BIMETRIC GRAVITY AND PHENOMENOLOGY OF DARK MATTER

Transcription

1 Observatoire de Paris-Meudon/LUTH BIMETRIC GRAVITY AND PHENOMENOLOGY OF DARK MATTER Luc Blanchet Gravitation et Cosmologie (GRεCO) Institut d Astrophysique de Paris 13 novembre 2014 Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 1 / 44

2 Plan of the talk 1 Phenomenology of dark matter 2 Modified gravity theories 3 Modified (dipolar) dark matter 4 Bimetric approach to dipolar dark matter Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 2 / 44

3 Phenomenology of dark matter PHENOMENOLOGY OF DARK MATTER Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 3 / 44

4 Phenomenology of dark matter Evidence for dark matter in astrophysics 1 Oort [1932] noted that the sum of observed mass in the vicinity of the Sun falls short of explaining the vertical motion of stars in the Milky Way 2 Zwicky [1933] reported that the velocity dispersion of galaxies in galaxy clusters is far too high for these objects to remain bound for a substantial fraction of cosmic time 3 Ostriker & Peebles [1973] showed that to prevent the growth of instabilities in cold self-gravitating disks like spiral galaxies, it is necessary to embed the disk in the quasi-spherical potential of a huge halo of dark matter 4 Bosma [1981] and Rubin [1982] established that the rotation curves of galaxies are approximately flat, contrarily to the Newtonian prediction based on ordinary baryonic matter Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 4 / 44

5 Phenomenology of dark matter Rotation curves of galaxies are approximately flat For a circular orbit we expect G M(r) v(r) = r The fact that v(r) is constant implies that beyond the optical disk M halo (r) r ρ halo (r) 1 r 2 Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 5 / 44

6 Phenomenology of dark matter The cosmological concordance model Λ-CDM F 2.8. The left panel shows a realisation of the CMB power spectrum of the concordance ΛCDM model (red This model brilliantly accounts for: 1 The mass discrepancy between the dynamical and luminous masses of clusters of galaxies 2 The precise measurements of the anisotropies of the cosmic microwave background (CMB) 3 The formation and growth of large scale structures as seen in deep redshift and weak lensing surveys 4 The fainting of the light curves of distant supernovae Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 6 / 44

7 Phenomenology of dark matter Problem of the fundamental constituents of the Universe Λ-CDM assumes General Relativity is the correct theory of gravity but: No known particle in the standard model of particle physics could be the particle of dark matter Extensions of the standard model of particle physics provide well-motivated but yet to be discovered candidates The cosmological constant Λ does not have the right value when interpreted as a vacuum energy associated with the fluctuations of the gravitational field Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 7 / 44

8 Phenomenology of dark matter The WIMP miracle [Lee & Weinberg 1977, Kolb & Turner 1988] 1 A new (heavy) particle X is initially in thermal equilibrium. Its relic density is Ω X 1 σv m2 X g 4 X 2 With m X 100 Gev and g X 0.6 (electroweak scale) Ω X 0.1 Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 8 / 44

9 Phenomenology of dark matter Challenges with CDM at galactic scales The CDM paradigm faces severe challenges when compared to observations at galactic scales [McGaugh & Sanders 2004, Famaey & McGaugh 2012] 1 Unobserved predictions Numerous but unseen satellites of large galaxies Phase-space correlation of galaxy satellites Generic formation of dark matter cusps in galaxies Tidal dwarf galaxies dominated by dark matter 2 Unpredicted observations Correlation between mass discrepancy and acceleration Surface brightness of galaxies and the Freeman limit Flat rotation curves of galaxies Baryonic Tully-Fisher relation for spirals Faber-Jackson relation for ellipticals All these challenges are mysteriously solved (sometimes with incredible success) by the MOND empirical formula [Milgrom 1983] Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 9 / 44

10 Phenomenology of dark matter Correlation between mass discrepancy and acceleration Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 10 / 44

11 Phenomenology of dark matter Baryonic Tully-Fisher relation [Tully & Fisher 1977, McGaugh 2011] We have approximately V f ( ) 1/4 G M b a 0 where a m/s 2 is very close (mysteriously enough) to typical cosmological values a a Λ with a Λ = c2 Λ 2π 3 Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 11 / 44

12 Phenomenology of dark matter Baryonic Tully-Fisher relation with Λ-CDM [Silk & Mamon 2012] Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 12 / 44

