Atelier vide quantique et gravitation DARK MATTER AND GRAVITATIONAL THEORY. Luc Blanchet. 12 décembre 2012

Transcription

1 Atelier vide quantique et gravitation DARK MATTER AND GRAVITATIONAL THEORY Luc Blanchet Gravitation et Cosmologie (GRεCO) Institut d Astrophysique de Paris 12 décembre 2012 Luc Blanchet (GRεCO) Atelier Vide Quantique 1 / 46

2 Plan of the talk 1 Dark matter in astrophysics 2 Dark matter or modified gravity? 3 Modified gravity approaches 4 Modified dark matter approach 5 Testing MOND in the Solar System Luc Blanchet (GRεCO) Atelier Vide Quantique 2 / 46

3 Dark matter in astrophysics DARK MATTER IN ASTROPHYSICS Luc Blanchet (GRεCO) Atelier Vide Quantique 3 / 46

4 Dark matter in astrophysics Evidence for Dark Matter in astrophysics 1 Oort [Oort 1932] noted that the sum of the observed matter in the vicinity of the Sun falls short of explaining the vertical motions of stars in the Milky Way 2 Zwicky [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 and Peebles [Ostriker & Peebles 1973] showed that in order to prevent the growth of instabilities in cold, self-gravitating disks like spiral galaxies, it was necessary to embed the disk in the quasi-spherical potential of a huge halo of dark matter 4 Bosma and Rubin [Bosma 1981, Rubin 1982] established that the rotation curves of galaxies are approximately flat, contrary to the Newtonian expectation without dark matter Luc Blanchet (GRεCO) Atelier Vide Quantique 4 / 46

5 Dark matter in astrophysics 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 (GRεCO) Atelier Vide Quantique 5 / 46

6 Dark matter in astrophysics Problem of the fundamental constituents of the Universe Λ-CDM assumes General Relativity is the correct theory of gravity But this poses the problem of the unknown constituents of the Universe: 1 No known particle in the standard model of particle physics could be the particle of dark matter 2 Extensions of the standard model of particle physics provide well-motivated but yet to be discovered candidates 3 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 (GRεCO) Atelier Vide Quantique 6 / 46

7 Dark matter in astrophysics Cosmic N-body simulations Thanks to high precision N-body simulations in cosmology the model Λ-CDM can be extrapolated and tested at the smaller scale of galaxies Luc Blanchet (GRεCO) Atelier Vide Quantique 7 / 46

8 Dark matter in astrophysics Problems of CDM with galactic halos The CDM paradigm faces severe challenges when compared to observations at galactic scales [McGaugh & Sanders 2004; Famaey & McGaugh 2012] 1 Prediction of numerous but unseen satellites of large galaxies 2 Generic formation of cusps of DM in central regions of galaxies while the rotation curves seem to favor a constant density profile in the core 3 Evidence that tidal dwarf galaxies are dominated by DM contrary to CDM predictions [Bournaud et al. 2007; Gentile et al. 2007] 4 Failure to explain in a natural way Milgrom s law, that DM arises only in regions where gravity falls below some universal acceleration scale a 0 5 Difficulty at explaining in a natural way the flat rotation curves of galaxies and the baryonic Tully-Fisher relation Luc Blanchet (GRεCO) Atelier Vide Quantique 8 / 46

9 Dark matter in astrophysics Rotation curves of galaxies are flat For a circular orbit v(r) = G M(r) r The fact that v(r) is approximately constant implies that beyond the optical disk M halo (r) r ρ halo (r) 1 r 2 Luc Blanchet (GRεCO) Atelier Vide Quantique 9 / 46

10 Dark matter in astrophysics Baryonic Tully-Fisher relation [Tully & Fisher 1977, McGaugh 2011] The relation between the asymptotic flat velocity and the mass of galaxies is V f ( G M b a 0 ) 1/4 where a m/s 2 Luc Blanchet (GRεCO) Atelier Vide Quantique 10 / 46

11 Dark matter or modified gravity? DARK MATTER OR MODIFIED GRAVITY? Luc Blanchet (GRεCO) Atelier Vide Quantique 11 / 46

12 Dark matter or modified gravity? Dark matter or modified gravity? [Le Verrier 1845] Dark matter Discovery of the planet Neptune from the Newtonian perturbations induced on the planet Uranus Luc Blanchet (GRεCO) Atelier Vide Quantique 12 / 46

