Within the landscape of modified theories of gravity, progress in understanding the behaviour of, and developing tests for, screening mechanisms has been hindered by the complexity of the field equations involved, which are nonlinear in nature and characterised by a large hierarchy of scales. This is especially true of Vainshtein screening, where the fifth force is suppressed by high-order derivative terms which dominate within a radius much larger than the size of the source, known as the Vainshtein radius.
In this work, we present the numerical code $φ$enics, building on the FEniCS library, to solve the full equations of motion from two theories of interest for screening: a model containing high-order derivative operators in the equation of motion and one characterised by nonlinear self-interactions in two coupled scalar fields. We also include functionalities that allow the computation of higher-order operators of the scalar fields in post-processing, enabling us to check that the profiles we find are consistent solutions within the effective field theory. These two examples illustrate the different challenges experienced when trying to simulate such theories numerically, and we show how these are addressed within this code. The examples in this paper assume spherical symmetry, but the techniques may be straightforwardly generalised to asymmetric configurations. This article therefore also provides a worked example of how the finite element method can be employed to solve the screened equations of motion. $φ$enics is publicly available and can be adapted to solve other theories of screening.
The Vainshtein screening mechanism relies on nonlinear interaction terms becoming dominant close to a compact source. However, theories displaying this mechanism are generally understood to be low-energy theories: it is unclear that operators emerging from UV completion do not interfere with terms inducing Vainshtein screening. In this work, we find a set of interacting massive Galileon theories that exhibit Vainshtein screening; examining potential UV completions of these theories, we determine that the screening does not survive the extension. We find that neglecting operators when integrati…
One of the most pressing questions in modified gravity is how deviations from general relativity can manifest in upcoming galaxy surveys. This is especially relevant for theories exhibiting Vainshtein screening, where such deviations are efficiently suppressed within a (typically large) Vainshtein radius. However, Vainshtein screening is known to be shape dependent: it is most effective around spherical sources, weaker around cylindrical objects and completely absent for planar sources. The Cosmic Web therefore offers a testing ground, as it displays many shapes in the form of clusters, filame…
Theories which modify general relativity to explain the accelerated expansion of the Universe often use screening mechanisms to satisfy constraints on Solar System scales. We investigate the effects of the cosmic web and the local environmental density of dark matter halos on the screening properties of the Vainshtein and chameleon screening mechanisms. We compare the cosmic web morphology of dark matter particles, mass functions of dark matter halos, mass and radial dependence of screening, velocity dispersions and peculiar velocities, and environmental dependence of screening mechanisms in $…
We study the Vainshtein mechanism in the context of slowly rotating stars in scalar-tensor theories. While the Vainshtein screening is well established for spherically symmetric spacetimes, we examine its validity in the axisymmetric case for slowly rotating sources. We show that the deviations from the general relativity solution are small in the weak-field approximation outside the star: the solution for the frame-dragging function is the same as in general relativity at leading order. Moreover, in most cases the corrections are suppressed by powers of the Vainshtein radius provided that the…
A generic second-order scalar-tensor theory contains a nonlinear derivative self-interaction of the scalar degree of freedom $ϕ$ à la Galileon models, which allows for the Vainshtein screening mechanism. We investigate this effect on subhorizon scales in a cosmological background, based on the most general second-order scalar-tensor theory. Our analysis takes into account all the relevant nonlinear terms and the effect of metric perturbations consistently. We derive an explicit form of Newton's constant, which in general is time-dependent and hence is constrained from observations, as suggeste…
Scalar field theories that possess a Vainshtein mechanism are able to dynamically suppress the associated fifth forces in the presence of massive sources through derivative non-linearities. The resulting equations of motion for the scalar are highly non-linear and therefore very few analytic solutions are known. Here we present a brief investigation of the structure of Vainshtein screening in symmetrical configurations, focusing in particular on the spherical, cylindrical and planar solutions that are relevant for observations of the cosmic web. We consider Vainshtein screening in both the Gal…
