Dipolar Dark Matter as an Effective Field Theory
Luc Blanchet Lavinia Heisenberg







   
     
       
  















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                                        ∇^2 FM_ susyḍ PΣσ G K L^(2)Ł* L_*Łℒ() i.e.  et.al. ∂_μνStückelberg ∂^μ_ ν⟨⟩ℰ̂ helicity∫^̣4k/(2π)^4k^2+m̃^2k^2+m^2 Trluc.blanchet@iap.fr𝒢ℝεℂ𝒪
                                          Institut d'Astrophysique de Paris — UMR 7095 du CNRS,
                                           Université Pierre & Marie Curie, 98bis
                                          boulevard Arago, 75014 Paris, Francelavinia.heisenberg@eth-its.ethz.chInstitute for Theoretical Studies, ETH Zurich, 
                                        
                                         Clausiusstrasse 47, 8092 Zurich, Switzerland
                                        Dipolar Dark Matter (DDM) is an alternative model motivated by the
                                        challenges faced by the standard cold dark matter model to describe
                                        the right phenomenology at galactic scales. A promising realisation of
                                        DDM was recently proposed in the context of massive bigravity
                                        theory. The model contains dark matter particles, as well as a vector
                                        field coupled to the effective composite metric of bigravity. This
                                        model is completely safe in the gravitational sector thanks to the
                                        underlying properties of massive bigravity. In this work we
                                        investigate the exact decoupling limit of the theory, including the
                                        contribution of the matter sector, and prove that it is free of ghosts
                                        in this limit. We conclude that the theory is acceptable as an
                                        Effective Field Theory below the strong coupling scale.
                                        95.35.+d, 04.50.KdDipolar Dark Matter as an Effective Field Theory
                                            Lavinia Heisenberg
                                            December 30, 2023
                                        ================================================
                                        
                                        
                                        
                                        
                                        
                                        § INTRODUCTION
                                        
                                        
                                        We are witnesses of centenaries. The year 2015 marked the 100th
                                        anniversary of Albert Einstein's elaborate theory of General
                                        Relativity (GR), while 2016 celebrated the centenary of the first
                                        paper on gravitational waves by the announcement of their experimental
                                        detection <cit.>. GR meets the requirements of the underlying
                                        physics in a broad range of scales, from black hole to solar system
                                        size. It stood up to intense scrutiny and prevailed against all
                                        alternative competitors. It constitutes the bedrock upon which our
                                        fundamental understanding of gravity relies. However,
                                        
                                        some important questions remain. 
                                        
                                        The lack of renormalizability motivates the modifications of gravity
                                        in the ultraviolet (UV), that incorporate the quantum nature of
                                        gravity. The singularities present in the classical theory could be
                                        regularized by the new physics <cit.>. The UV
                                        modifications might also dictate a different scenario for the early
                                        Universe as an alternative to inflation <cit.>. The
                                        inflaton field in the standard picture might be just a reminiscent of
                                        the modification of gravity in the UV.
                                        
                                        From a more observational point of view, GR faces additional
                                        challenges on cosmological scales. In order to account for the
                                        observed amount of ingredients of the Universe, it is necessary to
                                        introduce dark matter and dark energy despite of their unclear
                                        origin. Notwithstanding of remarkable efforts, the dark matter has so
                                        far not been directly detected. Concerning the dark energy,
                                        
                                        the standard model in form of a cosmological constant Λ
                                        accounts for most of the observations even though it faces the
                                        unnaturalness problem <cit.>. Combined with the
                                        non-baryonic cold dark matter (CDM) component, the model explains
                                        remarkably well the observed fluctuations of the cosmic microwave
                                        background and the formation of large scale structures.
                                        
                                        Albeit the many successes of the Λ-CDM model at large scales,
                                        it has difficulties to explain the observations of dark matter at
                                        galactic scales. For instance, it is not able to account for the tight
                                        correlations between dark and luminous matter in galaxy
                                        halos <cit.>. In this remark, the first
                                        unsatisfactory discrepancy comes from the observed Tully-Fisher
                                        relation between the baryonic mass of spiral galaxies and their
                                        asymptotic rotation velocity. Another discrepancy, perhaps more
                                        fundamental, comes from the correlation between the
                                        presence of dark matter and the acceleration scale <cit.>. The prevailing view regarding these problems is that they
                                        should be resolved once we understand the baryonic processes that
                                        affect galaxy formation and evolution <cit.>. However, this
                                        explanation is challenged by the fact that galactic data are in
                                        excellent agreement with the MOND (MOdified Newtonian Dynamics)
                                        empirical formula <cit.>. From a phenomenological
                                        point of view, this formula accommodates remarkably well all
                                        observations at galactic scales. Unfortunately, extrapolation of the
                                        MOND formula to the larger scale of galaxy clusters confronts an
                                        incorrect dark matter distribution <cit.>.
                                        
                                        The ideal scenario would be to have a hybrid model in which the
                                        properties of the Λ-CDM model are naturally incorporated on
                                        large scales, whereas the MOND formula would take place on galactic
                                        scales. There have been many attempts to embed the physics beyond the
                                        MOND formula into an approved relativistic theory, either via
                                        invoking new propagating fields without dark matter <cit.>, or by considering
                                        MOND as an emergent phenomenology <cit.>.
                                        
