Degenerate Higher-order Scalar-tensor Theories Research Articles

Merger estimates for a disformal Kerr black hole in quadratic degenerate higher-order scalar-tensor theories* *Supported by the National Key Research and Development Program of China (2020YFC2201400), and the National Natural Science Foundation of China (12275079, 12035005, 12275078)

We investigate the main features of a disformal Kerr black hole merger in quadratic degenerate higher-order scalar-tensor theories. In the ringdown stage of the black hole merger, for the prograde orbit, the real part of the quasinormal modes decreases with an increase in the disformal parameter, and the imaginary part also decreases, except in the Kerr case for a large spin parameter. However, for the retrograde orbit, the real part increases with an increase in the disformal parameter, and the imaginary part always decreases with it. For the approximate final spin, regardless of an equal spin, unequal spin, or generic spin configuration merger, the final black hole spin always increases with an increase in the disformal parameter. Our results show that the disformal parameter in the disformal Kerr solution and the MOG parameter in the Kerr-MOG case have obviously different effects on the black hole merger, which suggests the differences between these two spacetime structures.

Axial perturbations of black holes in scalar-tensor gravity: near-horizon behaviour

We consider axial (or odd-parity) perturbations of non-spinning hairy black holes (BH) in shift-symmetric DHOST (Degenerate Higher-Order Scalar-Tensor) theories, including terms quartic and cubic in second derivatives of the scalar field.We give a new formulation of the effective metric in which axial perturbations propagate as in general relativity. We then introduce a generic parametrization of the effective metric in the vicinity of the background BH horizon. Writing the dynamics of the perturbations in terms of a Schrödinger-like operator, we discuss in which cases the operator is (essentially) self-adjoint, thus leading to an unambiguous time evolution, according to the choice of parameters characterizing the near-horizon effective metric. This is in particular useful to investigate the stability of the perturbations. We finally illustrate our general analysis with two examples of BH solutions.

Early evolution of fully convective stars in scalar–tensor gravity

In this work, the early evolution of low-mass fully convective stars is studied in the context of DHOST (degenerate higher order scalar-tensor) theories of gravity. Although it is known that the hydrostatic equilibrium equation is modified for scalar-tensor gravity, the consequent modifications to the early evolution phases of a star were not explored in this framework. With this in mind, we consider three evolutionary phases—contraction to the main sequence, lithium burning and entrance to the main sequence—and investigate how each of these phases is affected by the theory’s parameter. Taking these effects into account, we are able to show, among other things, that the Hayashi tracks are shifted and the star’s age is considerably modified.

DHOST gravity in ultra-diffuse galaxies—Part II: NGC 1052-DF4 and Dragonfly 44

Ultra-Diffuse galaxies are a family of gravitational systems with quite varied properties, counting both objects highly deficient in dark matter and others which are dark matter-dominated. Such a variety of behaviors might be a problem for both the standard dark matter paradigm and for alternative theories of gravity. Here we consider two systems, NGC1052-DF4 and Dragonfly 44, which are respectively dark matter deficient and dark matter dominated, in the context of Degenerate Higher-Order Scalar Tensor theories exhibiting a partial breaking of the Vaishtein screening mechanism for which they might have an influence not only on cosmological scales but also on astrophysical ones, thus mimicking dark matter. For NGC1052-DF4 the galaxy dynamics might be successfully described only by a stellar component and, at least at the scales which are probed, the content of dark matter is quite low. We also show that our alternative gravity model is consistent with data and is statistically equivalent to a standard General Relativity dark matter scenario, and it might even replace dark matter. On the contrary, Dragonfly 44 requires dark matter both in General Relativity and in our extended theory. When the latter is considered only as a cosmological dark energy fluid, it is statistically fully reliable and equivalent to General Relativity. But when we try to use it to substitute dark matter, although we get good fits to the data, the constraints on the theoretical parameters are in sharp contrast with those derived from more stringent probes from the stellar scales.

Ostrogradsky mode in scalar–tensor theories with higher-order derivative couplings to matter

Abstract Metric transformation is a tool to find a new theory of gravity beyond general relativity. The gravity action is guaranteed to be free from the dangerous Ostrogradsky mode as long as the metric transformation is regular and invertible. Various degenerate higher-order scalar–tensor theories without extra degrees of freedom have been found through the metric transformation with a scalar field and its derivatives. In this work, we examine how a matter coupling changes the degeneracy for a theory generated from the Horndeski theory through the metric transformation with the second derivative of a scalar field, taking a minimally coupled free scalar field as the matter field. When the transformation is invertible, this theory is equivalent to the Horndeski theory with a higher-order derivative coupling to the matter scalar field. Working in this Horndeski frame and the unitary gauge, we find that the degeneracy conditions are solvable and the matter metric must have a certain structure to remove the Ostrogradsky mode.

