We give a detailed calculation for the Hamiltonian of single galileon field theory, keeping track of all the surface terms. We calculate the energy of static, spherically symmetric configuration of the single galileon field at cubic order coupled to a point-source and show that the 2-branches of the solution possess energy of equal magnitude and opposite sign, the sign of which is determined by the coefficient of the kinetic term $α_2$. Moreover the energy is regularized in the short distance (ultra-violet) regime by the dominant cubic term even though the source is divergent at the origin. We argue that the origin of the negativity is due to the ghost-like modes in the corresponding branch in the presence of the point source. This seems to be a non-linear manifestation of the ghost instability.
The galileon model was recently proposed to locally describe a class of modified gravity theories, including the braneworld DGP model. We discuss spontaneous symmetry breaking of the self-accelerating branch in a multi-galileon theory with internal global symmetries. We show a modified version of Goldstone's theorem is applicable to the symmetry breaking pattern and discuss its implications. We also derive the Hamiltonian of a general multi-galileon theory and discuss its implications.
We perform the Hamiltonian analysis of minimal massive gravity coupled to the Galileon tadpole term. We determine all constraints and we argue that the physical degrees of freedom correspond to ten modes of the massive gravity together with 2(D-3) Galileons so that given model is ghost free.
A ghost-free metric formulation of the recently proposed covariant Galileon model \cite{RPgal} which retains the internal shift symmetry has been constructed. This presents a new result because the covariant Galileon models introduced so far in the literature lacks the full Galileon symmetry. We demonstrate that the same general procedure is applicable to all the Galileon interaction terms and the resulting ghost-free Galileon action respecting internal shift symmetry cannot predict the late time acceleration of the universe. However, weakly breaking the shift symmetry in the covariant Galileo…
We show how the coupling of SO(N) gauge fields to galileons arises from a probe brane construction. The galileons arise from the brane bending modes of a brane probing a co-dimension N bulk, and the gauge fields arise by turning on certain off-diagonal components in the zero mode of the bulk metric. By construction, the equations of motion for both the galileons and gauge fields remain second order. Covariant gauged galileons are derived as well.
We discuss the gauging of non-linearly realized symmetries as a method to systematically construct spontaneously broken gauge theories. We focus in particular on galileon fields and, using a coset construction, we show how to recover massive gravity by gauging the galileon symmetry. We then extend our procedure to the special galileon, and obtain a theory that couples a massive spin-2 field with a traceless symmetric field, and is free of pathologies at quadratic order around flat space.
The effective action of the recently proposed vector Galileon theory is considered. Using the background field method, we obtain the one-loop correction to the propagator of the Proca field from vector Galileon self-interactions. Contrary to the so-called scalar Galileon interactions, the two-point function of the vector field gets renormalized at the one-loop level, indicating that there is no non-renormalization theorem in the vector Galileon theory. Using dimensional regularization, we remove the divergences and obtain the counterterms of the theory. The finite term is analytically calculat…
Galileon interactions represent a class of effective field theories that have received much attention since their inception. They can be treated in their own right as scalar field theories with a specific global shift and Galilean symmetry or as a descendant of a more fundamental theory like massive gravity. It is well known that the Galileon theories are stable under quantum corrections thanks to the non-renormalization theorem which is not due to the symmetry. We consider different covariant couplings of this Galileon scalar field with the matter field: the conformal coupling, the disformal …
In this paper we study cosmological dynamics of phantom as well as non-phantom fields with linear potential in presence of Galileon correction $(\partial_μϕ\partial^μϕ) \Box ϕ$. We show that the Big Crunch singularity is delayed compared to the standard case; the delay crucially depends upon the strength of Galileon correction. As for the phantom Galileon, $ρ_ϕ$ is shown to grow more slowly compared to the standard phantom delaying the approach to singularity. In case, $V\sim ϕ^n, n>4$, Big Rip is also delayed, similar phenomenon is shown to take place for potentials steeper than the exponenti…
In this plenary talk delivered at the DISCRETE 2014 conference in London, I briefly summarize the ideas behind and attractive properties of the Galileon field theories, and describe a broad class of scalar field theories that share these properties. After describing how Galileons arise, and commenting on their fascinating properties, in the latter half of the talk I focus on novel ways of constructing Galileon-like theories, using both the probe brane construction, and the coset construction.
