Well tempering is one of the few classical field theory methods for solving the original cosmological constant problem, dynamically canceling a large (possibly Planck scale) vacuum energy and leaving the matter component intact, while providing a viable cosmology with late time cosmic acceleration and an end de Sitter state. We present the general constraints that variations of Horndeski gravity models with different combinations of terms must satisfy to admit an exact de Sitter spacetime that does not respond to an arbitrarily large cosmological constant. We explicitly derive several specific scalar-tensor models that well temper and can deliver a standard cosmic history including current cosmic acceleration. Stability criteria, attractor behavior of the de Sitter state, and the response of the models to pressureless matter are considered. The well tempered conditions can be used to focus on particular models of modified gravity that have special interest -- not only removing the original cosmological constant problem but providing relations between the free Horndeski functions and reducing them to a couple of parameters, suitable for testing gravity and cosmological data analysis.
Well-tempering is a promising classical method of dynamically screening an arbitrarily large vacuum energy and generating a late-time, low energy de Sitter vacuum state. In this paper, we study for the first time self-tuning in teleparallel gravity and obtain well-tempered cosmological models in the teleparallel gravity analogue of Horndeski theory. This broadens the scope of well-tempered cosmology and teases the potentially far richer cosmological dynamics that could be anchored on teleperallel gravity. We expand the well-tempered recipe to its most general form so far and use it to search f…
Well-tempering stands among the few classical methods of screening vacuum energy to deliver a late-time, low energy vacuum state. We build on the class of Horndeski models that admit a Minkowski vacuum state despite the presence of an arbitrarily large vacuum energy to obtain a much larger family of models in teleparallel Horndeski theory. We set up the routine for obtaining these models and present a variety of cases, all of which are able to screen a natural particle physics scale vacuum energy using degeneracy in the field equations. We establish that well-tempering is the unique method of …
Well tempered cosmology provides a well defined path for obtaining cosmology with a low energy cosmic acceleration despite a high (Planck) energy cosmological constant $Λ$, through a scalar field dynamically canceling $Λ$. We explore relations between the mass scales entering the various Horndeski gravity terms, and focus on the cases of only one or only two mass scales, obtaining general solutions for the form of the action. The resulting cosmology can be natural and viable, and as one of the only paths to dealing with the cosmological constant problem it has a rationale to be a benchmark cos…
When faced with two nigh intractable problems in cosmology -- how to remove the original cosmological constant problem and how to parametrize modified gravity to explain current cosmic acceleration -- we can make progress by counterposing them. The well tempered solution to the cosmological constant through degenerate scalar field dynamics also relates disparate Horndeski gravity terms, making them contrapuntal. We derive the connection between the kinetic term $K$ and braiding term $G_3$ for shift symmetric theories (including the running Planck mass $G_4$), extending previous work on monomia…
Zero point fluctuations of quantum fields should generate a large cosmological constant energy density in any spacetime. How then can we have anything other than de Sitter space without fine tuning? Well tempering -- dynamical cancellation of the cosmological constant using degeneracy within the field equations -- can replace a large cosmological constant with a much lower energy state. Here we give an explicit mechanism to obtain a Minkowski solution, replacing the cosmological constant with zero, and testing its attractor nature and persistence through a vacuum phase transition. We derive th…
The discovery of the electromagnetic counterpart to GW170817 severely constrains the tensor mode propagation speed, eliminating a large model space of Horndeski theory. We use the cosmic microwave background data from Planck and the joint analysis of the BICEP2/Keck Array and Planck, galaxy clustering data from the SDSS LRG survey, BOSS baryon acoustic oscillation data, and redshift space distortion measurements to place constraints on the remaining Horndeski parameters. We evolve the Horndeski parameters as power laws with both the amplitude and power law index free. We find a 95% CL upper bo…
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-accelera…
We study a general scalar field Lagrangian coupled with matter and linear in $\Boxϕ$ (also called KGB model). Within this class of models, we find the most general form of the Lagrangian that allows for cosmological scaling solutions, i.e. solutions where the ratio of matter to field density and the equation of state remain constant. Scaling solutions of this kind may help solving the coincidence problem since in this case the presently observed ratio of matter to dark energy does not depend on initial conditions, but rather on the theoretical parameters. Extending previous results we find tha…
Gravitational wave propagation encounters a spacetime friction from a running Planck mass in modified gravity, causing the luminosity distance to deviate from that in general relativity (or given by the photon luminosity distance to the source), thus making it a valuable cosmological probe. We present the exact expression for the cosmological distance deviation in Horndeski gravity including theories that have a $G_5$ term yet propagate at the speed of light. An especially simple result ensues for coupled Gauss-Bonnet gravity, which we use to show it does not give a viable cosmology. We also g…
We study the third order solutions of the cosmological density perturbations in the Horndeski's most general scalar-tensor theory under the condition that the Vainshtein mechanism is at work. In this work, we thoroughly investigate the independence property of the functions describing the nonlinear mode-couplings, which is also useful for models within the general relativity. Then, we find that the solutions of the density contrast and the velocity divergence up to the third order ones are characterized by 6 parameters. Furthermore, the 1-loop order power spectra obtained with the third order …
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 parametri…
Horndeski models with a de Sitter critical point for any kind of material content may provide a mechanism to alleviate the cosmological constant problem. We study the cosmological evolution of two classes of families - the linear models and the non-linear models with shift symmetry. We conclude that the latter models can deliver a background dynamics compatible with the latest observational data.
