Horndeski Scalar-tensor Theory Research Articles

Exploring the self-tuning of the cosmological constant from Planck mass variation

Recently, the variation of the Planck mass in the general relativistic Einstein–Hilbert action was proposed as a self-tuning mechanism of the cosmological constant, preventing standard model vacuum energy from freely gravitating and enabling an estimation of the magnitude of its observed value. We explore here new aspects of this proposal. We first develop an equivalent Einstein-frame formalism to the current Jordan-frame formulation of the mechanism and use this to highlight similarities and differences of self-tuning to the sequestering mechanism. We then show how with an extension of the local self-tuning action by a coupled Gauss–Bonnet term and a companion four-form field strength, graviton loops can be prevented from incapacitating the degravitation of the standard model vacuum energy. For certain cases, we furthermore find that this extension can be recast as a Horndeski scalar–tensor theory and be embedded in the conventional local self-tuning formalism. We then explore the possibility of a unification of inflation with self-tuning. The resulting equations can alternatively be used to motivate a multiverse interpretation. In this context, we revisit the coincidence problem and provide an estimation for the probability of the emergence of intelligent life in our Universe as a function of cosmic age, inferred from star and terrestrial planet formation processes. We conclude that we live at a very typical epoch, where we should expect the energy densities of the cosmological constant and matter to be of comparable size. For a dimensionless quantity to compare the emergence of life throughout the cosmic history of different universes in an anthropic analysis of the multiverse, we choose the order of magnitude difference of the evolving horizon size of a Universe to the size of its proton as the basic building block of atoms, molecules, and eventually life. For our Universe we find this number to form peak at approximately 42. We leave the question of whether the same number is frequently assumed for the emergence of life across other universes or singles out a special case to future exploration.

Spherically symmetric, static black holes with scalar hair, and naked singularities in nonminimally coupled k -essence

We apply a recently developed 2+1+1 decomposition of spacetime, based on a nonorthogonal double foliation for the study of spherically symmetric, static black hole solutions of Horndeski scalar-tensor theory. Our discussion proceeds in an effective field theory (EFT) of modified gravity approach, with the action depending on metric and embedding scalars adapted to the nonorthogonal 2+1+1 decomposition. We prove that the most generic class of Horndeski Lagrangians compatible with observations can be expressed in this EFT form. By studying the first order perturbation of the EFT action we derive three equations of motion, which reduce to those derived earlier in an orthogonal 2+1+1 decomposition, and a fourth equation for the metric parameter N related to the nonorthogonality of the foliation. For the Horndeski class of theories with vanishing $G_3$ and $G_5$, but generic functions $G_2(\phi,X)$ (k-essence) and $G_4(\phi)$ (nonminimal coupling to the metric) we prove the unicity theorem that no action beyond Einstein--Hilbert allows for the Schwarzschild solution. Next we integrate the EFT field equations for the case with only one independent metric function obtaining new solutions characterized by a parameter interpreted as either mass or tidal charge, the cosmological constant and a third parameter. These solutions represent naked singularities, black holes with scalar hair or have the double horizon structure of the Schwarzschild--de Sitter spacetime. Solutions with homogeneous Kantowski--Sachs type regions also emerge. Finally, one of the solutions obtained for the function $G_4$ linear in the curvature coordinate, in certain parameter range exhibits an intriguing logarithmic singularity lying outside the horizon. The newly derived hairy black hole solutions evade previously known unicity theorems by being asymptotically nonflat, even in the absence of the cosmological constant.