We provide a complete classification of all abelian gauge invariant $p$-form theories with equations of motion depending only on the second derivative of the field---the $p$-form analogues of the Galileon scalar field theory. We construct explicitly the nontrivial actions that exist for spacetime dimension $D\leq11$, but our methods are general enough and can be extended to arbitrary $D$. We uncover in particular a new $4$-form Galileon cubic theory in $D\geq8$ dimensions. As a by-product we give a simple proof of the fact that the equations of motion depend on the $p$-form gauge fields only t…
In the decoupling limit, the DGP model reduces to the theory of a scalar field pi, with interactions including a specific cubic self-interaction - the galileon term. This term, and its quartic and quintic generalizations, can be thought of as arising from a probe 3-brane in a 5-dimensional bulk with Lovelock terms on the brane and in the bulk. We study multi-field generalizations of the galileon, and extend this probe brane view to higher co-dimensions. We derive an extremely restrictive theory of multiple galileon fields, interacting through a quartic term controlled by a single coupling, and…
We consider the classical equations of motion for a single Galileon field with generic parameters in the presence of non-relativistic sources. We introduce the concept of absolute stability of a theory: if one can show that a field at a single point---like infinity for instance---in spacetime is stable, then stability of the field over the rest of spacetime is guaranteed for any positive energy source configuration. The Dvali-Gabadadze-Porrati (DGP) model is stable in this manner, and previous studies of spherically symmetric solutions suggest that certain classes of the single field Galileon …
We study a scalar-tensor version of Lovelock theory with a non trivial higher order galileon term involving the coupling of the Lovelock two tensor with derivatives of the scalar galileon field. For a static and spherically symmetric spacetime we extend the Boulware-Deser solution to the presence of a Galileon field. The hairy solution has a regular scalar field on the black hole event horizon and presents certain self tuning properties for the bulk cosmological constant and the Gauss-Bonnet coupling. The combined time and radial dependence of the galileon field permits its horizon regularity.…
We study spherically symmetric solutions of the cubic covariant Galileon model in curved spacetime in presence of a matter source, in the test scalar field approximation. We show that a cosmological time evolution of the Galileon field gives rise to an induced matter-scalar coupling, due to the Galileon-graviton kinetic braiding, therefore the solution for the Galileon field is non trivial even if the bare matter-scalar coupling constant is set to zero. The local solution crucially depends on the asymptotic boundary conditions, and in particular, Minkowski and de Sitter asymptotics correspond …
Horndeski/Galileons may be considered as a proper generalization of General Relativity in high energy regime. Thus one may search for manifestation of Galileons interaction in collision experiments. In this paper we give arguments supporting this thesis. Galileon scalar field do not interact with matter via Standard Model interactions, we discuss a mechanism that allows Galileons to have influence on particle collisions. We give reasons to narrow the whole class of Horndeski/Galileons models to one particular term - John term from Fab Four subclass - for this particular issue. We were able to …
The recent progress in the study of Galileons, i.e. equations of second order with an action invariant under a Galilean transformation is related to work on `Universal Field Equations' \cite{dbfgov} which are second order equations arising by an iterative procedure from arbitrary Lagrangians of weight one in their first derivatives. It is pointed out that the Galileon is simply a Kaluza-Klein reduction of a Universal Field Equation. An implicit solution to the equation of motion is presented, and a class of explicit solutions pointed out. The multi-field extensions of both types of equations a…
A Poincarè invariant, local scalar field theory in which the Lagrangian and the equation of motion contain only up to second-order derivatives of the fields is called generalized Galileon. The covariant version of it in four dimensions is called Horndeski theory, and has been vigorously studied in applications to inflation and dark energy. In this paper, we study a class of multi-field extensions of the generalized Galileon theory. By imposing shift and $SO(N)$ symmetries on all the currently known multi-Galileon terms in general dimensions, we find that the structure of the Lagrangian is uniq…
We extend to curved backgrounds all flat-space scalar field models that obey purely second-order equations, while maintaining their second-order dependence on both field and metric. This extension simultaneously restores to second order the, originally higher derivative, stress tensors as well. The process is transparent and uniform for all dimensions.
