We study the role of field redefinitions in general scalar-tensor theories. In particular, we first focus on the class of field redefinitions linear in the spin-2 field and involving derivatives of the spin-0 mode, generically known as disformal transformations. We start by defining the action of a disformal transformation in the tangent space. Then, we take advantage of the great economy of means of the language of differential forms to compute the full transformation of Horndeski's theory under general disformal transformations. We obtain that Horndeski's action maps onto itself modulo a red…
We show that very general scalar-tensor theories of gravity (including, e.g., Horndeski models) are generically invariant under disformal transformations. However there is a special subset, when the transformation is not invertible, that yields new equations of motion which are a generalization of the so-called "mimetic" dark matter theory recently introduced by Chamsedinne and Mukhanov. These conclusions hold true irrespective of whether the scalar field in the action of the assumed scalar-tensor theory of gravity is the same or different than the scalar field involved in the transformation. …
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 analyzing 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 la Brans-Dicke.
Effective description of generalized disformal theories
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Centro de Astrofísica e Gravitação - CENTRA, Departamento de Física, Instituto Superior Técnico - IST, Universidade de Lisboa - UL（里斯本大学技术研究所天体物理学与引力中心）；Division of Liberal Arts, Kogakuin University（工学院大学人文学部）
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…
In the wake of interest to find black hole solutions with scalar hair, we investigate the effects of disformal transformations on static spherically symmetric space-times with a non-trivial scalar field. In particular, we study solutions that have a singularity in a given frame, while the action is regular. We ask if there exists a different choice of field variables such that the geometry and the fields are regular. We find that in some cases disformal transformations can remove a singularity from the geometry or introduce a new horizon. This is possible since the Weyl tensor is not invariant…
We study disformal transformations in the context of scalar extensions to teleparallel gravity, in which the gravitational interaction is mediated by the torsion of a flat, metric compatible connection. We find a generic class of scalar-torsion actions which is invariant under disformal transformations, and which possesses different invariant subclasses. For the most simple of these subclasses we explicitly derive all terms that may appear in the action. We propose to study actions from this class as possible teleparallel analogues of healthy beyond Horndeski theories.
We study the structure of scalar-tensor theories of gravity based on derivative couplings between the scalar and the matter degrees of freedom introduced through an effective metric. Such interactions are classified by their tensor structure into conformal (scalar), disformal (vector) and extended disformal (traceless tensor), as well as by the derivative order of the scalar field. Relations limited to first derivatives of the field ensure second order equations of motion in the Einstein frame and hence the absence of Ostrogradski ghost degrees of freedom. The existence of a mapping to the Jor…
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.
Primordial cosmological perturbations are the seeds that were cultivated by inflation and the succeeding dynamical processes, eventually leading to the current Universe. In this work, we investigate the behavior of the gauge-invariant scalar and tensor perturbations under the general extended disformal transformation, namely, $g_{μν} \rightarrow A(X,Y,Z)g_{μν} + Φ_μΦ_ν$, where $X \equiv -\tfrac{1}{2}ϕ^{;μ}ϕ_{;μ}, Y \equiv ϕ^{;μ}X_{;μ}, Z \equiv X^{;μ}X_{;μ} $ and $Φ_μ\equiv Cϕ_{;μ} + DX_{;μ}$, with $C$ and $D$ being a general functional of $(ϕ,X,Y,Z)$. We find that the tensor perturbation is i…
Leibniz University Hannover（汉诺威莱布尼茨大学）；Max Planck Institute for Gravitational Physics, Albert Einstein Institute（马克斯·普朗克引力物理研究所，爱因斯坦研究所）；Center for Physical Sciences and Technology（物理与技术科学中心）
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…
Determining the most general, consistent scalar tensor theory of gravity is important for building models of inflation and dark energy. In this work we investigate the number of degrees of freedom present in the theory of beyond Horndeski. We discuss how to construct the theory from the extrinsic curvature of the constant scalar field hypersurface, and find a simple expression for the action which guarantees the existence of the primary constraint necessary to avoid the Ostrogradsky instability. Our analysis is completely gauge-invariant. However we confirm that, mixing together beyond Horndes…
We analyse in all generality beyond Horndeski theories of shift symmetry in a static and spherically symmetric spacetime. By introducing four auxiliary functions, we write the field equations in a particularly compact form. We show that assuming additionally parity symmetry renders the system directly integrable giving multiple families of black-hole solutions. These have typically an asymptotically-flat Reissner-Nordstrom behaviour, and emerge in the presence of a canonical kinetic term for the scalar field. In the absence of parity symmetry, we present a general method which allows us to int…
The disformal transformation of metric $g_{μν} \to Ω^2 (ϕ)g_{μν}+Γ(ϕ,X) \partial_μϕ\partial_νϕ$, where $ϕ$ is a scalar field with the kinetic energy $X= \partial_μϕ\partial^μϕ/2$, preserves the Lagrangian structure of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories (which is the minimum extension of Horndeski theories). In the presence of matter, this transformation gives rise to a kinetic-type coupling between the scalar field $ϕ$ and matter. We consider the Einstein frame in which the second-order action of tensor perturbations on the isotropic cosmological background is of the same form as…
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…
It is commonly believed that the dRGT theory is the unique way to describe a massive spin-2 field without ghosts. While dRGT is arguably the most elegant massive gravity theory, it seems that it may not be the unique ghost-free one if one relaxes the assumption of locality. In this work, we derive a new massive gravity theory by using a disformal transformation of the metric tensor in the dRGT action. Its decoupling limit lives inside the class of beyond-Horndeski Lagrangians as long as the transformation of the metric remains purely disformal. This proves the absence of ghosts in this decoupl…