We demonstrate that the instabilities in linear cosmological perturbations in bimetric theory are the manifestation of the non-linear Vainshtein mechanism on an FRW background. The spin-2 mass serves as the cosmological Vainshtein scale in this case. This allows us to quantitatively address early universe cosmology. In particular, in a global analysis, we study data from the cosmic microwave background radiation and local measurements of the Hubble flow. We show that bimetric cosmology resolves the discrepancy in the local and early-time measurements of the Hubble scale via an effective phanto…
We study the Vainshtein screening mechanism in Horndeski theories in the presence of a scalar field $ϕ$ nonminimally and kinetically coupled to ordinary matter field. A general interacting Lagrangian describing this coupling is characterized by energy transfer $f_1$ and momentum exchange $f_2$. For a spherically symmetric configurations on top of the cosmological background, we investigate the static perturbations in linear and nonlinear regimes with respect to the scalar field perturbation. In the former regime, the parametrized post-Newtonian parameter generally deviates from unity as long a…
We study the stability of the Vainshtein screening solution of the massive/bi-gravity based on the massive nonlinear sigma model as the effective action inside the Vainshtein radius. The effective action is obtained by taking the $Λ_2$ decoupling limit around a curved spacetime. First we derive a general consequence that any Ricci flat Vainshtein screening solution is unstable when we take into account the excitation of the scalar graviton only. This instability suggests that the nonlinear excitation of the scalar graviton is not sufficient to obtain a successful Vainshtein screening in massiv…
Adding a light scalar degree of freedom to General Relativity often induces a fifth force whose magnitude is strongly constrained by laboratory experiments and solar system tests. The Vainshtein screening mechanism ensures that the effects of this supplementary force are suppressed in dense environments. However, the field solution of theories exhibiting Vainshtein screening is only known in spherically symmetric situations. In this article we examine in different configurations the two-body potential energy of pointlike particles in a specific $P(X)$ theory with Vainshtein screening. We use i…
We investigate the dependence of the Vainshtein screening mechanism on the cosmic web morphology of both dark matter particles and halos as determined by ORIGAMI. Unlike chameleon and symmetron screening, which come into effect in regions of high density, Vainshtein screening instead depends on the dimensionality of the system, and screened bodies can still feel external fields. ORIGAMI is well-suited to this problem because it defines morphologies according to the dimensionality of the collapsing structure and does not depend on a smoothing scale or density threshold parameter. We find that h…
We propose a dark energy model where a scalar field $ϕ$ has nonlinear self-interactions in the presence of a dilatonic coupling with the Ricci scalar. This belongs to a sub-class of theories beyond Horndeski, which accommodates covariant Galileons and Brans-Dicke theories as specific cases. We derive conditions under which the scalar sound speed squared $c_{\rm s}^2$ is positive from the radiation era to today. Since $c_{\rm s}^2$ remains to be smaller than the order of 1, the deviation from Horndeski theories does not cause heavy oscillations of gauge-invariant gravitational potentials. In th…
By using observations of the Hulse-Taylor pulsar we constrain the gravitational wave (GW) speed to the level of $10^{-2}$. We apply this result to scalar-tensor theories that generalize Galileon 4 and 5 models, which display anomalous propagation speed and coupling to matter for GWs. We argue that this effect survives conventional screening due to the persistence of a scalar field gradient inside virialized overdensities, which effectively "pierces" the Vainshtein screening. In specific branches of solutions, our result allows to directly constrain the cosmological couplings in the effective f…
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…
In this paper we study long range modifications of gravity in the consistent framework of bigravity, which introduces a second massive spin-2 field and allows to continuously interpolate between the regime of General Relativity (mediated by a massless spin-2 field) and massive gravity (mediated by a massive spin-2 field). In particular we derive for the first time the equations for light deflection in this framework and study the effect on the lensing potential of galaxy clusters. By comparison of kinematic and lensing mass reconstructions, stringent bounds can be set on the parameter space of…
We explore the space of spherically symmetric, static solutions in the decoupling limit of a class of non-linear covariant extensions of Fierz-Pauli massive gravity obtained recently in arXiv:1007.0443. In general, several such solutions with various asymptotic limits exist. We find their approximate short and long-distance behaviour and use numerical analysis to match them at the Vainshtein radius, $r_*$. Our findings indicate, that for a broad range of parameters, the theory does possess the Vainshtein mechanism, screening the scalar contribution to the gravitational force within $r_*$. In a…
Scalar forces "screened" by the Vainshtein mechanism may hold the key to understanding the cosmological expansion of our universe, while predicting new and exciting features in the interaction between massive bodies. Here we explore the dynamics of the Vainshtein screening mechanism, focusing on the decoupling limit of the DGP braneworld scenario and dRGT massive gravity. We show that there is a vast set of initial conditions whose evolution is well defined and which are driven to the static, screening solutions of these theories. Screening solutions are stable and behave coherently under smal…
