Generalised matter couplings in massive bigravity

We investigate matter couplings in massive bigravity. We find a new family of such consistent couplings, including and extending known consistent matter couplings, and we investigate their decoupling limits, ADM decompositions, Higuchi bounds and further aspects. We show that differences to previous known consistent couplings only arise beyond the $\Lambda_3$ decoupling limit and discuss the uniqueness of consistent matter couplings and how this is related to the so-called symmetric vielbein condition. Since we work in a vielbein formulation, these results easily generalise to multi-gravity.


Introduction
Interest in massive gravity has experienced a remarkable resurgence in recent years. Reasons for this are manifold, but here we emphasise two: Firstly the cosmologically motivated hope that such theories will have something to say about the (late-time) accelerated expansion of the universe, providing a satisfactory model of 'dark energy'/modified gravity. Secondly the theoretically motivated desire to understand the complete space of consistent spin-2 theories. With a spin-2 field, or a graviton, being the quintessential gravitational degree of freedom (dof), understanding what possible consistent interactions and theories one can build for such a field promises to teach us much about gravity.
In this paper we will focus on the second motivation for studying massive gravity and related theories. Until roughly five years ago, General Relativity (GR) -the unique consistent classical theory of a massless spin-2 field -was widely believed to be the only classically consistent spin-2 theory. However, following the initial discovery of a set of consistent massive spin-2 theories -so-called ghost-free massive gravity [1][2][3] and bigravity [4,5] -much progress has been made in understanding the landscape of consistent spin-2 field theories.
Outline: The outline for this paper is as follows. In section 2 we introduce possible candidate matter couplings in the vielbein formulation of bigravity (which we also briefly review). In the following four sections we subject these candidates to several tests: in section 3 we check whether the pure spin-2 interactions generated by the matter coupling (through the presence of an effective cosmological constant, e.g. generated via matter loops) are stable for the candidate couplings; in section 4 we perform mini-superspace and full ADM analyses to check for the presence of additional propagating (ghost-like) dof; in section 5 we then perform a decoupling limit analysis, investigating whether such couplings contribute to a valid low-energy effective field theory (EFT) (even where the full ADM analysis revealed a ghost, which may propagate outside the EFT's regime of validity); in section 6 we check for the potential presence of Higuchi ghosts and derive the associated Higuchi bounds on the theory. Finally, we discuss the uniqueness of our new consistent matter couplings in section 7, before concluding in section 8 and collecting and discussing further computations in the appendices.
Conventions: Throughout this paper we use the following conventions. D refers to the number of spacetime dimensions, frequently taken to be four for simplicity, and we use Greek letters µ, ν, . . . and lower case Latin letters a, b, c, . . . to denote spacetime indices, which are raised and lowered as specified (this issue is not trivial in theories with several vielbeins/spin-2 fields/'metrics'). Capital Latin letters A, B, . . . are reserved for Lorentz indices and are raised and lowered with the Minkowski metric η AB . Bracketed indices (i), (j), . . ., label the different vielbeins/spin-2 fields -label indices are not automatically summed over and whether they are upper or lower indices carries no meaning. We denote the completely anti-symmetric epsilon symbol by˜ and define it such that˜ 012···D = 1 regardless of the signature of the metric or the position (up/down) of indices (hence˜ 012···D =˜ 012···D = 1).

Candidate couplings to matter
In this section we lay out the structure of the couplings between spin-2 fields and matter fields that we will consider. Our construction aims to find the general form of allowed couplings in the vielbein formulation.

The vielbein picture for massive and bi-gravity
Vielbeins: Throughout this paper we will primarily be working in the vielbein formulation for gravity. Corresponding to each spin-2 field/metric g (i) and its inverse g −1 (i) we have a vielbein E (i) and an inverse vielbein E −1 (i) satisfying where we emphasise that the vielbein is not in general symmetric in its two indices and that it satisfies the following conditions Lorentz indices (capital Latin letters) are raised/lowered with the flat Minkowski metric η AB and spacetime indices (lower case Greek indices) on E (i) are raised/lowered with the full metric g (i) . However, since there are in principle several distinct spacetime 'metrics' in the theory, here we choose the convention to raise/lower spacetime indices with the flat spacetime metric η µν and to explicitly write out any metric g (i) whenever it is used to raise/lower spacetime indices instead.
In terms of vielbeins they can all be cast in the unified format [27] (in D dimensions) where the indices (i 1 i 2 . . . i D ) keep track of which fields are interacting. 2 Ghost-free massive gravity potential interactions then consist of all the ways to build (2.3) with a single dynamical vielbein E (1) and a non-dynamical reference vielbein, which is held fixed (in the case of a flat reference metric this non-dynamical vielbein is E (0) A µ = δ A µ ). Ghost-free Bigravity consists of all interactions (2.3) that can be built with two dynamical vielbeins E (1) and E (2) , and so on. The most general known, fully ghost-free potential interaction for N spin-2 fields and in D dimensions can therefore be written as

Possible effective (matter) vielbeins
Throughout this paper we consider theories of the form

6)
2 Equivalently one can write this in terms of vielbein one-forms E A (i) = E (i) A µ dx µ (i) . The interaction terms can then be written i.e. theories of multiple spin-2 fields in the vielbein formulation, whose kinetic sector is a superposition of Einstein-Hilbert terms and that have potential interactions of the type (2.5). The coupling to matter fields Φ i occurs in accordance with the weak equivalence principle, i.e. matter uniformly and minimally couples to a single effective matter metric that is a function of the vielbeins and inverse vielbeins living in the theory. 3 More specifically we write where the effective matter vielbeinẼ will be a general function of the vielbeins and inverse vielbeins in the theory.
Effective vielbeins: We will assume that the effective matter vielbein can be expressed as a power-law expansion in all the fundamental vielbeins (and their inverses) in the theory. 4 The effective vielbeinẼ must have one free spacetime and one free Lorentz index, so we write the possible building blocks ofẼ order-by-order as contractions of vielbeins and inverse vielbeins (labelled by a 'species' index (i)) as Here we have suppressed any combinations that become Kronecker deltas or reduce to one of the combinations listed above by virtue of the relation (2.2). In general the effective vielbein can be an arbitrary linear combination of the above building blocks, with scalar functions of the E (i) as prefactors. We discuss some additional details of the structure of these terms in appendix A. As a concrete example consider the case when the effective vielbein simply is one of the building blocks in (2.8) (i.e. not a superposition). While no effective 2-or 4-vielbein couplings are possible, effective 3-and 5-vielbein couplings respectively lead to effective matter metrics where α and β are arbitrary scalar functions of the E (i) . For simplicity we will mostly consider single terms throughout, but we will return to superpositions in section 5 when discussing the decoupling limit and the symmetric vielbein condition. Note that, in the massive gravity case where one of the two vielbeins is non-dynamical, any superposition like (2.11) will automatically reduce back to the massive gravity limit of the construction of [6,7] (since the 'Minkowski vielbein' is simply a Kronecker delta).

