Black hole solutions in general scalar-tensor theories are known to permit hair, i.e. non-trivial scalar profiles and/or metric solutions different from the ones of General Relativity (GR). Imposing that some such solutions—e.g. Schwarzschild or de Sitter solutions motivated in the context of black hole physics or cosmology—should exist, the space of scalar-tensor theories is strongly restricted. Here we investigate precisely what these restrictions are within general quadratic/cubic higher-order scalar-tensor theories for stealth solutions, whose metric is given by that in GR, supporting time-dependent scalar hair with a constant kinetic term. We derive, in a fully covariant approach, the conditions under which the Euler-Lagrange equations admit all (or a specific set of) exact GR solutions, as the first step toward our understanding of a wider class of theories that admit approximately stealth solutions. Focusing on static and spherically symmetric black hole spacetimes, we study the dynamics of linear odd-parity perturbations and discuss possible deviations from GR. Importantly, we find that requiring the existence of all stealth solutions prevents any deviations from GR in the odd-parity sector. In less restrictive scenarios, in particular for theories only requiring the existence of Schwarzschild(-de Sitter) black holes, we identify allowed deviations from GR, derive the stability conditions for the odd modes, and investigate the generic deviation of a non-trivial speed of gravitational waves. All calculations performed in this paper are reproducible via companion Mathematica notebooks.

We study linear even-parity perturbations of static and spherically symmetric black holes with a timelike scalar profile by use of the effective field theory (EFT) approach. For illustrative purposes, we consider a simple subclass of the EFT that accommodates ghost condensate, namely the k-essence model along with the so-called scordatura term, and focus on the spherical (monopole) perturbations about an approximately stealth Schwarzschild solution. The scordatura effect is introduced to avoid the strong coupling problem that typically happens in the scalar sector around stealth solutions with a timelike scalar profile. We argue that the scalar perturbation is decoupled from the metric perturbations at the leading order in the scordatura effect under a particular gauge choice. We stress that this is an important step in understanding the dynamics of even-parity perturbations, paving the way towards deriving a set of master equations—the generalized Zerilli and the scalar-field equations—for generic multipoles.

Typically, constraints on parameters of the effective field theory (EFT) of dark energy have been obtained in the Jordan frame, where matter fields are minimally coupled to gravity. To connect these constraints with those of the EFT of black hole perturbations with a timelike scalar profile, it is necessary to perform a frame transformation on the EFT in general. In this paper, we study the conformal/disformal transformation of EFT parameters on an arbitrary background. Furthermore, we explore the effect of an EFT operator $M_6(r) \bar{\sigma}^{\mu}_{\nu} \delta K^{\nu}_{\alpha} \delta K^{\alpha}_{\mu}$, which is elusive to the LIGO/Virgo bound on gravitational-wave speed, on the dynamics of odd-parity black hole perturbations. Intriguingly, a deviation from luminal propagation shows up only in the vicinity of the black hole, and the speeds of perturbations in the radial and angular directions are different in general due to the traceless part $\bar{\sigma}^\mu_\nu$ of the background extrinsic curvature. This study establishes an important link between cosmological constraints and those obtained in the black hole regime.

Recently, it has been proposed that the black hole greybody factors can be important to model ringdown spectral amplitudes. We study the stability of greybody factors against a small-bump correction in the perturbation equation. We find (I) that the greybody factor is stable in the frequency region relevant to ringdown and (II) that it is destabilized at higher frequencies, especially for a sharper bump correction. This behavior is similar to the case of higher overtones, which is also very sensitive to a small correction. We clarify this (in)stability with the WKB analysis. As the greybody factor is stable at the frequency region relevant to the main part of ringdown, we conclude that the greybody factor is suitable to model ringdown amplitude. In order to investigate a bump correction in a self-consistent manner, we consider the small-bump correction that can be realized in the general framework of effective field theory of black hole perturbations.