                                        Here we consider a model of the latter class, called Dipolar Dark
                                        Matter (DDM) <cit.>. The most compelling
                                        version of DDM has been recently developed, based on the formalism of
                                        massive bigravity theory <cit.>. To
                                        describe the potential interactions between the two metrics of
                                        bigravity the model uses the effective composite metric introduced in
                                        Refs. <cit.>. Two species of dark
                                        matter particles are separately coupled to the two metrics, and an
                                        internal vector field that links the two dark matter species is
                                        coupled to the effective composite metric. The MOND formula is
                                        recovered from a mechanism of gravitational polarization in the non
                                        relativistic approximation. The model has the potential to
                                        reproduce the physics of the Λ-CDM model at large
                                        cosmological scales.
                                        
                                        In the present paper we address the problem of whether there are ghost
                                        instabilities in this model. The model itself <cit.> will be reviewed in Sec. <ref>. The model is safe 
                                         in the gravitational sector because it uses the ghost-free framework of massive bigravity. 
                                         The interactions of the matter fields with the effective metric reintroduce a ghost
                                        in the matter sector beyond the strong coupling scale, as found in <cit.>. 
                                        In our model, apart from this effective coupling the different
                                          species of matter fields interact with each other via an internal vector field.
                                          This additional coupling might spoil the property of ghost freedom within the strong
                                          coupling scale.
                                          We therefore investigate, in Sec. <ref>,
                                        the exact decoupling limit (DL) of our model, crucially including the
                                        contributions coming from the matter sector and notably from the
                                        internal vector field. The model dictates what are the relevant
                                        scalings of the matter fields in terms of the Planck mass in the
                                        DL. Using that, we shall prove that the theory is free of ghosts in the
                                        DL and conclude that it is acceptable as an Effective Field Theory
                                        below the strong coupling scale. We end the paper with a few
                                        concluding remarks in Sec. <ref>.
                                        
                                        
                                        
                                        
                                        
                                        § DIPOLAR DARK MATTER
                                        
                                        
                                        The model that we would like to study in this work is the dark matter
                                        model proposed in Ref. <cit.> where the Dipolar Dark
                                        Matter (DDM) at small galactic scales is connected to bimetric gravity
                                        based on the ghost-free bimetric formulation of massive
                                        gravity <cit.>. The action of a successful
                                        realisation was investigated in <cit.> and we would
                                        like to push forward the analysis performed there. The Lagrangian is
                                        the sum of a gravitational part, based on massive bigravity theory,
                                        plus a matter part: ℒ = ℒ_grav +
                                        ℒ_mat. The gravitational part reads
                                        
                                        
                                            ℒ_grav = M_g^2/2√(-g) R_g +
                                            M_f^2/2√(-f) R_f + m^2M_eff^2
                                             √(-g_eff) ,
                                        
                                        where R_g and R_f denote the Ricci scalars of the two metrics
                                        g_μν and f_μν, with the corresponding Planck scales
                                        M_g and M_f and the interactions carrying another Planck scale
                                        M_eff, together with the graviton's mass m. In this
                                        formulation, the ghost-free potential interactions between the two
                                        metrics are defined as the square root of the determinant of the
                                        effective composite metric <cit.>
                                        
                                            g^eff_μν=α^2 g_μν +2αβ 𝒢^eff_μν +β^2 f_μν ,
                                        
                                        with the arbitrary dimensionless parameters α and β
                                        (typically of the order of one). Here
                                        𝒢^eff_μν denotes the effective metric in the
                                        previous DDM model <cit.>, given by
                                        𝒢^eff_μν = g_μρX^ρ_ν where
                                        X=√(g^-1f), or equivalently 𝒢^eff_μν =
                                        f_μρY^ρ_ν where Y=√(f^-1g). It is trivial to see
                                        that the square root of the determinant of this effective metric
                                        g^eff_μν corresponds to the allowed ghost-free
                                        potential interactions <cit.>.
                                        
                                        The matter part of the model will consist of ordinary baryonic matter
                                        and a dark sector including dark matter particles. The crucial feature
                                        of the model is the presence of a vector field 𝒜_μ in
                                        the dark sector, that is sourced by the mass currents of dark matter
                                        particles and represents a “graviphoton” <cit.>. This
                                        vector field stabilizes the DDM medium and ensures a mechanism of
                                        “gravitational polarisation”. The matter action reads
                                        
                                        
                                            ℒ_mat =    -
                                            √(-g)(ρ_bar+ρ_g) - √(-f) ρ_f
                                            
                                                + √(-g_eff)[
                                              𝒜_μ(j_g^μ-j_f^μ) + λ
                                              M_eff^2 𝒲(𝒳) ] .
                                        
                                        Note the presence of a non-canonical kinetic term for the vector field
                                        in form of a function 𝒲(𝒳) of
                                        
                                        
                                            𝒳 = -
                                            ℱ^μνℱ_μν/4λ ,
                                        
                                        with the field strength defined by ℱ^μν≡
                                        g_eff^μρ
                                        g_eff^νσℱ_ρσ where
                                        ℱ_μν = ∂_μ𝒜_ν -
                                        ∂_ν𝒜_μ. The form of the function
                                        𝒲(𝒳) has been determined by demanding that the
                                        model reproduces the MOND phenomenology at galactic
                                        scales <cit.>. This
                                        corresponds to the limit 𝒳→ 0 and we have
                                        