Metric representation and symmetric vielbein condition
At the level of kinetic (Einstein-Hilbert) terms, a multi-gravity theory with N dynamical fields possesses N copies of local Lorentz transformations LLT (i) . These symmetries are generically broken down to their diagonal subgroups by the potential interactions S pot (2.5) (for details see [27][28][29]). They may be re-introduced at the expense of introducing a (gauge) Stückelberg field Λ through the following replacement Clearly the kinetic (Einstein-Hilbert) terms are gauge-invariant under this replacement and therefore do not contribute to the Λ equations of motion. Λ is then a non-dynamical, auxiliary field which may simply be integrated out. For bigravity, in the absence of coupling to matter fields Φ i , the Λ eom implies (in matrix notation) 5 [32] (1) ). (2.13) This is known as the Deser-von-Nieuwenhuizen (DvN) condition and ensures that an equivalent metric formulation exists (where S pot may be written in terms of elementary symmetric polynomials). Significantly, this condition is independent of the coefficients c (i 1 ,i 2 ...i D ) . The DvN condition allows us to choose a gauge in which the vielbeins commute. In particular this means we can choose (i.e. gauge fix using a copy of LLT) a Λ such that If this commutativity holds for all the vielbeins in the theory, the vielbein strings in (2.8) may be freely reordered so that vielbeins cancel against inverse vielbeins and all building blocks in (2.8) simply collapse to single 1-vielbein strings. If the symmetric vielbein condition holds for all vielbeins, all the (superpositions of) vielbein building blocks (2.8) discussed in the previous subsection collapse to the couplings proposed by [6,7]. As should already be clear from the above, in general the DvN gauge condition is not imposed dynamically. When a coupling to matter fields Φ i is introduced, we have another contribution to the Lorentz eoms, As pointed out in [7,13,14], this means we are no longer guaranteed that an equivalent metric formulation of the theory exists. However, even when this is the case and the DvN condition therefore does not hold for the full theory, particular solutions of the theory (e.g. ones considering highly symmetric FLRW backgrounds) or particular (low-energy) scaling limits (e.g. the Λ 3 decoupling limit discussed in section 5) can recover this condition. In these limits an equivalent metric formulation can therefore still exist. We will discuss these points further in what follows.

Effective cosmological constants and matter loops
Healthy spin-2 interactions: An important and very powerful check for any candidate coupling to matter is the requirement that any effective cosmological constant contribution it generates must be healthy. By this we mean that matter loops should not break the healthy form of the pure spin-2 interactions at tree level and, since they will (also) generate a contribution to the overall action of the form we would like this to be of the ghost-free form (2.3), whereΛ is a constant. This is a pure spin-2 interaction term by virtue of the det(Ẽ). Equivalently (3.1) can be seen as the zeroth order Lovelock invariant, or a cosmological constant, for g eff . There are a number of ways in which to achieve this. It will prove useful to divide the couplings which generate pure spin-2 interactions of the form (2.4) into three classes: (a) Rank-2 construction: The effective matter vielbeinẼ is a (superposition of) combinations of N vielbeins and inverse vielbeins, with constant coefficients. There are n (i) copies of vielbein E (i) andn (i) copies of its inverse involved in a single vielbein string. If the effective matter vielbeinẼ only consists of such a single string as listed in (2.8), where i, j ∈ {1, . . . , N } and we have required that there exists one and only one vielbein j that satisfies n (j) −n (j) = 1. In the vielbein language a coupling of this kind was recently introduced in [7], where the effective vielbeinẼ was constructed from vielbeins only (as a linear superposition of 1-vielbein strings). If the symmetric vielbein condition holds, all the rank-2 couplings reduce to such a couplingẼ = p a p E (p) and furthermore the vielbein coupling (in the bi-and massive gravity cases) becomes equivalent to the metric-language matter coupling proposed by [6].
(b) Rank-0 construction: The 'index-carrying' part of the matter coupling is some combination of vielbeins and inverse vielbeins with unit determinant 6 , premultiplied by a scalar function of (inverse) vielbeins that encodes spin-2 interactions of the type (2.4). More specifically this means the matter vielbein takes on the form where the constants d (i 1 i 2 ...i D ) are independent of the coefficients c (i 1 i 2 ...i D ) in the potential S pot (2.5). In (massive) bi-gravity a specific instance of this case was recently considered in [33].
(c) Mixed construction: The effective matter coupling non-trivially depends on the dynamical vielbein dofs through both its rank-0 coefficients and its rank-2 constituents, where M is an arbitrary rank-2 function of the vielbeins. In the massive (bi-)gravity case a coupling of this kind was recently considered in [33], in which α D contained the zeroth order and Dth order elementary symmetric polynomials only. 7 All of the above cases yield healthy pure spin-2 interactions of the form (3.1) by construction (when an effective cosmological constant is present/via matter loops). In the following sections, we assess the ghost-freedom of all these constructions in different limits of the theory -the results are surmised in Table 1.

Constraint analysis
Having passed the check of generating acceptable pure spin-2 interactions in the section above, a healthy matter coupling should pass a number of other checks. In this section, we will subject our candidate matter couplings to a constraint analysis. If the correct number of constraints is present to allow only the dofs of a (massive) spin-2 field (and the matter fields it couples to) to propagate, then we know that the full theory is free of additional dangerous ghost-like dofs (so-called Boulware-Deser ghosts [34]). If the theory passes this test and the spin-2 dofs are all healthy (this is related to the Higuchi bound discussion in section 6), it is healthy at all energy scales. 8 Here we will explicitly check for the presence of the correct number of primary constraints -for a detailed discussion of secondary constraints in related matter couplings see [14].

ADM decomposition and mini-superspace
In order to make the Hamiltonian constraint structure of the theory explicit, we would like to express the dof of the theory in terms of canonical coordinates and their momenta. For the vielbeins we consequently follow [27] and perform a boosted ADM foliation [35], 1) where the Lorentz γ ≡ √ 1 + P a P a and the D 2 independent components of a general vielbein E (1) A µ are expressed in terms of (D − 1) 2 components of the spatial vielbein e a i , the lapse N and (D − 1) shifts N i , and the (D − 1) momenta P a (we use the shorthand P a (1) ≡ P a ). Lorentz gauge invariance has been used to fix the momentum for E (2) to P a (2) = 0. 9 Forming the corresponding metric from a general vielbein, we find that the dependence on P a and the D(D − 1)/2 antisymmetric components of e a i drop out (these are the gauge fields corresponding to Lorentz boosts and rotations). This leaves only the D(D +1)/2 components of a symmetric rank-2 tensor where g ij ≡ e a i e b j δ ab .
General Relativity and dof counting: In ADM variables, the Einstein-Hilbert kinetic term may be written [35], where the lapse, N , and shift, N k , act as Lagrange multipliers, π ij is the momentum conjugate to the spatial metric g ij and the precise form of C and C k is not of importance here. In General Relativity (GR), the usual D = 4 story is that diffeomorphism invariance can be used to fixed four of the ten independent components of g µν (effectively this is the statement that the lapse and shift are non-dynamical), leaving six potentially propagating dofs (associated to the spatial metric g ij ). Another four are rendered non-dynamical by the fact that the lapse and shift in fact enter linearly and generate constraints, C and C k , leaving only two propagating dofs (around a flat background spacetime). Moving from the ghost-free theory of a massless to that of a massive spin-2 field by including the dRGT self-interactions, the lapse and shift still enter non-dynamically leaving six potentially propagating dofs, but superficially all four shift and lapse constraints are now broken. However, it can be shown that the four N, N i eoms are not in fact independent [25]. This allows a single constraint to remain intact, which eliminates the sixth metric degree of freedom, leaving us with the correct number of dofs for a massive spin-2 field (2s + 1 = 5). When coupling matter directly to the vielbeins, in principle one must verify that enough residual gauge symmetry/constraints exist to ensure none of the D 2 components of the general vielbein are ghost-like. In the remainder of this section, we will explicitly consider the lapse and shift constraints. If all D of these are irreparably broken, then (regardless of the Lorentz gauge invariance), we expect the Boulware-Deser ghost [34] to propagate.
Mini-superspaces: Before discussing the full ADM analysis, it is instructive to consider the problem in the mini-superspace. The mini-superspace describes a consistent subset of solutions of the full ADM decomposition, which are spatially isotropic with N i and P a set to zero. Ghost-freedom in this simplified setup represents a check, which any candidate coupling must satisfy in order to have the required number of constraints in the full ADM decomposition. Explicitly, we may write the original vielbeins and the analytic expansion of E as, The α nm are fully determined onceẼ is given in terms of E (1) and E (2) . We emphasise that this implies that the α nm in general result in an explicit lapse dependence forã, i.e. all components of the effective matter vielbein in the mini-superspace decomposition then have explicit lapse dependence. As a concrete example, consider, In other words, whileã for the rank-2 construction (4.5) never has any lapse-dependence (as multiplying a string of diagonal matrices cannot mix time-time and space-space entries), it does for the rank-0 construction (4.6) (a non-trivial scalar prefactor must be a fully contracted object, and so contain both N and a).