13 Dark matter or modified gravity? Planet Neptune also discovered by Adams [unpublished] Luc Blanchet (GRεCO) Atelier Vide Quantique 13 / 46

14 Dark matter or modified gravity? Dark matter or modified gravity? [Le Verrier 1846] Modified gravity Anomalous orbital precession of the planet Mercury finally explained by general relativity Luc Blanchet (GRεCO) Atelier Vide Quantique 14 / 46

15 Dark matter or modified gravity? The MOND equation [Bekenstein & Milgrom 1984] The MOND equation can be written as the modified Poisson equation [ ( ) ] g µ g = 4π G ρ b a 0 }{{} MOND function where g = U is the gravitational field and ρ b the density of ordinary matter Newtonian regime g a MOND regime g a g a In the MOND regime g a 0 we have µ = g/a 0 + O(g 2 ) Luc Blanchet (GRεCO) Atelier Vide Quantique 15 / 46

16 Dark matter or modified gravity? Recovering flat rotation curves and the Tully-Fisher law 1 For a spherical mass in the MOND regime g 2 /a 0 g Newton hence G M a0 g or U G M a 0 ln r r 2 For circular motion V 2 /r g thus V is constant and we get V flat (G M a 0 ) 1/4 which is the baryonic Tully-Fisher relation The numerical value of a m/s 2 happens to be very close to the acceleration scale associated with the cosmological constant Λ a a Λ where a Λ 1 Λ 2π 3 Luc Blanchet (GRεCO) Atelier Vide Quantique 16 / 46

17 Dark matter or modified gravity? The MOND fit of galactic rotation curves = the solid line is the MOND fit Many galaxies are accurately fitted with MOND [Sanders 1996, McGaugh & de Blok 1998] Luc Blanchet (GRεCO) Atelier Vide Quantique 17 / 46

18 Dark matter or modified gravity? Problems of MOND in galaxy clusters [Gerbal, Durret et al. 1992] The bullet cluster [Clowe et al. 2006] and more generally X-ray emitting galaxy clusters can be fitted with MOND and a component of baryonic dark matter and hot/warm neutrinos [Angus, Famaey & Buote 2008] Luc Blanchet (GRεCO) Atelier Vide Quantique 18 / 46

19 Dark matter or modified gravity? Different approaches to the DM problem Faced with the unreasonable effectiveness of MOND, three solutions are possible 1 Standard: MOND could be explained within the CDM paradigm 2 Modified Gravity: There is a fundamental modification of the law of gravity in a regime of weak gravity (this is the traditional approach of MOND) 3 Modified Dark Matter: The law of gravity is unchanged but DM is endowed with a property which makes it able to explain the MOND phenomenology Luc Blanchet (GRεCO) Atelier Vide Quantique 19 / 46

20 Modified gravity approaches MODIFIED GRAVITY APPROACHES Luc Blanchet (GRεCO) Atelier Vide Quantique 20 / 46

21 Modified gravity approaches Scalar-Tensor theory [Bekenstein & Sanders 1994] Using the framework of scalar-tensor theories [ g L Scalar-Tensor = R + 2a 2 0 F 16π ( g µν µφ ν φ a 2 0 )] + L m [ g µν, Ψ] This is a scalar k-essence theory (non-standard kinetic term) and the matter fields are universally coupled to the physical metric g µν = e 2φ g µν The theory is not viable By relating F to the MOND function µ this theory reproduces MOND for the motion of stars in galaxies However light signals do not feel the presence of the scalar field φ so the theory does not explain the dark matter in galaxy clusters seen by gravitational lensing Luc Blanchet (GRεCO) Atelier Vide Quantique 21 / 46