We study static tidal Love numbers (TLNs) of a static and spherically symmetric black hole for odd-parity metric perturbations. We describe black hole perturbations using the effective field theory (EFT), formulated on an arbitrary background with a timelike scalar profile in the context of scalar-tensor theories. In particular, we obtain a static solution for the generalized Regge-Wheeler equation order by order in a modified-gravity parameter and extract the TLNs uniquely by analytic continuation of the multipole index $\ell$ to non-integer values. For a stealth Schwarzschild black hole, the TLNs are vanishing as in the case of Schwarzschild solution in general relativity. We also study the case of Hayward black hole as an example of non-stealth background, where we find that the TLNs are non-zero (or there is a logarithmic running). This result suggests that our EFT allows for non-vanishing TLNs and can in principle leave a detectable imprint on gravitational waves from inspiralling binary systems, which opens a new window for testing gravity in the strong-field regime.

We study the cosmic microwave background (CMB) radiation in the unified description of the effective field theory (EFT) of dark energy that accommodates both scalar-tensor and vector-tensor theories. The boundaries of different classes of theories are universally parameterised by a new EFT parameter $\alpha_V$ characterising the vectorial nature of dark energy and a set of consistency relations associated with the global/local shift symmetry. After implementing the equations of motion in a Boltzmann code, as a demonstration, we compute the CMB power spectrum based on the $w$CDM background with the EFT parameterisation of perturbations and a concrete Horndeski/generalised Proca theory. We show that the vectorial nature generically prevents modifications of gravity in the CMB spectrum. On the other hand, while the shift symmetry is less significant in the perturbation equations unless the background is close to the $\Lambda$CDM, it requires that the effective equation of state of dark energy is in the phantom region $w_{\rm DE}<-1$. The latter is particularly interesting in light of the latest result of the DESI+CMB combination as the observational verification of $w_{\rm DE}>-1$ can rule out shift-symmetric theories including vector-tensor theories in one shot.

The Standard Model of particle physics predicts the speed of light to be a universal speed of propagation of massless carriers. However, other possibilities exist—including Lorentz-violating theories—where different fundamental fields travel at different speeds. Black holes are interesting probes of such physics, as distinct fields would probe different horizons. Here, we build an exact spacetime for two interacting scalar fields which have different propagation speeds. One of these fields is able to probe the black hole interior of the other, giving rise to energy extraction from the black hole and a characteristic late-time relaxation. Our results provide further stimulus to the search for extra degrees of freedom, black hole instability, and extra ringdown modes in gravitational-wave events.

It has recently been pointed out that one can construct invertible conformal transformations with a parity-violating conformal factor, which can be employed to generate a novel class of parity-violating ghost-free metric theories from general relativity. We obtain exact solutions for rotating black holes in such theories by performing the conformal transformation on the Kerr solution in general relativity, which we dub conformal Kerr solutions. We explore the geodesic motion of a test particle in the conformal Kerr spacetime. While null geodesics remain the same as those in the Kerr spacetime, timelike geodesics exhibit interesting differences due to an effective external force caused by the parity-violating conformal factor.

We formulate the effective field theory (EFT) of vector-tensor gravity for perturbations around an arbitrary background with a timelike vector profile, which can be applied to study black hole perturbations. The vector profile spontaneously breaks both the time diffeomorphism and the $U(1)$ symmetry, leaving their combination and the spatial diffeomorphism as the residual symmetries in the unitary gauge. We derive two sets of consistency relations which guarantee the residual symmetries of the EFT. Also, we provide the dictionary between our EFT coefficients and those of generalized Proca (GP) theories, which enables us to identify a simple subclass of the EFT that includes the GP theories as a special case. For this subclass, we consider the stealth Schwarzschild(-de Sitter) background solution with a constant temporal component of the vector field and study the decoupling limit of the longitudinal mode of the vector field, explicitly showing that the strong coupling problem arises due to vanishing sound speeds. This is in sharp contrast to the case of gauged ghost condensate, in which perturbations are weakly coupled thanks to certain higher-derivative terms, i.e., the scordatura terms. This implies that, in order to consistently describe this type of stealth solutions within the EFT, the scordatura terms must necessarily be taken into account in addition to those already included in the simple subclass.