Mini-superspaces for Matter Lagrangians
Depending on the precise matter content, the matter Lagrangian L M [g µν , Φ i ] can affect the constraint structure of the theory in different ways. It is natural to consider a number of known L M as present in the Standard Model. In this subsection we therefore collect matter Lagrangians (and their corresponding Hamiltonians) for four different kinds of fields, and consider what restrictions each implies on the allowed matter metrics. Note that we will use the shorthand for the effective matter metric g eff µν ≡g µν throughout.
Vacuum energy: First up is the action for an effective cosmological constant,Λ, (3.1), as considered in section 3 and as will be generated by matter loops, even if there is no bare cosmological constant present in the theory. Trivially this introduces no new (matter) degrees of freedom/conjugate momenta, and so the corresponding Hamiltonian is simply where we recall thatã andÑ are the scale factor and lapse of the effective matter vielbein respectively.

Scalar field:
The action for a scalar field χ, with mass M and minimally coupled tog, is Accordingly the conjugate momentum for χ is given by 9) and the Hamiltonian corresponding to the matter action (4.8) consequently reads, (4.10) Finally, in the minisuperspace, the effective metricg is diagonal and explicitly substituting in the mini-superspace decomposition (4.4) we find Yang-Mills field: Next we consider a matter Yang-Mills field. For simplicity, first consider the Abelian theory of electromagnetism minimally coupled to an effective matter metricg, The gauge field, A µ , enters as F µν = ∇ µ A ν − ∇ ν A µ , making the gauge symmetry and explicitly substituting a diagonalg via (4.4), gives a classical Hamiltonian, (4.14) The extension to a non-Abelian theory is straightforward. The U (1) symmetry is replaced by another group G, with multiple generators T A (one for each dimension of G). The above Hamiltonian is modified, where f abc are the structure constants of the symmetry algebra, and the gauge fields transform under the adjoint representation of the algebra. However, the mini-superspace lapse dependence remains precisely as in (4.14).
Dirac field: For a fermionic field, ψ, of mass m, we have a matter action [37, eqn (2. 2)], where we have suppressed the 4-component spinor indices on ψ (they are contracted together via the 4 × 4 Dirac γ matrices). In the mini-superspace, we have, where an overdot denotes a time derivative, and a prime denotes a spatial derivative. Using the conjugate momenta, π = i 2Ñψ γ 0 detẼ, we find a Hamiltonian, Resulting conditions from matter Hamiltonians: In the minisuperspace picture (4.4), a necessary condition for (Boulware-Deser) ghost freedom is that the lapse constraints be preserved by the coupling. There is no shift to integrate out, and so this amounts to the lapse N appearing linearly in the Hamiltonian as a Lagrange multiplier. Via the matter Hamiltonians (4.7), (4.11), (4.14) and (4.18) this puts a constraint on the form taken byÑ andã, i.e. the mini-superspace form of the effective matter vielbein. Therefore, to have ghost-free couplings with each of these Standard Model-like matter Hamiltonians requires Vacuum energy: Scalar or Yang-Mills: Dirac spinor field: where we recall thatÑ andã are in general functions of (N, a, M, b). Analogous equations hold with derivatives with respect to M 2 and N M . Only rank-2 constructions (for whichã does not depend on N ) can satisfy all conditions simultaneously -this can be seen by inspection of equations (4.19)-(4.21). Rank-0 and mixed constructions (for whichã is generically not independent of N ) cannot be healthy for a general H M containing all of the above fields, so cannot be used to couple to matter while respecting the weak equivalence principle. 10 In appendix B we discuss how (4.19) can also be used to motivate the split into rank-2, rank-0 and mixed matter couplings, which are all 'solutions' of (4.19).

Full ADM primary constraints for scalar field matter
Performing a full ADM decomposition for generic types of matter and general matter couplings is beyond the scope of this paper. However, here we wish to briefly sketch the full ADM decomposition of an effective metric coupling to a massive scalar field (4.10). This simple case already suggests that non-trivial coupling will break some of the constraints in the theory and hence introduce a ghost at some energy scale.

Rank-2 construction
By (3.2) the single building block rank-2 constructions (i.e. those consisting of a single string from (2.8) and not of superpositions of those strings) satisfy the following property for some label index (p) by construction. Thus we define N ≡ N (p) and √ h ≡ det e (p) a i and, in terms of those variables, write the matter Hamiltonian for a massive scalar field in a full ADM decomposition (4.10) as where p χ labels the canonical momentum associated to the scalar field χ as before. For a non-linear single string coupling as in (2.8), we can separate the free spacetime index explicitly,Ẽ 10 Note that, if we were to temporarily ignore the vielbein origins ofÑ andã and treat them as independent functions of N, M, a, b, one could find a number of solutions satisfying subsets of matter Hamiltonians. For example, in our construction the coupling to vacuum energy is guaranteed, and in addition we could impose the conditions, but clearly we may not impose these simultaneously.
where κ is constructed from a string of vielbeins. Considering the square brackets of (4.24), we find 11 , The non-linearity in the shift (1 + κ 0 B κ 0B )∂ i χ∂ j χN i N j threatens to re-introduce a Boulware-Deser ghost 12 . This is because, schematically, the shift equation of motion gives, where the power t is not even necessarily integer, let alone unity or zero. If κ does not contain any N j dependence (and is not the identity), then it must mix shifts from the other vielbein species with N i . In both cases, when N i is integrated out, the resulting Hamiltonian has a non-trivial dependence on the remaining shifts. It is integrating out all these shifts, which then generically introduces a non-linearity in the lapses, eventually leading to the propagation of (an) extra ghost-like degree(s) of freedom.
For the matter couplings of [7] -a linear superposition of vielbeins, which in the presence of a symmetric vielbein condition corresponds to the metric construction of [6] - [9,14] concluded that there is a ghost on integrating out the shift as well. 13 As single non-linear vielbein strings are typically ghostly, and at least linear superpositions also introduce a ghost, this suggests (although of course it does not prove) that all non-minimal rank-2 constructions contain a ghost when coupled to a scalar field outside of the mini-superspace. In the following section 5, we shall show that this ghost lies above the Λ 3 decoupling limit scale for the rank-2 constructions and a valid low-energy effective field theory consequently exists, where the mass of the ghost or the energy scale where unitarity breaks down (whichever is lower) will set the cutoff of the theory.

