Horndeski Lagrangian Research Articles

In hot pursuit of a stable wormhole in beyond Horndeski theory

We consider the issue of stability at the linearized level for static, spherically symmetric wormhole solutions within a subclass of scalar-tensor theories of beyond Horndeski type. In this class of theories we derive a set of stability conditions ensuring the absence of ghosts and both radial and angular gradient instabilities about a static, spherically symmetric background. This set of constraints extends the existing one and completes the stability analysis for high energy modes in both parity odd and parity even sectors, while ``slow'' tachyonic instabilities remain unconstrained. We give an example of beyond Horndeski Lagrangian admitting a wormhole solution which complies with all stability constraints for the high energy modes.

Stability of hairy black holes in shift-symmetric scalar-tensor theories via the effective field theory approach

Shift-symmetric Horndeski theories admit an interesting class of Schwarzschild-de Sitter black hole solutions exhibiting time-dependent scalar hair. The properties of these solutions may be studied via a bottom-up effective field theory (EFT) based on the background symmetries. This is in part possible by making use of a convenient coordinate choice — Lemaître-type coordinates — in which the profile of the Horndeski scalar field is linear in the relevant time coordinate. We construct this EFT, and use it to understand the stability of hairy black holes in shift-symmetric Horndeski theories, providing a set of constraints that the otherwise-free functions appearing in the Horndeski Lagrangian must satisfy in order to admit stable black hole solutions. The EFT is analyzed in the decoupling limit to understand potential sources of instability. We also perform a complete analysis of the EFT with odd-parity linear perturbations around general spherically symmetric space-time.

Gravitational-wave propagation and polarizations in the teleparallel analog of Horndeski gravity

Gravitational waves (GWs) have opened a new window on fundamental physics in a number of important ways. The next generation of GW detectors may reveal more information about the polarization structure of GWs. Additionally, there is growing interest in theories of gravity beyond GR. One such theory which remains viable within the context of recent measurements of the speed of propagation of GWs is the teleparallel analogue of Horndeski gravity. In this work, we explore the polarization structure of this newly proposed formulation of Horndeski theory. In curvature-based gravity, Horndeski theory is almost synonymous with extensions to GR since it spans a large portion of these possible extensions. We perform this calculation by taking perturbations about a Minkowski background and consider which mode propagates. The result is that the polarization structure depends on the choice of model parameters in the teleparallel Horndeski Lagrangian with a maximum of seven propagating degrees of freedom. While the curvature-based Horndeski results follows as a particular limit within this setup, we find a much richer structure of both massive and massless cases which produce scalar--vector--tensor propagating degrees of freedom. We also find that the GW polarization that emerges from the teleparallel analogue of Horndeski gravity results in analogous massive and massless modes which take on at most four polarizations in the massless sector and two scalar ones in the massive sector. In none of the cases do we find vector polarizations.

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.

Extremal Cosmological Black Holes in Horndeski Gravity and the Anti-Evaporation Regime

Extremal cosmological black holes are analysed in the framework of the most general second order scalar-tensor theory, the so-called Horndeski gravity. Such extremal black holes are a particular case of Schwarzschild-De Sitter black holes that arises when the black hole horizon and the cosmological one coincide. Such metric is induced by a particular value of the effective cosmological constant and is known as Nariai spacetime. The existence of this type of solutions is studied when considering the Horndeski Lagrangian and its stability is analysed, where the so-called anti-evaporation regime is studied. Contrary to other frameworks, the radius of the horizon remains stable for some cases of the Horndeski Lagrangian when considering perturbations at linear order.

General formulation of cosmological perturbations in scalar-tensor dark energy coupled to dark matter

For a scalar field ϕ coupled to cold dark matter (CDM), we provide a general framework for studying the background and perturbation dynamics on the isotropic cosmological background. The dark energy sector is described by a Horndeski Lagrangian with the speed of gravitational waves equivalent to that of light, whereas CDM is dealt as a perfect fluid characterized by the number density nc and four-velocity ucμ. For a very general interacting Lagrangian f(nc, ϕ, X, Z), where f depends on nc, ϕ, X=−∂μ ϕ ∂μ ϕ/2, and Z=ucμ ∂μ ϕ, we derive the full linear perturbation equations of motion without fixing any gauge conditions. To realize a vanishing CDM sound speed for the successful structure formation, the interacting function needs to be of the form f=−f1(ϕ, X, Z)nc+f2(ϕ, X, Z). Employing a quasi-static approximation for the modes deep inside the sound horizon, we obtain analytic formulas for the effective gravitational couplings of CDM and baryon density perturbations as well as gravitational and weak lensing potentials. We apply our general formulas to several interacting theories and show that, in many cases, the CDM gravitational coupling around the quasi de-Sitter background can be smaller than the Newton constant G due to a momentum transfer induced by the Z-dependence in f2.

