Horndeski Gravity Research Articles

Chromo-natural inflation supported by enhanced friction from Horndeski gravity

We study the extension of the chromo-natural inflation model by incorporating nonminimal coupling between the axion field and gravity.Nonminimal coupling is introduced so that it enhances friction in the axion's equation of motion and thus supports slow-roll inflation.This enhanced friction effectively delays the activation of the gauge field, thereby preventing the overproduction of gravitational waves in the CMB scale.We extend previous results by describing the nonminimal coupling in a general and unifying way utilizing Horndeski gravity.This allows us to explore systematically and comprehensively possible enhanced friction models of chromo-natural inflation consistent with observations.We find a novel enhanced friction model that shows better agreement (within 1σ) with CMB measurements than the previous nonminimally coupled chromo-natural inflation model.The gravitational-wave spectrum starts to rise at some wavenumber due to retarded activation of the gauge field in the late stage of inflation.We show how one can identify the wavenumber at which this occurs based on the background evolution and present a universal analytic formula for the gravitational-wave spectrum that can be used for any enhanced friction model of chromo-natural inflation.

Mochi_class: Modelling Optimisation to Compute Horndeski In class

We introduce mochi_class, an extension of the Einstein-Boltzmann solver hi_class, designed to unlock the full phenomenological potential of Horndeski gravity. This extension allows for general input functions of time without the need for hard-coded parametrisations or covariant Lagrangians. By replacing the traditional α-parametrisation with a set of stable basis functions, mochi_class ensures that the resulting effective theories are inherently free from gradient and ghost instabilities. Additionally, mochi_class features a quasi-static approximation implemented at the level of modified metric potentials, enhancing prediction accuracy, especially for models transitioning between a super- and sub-Compton regime. mochi_class can robustly handle a wide range of models without fine-tuning, and introduces a new approximation scheme that activates modifications to the standard cosmology deep in the matter-dominated era. Furthermore, it incorporates viability conditions on the equation of motion for the scalar field fluctuations, aiding in the identification of numerical instabilities. Through comprehensive validation against other Einstein-Boltzmann solvers, mochi_class demonstrates excellent performance and accuracy, broadening the scope of hi_class by facilitating the study of specific modified gravity models and enabling exploration of previously inaccessible regions of the Horndeski landscape. The code is publicly available at this URL (https://github.com/mcataneo/mochi_class_public)

General analysis of Noether symmetries in Horndeski gravity

We explore Noether symmetries of Horndeski gravity, extending the classification of general scalar–tensor theories. Starting from the minimally coupled scalar field and the first-generation scalar–tensor gravity, the discussion is generalised to kinetic gravity braiding and Horndeski gravity. We highlight the main findings by focusing on the non-minimally coupled Gauss–Bonnet term and the extended cuscuton model. Finally, we discuss how the presence of matter can influence Noether symmetries. It turns out that the selected Horndeski functions are unchanged with respect to the vacuum case.

Phenomenology of Horndeski gravity under positivity bounds

A set of conditions that any effective field theory needs to satisfy in order to allow for the existence of a viable UV completion, has recently gained attention in the cosmological context under the name of positivity bounds. In this paper we revisit the derivation of such bounds for Horndeski gravity, highlighting the limitations that come from applying the traditional methodology to a theory of gravity on a cosmological background. We then translate these bounds into a complete set of viability conditions in the language of effective field theory of dark energy. We implement the latter into EFTCAMB and explore the large scale structure phenomenology of Horndeski gravity under positivity bounds. We build a statistically significant sample of viable Horndeski models, and derive the corresponding predictions for the background evolution, in terms of w DE, and the dynamics of linear perturbations, in terms of the phenomenological functions μ and Σ, associated to clustering and weak lensing, respectively. We find that the addition of positivity bounds to the traditional no-ghost and no-gradient conditions considerably tightens the theoretical constraints on all these functions. The most significant feature is a strengthening of the correlation μ ≃ Σ, and a related tight constraint on the luminal speed of gravitational waves c 2 T ≃ 1. In this work we demonstrate the strong potential of positivity bounds in shaping the viable parameter space of scalar-tensor theories. This is certainly promising, but it also highlights the importance of overcoming all issues that still plague a rigorous formulation of the positivity bounds in the cosmological context.