We study odd-parity perturbations about static and spherically symmetric black hole solutions with a linearly time-dependent scalar field in higher-order scalar-tensor theories. In particular, we consider stealth Schwarzschild and stealth Schwarzschild-de Sitter solutions, where the deviation from the general relativity case is controlled by a single parameter. We find that complex frequencies of quasinormal modes (QNMs) are given by a simple scaling of those in general relativity. We also show that there is a degeneracy between the parameter characterizing the modification from general relativity and the black hole mass. We then consider a physically sensible initial value problem by taking into account the fact that the effective metric for the odd-parity perturbations is in general different from the background metric. We confirm that damped oscillations appearing at late times are indeed dominated by the QNMs. Our analysis includes the case where the perturbations are superluminal, and we demonstrate in this case that the perturbations can escape from the region inside the horizon for the background metric.

Invertible disformal transformations serve as a useful tool to explore ghost-free scalar-tensor theories. In this paper, we construct a generalization of invertible disformal transformations that involves arbitrary higher-order covariant derivatives of the scalar field. As a result, we obtain a more general class of ghost-free scalar-tensor theories than ever. Notably, our generalization is such that matter fields can be consistently coupled to these theories without introducing an unwanted extra degree of freedom in the unitary gauge.

The Effective Field Theory (EFT) of perturbations on an arbitrary background geometry with a timelike scalar profile was recently constructed in the context of scalar-tensor theories. In this paper, we use this EFT to study quasinormal frequencies of odd-parity perturbations on a static and spherically symmetric black hole background. Keeping a set of operators that can accommodate shift-symmetric quadratic higher-order scalar-tensor theories, we demonstrate the computation for two examples of hairy black holes, of which one is the stealth Schwarzschild solution and the other is the Hayward metric accompanied by a non-trivial scalar field. We emphasize that this is the first phenomenological application of the EFT, opening a new possibility to test general relativity and modified gravity theories in the strong gravity regime.

Generalized disformal transformations enable us to construct the generalized disformal Horndeski theories, which form the most general class of ghost-free scalar-tensor theories to this date. We extend the effective field theory (EFT) of cosmological perturbations to incorporate these generalized disformal Horndeski theories. The main difference from the conventional EFT is that our extended EFT involves operators with higher spatial derivatives of the lapse function. Our EFT also accommodates the generalized disformal transformation of U-DHOST theories.

Recently, a generalization of invertible disformal transformations containing higher-order derivatives of a scalar field has been proposed in the context of scalar-tensor theories of gravity. By applying this generalized disformal transformation to the Horndeski theory, one can obtain the so-called generalized disformal Horndeski theories which are more general healthy scalar-tensor theories than ever. However, it is unclear whether or not the generalized disformal Horndeski theories can be coupled consistently to matter fields because introducing a matter field could break the degeneracy conditions of higher-order scalar-tensor theories and hence yield the unwanted Ostrogradsky ghost. We investigate this issue and explore the conditions under which a minimal coupling to a matter field is consistent in the generalized disformal Horndeski theories without relying on any particular gauge such as the unitary gauge. We find that all the higher derivative terms in the generalized disformal transformation are prohibited to avoid the appearance of the Ostrogradsky ghost, leading to the conclusion that only the theories that are related to the Horndeski theory through a conventional disformal transformation remain ghost-free in the presence of minimally coupled matter fields.

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 theories. We develop the transformation law for the tetrad associated with the generalized disformal transformation to see how it affects the fermionic matter coupling. We find that the consistency of the fermionic matter coupling requires an additional condition on top of the one required for the bosonic case. As a result, we identify a subclass of the generalized disformal Horndeski class which allows for consistent coupling of ordinary matter fields, including the standard model particles.