Invertible disformal transformations are a useful tool to investigate ghost-free scalar-tensor theories. By performing a higher-derivative generalization of the invertible disformal transformation on Horndeski theories, we construct a novel class of ghost-free scalar-tensor theories, which we dub generalized disformal Horndeski theories. Specifically, these theories lie beyond the quadratic/cubic DHOST class. We explore cosmological perturbations to identify a subclass where gravitational waves propagate at the speed of light and clarify the conditions for the absence of ghost/gradient instabi…
We study observational constraints on a specific dark energy model in the framework of Gleyzes-Langlois-Piazza-Vernizzi theories, which extends the Galileon ghost condensate (GGC) to the domain of beyond Horndeski theories. In this model, we show that the Planck cosmic microwave background (CMB) data, combined with datasets of baryon acoustic oscillations, supernovae type Ia, and redshift-space distortions, give the tight upper bound $|α_{\rm H}^{(0)}| \le {\cal O}(10^{-6})$ on today's beyond-Horndeski (BH) parameter $α_{\rm H}$. This is mostly attributed to the shift of CMB acoustic peaks ind…
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 i…
Gravitational waves (GW) are generally affected by modification of a gravity theory during propagation in cosmological distance. We numerically perform a quantitative analysis on Horndeski theory at cosmological scale to constrain the Horndeski theory by GW observations in model-independent way. We formulate a parameterization for a numerical simulation based on the Monte Carlo method and obtain the classification of the models that agrees with cosmic accelerating expansion within observational errors of the Hubble parameter. As a result, we find that a large group of the models in the Horndes…
We consider the general scalar field Horndeski Lagrangian coupled to matter. Within this class of models, we present two results that are independent of the particular form of the model. First, we show that in a Friedmann-Robertson-Walker metric the Horndeski Lagrangian coincides with the pressure of the scalar field. Second, we employ the previous result to identify the most general form of the Lagrangian that allows for cosmological scaling solutions, i.e. solutions where the ratio of matter to field density and the equation of state remain constant. Scaling solutions of this kind may help s…
We show that the generalized McVittie spacetime, which represents a black hole with time-dependent mass in an expanding universe, is an exact solution of a subclass of the Horndeski family of actions. The heat-flow term responsible for the energy transfer between the black hole and the cosmological background is generated by the higher-order kinetic gravity braiding term, which generalizes the cuscuton action that yields McVittie with constant mass as a solution. Finally, we show that this generalization can be understood in terms of a duality realized by a disformal transformation, connecting…
We consider Horndeski cosmological models, with a minisuperspace Lagrangian linear in the field derivative, that are able to screen any vacuum energy and material content leading to a spatially flat de Sitter vacuum fixed by the theory itself. Furthermore, we investigate particular models with a cosmic evolution independent of the material content and use them to understand the general characteristics of this framework. We also consider more realistic models, which we denote the "term-by-term" and "tripod" models, focusing attention on cases in which the critical point is indeed an attractor s…
We study linear scalar perturbations around a flat FLRW background in mimetic Horndeski gravity. In the absence of matter, we show that the Newtonian potential satisfies a second-order differential equation with no spatial derivatives. This implies that the sound speed for scalar perturbations is exactly zero on this background. We also show that in mimetic $G^3$ theories the sound speed is equally zero. We obtain the equation of motion for the comoving curvature perturbation (first order differential equation) and solve it to find that the comoving curvature perturbation is constant on all sc…
In light of the statistical performance of cosmological observations, in this work we present an improvement on the Gaussian reconstruction of the Hubble parameter data $H(z)$ from Cosmic Chronometers, Supernovae Type Ia and Clustering Galaxies in a model-independent way in order to use them to study new constraints in the Horndeski theory of gravity. First, we have found that the prior used to calibrate the Pantheon supernovae data significantly affects the reconstructions, leading to a 13$σ$ tension on the $H_0$ value. Second, according to the $χ^{2}$-statistics, the reconstruction carried o…