The Galileons are a set of terms within four-dimensional effective field theories, obeying symmetries that can be derived from the dynamics of a 3+1-dimensional flat brane embedded in a 5-dimensional Minkowski Bulk. These theories have some intriguing properties, including freedom from ghosts and a non-renormalization theorem that hints at possible applications in both particle physics and cosmology. In this brief review article, we will summarize our attempts over the last year to extend the Galileon idea in two important ways. We will discuss the effective field theory construction arising f…
We introduce bi-galileon theory, the generalisation of the single galileon model introduced by Nicolis et al. The theory contains two coupled scalar fields and is described by a Lagrangian that is invariant under Galilean shifts in those fields. This paper is the first of two, and focuses on the motivation and formulation of the theory. We show that the boundary effective theory of the cascading cosmology model corresponds to a bi-galileon theory in the decoupling limit, and argue that this is to be expected for co-dimension 2 braneworld models exhibiting infra-red modification of gravity. We …
Cubic Galileon massive gravity is a development of de Rham-Gabadadze-Tolly (dRGT) massive gravity theory is which the space of the Stueckelberg field is broken. We consider the cubic Galileon term as a scalar field coupled to the graviton filed. We present a detailed study of the cosmological aspects of this theory of gravity. We analyze self-accelerating solutions of the background equations of motion to explain the accelerated expansion of the Universe. Exploiting the latest Union2 Type Ia Supernovea (SNIa) dataset, which consists of 557 SNIa, we show that cubic Galileon massive gravity theo…
We show that scalar, 0-form, Galileon actions --models whose field equations contain only second derivatives-- can be generalized to arbitrary even p-forms. More generally, they need not even depend on a single form, but may involve mixed p combinations, including equal p multiplets, where odd p-fields are also permitted: We construct, for given dimension D, general actions depending on scalars, vectors and higher p-form field strengths, whose field equations are of exactly second derivative order. We also discuss and illustrate their curved-space generalizations, especially the delicate non-m…
We discuss non-renormalization theorems applying to galileon field theories and their generalizations. Galileon theories are similar in many respects to other derivatively coupled effective field theories, including general relativity and $P(X)$ theories. In particular, these other theories also enjoy versions of non-renormalization theorems that protect certain operators against corrections from self-loops. However, we argue that the galileons are distinguished by the fact that they are not renormalized even by loops of other heavy fields whose couplings respect the galileon symmetry.
We consider the recently introduced "galileon" field in a dynamical spacetime. When the galileon is assumed to be minimally coupled to the metric, we underline that both field equations of the galileon and the metric involve up to third-order derivatives. We show that a unique nonminimal coupling of the galileon to curvature eliminates all higher derivatives in all field equations, hence yielding second-order equations, without any extra propagating degree of freedom. The resulting theory breaks the generalized "Galilean" invariance of the original model.
We derive the 1-loop effective action of the cubic Galileon coupled to quantum-gravitational fluctuations in a background and gauge-independent manner, employing the covariant framework of DeWitt and Vilkovisky. Although the bare action respects shift symmetry, the coupling to gravity induces an effective mass to the scalar, of the order of the cosmological constant, as a direct result of the non-flat field-space metric, the latter ensuring the field-reparametrization invariance of the formalism. Within a gauge-invariant regularization scheme, we discover novel, gravitationally induced non-Gal…
We consider a galileon field coupled to gravity. The standard no-hair theorems do not apply, because of the galileon's peculiar derivative interactions. We prove that, nonetheless, static spherically symmetric black holes cannot sustain non-trivial galileon profiles. Our theorem holds regardless of whether there are non-minimal couplings between the galileon and gravity of the covariant galileon type.
We study whether it is possible to design a "classical" spatially flat bouncing cosmology or a static, spherically symmetric asymptotically flat Lorentzian wormhole in cubic Galileon theories interacting with an extra scalar field. We show that bouncing models are always plagued with gradient instabilities, while there are always ghosts in wormhole backgrounds.