Neutron stars in degenerate higher-order scalar-tensor theories

We study neutron star configurations in a simple shift-symmetric subfamily of degenerate higher-order scalar-tensor (DHOST) theories, whose deviations from General Relativity (GR) are characterized by a single parameter. We compute the radial profiles of neutron stars in these theories of modified gravity, using several realistic equations of state for the neutron star matter. We find neutron stars with masses and radii significantly larger than their GR counterparts. We then consider slowly-rotating solutions and determine the relation between the dimensionless moment of inertia and the compactness, relation that has the property to be almost insensitive to the equation of state.

CMB constraints on DHOST theories

We put constraints on the degenerate higher-order scalar-tensor (DHOST) theories using the Planck 2018 likelihoods. In our previous paper, we developed a Boltzmann solver incorporating the effective field theory parameterised by the six time-dependent functions, αi (i = B, K, T, M, H) and β 1, which can describe the DHOST theories. Using the Markov-Chain Monte-Carlo method with our Boltzmann solver, we find the viable parameter region of the model parameters characterising the DHOST theories and the other standard cosmological parameters. First, we consider a simple model with α = ΩDE(t)/ΩDE(t 0), α B = α T = α M = α H = 0 and β 1 = β 1,0ΩDE(t)/ΩDE(t 0) in the ΛCDM background where t 0 is the present time and obtain β 1,0 = 0.032-0.016 +0.013 (68% c.l.). Next, we focus on another theory given by ℒDHOST = X + c 3 X□ϕ/Λ3 + (M pl 2/2 + c 4 x 2/Λ6)R + 48c 4 2 X 2/ (M pl 2Λ12 + 2c 4Λ6 X 2)ϕμ ϕμρ ϕρν ϕν with X := ∂μϕ∂μϕ and two positive constant parameters, c 3 and c 4. In this model, we consistently treat the background and the perturbations, and obtain c 3 = 1.59+0.26 -0.28 and the upper bound on c 4, c 4 < 0.0088 (68% c.l.).

Black hole perturbations in higher-order scalar–tensor theories: initial value problem and dynamical stability

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.

On the effective metric of axial black hole perturbations in DHOST gravity

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 caseof quadratic DHOST theories, this effective metric is derived from the background BH metric via a disformal transformation. We illustrate our general study with three examples of BH solutions.In some so-called stealth solutions, the effective metric is Schwarzschild with a shifted horizon. We also give an example of BH solution for which the effective metric is associated with a naked singularity.

Avoidance of Strong Coupling in General Relativity Solutions with a Timelike Scalar Profile in a Class of Ghost-Free Scalar-Tensor Theories

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.

Cosmic evolution in DHOST theory with scaling solutions

We study cosmic evolution based on the fixed points in the dynamical analysis of the degenerate higher-order scalar-tensor (DHOST) theories. We consider the DHOST theory in which the propagation speed of gravitational waves is equal to the speed of light, the tensor perturbations do not decay to dark energy perturbations, and the scaling solutions exist. The scaling fixed point associated with late-time acceleration of the universe can be either stable or saddle depending on the parameters of the theory. For some ranges of the parameters, this scaling fixed point and field-dominated fixed point can be simultaneously stable. Cosmic evolution will reach either the scaling attractor or the field-dominated attractor depending on the sign of the time derivative of the scalar field in the theory during the matter domination. The density parameter of dark matter can be larger than unity before reaching the scaling attractor if the deviation from Einstein’s theory of gravity is too large. For this DHOST theory, stabilities of {phi }-matter-dominated epoch (phi MDE) and field-dominated solutions are similar to the coupled dark energy models in Einstein gravity even though gravity is described by different theories. In our consideration, the universe can only evolve from the phi MDE regime to the field-dominated regime. The ghost and gradient instabilities up to linear order in cosmological perturbations have been investigated. There is no gradient instability, while the ghost instability can be avoided for some range of the model parameters.