13 Phenomenology of dark matter Mass velocity dispersion relation [Faber & Jackson 1976] Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 13 / 44

14 Phenomenology of dark matter MOND versus Λ-CDM Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 14 / 44

15 Phenomenology of dark matter MOND problem with galaxy clusters [Gerbal, Durret et al. 1992, Sanders 1999] The mass discrepancy is 4 5 with Newton and 2 with MOND The bullet cluster and more generally X-ray emitting galaxy clusters can be fitted with MOND only with a component of baryonic dark matter and/or hot/warm neutrinos [Angus, Famaey & Buote 2008] Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 15 / 44

16 Modified gravity theories MODIFIED GRAVITY THEORIES Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 16 / 44

17 Modified gravity theories Modified Poisson equation [Milgrom 1983; Bekenstein & Milgrom 1984] [ ( g µ a 0 ) }{{} MOND function ] U = 4π G ρ }{{} b baryons Newtonian regime g a MOND regime g a g a The Newtonian regime is recoved when g a 0 In the MOND regime g a 0 we have µ = g a 0 + O(g 2 ) Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 17 / 44

18 Modified gravity theories Generalized scalar-tensor (RAQUAL) [Bekenstein & Sanders 1994] L scalar-tensor = [ ( g R + 2a 2 0 F g µν )] µφ ν φ 16π a 2 + L m [ g µν, Ψ] 0 This is a scalar k-essence theory (non-standard kinetic term) with matter fields universally coupled to the physical metric g µν = e 2φ g µν By specifying F this theory reproduces MOND in galaxies However light signals do not feel the presence of the scalar field so the theory does not explain the dark matter seen by gravitational lensing Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 18 / 44

19 Modified gravity theories Tensor-vector-scalar (TeVeS) [Bekenstein 2004, Sanders 2005] Disformal coupling postulated in the matter action Scalar-tensor part similar as RAQUAL L scalar-tensor = g µν = e 2φ (g µν + A µ A ν ) e 2φ A µ A ν [ ( g R + 2a 2 0 F (g µν A µ A ν ) )] µφ ν φ 16π a 2 + L m [ g µν, Ψ] 0 This requires adding a vector part L vector = g 16π [kf µν F µν + λ ( A µ A µ + 1 ) ] }{{} Lagrange constraint Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 19 / 44

20 Modified gravity theories Generalized Einstein-Æther [Zlosnik et al. 2007, Halle et al. 2008] Phenomenological approach to Lorentz-invariance violation [Jacobson & Mattingly 2001] g L Æ = [R + K + λ ( g µν n µ n ν + 1 ) ] 16π }{{} Lagrange constraint Most general Lagrangian quadratic in derivatives of the vector field n µ K = K µνρσ µ n ρ ν n σ K µνρσ = c 1 g µν g ρσ + c 2 g µρ g νσ + c 3 g µσ g νρ + c 4 n µ n ν g ρσ Vector k-essence generalization of Einstein-Æther theories g L GÆ = [R + F (K) + λ ( g µν n µ n ν + 1 )] 16π No problem with light deflection in these theories Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 20 / 44

21 Modified gravity theories Generalized Einstein-Æther [Zlosnik et al. 2007, Halle et al. 2008] Motivation from Lorentz-invariance violation [Jacobson & Mattingly 2001] g L Æ = [R + K + λ ( g µν n µ n ν + 1 ) ] 16π }{{} Lagrange constraint Most general Lagrangian quadratic in derivatives of the vector field n µ K = K µνρσ µ n ρ ν n σ K µνρσ = c 1 g µν g ρσ + c 2 g µρ g νσ + c 3 g µσ g νρ + c 4 n µ n ν g ρσ Vector k-essence generalization of Einstein-Æther theories g L GÆ = [R + F (K) + λ ( g µν n µ n ν + 1 )] 16π No problem with light deflection in these theories Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 21 / 44

22 Modified gravity theories From the Æther to the Khronon n μ A way to get rid of the Lagrange constraint is to choose n µ to be hypersurface orthogonal Στ n µ = N µ τ where τ is a dynamical scalar field called the Khronon and where N = 1 g ρσ ρ τ σ τ MOND is a modification of gravity in the weak-acceleration regime so it is natural to look for a modification of GR for weak accelerations. We can take the acceleration of the congruence of worldlines orthogonal to the foliation a µ = n ν ν n µ = D µ ln N Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 22 / 44

