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Rencontres de Moriond Dark Matter with Massive Bigravity Theory Luc Blanchet Gravitation et Cosmologie (GRεCO) Institut d’Astrophysique de Paris 27 mars 2017 Luc Blanchet (IAP) DM with massive bigravity Moriond 1 / 25 Mass discrepancy versus acceleration [Milgrom 1983] A critical acceleration scale a0 ' 1.2 × 10−10 m/s2 is present in the data Luc Blanchet (IAP) DM with massive bigravity Moriond 2 / 25 The critical acceleration scale The measured value of a0 is very close (mysteriously enough) to typical cosmological values a0 ' 1.3 aΛ with aΛ = c2 2π r Λ 3 The mass discrepancy is given by Mdyn ' Mb V Vb 2 r ∼ a0 g The scale a0 gives the transition between galaxies with high and low central surface brightness a0 Σ & G for HSB galaxies (baryon dominated) a Σ . 0 for LSB galaxies (DM dominated) G Luc Blanchet (IAP) DM with massive bigravity Moriond 3 / 25 HSB versus LSB galaxies High Surface Brightness Luc Blanchet (IAP) Low Surface Brightness DM with massive bigravity Moriond 4 / 25 The MOND formula [Milgrom 1983, Bekenstein & Milgrom 1984] Rotation curves of galaxies are extremely well reproduced by the MOND formula ∇· g g = −4π G ρbaryon µ a0 | {z } avec g = ∇U MOND function The Newtonian regime is recovered when g a0 In the MOND regime g a0 we have µ ' g/a0 Newtonian regime g≫ a MOND regime g≪ a Luc Blanchet (IAP) g a DM with massive bigravity Moriond 5 / 25 Baryonic Tully-Fisher relation [Tully & Fisher 1977, McGaugh 2011] The asymptotic flat rotation velocity is approximately Vf ' G Mb a0 Luc Blanchet (IAP) DM with massive bigravity 1/4 Moriond 6 / 25 BTF relation fitted with Λ-CDM Luc Blanchet (IAP) DM with massive bigravity [Silk & Mamon 2012] Moriond 7 / 25 Modified gravity versus modified dark matter Extra fields are postulated in addition to the spin-2 metric field of GR 1 Modified gravity: The stress-energy tensor of these fields is negligible with respect to the baryonic matter density µν µν µ ν G = 8π Tb + |∇ Φ∇ {z Φ} + · · · Φ 2 = .. . 8π Tb negligible w.r.t. baryonic density Modified dark matter: The stress-energy tensor of the extra fields is comparable to the baryonic matter density µν µν (µ ν) G = 8π Tb + ρ u Ω + · · · | {z } ν µ same order as u ∇ν Ω = ··· baryonic density .. . Luc Blanchet (IAP) DM with massive bigravity Moriond 8 / 25 Pure modified gravity: TeVeS 1 [Bekenstein & Sanders 1994; Bekenstein 2004] A disformal coupling between the physical (Jordan) and Einstein metrics is postulated in the matter action g̃µν = e2φ (gµν + Uµ Uν ) − e−2φ Uµ Uν 2 The vector field is a dynamical field Z SU = C √ d4 x −g g µρ g νσ Uµν Uρσ | {z } U (1) like kinetic term 3 The scalar part is a k-essence theory Z Sφ = K Luc Blanchet (IAP) d4 x √ −g f (X) with X = ` (g µν − U µ U ν ) ∂µ φ∂ν φ DM with massive bigravity Moriond 9 / 25 Tentative conclusions on modified gravity 1 Complicated theories modifying GR with ad-hoc extra fields 2 No physical explanation for the origin of the MOND effect 3 Non-standard kinetic terms depending on an arbitrary function which is linked in fine to the MOND function 4 Stability problems associated with the fact that the Hamiltonian is not bounded from below [Clayton 2001, Bruneton & Esposito-Farèse 2007] 5 Generic problems to recover the cosmological model Λ-CDM at large scales and in particular the spectrum of CMB anisotropies Luc Blanchet (IAP) DM with massive bigravity Moriond 10 / 25 TeVeS and the CMB Luc Blanchet (IAP) [Skordis, Mota, Ferreira & Boehm 2006] DM with massive bigravity Moriond 11 / 25 Dipolar dark matter? 