We investigate a generic quadratic higher-order scalar-tensor theory with a scordatura term, which is expected to provide a consistent perturbative description of stealth solutions with a timelike scalar field profile. In the DHOST subclass, exactly stealth solutions are known to yield perturbations infinitely strongly coupled and thus cannot be trusted. Beyond DHOST theories with the scordatura term, such as in ghost condensation and U-DHOST, we show that stealth configurations cannot be realized as exact solutions but those theories instead admit approximately stealth solutions where the deviation from the exactly stealth configuration is controlled by the mass scale $M$ of derivative expansion. The approximately stealth solution is time-dependent, which can be interpreted as the black hole mass growth due to the accretion of the scalar field. From observed astrophysical black holes, we put an upper bound on $M$ as $\hat{c}_{\rm D1}^{1/2} M\lesssim 2\times 10^{11}$ GeV, where $\hat{c}_{\rm D1}$ is a dimensionless parameter of order unity that characterizes the scordatura term. As far as $M$ is sufficiently below the upper bound, the accretion is slow and the approximately stealth solutions can be considered as stealth at astrophysical scales for all practical purposes while perturbations are weakly coupled all the way up to the cutoff $M$ and the apparent ghost is as heavy as or heavier than $M$.

Since the discovery of the accelerated expansion of the present Universe, significant theoretical developments have been made in the area of modified gravity. In the meantime, cosmological observations have been providing more high-quality data, allowing us to explore gravity on cosmological scales. To bridge the recent theoretical developments and observations, we present an overview of a variety of modified theories of gravity and the cosmological observables in the cosmic microwave background and large-scale structure, supplemented with a summary of predictions for cosmological observables derived from cosmological perturbations and sophisticated numerical studies. We specifically consider scalar-tensor theories in the Horndeski and DHOST family, massive gravity/bigravity, vector-tensor theories, metric-affine gravity, and cuscuton/minimally-modified gravity, and discuss the current status of those theories with emphasis on their physical motivations, validity, appealing features, the level of maturity, and calculability. We conclude that the Horndeski theory is one of the most well-developed theories of modified gravity, although several remaining issues are left for future observations. The paper aims to help to develop strategies for testing gravity with ongoing and forthcoming cosmological observations.

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 instabilities for tensor and scalar perturbations. We also investigate the conditions under which a matter field can be consistently coupled to these theories without introducing unwanted extra degrees of freedom.

Recently, the Effective Field Theory (EFT) of perturbations on an arbitrary background metric with a timelike scalar profile was formulated in the context of scalar-tensor theories. Here, we generalize the dictionary between the coefficients in the EFT action and those in covariant theories to accommodate shift- and reflection-symmetric quadratic higher-order scalar-tensor theories, including DHOST as well as U-DHOST. We then use the EFT action to study the dynamics of odd-parity perturbations on a static and spherically symmetric black hole background with a timelike scalar profile. As a result, we find that the sound speed in the radial direction has a non-trivial dependence on the multipole index in general, which also characterizes the deviation from the sound speed in angular directions. Finally, we obtain the generalized Regge-Wheeler equation that can be used, e.g., to determine the spectrum of quasinormal modes and tidal Love numbers.