23 Modified gravity theories MOND theory based on the Khronon [Blanchet & Marsat 2011] Like for Einstein-Æther the theory admits a covariant formulation g [ ] L Einstein-Khronon = R 2f(a) + L m [g µν, Ψ] 16π but its content is best understood in adapted coordinates where t = τ L Einstein-Khronon = γ [ ] 16π N R + K ij K ij K 2 2f(a) + L m [N, N i, γ ij, Ψ] where a µ = D µ ln N and N, N i, γ ij are purely geometrical quantities For a certain choice of function f we recover GR+Λ in the strong-field regime a a 0 and MOND in the weak-field regime a a 0 No problem with light deflection But no viable cosmology Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 23 / 44

24 Modified gravity theories Bimetric theory (BiMOND) [Milgrom 2009] S BiMOND = 1 16π [ d 4 x + S m [g µν, ψ] }{{} ordinary matter [ Z ( g) 1/2 R[g] + ( ĝ) 1/2 R[ĝ] + 2a 2 0(gĝ) 1/4 F a 2 0 }{{} interaction term + Ŝm[ĝ µν, ˆψ] }{{} twin matter ] ] The two metrics g µν and ĝ µν interact through the difference of Christoffel symbols C λ µν = Γ λ µν ˆΓ λ µν (which is a tensor) Z = 1 [ ] 2 (gµν + ĝ µν ) C γ µλ Cλ νγ CµνC γ λγ λ The twin matter is not dark matter but is a replica of ordinary matter living in the other sector Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 24 / 44

25 Modified (dipolar) dark matter MODIFIED (DIPOLAR) DARK MATTER Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 25 / 44

26 Modified (dipolar) dark matter Dielectric analogy of MOND [Blanchet 2007] In electrostratics the Gauss equation is modified by the polarization of the dielectric (dipolar) material [ ] (1 + χ e )E }{{} D field = ρ e E = ρ e + ρ polar e ε 0 Similarly MOND can be viewed as a modification of the Poisson equation by the polarization of some dipolar medium [ ( ) ] ( ) g µ g = 4π G ρ b g = 4π G ρ b +ρ polar a 0 }{{} dark matter The MOND function can be written µ = 1 + χ where χ appears as a susceptibility coefficient of some dipolar DM medium ε 0 Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 26 / 44

27 Modified (dipolar) dark matter Dark matter made of a polarizable medium? The DM medium by individual dipole moments p and a polarization field P P = n p with p = m ξ The polarization is induced by the gravitational field of ordinary masses P = χ 4π G g Suppose that the dipole moments are made by particles with positive and negative gravitational masses (m i, m g ) = (m, ±m) Because like masses attract and unlike ones repel we have anti-screening of ordinary masses by polarization masses χ < 0 This is in agreement with DM and MOND Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 27 / 44

28 Modified (dipolar) dark matter Need of a non-gravitational internal force The constituents of the dipole will repel each other so we need to a non-gravitational force to stabilize the dipolar medium dv ± dt = ± ( U + φ ) The internal force is generated by the gravitational charge i.e. the mass φ = 4πGm ( ) n+ n χ The DM medium appears as a polarizable plasma of particles (m, ±m) oscillating at the natural plasma frequency d 2 ξ dt 2 + ω2 ξ = 2g with ω = 8π G m n χ Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 28 / 44

29 Modified (dipolar) dark matter Dipolar dark matter (DDM) [Blanchet & Le Tiec 2008; 2009] Implements in a relativistic way the dielectric analogy of MOND The DDM action in standard general relativity is S DDM = d 4 x [ ] g ρ + J µ ξµ V (P ) An interaction term couples the relativistic matter current J µ = ρu µ to a vector field ξ µ called the dipole moment A potential term V is built from the norm of the associated polarization field P = ρ ξ and projected orthogonally to the four-velocity u µ The only physical components of the dipole moment are those orthogonal to the four-velocity, i.e. ξ µ = µ ν ξ ν Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 29 / 44

30 Modified (dipolar) dark matter The internal potential V MOND regime Newtonian regime 0 a 0 P The potential V is phenomenologically determined through third order V = Λ 8π + 2π P π2 3a 0 P 3 + O ( P 4 ) The natural order of magnitude of the cosmological constant Λ is comparable with a 0 namely Λ a 2 0 in good agreement with observations Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 30 / 44