22 Modified gravity approaches Scalar-Vector-Tensor theory [Bekenstein 2004, Sanders 2005] To explain the dark matter seen in clusters by gravitational lensing one invokes a disformal coupling in the matter action The scalar-tensor part of the action is [ g L Scalar-Tensor = R + 2a 2 0 F 16π g µν = e 2φ (g µν + U µ U ν ) e 2φ U µ U ν ( (g µν U µ U ν ) µφ ν φ a 2 0 )] + L m [ g µν, Ψ] This requires adding a vector part to the action g L Vector = [K g µρ g νσ [µ U ν] [ρ U σ] + λ ( g µν U µ U ν + 1 )] 16π This theory has played an important role of proof of concept Because of the constraint in the action the Hamiltonian is not bounded from below [Clayton 2001, Bruneton & Esposito-Farèse 2007] The agreement with the peaks of the CMB in cosmology is problematic Luc Blanchet (GRεCO) Atelier Vide Quantique 22 / 46

23 Modified gravity approaches Non-canonical Einstein-Æther theories [Zlosnik et al. 2007, Halle et al. 2008] These theories were introduced as a phenomenological approach to Lorentz-invariance violation [Jacobson & Mattingly 2001] g L Æ = [R + K + λ ( g µν n µ n ν + 1 )] 16π where K represents the most general Lagrangian density that is quadratic in the 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 k-essence Æ = [R + F (K) + λ ( g µν n µ n ν + 1 )] 16π Luc Blanchet (GRεCO) Atelier Vide Quantique 23 / 46

24 Modified gravity approaches From the Æther to the Khronon [Jacobson 2010, Blanchet & Marsat 2011] n μ Στ A way to get rid of the Lagrange constraint in the action is to choose the vector field to be hypersurface orthogonal 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-field 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 (GRεCO) Atelier Vide Quantique 24 / 46

25 Modified gravity approaches Specific example of MOND theory [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 However 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. The theory exhibits a violation of Lorentz invariance It reduces to GR with a cosmological constant in the strong-field regime a and so is consistent with Solar System tests For a certain choice of function f in the weak-field regime a 0 the theory reduces to MOND Luc Blanchet (GRεCO) Atelier Vide Quantique 25 / 46

26 Modified dark matter approach MODIFIED DARK MATTER APPROACH Luc Blanchet (GRεCO) Atelier Vide Quantique 26 / 46

27 Modified dark matter approach Analogy with the physics of dielectrics [Blanchet 2007] In electrostratics the Gauss equation is modified by the polarization of the dielectric 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 digravitational medium ε 0 [ ( ) ] ( ) g µ g = 4π G ρ b g = 4π G ρ b +ρ polar a 0 }{{} dark matter We may write µ = 1 + χ where χ appears to be the gravitational susceptibility of the DM medium Luc Blanchet (GRεCO) Atelier Vide Quantique 27 / 46

28 Modified dark matter approach Dark matter made of gravitational dipole moments? 1 Density of DM takes a dipolar form ρ DM = P 2 The polarization field P is aligned with the gravitational field P = χ 4π G g 3 The gravitational susceptibility coefficient is related to the MOND function by χ = µ 1 The MOND function µ appears as a digravitational coefficient characterizing the microscopic physics of the hypothetical dipolar DM medium Luc Blanchet (GRεCO) Atelier Vide Quantique 28 / 46

29 Modified dark matter approach Newtonian-like model of dipolar dark matter 1 Each gravitational dipole would be interpreted as a doublet of particles ordinary mass (M, M) (m i, m g ) = (m, ±m) m g 2 The dipole moments tend to align in the same direction as the gravitational field thus (m, m) dipole moment (m, - m) - m g χ < 0 3 The constituents of the dipole repel each other so we invoke a non-gravitational internal force 4 The DM medium then appears to be a plasma of particles (m, ±m) oscillating at the natural plasma frequency ω = 8π G m n χ Luc Blanchet (GRεCO) Atelier Vide Quantique 29 / 46

30 Modified dark matter approach Dipolar dark matter in general relativity The DDM action in standard general relativity is of the type S DDM = d 4 x [ g L DDM J µ, ξ µ, ξ ] µ, g µν where the current density J µ and the dipole moment ξ µ are two independent dynamical variables u x The current density J µ = ρu µ is conserved µ J µ = 0 t The covariant time derivative is denoted ξ µ Dξµ dτ = uν ν ξ µ Luc Blanchet (GRεCO) Atelier Vide Quantique 30 / 46