Screening mechanisms are essential features of dark energy models mediating a fifth force on large scales. We study the regime of strong scalar field nonlinearities, known as Vainshtein screening, in the most general scalar-tensor theories propagating a single scalar degree of freedom. We first develop an effective approach to parameterize cosmological perturbations beyond linear order for these theories. In the quasi-static limit, the fully nonlinear effective Lagrangian contains six independent terms, one of which starts at cubic order in perturbations. We compute the two gravitational poten…
In Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories, it is known that the conical singularity arises at the center of a spherically symmetric body ($r=0$) in the case where the parameter $α_{{\rm H}4}$ characterizing the deviation from the Horndeski Lagrangian $L_4$ approaches a non-zero constant as $r \to 0$. We derive spherically symmetric solutions around the center in full GLPV theories and show that the GLPV Lagrangian $L_5$ does not modify the divergent property of the Ricci scalar $R$ induced by the non-zero $α_{{\rm H}4}$. Provided that $α_{{\rm H}4}=0$, curvature scalar quantities can…
We show how to generalize the classical duals found by Gabadadze {\it et al} to a very large class of self-interacting theories. This enables one to adopt a perturbative description beyond the scale at which classical perturbation theory breaks down in the original theory. This is particularly relevant if we want to test modified gravity scenarios that exhibit Vainshtein screening on solar system scales. We recognise the duals as being related to the Legendre transform of the original Lagrangian, and present a practical method for finding the dual in general; our methods can also be applied to…
We present a non-linear analysis of perturbations around cosmological solutions in Generalised Massive gravity. This Lorentz invariant theory is an extension of de Rham, Gabadadze, Tolley massive gravity that propagates $5$ degrees of freedom while allowing stable open FLRW cosmologies. For a minimal model that supports a self-accelerating background, we study the dynamics of non-linear perturbations. We find that the equation for the scalar graviton is distinct from the analogues in Horndeski and DHOST theories. We numerically solve the equation to find a new type of nonlinear solution for th…
We study how the Vainshtein mechanism operates in the most general scalar-tensor theories with second-order equations of motion. The field equations of motion, which can be also applicable to most of other screening scenarios proposed in literature, are generally derived in a spherically symmetric space-time with a matter source. In the presence of a field coupling to the Ricci scalar, we clarify conditions under which the Vainshtein mechanism is at work in a weak gravitational background. We also obtain the solutions of the field equation inside a spherically symmetric body and show how they …
In the first part of this paper we critically examine the ultra-violet implications of theories that exhibit Vainshtein screening, taking into account both the standard Wilsonian perspective as well as more exotic possibilities. Aspects of this discussion draw on results from the second part of the paper in which we perform a general study of derivatively coupled scalar theories using non-perturbative exact renormalisation group techniques, which are of interest independently of their application to modified gravity. In this context, we demonstrate the suppression of quantum corrections within…
We show that the current bounds on the time variation of the Newton constant G can put severe constraints on many interesting scalar-tensor theories which possess a shift symmetry and a nonminimal matter-scalar coupling. This includes, in particular, Galileon-like models with a Vainshtein screening mechanism. We underline that this mechanism, if efficient to hide the effects of the scalar field at short distance and in the static approximation, can in general not alter the cosmological time evolution of the scalar field. This results in a locally measured time variation of G which is too large…
The Vainshtein mechanism is known as an efficient way of screening the fifth force around a matter source in modified gravity. This has been verified mainly in highly symmetric matter configurations. To study how the Vainshtein mechanism works in a less symmetric setup, we numerically solve the scalar field equation around a disk with a hole at its center in the cubic Galileon theory. We find, surprisingly, that the Galileon force is enhanced, rather than suppressed, in the vicinity of the hole. This anti-screening effect is larger for a thinner, less massive disk with a smaller hole. At this …
Recently, an extension to the Parameterized Post-Newtonian (PPN) formalism has been proposed. This formalism, the Parameterized Post-Newtonian-Vainshteinian (PPNV) formalism, is well suited to theories which exhibit Vainshtein screening of scalar fields. In this paper we apply the PPNV formalism to the Quartic and Quintic Galileon theories for the first time. As simple generalizations of standard scalar-tensor field theories they are important guides for the generalization of parameterized approaches to the effects of gravity beyond General Relativity. In the Quartic case, we find new PPNV pot…