1 [Blanchet 2006; Blanchet & Le Tiec 2008, 2009] In electrostratics the Gauss equation is modified by the polarization of the dielectric (dipolar) material h i ρ e ∇ · (1 + χe )E = | {z } ε0 ⇐⇒ ∇·E = ρe + ρpolar e ε0 D field 2 Similarly MOND can be viewed as a modification of the Poisson equation by the polarization of some dipolar medium g ∇· µ g = −4π G ρb ⇐⇒ ∇ · g = −4π G ρb +ρpolar | {z } a0 dark matter 3 The MOND function can be written µ = 1 + χ where χ appears as a susceptibility coefficient of some “dipolar dark matter” medium Luc Blanchet (IAP) DM with massive bigravity Moriond 12 / 25 Microscopic description of DDM? 1 The DM medium by individual dipole moments p and a polarization field P P = np 2 p = mξ The polarization is induced by the gravitational field of ordinary masses P =− 3 with χ g 4π G ρDDM = −∇ · P Because like masses attract and unlike ones repel we have anti-screening of ordinary masses by polarization masses in agreement with DM and MOND χ<0 Luc Blanchet (IAP) DM with massive bigravity Moriond 13 / 25 Anti-screening by polarization masses + - + - pe + - lines of - + E + + + + p - + g + - charge Q mass M Anti-screening by polarization masses χe > 0 Luc Blanchet (IAP) - + - - + Screening by polarization charges lines of χ<0 DM with massive bigravity Moriond 14 / 25 Need of a non-gravitational internal force 1 The constituents of the dipole will repel each other so we need a non-gravitational force dv 0 = −∇ U + φ dt and look for an equilibrium when ∇ U + φ = 0 dv =∇ U +φ dt 2 The internal force is generated by the gravitational charge i.e. the mass ∆φ = − 3 4πG ρ − ρ0 χ The DM medium appears as a polarizable plasma (similar to e+ e− plasma) oscillating at the natural plasma frequency d2 ξ + ω 2 ξ = 2g dt2 Luc Blanchet (IAP) s with DM with massive bigravity ω= − 8π G ρ0 χ Moriond 15 / 25 DDM via a bimetric extension of GR 1 [Bernard & Blanchet 2014] To describe relativistically some microscopic DM particles with positive or negative gravitational masses one needs two metrics gµν obeyed by ordinary particles (including baryons) fµν obeyed by “dark” particles 2 In addition the DM particles forming the dipole moment should interact via a non-gravitational force field, e.g. a (spin-1) “graviphoton” vector field Aµ [Scherk 1968] 3 One needs to introduce into the action the kinetic terms for all these fields, and one must be careful to properly define the interaction between the two metrics gµν and fµν 4 A model was proposed but the interaction between the two metrics implies the presence of ghosts in the gravitational sector [Blanchet & Heisenberg 2015] Luc Blanchet (IAP) DM with massive bigravity Moriond 16 / 25 Nonlinear ghost-free massive bigravity theory [de Rham, Gabadadze & Tolley 2011; Hassan & Rosen 2012] The theory (called dRGT) is defined non-perturbatively as ghost-free interaction mass term Z S= z }| { 4 2 X Mf2 p M √ √ g 2 4 −gRg + m −g αn en (X) + −f Rf d x 2 2 n=0 where the interaction between the two metrics is algebraic and defined from p the elementary symmetric polynomials en (X) of the square root matrix X = g −1 f Luc Blanchet (IAP) DM with massive bigravity Moriond 17 / 25 DDM via massive bigravity theory [Blanchet & Heisenberg 2015] 1 The two species of DM particles ρg and ρf are coupled to the two metrics gµν and fµν of bigravity 2 The ghost-free potential interactions take the form of the square root of the determinant of the effective metric [de Rham, Heisenberg & Ribeiro 2014] eff gµν = α2 gµν + 2αβ gµρ Xνρ + β 2 fµν 3 eff The graviphoton Aµ is coupled to the effective metric gµν and has a µρ νσ non-standard kinetic term (with X = geff geff Fµν Fρσ ) Z S= Luc Blanchet (IAP) 2 p Mf Mg2 −g d x Rg −ρbar − ρg + −f Rf −ρf 2 2 2 √ m α µ a20 µ + −geff + Aµ jg − jf + W X 4π β 8π 4 √ DM with massive bigravity Moriond 18 / 25 General structure of the model bar Luc Blanchet (IAP) GR g DM with massive bigravity Moriond 19 / 25 General structure of the model bar g Luc Blanchet (IAP) GR g geff A DM with massive bigravity f f Moriond 19 / 25 General structure of the model m G bar g GR g geff A f f a0 Luc Blanchet (IAP) DM with massive bigravity Moriond 19 / 25 General structure of the model m G bar g GR g geff A f f a0 m Luc Blanchet (IAP) DM with massive bigravity Moriond 19 / 25 Gravitational polarization & MOND 1 Equations of motion of DM particles in the non-relativistic limit c → ∞ dvg = ∇ Ug + φ dt 2 α dvf = ∇ Uf − φ dt β With massive bigravity the two g and f sectors are linked together by a constraint equation coming from the Bianchi