We introduce a general class of four-dimensional effective field theories which include curved space Galileons and DBI theories possessing nonlinear shift-like symmetries. These effective theories arise from purely gravitational actions for 3-branes probing higher dimensional spaces. In the simplest case of a Minkowski brane embedded in a higher dimensional Minkowski background, the resulting four-dimensional effective field theory is the Galileon one, with its associated Galilean symmetry and second order equations. However, much more general structures are possible. We construct the general …
Could a conformal Galileon be describing a gauge mode of a broader diffeomorphism invariant theory? We answer this question affirmatively in 3D by using a coset construction for a nonlinearly realized conformal symmetry. In particular, we show that the conformal Galileon emerges as a Stückelberg field of a local Weyl symmetry. The coset construction gives us a diffeomorphism and Weyl invariant 3D theory containing first, second, and third powers of the curvatures, their couplings to certain tensors made of the Galileon and its derivatives, and the conformal Galileon terms. In a theory with a b…
We review different constructions of Galileon theories in both flat and curved space, and for both single scalar field models as well as multi-field models. Our main emphasis is on the formal mathematical properties of these theories and their construction.
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…
We find and study the properties of black hole solutions for a subclass of Horndeski theory including the cubic Galileon term. The theory under study has shift symmetry but not reflection symmetry for the scalar field. The Galileon is assumed to have linear time dependence characterized by a velocity parameter. We give analytic 3-dimensional solutions that are akin to the BTZ solutions but with a non-trivial scalar field that modifies the effective cosmological constant. We then study the 4-dimensional asymptotically flat and de Sitter solutions. The latter present three different branches acc…
A simple way to construct models with early cosmological Genesis epoch is to employ bosonic fields whose Lagrangians transform homogeneously under scaling transformation. We show that in these theories, for a range of parameters defining the Lagrangian, there exists a homogeneous power-law solution in flat space-time, whose energy density vanishes, while pressure is negative (power-law Genesis). We find the condition for the legitimacy of the classical field theory description of such a situation. We note that this condition does not hold for our earlier Genesis model with vector field. We con…
We consider a particular truncation of the generalized Proca field theory in four dimensions for which we construct a static and axisymmetric rotating black hole "stealth solutions", namely solutions with (Anti) de Sitter or Kerr metric but non-trivial vector field. The static configuration is promoted to a backreacting black hole with asymptotic (Anti) de Sitter behavior by turning on a nonlinear electrodynamic source given as a fixed power of the Maxwell invariant. Finally we extend our solutions to arbitrary dimensions.
We consider an Abelian gauge field coupled to a particular truncation of Horndeski theory. The Galileon field has translation symmetry and couples non minimally both to the metric and the gauge field. When the gauge-scalar coupling is zero the gauge field reduces to a standard Maxwell field. By taking into account the symmetries of the action, we construct charged black hole solutions. Allowing the scalar field to softly break symmetries of spacetime we construct black holes where the scalar field is regular on the black hole event horizon. Some of these solutions can be interpreted as the equ…
We review the different frameworks in which Galileon scalar fields have been seen to emerge such an in DGP, New Massive Gravity and Ghost-free massive Gravity and emphasize their relation with the Lovelock invariant in braneworld models. The existence of a non-renormalization theorem for Galileon scalar fields makes them especially attractive candidates for inflation as well as for late-time acceleration. In particular we review the self-accelerating and degravitating branches of solutions present in Galileon models when arising from Massive Gravity and discuss their phenomenology.
We modify the scalar Einstein-aether theory by breaking the Lorentz invariance of a gravitational theory coupled to a Galileon type scalar field. This is done by introducing a Lagrange multiplier term into the action, thus ensuring that the gradient of the scalar field is time-like, with unit norm. The theory can also be considered as an extension to the mimetic dark matter theory, by adding some derivative self interactions to the action, which keeps the equation of motion at most second order in time derivatives. The cosmological implications of the model are discussed in detail. In particul…
In this paper we investigate the decoupling limit of a particular class of multi-gravity theories, i.e. of theories of interacting spin-2 fields. We explicitly compute the interactions of helicity-0 modes in this limit, showing that they take on the form of multi-Galileons and dual forms. In the process we extend the recently discovered Galileon dualities, deriving a set of new multi-Galileon dualities. These are also intrinsically connected to healthy, but higher-derivative, multi-scalar field theories akin to `beyond Horndeski' models.
