Dipolar Dark Matter with Massive Bigravity

Abstract

Massive gravity theories have been developed as viable IR modifications of
gravity motivated by dark energy and the problem of the cosmological constant.
On the other hand, modified gravity and modified dark matter theories were
developed with the aim of solving the problems of standard cold dark matter at
galactic scales. Here we propose to adapt the framework of ghost-free massive
bigravity theories to reformulate the problem of dark matter at galactic
scales. We investigate a promising alternative to dark matter called dipolar
dark matter (DDM) in which two different species of dark matter are separately
coupled to the two metrics of bigravity and are linked together by an internal
vector field. We show that this model successfully reproduces the phenomenology
of dark matter at galactic scales (MOND) as a result of a mechanism of
gravitational polarisation. The model is safe in the gravitational sector, but
because of the particular couplings of the matter fields and vector field to
the metrics, a ghost in the decoupling limit is present in the dark matter
sector. However, it might be possible to push the mass of the ghost beyond the
strong coupling scale by an appropriate choice of the parameters of the model.
Crucial questions to address in future work are the exact mass of the ghost,
and the cosmological implications of the model.

Full text

JCAP12(2015)026
ournal of
Cosmology and Astroparticle Physics
An IOP and SISSA journal
J
Dipolar dark matter with massive
bigravity
Luc Blanchetaand Lavinia Heisenbergb,c
aGRεCOInstitut d’Astrophysique de Paris — UMR 7095 du CNRS,
Universit´e Pierre & Marie Curie,
98bis boulevard Arago, 75014 Paris, France
bNordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, 10691 Stockholm, Sweden
cDepartment of Physics & The Oskar Klein Centre, AlbaNova University Centre,
Roslagstullsbacken 21, 10691 Stockholm, Sweden
E-mail: blanchet@iap.fr,laviniah@kth.se
Received May 22, 2015
Revised October 31, 2015
Accepted November 11, 2015
Published December 14, 2015
Abstract. Massive gravity theories have been developed as viable IR modifications of gravity
motivated by dark energy and the problem of the cosmological constant. On the other hand,
modified gravity and modified dark matter theories were developed with the aim of solving
the problems of standard cold dark matter at galactic scales. Here we propose to adapt
the framework of ghost-free massive bigravity theories to reformulate the problem of dark
matter at galactic scales. We investigate a promising alternative to dark matter called dipolar
dark matter (DDM) in which two different species of dark matter are separately coupled to
the two metrics of bigravity and are linked together by an internal vector field. We show
that this model successfully reproduces the phenomenology of dark matter at galactic scales
(i.e. MOND) as a result of a mechanism of gravitational polarisation. The model is safe in the
gravitational sector, but because of the particular couplings of the matter fields and vector
field to the metrics, a ghost in the decoupling limit is present in the dark matter sector.
However, it might be possible to push the mass of the ghost beyond the strong coupling scale
by an appropriate choice of the parameters of the model. Crucial questions to address in
future work are the exact mass of the ghost, and the cosmological implications of the model.
Keywords: modified gravity, dark matter theory, gravity, dark energy theory
ArXiv ePrint: 1505.05146
Article funded by SCOAP3. Content from this work may be used
under the terms of the Creative Commons Attribution 3.0 License.
Any further distribution of this work must maintain attribution to the author(s)
and the title of the work, journal citation and DOI.
doi:10.1088/1475-7516/2015/12/026

JCAP12(2015)026
Contents
1 Introduction 1
1.1 Massive gravity context 1
1.2 Dark matter context 3
2 The covariant theory 4
3 Linear perturbations 7
4 Polarisation mechanism and MOND 10
5 The decoupling limit 14
6 Conclusions 18
1 Introduction
In the last century, cosmology has progressively developed from a philosophical to an empiri-
cal scientific discipline, witnessing high precision cosmological observations, which culminated
with the standard model of cosmology, the Λ-CDM model [1]. The standard model is based
on General Relativity (GR) and is in great agreement with the abundance of the light ele-
ments in the big bang nucleosynthesis, the anisotropies of the cosmic microwave background
(CMB), the baryon acoustic oscillations, lensing and the observed large scale structures. It
notably relies on the presence of dark energy in form of a cosmological constant Λ, giving
rise to the accelerated expansion of the universe.
1.1 Massive gravity context
In the prevailing view, the cosmological constant corresponds to the constant vacuum energy
density which receives large quantum corrections. Unfortunately, so far there is no successful
mechanism to explain the observed unnatural tiny value of the cosmological constant, see
e.g. [2,3]. This difficulty has sparked a whole industry studying modifications of gravity in
the infra-red (IR) invoking new dynamical degrees of freedom.
One tempting route is massive gravity, motivated by the possibility that the graviton
has a mass. It was a challenge over forty years to construct a covariant non-linear theory
for massive gravity. The foregathered remarkable amount of effort finally rose to the chal-
lenge [4–11], to construct the potential interactions in a way that got rid of the Boulware-
Deser (BD) ghost [12]. This theory has been further extended to more general models
by adding additional degrees of freedom. A very important extension is massive bigravity
theory [13].
Once the interactions in the gravitational sector were guaranteed to be ghost-free, a nat-
ural follow up question was how to couple this theory to the matter fields without spoiling the
ghost-freedom. First attempts were already discussed in the original paper of bigravity [13].
If one couples the matter fields to both metrics simultaneously, this reintroduces the BD
ghost [14,15]. Furthermore, the one-loop quantum corrections detune the special potential
interactions at an unacceptable low scale and hence this way of coupling is not a consistent
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one [15]. The safer way is to couple the matter fields to just one metric. In this way, the
quantum corrections give rise to contributions in form of cosmological constants. One could
also try to couple the matter field to the massless mode, which is unfortunately also not ghost
free [16]. Another possible way of coupling the matter fields to both metrics is through a
composite effective metric constructed out of both metrics [15,17–19], which is unique in the
sense that it is the only non-minimal matter coupling that maintains ghost-freedom in the
decoupling limit [20–22]. Furthermore, the quantum corrections are guaranteed to maintain
the nice potential structure. Other important consequence of this new matter coupling is the
fact that it helps to evade the no-go result [23] for the flat Friedmann-Lemaˆıtre-Robertson-
Walker (FLRW) background. A detailed perturbed ADM analysis revealed the existence of
a BD ghost originating from an operator involving spatial derivatives [15,17]. Therefore, the
ghost will probably reappear for highly anisotropic solutions. Since the ghost remains absent
up to the strong coupling scale, the matter coupling can be considered in an effective field
theory sense at the very least till the strong coupling scale. The precise cut-off of the theory
or mass of the ghost has still to be established.
