## Publications

92 results found

Alberte L, de Rham C, Momeni A,
et al., 2020, EFT of interacting spin-2 fields, *The Journal of High Energy Physics*, Vol: 2020, Pages: 1-57, ISSN: 1029-8479

We consider the effective field theory of multiple interacting massive spin-2 fields. We focus on the case where the interactions are chosen so that the cutoff is the highest possible, and highlight two distinct classes of theories. In the first class, the mass eigenstates only interact through potential operators that carry no derivatives in unitary gauge at leading order. In the second class, a specific kinetic mixing between the mass eigenstates is included non-linearly. Performing a decoupling and ADM analysis, we point out the existence of a ghost present at a low scale for the first class of interactions. For the second class of interactions where kinetic mixing is included, we derive the full Λ3-decoupling limit and confirm the absence of any ghosts. Nevertheless both formulations can be used to consistently describe an EFT of interacting massive spin-2 fields which, for a suitable technically natural tuning of the EFT, have the same strong coupling scale Λ3. We identify the generic form of EFT corrections in each case. By using Galileon Duality transformations for the specific case of two massive spin-2 fields with suitable couplings, the decoupling limit theory is shown to be a bi-Galileon.

Alberte L, Rham CD, Momeni A, et al., 2019, Positivity constraints on interacting pseudo-linear spin-2 fields, Publisher: arXiv

We explore the effective field theory for single and multiple interactingpseudo-linear spin-2 fields. By applying forward limit positivity bounds, weshow that among the parameters contributing to elastic tree level scatteringamplitude, there is no region of compatibility of the leading interactions witha standard local UV completion. Our result generalizes to any number ofinteracting pseudo-linear spin-2 fields. This results have significantimplications for the organization of the effective field theory expansion forpseudo-linear fields.

de Rham C, Zhang J, 2019, Perturbations of stealth black holes in degenerate higher-order scalar-tensor theories, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 100, Pages: 124023-1-124023-12, ISSN: 1550-2368

Among the scalar-tensor modified theories of gravity, degenerate higher-order scalar-tensor (DHOST) models could play a special role for dark energy while being consistent with current observations, notably those constraining the speed of gravitational waves. Schwarzschild–de Sitter black holes were shown to be exact solutions of a particular subclass of quadratic DHOST theories, while carrying a nontrivial scalar profile that linearly evolves in time and hence potentially providing exciting new phenomenological windows to explore this model. We investigate the physical perturbations about such black holes and find that the odd-parity tensor perturbations behave in a way indistinguishable to general relativity. On the other hand, the effective metric for the (even-parity) scalar perturbations is singular, indicating that those exact black hole solutions are infinitely strongly coupled and cannot be trusted within the regime of validity of the DHOST effective field theory. We show how this strong coupling result is generalizable to a whole class of solutions with arbitrary manifolds both for DHOST and Horndeski.

de Rham C, Heisenberg L, Tolley AJ, 2019, Spin-2 fields and the weak gravity conjecture, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 100, Pages: 1-20, ISSN: 1550-2368

Recently, it has been argued that application of the weak gravity conjecture (WGC) to spin-2 fields implies a universal upper bound on the cutoff of the effective theory for a single spin-2 field. We point out here that these arguments are largely spurious, because of the absence of states carrying spin-2 Stückelberg U(1) charge, and because of incorrect scaling assumptions. Known examples such as Kaluza-Klein theory that respect the usual WGC do so because of the existence of a genuine U(1) field under which states are charged, as in the case of the Stückelberg formulation of spin-1 theories, for which there is an unambiguously defined U(1) charge. Theories of bigravity naturally satisfy a naive formulation of the WGC, MW<MPl, since the force of the massless graviton is always weaker than the massive spin-2 modes. It also follows that theories of massive gravity trivially satisfies this form of the WGC. We also point out that the identification of a massive spin-2 state in a truncated higher derivative theory, such as Einstein-Weyl-squared or its supergravity extension, bears no relationship with massive spin-2 states in the UV completion, contrary to previous statements in the literature. We also discuss the conjecture from a swampland perspective and show how the emergence of a universal upper bound on the cutoff relies on strong assumptions on the scale of the couplings between the spin-2 and other fields, an assumption which is known to be violated in explicit examples.