In this work we have considered a model that includes the interaction of gravity and matter fields with Galilean invariance (the so-called derivative coupling) as well as some corresponding black hole type solutions. Quasinormal perturbations of two kinds of matter fields have been computed by different methods. The effect of the derivative coupling in the quasinormal spectrum has been analyzed and evaluated.
We propose a simple field theory reproducing the MOND phenomenology at galaxy scale, while predicting negligible deviations from general relativity at small scales thanks to an extended Vainshtein ("k-mouflage") mechanism induced by a covariant Galileon-type Lagrangian. The model passes solar-system tests at the post-Newtonian order, including those of local Lorentz invariance, and its anomalous forces in binary-pulsar systems are orders of magnitude smaller than the tightest experimental constraints. The large-distance behavior is obtained as in Bekenstein's tensor-vector-scalar (TeVeS) model…
We construct four-dimensional effective field theories of a generalized DBI galileon field, the dynamics of which naturally take place on a Friedmann-Robertson-Walker spacetime. The theories are invariant under non-linear symmetry transformations, which can be thought of as being inherited from five-dimensional bulk Killing symmetries via the probe brane technique through which they are constructed. The resulting model provides a framework in which to explore the cosmological role that galileons may play as the universe evolves.
We discuss the possibility of constructing stable, static, spherically symmetric, asymptotically flat Lorentzian wormhole solutions in General Relativity coupled to a generalized Galileon field $π$. Assuming that Minkowski space-time is obtained at $\partial π=0$, we find that there is tension between the properties of the energy-momentum tensor required to support a wormhole (violation of average null energy conditions) and stability of the Galileon perturbations about the putative solution (absence of ghosts and gradient instabilities). In 3-dimensional space-time, this tension is strong eno…
We investigate a large class of infinitesimal, but fully nonlinear in the field, transformations of the Galileon and search for extended symmetries. The transformations involve powers of the coordinates $x$ and the field $π$ up to any finite order $N$. Up to quadratic order the structure of these symmetry transformations is the unique generalisation of both the infinitesimal version of the standard Galileon shift symmetry as well as a recently discovered infinitesimal extension of this symmetry. The only higher-order extensions of this symmetry we recover are (`Galileon dual' versions of) symm…
We construct simple Lagrangians of vector fields which involve second derivatives, but nevertheless lead to second order field equations. These vector fields are, therefore, analogs of generalized Galileons. Our construction is given first in Minkowski space, and then generalizied to include dynamical gravity. We present examples of backgrounds that are stable and ghost-free despite the absence of gauge invariance. Some of these backgrounds violate the Null Energy Condition.
In this brief article, I summarize attempts with collaborators over the last couple of years to extend the Galileon idea in two important ways. I discuss the effective field theory construction arising from co-dimension greater than one flat branes embedded in a flat background - the multi-Galileons - and then describe symmetric covariant versions of the Galileons, more suitable for general cosmological applications. These generalized Galileons can be thought of as interesting four-dimensional field theories in their own rights, but the work described here may also make it easier to embed them…
In this work we study various aspects of supersymmetric three-dimensional higher-derivative field theories. We classify all possible models without derivative terms in the auxiliary field of the fermionic sector and find that scalar field theories of the form $P(X,ϕ)$, where $X=-(\partialϕ)^2/2$, belong to this kind of models. A ghost-free supersymmetric extension of Galileon models is found in three spacetime dimensions. Finally, the auxiliary field problem is discussed.