In full Horndeski theories, we show that the static and spherically symmetric black hole (BH) solutions with a static scalar field $\phi$ whose kinetic term $X$ is nonvanishing on the BH horizon are generically prone to ghost/Laplacian instabilities. We then search for asymptotically flat hairy BH solutions with a vanishing $X$ on the horizon free from ghost/Laplacian instabilities. We show that models with regular coupling functions of $\phi$ and $X$ result in no-hair Schwarzschild BHs in general. On the other hand, the presence of a coupling between the scalar field and the Gauss-Bonnet (GB) term $R_{\rm GB}^2$, even with the coexistence of other regular coupling functions, leads to the realization of asymptotically flat hairy BH solutions without ghost/Laplacian instabilities. Finally, we find that hairy BH solutions in power-law $F(R_{\rm GB}^2)$ gravity are plagued by ghost instabilities. These results imply that the GB coupling of the form $\xi(\phi)R_{\rm GB}^2$ plays a prominent role for the existence of asymptotically flat hairy BH solutions free from ghost/Laplacian instabilities.

We investigate the relativistic effective field theory (EFT) describing a non-dissipative gravitating continuum. In addition to ordinary continua, namely solids and fluids, we find an extraordinary more symmetric continuum, aether. In particular, the symmetry of the aether concludes that a homogeneous and isotropic state behaves like a cosmological constant. We formulate the EFT in the unitary/comoving gauge in which the dynamical degrees of freedom of the continuum (phonons) are eaten by the spacetime metric. This gauge choice, which is interpreted as the Lagrangian description in hydrodynamics, offers a neat geometrical understanding of continua. We examine a thread-based spacetime decomposition with respect to the four-velocity of the continuum which is different from the foliation-based Arnowitt-Deser-Misner one. Our thread-based decomposition respects the symmetries of the continua and, therefore, makes it possible to systematically find invariant building blocks of the EFT for each continuum even at higher orders in the derivative expansion. We also discuss the linear dynamics of the system and show that both gravitons and phonons acquire "masses" in a gravitating background.

We propose a physically sensible formulation of initial value problem for black hole perturbations in higher-order scalar-tensor theories. As a first application, we study monopole perturbations around stealth Schwarzschild solutions in a shift- and reflection-symmetric subclass of degenerate higher-order scalar-tensor (DHOST) theories. In particular, we investigate the time evolution of the monopole perturbations by solving a two-dimensional wave equation and analyze the Vishveshwara's classical scattering experiment, i.e., the time evolution of a Gaussian wave packet. As a result, we confirm that stealth Schwarzschild solutions in the DHOST theory are dynamically stable against the monopole perturbations with the wavelength comparable or shorter than the size of the black hole horizon. We also find that the damped oscillations at the late time do not show up unlike the ringdown phase in the standard case of general relativity. Moreover, we investigate the characteristic curves of the monopole perturbations as well as a static spherically symmetric solution with monopole hair.

Modified gravity theories can accommodate exact solutions, for which the metric has the same form as the one in general relativity, i.e., stealth solutions. One problem with these stealth solutions is that perturbations around them exhibit strong coupling when the solutions are realized in degenerate higher-order scalar-tensor theories. We show that the strong coupling problem can be circumvented in the framework of the so-called U-DHOST theories, in which the degeneracy is partially broken in such a way that higher-derivative terms are degenerate only in the unitary gauge. In this sense, the scordatura effect is built-in in U-DHOST theories in general. There is an apparent Ostrogradsky mode in U-DHOST theories, but it does not propagate as it satisfies a three-dimensional elliptic differential equation on a spacelike hypersurface. We also clarify how this nonpropagating mode, i.e., the "shadowy" mode shows up at the nonlinear level.

We study linear perturbations about static and spherically symmetric black holes with a time-independent background scalar field in shift-symmetric Horndeski theories, whose Lagrangian is characterized by coupling functions depending only on the kinetic term of the scalar field $X$. We clarify conditions for the absence of ghosts and Laplacian instabilities along the radial and angular directions in both odd- and even-parity perturbations. For reflection-symmetric theories described by a k-essence Lagrangian and a nonminimal derivative coupling with the Ricci scalar, we show that black holes endowed with nontrivial scalar hair are unstable around the horizon in general. This includes non-asymptotically-flat black holes known to exist when the nonminimal derivative coupling to the Ricci scalar is a linear function of $X$. We also investigate several black hole solutions in non-reflection-symmetric theories. For cubic Galileons with the Einstein-Hilbert term, there exists a non-asymptotically-flat hairy black hole with no ghosts/Laplacian instabilities. Also, for the scalar field linearly coupled to the Gauss-Bonnet term, asymptotically-flat black hole solutions constructed perturbatively with respect to a small coupling are free of ghosts/Laplacian instabilities.