Alberte L, Rham CD, Momeni A, et al., 2019, Positivity constraints on interacting spin-2 fields, Publisher: arXiv

The consistency of the EFT of two interacting spin-2 fields is checked byapplying forward limit positivity bounds on the scattering amplitudes toexclude the region of parameter space devoid of a standard UV completion. Wefocus on two classes of theories that have the highest possible EFT cutoff,namely those theories modelled on ghost-free interacting theories of a singlemassive spin-2 field. We find that the very existence of interactions betweenthe spin-2 fields implies more stringent bounds on all the parameters of theEFT, even on the spin-2 self-interactions. This arises for two reasons. First,with every new field included in the low-energy EFT, comes the `knowledge' ofan extra pole to be subtracted, hence strengthening the positivity bounds.Second, while adding new fields increases the number of free parameters fromthe new interactions, this is rapidly overcome by the increased number ofpositivity bounds for different possible scattering processes. We also discusshow positivity bounds appear to favour relations between operators thateffectively raise the cutoff of the EFT.

Rham CD, Tolley AJ, 2019, The speed of gravity, Publisher: arXiv

Within the standard effective field theory of General Relativity, we showthat the speed of gravitational waves deviates, ever so slightly, fromluminality on cosmological and other spontaneously Lorentz-breakingbackgrounds. This effect results from loop contributions from massive fields ofany spin, including Standard Model fields, or from tree level effects frommassive higher spins $s \ge 2$. We show that for the choice of interactionsigns implied by S-matrix and spectral density positivity bounds suggested byanalyticity and causality, the speed of gravitational waves is in generalsuperluminal at low-energies on NEC preserving backgrounds, meaninggravitational waves travel faster than allowed by the metric to which photonsand Standard Model fields are minimally coupled. We show that departure of thespeed from unity increases in the IR and argue that the speed inevitablyreturns to luminal at high energies as required by Lorentz invariance.Performing a special tuning of the EFT so that renormalization sensitivecurvature-squared terms are set to zero, we find that finite loop correctionsfrom Standard Model fields still lead to an epoch dependent modification of thespeed of gravitational waves which is determined by the precise field contentof the lightest particles with masses larger than the Hubble parameter today.Depending on interpretation, such considerations could potentially havefar-reaching implications on light scalar models, such as axionic or fuzzy colddark matter.

Rham CD, Zhang J, 2019, Perturbations of stealth black holes in DHOST theories, Publisher: arXiv

Among the Scalar-Tensor modified theories of gravity, DHOST models could playa special role for dark energy while being consistent with currentobservations, notably those constraining the speed of gravitational waves.Schwarzschild-de Sitter black holes were shown to be exact solutions of aparticular subclass of quadratic DHOST theories, while carrying a nontrivialscalar profile that linearly evolves in time and hence potentially providingexciting new phenomenological windows to explore this model. We investigate thephysical perturbations about such black holes and find that the odd-paritytensor perturbations behave in a way indistinguishable to GR. On the otherhand, the effective metric for the (even-parity) scalar perturbations issingular, indicating that those exact black hole solutions are infinitelystrongly coupled and cannot be trusted within the regime of validity of theDHOST effective field theory. We show how this strong coupling result isgeneralizable to a whole class of solutions with arbitrary manifolds both forDHOST and Horndeski.

JimĂ©nez JB, Rham CD, Heisenberg L, 2019, Generalized proca and its constraint algebra, Publisher: arXiv

We reconsider the construction of general derivative self-interactions for amassive Proca field. The constructed Lagrangian is such that the vector fieldpropagates at most three degrees of freedom, thus avoiding the ghostly natureof a fourth polarisation. The construction makes use of the well-knowncondition for constrained systems of having a degenerate Hessian. We brieflydiscuss the casuistry according to the nature of the existing constraintsalgebra. We also explore various classes of interesting new interactions thathave been recently raised in the literature. For the sixth order Lagrangianthat satisfies the constraints by itself we prove its topological character,making such a term irrelevant. There is however a window of opportunity forexploring other classes of fully-nonlinear interactions that satisfy theconstraint algebra by mixing terms of various order.

de Rham C, 2019, The gravitational rainbow beyond Einstein gravity, *International Journal of Modern Physics D: Gravitation, Astrophysics and Cosmology*, Vol: 28, ISSN: 0218-2718