Constraining scalar-tensor modified gravity with gravitational waves and large scale structure surveys

The first multi-messenger gravitational wave event has had a transformative effect on the space of modified gravity models. In this paper we study the enhanced tests of gravity that are possible with a future set of gravitational wave standard siren events. We perform MCMC constraint forecasts for parameters in Horndeski scalar-tensor theories. In particular, we focus on the complementarity of gravitational waves with electromagnetic large-scale structure data from galaxy surveys. We find that the addition of fifty low redshift (z ≲ 0.2) standard sirens from the advanced LIGO network offers only a modest improvement (a factor 1.1–1.3, where 1.0 is no improvement) over existing constraints from electromagnetic observations of large-scale structures. In contrast, high redshift (up to z ∼ 10) standard sirens from the future LISA satellite will improve constraints on the time evolution of the Planck mass in Horndeski theories by a factor ∼ 5. By simulating different scenarios, we find this improvement to be robust to marginalisation over unknown merger inclination angles and to variation between three plausible models for the merger source population.

Parameterised post-Newtonian formalism for the effective field theory of dark energy via screened reconstructed Horndeski theories

We bring together two popular formalisms which generically parameterise deviations from General Relativity on astrophysical and cosmological scales, namely the parameterised post-Newtonian (PPN) formalism and the effective field theory (EFT) of dark energy and modified gravity. These separate formalisms are successfully applied to independently perform tests of gravity in their respective regimes of applicability on vastly different length scales. Nonlinear screening mechanisms indeed make it imperative to probe General Relativity across a wide range of scales. For a comprehensive interpretation of the complementary measurements it is important to connect them to effectively constrain the vast gravitational model space. We establish such a connection within the framework of Horndeski scalar-tensor theories restricted to a luminal propagation speed of gravitational waves. This is possible via the reconstruction of the family of linearly degenerate covariant Horndeski actions from the set of EFT functions and the subsequent derivation of the PPN parameters from the reconstructed theory. We outline the required conditions which ensure a reconstructed Horndeski model possesses a screening mechanism that enables significant modifications on cosmological scales while respecting stringent astrophysical bounds. Employing a scaling method, we then perform the general post-Newtonian expansion of the reconstructed models to derive their PPN parameters γ and β in their screened regimes.

Radiative stability and observational constraints on dark energy and modified gravity

We investigate the radiative stability of Horndeski scalar-tensor theories with luminally propagating gravitational waves (as extensively discussed in the wake of GW170817) and show that in general there is a tension between obtaining observable deviations from General Relativity (GR) in cosmology and the requirement of radiative stability. Using this as a constraint, we discuss the subsets of theories that are capable of evading this conclusion and yielding observable, radiatively stable departures from GR. We find several classes of theories that can do so, recovering known cases and identifying several additional radiatively stable cases. Finally, we also extract the cosmological signatures of two particularly well-motivated radiatively stable classes of theories: shift-symmetric theories and theories with a conformal coupling between the scalar and gravity. We find that cosmological parameter constraints on dark energy and modified gravity parameters for both of these two classes, which we explicitly compute using data from the Planck, SDSS/BOSS and 6dF surveys, are significantly tightened with respect to generic Horndeski theories.

Measuring Gravity at Cosmological Scales

This review is a pedagogical introduction to models of gravity and how they are constrained through cosmological observations. We focus on the Horndeski scalar-tensor theory and on the quantities that can be measured with a minimum of assumptions. Alternatives or extensions of general relativity have been proposed ever since its early years. Because of the Lovelock theorem, modifying gravity in four dimensions typically means adding new degrees of freedom. The simplest way is to include a scalar field coupled to the curvature tensor terms. The most general way of doing so without incurring in the Ostrogradski instability is the Horndeski Lagrangian and its extensions. Testing gravity means therefore, in its simplest term, testing the Horndeski Lagrangian. Since local gravity experiments can always be evaded by assuming some screening mechanism or that baryons are decoupled, or even that the effects of modified gravity are visible only at early times, we need to test gravity with cosmological observations in the late universe (large-scale structure) and in the early universe (cosmic microwave background). In this work, we review the basic tools to test gravity at cosmological scales, focusing on model-independent measurements.