31 Modified (dipolar) dark matter Agreement with Λ-CDM at cosmological scales In a cosmological perturbation around a FLRW background the space-like dipole moment must belong to the first-order perturbation ξ µ = O(1) The stress-energy tensor reduces to T µν = T µν DE + T µν DDM where the DE is given by the cosmological constant Λ and the DDM takes the form of a perfect fluid with zero pressure T µν DDM = ε uµ u ν + O(2) where ε = ρ µ P µ is a dipolar energy density The dipolar fluid is undistinguishable from standard Λ-CDM at the level of first-order cosmological perturbations At second-order cosmological perturbation it induces some non-gaussianity in the bispectrum of curvature perturbation [Blanchet, Langlois, Le Tiec & Marsat 2012] Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 31 / 44

32 Modified (dipolar) dark matter Hypothesis of weak clustering of DDM Baryonic matter follows the geodesic equation u µ = 0 therefore collapses in regions of overdensity Dipolar dark matter obeys u µ = F µ with the internal force balancing the gravitational field created by an overdensity The mass density of dipolar dark matter in a galaxy at low redshift should be smaller than the baryonic density and maybe close to its mean cosmological value ρ ρ ρ b and v 0 Using this hypothesis one can show that the phenomenology of MOND is recovered at galactic scales However the model involves an instability and does not reduces to the polarizable DM medium of masses (m, ±m) in the Newtonian limit Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 32 / 44

33 Bimetric approach to dipolar dark matter BIMETRIC APPROACH TO DIPOLAR DARK MATTER Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 33 / 44

34 Bimetric approach to dipolar dark matter Bimetric extension of GR with dark matter [Bernard & Blanchet 2014] 1 Baryons are coupled to the ordinary metric g µν 2 Two species of DM particles are coupled to different metrics g µν and g µν 3 The two metrics interact via an auxiliary metric f µν defined by f µν = f ρσ g ρµ g νσ = f ρσ g ρν g µσ 4 An internal gauge vector field A µ links the DM particles together S = ordinary sector dark sector { { ( }} ){{ ( }} ){ d 4 R 2λ x g 32π ρ R 2λ b ρ + g 32π ρ + [ ]} R 2λf f + (j µ j µ )A µ + a2 0 16πε 8π W (X) }{{} interaction sector Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 34 / 44

35 Bimetric approach to dipolar dark matter Vector k-essence internal force field 1 The field A µ is generated by the mass of the DM particles ( graviphoton ) 2 Assume a non-standard kinetic term W (X) where X = F µν F µν 2a Adjust phenomenologically the function W so as to recover MOND in the weak-acceleration regime a 0 the 1PN limit of GR in the strong-acceleration regime a 0 X 2 3 X3/2 + O ( X 2) when X 0 W (X) = A + B ( ) 1 X b + o X b when X Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 35 / 44

36 Bimetric approach to dipolar dark matter Plasma-like solution for the dark matter medium Field equation for the graviphoton D ν [ W F µν] = 4π ( j µ j µ) The two DM fluids differ by small displacements y µ and y µ from a common equilibrium configuration j µ = j µ 0 + D ν (j0 ν y µ jµ 0 yν ) + O (2), j µ = j µ 0 + D ( ν j ν 0 y µ ) jµ 0 yν + O (2) At the basis of the model is the plasma-like solution for the internal field W F µν = 4π ( j µ 0 ξν j ν 0 ξ µ ) + O (2) where ξ µ = yµ yµ is the relative displacement vector or dipole vector Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 36 / 44

37 Bimetric approach to dipolar dark matter Perturbative solution for the interaction metric f µν Define the matrices G = (f νρ g µρ ) and G = (f νρ g µρ ) GG = GG = 1 Perturbative solution in H = 1 2 (G G) is The matrix expansion series 1 + H2 = + p=0 G = H H 2 G = H H 2 γ p H 2p with γ p = ( )p+1 (2p 3)!! 2 p p! also plays a crucial role in the definition of the mass term in resummed ghost-free massive gravity theories [de Rham, Gabadadze & Tolley 2012] Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 37 / 44

38 Bimetric approach to dipolar dark matter Non-relativistic limit of the model 1 The fluids of DM particles slightly differ from an equilibrium configuration ρ = ρ P ρ = ρ P where the polarization P = ρ 0 ξ is proportional to the internal force φ 2 The two Newtonian potentials U and U obey when ε 1 U + U = 0 3 The remaining Poisson equation in the ordinary sector reduces to [ ] U = 4π ρ b + ρ ρ }{{} dark matter ρ DM = ρ ρ = P In the limit ε 1 one obtains a mechanism of gravitational polarization Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 38 / 44