The recent direct detection of gravitational waves have been successfully used to examine the basic properties of the gravitational degrees of freedom. They set an upper bound on their mass and constrain their speed of propagation with unprecedented accuracy. Within the current realm of observational and theoretical constraints, we explore the possibility for gravity to depart from general relativity (GR) in the infrared and derive the implications on our observable Universe. We also investigate whether these types of models could ever enjoy a standard analytic UV completion.

de Rham C, Melville S, Tolley AJ,
et al., 2019, Positivity bounds for massive spin-1 and spin-2 fields, *The Journal of High Energy Physics*, Vol: 2019, ISSN: 1029-8479

We apply the recently developed positivity bounds for particles with spin, applied away from the forward limit, to the low energy effective theories of massive spin-1 and spin-2 theories. For spin-1 theories, we consider the generic Proca EFT which arises at low energies from a heavy Higgs mechanism, and the special case of a charged Galileon for which the EFT is reorganized by the Galileon symmetry. For spin-2, we consider generic Λ5 massive gravity theories and the special ‘ghost-free’ Λ3 theories. Remarkably we find that at the level of 2-2 scattering, the positivity bounds applied to Λ5 massive gravity theories, impose the special tunings which generate the Λ3 structure. For Λ3 massive gravity theories, the island of positivity derived in the forward limit appears relatively stable against further bounds.

de Rham C, Melville S, Tolley AJ,
et al., 2019, UV complete me: positivity bounds for particles with spin, *Journal of High Energy Physics*, Vol: 2018, ISSN: 1029-8479

For a low energy effective theory to admit a standard local, unitary, analytic and Lorentz-invariant UV completion, its scattering amplitudes must satisfy certain inequalities. While these bounds are known in the forward limit for real polarizations, any extension beyond this for particles with nonzero spin is subtle due to their non-trivial crossing relations. Using the transversity formalism (i.e. spin projections orthogonal to the scattering plane), in which the crossing relations become diagonal, these inequalities can be derived for 2-to-2 scattering between any pair of massive particles, for a complete set of polarizations at and away from the forward scattering limit. This provides a set of powerful criteria which can be used to restrict the parameter space of any effective field theory, often considerably more so than its forward limit subset alone.

Dar F, de Rham C, Deskins JT,
et al., 2019, Scalar gravitational radiation from binaries: Vainshtein mechanism in time-dependent systems, *Classical and Quantum Gravity*, Vol: 36, ISSN: 0264-9381

We develop a full four-dimensional numerical code to study scalar gravitational radiation emitted from binary systems and probe the Vainshtein mechanism in situations that break the static and spherical symmetry, relevant for binary pulsars as well as black holes and neutron stars binaries. The present study focuses on the cubic Galileon which arises as the decoupling limit of massive theories of gravity. Limitations associated with the numerical methods prevent us from reaching a physically realistic hierarchy of scales; nevertheless, within this context we observe the same power law scaling of the radiated power as previous analytic estimates, and confirm a strong suppression of the power emitted in the monopole and dipole as compared with quadrupole radiation. Following the trend to more physically realistic parameters, we confirm the suppression of the power emitted in scalar gravitational radiation and the recovery of general relativity with good accuracy. This paves the way for future numerical work, probing more generic, physically relevant situations and sets of interactions that may exhibit the Vainshtein mechanism.

Amendola L, Appleby S, Avgoustidis A,
et al., 2018, Cosmology and fundamental physics with the Euclid satellite, *Living Reviews in Relativity*, Vol: 21, Pages: 1-345, ISSN: 1433-8351

Euclid is a European Space Agency medium-class mission selected for launch in 2020 within the cosmic vision 2015–2025 program. The main goal of Euclid is to understand the origin of the accelerated expansion of the universe. Euclid will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and red-shifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky. Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis. This review has been planned and carried out within Euclid’s Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission.

de Rham C, Melville S, 2018, Gravitational rainbows: LIGO and dark energy at its cutoff, *Physical Review Letters*, Vol: 121, Pages: 221101-1-221101-6, ISSN: 0031-9007