Positivity in the Sky: Constraining dark energy and modified gravity from the UV

Positivity bounds - the consequences of requiring a unitary, causal, local UV completion - place strong restrictions on theories of dark energy and/or modified gravity. We derive and investigate such bounds for Horndeski scalar-tensor theories and for the first time pair these bounds with a cosmological parameter estimation analysis, using CMB, redshift space distortion, matter power spectrum and BAO measurements from the Planck, SDSS/BOSS and 6dF surveys. Using positivity bounds as theoretical priors, we show that their inclusion in the parameter estimation significantly improves the constraints on dark energy/modified gravity parameters. Considering as an example a specific class of models, which are particularly well-suited to illustrate the constraining power of positivity bounds, we find that these bounds eliminate over 60% of the previously allowed parameter space. We also discuss how combining positivity requirements with additional theoretical priors has the potential to further tighten these constraints: for instance also requiring a subluminal speed of gravitational waves eliminates all but 1% of the previously allowed parameter space.

Strong cosmic censorship in Horndeski theory

The strong cosmic censorship hypothesis has recently regained a lot of attention in charged and rotating black holes immersed in de Sitter space. Although the picture seems to be clearly leaning towards the validity of the hypothesis in Kerr-de Sitter geometries, Reissner-Nordström-de Sitter black holes appear to be serious counter-examples. Here, we perform another test to the hypothesis by using a scalar field perturbation non-minimally coupled to the Einstein tensor propagating on Reissner-Nordström-de Sitter spacetimes. Such non-minimal derivative coupling is characteristic of Horndeski scalar-tensor theories. Although the introduction of higher-order derivative couplings in the energy-momentum tensor increases the regularity requirements for the existence of weak solutions beyond the Cauchy horizon, we are still able to find a small finite region in the black hole’s parameter space where strong cosmic censorship is violated.

Screening and degenerate kinetic self-acceleration from the nonlinear freedom of reconstructed Horndeski theories

We have previously presented a reconstruction of Horndeski scalar-tensor theories from linear cosmological observables. It includes free nonlinear terms which can be added onto the reconstructed covariant theory without affecting the background and linear dynamics. After discussing the uniqueness of these correction terms, we apply this nonlinear freedom to a range of different applications. First we demonstrate how the correction terms can be configured to endow the reconstructed models with screening mechanisms such as the chameleon, k-mouflage and Vainshtein effects. A further implication is the existence of classes of Horndeski models that are degenerate with standard cosmology to an arbitrary level in the cosmological perturbations. Particularly interesting examples are kinetically self-accelerating models that mimic the dynamics of the cosmological constant to an arbitrary degree in perturbations. Finally, we develop the reconstruction method further to the level of higher-order effective field theory, which under the restriction to a luminal propagation speed of gravitational waves introduces two new free functions per order. These functions determine the corresponding correction terms in the linearly reconstructed action at the same order. Our results enable the connection of linear cosmological constraints on generalised modifications of gravity and dark energy with the nonlinear regime and astrophysical probes for a more global interpretation of the wealth of forthcoming cosmological survey data.

Cosmological parameter constraints for Horndeski scalar-tensor gravity

We present new cosmological parameter constraints for general Horndeski scalar-tensor theories, using CMB, redshift space distortion, matter power spectrum and BAO measurements from the Planck, SDSS/BOSS and 6dF surveys. We focus on theories with cosmological gravitational waves propagating at the speed of light, $c_{\rm GW} = c$, implementing and discussing several previously unaccounted for aspects in the constraint derivation for such theories, that qualitatively affect the resulting constraints. In order to ensure our conclusions are robust, we compare results for three different parametrisations of the free functions in Horndeski scalar-tensor theories, identifying several parametrisation-independent features of the constraints. We also consider models, where $c_{\rm GW} \neq c$ in cosmological settings (still allowed after GW170817 for frequency-dependent $c_{\rm GW}$) and show how this affects cosmological parameter constraints.