Magnetized AdS/BCFT Correspondence in Horndeski Gravity

AbstractThis work examines the thermodynamics and hydrodynamics behaviors of a five‐dimensional black hole under the influence of an external magnetic field. The solution is the gravity dual to the Anti‐de Sitter/Boundary Conformal Field Theory correspondence, enabling the study of properties within an anisotropic fluid framework. Utilizing holographic renormalization, the free energy and the holographic stress tensor residing on the boundary denoted as are computed. Within the fluid/gravity correspondence framework, authors have a class of boundary extensions in , where the stress‐energy tensor describes a magnetizing conformal fluid. The characteristics of this special solution as well as its thermodynamic properties, including the bulk and shear viscosity, the square of the speed of sound, as well as the anisotropic effects induced by the magnetic field in the magnetized conformal plasma are discussed.

Parameter constraints on traversable wormholes within beyond Horndeski theories through quasiperiodic oscillations

compact astrophysical objects such as black holes and wormholes, as well as testing gravity theories, are important issues in relativistic astrophysics. In this sense, theoretical and observational studies of quasiperiodic oscillations (QPOs) observed in (micro)quasars become helpful in exploring their central object, which can be a black hole or a wormhole. In the present work, we study the throat properties of traversable wormholes beyond Horndeski theory. Also, we investigate the circular motion of test particles orbiting the wormhole. We analyze the test particle’s effective potential and angular momentum for circular orbits. Frequencies of radial and vertical oscillations of the particles around stable circular orbits have also been studied and applied in explaining the quasiperiodic oscillations mechanism in the relativistic precession (RP) model. Finally, we obtain constraint values for the parameters of beyond Horndeski gravity and the mass of the wormhole candidates using QPOs observed in the microquasars GRO J1655-40, GRS 1915+105 & XTE J1550-564 and at the center of Milky Way galaxy through Monte-Carlo-Markovian-chain (MCMC) analyses. Published by the American Physical Society 2024

Holographic renormalization of Horndeski gravity

We study the renormalization of a particular sector of Horndeski theory. In particular, we focus on the nonminimal coupling of a scalar field to the Gauss-Bonnet term and its kinetic coupling to the Einstein tensor. Adopting a power expansion on the scalar function that couples the Gauss-Bonnet term, we find specific conditions on their coefficients such that the action and charges are finite. To accomplish the latter, we add a finite set of intrinsic boundary terms. The contribution of the nonminimal coupling generates an effective scalar mass, allowing us to recover a modified Breitenlohner-Freedman bound. Furthermore, we compute the holographic 1-point functions and Ward identities associated with the scalar field and the metric. We constrain the parameter space of the theory by taking into account the preservation of scaling symmetry at the boundary.

Notes on thermodynamics of Schwarzschild-like bumblebee black hole

In this work, we revisited the thermodynamics of Schwarzschild-like bumblebee black hole using Iyer–Wald covariant phase space formalism. With a non zero vacuum expectation value, the bumblebee field is responsible for spontaneously breaking the Lorentz symmetry. As the bumblebee field is non-minimally coupled to gravity, we showed that the thermodynamic variables will be different from the counterpart in Einstein gravity. Especially, by using Iyer–Wald formalism, we found that the black hole entropy also differs from the result obtained from Wald entropy formula. Like Horndeski gravity, this mismatch is due to the divergence of bumblebee one-form field at the horizon. After figuring out the thermodynamics, we also briefly discussed the evaporation behavior of Schwarzschild like bumblebee black hole. We found that although bumblebee field has no influence on the critical impact factor, it can influence the black hole evaporation time.

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.

Unifying ordinary and null memory

Based on a recently proposed reinterpretation of gravitational wave memory that builds up on the definition of gravitational waves pioneered by Isaacson, we provide a unifying framework to derive both ordinary and null memory from a single well-defined equation at leading order in the asymptotic expansion. This allows us to formulate a memory equation that is valid for any unbound asymptotic energy-flux that preserves local Lorentz invariance. Using Horndeski gravity as a concrete example metric theory with an additional potentially massive scalar degree of freedom in the gravitational sector, the general memory formula is put into practice by presenting the first account of the memory correction sourced by the emission of massive field waves. Throughout the work, physical degrees of freedom are identified by constructing manifestly gauge invariant perturbation variables within an SVT decomposition on top of the asymptotic Minkowski background, which will in particular prove useful in future studies of gravitational wave memory within vector tensor theories.