The hidden scalar Lagrangians within Horndeski theory

In this paper, I show that there exists a new way to obtain scalar–tensor field theories by combining a special scalar field on the cotangent bundle with a scalar field on spacetime. These two scalar fields act as a generating function for the metric tensor. When using these two scalar fields in the Horndeski Lagrangians, we discover, while seeking Friedmann–Lemaître–Robertson–Walker-type cosmological solutions, that hidden in the Horndeski Lagrangians are nondegenerate second-order scalar Lagrangians. In accordance with Ostrogradsky’s work, these hidden scalar Lagrangians lead to multiple vacuum solutions, and thereby predict the existence of the multiverse. The multiverse is comprised of numerous different types of individual universes. For example, some begin explosively, and then coast along exponentially forever at an accelerated rate, while others begin in that manner, and then stop expanding and contract.

General constraints on Horndeski wormhole throats

In this work, we consider the full Horndeski Lagrangian applied to wormhole geometries and present the full gravitational field equations. We analyse the general constraints imposed by the flaring-out conditions at the wormhole throat and consider a plethora of specific subclasses of the Horndeski Lagrangian, namely, quintessence/phantom fields, $k$-essence, scalar-tensor theories, covariant galileons, nonminimal kinetic coupling, kinetic gravity braiding, and the scalar-tensor representation of Gauss-Bonnet couplings, amongst others. The generic constraints analysed in this work serve as a consistency check of the main solutions obtained in the literature and draws out new avenues of research in considering applications of specific subclasses of the Horndeski theory to wormhole physics.

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.

Varying the Horndeski Lagrangian within the Palatini approach

We analyse what happens when the Horndeski Lagrangian is varied within the Palatini approach by considering the metric and connection as independent variables. Assuming the connection to be torsionless, there can be infinitely many metric-affine versions LP of the original Lagrangian which differ from each other by terms proportional to the non-metricity tensor. After integrating out the connection, each LP defines a metric theory, which can either belong to the original Horndeski family, or it can be of a more general DHOST type, or it shows the Ostrogradsky ghost. We analyse in detail the subclass of the theory for which the equations are linear in the connection and find that its metric-affine version is ghost-free. We present a detailed classifications of homogeneous and isotropic cosmologies in these theories. Taking into consideration other pieces of the Horndeski Lagrangian which are non-linear in the connection leads to more complex metric-affine theories which generically show the ghost. In some special cases the ghost can be removed by carefully adjusting the non-metricity contribution, but it is unclear if this is always possible. Therefore, the metric-affine generalisations of the Horndeski theory can be ghost-free, but not all of them are ghost-free, neither are they the only metric-affine theories for a gravity-coupled scalar field which can be ghost-free.

Horndeski model in nonlinearly realized supergravity

We construct the Horndeski Lagrangian within non-linearly realized super- gravity. We will show that the bosonic part of the Horndeski Lagrangian can be realized. Gravitino naturally couples to Horndeski sector in a super-covariant way. Such gravitino couplings are also free from ghosts.

Nonlinear matter terms in general scalar–tensor braneworld cosmology

A five-dimensional braneworld cosmological model in general scalar–tensor action that is comprised of various Horndeski Lagrangians is considered. The Friedmann equations in the case of strongly and weakly coupled [Formula: see text] Horndeski Lagrangians have been obtained. The strongly coupled [Formula: see text] model produces the Cardassian term [Formula: see text] with [Formula: see text], which can serve as an alternative explanation for the accelerated expansion phase of the universe. Furthermore, the latest combined observational facts from BAO, CMB, SNIa, [Formula: see text] and [Formula: see text] value observation suggest that the [Formula: see text] term lies quite close to the constrained value. On the other hand, the weakly coupled [Formula: see text] case has several new correction terms which are omitted in the braneworld Einstein–Hilbert model, e.g. the cubic [Formula: see text] and the dark radiation–matter interaction term [Formula: see text]. Furthermore, this model provides a cosmological constant constructed from the bulk scalar field, requires no brane tension and supports the big bang nucleosynthesis (BBN) constraint naturally.