The absence of the BD ghost is not only important at the classical level, but also at
the quantum level. For the classical ghost-freedom the relative tuning of the potential inter-
actions is the key point. Therefore one has to ensure that the quantum corrections do not
detune the potential interactions. Concerning the decoupling limit, it is easy to show that
the theory receives no quantum corrections via the non-renormalization theorem due to the
antisymmetric structure of the interactions [24] (in fact the same antisymmetric structure
of the Galileon interactions protect them from quantum corrections [25–27]). Beyond the
decoupling limit, the quantum corrections of the matter loops maintain the potential inter-
actions provided that the above criteria are fulfilled. Concerning the graviton loops, they
do destroy the relative tuning of the potential interactions. Nevertheless, this detuning is
harmless since the mass of the corresponding BD ghost is never below the cut-off scale of the
theory [28]. The bimetric version of the theory shares the same property [29].
Massive gravity/bigravity theory has a rich phenomenology. The decoupling limit ad-
mits stable self-accelerating solutions [30], where the helicity-0 degree of freedom of the mas-
sive graviton plays the role of a condensate whose energy density sources self-acceleration.
Unfortunately these solutions suffer from strong coupling issues due to the vanishing kinetic
term of the vector modes [30,31]. With the original massive gravity and the restriction of
flat reference metric, one faces the no-go result, namely that there are no flat FLRW so-
lution [23]. One can construct self-accelerating open FLRW solutions [32], which however
have three instantaneous modes [33] and suffer from a nonlinear ghost instability [34]. The
attempt to promote the reference metric to de Sitter [35,36] also failed due to the presence of
the Higuchi ghost [35]. In order to avoid these difficulties, one either gives up on the FLRW
symmetries [23], or invokes new additional degrees of freedom [13,37,38].
Thanks to the freedom gained in the inclusion of the second kinetic term in the bi-
metric extension [13], there now exists many elaborate works concerning the cosmology of
the bigravity theory, see e.g. [39–41]. In the case of minimally coupled matter fields and
small graviton mass, the theory admits several interesting branches of solutions. Unfortu-
nately, among them the self-accelerating branch is unstable due to the presence of three
instantaneous modes, and a second branch of solutions admits an early time gradient insta-
bility [42]. Nevertheless, there exists attempts to overcome the gradient instability either by
viable though finely tuned solutions in the case of a strongly interacting bimetric theory with
m≫H0[43,44], or by demanding that Mg≫Mfas was proposed in [49]. See also other
recent works concerning the phenomenology of bimetric gravity [45–48,50].
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1.2 Dark matter context
On a quite different but equally fascinating topic is the phenomenology of dark matter at
galactic scales. The evidence for dark matter is through the measurement of the rotation
curves of spiral galaxies which turn out to be approximately flat, contrary to the Newtonian
prediction based on ordinary baryonic matter [51,52]. The standard explanation is that the
disk of galaxies is embedded into the quasi-spherical potential of a huge halo of dark matter,
and that this dark matter is the same as the cold dark matter (CDM) which is evidenced at
large cosmological scales notably with the fit of Λ-CDM to the CMB anisotropies [53,54].
Unfortunately this explanation faces severe challenges when compared to observations at
galactic scales [55,56]. There are predictions of the Λ-CDM model that are not observed,
like the phase-space correlation of galaxy satellites and the generic formation of dark matter
cusps in the central regions of galaxies. Even worse, there are observations which are not
predicted by Λ-CDM, such as the tight correlation between the mass discrepancy (luminous
vs. dynamical mass) which measures the presence of dark matter and the involved scale of
acceleration, the famous baryonic Tully-Fisher (BTF) relation for spiral galaxies [57–59], and
its equivalent for elliptical galaxies, the Faber-Jackson relation [60].
Instead of additional, non-visible mass, Milgrom [61–63] proposed an amendment to
the Newtonian laws of motion in order to account for the phenomenology of dark matter
in galaxies, dubbed MOND for MOdified Newtonian Dynamics. According to the MOND
hypothesis the change has a relevant influence on the motion only for very small accelerations,
as they occur at astronomical scales, below the critical value a0≃1.2×10−10 m/s2measured
for instance from the BTF relation.1A more elaborate version of MOND is the Bekenstein-
Milgrom [64] modification of the Poisson equation of Newtonian gravity. All the challenges
met by Λ-CDM at galactic scales are then solved — sometimes with incredible success —
by the MOND formula [55,56]. Unfortunately, MOND faces difficulties in explaining the
dark matter distribution at the larger scales of galaxy clusters [65–69]. It is also severely
constrained by observations in the solar system [70,71].
Reconciling Λ-CDM at cosmological scales and MOND at galactic scales into a single
relativistic theory is a great challenge. There has been extensive works on this. One ap-
proach is to modify gravity by invoking new gravitational degrees of freedom without the
presence of dark matter [72–81]. Primary examples are the tensor-vector-scalar (TeVeS) the-
ory [73,74] and generalized Einstein-Æther theories [75,76]. Another approach is a new form
of dark matter `a la MOND, called dipolar dark matter (DDM). It is based on the dielec-
tric analogy of MOND [82] — a remarkable property of MOND with possible far-reaching
implications. Indeed the MOND equation represents exactly the gravitational analogue (in
the non-relativistic limit) of the Gauss equation of electrostatics modified by polarisation
effects in non-linear dielectric media. Some early realizations of this analogy were proposed
in [83–85] and shown to also reproduce the cosmological model Λ-CDM. The best way to
interpret this property is by a mechanism of gravitational polarisation, involving two differ-
ent species of particles coupled to different gravitational potentials. This was the motivation
for pushing forward a more sophisticated model in the context of a bimetric extension of
GR [86,87]. In this model the two species of dark matter particles that couple to the two
metrics are linked through an internal vector field. The model [87] is able to recover MOND
at galactic scales and the cosmological model Λ-CDM at large scales. Furthermore the post-
Newtonian parameters (PPN) in the solar system are the same as those of GR. Unfortunately,
1A striking observation is the numerical coincidence that a0∼√Λ [55,56].
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even if this model delivers a very interesting phenomenology, it is plagued by ghost instabil-
ities in the gravitational sector already at the linear order of perturbations and within the
decoupling limit [88]. A more promising model was then proposed in [88], with the hope to
cure these instabilities in the gravitational sector.
In the present work we will study in detail the theoretical and phenomenological con-
sistency of this recently proposed model for dipolar dark matter in the context of bimetric
gravity [88]. We will first work out the covariant field equations in section 2and analyze the
linear field equations in section 3. As next, we will investigate in section 4the mechanism
of gravitational polarisation in the non relativistic approximation, and show how it success-
fully recovers the MOND phenomenology on galactic scales. Finally we will pay attention in
section 5to the matter sector and investigate the interactions in the decoupling limit. The
outcome of section 5is that a ghost instability is still present in the (dark) matter sector of
the model. The paper ends with some short conclusions in section 6.