Rank-0 construction
Using the rank-0 construction (3.3), we have that (4.10) becomes, (4.29) Unless this expression is linear in the lapse, or can be made so by a field redefinition, then we have a propagating potentially ghost-like dof again. Outside of the mini-superspace, both 11 Here E (s) is in an upper triangular gauge, P (s) = 0. 12 Unless of course κ B A = δ B A (and we have a healthy minimal coupling) or ∂iχ = 0 (and the matter Lagrangian becomes that of an effective cosmological constant, for which all our couplings are healthy by construction). 13 In this case, the Hamiltonian is superficially linear in lapse and shift, however when integrating out the shift, non-linearities in the lapse are induced. This is more subtle than the emergence of a ghost identified for the more complicated (2.8) couplings, which leads to non-linear shift dependence immediately in the Hamiltonian.
α andM generally have complicated dependences on N, N i , enforced by the condition (3.3). At least one of the Hamiltonian coefficients will then be irreparably non-linear in lapse and shift, (4.30) As these three coefficients multiply different orders of χ and p χ , they cannot cancel among themselves. The constraints are broken and the BD ghost is reintroduced. The special case of a diagonalM warrants attention (for example, if n (p) = 0 andM = η), as then we have, This is manifestly healthy for M = 0, D = 2. In Section 5, we will see that this is consequence of conformal symmetry (a massless scalar field in D = 2 is conformally invariant [38]). However, in the absence of conformal symmetry, all the rank-0 constructions appear to suffer from a ghost when coupled to a scalar field. In the following section, we shall show that this ghost is already present in the Λ 3 decoupling limit (for rank-0 constructions), and is therefore fatal even in a low-energy EFT sense.

Mixed construction
The ghost which exists in the scalar field mini-superspace (4.11) for a mixed construction coupling (cf. equations (4.19)-(4.21)) cannot be healed by introducing shifts or momenta (leaving the mini-superspace). Therefore without performing the full computation, we can immediately conclude that a mixed construction will have a ghost when coupled to a scalar field.

The Λ 3 decoupling limit
Given that every non-minimal coupling we have considered so far introduces a ghost in the full ADM decomposition when coupled to a simple massive scalar field, a most pressing question becomes: at what scale does this ghost enter? If the ghost enters at a large enough scale (ideally above the strong coupling scale of the theory), a healthy low-energy EFT below that scale is still of interest. If, on the other hand, the ghost is already present at very low scales, one should discard the theory in question.
In order to answer this question in the context of general matter couplings, we now investigate the Λ 3 decoupling limit (for the different matter couplings considered so far). For simplicity we will set M (1) = M (2) ≡ M Pl and work in D = 4 throughout this section, but the conclusions readily generalise. This calculation was carried out for the linear rank-2 linear couplings in the metric formulation by [6], and here we extend their approach to the entire set of candidate couplings discussed here. The Λ 3 decoupling limit corresponds to performing the following scaling of coupling constants in the theory Since we are primarily interested in whether the matter coupling reintroduces a Boulware-Deser like instability, we will focus on the gravitational dof and hence not scale the matter Scale Pure spin-2 interactions Matter-spin-2 mixing The presence of the dimensionful overall coupling constant m 2 M 2 Pl in front of the pure spin-2 interactions and the lack of such a coupling constant in front of the matter action is the reason for the shifted hierarchies. All matter fields Φ i minimally couple to the effective metricg and do not scale with m or M Pl . In ghost-free massive (multi/bi)-gravity theories, the interactions of the first two rows vanish up to total derivatives (denoted by brackets in the table), so the least suppressed non-linear interactions that enter from all parts of the action all come in at Λ 3 .
fields. 14 If interactions in the Λ 3 decoupling limit are healthy (i.e. ghost-free), then the model in question is a consistent EFT (at least) up to the scale Λ 3 . Whether the theory is ghost-free beyond that scale is only a meaningful question if the cutoff of the theory is parametrically larger than Λ 3 .
In order to make the interactions of different dofs more explicit, we will employ the Stückelberg trick, that results in an equivalent formulation of the theory with more fields and more symmetry, but the same propagating dofs (the Stückelberg trick is essentially the inverse of gauge-fixing -for details in the context of massive, bi-and multi-gravity see [29,39]).

2)
14 Technically this means we keep all further coupling constants that only appear in the matter sector fixed. Note that one may be interested in considering different scaling/decoupling limits to zoom in on interactions at different scales. For example the scale of the high-energy ghost (that lives above the Λ3 decoupling limit and is hence not dangerous) in the matter couplings of [6,7] could in principle be identified in this way and the same is true for the matter couplings we consider throughout this paper. See [18] for a more detailed related discussion. We thank Claudia de Rham and Andrew Tolley for explaining this point to us. 15 For a discussion of various equivalent ways of introducing Stückelberg fields see [29]. This redundancy is directly related to the existence of Galileon dualities [41][42][43] and their generalisation to Multi-Galileon dualities [39].
After performing the Stückelberg trick, in the decoupling limit h, A and π capture the dynamics of the helicity-2, -1 and -0 modes respectively. The scaling of A and π in (5.3) is set by the coupling constant in front of the graviton potential interactions and their canonical normalisation. The scaling of λ is set by the requirement that it enters in the decoupling limit in a non-vanishing and non-divergent way [32] -note that this results in the same scaling for λ as for A. Performing the Stückelberg trick also effectively restores the copies of Lorentz and diffeomorphism invariance broken by the massive interaction in the original action. Finally, we will also expand both vielbeins around a (flat) reference vielbein