Self tuning is one of the few methods for dynamically cancelling a large cosmological constant and yet giving an accelerating universe. Its drawback is that it tends to screen all sources of energy density, including matter. We develop a model that tempers the self tuning so the dynamical scalar field still cancels an arbitrary cosmological constant, including the vacuum energy through any high energy phase transitions, without affecting the matter fields. The scalar-tensor gravitational action is simple, related to cubic Horndeski gravity, with a nonlinear derivative interaction plus a tadpol…
We derive the cosmological matching conditions for the homogeneous and isotropic background and for linear perturbations in Horndeski's most general second-order scalar-tensor theory. In general relativity, the matching is done in such a way that the extrinsic curvature is continuous across the transition hypersurface. This procedure is generalized so as to incorporate the mixing of scalar and gravity kinetic terms in the field equations of Horndeski's theory. Our matching conditions have a wide range of applications including the galilean genesis and the bounce scenarios, in which stable, nul…
Cosmic acceleration may be due to modified gravity, with effective field theory or property functions describing the theory. Connection to cosmological observations through practical parametrization of these functions is difficult and also faces the issue that not all assumed time dependence or parts of parameter space give a stable theory. We investigate the relation between parametrization and stability in Horndeski gravity, showing that the results are highly dependent on the function parametrization. This can cause misinterpretations of cosmological observations, hiding and even ruling out…
Horndeski gravity was highly constrained from the recent gravitational wave observations by the LIGO Collaboration down to $|c_{g}/c-1|\gtrsim 10^{-15}$. In this Letter we study the tensorial perturbations in a flat cosmological background for an analogue version of Horndenki gravity which is based in Teleparallel Gravity constructed from a flat manifold with a nonvanishing torsion tensor. It is found that in this approach, one can construct a more general Horndeski theory satisfying $c_T=c_g/c=1$ without eliminating the coupling functions $G_5(ϕ,X)$ and $G_4(ϕ,X)$ that were highly constrained…
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 interpretati…
This paper is a pedagogical introduction to models of gravity and how to constrain them 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 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 Ostr…
We consider Horndeski cosmological models able to screen the vacuum energy coming from any field theory assuming that after this screening the space should be in a de Sitter vacuum with a particular value of the cosmological constant specified by the theory of gravity itself. The most general scalar-tensor cosmological models without higher than second order derivatives in the field equations that have a spatially flat de Sitter critical point for any kind of material content or vacuum energy are, therefore, presented. These models could allow us to understand the current accelerated expansion…
We investigate the production of primordial black holes (PBHs) and scalar-induced gravitational waves (GWs) for cosmological models in the Horndeski theory of gravity. The cosmological models of our interest incorporate the derivative self-interaction of the scalar field and the kinetic coupling between the scalar field and gravity. We show that the scalar power spectrum of the primordial fluctuations can be enhanced on small scales due to these additional interactions. Thus, the formation of PBHs and the production of induced GWs are feasible for our model. Parameterizing the scalar power spe…
The tachyonic instability is associated with the unboundedness of the Hamiltonian from below and results in an unstable low-$k$ regime. In the cosmological exploration of modified gravity, it is seldom taken into account, with more focus given to the popular no-ghost and no-gradient conditions. The latter though are intrinsically high-$k$ statements. Here we combine all three conditions into a full set of requirements that we show to guarantee stability on the whole range of cosmological scales. We then explore the impact of the different conditions on the parameter space of scalar-tensor grav…
Cosmological constraints and standard sirens forecasts for non-dynamical dark energy in Horndeski gravity
Scuola Superiore Meridionale（南方高等学院）；Istituto Nazionale di Fisica Nucleare (INFN)（意大利国家核物理研究所）；Universidad de Salamanca（萨拉曼卡大学）；Instituto Universitario de Física Fundamental y Matemáticas (IUFFyM)（基础物理与数学大学研究所）；Università degli Studi di Napoli “Federico II”（那不勒斯费德里科二世大学）；Università di Salerno（萨莱诺大学）