                                        
                                            𝒲(𝒳)=𝒳-2/3 (α+β)^2
                                            𝒳^3/2+𝒪(𝒳^2)  ,
                                        
                                        so that the leading term in the action (<ref>) is
                                        
                                        
                                            λ M_eff^2 𝒲(𝒳) = -
                                            M_eff^2/4ℱ^μνℱ_μν +
                                            𝒪(ℱ^3)  .
                                        
                                        Hence, we observe that the coupling scale of the vector field is
                                        dictated by M_eff, while the parameter λ enters into
                                        higher-order corrections. In order to recover the correct MOND regime
                                        for very weak accelerations of baryons in the ordinary g sector,
                                        i.e. below the MOND acceleration scale a_0, these constants
                                        have been determined as <cit.>[Recall also
                                          that the MOND acceleration a_0 is of the order of the cosmological
                                          parameters, and thus is extremely small in Planck units, a_0∼
                                          10^-63 M_Pl.]
                                        
                                            M_eff=√(2) r_g
                                            M_Pl  and  λ=a_0^2/2 .
                                        
                                        Here M_Pl represents the standard Planck constant of GR and
                                        the constant r_g is defined below. It is worth mentioning that the
                                        standard Newtonian limit in the ordinary g sector is obtained by
                                        imposing the relation
                                        
                                        
                                            M_g^2+α^2/β^2M_f^2= M_Pl^2 .
                                        
                                        Thus, in this model the three mass scales M_g, M_f and
                                        M_eff are of the order of the Planck mass.
                                        
                                        We represent the scalar energy densities of the ordinary pressureless
                                        baryons, and the two species of pressureless dark matter particles by
                                        ρ_bar, ρ_g and ρ_f respectively. Such densities
                                        are conserved in the usual way with respect to their respective
                                        metrics, hence ∇^g_μ(ρ_bar u_bar^μ)=0,
                                        ∇^g_μ(ρ_g u_g^μ)=0 and ∇^f_μ(ρ_f u_f^μ)=0,
                                        with the four velocities being normalized as
                                        g_μνu_bar^μ u_bar^ν=-1, g_μνu_g^μ
                                        u_g^ν=-1 and f_μνu_f^μ u_f^ν=-1. The respective
                                        stress-energy tensors are defined as
                                        T_bar^μν=ρ_bar u_bar^μ
                                        u_bar^ν, T_g^μν=ρ_g u_g^μ u_g^ν and
                                        T_f^μν=ρ_f u_f^μ u_f^ν. The pressureless baryonic fluid
                                        obeys the geodesic law of motion a^bar_μ≡
                                        u_bar^ν∇^g_ν u^bar_μ=0, hence
                                        ∇_g^ν T^bar_μν=0. On the other hand, because of
                                        their coupling to the vector field, the dark matter fluids pursue a
                                        non-geodesic motion:
                                        
                                        
                                            ∇_g^ν T^g_μν   = J_g^νℱ_μν ,
                                            ∇_f^ν T^f_μν   = - J_f^νℱ_μν ,
                                        
                                        where the dark matter currents J_g^μ and J_f^μ are related to
                                        those appearing in Eq. (<ref>) by
                                        
                                        
                                            J_g^μ =
                                            √(-g_eff)/√(-g) j_g^μ  and  
                                            J_f^μ = √(-g_eff)/√(-f) j_f^μ .
                                        
                                        It remains to specify the link between these currents and the scalar
                                        densities ρ_g and ρ_f of the particles. This is provided by
                                        J_g^μ=r_g ρ_g u_g^μ and J_f^μ=r_f ρ_f u_f^μ, where
                                        r_g and r_f are two constants of the order of one, which can be
                                        interpreted as the ratios between the “charge” of the particles
                                        (with respect to the vector interaction) and their inertial mass. For
                                        correctly recovering MOND we must have α r_g=β
                                        r_f <cit.>.
                                        
                                        Whereas, the stress-energy tensor of the vector field
                                        𝒜_μ is obtained by varying (<ref>) with respect to
                                        g_μν^eff (holding the g and f metrics fixed) and
                                        corresponds to
                                        
                                        
                                            T_g_eff^μν = M_eff^2[𝒲_𝒳 ℱ^μρℱ^ν_νρ +
                                              λ𝒲 g_eff^μν] ,
                                        
                                        where 𝒲_𝒳≡𝒲/𝒳.
                                        The evolution of the vector field is dictated by the Maxwell law
                                        
                                        
                                            ∇^g_eff_ν[
                                              𝒲_𝒳ℱ^μν] =
                                            1/M_eff^2(j^μ_g-j^μ_f) ,
                                        
                                        where the covariant derivative associated with g_eff is
                                        denoted by ∇^g_eff_ν. Together with the conservation
                                        of the currents, ∇^g_eff_μ j_g^μ=0 and
                                        ∇^g_eff_μ j_f^μ=0, the equations of motion for the
                                        vector field can also be expressed as
                                        
                                        
                                            ∇_g_eff^ν T^g_eff_μν = -
                                            (j^ν_g-j^ν_f )ℱ_μν ,
                                        