Electromagnetic field and chaotic charged-particle motion around hairy black holes in Horndeski gravity

The Wald vector potential is an exact solution of the source-less Maxwell equations regarding an electromagnetic field of a vacuum uncharged black hole like the Kerr background black hole in an asymptotically uniform magnetic field. However, it is not if the black hole is a nonvacuum solution in a theory of modified gravity with extra fields or a charged Kerr–Newman spacetime. To satisfy the source-less Maxwell equations in this case, the Wald vector potential must be modified and generalized appropriately. Following this idea, we derive an expression for the vector potential of an electromagnetic field surrounding a hairy black hole in the Horndeski modified gravity theory. Explicit symplectic integrators with excellent long-term behaviour are used to simulate the motion of charged particles around the hairy black hole immersed in the external magnetic field. The recurrence plot method based on the recurrence quantification analysis uses diagonal structures parallel to the main diagonal to show regular dynamics, but adopts no diagonal structures to indicate chaotic dynamics. The method is efficient to detect chaos from order in the curved spacetime, as the Poincaré map and the fast Lyapunov indicator are.

Scaling symmetry, Smarr relation, and the extended first law in lower-dimensional Lovelock gravity

Recently, it was discovered that lower-dimensional versions of Lovelock gravity exist as scalar-tensor theories that are examples of Horndeski gravity. We study the thermodynamics of the static black hole solutions in these theories up to cubic order through Euclidean methods. Considering solutions with spherical, planar and hyperbolic event horizons (k=+1,0,−1), we show that the universality of the thermodynamics for planar black holes (k=0) and the extended first law that include the variation of the couplings together with their associated potentials hold also in lower dimensions. We find that in D=4,6 where the 2nd- and the 3rd-order Lovelock Lagrangians are boundary terms respectively, the Smarr relation is modified since the entropy is not a homogenous function in these dimensions. We also present a derivation of the Smarr relation and its modified version based on the global scaling properties of the reduced action that is used to obtain the solutions consistently. Unlike the other hairy black hole solutions that have been analyzed before, despite the terms in the reduced action that break the scaling symmetry, the derivation still follows from a conserved Noether charge.

Influences of tilted thin accretion disks on the observational appearance of hairy black holes in Horndeski gravity

Research on the observational appearance of black holes, both in general relativity and modified gravity, has been in full swing since the Event Horizon Telescope Collaboration announced photos of M87* and Sagittarius A*. Nevertheless, limited attention has been given to the impact of tilted accretion disks on black hole images. This paper investigates the 230 GHz images of non-rotating hairy black holes illuminated by tilted, thin accretion disks in Horndeski gravity with the aid of a ray tracing method. The results indicate that reducing the scalar hair parameter effectively diminishes image luminosity and extends both the critical curve and the inner shadow. This trend facilitates the differentiation between hairy black holes and Schwarzschild black holes, especially in certain parameter spaces where the current Event Horizon Telescope array is capable of capturing such variations. Furthermore, we observe that the inclination of the tilted accretion disk can mimic the observation angle, consequently affecting image brightness and the morphology of the inner shadow. In specific parameter spaces, alterations in the tilt or position of the accretion disk can lead to a drift in the light spot within the images of hairy black holes. This finding may establish a potential correlation between the precession of the tilted accretion disk and image features. Additionally, through an examination of images depicting hairy black holes surrounded by two thin accretion disks, we report the obscuring effect of the accretion environment on the inner shadow of the black hole.