More about stable wormholes in beyond Horndeski theory

It is known that Horndeski theories, like many other scalar-tensor gravities, do not support static, spherically symmetric wormholes: they always have either ghosts or gradient instabilities among parity-even linearized perturbations. Here we address the issue of whether or not this no-go theorem is valid in ‘beyond Horndeski’ theories. We derive, in the latter class of theories, the conditions for the absence of ghost and gradient instabilities for non-spherical parity even perturbations propagating in radial direction. We find, in agreement with existing arguments, that the proof of the above no-go theorem does not go through beyond Horndeski. We also obtain conditions ensuring the absence of ghosts and gradient instabilities for all parity odd modes. We give an example of beyond Horndeski Lagrangian which admits a wormhole solution obeying our (incomplete set of) stability conditions. Even though our stability analysis is incomplete, as we do not consider spherically symmetric parity even modes and parity even perturbations propagating in angular directions, as well as ‘slow’ tachyonic instabilities, our findings indicate that beyond Horndeski theories may be viable candidates to support traversable wormholes.

Horndeski gravity in the swampland

We investigate the implications of string Swampland criteria for alternative theories of gravity. Even though this has not rigorously been proven, there is some evidence that exact de-Sitter solutions with a positive cosmological constant cannot be successfully embedded into string theory, and that the low energy effective field theories containing a scalar field $\pi$ with a potential $V$ in the habitable landscape should satisfy the Swampland criteria $|V'|/V\ge c\sim\mathcal{O}(1)$. As a paradigmatic class of modified gravity theories for inflation and dark energy, we consider the extensively studied family of Horndeski Lagrangians in view of cosmological observations. Apart from a possible potential term, they contain derivative self-interactions as the Galileon and non-minimal couplings to the gravity sector. Hence, our conclusions on the Galileon sector can be also applied to many other alternative theories with scalar helicity modes containing derivative interactions such as massive gravity and generalized Proca. In the presence of such derivative terms, the dynamics of the scalar mode is substantially modified, and imposing the cosmological evolution constrained by observations places tight constraints on $c$ within the Swampland conjecture.

Tracker and scaling solutions in DHOST theories

In quadratic-order degenerate higher-order scalar–tensor (DHOST) theories compatible with gravitational-wave constraints, we derive the most general Lagrangian allowing for tracker solutions characterized by ϕ˙/Hp=constant, where ϕ˙ is the time derivative of a scalar field ϕ, H is the Hubble expansion rate, and p is a constant. While the tracker is present up to the cubic-order Horndeski Lagrangian L=c2X−c3X(p−1)/(2p)□ϕ, where c2,c3 are constants and X is the kinetic energy of ϕ, the DHOST interaction breaks this structure for p≠1. Even in the latter case, however, there exists an approximate tracker solution in the early cosmological epoch with the nearly constant field equation of state wϕ=−1−2pH˙/(3H2). The scaling solution, which corresponds to p=1, is the unique case in which all the terms in the field density ρϕ and the pressure Pϕ obey the scaling relation ρϕ∝Pϕ∝H2. Extending the analysis to the coupled DHOST theories with the field-dependent coupling Q(ϕ) between the scalar field and matter, we show that the scaling solution exists for Q(ϕ)=1/(μ1ϕ+μ2), where μ1 and μ2 are constants. For the constant Q, i.e., μ1=0, we derive fixed points of the dynamical system by using the general Lagrangian with scaling solutions. This result can be applied to the model construction of late-time cosmic acceleration preceded by the scaling ϕ-matter-dominated epoch.

Most general cubic-order Horndeski Lagrangian allowing for scaling solutions and the application to dark energy

In cubic-order Horndeski theories where a scalar field $\phi$ is coupled to nonrelativistic matter with a field-dependent coupling $Q(\phi)$, we derive the most general Lagrangian having scaling solutions on the isotropic and homogenous cosmological background. For constant $Q$ including the case of vanishing coupling, the corresponding Lagrangian reduces to the form $L=Xg_2(Y)-g_3(Y)\square \phi$, where $X=-\partial_{\mu}\phi\partial^{\mu}\phi/2$ and $g_2, g_3$ are arbitrary functions of $Y=Xe^{\lambda \phi}$ with constant $\lambda$. We obtain the fixed points of the scaling Lagrangian for constant $Q$ and show that the $\phi$-matter-dominated-epoch ($\phi$MDE) is present for the cubic coupling $g_3(Y)$ containing inverse power-law functions of $Y$. The stability analysis around the fixed points indicates that the $\phi$MDE can be followed by a stable critical point responsible for the cosmic acceleration. We propose a concrete dark energy model allowing for such a cosmological sequence and show that the ghost and Laplacian instabilities can be avoided even in the presence of the cubic coupling.