                                        and we can combine these equations of motion all together into a
                                        “global” conservation law
                                        
                                        
                                            √(-g_eff) ∇_g_eff^ν
                                            T^g_eff_μν + √(-g) ∇_g^ν T^g_μν +
                                            √(-f) ∇_f^ν T^f_μν = 0 .
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        § DECOUPLING LIMIT
                                        
                                        
                                        Being based on massive bigravity theory, the gravitational sector of
                                        the model, Eq. (<ref>), is ghost-free up to any order in
                                        perturbation theory <cit.>. In addition, the
                                        baryonic and dark matter particles can be coupled separately to either
                                        the g metric or f metric without changing this
                                        property <cit.>. The case of the pure matter coupling between
                                        the vector field 𝒜_μ and the effective composite metric
                                        g_eff in Eqs. (<ref>)–(<ref>), is not trivial. In
                                        that case, it was shown in Ref. <cit.> that the coupling is
                                        ghost-free in the mini-superspace and in the decoupling
                                        limit. Furthermore it is known that such coupling to the composite
                                        metric is unique in the sense that it is the only non-minimal matter
                                        coupling that maintains ghost-freedom in the decoupling
                                        limit <cit.>.
                                        
                                        
                                        However, in our model the vector field is also coupled to the g and
                                        f particles, through the standard interaction term
                                        ∝𝒜_μ(j_g^μ-j_f^μ). This term plays a crucial
                                        role for the dark matter model to work. This coupling introduces a
                                        suplementary, indirect interaction between the two metrics of
                                        bigravity, via the g and f particles coupled together by
                                        the term 𝒜_μ(j_g^μ-j_f^μ). See Fig. <ref> for a
                                        schematic illustration of the interactions in the model. As a result
                                        it was found in Ref. <cit.> that a ghost is
                                        reintroduced in the dark matter sector in the full theory. The aim of
                                        this paper is to investigate the occurence and mass of this ghost, and
                                        whether or not the decoupling limit (DL) is maintained ghost-free. If
                                        the latter is true, then the model can be used in a consistent way as
                                        an Effective Field Theory valid below the strong coupling scale.
                                        
                                        We now detail the analysis of the DL interactions in the graviton and
                                        matter sectors. We follow the preliminary work <cit.>
                                        and investigate the scale of the reintroduced Boulware-Deser (BD)
                                        ghost <cit.>. We first decouple the interactions below
                                        the strong coupling scale from those entering above it, and
                                        concentrate on the pure interactions of the helicity-0 mode of the
                                        massive graviton. Using the Stückelberg trick, we restore the broken
                                        gauge invariance in the f metric by replacing it by
                                        
                                        
                                            f̃_μν = f_ab∂_μϕ^a ∂_νϕ^b ,
                                        
                                        where a, b=0,1,2,3 and the four Stückelberg fields ϕ^a are
                                        decomposed into the helicity-0 mode π and the helicity-1 mode
                                        A^a,
                                        
                                        
                                            ϕ^a=x^a-m A^a/Λ^3_3 -f^ab∂_b
                                              π/Λ^3_3 .
                                        
                                        Here Λ_3 ≡ (m^2 M_eff)^1/3 denotes the strong
                                        coupling scale. Note, that we define it with respect to M_eff
                                        here, since the potential interactions scale with m^2M_eff^2
                                        in our case.  
                                        
                                        It is well known that the would-be BD ghost in the DL, would come in
                                        the form of higher derivative interactions of the helicity-0 mode at
                                        the level of the equations of motion. Therefore we shall only follow
                                        the contributions of the helicity-0 mode π and neglect the
                                        interactions of the helicity-1 mode A^a. For simplicity we do not
                                        write the tilde symbol over the Stükelbergized version of the
                                        metric (<ref>). Thus, considering also the helicity-2 mode in
                                        the g metric, we have[If there is a BD ghost in the DL,
                                          then it will manifest itself in the higher-order equations of motion
                                          of the helicity-0 mode. For this purpose, it will be enough to
                                          follow closely the contributions of the matter couplings to the
                                          helicity-0 mode equations of motion and decouple the dynamics of the
                                          secondary helicity-2 mode in f. The contributions of the latter,
                                          as derived in <cit.>, will not play any role in our
                                          analysis and will not change the final conclusions. The same is true
                                          for the contributions of the helicity-1 mode.]
                                        
                                            g_μν   = (η_μν+h_μν/M_g)^2 ,
                                            
                                             f_μν   =
                                            (η_μν-Π_μν/Λ^3_3)^2 ,
                                        
                                        where we introduced the notation Π_μν≡∂_μ∂_νπ for convenience, and raised and lowered indices with
                                        the Minkowski metric η_μν. The effective metric reads then
                                        
                                        
                                            g_μν^eff = ((α+β)η_μν +
                                            K_μν)^2 ,
                                        
                                        in which we have introduced as a short-cut notation the linear combination
                                        
                                        
                                            K_μν = α/M_gh_μν -
                                             β/Λ^3_3Π_μν .
                                        
                                        We will as next investigate the different contributions in the
                                        gravitational and matter sectors.
                                        
                                        
                                        