We investigate invariant canonical transformations of a spatially covariant scalar-tensor theory of gravity, called the XG theory, by which the action or the Hamiltonian and the primary constraints keep their forms invariant. We derive the Hamiltonian in a non perturbative manner and complete the Hamiltonian analysis for all regions of the theory. We confirm that the theory has at most 3 degrees of freedom in all regions of the theory as long as the theory has the symmetry under the spatial diffeormorphism. Then, we derive the invariant canonical transformation by using the infinitesimal trans…
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）
Unique gravitational wave signatures of GLPV scalar-tensor theories
Leibniz University Hannover（汉诺威莱布尼茨大学）；Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut（马克斯·普朗克引力物理研究所，阿尔伯特·爱因斯坦研究所）；Institute for Basic Science (IBS)（基础科学研究院）；Institute of Science Tokyo（东京科学大学）；Yonsei University（延世大学）
We explore general scalar-tensor models in the presence of a kinetic mixing between matter and the scalar field, which we call Kinetic Matter Mixing. In the frame where gravity is de-mixed from the scalar this is due to disformal couplings of matter species to the gravitational sector, with disformal coefficients that depend on the gradient of the scalar field. In the frame where matter is minimally coupled, it originates from the so-called beyond Horndeski quadratic Lagrangian. We extend the Effective Theory of Interacting Dark Energy by allowing disformal coupling coefficients to depend on t…
Kinetic mixing between the metric and scalar degrees of freedom is an essential ingredient in contemporary scalar-tensor theories. This often makes hard to understand their physical content, especially when derivative mixing is present, as it is the case for Horndeski action. In this work we develop a method that allows to write a Ricci curvature-free scalar field equation and discuss some of the advantages of such rephrasing in the study of stability issues in the presence of matter, the existence of an Einstein frame and the generalization of the disformal screening mechanism. For quartic Ho…
We study a higher order conformally coupled scalar tensor theory endowed with a covariant geometric constraint relating the scalar curvature with the Gauss-Bonnet scalar. It is a particular Horndeski theory including a canonical kinetic term but without shift or parity symmetry for the scalar. The theory also stems from a Kaluza-Klein reduction of a well defined higher dimensional metric theory. Properties of an asymptotically flat spherically symmetric black hole are analyzed, and new slowly rotating and radiating extensions are found. Through disformal transformations of the static configura…
We study new consistent scalar-tensor theories of gravity recently introduced by Langlois and Noui with potentially interesting cosmological applications. We derive the conditions for the existence of a primary constraint that prevents the propagation of an additional dangerous mode associated with higher order equations of motion. We then classify the most general, consistent scalar-tensor theories that are at most quadratic in the second derivatives of the scalar field. In addition, we investigate the possible connection between these theories and (beyond) Horndeski through conformal and dis…
We study axial (or odd-parity) perturbations about static and spherically symmetric hairy black hole (BH) solutions in shift-symmetric DHOST (Degenerate Higher-Order Scalar-Tensor) theories. We first extend to the family of DHOST theories the first-order formulation that we recently developed for Horndeski theories. Remarkably, we find that the dynamics of DHOST axial perturbations is equivalent to that of axial perturbations in general relativity (GR) evolving in a, distinct, effective metric. In the particular case of quadratic DHOST theories, this effective metric is derived from the backgr…
Scalar fields have played an important role in the development of the fundamental theories of physics as well as in other branches of physics such as gravitation and cosmology. For a long time these escaped detection until 2012 year when the Higgs boson was observed for the first time. Since then alternatives to the general theory of relativity like the Brans-Dicke theory, scalar-tensor theories of gravity and their higher derivative generalizations -- collectively known as Horndeski theories -- have acquired renewed interest. In the present review we discuss on several selected topics regardi…
In the approach of the effective field theory of modified gravity, we derive the second-order action and the equation of motion for tensor perturbations on the flat isotropic cosmological background. This analysis accommodates a wide range of gravitational theories including Horndeski theories, its generalization, and the theories with spatial derivatives higher than second order (e.g., Horava-Lifshitz gravity). We obtain the inflationary power spectrum of tensor modes by taking into account corrections induced by higher-order spatial derivatives and slow-roll corrections to the de Sitter back…
The LIGO/VIRGO collaboration has recently announced the detection of gravitational waves from a neutron star-neutron star merger (GW170817) and the simultaneous measurement of an optical counterpart (the gamma-ray burst GRB 170817A). The close arrival time of the gravitational and electromagnetic waves limits the difference in speed of photons and gravitons to be less than about one part in $10^{15}$. This has three important implications for cosmological scalar-tensor gravity theories that are often touted as dark energy candidates and alternatives to $Λ$CDM. First, for the most general scala…
Low-energy effective field theories containing a light scalar field are used extensively in cosmology, but often there is a tension between embedding such theories in a healthy UV completion and achieving a phenomenologically viable screening mechanism in the IR. Here, we identify the range of interaction couplings which allow for a smooth resummation of classical non-linearities (necessary for kinetic/Vainshtein-type screening), and compare this with the range allowed by unitarity, causality and locality in the underlying UV theory. The latter region is identified using positivity bounds on t…
AI--Assisted Exploration: DHOST Theories without Quantum Ghosts
Gran Sasso Science Institute（格兰萨索科学研究所）；INFN-Laboratori Nazionali del Gran Sasso（意大利国家核物理研究所大萨索国家实验室）；ICC, University of Barcelona（巴塞罗那大学天体物理学与宇宙学研究所）；ICREA（加泰罗尼亚高级研究学院）；Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi di Padova（帕多瓦大学伽利莱物理与天文系）；INFN, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）；INAF-Osservatorio Astronomico di Padova（意大利国家天体物理研究所帕多瓦天文台）