We consider a higher-derivative generalization of disformal transformations and clarify the conditions under which they form a group with respect to the matrix product and the functional composition. These conditions allow us to systematically construct the inverse transformation in a fully covariant manner. Applying the invertible generalized disformal transformation to known ghost-free scalar-tensor theories, we obtain a novel class of ghost-free scalar-tensor theories, whose action contains the third- or higher-order derivatives of the scalar field as well as nontrivial higher-order derivative couplings to the curvature tensor.

We investigate a systematic formulation of vector-tensor theories based on the effective field theory (EFT) approach. The input of our EFT is that the spacetime symmetry is spontaneously broken by the existence of a preferred timelike direction in accordance with the cosmological principle. After clarifying the difference of the symmetry breaking pattern from the conventional EFT of inflation/dark energy, we find an EFT description of vector-tensor theories around the cosmological background. This approach not only serves as a unified description of vector-tensor theories but also highlights universal differences between the scalar-tensor theories and the vector-tensor theories. The theories having different symmetry breaking patterns are distinguished by a phenomenological function and consistency relations between the EFT coefficients. We study the linear cosmological perturbations within our EFT framework and discuss the characteristic properties of the vector-tensor theories in the context of dark energy. In particular, we compute the effective gravitational coupling and the slip parameter for the matter density contrast in terms of the EFT coefficients.

We study U-DHOST theories, i.e., higher-order scalar-tensor theories which are degenerate only in the unitary gauge and yield an apparently unstable extra mode in a generic coordinate system. We show that the extra mode satisfies a three-dimensional elliptic differential equation on a spacelike hypersurface, and hence it does not propagate. We clarify how to treat this "shadowy" mode at both the linear and the nonlinear levels.

We study inflationary universes with an SU(3) gauge field coupled to an inflaton through a gauge kinetic function. Although the SU(3) gauge field grows at the initial stage of inflation due to the interaction with the inflaton, nonlinear self-couplings in the kinetic term of the gauge field become significant and cause nontrivial dynamics after sufficient growth. We investigate the evolution of the SU(3) gauge field numerically and reveal attractor solutions in the Bianchi type I spacetime. In general cases where all the components of the SU(3) gauge field have the same magnitude initially, they all tend to decay eventually because of the nonlinear self-couplings. Therefore, the cosmic no-hair conjecture generically holds in a mathematical sense. Practically, however, the anisotropy can be generated transiently in the early universe. Moreover, we find particular cases for which several components of the SU(3) gauge field survive against the nonlinear self-couplings. It occurs due to flat directions in the potential of a gauge field for Lie groups whose rank is higher than one. Thus, an SU(2) gauge field has a specialty among general non-Abelian gauge fields.

We study linear perturbations about static and spherically symmetric black hole solutions with stealth scalar hair in degenerate higher-order scalar-tensor (DHOST) theories. We clarify master variables and derive the quadratic Lagrangian for both odd- and even-parity perturbations. It is shown that the even modes are in general plagued by gradient instabilities, or otherwise the perturbations would be strongly coupled. Several possible ways out are also discussed.