                                         §.§ Gravitational sector
                                        
                                        
                                        There is no contribution of the Einstein-Hilbert term to the
                                        helicity-0 mode, since this is invariant under diffeomorphisms. On the
                                        other hand, there will be different contributions coming from the
                                        ghost-free potential interactions. The allowed potential interactions
                                        between the metrics g and f have been chosen in our model to be
                                        given by the square root of the determinant of the composite
                                        metric (<ref>), which becomes in this case
                                        
                                        
                                            √(-g_eff) = ∑_n=0^4 (α+β)^4-n e_(n)(K) ,
                                        
                                        where e_(n)(K) denote the usual symmetric polynomials associated
                                        with the matrix K_μ^ρ≡η^ρνK_μν, and given
                                        by products of antisymmetric Levi-Cevita tensors,
                                        
                                        
                                            e_(0)(K)    = - 1/24ε^μνρσε_μνρσ ,
                                            
                                             e_(1)(K)    = - 1/6ε^μνρσε_μνρλ
                                            K^λ_σ , 
                                             e_(2)(K)    = - 1/4ε^μνρσε_μντλ
                                            K^τ_ρ K^λ_σ , 
                                             e_(3)(K)    = - 1/6ε^μνρσε_μπτλ
                                            K^π_ν K^τ_ρ K^λ_σ , 
                                             e_(4)(K)    = - 1/24ε^μνρσε_ϵπτλ
                                            K^ϵ_μ K^π_ν K^τ_ρ K^λ_σ .
                                        
                                        In particular, we see that e_(4)(K)=det(K).
                                        
                                        First of all, the pure helicity-0 mode in the ghost-free potential
                                        interactions (<ref>) will come in the form of total
                                        derivatives <cit.>. Indeed, as is clear from their
                                        definitions (<ref>) in terms of antisymmetric
                                        Levi-Cevita tensors, the symmetric polynomials
                                        e_(n)(Π)≡ℒ^der_(n)(Π) fully encode
                                        the total derivatives at that order, and thus will not contribute to
                                        the equation of motion of the helicity-0 mode. In fact, in
                                        Ref. <cit.>, this very same property of total derivatives of
                                        the leading contributions at each order was used to build the
                                        ghost-free interactions away from h=0. Secondly, there will be the
                                        pure interactions of the helicity-2 mode, obtained by setting Π=0,
                                        and these will come with the corresponding inverse powers of
                                        M_g. Finally, there will be the mixed interactions between the
                                        helicity-2 and helicity-0 modes.
                                        
                                        We are after the leading interactions in the DL, which correspond to
                                        sending all the Planck scales to infinity,
                                        
                                        
                                            M_Pl→∞ ,   M_g→∞ ,   M_eff→∞ ,   M_f →∞ ,
                                        
                                        together with the graviton's mass m →0, while keeping
                                        
                                        
                                            {Λ^3_3 = m^2 M_eff ,   M_g/M_Pl ,
                                              M_eff/M_Pl ,   M_f/M_Pl} = const .
                                        
                                        Taking into account the factor m^2 M_eff^2 in front of the
                                        potential interactions, one immediately observes that the pure
                                        non-linear interactions of the helicity-2 modes do not contribute to
                                        the DL. As we already mentioned, the pure helicity-0 mode interactions
                                        do not contribute either. So it remains the mixed terms, for which the
                                        only surviving terms will be linear in the helicity-2 mode, and we
                                        finally obtain
                                        
                                        
                                            m^2 M_eff^2 √(-g_eff) = ∑_n=1^3
                                            a_n/Λ_3^3(n-1) h^μν P_μν^(n)(Π) +
                                            𝒪(1/M_g) ,
                                        
                                        where a_n≡
                                        (M_eff/M_g)^n+1α(-β)^n(α+β)^3-n
                                        and we posed
                                        
                                        
                                            P_μν^(n-1)(Π) ≡∂ e_(n)(Π)/∂Π^μν .
                                        
                                        In arriving at Eq. (<ref>) we have removed the trivial
                                        constant term in (<ref>), and ignored the “tadpole” which is
                                        simply proportional to the trace [h]=h^μ_μ and can be
                                        eliminated by choosing an appropriate de Sitter background (see,
                                        e.g., a discussion in <cit.>).
                                        
                                        We can then write the total contribution of the gravitational sector
                                        in the DL, including that coming from the Einstein-Hilbert term of the
                                        g metric, which enters only at the leading quadratic order in
                                        h_μν,
                                        
                                        
                                            ℒ^DL_grav = -
                                            h^μνℰ^ρσ_μνh_ρσ +
                                            ∑_n=1^3 a_n/Λ_3^3(n-1) h^μν
                                            P_μν^(n)(Π) ,
                                        
                                        where ℰ^ρσ_μν is the usual Lichnerowicz
                                        operator on a flat background as defined by
                                        
                                        
                                            -2ℰ_μν^ρσh_ρσ   =
                                            (h_μν-η_μνh) + ∂_μ∂_ν h
                                            
                                               - 2 ∂_(μ H_ν) +
                                            η_μν∂_ρ H^ρ ,
                                        
                                        with h=[h]=h^μ_μ and H_μ=∂_ν
                                        h^ν_μ. 
                                         The symmetric tensors P^(n)_μν
                                        are conserved, i.e.∂_ν P_(n)^μν=0. For
                                        an easier comparison with the literature we give them as the product
                                        of two Levi-Cevita tensors appropriately contracted with the second
                                        derivative of the helicity-0 field,
                                        