The recent direct detection of gravitational waves from a neutron star merger with optical counterpart has been used to severely constrain models of dark energy that typically predict a modification of the gravitational wave speed. However, the energy scales observed at LIGO, and the particular frequency of the neutron star event, lie very close to the strong coupling scale or cutoff associated with many dark energy models. While it is true that at very low energies one expects gravitational waves to travel at a speed different than light in these models, the same is no longer necessarily true as one reaches energy scales close to the cutoff. We show explicitly how this occurs in a simple model with a known partial UV completion. Within the context of Horndeski, we show how the operators that naturally lie at the cutoff scale can affect the speed of propagation of gravitational waves and bring it back to unity at LIGO scales. We discuss how further missions including LISA and PTAs could play an essential role in testing such models.

de Rham C, Hinterbichler K, Johnson LA, 2018, On the (A)dS decoupling limits of massive gravity, *The Journal of High Energy Physics*, Vol: 154, Pages: 1-38, ISSN: 1029-8479

We consider various decoupling limits of ghost-free massive gravity on (A)dS. The first is a decoupling limit on AdS space where the mass goes to zero while the AdS radius is held fixed. This results in an interacting massive Proca vector theory with a Λ2 ∼ (MPlm)1/2 strong coupling scale which is ghost-free by construction and yet can not be put in the form of the generalized Proca theories considered so far. We comment on the existence of a potential duality between this Proca theory and a CFT on the boundary. The second decoupling limit we consider is a new limit on dS, obtained by sending the mass towards the finite partially massless value. We do this by introducing the scalar Stückelberg field which restores the partially massless symmetry. For generic values of the parameters, only a finite number of operators enter the partially massless decoupling limit and take the form of dS Galileons. If the interactions are chosen to be precisely those of the ‘candidate’ non-linear partially massless theory, the resulting strong coupling scale has a higher value and the resulting decoupling limit includes an infinite number of interactions which we give in closed form. These interactions preserve both the linear partially massless symmetry and the dS version of the Galileon shift symmetry.

de Rham C, Melville S, Tolley AJ, 2018, Improved positivity bounds and massive gravity, *Journal of High Energy Physics*, Vol: 2018, ISSN: 1029-8479

Theories such as massive Galileons and massive gravity can satisfy the presently known improved positivity bounds provided they are weakly coupled. We discuss the form of the EFT Lagrangian for a weakly coupled UV completion of massive gravity which closely parallels the massive Galileon, and perform the power counting of corrections to the scattering amplitude and the positivity bounds. The Vainshtein mechanism which is central to the phenomenological viability of massive gravity is entirely consistent with weak coupling since it is classical in nature. We highlight that the only implication of the improved positivity constraints is that the EFT cutoff is lower than previous assumed, and discuss the observable implications, emphasizing that these bounds are not capable of ruling out the model contrary to previous statements in the literature.

de Rham C, Melville S, Tolley AJ,
et al., 2017, Positivity bounds for scalar field theories, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, Vol: 96, ISSN: 1550-2368

Assuming the existence of a local, analytic, unitary UV completion in a Poincaré invariant scalar field theory with a mass gap, we derive an infinite number of positivity requirements using the known properties of the amplitude at and away from the forward scattering limit. These take the form of bounds on combinations of the pole subtracted scattering amplitude and its derivatives. In turn, these positivity requirements act as constraints on the operator coefficients in the low energy effective theory. For certain theories these constraints can be used to place an upper bound on the mass of the next lightest state that must lie beyond the low energy effective theory if such a UV completion is to ever exist.

de Rham C, Melville S, Tolley AJ,
et al., 2017, Massive Galileon positivity bounds, *Journal of High Energy Physics*, Vol: 2017, ISSN: 1029-8479

The EFT coefficients in any gapped, scalar, Lorentz invariant field theory must satisfy positivity requirements if there is to exist a local, analytic Wilsonian UV completion. We apply these bounds to the tree level scattering amplitudes for a massive Galileon. The addition of a mass term, which does not spoil the non-renormalization theorem of the Galileon and preserves the Galileon symmetry at loop level, is necessary to satisfy the lowest order positivity bound. We further show that a careful choice of successively higher derivative corrections are necessary to satisfy the higher order positivity bounds. There is then no obstruction to a local UV completion from considerations of tree level 2-to-2 scattering alone. To demonstrate this we give an explicit example of such a UV completion.