We propose a class of theories that can limit scalars constructed from the extrinsic curvature. Applied to cosmology, this framework allows us to control not only the Hubble parameter but also anisotropies without the problem of Ostrogradsky ghost, which is in sharp contrast to the case of limiting spacetime curvature scalars. Our theory can be viewed as a generalization of mimetic and cuscuton theories (thus clarifying their relation), which are known to possess a structure that limits only the Hubble parameter on homogeneous and isotropic backgrounds. As an application of our framework, we construct a model where both anisotropies and the Hubble parameter are kept finite at any stage in the evolution of the universe in the diagonal Bianchi type I setup. The universe starts from a constant-anisotropy phase and recovers Einstein gravity at low energies. We also show that the cosmological solution is stable against a wide class of perturbation wavenumbers, though instabilities may remain for arbitrary initial conditions.

We explore General Relativity solutions with stealth scalar hair in general quadratic higher-order scalar-tensor theories. Adopting the assumption that the scalar field has a constant kinetic term, we derive in a fully covariant manner a set of conditions under which the Euler-Lagrange equations allow General Relativity solutions as exact solutions in the presence of a general matter component minimally coupled to gravity. The scalar field possesses a nontrivial profile, which can be obtained by integrating the condition of constant kinetic term for each metric solution. We demonstrate the construction of the scalar field profile for several cases including the Kerr-Newman-de Sitter spacetime as a general black hole solution characterized by mass, charge, and angular momentum in the presence of a cosmological constant. We also show that asymptotically anti-de Sitter spacetimes cannot support nontrivial scalar hair.

Late-time cosmology in the extended cuscuton theory is studied, in which gravity is modified while one still has no extra dynamical degrees of freedom other than two tensor modes. We present a simple example admitting analytic solutions for the cosmological background evolution that mimics $\Lambda$CDM cosmology. We argue that the extended cuscuton as dark energy can be constrained, like usual scalar-tensor theories, by the growth history of matter density perturbations and the time variation of Newton's constant.

We find that the recently-proposed ghost-free interaction of a 2-form gauge field in four dimensions, which contains derivative couplings in a nonperturbative manner, can be regarded as a resummation of ghostly interaction terms. We investigate the higher derivative structure of this model in a minisuperspace description and demonstrate that the higher derivative terms can be removed by taking appropriate combinations of the Euler-Lagrange equations, while a truncation at a finite order spoils this structure. We also show that this nature is peculiar to four dimensions.

We develop an effective-field-theory (EFT) framework for inflation with various symmetry breaking pattern. As a prototype, we formulate anisotropic inflation from the perspective of EFT and construct an effective action of the Nambu-Goldstone bosons for the broken time translation and rotation symmetries. We also calculate the statistical anisotropy in the scalar two-point correlation function for concise examples of the effective action.

We study static spherically symmetric black hole solutions with a linearly time-dependent scalar field and discuss their linear stability in the shift- and reflection-symmetric subclass of quadratic degenerate higher-order scalar-tensor (DHOST) theories. We present the explicit forms of the reduced system of background field equations for a generic theory within this subclass. Using the reduced equations of motion, we show that in several cases the solution is forced to be of the Schwarzschild or Schwarzschild–(anti-)de Sitter form. We consider odd-parity perturbations around general static spherically symmetric black hole solutions and derive the concise criteria for the black holes to be stable. Our analysis also covers the case with a static or constant profile of the scalar field.

We show explicitly that the nonminimal coupling between the scalar field and the Ricci scalar in 2D dilaton gravity can be recast in the form of kinetic gravity braiding (KGB). This is as it should be, because KGB is the 2D version of the Horndeski theory. We also determine all the static solutions with a linearly time-dependent scalar configuration in the shift-symmetric KGB theories in 2D.

Among single-field scalar-tensor theories, there is a special class called "cuscuton", which is represented as some limiting case of k-essence in general relativity. This theory has a remarkable feature that the number of propagating degrees of freedom is only two in the unitary gauge in contrast to ordinary scalar-tensor theories with three degrees of freedom. We specify a general class of theories with the same property as the cuscuton in the context of the beyond Horndeski theory, which we dub as the extended cuscuton. We also study cosmological perturbations in the presence of matter in these extended cuscuton theories.