Einstein-Gauss-Bonnet Gravity with Extra Dimensions

We consider a theory of modified gravity possessing d extra spatial dimensions with a maximally symmetric metric and a scale factor, whose ( 4 + d ) -dimensional gravitational action contains terms proportional to quadratic curvature scalars. Constructing the 4D effective field theory by dimensional reduction, we find that a special case of our action where the additional terms appear in the well-known Gauss-Bonnet combination is of special interest as it uniquely produces a Horndeski scalar-tensor theory in the 4D effective action. We further consider the possibility of achieving stabilised extra dimensions in this scenario, as a function of the number and curvature of extra dimensions, as well as the strength of the Gauss-Bonnet coupling. Further questions that remain to be answered such as the influence of matter-coupling are briefly discussed.

Reconstructing Horndeski theories from phenomenological modified gravity and dark energy models on cosmological scales

Recently we have derived a set of mapping relations that enables the reconstruction of the family of Horndeski scalar-tensor theories which reproduce the background dynamics and linear perturbations of a given set of effective field theory of dark energy coefficients. In this paper we present a number of applications of this reconstruction. We examine the form of the underlying theories behind different phenomenological parameterizations of modified gravity and dark energy used in the literature, as well as examine theories that exhibit weak gravity, linear shielding, and minimal self-acceleration. Finally, we propose a new inherently stable parametrization basis for modified gravity and dark energy models.

Matter bispectrum beyond Horndeski theories

The Horndeski scalar-tensor theory and its recent extensions allow nonlinear derivative interactions of the scalar degree of freedom. We study the matter bispectrum of large scale structure as a probe of these modified gravity theories, focusing in particular on the effect of the terms that newly appear in the so-called "beyond Horndeski" theories. We derive the second-order solution for the matter density perturbations and find that the interactions beyond Horndeski lead to a new time-dependent coefficient in the second-order kernel which differs in general from the standard value of general relativity and the Horndeski theory. This can deform the matter bispectrum at the folded triangle configurations ($k_1+k_2=k_3$), while it is never possible within the Horndeski theory.

Covariant conserved currents for scalar-tensor Horndeski theory

The scalar-tensor theories have become popular recently in particular in connection with attempts to explain present accelerated expansion of the universe, but they have been considered as a natural extension of general relativity long time ago. The Horndeski scalar-tensor theory involving four invariantly defined Lagrangians is a natural choice since it implies field equations involving at most second derivatives. Following the formalisms of defining covariant global quantities and conservation laws for perturbations of spacetimes in standard general relativity, we extend these methods to the general Horndeski theory and find the covariant conserved currents for all four Lagrangians. The current is also constructed in the case of linear perturbations involving both metric and scalar fields. As a specific illustration, we derive a superpotential that leads to the covariantly conserved current in the Branse-Dicke theory.

Reconstructing Horndeski models from the effective field theory of dark energy

Studying the effects of dark energy and modified gravity on cosmological scales has led to a great number of physical models being developed. The effective field theory (EFT) of cosmic acceleration allows an efficient exploration of this large model space, usually carried out on a phenomenological basis. However, constraints on such parametrized EFT coefficients cannot be trivially connected to fundamental covariant theories. In this paper we reconstruct the class of covariant Horndeski scalar-tensor theories that reproduce the same background dynamics and linear perturbations as a given EFT action. One can use this reconstruction to interpret constraints on parametrized EFT coefficients in terms of viable covariant Horndeski theories. We demonstrate this method with a number of well-known models and discuss a range of future applications.

Warm inflation in Horndeski gravity

In this paper, we investigate a class of Horndeski scalar-tensor theory of gravity for warm inflation. We present some models where the early-time acceleration is realized in the weak and in the strong dissipation regime. Cosmological perturbations are analyzed.