de Rham C, Melville S, 2017, Unitary null energy condition violation in P(X) cosmologies, *Physical Review D: Particles, Fields, Gravitation and Cosmology*, Vol: 95, ISSN: 1550-2368

A non singular cosmological bounce in the Einstein frame can only take place if the null energy condition (NEC) is violated. We explore situations where a single scalar field drives the NEC violation and derive the constraints imposed by demanding tree level unitarity on a cosmological background. We then focus on the explicit constraints that arise in P(X) theories and show that constraints from perturbative unitarity make it impossible for the NEC violation to occur within the region of validity of the effective field theory without also involving irrelevant operators that arise at a higher scale that would enter from integrating out more massive degrees of freedom. Within the context of P(X) theories we show that including such operators allows for a bounce that does not manifestly violate tree level unitarity, but at the price of either imposing a shift symmetry or involving technically unnatural small operator coefficients within the low-energy effective field theory.

de Rham C, Deskins JT, Tolley AJ,
et al., 2017, Graviton mass bounds, *Reviews of Modern Physics*, Vol: 89, ISSN: 0034-6861

Recently, aLIGO announced the first direct detections of gravitational waves, a direct manifestation of the propagating degrees of freedom of gravity. The detected signals GW150914 and GW151226 have been used to examine the basic properties of these gravitational degrees of freedom, particularly setting an upper bound on their mass. It is timely to review what the mass of these gravitational degrees of freedom means from the theoretical point of view, particularly taking into account the recent developments in constructing consistent massive gravity theories. Apart from the GW150914 mass bound, a few other observational bounds have been established from the effects of the Yukawa potential, modified dispersion relation, and fifth force that are all induced when the fundamental gravitational degrees of freedom are massive. These different mass bounds are reviewed, how they stand in the wake of recent theoretical developments and how they compare to the bound from GW150914 are examined.

de Rham C, Motohashi H, 2017, Caustics for spherical waves, *Physical Review D - Particles, Fields, Gravitation and Cosmology*, Vol: 95, ISSN: 1550-2368

We study the development of caustics in shift-symmetric scalar field theories by focusing on simple waves with an SO(p)-symmetry in an arbitrary number of space dimensions. We show that the pure Galileon, the DBI–Galileon, and the extreme-relativistic Galileon naturally emerge as the unique set of caustic-free theories, highlighting a link between the caustic-free condition for simple SO(p)-waves and the existence of either a global Galilean symmetry or a global (extreme-)relativistic Galilean symmetry.

de Rham C, Tolley AJ, Zhou S-Y, 2016, Non-compact nonlinear sigma models, *Physics Letters B*, Vol: 760, Pages: 579-583, ISSN: 0370-2693

The target space of a nonlinear sigma model is usually required to be positive definite to avoid ghosts. We introduce a unique class of nonlinear sigma models where the target space metric has a Lorentzian signature, thus the associated group being non-compact. We show that the would-be ghost associated with the negative direction is fully projected out by 2 second-class constraints, and there exist stable solutions in this class of models. This result also has important implications for Lorentz–invariant massive gravity: There exist stable nontrivial vacua in massive gravity that are free from any linear vDVZ-discontinuity and a Λ2 decoupling limit can be defined on these vacua.

de Rham C, Matas A, 2016, Ostrogradsky in theories with multiple fields, *Journal of Cosmology and Astroparticle Physics*, Vol: 2016, ISSN: 1475-7516

We review how the (absence of) Ostrogradsky instability manifests itself in theories with multiple fields. It has recently been appreciated that when multiple fields are present, the existence of higher derivatives may not automatically imply the existence of ghosts. We discuss the connection with gravitational theories like massive gravity and beyond Horndeski which manifest higher derivatives in some formulations and yet are free of Ostrogradsky ghost. We also examine an interesting new class of Extended Scalar-Tensor Theories of gravity which has been recently proposed. We show that for a subclass of these theories, the tensor modes are either not dynamical or are infinitely strongly coupled. Among the remaining theories for which the tensor modes are well-defined one counts one new model that is not field-redefinable to Horndeski via a conformal and disformal transformation but that does require the vacuum to break Lorentz invariance. We discuss the implications for the effective field theory of dark energy and the stability of the theory. In particular we find that if we restrict ourselves to the Extended Scalar-Tensor class of theories for which the tensors are well-behaved and the scalar is free from gradient or ghost instabilities on FLRW then we recover Horndeski up to field redefinitions.