We propose a novel class of degenerate higher-order scalar-tensor theories as an extension of mimetic gravity. By performing a noninvertible conformal transformation on "seed" scalar-tensor theories which may be nondegenerate, we can generate a large class of theories with at most three physical degrees of freedom. We identify a general seed theory for which this is possible. Cosmological perturbations in these extended mimetic theories are also studied. It is shown that either of tensor or scalar perturbations is plagued with gradient instabilities, except for a special case where the scalar perturbations are presumably strongly coupled, or otherwise there appear ghost instabilities.

In shift-symmetric Horndeski theories, a static and spherically symmetric black hole can support linearly time-dependent scalar hair. However, it was shown that such a solution generically suffers from ghost or gradient instability in the vicinity of the horizon. In the present paper, we explore the possibility to avoid the instability, and present a new example of theory and its black hole solution with a linearly time-dependent scalar configuration. We also discuss the stability of solutions with static scalar hair for a special case where nonminimal derivative coupling to the Einstein tensor appears.

An invertible field transformation is such that the old field variables correspond one-to-one to the new variables. As such, one may think that two systems that are related by an invertible transformation are physically equivalent. However, if the transformation depends on field derivatives, the equivalence between the two systems is nontrivial due to the appearance of higher derivative terms in the equations of motion. To address this problem, we prove the following theorem on the relation between an invertible transformation and Euler-Lagrange equations: If the field transformation is invertible, then any solution of the original set of Euler-Lagrange equations is mapped to a solution of the new set of Euler-Lagrange equations, and vice versa. We also present applications of the theorem to scalar-tensor theories.

We analyze the mode stability of odd-parity perturbations of black holes with linearly time-dependent scalar hair in shift-symmetric Horndeski theories. We show that a large class of black hole solutions in these theories suffer from ghost or gradient instability, while there are some classes of solutions that are stable under linear odd-parity perturbations in the context of mode analysis.

Regardless of the long history of gauge theories, it is not well recognized under which condition gauge fixing at the action level is legitimate. We address this issue from the Lagrangian point of view, and prove the following theorem on the relation between gauge fixing and Euler-Lagrange equations: In any gauge theory, if a gauge fixing is complete, i.e., the gauge functions are determined uniquely by the gauge conditions, the Euler-Lagrange equations derived from the gauge-fixed action are equivalent to those derived from the original action supplemented with the gauge conditions. Otherwise, it is not appropriate to impose the gauge conditions before deriving Euler-Lagrange equations as it may in general lead to inconsistent results. The criterion to check whether a gauge fixing is complete or not is further investigated. We also provide applications of the theorem to scalar-tensor theories and make comments on recent relevant papers on theories of modified gravity, in which there are confusions on gauge fixing and counting physical degrees of freedom.

We analyze spherically symmetric black hole solutions with time-dependent scalar hair in a class of Lovelock-Galileon theories, which are the scalar-tensor theories with second-order field equations in arbitrary dimensions. We first show that known black hole solutions in five dimensions are always plagued by the ghost/gradient instability in the vicinity of the horizon. We then generalize such black hole solutions to higher dimensions and show that the same instability found in five dimensions appears universally in any number of dimensions.

$f(R)$ gravity is one of the simplest generalizations of general relativity, which may explain the accelerated cosmic expansion without introducing a cosmological constant. Transformed into the Einstein frame, a new scalar degree of freedom appears and it couples with matter fields. In order for $f(R)$ theories to pass the local tests of general relativity, it has been known that the chameleon mechanism with a so-called thin-shell solution must operate. If the thin-shell constraint is applied to a cosmological situation, it has been claimed that the equation-of-state parameter of dark energy $w$ must be extremely close to $-1$. We argue this is due to the incorrect use of the Poisson equation, which is valid only in the static case. By solving the correct Klein-Gordon equation perturbatively, we show that a thin-shell solution exists even if $w$ deviates appreciably from $-1$.