Horndeski Theory Research Articles

Stability of neutron stars in Horndeski theories with Gauss-Bonnet couplings

In Horndeski theories containing a scalar coupling with the Gauss-Bonnet (GB) curvature invariant ${R}_{\mathrm{GB}}^{2}$, we study the existence and linear stability of neutron star (NS) solutions on a static and spherically symmetric background. For a scalar-GB coupling of the form $\ensuremath{\alpha}\ensuremath{\xi}(\ensuremath{\phi}){R}_{\mathrm{GB}}^{2}$, where $\ensuremath{\xi}$ is a function of the scalar field $\ensuremath{\phi}$, the existence of linearly stable stars with a nontrivial scalar profile without instabilities puts an upper bound on the strength of the dimensionless coupling constant $|\ensuremath{\alpha}|$. To realize maximum masses of NSs for a linear (or dilatonic) GB coupling ${\ensuremath{\alpha}}_{\mathrm{GB}}\ensuremath{\phi}{R}_{\mathrm{GB}}^{2}$ with typical nuclear equations of state, we obtain the theoretical upper limit $\sqrt{|{\ensuremath{\alpha}}_{\mathrm{GB}}|}<0.7\text{ }\text{ }\mathrm{km}$. This is tighter than those obtained by the observations of gravitational waves emitted from binaries containing NSs. We also incorporate cubic-order scalar derivative interactions, quartic derivative couplings with nonminimal couplings to a Ricci scalar besides the scalar-GB coupling, and show that NS solutions with a nontrivial scalar profile satisfying all the linear stability conditions are present for certain ranges of the coupling constants. In regularized four-dimensional Einstein-GB gravity obtained from a Kaluza-Klein reduction with an appropriate rescaling of the GB coupling constant, we find that NSs in this theory suffer from a strong coupling problem as well as Laplacian instability of even-parity perturbations. We also study NS solutions with a nontrivial scalar profile in power-law $F({R}_{\mathrm{GB}}^{2})$ models, and show that they are pathological in the interior of stars and plagued by ghost instability together with the asymptotic strong coupling problem in the exterior of stars.

Probing modified gravity with integrated Sachs-Wolfe CMB and galaxy cross-correlations

We use the cross-correlation power spectrum of the integrated Sachs-Wolfe (ISW) effect in the cosmic microwave background (CMB) temperature anisotropy and galaxy fluctuations to probe the physics of late-time cosmic acceleration. For this purpose, we focus on three models of dark energy that belong to a sub-class of Horndeski theories with the speed of gravity equivalent to that of light: Galileon Ghost Condensate (GGC), Generalized Cubic Covariant Galileon (GCCG), and K-mouflage. In the GGC and GCCG models, the existence of cubic-order scalar self-interactions allows a possibility for realizing negative ISW-galaxy cross-correlations, while the K-mouflage model predicts a positive correlation similar to the Λ-cold-dark-matter (ΛCDM) model. In our analysis, we fix the parameters of each model to their best-fit values derived from a baseline likelihood analysis with observational data from CMB, baryon acoustic oscillations, and supernovae type Ia. Then we fit those best-fit models to the ISW-galaxy cross-correlation power spectrum extracted from a collection of photometric redshift surveys. We find that both GGC and GCCG best-fit models degrade the fit to the ISW-galaxy cross-correlation data compared to ΛCDM best-fit model. This is attributed to the fact that, for their best-fit values constrained from the baseline likelihood, the cubic-order scalar self-interaction gives rise to suppressed ISW tails relative to ΛCDM. The K-mouflage best-fit model is largely degenerate with the ΛCDM best-fit model and has a positively correlated ISW-galaxy power close to that of ΛCDM.