In this paper, we show how the proper choice of gauge is critical in analyzing the stability of non-singular cosmological bounce solutions based on Horndeski theories. We show that it is possible to construct non-singular cosmological bounce solutions with classically stable behavior for all modes with wavelengths above the Planck scale where: (a) the solution involves a stage of null-energy condition violation during which gravity is described by a modification of Einstein's general relativity; and (b) the solution reduces to Einstein gravity both before and after the null-energy condition vi…
The Horndeski theory of gravity is known as the most general scalar-tensor theory with second-order field equations. Recently, it was demonstrated by Gleyzes et al. that the Horndeski theory can further be generalized in such a way that although field equations are of third order, the number of propagating degrees of freedom remains the same. We study small-scale gravity in the generalized Horndeski theory, focusing in particular on an impact of the new derivative interaction beyond Horndeski on the Vainshtein screening mechanism. In the absence of the quintic galileon term and its generalizat…
It is known that Horndeski theories can be transformed to a sub-class of Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories under the disformal transformation of the metric $g_{μν} \to Ω^2(ϕ)g_{μν}+Γ(ϕ,X) \nabla_μ ϕ\nabla_ν ϕ$, where $Ω$ is a function of a scalar field $ϕ$ and $Γ$ is another function depending on both $ϕ$ and $X=g^{μν}\nabla_μ ϕ\nabla_ν ϕ$. We show that, with the choice of unitary gauge, both curvature and tensor perturbations on the flat isotropic cosmological background are generally invariant under the disformal transformation. By means of the effective field theories encompa…
Adopting Noether point symmetries, we classify and integrate dynamical systems coming from Horndeski cosmologies. The method is particularly effective both to select the form of Horndeski models and to derive exact cosmological solutions. Starting from the Lagrangians selected by the Noether symmetries, it is possible to derive several modified theories of gravity like $f(R)$ gravity, Brans-Dicke gravity, string inspired gravity and so on. In any case, exact solutions are found out.
Horndeski models with a de Sitter critical point for any kind of material content may provide a mechanism to alleviate the cosmological constant problem. Moreover, they could allow us to understand the current accelerated expansion of the universe as the result of the dynamical approach to the critical point when it is an attractor. We show that this critical point is indeed an attractor for the shift- symmetric subfamily of models with these characteristics. We study the cosmological scenario that results when considering radiation and matter content, and conclude that their background dynami…
Reconstructions of Horndeski Subclasses by Gaussian Processes
Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla（普埃布拉自治大学物理数学院）；Centro Internacional de Física Fundamental, Benemérita Universidad Autónoma de Puebla（普埃布拉自治大学基础物理国际中心）
We study the cosmological evolution of a scalar field in Horndeski gravity, assuming that the scalar field is subdominant with respect to the cosmic fluid. We first analyse the most general shift-symmetric action that respects local Lorentz symmetry. We show that the bound on the speed of gravitational waves set by GW170817+GRB170817A imposes a constraint only on the linear coupling between the scalar and the Gauss-Bonnet invariant and this constraint is rather mild. Then, we consider some interesting examples of theories that break shift-symmetry, such as the Damour-Esposito-Farèse model of s…
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 $π$ 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…
Standard sirens have been proposed as probes of alternative theories of gravity, such as Horndeski models. Hitherto, all studies have been conducted on a homogeneous-isotropic cosmological background, which is unable to consistently account for realistic distributions of matter, and for inhomogeneities in the Horndeski scalar field. Yet, the latter are essential for screening mechanisms. In this article, we comprehensively analyze the propagation of Horndeski gravitational waves in an arbitrary background spacetime and scalar field. We restrict to the class of theories in which gravitational w…
Gravity theories beyond General Relativity typically predict dipolar gravitational emission by compact-star binaries. This emission is sourced by "sensitivity" parameters depending on the stellar compactness. We introduce a general formalism to calculate these parameters, and show that in shift-symmetric Horndeski theories stellar sensitivities and dipolar radiation vanish, provided that the binary's dynamics is perturbative (i.e. the post-Newtonian formalism is applicable) and cosmological-expansion effects can be neglected. This allows reproducing the binary-pulsar observed orbital decay.