31 Modified dark matter approach Lagrangian for dipolar dark matter [Blanchet & Le Tiec 2008; 2009] We propose a modification of the CDM Lagrangian L DDM = ρ +J µ ξµ V (P }{{} ) CDM 1 Mass term ρ in an ordinary sense (like for ordinary CDM) 2 Interaction term between the fluid s mass current J µ = ρu µ and the covariant time derivative of the dipole moment ξ µ 3 Potential term V describing an internal force and depending on the norm of the polarization field P = ρ ξ Luc Blanchet (GRεCO) Atelier Vide Quantique 31 / 46

32 Modified dark matter approach 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 ) 1 The minimum of that potential is the cosmological constant Λ and the third-order deviation from the minimum contains the MOND scale a 0 2 In this unification scheme the natural order of magnitude of the cosmological constant is comparable with a 0 namely Λ a 2 0 Luc Blanchet (GRεCO) Atelier Vide Quantique 32 / 46

33 Modified dark matter approach Viability of the model in cosmology and in galaxies In a cosmological perturbation around a FLRW background, the dipole moment, which is space-like, must belong to the first-order perturbation ξ µ = O(1) The stress-energy tensor reduces to T µν = T µν DE + T µν DM where 1 the DE is given by the cosmological constant Λ 2 the DM takes the form of a perfect fluid with zero pressure T µν DM = 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 Using an hypothesis of weak clustering of dipolar dark matter the phenomenology of MOND is recovered at galactic scales Luc Blanchet (GRεCO) Atelier Vide Quantique 33 / 46

34 Modified dark matter approach Non-Gaussianity in the CMB induced by DDM [Blanchet, Langlois, Le Tiec & Marsat 2012] 1 At second-order cosmological perturbation around FLRW the dipolar dark matter deviates from standard CDM T µν DDM = (ε + P ) uµ u ν + P g µν } {{ } perfect fluid with internal energy W + }{{} Σ µν +O(3) anisotropic stresses 2 As a result the curvature perturbation [Langlois & Vernizzi 2005, 2006] is modified by a term quadratic in the dipole moment ζ DDM = ζ CDM + W 3 3 This term induces a new-type of non-gaussianity in the bispectrum of the curvature perturbation which could provide a specific signature of DDM in the CMB or in the LSS Luc Blanchet (GRεCO) Atelier Vide Quantique 34 / 46

35 Testing MOND in the Solar System TESTING MOND IN THE SOLAR SYSTEM Luc Blanchet (GRεCO) Atelier Vide Quantique 35 / 46

36 Testing MOND in the Solar System Usual MOND effect in the Solar System 1 In spherical symmetry the MOND equation becomes ( ) g µ g = g Newtonian GM a 0 r 2 2 Suppose MOND approaches the Newtonian regime like ( ) ( ) λ g a0 µ = 1 ε when g a 0 g 3 With r 0 = GM /a 0 the MOND transition radius for the Sun g = g Newtonian + ε a 0 ( r r 0 ) 2λ 2 When λ = 1 this gives a Pioneer-like anomaly a P = ε a 0 Luc Blanchet (GRεCO) Atelier Vide Quantique 36 / 46

37 Testing MOND in the Solar System Solar System data and Pioneer anomaly [Fienga et al. 2009] The data exclude a Pioneer-like anomaly at the level m/s 2 Luc Blanchet (GRεCO) Atelier Vide Quantique 37 / 46

38 Testing MOND in the Solar System The external field effect in MOND [Milgrom 1983] 1 Open star clusters in our Galaxy do not show evidence for dark matter despite their typical low internal gravity g i a 0 2 In the presence of the external Galactic field g e the MOND equation which is non-linear can be approximated by ( ) gi +g e µ g i gi Newtonian a 0 When a 0 g e the sub-system exhibits Newtonian behaviour When g i g e a 0 the system is still Newtonian but with an effective Newton s constant G/µ e The EFE results from a violation of the strong version of the equivalence principle The gravitational dynamics of a system is influenced by the external gravitational field in which the system is embedded Luc Blanchet (GRεCO) Atelier Vide Quantique 38 / 46

39 Testing MOND in the Solar System Deformation of the field of the Sun by the Galactic field The external field effect is a prediction of the non-linear Poisson equation [ ( ) ] g µ U = 4π G ρ b a 0 z ϕ θ r x M g e y The MOND field of the Sun, in the presence of the external field of the Galaxy, is deformed along the direction of the Galactic center ( ) GM /µ e 1 U = g e x + r 1 + λ e sin 2 θ + O r 2 This effect influences the motion of inner planets of the Solar System x Luc Blanchet (GRεCO) Atelier Vide Quantique 39 / 46