Effective field theory of black hole perturbations with timelike scalar profile: formulation

We formulate the Effective Field Theory (EFT) of perturbations within scalar-tensor theories on an inhomogeneous background. The EFT is constructed while keeping a background of a scalar field to be timelike, which spontaneously breaks the time diffeomorphism. We find a set of consistency relations that are imposed by the invariance of the EFT under the 3d spatial diffeomorphism. This EFT can be generically applied to any inhomogeneous background metric as long as the scalar profile is everywhere timelike. For completeness, we report a dictionary between our EFT parameters to those of Horndeski theories. Finally, we compute background equations for a class of spherically symmetric, static black hole backgrounds, including a stealth Schwarzschild-de Sitter solution.

Linear perturbations of Einstein-Gauss-Bonnet black holes

We study linear perturbations about non rotating black hole solutions in scalar-tensor theories, more specifically Horndeski theories. We consider two particular theories that admit known hairy black hole solutions. The first one, Einstein-scalar-Gauss-Bonnet theory, contains a Gauss-Bonnet term coupled to a scalar field, and its black hole solution is given as a perturbative expansion in a small parameter that measures the deviation from general relativity. The second one, known as 4-dimensional-Einstein-Gauss-Bonnet theory, can be seen as a compactification of higher-dimensional Lovelock theories and admits an exact black hole solution. We study both axial and polar perturbations about these solutions and write their equations of motion as a first-order (radial) system of differential equations, which enables us to study the asymptotic behaviours of the perturbations at infinity and at the horizon following an algorithm we developed recently. For the axial perturbations, we also obtain effective Schrödinger-like equations with explicit expressions for the potentials and the propagation speeds. We see that while the Einstein-scalar-Gauss-Bonnet solution has well-behaved perturbations, the solution of the 4-dimensional-Einstein-Gauss-Bonnet theory exhibits unusual asymptotic behaviour of its perturbations near its horizon and at infinity, which makes the definition of ingoing and outgoing modes impossible. This indicates that the dynamics of these perturbations strongly differs from the general relativity case and seems pathological.

Compact objects of spherical symmetry in beyond Horndeski theories

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 integrate the field equations by choosing the form of only one coupling function and an auxiliary quantity. This method leads to asymptotically flat and AdS black hole solutions with differing properties. We finally discuss disformal transformations within this context as a means of obtaining wormhole and black hole solutions in different theories.

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.

Linear stability of black holes with static scalar hair in full Horndeski theories: Generic instabilities and surviving models

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 Minkowski 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 Minkowski 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 Minkowski hairy BH solutions free from ghost/Laplacian instabilities.

Magnetic and electric black holes in the vector-tensor Horndeski theory

We construct exact solutions of magnetically charged black holes in the vector-tensor Horndeski gravity and discuss their main features. Unlike the analogous electric case, the field equations are linear in a simple (quite standard) parametrization of the metric tensor, and they can be solved analytically even when a cosmological constant is added. The solutions are presented in terms of hypergeometric functions, which makes the analysis of the black hole properties relatively straightforward. Some of the aspects of these black holes are quite ordinary, like the existence of extremal configurations with maximal magnetic charge for a given mass or the existence of a mass with maximal temperature for a given charge, but others are somewhat unexpected, like the existence of black holes with a repulsive gravitational field. We perform our analysis for both signs of the nonminimal coupling constant and find black hole solutions in both cases but with significant differences between them. The most prominent difference is the fact that the black holes for the negative coupling constant have a spherical surface of curvature singularity rather than a single point. On the other hand, the gravitational field produced around this kind of black holes is always attractive. Also, for small enough magnetic charge and negative coupling constant, extremal black holes do not exist, and all magnetic black holes have a single horizon. In addition, we study the trajectories around these magnetic black holes for light as well as massive particles either neutral or electrically charged. Finally, we compare the main features of these black holes with their electric counterparts, adding some aspects that have not been discussed before, like temperature, particle trajectories, and light deflection by electrically charged Horndesky black holes.