New fifth forces have not yet been detected in the laboratory or in the solar system, hence it is typically difficult to introduce new light scalar fields that would mediate such forces. In recent years it has been shown that a number of non-linear scalar field theories allow for a dynamical mechanism, such as the Vainshtein and chameleon ones, that suppresses the strength of the scalar fifth force in experimental environments. This is known as screening, however it is unclear how common screening is within non-linear scalar field theories. k-essence models are commonly studied examples of non…
Assuming static and spherically symmetric spacetimes in the ghost-free bigravity theory, we find a relativistic star solution, which is very close to that in general relativity. The coupling constants are classified into two classes: Class [I] and Class [II]. Although the Vainshtein screening mechanism is found in the weak gravitational field for both classes, we find that there is no regular solution beyond the critical value of the compactness in Class [I]. This implies that the maximum mass of a neutron star in Class [I] becomes much smaller than that in GR. On the other hand, for the solut…
We investigate low energy limits of massive gauge theories that feature the Vainshtein mechanism, focussing on the effects of the UV modes that are integrated out. It turns out that the Goldstone sectors are significantly influenced by the effects from such modes relative to the effective field theories where the irrelevant operators induced by heavy modes are simply cast aside. The effects of the consistently retained higher order corrections affect the strong coupling and show that the nature of the UV completion influences the low energy theory significantly. This casts doubts on the naivel…
The Brans-Dicke theory of gravity is one of the oldest ideas to extend general relativity by introducing a non-minimal coupling between the scalar field and gravity. The Solar System tests put tight constraints on the theory. In order to evade these constraints, various screening mechanisms have been proposed. These screening mechanisms allow the scalar field to couple to matter as strongly as gravity in low density environments while suppressing it in the Solar System. The Vainshtein mechanism, which is found in various modified gravity models such as massive gravity, braneworld models and sc…
We study cosmic structures in the quadratic Degenerate Higher Order Scalar Tensor (qDHOST) model, which has been proposed as the most general scalar-tensor theory (up to quadratic dependence on the covariant derivatives of the scalar field), which is not plagued by the presence of ghost instabilities. We then study a static, spherically symmetric object embedded in de Sitter space-time for the qDHOST model. This model exhibits breaking of the Vainshtein mechanism inside the cosmic structure and Schwarzschild-de Sitter space-time outside, where General Relativity (GR) can be recovered within th…
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 th…
We consider the challenging problem of obtaining an analytic understanding of realistic astrophysical dynamics in the presence of a Vainshtein screened fifth force arising from infrared modifications of General Relativity. In particular, we attempt to solve -- within the most general flat spacetime galileon model -- the scalar force law between well separated bodies located well within the Vainshtein radius of the Sun. To this end, we derive the exact static Green's function of the galileon wave equation linearized about the background field generated by the Sun, for the minimal cubic and maxi…
We study the screening mechanism in the most general scalar-tensor theories that leave gravitational waves unaffected and are thus compatible with recent LIGO/Virgo observations. Using the effective field theory of dark energy approach, we consider the general action for perturbations beyond linear order, focussing on the quasi-static limit. When restricting to the subclass of theories that satisfy the gravitational wave constraints, the fully nonlinear effective Lagrangian contains only three independent parameters. One of these, $β_1$, is uniquely present in degenerate higher-order theories.…
This paper is concerned with theories of gravity that contain a scalar coupled both conformally and disformally to matter through the metric. By systematically deriving the non-relativistic limit, it is shown that no new non-linear screening mechanisms are present beyond the Vainshtein mechanism and chameleon-like screening. If one includes the cosmological expansion of the universe, disformal effects that are usually taken to be absent can be present in the solar system. When the conformal factor is absent, fifth-forces can be screened on all scales when the cosmological field is slowly-rolli…
We test the subclasses of Horndeski gravity without Vainshtein mechanism in the strong field regime of binary pulsars. We find the rate of energy losses via the gravitational radiation predicted by such theories and compare our results with observational data from quasi-circular binaries PSR J1738+0333, PSR J0737-3039, PSR J1012+5307. In addition, we consider few specific cases: the hybrid metric-Palatini f(R)-gravity and massive Brans-Dicke theory.