Recent anomalies found in cosmological datasets such as the low multipoles of the Cosmic Microwave Background or the low redshift amplitude and growth of clustering measured by e.g., abundance of galaxy clusters and redshift space distortions in galaxy surveys, have motivated explorations of models beyond standard $Λ$CDM. Of particular interest are models where general relativity (GR) is modified on large cosmological scales. Here we consider deviations from $Λ$CDM+GR within the context of Horndeski gravity, which is the most general theory of gravity with second derivatives in the equations o…
Attempts at constraining theories of late time accelerated expansion often assume broad priors for the parameters in their phenomenological description. Focusing on shift-symmetric scalar-tensor theories with standard gravitational wave speed, we show how a more careful analysis of their dynamical evolution leads to much narrower priors. In doing so, we propose a simple and accurate parametrisation of these theories, capturing the redshift dependence of the equation of state, $w(z)$, and the kinetic braiding parameter, $α_{\rm B}(z)$, with only two parameters each, and derive their statistical…
We study the perturbations to General Relativistic black holes (i.e. those without scalar hair) in Horndeski scalar-tensor gravity. First, we derive the equations of odd and even parity perturbations of both the metric and scalar field in the case of a Schwarzschild black hole, and show that the gravitational waves emitted from such a system contain a mixture of quasi-normal mode frequencies from the usual General Relativistic spectrum and those from the new scalar field spectrum, with the new scalar spectrum characterised by just two free parameters. We then specialise to the sub-family of Ho…
We review the effective field theory of modified gravity in which the Lagrangian involves three dimensional geometric quantities appearing in the 3+1 decomposition of space-time. On the flat isotropic cosmological background we expand a general action up to second order in the perturbations of geometric scalars, by taking into account spatial derivatives higher than two. Our analysis covers a wide range of gravitational theories-- including Horndeski theory/its recent generalizations and the projectable/non-projectable versions of Hořava-Lifshitz gravity. We derive the equations of motion for …
Since the rediscovery of Horndeski gravity, a lot of work has been devoted to the exploration of its properties, especially in the context of dark energy. However, one sector of this theory, namely the one containing the coupling of the Einstein tensor to the kinetic term of the scalar field, shows some surprising features in the construction of black holes and neutron stars. Motivated by these new results, I explore the possibility that this sector of Horndeski gravity can mimic cold dark matter at cosmological level and also explain the flattening of galactic rotation curves. I will show tha…
Since the discovery of the accelerated expansion of the present Universe, significant theoretical developments have been made in the area of modified gravity. In the meantime, cosmological observations have been providing more high-quality data, allowing us to explore gravity on cosmological scales. To bridge the recent theoretical developments and observations, we present an overview of a variety of modified theories of gravity and the cosmological observables in the cosmic microwave background and large-scale structure, supplemented with a summary of predictions for cosmological observables …
The gravitational wave event GW170817 together with its electromagnetic counterparts constrains the speed of gravity to be extremely close to that of light. We first show, on the example of an exact Schwarzschild-de Sitter solution of a specific beyond-Horndeski theory, that imposing the strict equality of these speeds in the asymptotic homogeneous Universe suffices to guarantee so even in the vicinity of the black hole, where large curvature and scalar-field gradients are present. We also find that the solution is stable in a range of the model parameters. We finally show that an infinite cla…
In the context of Horndeski cosmologies, we consider a dynamical adjustment mechanism able to screen any value of the vacuum energy of the matter fields leading to a fixed de Sitter geometry. Thus, we present the most general scalar-tensor cosmological models without higher than second order derivatives in the field equation that have a fixed spatially flat de Sitter critical point for any kind of material content or vacuum energy. These models allow us to understand the current accelerated expansion of the universe as the result of the evolution towards the critical point when it is an attrac…
Requiring the existence of a unitary, causal and local UV-completion places a set of positivity bounds on the corresponding effective field theories (EFTs). We discuss the obstructions and possibility in applying the positivity bound to cosmology, in particular the EFT of cosmological perturbations. Taking a $c_T=1$ beyond-Horndeski EFT as an illustrative example, we derive such bounds, which incorporate the cosmological correction of order $H^2/Λ^2$, $Λ$ being the cutoff scale. The derived bounds are applied to slow-roll inflation with beyond Horndeski operators. It is found that the cosmolog…
We present a systematic analysis of homogeneous and isotropic cosmologies in a particular Horndeski model with Galileon shift symmetry, containing also a $Λ$-term and a matter. The model, sometimes called Fab Five, admits a rich spectrum of solutions. Some of them describe the standard late time cosmological dynamic dominated by the $Λ$-term and matter, while at the early times the universe expands with a constant Hubble rate determined by the value of the scalar kinetic coupling. For other solutions the $Λ$-term and matter are screened at all times but there are nevertheless the early and lat…