2 The covariant theory
A new relativistic model for dipolar dark matter has been recently proposed in [88]. Basically
this model is defined by combining the specific dark matter sector of a previous model [87]
with gravity in the form of ghost-free bimetric extensions of GR [13]. The dark matter in this
model consists of two types of particles respectively coupled to the two dynamical metrics
gµν and fµν of bigravity. In addition, these two sectors are linked together via an additional
internal field in the form of an Abelian U(1) vector field Aµ. This vector field is coupled to
the effective composite metric geff of bigravity which is built out of the two metrics gand
f, and which has been shown to be allowed in the matter Lagrangian in the effective field
theory sense at least up to the strong coupling scale [15,17]. The total matter-plus-gravity
Lagrangian reads2
L=√−gM2
g
2Rg−ρbar −ρg+p−fM2
f
2Rf−ρf
+√−geffm2M2
eff +Aµjµ
g−jµ
f+λM2
eff WX.(2.1)
Here Rgand Rfare the Ricci scalars of the two metrics, and the scalar energy densities of the
ordinary matter modelled simply by pressureless baryons,3and the two species of pressureless
dark matter particles are denoted by ρbar,ρgand ρfrespectively. In addition jµ
g,jµ
fstand
for the mass currents of the dark matter, defined by
jµ
g=√−g
√−geff
Jµ
gand jµ
f=√−f
√−geff
Jµ
f,(2.2)
where Jµ
g=rgρguµ
gand Jµ
f=rfρfuµ
fare the corresponding conserved dark matter currents
associated with the respective metrics gand f, thus satisfying ∇g
µJµ
g= 0 and ∇f
µJµ
f= 0.
Here ρgand ρfdenote the scalar densities and uµ
g,uµ
fare the four velocities normalized to
gµν uµ
guν
g=−1 and fµν uµ
fuν
f=−1. We also introduced two constants rgand rfspecifying
the ratios between the charge and the inertial mass of the dark matter particles.
2The metric signature convention is (−,+,+,+). We adopt geometrical units with the gravitational con-
stant, the Planck constant and the speed of light being unity, G=~=c= 1, unless specified otherwise.
3We have in mind that the baryons actually represent the full standard model of particle physics.
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The metric independent matter degrees of freedom are the coordinate densities ρ∗
g=
√−gρgu0
gand ρ∗
f=√−fρfu0
f, and the coordinate velocities vµ
g=uµ
g/u0
gand vµ
f=uµ
f/u0
f.
The associated metric independent currents are J∗µ
g=rgρ∗
gvµ
gand J∗µ
f=rfρ∗
fvµ
f. They are
conserved in the ordinary sense, ∂µJ∗µ
g= 0 and ∂µJ∗µ
f= 0. They relate to the classical
currents Jµ
g,Jµ
for jµ
g,jµ
fby
J∗µ
g=√−g Jµ
g=√−geff jµ
gand J∗µ
f=p−f Jµ
f=√−geff jµ
f.
Notice that the coupling term √−geff Aµ(jµ
g−jµ
f) in the action (2.1) is actually independent
of any metric, neither gnor fnor geff. We shall study in detail in section 5the implications
of the term √−geff Aµ(jµ
g−jµ
f) for the decoupling limit of the theory.
The vector field Aµis generated by the dark matter currents and plays the role of a
“graviphoton”. The presence of this internal field is necessary to stabilize the dipolar medium
and will yield the wanted mechanism of gravitational polarisation, as we shall see in section 4.
It has a non-canonical kinetic term W(X), where Wis a function which is left unspecified at
this stage, and
X=−Fµν Fµν
4λ.(2.3)
Here λis a constant and the field strength is constructed with the effective composite metric
geff
µν given by (2.4) below, i.e. Fµν =gµρ
eff gνσ
eff Fρσ with Fµν =∂µAν−∂νAµnot depending on
the metric as usual.
The model (2.1) is defined by several constant parameters, the coupling constants Mg,
Mfand Meff, the mass of the graviton m, the charge to mass ratios rgand rfof dark
matter, the constant λassociated with the vector field, and the arbitrary constants αand β
entering the effective metric (2.4). Parts of these constants, as well as the precise form of the
function W(X), will be determined in section 4when we demand that the model reproduces
the required physics of dark matter at galactic scales. In particular λwill be related to the
MOND acceleration scale a0, and the model will finally depend on a0and the mass of the
graviton m. Additionally, there will be still some remaining freedom in the parameters α
and β, and in the coupling constants Mgand Mf.
In the model (2.1) the ghost-free potential interactions between the two metrics g
and ftake the particular form of the square root of the determinant of the effective met-
ric [15,17,19]
geff
µν =α2gµν + 2αβ Geff
µν +β2fµν ,(2.4)
where αand βare arbitrary constants, and we have defined
Geff
µν =gµρXρ
ν=fµρYρ
ν,(2.5)
where the square root matrix is defined by X=pg−1f, together with its inverse Y=pf−1g.
Interchanging the two metrics gand fdoes not change the form of the metric (2.4)–(2.5)
except for a redefinition of the parameters αand β. Notice that Geff
µν can be proved to
be automatically symmetric [16,89], and in [88] it was shown, that it corresponds to the
composite metric considered in the previous model [87]. The square root of the determinant
of geff
µν is given in either forms respectively associated with the gor fmetrics, by
√−geff =√−gdetα
1
+βX =p−fdetβ
1
+αY .(2.6)
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JCAP12(2015)026
More explicitly, introducing the elementary symmetric polynomials en(X) and en(Y) of the
square root matrices Xor Y,4
e0(X) = 1 ,(2.7a)
e1(X) = X,(2.7b)
e2(X) = 1
2X2−X2,(2.7c)
e3(X) = 1
6X3−3XX2+ 2X3,(2.7d)
e4(X) = 1
24X4−6X2X2+ 3X22+ 8XX3−6X4,(2.7e)
we can write, still in symmetric guise,
√−geff =√−g
4
X
n=0
α4−nβnen(X) = p−f
4
X
n=0
αnβ4−nen(Y).(2.8)
In this form we see that (2.6) corresponds to the right form of the acceptable potential
interactions between the metrics gand f.
We can first vary the Lagrangian (2.1) with respect to the two metrics gand f, which
yields the following two covariant Einstein field equations [90]
M2
gY(µ
ρGν)ρ
g−m2M2
eff
3
X
n=0
α4−nβnY(µ
ρUν)ρ
(n)
=Y(µ
ρTν)ρ
bar +Tν)ρ
g+α√−geff
√−gαY (µ
ρTν)ρ
geff +βT µν
geff ,(2.9a)
M2
fX(µ
ρGν)ρ
f−m2M2
eff
3
X
n=0
αnβ4−nX(µ
ρVν)ρ
(n)
=X(µ
ρTν)ρ
f+β√−geff
√−fβX (µ
ρTν)ρ
geff +αT µν
geff ,(2.9b)
where Gµν
gand Gµν
fare the Einstein tensors for the gand fmetrics. The tensors Uµν
(n)and
Vµν
(n)are defined from the following sums of matrices [8]5
U(n)=
n
X
p=0
(−)pen−p(X)Xp,(2.10a)
V(n)=
n
X
p=0
(−)pen−p(Y)Yp,(2.10b)
by raising or lowering indices with their respective metrics, thus Uµν
(n)=gµρUν
(n)ρand Vµν
(n)=
fµρVν
(n)ρ. By the same property which makes (2.5) to be symmetric, one can show that Uµν
(n)
and Vµν
(n)are indeed automatically symmetric.