For a massive vector field with derivative self-interactions, the breaking of the gauge invariance allows the propagation of a longitudinal mode in addition to the two transverse modes. We consider generalized Proca theories with second-order equations of motion in a curved space-time and study how the longitudinal scalar mode of the vector field gravitates on a spherically symmetric background. We show explicitly that cubic-order self-interactions lead to the suppression of the longitudinal mode through the Vainshtein mechanism. Provided that the dimensionless coupling of the interaction is n…
In this paper, we examine the cosmological viability of a light mass galileon field consistent with local gravity constraints. The minimal, L_3=\Boxϕ(\partial_μϕ)^2, massless galileon field requires an additional term in order to give rise to a viable ghost free late time acceleration of Universe. The desired cosmological dynamics can either be achieved by incorporating an additional terms in the action such as (L_4,L_5) - the higher order galileon Lagrangians or by considering a light mass field a la galileon field potential. We analyse the second possibility and find that: (1) The model prod…
Many theories of modified gravity, including the well studied Horndeski models, are characterized by a screening mechanism that ensures that standard gravity is recovered near astrophysical bodies. In a recently introduced class of gravitational theories that goes beyond Horndeski, it has been found that new derivative interactions lead to a partial breaking of the Vainshtein screening mechanism inside any gravitational source, although not outside. We study the impact of this new type of deviation from standard gravity on the density profile of a spherically symmetric matter distribution, in …
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…
Low-energy effective field theories containing a light scalar field are used extensively in cosmology, but often there is a tension between embedding such theories in a healthy UV completion and achieving a phenomenologically viable screening mechanism in the IR. Here, we identify the range of interaction couplings which allow for a smooth resummation of classical non-linearities (necessary for kinetic/Vainshtein-type screening), and compare this with the range allowed by unitarity, causality and locality in the underlying UV theory. The latter region is identified using positivity bounds on t…
We study relativistic stars in degenerate higher-order scalar-tensor theories that evade the constraint on the speed of gravitational waves imposed by GW170817. It is shown that the exterior metric is given by the usual Schwarzschild solution if the lower order Horndeski terms are ignored in the Lagrangian and a shift symmetry is assumed. However, this class of theories exhibits partial breaking of Vainshtein screening in the stellar interior and thus modifies the structure of a star. Employing a simple concrete model, we show that for high-density stars the mass-radius relation is altered sig…
Scalar fields have played an important role in the development of the fundamental theories of physics as well as in other branches of physics such as gravitation and cosmology. For a long time these escaped detection until 2012 year when the Higgs boson was observed for the first time. Since then alternatives to the general theory of relativity like the Brans-Dicke theory, scalar-tensor theories of gravity and their higher derivative generalizations -- collectively known as Horndeski theories -- have acquired renewed interest. In the present review we discuss on several selected topics regardi…
PySCo-EFT and ECOSMOG-EFT: a tandem of N-body simulation codes for the Effective Field Theory of Dark Energy
Observatoire de Paris, Université PSL, Sorbonne Université, CNRS（巴黎天文台，巴黎文理研究大学，索邦大学，法国国家科学研究中心）；Université Paris Cité（巴黎西岱大学）；Institut Universitaire de France（法兰西学院）；University of Nova Gorica（新戈里察大学）；Institut d’Astrophysique de Paris, CNRS and Sorbonne Université（巴黎天体物理研究所，法国国家科学研究中心和索邦大学）