Scalar, vector and tensor modes
With a potential interaction term like (2.5) the least suppressed non-linear interactions arise at the scale Λ 3 . For our matter couplings to at least give rise to a valid low energy effective field theory with non-trivial dynamics for the graviton, it is therefore important to check that the matter coupling is healthy in the corresponding decoupling limit. In the Λ 3 decoupling limit the helicity-0, -1, and -2 modes transform as scalar, vector and (rank-2) tensor modes respectively. We will now consider which of these modes contribute to decoupling limit contribution of the matter action and what that contribution is.
Hierarchy of modes and decoupling limit vielbeins: Non-linear interactions of the different helicity modes always enter in the same hierarchy. For a given overall order n in the fields, pure helicity-0 interactions π n enter at the lowest energy scale, followed by Aπ n−1 interaction terms which contribute at the next lowest energy scale and which in turn are followed by hπ n−1 and A 2 π n−2 interaction terms. In particular, this is the case for ghost-free massive (bi-)gravity potential interactions, where the first two sets of terms are tuned to vanish up to total derivatives and the last set of terms comes in at the scale Λ 3 . The Λ 3 terms then are the least suppressed non-vanishing non-linear interactions in these models. When considering a coupling to matter with non-scaling matter fields (as we do here), the hierarchy of scales is still completely set by the scaling of gravitational (helicity) modes. However, two important differences arise. Firstly the graviton potential interaction comes with a m 2 M 2 Pl coupling constant, whereas our matter action has no such m-dependent coupling constant. This means identical interactions coming from potential vs matter parts of the action can enter at different scales. Secondly, interactions which were tuned to vanish up to total derivatives in the pure spin-2 case, e.g. pure helicity-0 π n interaction terms, can become relevant when mixing with matter dofs. For example, π n Φ m i terms no longer vanish due to the mixing with matter fields Φ i . These two observations taken in conjunction mean that, classically and in the absence of an explicit cosmological constant term in the matter action, the non-linear interactions that survive the Λ 3 decoupling limit are hπ n−1 and A 2 π n−2 terms from the spin-2 potential part of the action and π n Φ m i from the matter part of the action.
No other terms contribute at Λ 3 and in particular helicity-1 and helicity-2 modes mix with matter fields only at higher energy scales (beyond the decoupling limit). These findings are summarised in table 2. Note that, if we were to introduce an overall m-dependent coupling constant for the matter action, this would either make the matter contribution to the Λ 3 decoupling limit vanish or it would introduce non-linear graviton interactions below Λ 3 , in which case one would have to re-do the decoupling limit analysis considered here (non-vanishing non-linear interactions below Λ 3 diverge in the Λ 3 limit). Having non-scaling matter fields in the sense discussed above is consequently a straightforward way to have matter fields enter the Λ 3 decoupling limit in a non-vanishing and non-divergent way. When computing the matter action contribution to non-linear decoupling limit interactions, we can therefore suppress most of the involved helicity dependence of the vielbeins post-Stückelberging (since this will drop out by the arguments presented above) and instead just work with the 'matter decoupling limit' vielbeins For an effective matter vielbeinẼ = αE (1) + βE (2) , the 'matter decoupling limit' of that effective matter vielbein would consequently beẼ MDL = (α + β)I − βΠ, in matrix notation.
Helicity-0 mode eoms: Seeing as there will be explicit helicity-0 dependence of the decoupling limit matter action, one has to ensure that the dynamics of this mode remains ghost-free in this limit while taking into account the new (matter action) contributions. The equation of motion for the helicity-0 mode π in this limit takes on the schematic form In the absence of a coupling to matter, the uncoupled dRGT or Bigravity theory yields the ghost-free equations of motion, where the π eoms are those of a Galileon on flat space [44]. Once a coupling to matter is included, the additional contribution from the matter sector may be written, In general, the effective energy-momentumT µν will depend on derivatives of π, and one might expect that the π eom will consequently depend on higher-than-second-order derivatives of π, generically introducing an Ostrogradski ghost. While this is true for arbitrary couplings, we shall see that rank-2 and (to some extent also) rank-0 matter couplings investigated here have interesting structure, that allows them to evade the appearance of an (Ostrogradski) ghost in the Λ 3 decoupling limit.

Symmetric vielbein condition in the decoupling limit
We have already argued above that the 'matter decoupling vielbeins' (by which we mean the piece of the matter vielbein which survives in the Λ 3 decoupling limit) take on a vastly simplified form, with the Lorentz Stückelberg fields, helicity-1 and helicity-2 modes all dropping out. Here we explicitly show this for some specific couplings, demonstrate how this is related to the restoration of the symmetric vielbein condition in the decoupling limit and what this means for superpositions of the vielbein building blocks (2.8) discussed in this paper.
Matter vielbeins in the decoupling limit: To begin, let us consider three specific effective matter vielbeinsẼ We now perform the Stückelberg replacements (5.2), (5.4) on these matter vielbeins. In order to do so, it is useful to write out the inverse vielbeins (up to quadratic order in the fields) As far as the matter decoupling limit is concerned (in fact scaling M Pl → ∞ is sufficient), this means the inverse vielbeins in this limit become Using the explicit examples from (5.10), we can also see that performing the decoupling limit scaling in the full effective matter vielbeins has the same effect as substituting using (5.13), as it should. Upon performing the Stückelberg replacement, the three specific matter vielbeins from (5.10) become (up to quadratic order in the fields) In the Λ 3 decoupling limit the effective matter vielbeins for the three cases considered therefore becomeẼ This agrees with the expressions shown in (5.6), suggesting that the matter decoupling limit version of all the different vielbein building blocks reduce to the matter decoupling limit version of just the linear superposition of 1-vielbein strings, i.e.Ẽ = αE 1 + βE 2 . We will see precisely why this happens when discussing the symmetric vielbein condition below.
Restoration of the symmetric vielbein condition: When inspecting the matter decoupling limit vielbeins in (5.17), it is important to notice that the Lorentz Stückelberg field λ completely drops out of the vielbein. Consider what this means for the Lorentz Stückelberg eom. This can schematically be written as Now the kinetic part of the action, the two Einstein-Hilbert terms and their linearised pieces that survive in the decoupling limit, are gauge-invariant, so never contribute to this eom.
What the above scaling arguments show is that, in the decoupling limit, neither does the matter part of the action, so that the decoupling limit version of the λ eom is while the general eom is given by The λ eom in the decoupling limit is therefore insensitive to the difference between any of the rank-2 matter couplings (including a minimal coupling) considered throughout this paper. As a result, the well-known result for standard massive (bi-)gravity -that the symmetric vielbein condition is recovered as a consequence of the λ eom 16 -still holds in the decoupling limit version of the theories considered here and equivalence with the metric formulation is obtained as a direct consequence as well. As (5.20) shows, this will not be the case beyond the decoupling limit. 17 More specifically the symmetric vielbein (or DvN condition) for two vielbeins takes on the form For details on the derivation of this condition from the Lorentz Stückelberg eom and on why this leads to equivalence between the metric and vielbein formulations, we refer to [32].
What is important here is that the commutativity enforced by (5.21) collapses the building blocks of (2.8) to their simplest form. Consider the following examplẽ where we have used the symmetric vielbein condition in going to the second line. Similarly all the building blocks of (2.8) collapse in the same way when the symmetric vielbein condition is enforced In other words, all the vielbein building blocks considered collapse to a single vielbein E (p) , so coupling to an effective vielbein made up of a single one of any of these building blocks in the decoupling limit reduces to a minimal coupling to E (p) .
Superpositions: Earlier we alluded to superpositions of the different vielbein building blocks (2.8), but did not consider them in any further detail in sections 3 and 4. The considerations in this section now show that such superpositions are in fact also healthy in the decoupling limit. This is because, from (5.23), we have that any superposition of the matter vielbein building blocks (2.8) in the decoupling limit will take on the form of the (matter decoupling limit version of the) linear vielbein superposition of [7] which is known to be healthy at least up to the Λ 3 scale [6,14]. The linear vielbein coupling consequently emerges as the unique form taken by all the rank-2 matter couplings considered in this paper in the decoupling limit, with differences between the different matter couplings only becoming relevant outside of the decoupling limit when the symmetric vielbein condition is no longer enforced (since the λ eom now is (5.20) and no longer (5.19)). Superpositions of rank-0 couplings (3.3) in the decoupling limit are also particularly simple. Using the commutativity of E (1) , E (2) , from whichM is formed, we can annihilate all inverse vielbeins to be left with simply, where we recall that α is a scalar function of the two vielbeins here and all indices are contracted with η µ and η AB . Superpositions of such couplings simply redefine α.