Effect of noncircularity on the dynamic behaviors of particles in a disformal rotating black-hole spacetime

A disformal rotating black-hole solution is a black-hole solution in quadratic degenerate higher-order scalar-tensor theories. It breaks the circular condition of spacetime different from the case of the usual Kerr spacetime. This study investigated the dynamic behaviors of the motion of timelike particles in such disformal black-hole spacetime with an extra deformation parameter. Results showed that the characteristics of the particle's motion depend on the sign of the deformation parameter. For the positive deformation parameter, the motion is regular and orderly. For the negative one, as the deformation parameter changes, the motion of the particles undergoes a series of transitions between the chaotic motion and the regular motion and falls into the horizon or escapes to spatial infinity. This means that the dynamic behavior of timelike particles in the disformal Kerr black-hole spacetime with noncircularity becomes richer than that in the usual Kerr black-hole case.

Second-order peculiar velocity field as a novel probe of scalar-tensor theories

We investigate the galaxy bispectrum induced by the nonlinear gravitational evolution as a possible probe to constrain degenerate higher-order scalar tensor (DHOST) theories. We find that the signal obtained from the leading kernel of second-order density fluctuations is partially hidden by the uncertainty in the nonlinear galaxy bias, and that the kernel of second-order velocity fields instead provides unbiased information on the modification of gravity theory. Based on this fact, we propose new phenomenological time-dependent functions, written as a combination of the coefficients of the second-order kernels, which is expected to trace the higher-order growth history. We then present approximate expressions for these variables in terms of parameters that characterize the DHOST theories. We also show that the resultant formulae provides new constraints on the parameter space of the DHOST theories.

Generalised scalar-tensor theories of gravity and pressure profiles of galaxy clusters

In the current proceedings, we summarise the results presented during the mm Universe@NIKA2 conference, taken from our main results in [1]. We test the Degenerate higher-order scalar-tensor(DHOST) theory as a generalised platform for scalar-tensor theory at galaxy cluster scales to predict in such static systems small scale modification to the gravitational potential. DHOST theory is not only a good alternative to ΛCDM for the background evolution but also predicts small-scale modification to the gravitational potential in static systems such as galaxy clusters. With a sample of 12 clusters with accurate Xray Intra Cluster Medium (ICM) data (X-COP project) and Sunyaev-Zel’dovich (SZ) ICM pressure (Planck satellite), we place preliminary constraints on the DHOST parameter (Ξ1) defining the deviation from GR. Moreover, we also collect a few supplementary analyses we have performed during the course: i) Gaussian process reconstruction without parametric assumptions, ii) PSZ-only data analysis not aided by the X-ray data. Finally, we present possible extensions to the current work which may benefit from future high sensitivity and spatial resolution observations.

Black hole perturbations in modified gravity

We study the linear perturbations about nonrotating black holes in the context of degenerate higher-order scalar-tensor (DHOST) theories, using a systematic approach that extracts the asymptotic behaviour of perturbations (at spatial infinity and near the horizon) directly from the first-order radial differential system governing these perturbations. For axial (odd-parity) modes, this provides an alternative to the traditional approach based on a second-order Schr\"odinger-like equation with an effective potential, which we also discuss for completeness. For polar (even-parity) modes, which contain an additional degree of freedom in DHOST theories, and are thus more complex, we use a direct treatment of the four-dimensional first-order differential system (without resorting to a second order reformulation). We illustrate our study with two specific types of black hole solutions: 'stealth' Schwarzschild black holes, with a non trivial scalar hair, as well as a class of non-stealth black holes whose metric is distinct from Schwarzschild. The knowledge of the asymptotic behaviours of the perturbations enables us to compute numerically quasi-normal modes, as we show explicitly for the non-stealth solutions. Finally, the asymptotic form of the modes also signals some pathologies in the stealth and non-stealth solutions considered here.

Testing gravity of a disformal Kerr black hole in quadratic degenerate higher-order scalar-tensor theories by quasi-periodic oscillations

By using the relativistic precession model, we have studied frequencies of quasi-periodic oscillations in the spacetime of a disformal Kerr black hole. This black hole owns an extra disformal parameter and belongs to a class of non-stealth solutions in quadratic degenerate higher-order scalar-tensor (DHOST) theories. Our result shows that only the periastron precession frequency is related to the disformal parameter, while the azimuthal frequency and the nodal precession frequency are identical with those in the usual Kerr black hole in general relativity. Combing with the observation data of GRO J1655-40, we fit parameters of the disformal Kerr black hole, and find that the disformal parameter α is almost negative in the range of 1 σ, which implies the negative disformal parameter α is favored by the observation data of GRO J1655-40.