Effect of Horndeski theory on weak deflection angle using the Gauss–Bonnet theorem

The main goal of this paper is to study the weak gravitational lensing by Horndeski black hole in weak field approximation. In order to do so, we exploit the Gibbons–Werner method to the optical geometry of Horndeski black hole and implement the Gauss–Bonnet theorem to accomplish the deflection angle of light in weak field region. Furthermore, we have endeavored to extend the scale of our work by comprising the impact of plasma medium on the deflection angle as properly. Later, the graphical influence of the deflection angle of photon on Horndeski black hole in plasma and nonplasma medium is examined.

GUP-corrected thermodynamics and remnant of a black hole in Horndeski theory

This paper is devoted to study the effects of the generalized uncertainty principle (GUP) on Hawking radiation of scalar particles and fermions tunneling of Babichev–Charmousis black hole by using Hamilton–Jacobi method. To this end, we modify the Klein–Gordon equation (K-G equation) and Dirac equation firstly on account of GUP. Then, we apply the semiclassical WKB approximation to derive the corrected Hawking temperature of scalar particles and fermions tunneling process and corresponding corrected thermodynamics quantities of this black hole. We find that the corrected temperature depends on GUP parameter [Formula: see text]. Furthermore, the existence of minimal mass and minimal entropy implies that when the evaporation of the black hole stops, it will leave a remnant.

Reconstruction in the Horndeski theory within the scope of the Bianchi I cosmology

In this paper, [R. K. Muharlyamov and T. N. Pankratyeva, Eur. Phys. J. Plus 136, 590 (2021)] we have proposed a reconstruction method for the kinetic gravity braiding theory in the framework of the flat Friedman–Robertson–Walker spacetime. Here, we develop this method in the Bianchi I spacetime model for a subclass of the Horndeski theory: [Formula: see text], [Formula: see text], [Formula: see text]. The Hubble parameter [Formula: see text] and the canonical kinetic term [Formula: see text] are set a priori. The choice of the function [Formula: see text] determines the anisotropic properties of the Universe. This makes it possible to provide believable anisotropy. The presented method allows for a realistic model of the Universe to simply reconstruct some scalar field theory. Reconstruction example is given for anisotropic model of a post-inflationary transition to the radiation-dominated phase. The model is investigated for the absence ghosts and Laplacian instabilities.

First-order thermodynamics of Horndeski gravity

We extend to the Horndeski realm the irreversible thermodynamics description of gravity previously studied in "first generation" scalar-tensor theories. We identify a subclass of Horndeski theories as an out-of--equilibrium state, while general relativity corresponds to an equilibrium state. In this context, we identify an effective heat current, "temperature of gravity", and shear viscosity in the space of theories. The identification is accomplished by recasting the field equations as effective Einstein equations with an effective dissipative fluid, with Einstein gravity as the equilibrium state, following Eckart's first-order thermodynamics.

Black hole binaries in cubic Horndeski theories

We study black hole binary mergers in certain cubic Horndeski theories of gravity, treating them fully nonlinearly. In the regime of validity of effective field theory, the mismatch of the gravitational wave strain between Horndeski and general relativity (coupled to a scalar field) can be as large as 10%--13% in the Advanced LIGO mass range. The initial data and coupling constants are chosen such the theory remains in the weakly coupled regime throughout the evolution. In all cases that we have explored, we observe that the waveform in the Horndeski theory is shifted by an amount much larger than the smallness parameter that controls the initial data. This effect is generic and may be present in other theories of gravity involving higher derivatives.

Analysis of gravitational waves from inflation model with minimal, non-minimal, and non-minimal derivative coupling of scalar field from Horndeski theory

Inflation theory provides solutions for problems in cosmology, such as horizon problem and flatness problem. The study of Gravitational waves production in cosmic inflation era not prove the general inflation theory, but also to differentiate in detail among specific models. In this research, we use inflation model with minimal, non-minimal, and non-minimal derivative coupling of scalar field without potential from Horndeski Theory. From this model, we calculate scalar and tensor perturbation equations and then obtain its equation of gravitational waves, spectral index for each perturbation mode and tensor-to-scalar ratio. Spectral index and tensor-to-scalar ratio nearly scale-invariant and agree with observational data for some Ho, ζ, and ξ. Gravitational waves remain constant during inflation and start oscillates when its modes enter the horizon. Energy of this gravitational waves is scale-invariant for modes that re-enter horizon during radiation dominated era and rises toward lower frequencies.