Non-trivial gravitational waves and structure formation phenomenology from dark energy

The detection of the GW170817/GRB170817A event improved the constraints on the propagation speed of gravitational waves, thus placing possible variations caused by dark energy under restraint. For models based on scalar fields belonging to the family of Horndeski Lagrangians, non-minimal derivative couplings are now severely constrained, entailing a substantially limited phenomenology. In this work we want to stress that there is still a plethora of dark energy models that get around this obstacle while still providing interesting phenomenologies able to distinguish them from the standard cosmology. We focus on a class involving vector fields as a proxy, but our discussion is extensible to a broader class of models. In particular, we show the possibility of having a non-minimal derivative coupling giving a non-trivial effect on scalar modes without affecting gravitational waves and the possibility of having a second tensor mode that can oscillate into gravitational waves. We also present a novel class of configurations breaking rotational invariance but with an energy-momentum tensor that is isotropic on-shell. This peculiar feature makes the scalar and vector sectors of the perturbations mix so that, even in a perfectly isotropic background cosmology, preferred direction effects can appear in the perturbations. We also comment on models that give rise to isotropic solutions when averaging over rapid oscillations of the vector fields. The explored models are classified according to distinctive field configurations that provide inequivalent realisations of the Cosmological Principle.

Unification of dark matter-dark energy in generalized Galileon theories

We present a unified description of the dark matter and the dark energy sectors, in the framework of shift-symmetric generalized Galileon theories. Considering a particular combination of terms in the Horndeski Lagrangian in which we have not introduced a cosmological constant or a matter sector, we obtain an effective unified cosmic fluid whose equation of state wU is zero during the whole matter era, namely from redshifts z ∼ 3000 up to z ∼ 2–3. Then at smaller redshifts it starts decreasing, passing the bound wU = −1/3, which marks the onset of acceleration, at around z ∼ 0.5. At present times it acquires the value wU = −0.7. Finally, it tends toward a de-Sitter phase in the far future. This behaviour is in excellent agreement with observations. Additionally, confrontation with Supernovae type Ia data leads to a very efficient fit. Examining the model at the perturbative level, we show that it is free from pathologies such as ghosts and Laplacian instabilities, at both scalar and tensor sectors, at all times.

Nonminimal couplings, gravitational waves, and torsion in Horndeski’s theory

The Horndeski Lagrangian brings together all possible interactions between gravity and a scalar field that yield second-order field equations in four-dimensional spacetime. As originally proposed, it only addresses phenomenology without torsion, which is a non-Riemannian feature of geometry. Since torsion can potentially affect interesting phenomena such as gravitational waves and early universe inflation, in this paper we allow torsion to exist and propagate within the Horndeski framework. To achieve this goal, we cast the Horndeski Lagrangian in Cartan's first-order formalism and introduce wave operators designed to act covariantly on $p$-form fields that carry Lorentz indices. We find that nonminimal couplings and second-order derivatives of the scalar field in the Lagrangian are indeed generic sources of torsion. Metric perturbations couple to the background torsion, and new torsional modes appear. These may be detected via gravitational waves but not through Yang-Mills gauge bosons.

Integrability from point symmetries in a family of cosmological Horndeski Lagrangians

For a family of Horndeski theories, formulated in terms of a generalized Galileon model, we study the integrability of the field equations in a Friedmann–Lemaître–Robertson–Walker space-time. We are interested in point transformations which leave invariant the field equations. Noether’s theorem is applied to determine the conservation laws for a family of models that belong to the same general class. The cosmological scenarios with or without an extra perfect fluid with constant equation of state parameter are the two important cases of our study. The de Sitter universe and ideal gas solutions are derived by using the invariant functions of the symmetry generators as a demonstration of our result. Furthermore, we discuss the connection of the different models under conformal transformations while we show that when the Horndeski theory reduces to a canonical field the same holds for the conformal equivalent theory. Finally, we discuss how singular solutions provides nonsingular universes in a different frame and vice versa.