Higher-order Scalar-tensor Theories Research Articles

Nonlinear gravitational waves in Horndeski gravity: scalar pulse and memories

We present and analyze a new non-perturbative radiative solution of Horndeski gravity. This exact solution is constructed by a disformal mapping of a seed solution of the shift-symmetric Einstein-Scalar system belonging to the Robinson-Trautman geometry describing the gravitational radiation emitted by a time-dependent scalar monopole. After analyzing in detail the properties of the seed, we show that while the general relativity solution allows for shear-free parallel transported null frames, the disformed solution can only admit parallel transported null frames with a non-vanishing shear. This result shows that, at the nonlinear level, the scalar-tensor mixing descending from the higher-order terms in Horndeski dynamics can generate shear out of a pure scalar monopole. We further confirm this analysis by identifying the spin-0 and spin-2 polarizations in the disformed solution using the Penrose limit of our radiative solution. Finally, we compute the geodesic motion and the memory effects experienced by two null test particles with vanishing initial relative velocity after the passage of the pulse. This exact radiative solution offers a simple framework to witness nonlinear consequences of the scalar-tensor mixing in higher-order scalar-tensor theories.

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.

Quasinormal modes from EFT of black hole perturbations with timelike scalar profile

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.

New constraints on cosmological modified gravity theories from anisotropic three-point correlation functions of BOSS DR12 galaxies

ABSTRACT We report a new test of modified gravity theories using the large-scale structure of the Universe. This paper is the first attempt to (1) apply a joint analysis of the anisotropic components of galaxy two- and three-point correlation functions (2 and 3PCFs) to actual galaxy data and (2) constrain the non-linear effects of degenerate higher-order scalar-tensor (DHOST) theories on cosmological scales. Applying this analysis to the Baryon Oscillation Spectroscopic Survey (BOSS) data release 12, we obtain the lower bounds of −1.655 < ξt and −0.504 < ξs at the $95{{\ \rm per\ cent}}$ confidence level on the parameters characterizing the time evolution of the tidal and shift terms of the second-order velocity field. These constraints are consistent with GR predictions of ξt = 15/1144 and ξs = 0. Moreover, they represent a 35-fold and 20-fold improvement, respectively, over the joint analysis with only the isotropic 3PCF. We ensure the validity of our results by investigating various quantities, including theoretical models of the 3PCF, window function corrections, cumulative S/N, Fisher matrices, and statistical scattering effects of mock simulation data. We also find statistically significant discrepancies between the BOSS data and the Patchy mocks for the 3PCF measurement. Finally, we package all of our 3PCF analysis codes under the name hitomi and make them publicly available so that readers can reproduce all the results of this paper and easily apply them to ongoing future galaxy surveys.

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.

Approximately stealth black hole in higher-order scalar-tensor theories

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 ĉ D1 1/2 M ≲ 2 × 1011 GeV, where ĉ 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.

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.).

Generalized Regge-Wheeler equation from Effective Field Theory of black hole perturbations with a timelike scalar profile

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. 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.

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.

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.

Traversable wormholes in beyond Horndeski 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 theory. The particular class of wormholes we study are parametrised by their ADM mass and two coupling constants of the theory, one related to their higher dimensional Lovelock origin and one to the disformal transformation itself. The latter of the coupling constants affects the throat size of the wormhole solutions, thus giving them a compact or non-compact nature, as well as their properties.

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.

Coupled dark energy model inspired from general conformal transformation

We study the coupled dark energy model constructed from the general conformal transformation in which the coefficient of the conformal transformation depends on both the scalar field and its kinetic term. Under this conformal transformation, the action for subclass of Degenerate Higher-Order Scalar-Tensor (DHOST) theories is related to the Einstein-Hilbert action. The evolution of the background universe has the scaling fixed point which corresponds to acceleration of the universe at late time. For the choices of parameters which make the late-time scaling point stable, the fixed point corresponding to $\phi$-matter-dominated-era ($\phi$MDE) is a saddle point, and the universe can evolve from radiation dominated epoch through $\phi$MDE before reaching the scaling point at late time with the cosmological parameters which satisfy the observational bound. During the $\phi$MDE, the effective equation of state parameter is slightly positive, so that one of possible mechanisms for alleviating the $H_0$ tension can be achieved. In this coupled dark energy model, the effective gravitational coupling for dark matter perturbations on small scales can be smaller than that in the $\Lambda$CDM model. Therefore a growth rate of the dark matter perturbations is suppressed compare with the $\Lambda$CDM model, which implies that the $\sigma_8$ tension could be alleviated.

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.

Nonlinear definition of the shadowy mode in higher-order scalar-tensor theories

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.