Rank-2 construction
As discussed above, the symmetric vielbein condition allows us to commute the vielbeins E (1) and E (2) , and therefore any single vielbein string satisfying the valence condition (3.2) may be reordered so that all inverse vielbeins annihilate, leaving only a single E (1) or E (2) in the case of bigravity. Closely following the metric formulation argument of [6], we therefore here show that this coupling is healthy in the vielbein formulation. Allowing for the general linear vielbein superposition (5.24), we havẽ Taking the covariant derivative of the stress energy tensor (with respect to the effective metric), we find [10,11], where the Christoffel symbols are given by Γ α µν = −g βν ∂ µẼ β CẼ α D η CD [45]. In the decoupling limit, this gives, from (5.26), All rank-2 couplings, and their superpositions (as discussed above and in section 2), are therefore healthy at and below Λ 3 when coupled to an arbitrary L mat which does not scale with m or M Pl .
Covariant conservation of T µν : It is worth emphasising that the covariant conservation of the stress-energy tensor with respect to the effective matter metric (5.27) follows as a consequence of the weak equivalence principle-respecting matter coupling, and does not rely on any further details of the matter coupling. Take the following matter Lagrangian for explicitness The corresponding matter eom, i.e. in this case simply the eom for χ, is given by [6] The key observation is that the only way in which χ knows about the gravity sector is via the minimal coupling to g eff . So regardless of what form g eff takes, whether it is Stückelberged or not, whether it is replaced by a different effective matter metric or not, (5.30) will remain valid with respect to the effective metric that matter is minimally coupled to. We have explicitly shown this for a single scalar field here, but none of the reasoning depends on this assumption.

Rank-0 construction
Suppose the effective matter vielbein coupling takes a rank-0 form (3.3). In the decoupling limit, the effective vielbein becomes (5.25), and we now have This no longer vanishes in general on-shell. However, in special cases where the energymomentum is traceless, (5.31) trivially vanishes and the coupling (5.25) consequently does not introduce a ghost. Note that systems with traceless energy-momentum are scale invariant, and (together with a virial condition [38]) this implies conformal symmetry. While this means the rank-0 coupling is disqualified as a universal coupling for all matter, let us briefly consider a coupling to a system with conformal symmetry. As a concrete example consider a Klein-Gordon scalar field φ minimally coupled tog. This corresponds to the following matter Lagrangian, This shows that general scalar fields will not have vanishing T µν trace, and so a scalar field coupled to gravity via the rank-0 coupling as discussed here will generically give rise to a ghost, even in the Λ 3 decoupling limit. For a Klein-Gordon scalar the exception is now readily seen to be a massless scalar field around flat (1+1)-dimensional space-a well-known example of conformal symmetry. However, coupling individual matter species (in some given dimension) to gravity in this way, whilst others have to be coupled in some different way, of course does away with the weak equivalence principle altogether. Such an approach was already partially discussed in [18], but we shall leave fully exploring the consequences of opening this Pandora's box for future work. For general matter, without a traceless energy-momentum tensor, the non-vanishing contribution to the π eoms will generically be higher-derivative, introducing an Ostrogradski ghost. This is the case because ∂α/∂Π generically depends on Π and (5.31) consequently depends on up to fourth derivatives acting on π, leaving us with the conclusion that rank-0 matter couplings are generically ghostly in the decoupling limit, even if their 'effective cosmological constant'/one-(matter)loop contributions are ghost-free (see section 3).

Mixed construction
Mixed constructions do not have any helpful index structure, and typically (for general α and superpositions) have an Ostrogradski ghost in the helicity-0 components of the metrics below Λ 3 , The set of matter Lagrangians to which the mixed constructions can healthily couple is given by a restricted set of stress-energy tensors, namely those for which, is at most second order in field derivatives in the decoupling limit. This is more restrictive than either the rank-2 set (all L M ) or the rank-0 set (conformal L M ), and employing such a coupling would require doing away with the weak equivalence principle even more radically than in the rank-0 case. The exceptional cases are where the mixedẼ actually reduces to a rank-2 coupling in the decoupling limit, when the symmetric vielbein condition can be imposed. This occurs whenever, As a concrete example, consider, Indeed, any pair of rank-2 constructions from Section 3 can be used in place of E (1) , E (1) E (2) E −1 (1) and this kind of construction (and any constant superposition of such terms) will be healthy in the decoupling limit, as per the arguments of Section 5.4.

The Higuchi bound
Until this point we have performed various checks for the presence of a Boulware-Deser ghost in massive (bi-)gravity models with a non-trivial matter coupling. In this section we will consider the Higuchi ghost instead, i.e. we will check that the theories in question are well-defined in the sense that perturbations of the helicity-0 mode π are well-defined around cosmologically motivated FRW backgrounds. This was checked for minimally coupled bigravity in [40] and we extend this analysis to the case of the general matter couplings considered in this paper. The Higuchi bound, representing the condition for the absence of a Higuchi ghost, can be derived in various ways, namely via a full Hamiltonian analysis, via the mini-superspace or via a decoupling limit analysis 18 . Here we closely follow the minisuperspace approach of [40] in order to check whether one may simultaneously satisfy the Higuchi bound [46] and have a working Vainshtein mechanism [47,48] and what conditions this imposes.
In general there is a tension between these two requirements. This is because the Higuchi bound relates the value of the Hubble parameter H to the graviton mass m, while simultaneously the observed proximity to a GR-like solution (the Vainshtein mechanism allows this to take place in massive (bi-)gravity theories) on large scales means H-dependent corrections to the Friedmann equations should be small for large H. Generically the resulting bounds can be incompatible -indeed this is the case for FLRW solutions in massive gravity [49] so it is important to verify whether the bigravity theories discussed here can simultaneously be free of Higuchi ghosts and lead to acceptable cosmological (background) evolutions.
In this section we will consider two vielbeins, E (g) and E (f ) , in the D = 4 dimensional FLRW ansatz, The corresponding Hubble rates associated with each metric are then, This ansatz allows us to use the mini-superspace vielbeins (4.4) as discussed in Section 4. We recall that there we found that general rank-2 couplings are potentially healthy, rank-0 couplings may only be healthy if the matter has traceless stress-energy, and mixed constructions are generally not healthy (so will not be considered in this section). Schematically, one can consider (on introducing the Stückelberg fields as in Section 5) expanding to second order in helicity-0 perturbations about FLRW, where F is a function which depends on the choice of theory. Requiring that these perturbations in π are not ghost-like immediately gives the Higuchi bound, However, by restricting attention to a certain region of (H g , H f ) parameter space, it may be possible to remove the m dependence from the Higuchi bound-removing the Higuchi-Vainshtein tension in the process. This is certainly the case for massive bigravity with a simple single-metric coupling to matter [40]. In order to have valid FLRW solutions, both this bound and a complementary condition on m from observed cosmic expansion history must be satisfied. The latter requires a continuity with GR, which is only possible in massive gravity with a Vainshtein mechanism. [49] discusses a tension between these two requirements (Higuchi bound and Vainshtein regime) for massive gravity in detail, concluding that there are no stable and empirically viable FLRW solutions (for massive gravity). We will study how the form of F , and the allowed region of (H g , H f ) parameter space, varies when our different matter couplings are used as part of the action, β n e n E −1 where V 3 is a constant spatial volume, e n are the usual characteristic symmetric polynomials. For generic β parameters interesting distinct regimes where the bound is satisfied can be found. Exploring the full Higuchi parameter-space is beyond the scope of this paper (we do so in [50]). But to illustrate that several of the couplings discussed throughout this paper can be healthy, we here focus on models with β 0 = 0 = β 4 (hence the sum in (6.5) only runs 1 − 3 in other words, we set the cosmological constants for each metric/vielbein to zero). We emphasise that (6.5) this is simply rewriting (2.6) in order to enable an easier comparison with the derivation of [40]. In FLRW variables (6.1), this action becomes, where the dRGT β n coefficients enter as, For different D, these are simply modified. Note that we will define the dimensionless field, e χ ≡ b/a, such that δχ ∼ δπ. For details on how χ is related to the Stückelberg procedure and why it captures kinetic interactions of (perturbations of) the helicity-0 π in this way, we refer to [40]. In what follows we will now explicitly consider a matter Lagrangian for a free scalar field, 00φ2 if homogeneous, (6.8) and note when our results are independent of this choice.