4As usual we denote the traces of rank-2 tensors as Xµ
µ= [X], Xµ
νXν
µ= [X2], and so on. Recall in
particular that √−g en(X) = √−f e4−n(Y), with e4(X) = det(X) and e4(Y) = det(Y).
5Here we adopt a slight change of notation with respect to [8]. Our notation is related to the one of [8] by
U(n)= (−)nY(n)(X) and V(n)= (−)nY(n)(Y).
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JCAP12(2015)026
In the right sides of (2.9) the stress-energy tensors Tµν
bar and Tµν
gare defined with respect
to the metric g,Tµν
fis defined with respect to fand Tµν
geff with respect to geff. Thus, for
pressureless baryonic and dark matter fluids we have Tµν
bar =ρbaruµ
baruν
bar,Tµν
g=ρguµ
guν
gand
Tµν
f=ρfuµ
fuν
fin terms of the corresponding scalar densities and normalized four velocities,
while Tµν
geff represents in fact the stress-energy tensor of the vector field Aµ, i.e.
Tµν
geff =M2
effhWXFµρ Fνρ+λWgµν
eff i,(2.11)
where WXis the derivative of Wwith respect to its argument Xdefined by (2.3).
The equation of motion for the baryons follows a geodesic law abar
µ≡uν
bar∇g
νubar
µ= 0,
whereas the equations of motion for the dark matter particles are non-geodesic,
ag
µ=rguν
gFµν ,(2.12a)
af
µ=−rfuν
fFµν ,(2.12b)
where the accelerations ag
µ≡uν
g∇g
νug
µand af
µ≡uν
f∇f
νuf
µare defined with their respective
metrics. On the other hand, the equation for the vector field yields
∇geff
νhWXFµν i=1
M2
eff jµ
g−jµ
f,(2.13)
with ∇geff
νdenoting the covariant derivative associated with geff. This equation is obviously
compatible with the conservation of the currents, since by (2.2) we have also ∇geff
µjµ
g= 0 and
∇geff
µjµ
f= 0. The equation (2.13) can be also written as
∇ν
geff Tgeff
µν =−jν
g−jν
fFµν .(2.14)
Combining the equations of motion (2.12) with (2.14) we obtain the conservation law
√−g∇ν
gTg
µν +p−f∇ν
fTf
µν +√−geff∇ν
geff Tgeff
µν = 0 .(2.15)
Alternatively, such global conservation law can be obtained from the scalarity of the total
matter action under general diffeomorphisms.
3 Linear perturbations
In this section we will be interested in computing the linear perturbations of our model,
which will be extensively used in the following section for the study of the polarisation
mechanism. Following the ideas of [87] we will assume that the two fluids of dark matter
particles are slightly displaced from the equilibrium configuration by displacement vectors yµ
g
and yµ
f. This makes the dark matter medium to act as an analogue of a relativistic plasma in
electromagnetism. The dark matter currents will be slightly displaced from the equilibrium
current jµ
0=ρ0uµ
0as well, with jµ
0satisfying ∇geff
µjµ
0= 0. Furthermore, we will perturb the
two metrics around a general background in the following way6
gµν = (¯gµν +hµν )2,(3.1a)
fµν = ( ¯
fµν +ℓµν )2,(3.1b)
6By which we mean, for instance, gµν = (¯gµρ +hµρ)¯gρσ(¯gσν +hσν ).
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JCAP12(2015)026
with main interest in background solutions where ¯gµν =¯
fµν . In the latter case the potential
interactions take the form
√−geff =√−¯g
4
X
n=0
(α+β)4−nen(k),(3.2)
where kµν =αhµν +βℓµν , and traces are defined with the common background metric ¯g=¯
f.
Working at first order in the displacement of the dark matter particles and assuming
that their gradients are of the same order as the metric perturbations, the dark matter
currents can then be expanded as [87]
jµ
g=jµ
0+ 2∇geff
ν(j[ν
0⊥µ]
ρyρ
g) + O(2) ,(3.3a)
jµ
f=jµ
0+ 2∇geff
ν(j[ν
0⊥µ]
ρyρ
f) + O(2) ,(3.3b)
with the projector operator perpendicular to the four velocity of the equilibrium uµ
0given as
⊥µν =gµν
eff +uµ
0uν
0, and the remainder O(2) indicating that the expansion is at first order. The
desired plasma-like solution is achieved after plugging our ansatz (3.3) into the equations of
the vector field (2.13), which results in
WXFµν =−2
M2
eff
j[µ
0ξν]+O(2) ,(3.4)
where we have defined the projected relative displacement as ξµ≡⊥µ
ν(yν
g−yν
f). Note that
ξµis necessarily a space-like vector. In section 4we shall see that the spatial components
ξiof this vector, which can be called a dipole moment, define in the non-relativistic limit
the polarisation field of the dark matter medium as Pi=ρ∗
0ξi, where ρ∗
0is the coordinate
density associated with ρ0. But for the moment we only need to notice that ξµis a first
order quantity, therefore the field strength Fµν is itself a first order quantity, and hence the
stress-energy tensor (2.11) with respect to geff is already of second order,7
Tµν
geff =O(2) .(3.5)
For this reason, in the case of our desired plasma-like solution the contributions of the self-
interactions of the internal vector field will be at least third order in perturbation at the level
of the action. For more detail see the comprehensive derivations in [87].
Let us first expand the Lagrangian (2.1) to first order in perturbation around a com-
mon background ¯g=¯
f. We assume that there is no matter in the background, so the
matter interactions do not contribute at that order. Using (3.2) we obtain (modulo a total
divergence)
L=L¯g+√−¯gh−M2
ghµν +M2
fℓµν G¯g
µν +m2M2
eff(α+β)3ki+O(2) ,(3.6)
where G¯g
µν is the Einstein tensor for the background metric ¯g, and L¯gis the background
value of the Lagrangian. In order for the background ¯gµν in our interest to be a solution of
the theory, we have to impose that linear perturbations vanish. We find
G¯g
µν =m2M2
eff
M2
g
α(α+β)3¯gµν =m2M2
eff
M2
f
β(α+β)3¯gµν .(3.7)
7We are anticipating the form of Wgiven by (4.23) and which implies W=O(2) and WX= 1 + O(1).
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JCAP12(2015)026
This is only possible if
α
M2
g
=β
M2
f
.(3.8)
With this choice we guarantee that we are expanding around the background ¯gµν =¯
fµν that
is a solution to the background equations of motion.