39 Bimetric approach to dipolar dark matter Gravitational polarization and MOND 1 The DM medium appears to be made of particles (m i, m g ) = (m, ±m) dv dt = ( U + φ ) dv dt = ( U + φ ) 2 At equilibrium the polarization is aligned with the gravitational field P = W 4π U 3 The equation for the gravitational field is modified by gravitational polarization and becomes equivalent to the MOND equation [ ] U 4πP = 4π ρ b 4 The DM medium undergoes stable plasma-like oscillations Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 39 / 44

40 Bimetric approach to dipolar dark matter Post-Newtonian limit in the Solar System 1 Parametrize the two metrics at 1PN order by standard 1PN potentials [ V, V i ] and g 1PN [ V, V i ] g 1PN µν 2 Solve the algebraic equation defining the metric f µν to obtain [ V + V fµν 1PN 2 µν, V i + V i ] 2 3 When ε 1 the potentials V and V i in the ordinary sector obey the same standard 1PN equations as in GR The model has the same post-newtonian limit as general relativity and is thus viable in the Solar System (in particular β PPN = γ PPN = 1) Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 40 / 44

41 Bimetric approach to dipolar dark matter First-order cosmological perturbations 1 Start from isotropic and homogeneous background solutions g FLRW µν scale factor a g FLRW µν scale factor a } = f FLRW µν scale factor aa 2 Adjust a and a to the different matter contents in the two backgrounds and relate the cosmological constants in the action to the observed cosmological constant Λ in the ordinary sector 3 Compute the first-order perturbations using the standard SVT formalism and define effective gauge invariant DM variables in first-order perturbations as seen in the ordinary sector The model is undistinguishable from standard Λ-CDM at the level of first-order cosmological perturbations In particular it reproduces the observed anisotropies of the CMB Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 41 / 44

42 Bimetric approach to dipolar dark matter Could DM be due to vacuum quantum fluctuations? Suppose that the DM medium with masses (m, ±m) is made of virtual particle-antiparticle pairs The classical separation ξ between pairs should be ξ In the MOND regime the polarization is P a0 4πG n This corresponds to a typical separation a 0 c 4πG cm 3 ξ n 1/ cm m c with P = nm ξ so And a typical mass m 14 Mev Such a medium looks like the QCD vacuum! [Chardin 2008, Hajdukovic 2011] Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 42 / 44

43 Bimetric approach to dipolar dark matter Could DM be due to vacuum quantum fluctuations? A priori such an interpretation poses a lot of problems Assuming that anti-particles have mass (m i, m g ) = (m, m) is at odds with all theoretical expectations [Morrison & Gold 1957] This is ruled out by the K 0 - K 0 system which shows that the gravitational mass of K 0 is the same as that of K 0 [Good 1960] This is severely constrained by equivalence principle Eötvos-type experiments using the virtual e + e and q q pairs in ordinary materials [Schiff 1959] The description of vacuum fluctuations based on Compton s separation is merely semi-classical and probably oversimple Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 43 / 44

44 Bimetric approach to dipolar dark matter Could DM be due to vacuum quantum fluctuations? However the present model seems to allow such an interpretation There are no negative masses in the model All experiments testing the free-fall of anti-particles (e.g. GBAR at CERN) are done in the Earth gravitational field g a 0 in which the dipolar DM medium is inactive The asymmetry between the two sectors of the model would simply be the usual matter-antimatter asymmetry The internal U(1) force field A µ would have to be replaced by a non-abelian Yang-Mills vector field A a µ based on the color group SU(3) Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 43 / 44

45 Bimetric approach to dipolar dark matter Conclusions 1 Λ-CDM is an extremely successful cosmological model but poses the problem of the fundamental constituents of the Universe faces severe challenges when compared to observations at galactic scale 2 MOND is a successful alternative to dark matter at galactic scales but is based on an empirical formula not explained in terms of fundamental physics does not work at galaxy cluster scale and in cosmolgy 3 Reconciling Λ-CDM at cosmological scale and MOND at galactic scale into a single relativistic theory is a great challenge 4 A non-standard form of dark matter could exist, possibly coupled to a modification of gravity itself, which might explain the two antinomic aspects of DM that we see at cosmological and galactic scales Luc Blanchet (IAP) Bimetric gravity and DM Séminaire Meudon/LUTH 44 / 44