identities ∇ αUg + βUf = 0 3 showing that α/β is the ratio between gravitational and inertial masses of f particles with respect to g metric The DM medium is at equilibrium when the Coulomb force annihilates the gravitational force, ∇Ug + ∇φ = 0, at which point the polarization is aligned with the gravitational field P = Luc Blanchet (IAP) 1 W 0 ∇Ug 4π DM with massive bigravity Moriond 20 / 25 Gravitational polarization & MOND 1 From the massless combination of the two metrics combined with the Bianchi identity we get a Poisson equation for the ordinary Newtonian potential Ug α ∆Ug = −4π ρbar + ρg − ρf β | {z } DDM 2 With the plasma-like solution for the internal force and the mechanism of gravitational polarization this yields the MOND equation ∇· h i 1 − W 0 ∇Ug = −4πρbar | {z } MOND function 3 Finally the DM medium undergoes stable plasma-like oscillations in linear perturbations around the equilibrium Luc Blanchet (IAP) DM with massive bigravity Moriond 21 / 25 Status of the model regarding ghost instabilities 1 By construction the model is safe in the gravitational sector (no ghosts) 2 The particles ρbar , ρg and ρf are separately coupled to one and only one of the metrics of bigravity which is also safe 3 The vector field Aµ is coupled to the effective composite metric gµeffν and it has been shown that the decoupling limit is safe i.e. the mass of possible guosts is above the strong coupling scale [de Rham, Heisenberg & Ribeiro 2014] Λ3 = m2 MPlanck 1/3 4 However in our model the DM particles ρg , ρf are also coupled to the internal vector Aµ and this introduces an extra indirect interaction between the two metrics of bigravity 5 Thus we have to check the decoupling limit of the model consistently including all the interactions with matter fields Luc Blanchet (IAP) DM with massive bigravity Moriond 22 / 25 Decoupling limit of the theory 1 2 [Blanchet & Heisenberg 2017] We decouple the interactions below the strong coupling scale and concentrate on the pure interactions of the helicity-0 mode π of the massive graviton Using the Stückelberg trick, we restore the broken gauge invariance in the fµν metric by replacing it by fµν −→ fαβ ∂µ φα ∂ν φβ where 3 φα = x α − f αβ ∂β π Λ33 The two g and f metrics are given by (posing Πµν = ∂µ ∂ν π) helicity-2 mode gµν = ηµν fµν = ηµν z}|{ 2 hµν + Mg 2 Πµν − Λ33 |{z} helicity-0 mode Luc Blanchet (IAP) DM with massive bigravity Moriond 23 / 25 Decoupling limit of the theory 1 2 [Blanchet & Heisenberg 2017] We send MPl → ∞ and m → 0 while keeping Λ3 = (m2 MPl )1/3 constant In the DL limit the gravitational part of the action takes the ghost-free form Galileon terms LDL grav 3 µν = −ĥ ρσ ĥρσ Eµν z }| { 3 X bn a3 2 (3) + (∂π) (Π) e (Π) + 6 ĥµν Pµν (n) 3n Λ Λ 3 n=0 3 We must specify how the vector field Aµ will behave in the DL limit. Since it is a graviphoton, with scale given by the Planck mass, we must rescale it as Aµ = 4 µ MPl When this is done we find that the contribution of the matter to the equation of motion of the helicity-0 mode is perfectly safe a2 δLDL a1 µν mat =− T̂bar + T̂g + 3 T̂bar + T̂gµν ∂µ ∂ν π δπ 2 Λ3 | {z } second-order field equation Luc Blanchet (IAP) DM with massive bigravity Moriond 24 / 25 Conclusions 1 An important challenge is to reproduce within a single relativistic framework The concordance cosmological model Λ-CDM and its tremendous successes at cosmological scales and notably the fit of the CMB The phenomenology of MOND which is a basic set of phenomena relevant to galaxy dynamics and DM distribution at galactic scales 2 While Λ-CDM meets severe problems when extrapolated at the scale of galaxies, MOND fails on galaxy cluster and larger scales 3 Pure modified gravity theories (extending GR with extra fields) do not seem to be able to reproduce Λ-CDM and the CMB anisotropies at large scales 4 An hybrid approach (DM à la MOND) called DDM, defined within the powerful framework of the massive bigravity theory, could meet the challenges posed by MOND versus Λ-CDM Luc Blanchet (IAP) DM with massive bigravity Moriond 25 / 25