As next we shall compute the action to second order in perturbations. Let us emphasize
again that thanks to the plasma-like ansatz (3.3) the self-interaction of the internal vector field
does not contribute to this order, see (3.5). Our quadratic Lagrangian above the background
¯gµν simply becomes
L=L¯g+√−¯g−M2
ghρσ ¯
Eµν
ρσ hµν −M2
fℓρσ ¯
Eµν
ρσ ℓµν +hµν Tµν
bar +Tµν
g+ℓµν Tµν
f
+1
2m2M2
eff(α+β)2k2−k2+O(3) ,(3.9)
where ¯
Eρσ
µν is the Lichnerowicz operator on the background ¯ggiven in the general case by
−2¯
Eρσ
µν hρσ =¯ghµν −¯gµν h+∇¯g
µ∇¯g
νh−2∇¯g
(µHν)+ ¯gµν ∇¯g
ρHρ
−2C¯g
µρσν hρσ −2
3R¯ghµν −1
4¯gµν h,(3.10)
where h= [h] = ¯gµν hµν ,Hµ=∇ν
¯ghµν , with C¯g
µρσν denoting the Weyl curvature of the
background metric. Of course this expression is to be simplified using the background equa-
tions (3.7). Also bear in mind that αM 2
f=βM 2
gfor having a common background ¯g=¯
f.
The field equations obtained by varying with respect to hµν and ℓµν yield
−2M2
g¯
Eµν
ρσ hρσ +Tµν
bar +Tµν
g+α(α+β)2m2M2
eff¯gµν k−kµν =O(2) ,(3.11a)
−2M2
f¯
Eµν
ρσ ℓρσ +Tµν
f+β(α+β)2m2M2
eff¯gµν k−kµν =O(2) ,(3.11b)
where we recall that kµν =αhµν +βℓµν and k= [k] = ¯gµν kµν . Of course those equations can
also be recovered directly from the general Einstein field equations (2.9).
In the present paper we shall mostly make use of that combination of the two Einstein
field equations (3.11) which corresponds to the propagation of a massless spin-2 field. We
subtract the two equations from each other so as to cancel the mass term, resulting in
−2¯
Eµν
ρσ M2
g
αhρσ −M2
f
βℓρσ+1
αTµν
bar +Tµν
g−1
βTµν
f=O(2) .(3.12)
However we shall also use the contribution of the mass term in the linearised Bianchi identities
associated with (3.11). Thus, last but not least we can act on (3.11) with ∇¯g
ν, yielding
∇¯g
νkµν −¯gµν k=O(2) .(3.13)
In deriving this relation we use the result that, for instance, ∇¯g
νTµν
g=ρgaµ
g=O(2) which
comes from the equations of motion (2.12) and the fact that there is no matter in the
background. We shall see in the next section how important is this relation for the present
model to recover the looked-for phenomenology of dark matter at galactic scales.
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JCAP12(2015)026
4 Polarisation mechanism and MOND
We next consider the Newtonian or non-relativistic (NR) limit of our model. In the previous
section we derived the field equations at linear order around a vacuum background metric
¯g, which necessarily obeys the equations (3.7). Thus, the background is a de Sitter solution,
with cosmological constant given by the graviton’s mass, Λ ∼m2. When computing the
Newtonian limit applied to describe the physics of the local universe, e.g. the solar system
or a galaxy at low redshift, we shall neglect the cosmological constant and shall approximate
the de Sitter metric ¯gby a flat Minkowski background η.
The best way to implement (and justify) this approximation is to perform an expansion
on small scales, say r→0. When we solve for the gravitational field in the solar system or a
galaxy embedded in the de Sitter background, we get terms like 1−2M r−1−Λr2/3 (de Sitter-
Schwarzschild solution), where Mis the mass of the Sun or the galaxy. On small scales the
term 2Mr−1will dominate over Λr2/3, so we can ignore the influence of the cosmological
constant if the size of the system is ≪(6M/Λ)1/3. In the present case we shall require
that the size of our system includes the very weak field region far from the system where
the MOND formula applies. In that case the relevant scale is r0=pM/a0, where a0≃
1.2×10−10 m/s2is the MOND acceleration. So our approximation will make sense provided
that r0≪(6M/Λ)1/3. For a galaxy with mass M∼1011 M⊙the MOND transition radius is
r0∼10 kpc. With the value of the cosmological constant Λ ∼(3 Gpc)−2we find (6M/Λ)1/3∼
700 kpc, so we are legitimate to neglect the influence of the cosmological constant. For the
solar system, r0∼0.04 pc while (6M⊙/Λ)1/3∼150 pc and the approximation is even better.
Actually we shall make the approximation ¯g≃ηonly on the particular massless spin-2
combination of the two metrics given by (3.12), namely
−2¯
Eµν
ρσ M2
g
αhρσ −M2
f
βℓρσ=−1
αTµν
bar +Tµν
g+1
βTµν
f+O(2) ,(4.1)
where the Lichnerowicz operator (3.10) is now the flat one, and also in the constraint equa-
tion (3.13). We shall not need to consider the other, independent (massive) combination of
the two metrics. In the NR limit when c→ ∞ we parametrize the two metric perturbations
hµν and ℓµν by single Newtonian potentials Ugand Ufsuch that
h00 =Ug
c2+O1
c4, ℓ00 =Uf
c2+O1
c4,(4.2a)
h0i=O1
c3, ℓ0i=O1
c3,(4.2b)
hij =δij
Ug
c2+O1
c4, ℓij =δij
Uf
c2+O1
c4.(4.2c)
We shall especially pay attention to the potential Ugsince it represents the ordinary Newto-
nian potential felt by ordinary baryonic matter. Similarly we parametrize the internal vector
field Aµby a single Coulomb scalar potential φsuch that
A0=φ
c2+O1
c4,(4.3a)
Ai=O1
c3.(4.3b)
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JCAP12(2015)026
It is now straightforward to show that (4.1) reduces in the NR limit to a single scalar equation
for a combination of the Newtonian potentials Ugand Uf,8
∆ 2M2
g
αUg−2M2
f
βUf!=−1
αρ∗
bar +ρ∗
g+1
βρ∗
f,(4.4)
where ∆ = ∇2is the ordinary Laplacian, and ρ∗
bar,ρ∗
gand ρ∗
fare the ordinary Newtonian
(coordinate) densities, satisfying usual continuity equations such as ∂tρ∗
g+∇·(ρ∗
gvg) = 0.