Minimal matter couplings: A warm-up
We first briefly recap the derivation of the Higuchi bound for a minimal coupling of matter to a single metric/vielbein. We closely follow the derivation and notation of [40]. For a minimal couplingẼ = E (g) , We can then replace time derivatives with the canonically conjugate momenta (going to a Hamiltonian formulation), (6.11) The lapse constraints then correspond to the Friedmann equations, and can be used to integrate out p a and p b . The equation of motion for p φ is then, which can be used to integrate p φ out also. This leaves, in the convenient gauge a ∝ t, where g is a dimensionless field proportional to the matter energy density (not to be confused with any property of g µν ).
We can now perturb the action to second order, Note that ∂L/∂ġ = ∂ 2 L/∂χ 2 = 0, and that the 2(∂ 2 L/∂χ∂χ)δχδχ term can be integrated by parts, giving, This can be diagonalised by the linear, invertible field redefinition, δg = δḡ − ∂ 2 L/∂χ∂g ∂ 2 L/∂g 2 δχ, (6.17) giving the desired Higuchi bound, as we require the sign of the kinetic fluctuations for δπ to be positive. In order to write F solely in terms of (m, H g , H f ) we need to solve for χ(m, H g , H f ). The field equation of motion is, where the Y (n) µν are given in [10]. On taking the covariant derivative with respect to E (g) and imposing stress-energy conservation and the Bianchi identity, this gives [19], Further, we can absorb one of these constant coefficients into the definition of the fields in L mat , and consider the couplingẼ = E (1) + c 2 E (2) . This gives a matter action, As before, we go to a Hamiltonian formulation in order to integrate out the momenta, (6.29) The lapse constraints give Friedmann equations, and can be used to integrate out p a and p b . The equation of motion for p φ is then, in the gauge a ∝ t, which can be used to integrate p φ out also. This leaves, where g again is a dimensionless field proportional to the matter energy density (not to be confused with any property g µν ).
Before we perturb the final version of the action (with p φ integrated out), let us consider the field equations, T αβ δg αβ δg µν (6.33) where the Y (n) µν are given in [10]. On taking the covariant derivative with respect to E (g) and using stress-energy conservation and the Bianchi identity, this gives [19], Other regions may also admit stable FLRW solutions or interesting phenomenology, but we content ourselves with demonstrating that, for M f > |β 1 |/3 M g , there are at least some viable solutions. (6.35) shows that, for certain choices of H f /H g , a matter (g) dependence or a singular c 2 → 0 limit is introduced.
By considering the relative order of each of these terms, as well as the diagonalisation (6.17), we find that to leading order 22 , where, This gives, to leading order, the same Higuchi bound as the minimally coupled case (6.25), However, at next-to-leading order, this is a new result, with an explicit matter dependence entering viaχ. Note that the next-to-leading-order term vanishes in the c 2 → 0 limit, and so has not appeared in previous calculations of this bound for singly coupled matter. While this term (and the other parameter space regions we have not considered) may have interesting physical consequences, we will postpone a detailed discussion to future work. Significantly, in the context of our new matter couplings, we have shown that there exists a region of parameter space (H f H g , M f > |β 1 |/3M p ) in which our new rank-2 couplings may be expanded about FLRW without an Higuchi ghost interfering with a Vainshtein mechanism and a smooth GR m → 0 limit. 22 Note that the χ dependence of g (6.32) is subleading order, (6.40) and so the second order expansion of the action (6.15) is still valid.

Rank-0 constructions
Any rank-0 coupling in the FLRW ansatz may be written as, The field equations of motion are, (and analogously for the E (f ) variation). Note that the term proportional to the stress-energy no longer preserves the Mȧ = Nḃ branch studied in the rank-2 case, unless the stress-energy is traceless, in which case its contribution vanishes, and we are left with the same Bianchi constraint (6.20) as in the minimal coupling case. We will consider such a traceless energymomentum, and focus once more on the branch (6.21). Using the free scalar field (6.8) requires D = 2 in order to have conformal symmetry. In this case the Higuchi bound is always trivially satisfied because a massive spin-2 field carries no degrees of freedom, the theory is purely topological. Staying in D = 4 we instead consider the Maxwell action for a time-dependent vector field A µ (t) in temporal gauge A 0 = 0, and so once again we can change to canonical momenta, (6.48) Note that the matter sector has completely decoupled from the metric fields, and will therefore evolve independently. The calculation of the Higuchi bound then goes through exactly as before, this time with the trivial condition P 2 = 0. As such the rank-0 couplings (with β 0 = 0) may produce stable FLRW solutions in the specific case where this coupling is only to external matter with traceless energy-momentum (as would be implied by conformal symmetry). However, the ghost identified in the minisuperspace will enter in this kind of FRW setup whenever T µν is not traceless -and so, as discussed above, use of this coupling amounts to a violation of the strong equivalence principle.

New couplings and uniqueness
Arguments for the uniqueness of the matter couplings of [6,7] were recently presented in [14,15,17,18]. So how do the new couplings presented here fit into these arguments? Two aspects of the uniqueness arguments are important to highlight in this context and in order to explain how our new couplings escape these arguments: Metric vs vielbein formulation: In [15,17] the matter couplings of [6,7] were argued to be unique in the metric formulation, in the sense that no other matter couplings remain ghost-free at least up to the scale Λ 3 . All the new consistent couplings we have discussed in this paper (i.e. the rank-2 constructions) reduce back to the couplings of [6,7] whenever the symmetric vielbein condition is imposed, as we have shown above. In other words, when the symmetric vielbein condition holds and consequently an equivalent metric formulation exists for our couplings, our results are perfectly compatible with the claims of [15,17]. Only when an equivalent metric formulation ceases to exist, i.e. in settings where the analyses of [15,17] do not apply, do our new couplings become different.
Jordan frame: The linear vielbein coupling has the interesting feature that the associated field re-definition which maps the action from Einstein to Jordan frame -a linear field redefinition of the vielbeins -also leaves the form of the potential-like bigravity interactions invariant [10,55]. In other words, this field re-definition changes the values of coupling constants in the non-derivative bigravity interactions, but not their overall form. If (a superposition of) the vielbein building blocks in (2.8) is used, this will of course no longer be the case and the Jordan frame transformation will no longer leave the potential interactions invariant. This was pointed out in [14,18]. It is important to keep in mind that, in order to determine where in the energy spectrum, if anywhere, any potential ghosts are located, one now has to take into account the nature of the modified potential as well as of the now modified kinetic terms (the Jordan frame transformation is not a symmetry of the Einstein-Hilbert terms). The non-standard form of the potential interactions in the Jordan frame is not enough to conclude that another coupling is ghostly (at least up to the scale Λ 3 ). This is the case if one works in the Jordan frame -when working in the Einstein frame kinetic and potential interactions of course take on their usual form here. In any case, as we have explicitly shown above, the new rank-2 couplings introduced here only lead to differences to the couplings of [6,7] outside of the decoupling limit, so that the decoupling limit proof of the healthiness of the couplings of [6,7] in [6] also covers the couplings considered here (given that the metric and vielbein formulations are identical in this limit as discussed in sections 2 and 5).