We prove the existence of a fully nonlinear conserved curvature perturbation on large scales in Galileon-type scalar field models in two approaches. The first approach is based on the conservation of energy-momentum tensor of the Galileon field, which is also the familiar approach in understanding the conservation in $k$-essence or perfect fluid models. We show that the fluid corresponding to the Galileon field becomes perfect and barotropic on large scales, which is responsible to the conservation. The difference from $k$-essence model is that, besides the energy-momentum conservation, the Ei…
Vortex solutions are topologically stable field configurations that can play an important role in condensed matter, field theory, and cosmology. We investigate vortex configuration in a 2+1 dimensional Abelian Higgs theory supplemented by higher order derivative self-interactions, related with Galileons. Our vortex solutions have features that make them qualitatively different from well-known Abrikosov-Nielsen-Olesen configurations, since the derivative interactions turn on gauge invariant field profiles that break axial symmetry. By promoting the system to a 3+1 dimensional string configurati…
The particular structure of Galileon interactions allows for higher-derivative terms while retaining second order field equations for scalar fields and Abelian $p$-forms. In this work we introduce an index-free formulation of these interactions in terms of two sets of Grassmannian variables. We employ this to construct Galileon interactions for mixed-symmetry tensor fields and coupled systems thereof. We argue that these tensors are the natural generalization of scalars with Galileon symmetry, similar to $p$-forms and scalars with a shift-symmetry. The simplest case corresponds to linearised g…
We consider Galileon models on curved spacetime, as well as the counterterms introduced to maintain the second-order nature of the field equations of these models when both the metric and the scalar are made dynamical. Working in a gauge invariant framework, we first show how all the third-order time derivatives appearing in the field equations -- both metric and scalar -- of a Galileon model or one defined by a given counterterm can be eliminated to leave field equations which contain at most second-order time derivatives of the metric and of the scalar. The same is shown to hold for arbitrar…
Galileon radiation in the collapse of a thin spherical shell of matter is analyzed. In the framework of a cubic Galileon theory, we compute the field profile produced at large distances by a short collapse, finding that the radiated field has two peaks traveling ahead of light fronts. The total energy radiated during the collapse follows a power law scaling with the shell's physical width and results from two competing effects: a Vainshtein suppression of the emission and an enhancement due to the thinness of the shell.
We determine the most general scalar field theories which have an action that depends on derivatives of order two or less, and have equations of motion that stay second order and lower on flat space-time. We show that those theories can all be obtained from linear combinations of Lagrangians made by multiplying a particular form of the Galileon Lagrangian by an arbitrary scalar function of the scalar field and its first derivatives. We also obtain curved space-time extensions of those theories which have second order field equations for both the metric and the scalar field. This provide the mo…
It has been pointed out that non-singular cosmological solutions in second-order scalar-tensor theories generically suffer from gradient instabilities. We extend this no-go result to second-order gravitational theories with an arbitrary number of interacting scalar fields. Our proof follows directly from the action of generalized multi-Galileons, and thus is different from and complementary to that based on the effective field theory approach. Several new terms for generalized multi-Galileons on a flat background were proposed recently. We find a covariant completion of them and confirm that t…
We establish a correspondence between general relativity with diffeomorphism invariance and scalar field theories with Galilean invariance: notions such as the Levi-Civita connection and the Riemann tensor have a Galilean counterpart. This suggests Galilean theories as the unique nontrivial alternative to gauge theories (including general relativity). Moreover, it is shown that the requirement of first-order Palatini formalism uniquely determines the Galileon models with second-order field equations, similar to the Lovelock gravity theories. Possible extensions are discussed.
We study the cosmology of a covariant scalar field respecting a Galilean symmetry in flat space-time. We show the existence of a tracker solution that finally approaches a de Sitter fixed point responsible for cosmic acceleration today. The viable region of model parameters is clarified by deriving conditions under which ghosts and Laplacian instabilities of scalar and tensor perturbations are absent. The field equation of state exhibits a peculiar phantom-like behavior along the tracker, which allows a possibility to observationally distinguish the Galileon gravity from the Lambda-CDM model.
We propose a picture for the UV properties of Galileon field theories. We conjecture that Galileons, and all theories incorporating the Vainshtein mechanism, fall into Jaffe's class of `non-localizable' field theories characterized by an exponential growth in their Kallen-Lehmann spectral densities. Similar properties have been argued to arise for Little String Theories and M-theory. For such theories, the notion of micro-causality and the time ordering used to define the S-matrix and correlation functions must be modified, and we give a Lorentz invariant prescription for how this can be achie…
A generalized theory of multi-field galileons has been recently put forward. This model stems from the ongoing effort to embed generic galileon theories within brane constructions. Such an approach has proved very useful in connecting interesting and essential features of these theories with geometric properties of the branes embedding. We investigate the cosmological implications of a very restrictive multi-field galileon theory whose leading interaction is solely quartic in the scalar field and lends itself nicely to an interesting cosmology. The trispectrum of curvature fluctuations has fea…