Notably, all non-linear corrections O(2) in (4.1) are negligible. In principle, we should a priori
allow some parametrized post-Newtonian (PPN) coefficients in the spatial components ij of
the metrics (4.2), say γgand γf. However, when plugging (4.2) into (4.1) we determine
from the ij components of (4.1) that in fact γg= 1 and γf= 1. Similarly we find that the
equations (2.13) governing the internal vector field reduce to a single Coulomb type equation,
∇·hWX∇φi=1
M2
eff rgρ∗
g−rfρ∗
f.(4.5)
Notice that the constants αand βin the effective metric geff to which is coupled the internal
field intervene in the NR limit only in the expression
X=|∇φ|2
2λ(α+β)4.(4.6)
Next, the equations of the baryons which are geodesic in the metric greduce to
dvbar
dt=∇Ug,(4.7)
while the equations of motion (2.12) of the dark matter particles become
dvg
dt=∇Ug+rg∇φ , (4.8a)
dvf
dt=∇Uf−rf∇φ . (4.8b)
At this stage the important point is to recall that the two sectors associated with
the metrics gand fdo not evolve independently but are linked together by the constraint
equation (3.13). We now show that this constraint provides a mechanism of gravitational
polarisation in the NR limit and yields the MOND equation for the potential Ugfelt by
baryons. In flat space-time this constraint is
∂νkµν −ηµν k=O(2) ,(4.9)
where kµν =αhµν +βℓµν and k=ηρσ kρσ . Plugging (4.2) into (4.9) we readily obtain a
constraint on the gravitational forces felt by the DM particles, namely
∇αUg+βUf= 0 .(4.10)
8We shall adopt the usual boldface notation for ordinary three-dimensional Euclidean vectors. Also, from
now on we no longer write the neglected remainders O(1/c2).
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JCAP12(2015)026
Note that the constraint (4.10) comes from a combination between the 00 and ij components
of the metrics (4.2). Using (4.10) we express the equations of motion (4.8) solely in terms of
the potential Ugruling the baryons,
dvg
dt=∇Ug+rg∇φ , (4.11a)
dvf
dt=−α
β∇Ug−rf∇φ . (4.11b)
As we see, the gravitational to inertial mass ratio mg/miof the fparticles when measured
with respect to the gmetric is mg/mi=−α/β.
We look for explicit solutions of (4.11) in the form of plasma-like oscillations around
some equilibrium solution. We immediately see from (4.11) the possibility of an equilibrium
(i.e. for which dvg/dt= dvf/dt= 0) when we have the following relation between constants,
α
β=rf
rg
.(4.12)
The equilibrium holds when ∇Ug+rg∇φ= 0, i.e. when the Coulomb force annihilates the
gravitational force. To describe in the proper way the two DM fluids near or at equilibrium
we use the NR limits of the relations (3.3), with an appropriate choice of the equilibrium
configuration. From (4.11) we note that (α+β)v0=αvg+βvfis constant, hence we define
the equilibrium in such a way that the two displacement vectors ygand yfwith respect to
that equilibrium obey αyg+βyf=0, and in particular we choose the equilibrium velocity
to be v0=0. We shall now define the relative displacement or dipole moment vector by
ξ=yg−yf.9Hence (3.3) imply
rgρ∗
g=ρ∗
0−β
α+β∇·P,(4.13a)
rfρ∗
f=ρ∗
0+α
α+β∇·P,(4.13b)
where ρ∗
0is the common density of the two DM fluids at equilibrium in the absence of external
perturbations (far from any external mass), and P=ρ∗
0ξis the polarisation. The velocity
fields of the two fluids are
vg=β
α+β
dξ
dt,vf=−α
α+β
dξ
dt,(4.14)
where dξ/dt=∂tξ+v0·∇ξdenotes the convective derivative, which reduces here to the
ordinary derivative since v0=0.
By inserting (4.13) into (4.5) we can solve for the polarisation resulting in
P=−M2
eff WX∇φ . (4.15)
Furthermore, by combining (4.14) and (4.11) and making use of the previous solution for the
polarisation (4.15), we readily arrive at a simple harmonic oscillator describing plasma like
oscillations around equilibrium, namely
d2ξ
dt2+ω2
gξ=α+β
β∇Ug.(4.16)
9Here ygand yfdenote the spatial components of the displacement four vectors ⊥µ
νyν
gand ⊥µ
νyν
fconsidered
in section 3, see (3.3)–(3.4).
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JCAP12(2015)026
In this case the plasma frequency is given by
ωg=sα+β
β
rgρ∗
0
M2
eff WX
.(4.17)
Next, consider the equation for the Newtonian potential Ugin the ordinary sector.
Combining (4.10) with (4.4) we readily obtain
∆Ug=−1
2M2
g+α2
β2M2
fhρ∗
bar +ρ∗
g−α
βρ∗
fi.(4.18)
To recover the usual Newtonian limit we must impose, in geometrical units,
M2
g+α2
β2M2
f=1
8π.(4.19)
This condition being satisfied, the right-hand side of (4.18) can be rewritten with the help
of (4.13) and the relation (4.12). We obtain an ordinary Poisson equation but modified by
polarisation effects,
∇·∇Ug−4πP
rg=−4πρ∗
bar .(4.20)
We still have to show that the polarisation will be aligned with the gravitational field ∇Ug, i.e.
we grasp a mechanism of gravitational polarisation. This is a consequence of the constitutive
relation (4.15) taken at the equilibrium point, neglecting plasma like oscillations. At this
point we have ∇Ug+rg∇φ= 0. Thus,
P=M2
eff
rgWX∇Ug.(4.21)
Finally (4.20) with (4.21) takes exactly the form of the Bekenstein-Milgrom equation [64].
To recover the correct deep MOND regime we must impose that when X → 0
1−4πM2
eff
r2
gWX=|∇Ug|
a0
,(4.22)
where a0is the MOND acceleration scale. This is easily achieved with the choice
W(X) = X − 2
3(α+β)2X3/2+OX2,(4.23)
together with fixing the constants Meff and λto the values
M2
eff =r2
g
4π, λ =a2
0
2.(4.24)
We thus recovered gravitational polarisation and the MOND equation (in agreement
with the dielectric analogy of MOND [82]) as a natural consequence of bimetric gravity
since it is made possible by the constraint equation (4.9) linking together the two metrics of
bigravity.10 The polarisation mechanism is possible only if we can annihilate the gravitational
10In the previous model [87] we had to assume that some coupling constant εin the action tends to zero.
With bimetric gravity and its nice potential interactions we see that no particular assumption is required.
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JCAP12(2015)026
force by some internal force, here chosen to be a vector field, and therefore assume a coupling
between the two species of DM particles living in the gand fsectors.
Let us recapitulate the various constraints we have found on the parameters in the
original action (2.1). These are given by (4.12) for having a polarisation process, (4.19) for
recovering the Poisson equation and (4.24) for having the correct deep MOND regime. In
addition, we recall (3.8) which was imposed in order to be able to expand the two metrics
around the same background. Strictly speaking the relation (3.8) is not necessary for the
present calculation because we neglected the influence of the background, and (3.8) may be
relaxed for some applications. Finally, we note that the result can be simplified by absorbing
the remaining constant rgtogether with the sum α+βinto the following redefinitions of the
vector field and the effective metric: Aµ→rgAµand geff
µν →(α+β)−2geff
µν . When this is
done, and after redefining also the graviton mass m, we see that we can always choose rg= 1
and α+β= 1 without loss of generality.