de Rham C, Tolley AJ, Zhou SY, 2016, The Λ2 limit of massive gravity, *Journal of High Energy Physics*, Vol: 2016, ISSN: 1126-6708

Lorentz-invariant massive gravity is usually associated with a strong couplingscale Λ3. By including non-trivial effects from the St¨uckelberg modes, we show that aboutthese vacua, one can push the strong coupling scale to higher values and evade the linearvDVZ-discontinuity. For generic parameters of the theory and generic vacua for theSt¨uckelberg fields, the Λ2-decoupling limit of the theory is well-behaved and free of anyghost or gradient-like instabilities. We also discuss the implications for nonlinear sigmamodels with Lorentzian target spaces.

de Rham C, Matas A, Tolley AJ, 2015, New kinetic terms for massive gravity and multi-gravity: a no-go in vielbein form, *CLASSICAL AND QUANTUM GRAVITY*, Vol: 32, ISSN: 0264-9381

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- Citations: 23

de Rham C, Matas A, Ondo NA,
et al., 2015, Interactions of charged spin-2 fields, *Classical and Quantum Gravity*, Vol: 32, ISSN: 0264-9381

In light of recent progress in ghost-free theories of massive gravity and multi-gravity, we reconsider the problem of constructing a ghost-free theory of an interacting spin-2 field charged under a U(1) gauge symmetry. Our starting point is the theory originally proposed by Federbush, which is essentially Fierz–Pauli generalized to include a minimal coupling to a U(1) gauge field. We show the Federbush theory with a dynamical U(1) field is in fact ghost-free and can be treated as a healthy effective field theory to describe a massive charged spin-2 particle. It can even potentially have healthy dynamics above its strong-coupling scale. We then construct candidate gravitational extensions to the Federbush theory both by using dimensional deconstruction, and by constructing a general nonlinear completion. However, we find that the U(1) symmetry forces us to modify the form of the Einstein–Hilbert kinetic term. By performing a constraint analysis directly in the first-order form, we show that these modified kinetic terms inevitably reintroduce the Boulware–Deser ghost. As a by-product of our analysis, we present a new proof for ghost-freedom of bi-gravity in 2+1 dimensions (also known as Zwei-Dreibein gravity). We also give a complementary algebraic argument that the Einstein–Hilbert kinetic term is incompatible with a U(1) symmetry, for a finite number of gravitons.

de Rham C, Tolley AJ, 2015, Vielbein to the rescue? Breaking the symmetric vielbein condition in massive gravity and multigravity, *PHYSICAL REVIEW D*, Vol: 92, ISSN: 1550-7998

Nonminimal matter couplings have recently been considered in the context of massive gravity and multigravity. These couplings are free of the Boulware-Deser ghost in the decoupling limit and can thus be considered within an effective field theory setup. Beyond the decoupling limit the ghost was shown to reemerge in the metric formulation of the theory. Recently it was argued that this pathology is absent when formulated in terms of unconstrained vielbeins. We investigate this possibility and show that the Boulware-Deser ghost is always present beyond the decoupling limit in any dimension larger than 2. We also show that the metric and vielbein formulations have an identical ghost-free decoupling limit. Finally we extend these arguments to more generic multigravity theories and argue that for any dimension larger than 2 a ghost is also present in the vielbein formulation whenever the symmetric vielbein condition is spoiled and the equivalence with the metric formulation is lost.

de Rham C, Heisenberg L, Riberiro RH, 2015, On couplings to matter in massive (bi-)gravity, *CLASSICAL AND QUANTUM GRAVITY*, Vol: 32, ISSN: 0264-9381

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- Citations: 110

de Rham C, 2015, Introduction to Massive Gravity, Modifications of Einstein's Theory of Gravity at Large Distances, Publisher: Springer International Publishing, Pages: 139-159, ISBN: 9783319100692

de Rham C, Heisenberg L, Ribeiro RH, 2014, Ghosts and matter couplings in massive gravity, bigravity and multigravity, *PHYSICAL REVIEW D*, Vol: 90, ISSN: 1550-7998

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- Citations: 86

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