Scale-invariant perturbations from null-energy-condition violation: A new variant of Galilean genesis

We propose a novel branch of the Galilean Genesis scenario as an alternative to inflation, in which the universe starts expanding from Minkowski in the asymptotic past with a gross violation of the null energy condition (NEC). This variant, described by several functions and parameters within the Horndeski scalar-tensor theory, shares the same background dynamics with the existing Genesis models, but the nature of primordial quantum fluctuations is quite distinct. In some cases, tensor perturbations grow on superhorizon scales. The tensor power spectrum can be red, blue, or scale invariant, depending on the model, while scalar perturbations are nearly scale invariant. This is in sharp contrast to typical NEC-violating cosmologies, in which a blue tensor tilt is generated. Though the primordial tensor and scalar spectra are both nearly scale invariant as in the inflationary scenario, the consistency relation in our variant of Galilean Genesis is non-standard.

Observational future of cosmological scalar-tensor theories

The next generation of surveys will greatly improve our knowledge of cosmological gravity. In this paper we focus on how Stage IV photometric redshift surveys, including weak lensing and multiple tracers of the matter distribution and radio experiments combined with measurements of the cosmic microwave background will lead to precision constraints on deviations from General Relativity. We use a broad subclass of Horndeski scalar-tensor theories to forecast the accuracy with which we will be able to determine these deviations and their degeneracies with other cosmological parameters. Our analysis includes relativistic effects, does not rely on the quasi-static evolution and makes conservative assumptions about the effect of screening on small scales. We define a figure of merit for cosmological tests of gravity and show how the combination of different types of surveys, probing different length scales and redshifts, can be used to pin down constraints on the gravitational physics to better than a few percent, roughly an order of magnitude better than present probes. Future cosmological experiments will be able to constrain the Brans-Dicke parameter at a level comparable to Solar System and astrophysical tests

Challenges to self-acceleration in modified gravity from gravitational waves and large-scale structure

With the advent of gravitational-wave astronomy marked by the aLIGO GW150914 and GW151226 observations, a measurement of the cosmological speed of gravity will likely soon be realised. We show that a confirmation of equality to the speed of light as indicated by indirect Galactic observations will have important consequences for a very large class of alternative explanations of the late-time accelerated expansion of our Universe. It will break the dark degeneracy of self-accelerated Horndeski scalar–tensor theories in the large-scale structure that currently limits a rigorous discrimination between acceleration from modified gravity and from a cosmological constant or dark energy. Signatures of a self-acceleration must then manifest in the linear, unscreened cosmological structure. We describe the minimal modification required for self-acceleration with standard gravitational-wave speed and show that its maximum likelihood yields a 3σ poorer fit to cosmological observations compared to a cosmological constant. Hence, equality between the speeds challenges the concept of cosmic acceleration from a genuine scalar–tensor modification of gravity.

Relativistic stars in beyond Horndeski theories

This work studies relativistic stars in beyond Horndeski scalar–tensor theories that exhibit a breaking of the Vainshtein mechanism inside matter, focusing on a model based on the quartic beyond Horndeski Lagrangian. We self-consistently derive the scalar field profile for static spherically symmetric objects in asymptotically de Sitter space–time and show that the Vainshtein breaking branch of the solutions is the physical branch thereby resolving several ambiguities with non-relativistic frameworks. The geometry outside the star is shown to be exactly Schwarzschild-de Sitter and therefore the parameterised post-Newtonian parameter , confirming that the external screening works at the post-Newtonian level. The Tolman–Oppenheimer–Volkoff (TOV) equations are derived and a new lower bound on the Vainshtein breaking parameter is found by requiring the existence of static spherically symmetric stars. Focusing on the unconstrained case where , we numerically solve the TOV equations for polytropic and realistic equations of state and find stars with larger radii at fixed mass. Furthermore, the maximum mass can increase dramatically and stars with masses in excess of can be found for relatively small values of the Vainshtein breaking parameter. We re-examine white dwarf stars and show that post-Newtonian corrections are important in beyond Horndeski theories and therefore the bounds coming from previous analyses should be revisited.