Finally the fully reduced form of the action (2.1) reads
Lfinal =√−gM2
g
2Rg−ρbar −ρgh1− Aµuµ
gi+p−fM2
f
2Rf−ρfh1 + α
βAµuµ
fi
+√−geffm2
4π+a2
0
8πWX,(4.25)
where we have moved for convenience the mass currents Jµ
g=ρguµ
gand Jµ
f=α
βρfuµ
fto the g
and fsectors, where the effective metric geff
µν of bigravity is given by (2.4) but with α+β= 1,
and where the kinetic term of the vector field is defined by
X=−Fµν Fµν
2a2
0
,(4.26a)
W(X) = X − 2
3X3/2+OX2.(4.26b)
In addition the coupling constants M2
gand M2
fare constrained by (4.19). We can still further
impose (3.8) if we insist that the two metrics gand fcan be expanded around a common
background ¯g=¯
f.11 Finally the theory depends also on the graviton’s mass m, hopefully
to be related to the observed cosmological constant Λ, and the MOND acceleration scale
a0. Note that in such unified approach between the cosmological constant and MOND, it is
natural to expect that a0and Λ should have comparable orders of magnitudes, i.e. a0∼√Λ
which happens to be in very good agreement with observations.
5 The decoupling limit
In this section we would like to focus on the matter sector and address the question whether
or not the interactions between the matter fields reintroduces the BD ghost [12]. For this we
derive the decoupling limit and pay special attention to the helicity-0 mode. Before doing
11A simple choice satisfying all these requirement is α=β=1
2and M2
g=M2
f=1
16π, i.e. each coupling
constant takes half the GR value. With this choice the plasma frequency (4.17) becomes
ωg=r8πρ∗
0
WX
,
in agreement with the finding of the previous model [87].
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JCAP12(2015)026
so, we first restore the broken gauge invariance by introducing the Stueckelberg fields in the
fmetric (with a, b = 0,1,2,3)
fµν →˜
fµν =fab∂µφa∂νφb,(5.1)
where the Stueckelberg fields can be further decomposed into their helicity-0 πand helicity-1
Aacounterparts,
φa=xa−mAa
Λ3
3−fab∂bπ
Λ3
3
.(5.2)
As next we can take the decoupling limit by sending Mg, Mf→ ∞, while keeping the scale
Λ3
3=Mgm2constant. For our purpose, it is enough to keep track of the contributions of
the helicity-0 mode πto the matter interactions. The BD ghost is hidden behind the higher
derivative terms of the helicity-0 interactions after using all of the covariant equations of
motion and constraints. We will therefore neglect the contribution of the helicity-1 field and
assume gµν =ηµν and
fµν =ηµν →˜
fµν = (ηµν −Πµν )2,(5.3)
where Πµν stands for Πµν ≡∂µ∂νπ/Λ3
3. In the remaining of this section all indices are raised
and lowered with respect to the Minkowski metric ηµν. The effective metric in the decoupling
limit corresponds to
geff
µν →˜geff
µν =(α+β)ηµν −βΠµν 2.(5.4)
In [88] the required criteria for ghost freedom in the kinetic and potential terms were inves-
tigated in detail and hence the new model was constructed in such a way that it fulfils these
criteria. In the decoupling limit, the contribution to the equation of motion for the helicity-0
field coming from the potential term is at most second order in derivative for the allowed
potentials. Therefore, we can concentrate on the problematic terms coming from the matter
interactions. Let us for instance consider the coupling between jµ
˜
fand Aµ, thus12
Lmat =−q−˜
f ρ ˜
f−p−˜geff Aµjµ
˜
f+1
4p−˜geff F2
µν ,(5.5)
which we can also write as
Lmat =−q−˜
f ρ ˜
f1 + Aµuµ
˜
f+1
4p−˜geff F2
µν .(5.6)
Let us first compute the equation of motion with respect to the vector field, which is simply
given by
δLmat
δAµ
=−p˜geff ∇˜geff
νFµν −q−˜
f ρ ˜
fuµ
˜
f,(5.7)
which can be also expressed as
∇˜geff
νFµν =−q−˜
f
√−˜geff
ρ˜
fuµ
˜
f=−jµ
˜
f.(5.8)
12For simplicity we will drop the constants λ,rfand Meff and assume that W(X) = Xin the following.
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JCAP12(2015)026
As next we can compute the contribution of the matter to the equation of motion with respect
to the helicity-0 field,
δLmat
δπ =∂µ∂ν
Λ3
3 δLmat
δ˜
fρσ
δ˜
fρσ
δΠµν !+∂µ∂ν
Λ3
3 δLmat
δ˜geff
ρσ
δ˜geff
ρσ
δΠµν !