                                        
                                            P^(1)_μν(Π)    = -1/2ε_μ^μλρσε_νλρτΠ^τ_σ ,
                                             P^(2)_μν(Π)    = -1/2ε_μ^μλρσε_νλπτΠ^π_ρΠ^τ_σ ,
                                             P^(3)_μν(Π)    =
                                            -1/6ε_μ^μλρσε_νϵπτΠ^ϵ_λΠ^π_ρΠ^τ_σ .
                                        
                                        
                                        The first two interactions between the helicity-0 and helicity-2
                                        fields in the Lagrangian (<ref>) can be removed by the change
                                        of variable, defining
                                        
                                        
                                            ĥ_μν≡ h_μν - a_1/2π η_μν + a_2/2Λ_3^3∂_μπ∂_νπ .
                                        
                                        In this way the Lagrangian of the gravitational sector in the
                                        decoupling limit becomes <cit.>
                                        
                                            ℒ^DL_grav =    -
                                            ĥ^μνℰ^ρσ_μνĥ_ρσ
                                            + ∑_n=0^3 b_n/Λ_3^3n(∂π)^2
                                            e_(n)(Π)
                                                + a_3/Λ_3^6ĥ^μν P_μν^(3)(Π) .
                                        
                                        We see in the first line the appearance of the ordinary Galileon terms
                                        up to quintic order [we denote
                                          (∂π)^2≡∂_μπ∂^μπ]. The
                                        coefficients b_n are given by certain combinations of the
                                        a_n's.[Namely, b_0=-3/4a_1^2,
                                          b_1=-3/4a_1 a_2, b_2=-1/2a_2^2-1/3a_1
                                          a_3 and b_3=-5/4a_2 a_3.] The last term of
                                        Eq. (<ref>) is the remaining mixing between the helicity-0
                                        and helicity-2 modes and is not removable by any local field
                                        redefinition like in (<ref>).
                                        
                                        The contribution of the gravitational sector to the equation of motion
                                        of the helicity-2 field gives
                                        
                                        
                                            δℒ^DL_grav/δĥ^μν = -2 ℰ^ρσ_μνĥ_ρσ + a_3/Λ_3^6
                                            P^(3)_μν(Π)  ,
                                        
                                        while its contribution to the equation of motion of the helicity-0
                                        field reads
                                        
                                        
                                            δℒ^DL_grav/δπ = -2
                                            ∑_n=1^4 n b_n-1/Λ_3^3(n-1) e_(n)(Π) +
                                            a_3/Λ_3^6 Q_μν^(2)ρσ(Π)
                                            ∂_ρ∂_σĥ^μν ,
                                        
                                        where we posed
                                        
                                        
                                            Q_μν^(2)ρσ(Π) ≡∂
                                              P_μν^(3)/∂Π_ρσ = -1/2ε_μ^μρϵλε_νσπτ^νσΠ^π_ϵΠ^τ_λ .
                                        
                                        The second-order nature of the equations of motion in the gravity
                                        sector is apparent. This is the standard property of the ghost-free
                                        massive gravity interactions <cit.>.
                                        
                                        
                                        
                                         §.§ Matter sector
                                        
                                        
                                        As next, we shall control the contributions in the matter sector due
                                        to both the helicity-0 and helicity-2 fields. To this aim it is
                                        important to properly identify the matter degrees of freedom that are
                                        metric independent. These are provided by the coordinate densities
                                        defined as ρ^*_g = √(-g)ρ_g u^0_g and ρ^*_f =
                                        √(-f)ρ_f u^0_f, and by the ordinary (coordinate) velocities
                                        v^μ_g=u^μ_g/u^0_g and v^μ_f=u^μ_f/u^0_f. The associated
                                        currents J^*μ_g= ρ^*_g v^μ_g and J^*μ_f= ρ^*_f
                                        v^μ_f are conserved in the ordinary sense, ∂_μ
                                        J^*μ_g=0 and ∂_μ J^*μ_f=0, and are related to the
                                        classical currents by
                                        
                                        
                                            J^*μ_g = √(-g) J_g^μ  and   J^*μ_f =
                                            √(-f) J_f^μ .
                                        
                                        When varying the action we must carefully impose that the independent
                                        matter degrees of freedom are the metric independent currents
                                        J^*μ_g and J^*μ_f. After variation we may restore the
                                        manifest covariance by going back to the classical currents
                                        using (<ref>).
                                        
                                        Next we must specify how the matter variables will behave in the DL
                                        when we take the scaling
                                        limits (<ref>)–(<ref>). In the DL we want to
                                        keep intact the coupling between the helicity-2 mode h_μν and
                                        the particles living in the g sector, therefore we impose
                                        
                                        
                                            T_bar^μν = M_g
                                            T̂_bar^μν  and   T_g^μν = M_g
                                            T̂_g^μν ,
                                        
                                        with T̂_bar^μν and T̂_g^μν remaining
                                        constant in the DL. As for the f particles, 
                                         in a
                                        similar way we demand that T_f^μν = M_f T̂_f^μν
                                        with T̂_f^μν being constant.
                                        