40 Testing MOND in the Solar System Multipole expansion of the MOND field of the Sun 1 The Newtonian physicist measures from the motion of planets the internal gravitational potential u = U g e x and detects the anomaly δu = u u N = G d 3 x x x ρ pdm(x, t) 2 Since the phantom dark matter vanishes in the strong-field regime near the Sun δu is an harmonic function and admits the multipole expansion δu = + l=0 ( ) l x L Q L l! where Q L are trace-free multipolar coefficients 3 This expansion is valid in the region inside the MOND transition radius r 0 = GM a AU Luc Blanchet (GRεCO) Atelier Vide Quantique 40 / 46

41 Testing MOND in the Solar System Dominant effect in the Solar System [Milgrom 2009, Blanchet & Novak 2011] 1 The effect is dominantly quadrupolar and grows with the distance squared u = GM r xi x j Q ij 2 The quadrupole moment is aligned in the direction of the Galactic center Q ij = Q 2 (e i e j 1 ) 3 δ ij 3 The quadrupole moment is computed by solving numerically the MOND equation in the presence of the external galactic field. We find s 2 Q s 2 depending on the MOND function in use Luc Blanchet (GRεCO) Atelier Vide Quantique 41 / 46

42 Testing MOND in the Solar System Quadrupole moment as a function of distance Luc Blanchet (GRεCO) Atelier Vide Quantique 42 / 46

43 Testing MOND in the Solar System Quadrupole moment as a function of distance Luc Blanchet (GRεCO) Atelier Vide Quantique 42 / 46

44 Testing MOND in the Solar System Effect on the dynamics of inner Solar System planets The quadrupole effect yields a suplementary precession of the semi-major axis of planets of the Solar System [Blanchet & Novak 2011] de dt dl dt d ω dt = 5Q 2e 1 e 2 sin(2 ω) 4n = n Q [ ] e (1 + e 2 ) cos(2 ω) 12n = Q 2 1 e 2 [ ] cos(2 ω) 4n Luc Blanchet (GRεCO) Atelier Vide Quantique 43 / 46

45 Testing MOND in the Solar System Comparison with Solar System ephemerides Predicted values for the orbital precession Quadrupolar precession rate in mas/cy Mercury Venus Earth Mars Jupiter Saturn µ µ µ µ µ exp Best published residuals for orbital precession Postfit residuals for the precession rates in mas/cy Mercury Venus Earth Mars Jupiter Saturn [Pitjeva 2005] 3.6 ± ± ± ± ± 2 [Fienga et al. 2009] 10 ± 30 4 ± 6 0 ± ± ± ± 8 [Fienga et al. 2010] 0.4 ± ± ± ± ± ± 0.7 The MOND function is constrained by the precession of Saturn to be µ = g ( ) n g + o when g 0 a 0 a 0 Luc Blanchet (GRεCO) Atelier Vide Quantique 44 / 46

46 Testing MOND in the Solar System Testing MOND with LISA-Pathfinder? It has been suggested that LPF may visit the saddle point of the Earth-Sun system and test gravity in the weak field limit However the MOND sphere around the saddle point (for which g a 0 ) is only a few meters in diameter Thus LPF will only be able to test the transition between the Newtonian and MOND regimes in the specific TeVeS theory Luc Blanchet (GRεCO) Atelier Vide Quantique 45 / 46

47 Conclusions Testing MOND in the Solar System 1 Λ-CDM is an extremely powerful model in cosmology but poses the problem of the fundamental constituents of the Universe 2 MOND is a successful alternative for interpreting the galactic rotation curves and the baryonic Tully-Fisher relation without dark matter 3 Reconciling Λ-CDM at cosmological scale and MOND at galactic scale into a single relativistic modified gravity theory with connections to fundamental physics is a great challenge 4 A non-standard form of dark matter might exist, while keeping the standard law of gravity unchanged, and able to successfully address both cosmological and galactic scales 5 The external field effect in MOND can be checked in the Solar System dynamics by looking at precession anomalies in the motion of inner planets Luc Blanchet (GRεCO) Atelier Vide Quantique 46 / 46