=−∂µ∂ν
Λ3
3q−˜
f T µρ
˜
f(δν
ρ−Πν
ρ) + βp−˜geff Tµρ
˜geff (α+β)δν
ρ−βΠν
ρ,(5.9)
where we made use of
Tµν
˜
f=2
q−˜
f
δLmat
δ˜
fµν
,and Tµν
˜geff =2
√−˜geff
δLmat
δ˜geff
µν
.(5.10)
Applying one of the derivatives we have
δLmat
δπ =−∂ν
Λ3
3∂µq−˜
f T µρ
˜
f(δν
ρ−Πν
ρ)−q−˜
f T µρ
˜
f∂µΠν
ρ
+β∂µ(p−˜geffTµρ
eff )(α+β)δν
ρ−βΠν
ρ−β2p−˜geff Tµρ
˜geff ∂µΠν
ρ.(5.11)
The equation of motion for the vector field (5.8) can also be written as
∇µ
˜geff T˜geff
ρµ =jµ
˜
fFρµ ,(5.12)
which we can use to express
∇˜geff
µTµρ
˜geff =1
√−˜geff
∂µp−˜geffTµρ
˜geff + Γρ˜geff
µσ Tµσ
˜geff =jµ
˜
f˜gρκ
eff Fκµ ,(5.13)
hence we have
∂µp−˜geffTµρ
˜geff =p−˜geff jµ
˜
f˜gρκ
eff Fκµ −p−˜geff Γρ˜geff
µσ Tµσ
˜geff .(5.14)
Furthermore, the conservation law [see (2.15)] gives
p−˜geff∇˜geff
µTµρ
˜geff +q−˜
f∇˜
f
µTµρ
˜
f= 0 .(5.15)
Thus, we have further
∂µq−˜
f T µρ
˜
f=−p−˜geffjµ
˜
f˜
fρκFκµ −q−˜
fΓρ˜
f
µσ Tµσ
˜
f.(5.16)
Using the equation of motion for the vector field and the conservation equation, the equation
for the helicity-0 becomes
δLmat
δπ =−∂ν
Λ3
3−p−˜geff jµ
˜
f˜
fρκFκµ −q−˜
fΓρ˜
f
µσ Tµσ
˜
f(δν
ρ−Πν
ρ)−q−˜
f T µρ
˜
f∂µΠν
ρ(5.17)
+βp−˜geff jµ
˜
f˜gρκ
eff Fκµ −p−˜geff Γρ˜geff
µσ Tµσ
˜geff (α+β)δν
ρ−βΠν
ρ−β2p−˜geff Tµρ
˜geff ∂µΠν
ρ,
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JCAP12(2015)026
which we can rewrite as
δLmat
δπ =−∂ν
Λ3
3−p−˜geffjµ
˜
f˜
fρκFκµ (δν
ρ−Πν
ρ)−q−˜
f T µσ
˜
fRν
µσ
+βp−˜geffjµ
˜
f˜gρκ
eff Fκµ(α+β)δν
ρ−βΠν
ρ−β2p−˜geffTµσ
˜geff
˜
Rν
µσ,(5.18)
where we have introduced
Rν
µσ = Γρ˜
f
µσ (δν
ρ−Πν
ρ) + ∂νΠµσ ,(5.19a)
˜
Rν
µσ = Γρ˜geff
µσ (α+β)δν
ρ−Πν
ρ+β2∂νΠµσ .(5.19b)
Using the fact that the Christoffel symbol with respect to the ˜
fmetric is given by
Γρ˜
f
µσ =−˜
fρκ(δν
κ−Πν
κ)∂νΠµσ ,(5.20)
and further taking into account the following relation,
(δµ
ρ−Πµ
ρ)˜
fρσ(δν
σ−Πν
σ) = ηµν ,(5.21)
we see immediately that Rν
µσ = 0. Similarly, using the definition of the Christoffel symbol
with respect to the ˜geff metric,
Γρ˜geff
µσ =−˜gρκ
eff (α+β)δν
κ−βΠν
κ∂νΠµσ ,(5.22)
and the relation
(α+β)δµ
ρ−βΠµ
ρ˜gρσ
eff (α+β)δν
σ−βΠν
σ=ηµν ,(5.23)
we can show that also ˜
Rν
µσ = 0. Thus, the contribution of the matter part to the equation
of motion for the helicity-0 mode simplifies to
δLmat
δπ =−∂ν
Λ3
3n−p−˜geff jµ
˜
f˜
fρκ Fκµ(δν
ρ−Πν
ρ) + βp−˜geff jµ
˜
f˜gρκ
eff Fκµ(α+β)δν
ρ−βΠν
ρo.
(5.24)
Using (5.21) this can be equally written as
δLmat
δπ =−∂ν
Λ3
3n−jµ
∗Fκµ(ην κ −Πνκ )−1+βjµ
∗Fκµ(α+β)ην κ −βΠνκ −1o,(5.25)
where jµ
∗=√−˜geff jµ
˜
fis the coordinate current and represents the matter degrees of freedom
independent of the metric, while Fκµ is also independent of the metric. We can now also
apply the remaining derivative in front,
δLmat
δπ =−1
Λ3
3−∂ν(jµ
∗˜
fρκFκµ )(δν
ρ−Πν
ρ) + 1
Λ3
3
jµ
∗˜
fρκFκµ ∂ρπ
+β∂ν(jµ
∗˜gρκ
eff Fκµ)(α+β)δν
ρ−βΠν
ρ−1
Λ3
3
β2jµ
∗˜gρκ
eff Fκµ∂ρπ.(5.26)
The contribution of the matter interactions to the equation of motion for the πfield contains
higher derivative terms which we are not able to remove by invoking the covariant equations
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JCAP12(2015)026
of motion for the vector field and the dark matter particle. Hence, this reflects the presence
of a ghostly degree of freedom in the decoupling limit through the matter interaction term.
The scale Λ3is not the cut-off scale of this theory anymore, but rather given by the scale of
the mass of the introduced ghost mBD. For the precise plasma-like background solution that
we considered here, the contributions of the matter interactions and the internal vector field
will be at least third order in perturbation at the level of the action. Hence the ghost would
enter at non-linear order in perturbations, and does not show up in our linear perturbation
analysis of the gravitational polarisation in section 4. This indicates that the mass of the
ghost mBD could be large.
6 Conclusions
In this work we followed the philosophy of combining the different approaches used to tackle
the cosmological constant and dark matter problems. The origin of this dark sector consti-
tutes one of the most challenging puzzle of contemporary physics. Here we want essentially
to consider them on the same footing. The standard model of cosmology Λ-CDM, despites
many observational successes, fails to explain the observed tiny value of the cosmological
constant in the presence of large quantum corrections using standard quantum field the-
ory techniques. This has initiated the development of IR modifications of GR, like massive
gravity and bigravity. On the other hand, the model Λ-CDM does not account for many
observations of dark matter at galactic scales, being unable to explain without fine-tuning
the tight correlations between the dark and luminous matter in galaxy halos. Rather, the
dark matter phenomenology at galactic scales is in good agreement with MOND [61–63].
In the present work we aimed at addressing these two motivations under the same
umbrella using a common framework, namely the one of bigravity. An important clue in
this respect is the fact that the MOND acceleration scale is of the order of the cosmological
constant, a0∼√Λ. An additional hope was to be able to promote the MOND formula into a
decent relativistic theory in the context of bigravity. For this purpose we followed tightly the
same ingredients as in the model proposed in [88], with two species of dark matter particles
coupled to the two metrics of bigravity respectively, and linked via an internal vector field.
This paper was dedicated to explore the theoretical and phenomenological consistency
of this model and verify that it is capable to recover the MOND phenomenology on galactic
scales. We first worked out the covariant field equations with a special emphasis on the con-
tributions of the vector field and the dark matter particles. Because of the interaction term
between the dark matter particles and the vector field, the stress-energy tensors are not con-
served separately, but rather a combination of them, giving rise to a global conservation law.
The divergence of the stress-energy tensor with respect to the effective metric geff to which
is coupled the vector field [15,17], is given by the interaction of the dark matter particles
with the vector field, which has important consequences for the polarisation mechanism but
also for our decoupling limit analysis.
As next we computed the linear field equations around a de Sitter background where
the mass term shall play the role of the cosmological constant on large scales. On small scales
where the post-Newtonian limit applies, the de Sitter background can be approximated by a
flat Minkowski background. Considering a small perturbation of the Minkowski metric and
computing the Newtonian limit, we were able to show that the polarisation mechanism works
successfully and recovers the MOND phenomenology on galactic scales. We find that this
polarisation mechanism is a natural consequence of bimetric gravity since it is made possible
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JCAP12(2015)026
by the constraint equation (linearized Bianchi identity) linking together the two metrics of
bigravity. In addition it strongly relies on the internal vector field generated by the coupling
between the two species of DM particles