Numerical Analysis of Non-minimal Coupling and Non-minimal Derivative Coupling Models using Direct Random Method

In this work, we studied classes of Horndeski theory, specifically the Non-minimal Derivative Coupling (NMDC) ‘model and Non-minimal Coupling (NMC) model, against the observation of gravitational waves using the numerical framework given by Arai and Nishizawa. The framework takes randomized set of parameters of the theory, filters results that disagree with the late-time evolution of the universe and the stability condition, and outputs observable quantities of gravitational waves (GW) that were then compared to the quantities obtained from observational data of gravitational waves detection.

Stable nonsingular cosmologies in beyond Horndeski theory and disformal transformations

In this paper, we collect, systemize and generalize the existing results for relations between general Horndeski theories and beyond Horndeski theories via disformal transformations of metric. We focus on those subclasses of beyond Horndeski theories which are disformally related to Horndeski theories and derive additional transformation rules relating the Lagrangian functions of these two theories. Then we demonstrate that some of these transformation rules become singular at some moments(s) once one constructs a nonsingular cosmological solution in beyond Horndeski theory that is free from ghost, gradient instabilities and strong gravity regime during the entire evolution of the system. The key issue here is that such solutions are banned in Horndeski theory due to the existing no-go theorem. The proof of singular behavior of disformal relations in this case resolves the apparent contradiction between the fact that Horndeski and beyond Horndeski theories were initially implied to be related via healthy field redefinition and at the same time these theories describe different physics in the context of nonsingular cosmologies.

Vainshtein screening in Horndeski theories nonminimally and kinetically coupled to ordinary matter

We study the Vainshtein screening mechanism in Horndeski theories in the presence of a scalar field $\phi$ nonminimally and kinetically coupled to ordinary matter field. A general interacting Lagrangian describing this coupling is characterized by energy transfer $f_1$ and momentum exchange $f_2$. For a spherically symmetric configurations on top of the cosmological background, we investigate the static perturbations in linear and nonlinear regimes with respect to the scalar field perturbation. In the former regime, the parametrized post-Newtonian parameter generally deviates from unity as long as the matter coupling or $G_{4,\phi}$ exists. On the other hand, in the latter regime, we show that the nonlinear self-interaction term of scalar field successfully activates the Vainshtein mechanism even in the presence of the couplings $f_1$ and $f_2$. The gravitational potentials recover the Newtonian behavior deep inside the Vainshtein radius. The bounds on coupling terms not to substantially change the Vainshtein radius are also given.

Linear stability of black holes in shift-symmetric Horndeski theories with a time-independent scalar field

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 nonasymptotically 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 nonreflection-symmetric theories. For cubic Galileons with the Einstein-Hilbert term, there exists a nonasymptotically 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.

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.

Dynamical dimensional reduction in multivalued Hamiltonians

Several interesting physical systems, such as the Lovelock extension of general relativity in higher dimensions, classical time crystals, $k$-essence fields, Horndeski theories, compressible fluids, and nonlinear electrodynamics, have apparent ill-defined sympletic structures, due to the fact that their Hamiltonians are multivalued functions of the momenta. In this paper, the dynamical evolution generated by such Hamiltonians is described as a degenerate dynamical system, whose sympletic form does not have a constant rank, allowing novel features and interpretations not present in previous investigations. In particular, it is shown how the multivaluedness is associated with a dynamical mechanism of dimensional reduction, as some degrees of freedom turn into gauge symmetries when the system degenerates.