                                        The next important point concerns the internal vector field
                                        𝒜_μ. As we have seen this vector field is a
                                        graviphoton <cit.>, i.e. its scale is given by
                                        the Planck mass, witness the factor M_eff^2 in front of the
                                        kinetic term of the vector field (<ref>), see also the factor
                                        M_eff^2 in front of the stress-energy tensor of the vector
                                        field, Eq. (<ref>). For the model to work M_eff must be
                                        of the order of the Planck mass, as determined
                                        in (<ref>). This means that we have to canonically
                                        normalize the vector field 𝒜_μ according to
                                        
                                        
                                            𝒜_μ = 𝒜̂_μ/M_eff ,
                                        
                                        and keep 𝒜̂_μ constant in the DL. Thus
                                        T_g_eff^μν = T̂_g_eff^μν should
                                        be considered constant in that limit.
                                        
                                        A general variation of the matter action with respect to the two
                                        metrics reads
                                        
                                        
                                            δℒ_mat   = √(-g)/2(T_bar^μν + T_g^μν)δ g_μν +
                                            √(-f)/2 T_f^μνδ f_μν
                                               +
                                            √(-g_eff)/2 T_g_eff^μνδ
                                            g^eff_μν .
                                        
                                        We insert Eqs. (<ref>)–(<ref>) and change the
                                        helicity-2 variable according to (<ref>) to obtain the
                                        contribution of the matter action to the field equation for the
                                        helicity-2 field (in guise ĥ_μν) as
                                        
                                        
                                            δℒ_mat/δĥ_μν   =
                                            1/M_g√(-g) (T_bar^ρ(μ +
                                            T_g^ρ(μ)(δ^ν)_ρ +
                                            h^ν)_ρ/M_g)
                                                +
                                            α/M_g√(-g_eff) T_g_eff^ρ(μ((α+β)δ^ν)_ρ
                                            + K^ν)_ρ) .
                                        
                                        Taking the DL with the postulated
                                        scalings (<ref>)–(<ref>) we find that the
                                        helicity-2 mode of the massive graviton is just coupled in this limit
                                        to the baryons and g particles,
                                        
                                        
                                            δℒ^DL_mat/δĥ_μν   = T̂_bar^μν + T̂_g^μν ,
                                        
                                        where the (rescaled) stress-energy tensors
                                        T̂_bar^μν and T̂_g^μν in the DL are
                                        computed with the Minkowski background.
                                        
                                        We next consider the contributions of the matter sector to the
                                        equation of motion of the helicity-0 field. We find three
                                        contributions, two coming from the field
                                        redefinition (<ref>),
                                        
                                        
                                            δℒ_mat/δπ|_(1a) =
                                              a_1/2M_g√(-g) (T_bar^μν +
                                              T_g^μν)(η_μν +
                                              h_μν/M_g) 
                                                    +
                                              a_2/M_g Λ_3^3∂_ν[√(-g) (T_bar^μ(ν +
                                                T_g^μ(ν)(δ_μ^ρ) +
                                                h_μ^ρ)/M_g)∂_ρπ]
                                               ,
                                            
                                               δℒ_mat/δπ|_(1b) =
                                              α a_1/2M_g√(-g_eff) T_g_eff^μν((α+β)η_μν
                                              + K_μν) 
                                                    + α a_2/M_g
                                                Λ_3^3∂_ν[√(-g_eff) T_g_eff^μ(ν((α+β)δ^ρ)_μ
                                                + K^ρ)_μ)∂_ρπ]  ,
                                        
                                        and the third one being “direct”, and already investigated
                                        in <cit.> with result
                                        
                                        
                                            δℒ_mat/δπ|_(2) =   
                                            -1/Λ_3^3∂_μ∂_ν[√(-f) T_f^ρμ(δ^ν_ρ-Π^ν_ρ/Λ_3^3) 
                                                  + β√(-g_eff) T_g_eff^ρμ((α+β)δ^ν_ρ +
                                              K^ν_ρ)] .
                                        
                                        The latter contribution might look worrisome in the DL, but it becomes
                                        finite after using the equation of motion for the f particles,
                                        Eq. (<ref>), and that for the vector field,
                                        Eq. (<ref>). The calculation proceeds similarly to the one
                                        using Eqs. (3.29)–(3.32) in Ref. <cit.>. Finally the result can
                                        be brought into the form <cit.>
                                        
                                            δℒ_mat/δπ|_(2)   =
                                            1/Λ_3^3∂_ν[
                                              J^*ρ_f ℱ_μρ(η^μν -
                                              Π^μν/Λ_3^3)^-1
                                               +
                                              β(J^*ρ_g - J^*ρ_f)
                                              ℱ_μρ((α+β)η^μν +
                                              K^μν)^-1]  ,
                                        
                                        where we describe the matter degrees of freedom by means of the
                                        coordinate currents (<ref>).
                                        