Towards a possible solution to the Hubble tension with Horndeski gravity

The Hubble tension refers to the discrepancy in the value of the Hubble constant H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_0$$\\end{document} inferred from the cosmic microwave background observations, assuming the concordance Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model of the Universe, and that from the distance ladder and other direct measurements. In order to alleviate this tension, we construct a plausible dark energy scenario, within the framework of Horndeski gravity which is one of the most general scalar–tensor theories yielding second-order equations. In our set-up, we include the self-interactions and nonminimal coupling of the dynamical dark energy scalar field which enable very interesting dynamics leading to a phantom behaviour at low redshifts along with negative dark energy densities at high redshifts. These two features together make this model a promising scenario to alleviate the Hubble tension for appropriate choices of the model parameters. Towards a consistent model building, we show that this set-up is also free from both the gradient and ghost instabilities. Finally, we confront the predictions of the model with low redshift observations from Pantheon, SH0ES, cosmic chronometers and BAO, to obtain best fit constraints on model parameters.

Domain walls in Horndeski gravity

The Horndeski gravity yields domain wall solutions that connect asymmetric vacua for a certain region of parameters. In the thin wall limit there exist BPS domain walls that connect stable vacua. That is, the false vacua in this theory follows the same effect obtained in supergravity vacua, i.e., it does not decay according to Coleman-de Luccia theory and the BPS solutions interpolating different vacua, e.g., non-degenerate AdS vacua, are static infinite planar domain walls separating vacua associated with different spacetimes.

50 Years of Horndeski Gravity: Past, Present and Future

50 Years of Horndeski Gravity: Past, Present and Future

Thick Branes in Horndeski Gravity

We investigate thick brane solutions in the Horndeski gravity. In this setup, we found analytical solutions, applying the first-order formalism to two scalar fields where the first field comes from the nonminimal scalar-tensor coupling and the second is due to the matter contribution sector. With these analytical solutions, we evaluate the symmetric thick brane solutions in Horndeski gravity with four-dimensional geometry. In such a setup, we evaluate the gravity fluctuations to find “almost massless modes,” for any values of the Horndeski parameters. These modes were used to compute the corrections to the Newtonian potential and evaluate the limit four-dimensional gravity.

FeynGrav and Recent Progress in Computational Perturbative Quantum Gravity

This article reviews recent progress in computational quantum gravity caused by the framework that efficiently computes Feynman’s rules. The framework is implemented in the FeynGrav package, which extends the functionality of the widely used FeynCalc package. FeynGrav provides all the tools to study quantum gravitational effects within the standard model. We review the framework, provide the theoretical background for the efficient computation of Feynman rules, and present the proof of its completeness. We review the derivation of Feynman rules for general relativity, Horndeski gravity, Dirac fermions, Proca field, electromagnetic field, and SU(N) Yang–Mills model. We conclude with a discussion of the current state of the FeynGrav package and discuss its further development.

Post-Newtonian binary dynamics in the effective field theory of Horndeski gravity* *Supported by National Key Research and Development Program of China (2021YFC2201901) and National Natural Science Foundation of China (12147103, 11851302)

General relativity has been very successful since its proposal more than a century ago. However, various cosmological observations and theoretical consistency still motivate us to explore extended gravity theories. Horndeski gravity stands out as one attractive theory by introducing only one scalar field. Here we formulate the post-Newtonian effective field theory of Horndeski gravity and investigate the conservative dynamics of inspiral compact binary systems. We calculate the leading effective Lagrangian for a compact binary and obtain the periastron advance per period. In particular, we apply our analytical calculation to two binary systems, PSR B 1534+12 and PSR J0737-3039, and constrain the relevant model parameters. This theoretical framework can also be systematically extended to higher orders.

Total light bending in non-asymptotically flat black hole spacetimes

The gravitational deflection of light is a critical test of modified theories of gravity. A few years ago, Gibbons and Werner introduced a definition of the deflection angle based on the Gauss–Bonnet theorem. In more recent years, Arakida proposed a related idea for defining the deflection angle in non-asymptotically flat spacetimes. We revisit this idea and use it to compute the angular difference in the Kottler geometry and a non-asymptotically flat solution in Horndeski gravity. Our analytic and numerical calculations show that a triangular array of laser beams can be designed so that the proposed definition of the deflection angle is sensitive to different sources of curvature. Moreover, we find that near the photon sphere, the deflection angle in the Horndeski solution is similar to its Schwarzschild counterpart, and we confirm that the shadows seen by a static observer are identical.