Conclusions
In this paper we considered different ways of coupling matter to spin-2 fields. Until recently the minimal GR-like coupling of a metric/spin-2 field was the only known consistent way to do so without introducing extra dofs by hand, i.e. the only known consistent way of universally coupling a spin-2 field to all other fields in the theory. The recently proposed couplings of [6,7] add to the space of consistent couplings in a low-energy EFT sense -they are consistent couplings at least up to the scale Λ 3 .
Here we have systematically considered a variety of further couplings constructed orderby-order in the vielbein formulation. In particular we have identified a family of new consis-tent universal matter couplings where matter minimally couples to a metric where the effective matter vielbeinẼ is an arbitrary superposition of any of the following terms . . . (8.2) and so on. Here (p), (q) are simply label indices and the individual terms are all built from the two vielbeins E (1) and E (2) in the theory. As such, in index-free notation we can write the effective matter-coupling vielbein as This extends the known consistent matter couplings of [6,7], which took the formẼ eff = i α i E (i) and hence were the lowest order piece of (8.3). Matter couplings of this type, as presented in this paper, have the following key features • They are healthy at least up to the energy scale Λ 3 , as shown by the Λ 3 decoupling limit analysis in section 5, and are therefore part of a consistent low-energy EFT. In this limit, the symmetric vielbein is also restored and as a result matter couplings of the type (8.3) effectively reduce to the consistent couplings of [6,7] in this limit.
• In the massive gravity limit, where only one vielbein is dynamical, (8.3) automatically collapses back to the construction of [6,7], also beyond any decoupling/scaling limit. Extensions of our matter coupling construction to Multi-Gravity theories with more spin-2 fields are straightforward and discussed explicitly in appendix A.
• The effective cosmological constant generated by a matter coupling of this type, and consequently matter loop corrections to this coupling, generates healthy dRGT-type pure spin-2 interactions, whenever the effective vielbein is made up from just a single of the building blocks in (2.8). Equivalently, this is the case when only one of the coefficients in (8.3) is non-zero. For general superpositions like (8.3), this will not be the case. However, by the decoupling limit argument above, this should only significantly affect the theory above Λ 3 .
• The Higuchi bound for these couplings can be satisfied in a certain region of parameter space (for a more complete analysis see [50]) and in the space considered in this paper the bound agrees with that derived for minimal couplings [40] to leading order. At next-to-leading order we do find modifications (6.43), however.
• A minisuperspace analysis for these couplings shows them to be healthy, also indicating that healthy FLRW solutions exist at the background level. A full constraint analysis was not carried out here, but, just as for the linear vielbein coupling of [6,7], we expect there to be a ghost-like instability at some scale Λ ghost > Λ 3 , effectively imposing a cutoff for the theory.
In addition to the couplings (8.3), which we dubbed 'rank-2' couplings, we also briefly investigated what we called 'rank-0' and 'mixed' couplings, essentially effective vielbein couplings where there is an overall scalar function of the gravitational dofs multiplying some rank-2 object. For details we refer to sections 3-6, but to summarise we found that matter cannot be coupled to these constructions universally without introducing a ghost already at the Λ 3 scale and in the minisuperspace, unless the couplings reduce back to the rank-2 constructions in this decoupling limit (in the mixed case), or if the coupling is restricted to particular types of matter with conformal symmetry (in the rank-0 case). These findings raise several interesting questions for future research. First and foremost the cutoff scale Λ c for this whole class of spin-2 field theories coupled to matter has to be established along the lines of [18] in order to know how far beyond Λ 3 they represent a valid low-energy EFT. If Λ cut Λ 3 , it will be of great interest to investigate the phenomenological differences between the different couplings (8.3) presented here. In general this is also true of differences between the vielbein and metric formulation of massive (bi)-gravity theories with non-trivial matter coupling, as discussed throughout this paper and in [13,14]. Finally it would be interesting to prove whether the couplings presented here really are the unique set of couplings in the vielbein language that yield consistent field theories up to at least Λ 3 , or if further extensions exist (and if so, where they differ). For example, it may be possible to find additional matter coupling terms that simply vanish in the decoupling limit and only contribute above it. Answering some (or all) of these questions should move us closer to fully understanding the space of consistent spin-2 field theories, and in the process improve our understanding of the way gravity works.

A Extension to multi-gravity theories
Given that we have been working in the vielbein formulation throughout, a generalisation of our matter couplings to setups with N spin-2 fields is straightforward. For potential interactions the full consistent multi-gravity generalisation of the ghost-free potential massive and bigravity interactions was presented in [27] -features of multi-gravity theories are further discussed in [27,29,39,[56][57][58][59][60][61].
In the multi-gravity case the effective matter vielbein that matter can couple to is made up of the following building blocks just like in (2.8), except that, at higher than cubic order in the vielbeins and their inverses, we can now have more than two species of vielbeins. These are labelled by p, q, r, . . . and, except for p in the above expressions have to satisfy n q −n q = 0, n r −n r = 0 and so on, where n i labels the number of vielbeins E (i) in the building block andn i labels the number of inverse vielbeins E −1 (i) in the building block. We impose this restriction, because with it the mini-superspace action will be linear in the lapse associated with (p), N (p) . Otherwise we will have a dependence of the mini-superspace action on N n (p) M m (p) O o (p) . . . with the restriction that all the powers n+m+p+. . . = 1, which is imposed by the index structure of the effective vielbein (essentially by requiring one free Lorentz and one free space-time index to survive). If more than one power i non-zero, the mini-superspace will already be highly non-linear in the lapses and the ghost-free constraint structure of the theory will generically be destroyed.
The nature of the decoupling limit and symmetric vielbein condition is somewhat more complicated in multi-gravity. The decoupling limit was investigated in [29,39] and the symmetric vielbein condition is known to be broken in the presence of 'cycles of interactions' [14,60], leading to an inequivalence between the metric and vielbein formulation even at the level of the decoupling limit. In fact, in the metric formulation, they even lower the scale of the least suppressed non-linear interactions [60]. In the absence of such cycles of interactions -equivalently: loops in the theory graph -we expect the symmetric vielbein condition to be recovered. When building a tree graph of interactions for a multi-gravity theory with only bigravity type interactions, this is straightforward to see [27] and each interaction gives the relevant symmetric vielbein condition. Proving this statement rigorously in the general case, i.e. that the symmetric vielbein condition between any pair of vielbeins is recovered for any superposition of N-vielbein interaction vertices in the absence of interaction cycles, we leave for future work, however. Here we conjecture that a matter coupling using an arbitrary linear superposition of building blocks (A.1) of the form will lead to a multi-gravity theory with a healthy Λ 3 decoupling limit (for subtleties in defining such a limit and why it is no longer unique in multi-gravity, see [29]). i, j, k, . . . here label all the N vielbeins.
These are our rank-0 constructions. Finally, in D = 1 or D = 2, the polynomial in (B.4) has roots which are independent of b, and therefore we can construct solutions like, But otherwise, the coefficients c pn must be carefully (highly non-trivially) tuned to allow cancellation of the sum (B.4) for all b. These are our mixed constructions.