                                        The results (<ref>) and (<ref>) are general at
                                        this stage, and involve couplings between both the helicity-0 and
                                        helicity-2 modes with the matter fields —g and f particles, and
                                        the internal vector field 𝒜_μ. However, because of the
                                        scaling (<ref>), which we recall is appropriate to the
                                        graviphoton whose coupling scale is given by the Planck mass, the
                                        vector field strength actually scales like
                                        ℱ_μν=ℱ̂_μν/M_eff in the
                                        DL limit. This fact kills all the interactions between the helicity-0
                                        mode and the vector field in the DL, since they come with an inverse
                                        power of M_eff.[Note that if we do not impose the
                                          scaling
                                          ℱ_μν=ℱ̂_μν/M_eff the
                                          equation (<ref>) for the helicity-2 field diverges in the
                                          DL. Similarly for Eq. (<ref>).] Thus the direct
                                        contribution (<ref>) is identically zero in the DL, and
                                        only the contribution (<ref>) is surviving,
                                        while (<ref>) is also zero. After further simplification
                                        with the matter equations of motion, we obtain (with
                                        T̂_bar and T̂_g denoting the Minkowskian traces)
                                        
                                        
                                            δℒ^DL_mat/δπ =
                                            a_1/2 (T̂_bar + T̂_g) +
                                            a_2/Λ_3^3 (T̂_bar^μν +
                                            T̂_g^μν)∂_μ∂_νπ .
                                        
                                        
                                        Recapitulating, we find that the DL of the model consists of the
                                        following equation for the helicity-2 mode, i.e.δℒ^DL/δĥ_μν=0 or
                                        equivalently
                                        
                                        
                                            -2 ℰ_ρσ^μνĥ_ρσ +
                                            a_3/Λ_3^6 P_(3)^μν(Π) +
                                            T̂_bar^μν + T̂_g^μν = 0 ,
                                        
                                        which is of second-order nature. Thus, the contributions of the
                                        gravitational and matter sector to the equations of motion of the
                                        helicity-2 mode in the DL are ghost-free. Note, that the Bianchi
                                        identity of this equation (taking the divergence of it) is identically
                                        satisfied, since the particles actually follow geodesics in the
                                        DL. Indeed, using (<ref>)–(<ref>) together with the
                                        equations of motion [e.g. (<ref>)], we have
                                        ∂_νT̂_bar^μν=∂_νT̂_g^μν=0
                                        (the particles move on Minkowski straight lines).
                                        
                                        In addition we have the total equation of motion of the helicity-0
                                        mode, namely δℒ^DL/δπ=0 which reads
                                        
                                        
                                            -2 ∑_n=1^4 n b_n-1/Λ_3^3(n-1)
                                              e_(n)(Π) + a_3/Λ_3^6
                                              Q_μν^(2)ρσ(Π)
                                              ∂_ρ∂_σĥ^μν
                                                    = -
                                              a_1/2 (T̂_bar + T̂_g) -
                                              a_2/Λ_3^3 (T̂_bar^μν +
                                              T̂_g^μν)∂_μ∂_νπ .
                                        
                                        Since this equation is perfectly of second-order in the derivatives of
                                        the π field, we conclude our study by stating that the model is
                                        safe (ghost-free) up to the strong coupling scale. Below that scale
                                        the theory is perfectly acceptable as an Effective Field Theory, and
                                        its consequences can be worked out using perturbation theory as
                                        usual. For instance, solving at linear order the helicity-0
                                        equation (<ref>) we obtain the usual well-posed
                                        (hyperbolic-like) equation
                                        
                                        
                                            π = a_1/4b_0(T̂_bar +
                                            T̂_g)+𝒪(π^2) ,
                                        
                                        which can then be perturbatively iterated to higher order. With this
                                        we have proved, that the coupling of the dark matter particles with
                                        the internal vector field does not introduce any ghostly contribution
                                        in the DL.
                                        
                                        
                                        
                                        
                                        
                                        
                                        § CONCLUSIONS
                                        
                                        
                                        This work was dedicated to the detailed study of the decoupling limit
                                        interactions of the dark matter model proposed
                                        in <cit.>. This model is
                                        constructed via a specific coupling of two copies of dark
                                        matter particles to two metrics in the framework of massive
                                        bigravity. Furthermore, an internal vector field links the two dark
                                        matter species. This enables us to implement a mechanism of
                                        gravitational polarization, which induces the MOND phenomenology on
                                        galactic scales (with the specific choice of parameters studied
                                        in <cit.>). Note that, since our model successfully
                                        reproduces all aspects of that phenomenology, it will be in agreement
                                        with the recent observations of the MOND mass-discrepancy-acceleration
                                        relation in <cit.>.
                                        
                                        Some theoretical and phenomenological consequences of this model were
                                        studied in detail in Ref. <cit.>, but it was also
                                        pointed out that the decoupling limit of the theory may be
                                        problematic, with higher derivative terms occuring in the equation of
                                        motion of the helicity-0 mode of the massive graviton.
                                        
                                        In the present work, we studied the complete DL interactions crucially
                                        including the contributions of the matter sector, and we showed that
                                        by necessary rescaling of the vector field (as appropriate for a
                                        vector field with Planckian coupling constant) the theory is free from
                                        ghosts in the DL, and hence can be used as a valid Effective Field
                                        Theory up to the strong coupling scale.
                                        
                                        
                                        
                                         
                                        
                                        We would like to thank Claudia de Rham and Andrew Tolley for very
                                        useful and enlightening discussions. L.H. wishes to acknowledge the
                                        Institut d'Astrophysique de Paris for hospitality and support at the
                                        final stage of this work.