Quadratic degenerate higher-order scalar-tensor theories revisited

We present a novel and remarkably simple formulation of degenerate higher-order scalar-tensor (DHOST) theories whose Lagrangian is quadratic in second derivatives of some scalar field. Using disformal transformations of the metric, we identify a special "frame" (or metric) for which the Lagrangian of quadratic DHOST theories reduces to the usual Einstein-Hilbert term plus a few terms that depend on simple geometric quantities characterizing the uniform scalar field hypersurfaces. In particular, for quadratic DHOST theories in the physically interesting class Ia, the Lagrangian simply consists of the Einstein-Hilbert term plus a term proportional to the three-dimensional scalar curvature of the uniform scalar field hypersurfaces. The classification of all quadratic DHOST theories becomes particularly transparent in this geometric reformulation, which also applies to scalar-tensor theories that are degenerate only in the unitary gauge.

Jointly fitting weak lensing, x-ray, and Sunyaev-Zel’dovich data to constrain scalar-tensor theories with clusters of galaxies

Degenerate higher-order scalar-tensor (DHOST) theories are considered the most general class of scalar-tensor theories so that any constraints on them apply to the full set of scalar-tensor models. DHOST theories modify the laws of gravity even at galaxy clusters scale hence affecting the weak lensing (WL), X-ray and Sunyaev-Zel'dovich observables. We derive the theoretical expression for the lensing convergence $\kappa$, and the pressure profile $P$, of clusters in the framework of DHOST theories, and quantify how much they deviate from their General Relativity (GR) counterparts. We argue that combined measurements of $\kappa$, $P$, and of the electron number density, $n_e$, can constrain both the cluster and DHOST theory parameters. We carry on a Fisher matrix forecasts analysis to investigate whether this is indeed the case considering different scenarios for the spatial resolution and errors on the measured quantities.

Strong gravitational lensing by DHOST black holes

The gravitational deflection of light in the strong field limit is an important test for alternative theories of gravity. However, solutions for the metric that allow for analytic computations are not always available. We implement a hybrid analytic-numerical approximation to determine the deflection angle in static, spherically symmetric spacetimes. We apply this to a set of numerical black hole solutions within the class of modified gravity theories known as degenerate higher order scalar–tensor theories (DHOST). Comparing our results to a more time consuming full numerical integration, we find that we can accurately describe the deflection angle for light rays passing at arbitrary distances from the photon sphere with a combination of two analytic-numerical approximations. Furthermore, we find a range of parameters where our DHOST black holes predict strong lensing effects whose size is comparable with the uncertainty in the properties of the supermassive black hole in M87 reported by the event horizon telescope, showing that strong lensing is a viable alternative to put constraints on these models of modified gravity.

Stealth dark energy in scordatura DHOST theory

A stealth de Sitter solution in scalar-tensor theories has an exact de Sitter background metric and a nontrivial scalar field profile. Recently, in the context of Degenerate Higher-Order Scalar-Tensor (DHOST) theories it was shown that stealth de Sitter solutions suffer from either infinite strong coupling or gradient instability for scalar field perturbations. The sound speed squared is either vanishing or negative. In the first case, the strong coupling scale is zero and thus lower than the energy scale of any physical phenomena. From the viewpoint of effective field theory, this issue is naturally resolved by introducing a controlled detuning of the degeneracy condition dubbed scordatura, recovering a version of ghost condensation. In this paper we construct a viable dark energy model in the scordatura DHOST theory based on a stealth cosmological solution, in which the metric is the same as in the standard ΛCDM model and the scalar field profile is linearly time-dependent. We show that the scordatura mechanism resolves the strong coupling and gradient instability. Further, we find that the scordatura is also necessary to make the quasi-static limit well-defined, which implies that the subhorizon observables are inevitably affected by the scordatura. We derive the effective gravitational coupling and the correction to the friction term for the subhorizon evolution of the linear dark matter energy density contrast as well as the Weyl potential and the gravitational slip parameter. In the absence of the scordatura, the quasi-static approximation would break down at all scales around stealth cosmological solutions even if the issue of the infinite strong coupling is unjustly disregarded. Therefore previous estimations of the subhorizon evolution of matter density contrast in modified gravity in the literature need to be revisited by taking into account the scordatura effect.