Implication of Graviton-Graviton Interaction to dark matter

Transcription

1 Implication of Graviton-Graviton Interaction to dark matter A dark universe Dark matter (Dark energy) A. Deur Estimating Graviton-Graviton interaction effects Application to galaxy and cluster dynamics Theoretical parallels between gravity and QCD Empirical parallels between cosmology and hadronic physics A.D. Phys. Lett. B (2009).

2 Observational clues for Dark Matter Rotation curves of galaxies: Spiral galaxies spin too fast compared to expectation. With seen mass and Newton Law for gravity, stars should fly apart.

3 Observational clues for Dark Matter Rotation curves of galaxies: Spiral galaxies spin too fast compared to expectation. With seen mass and Newton Law for gravity, stars should fly apart. 1) Galaxies have more mass than seen. (Dark matter, favorite explanation) 2) The laws of gravity at large distances are modified. (Outsider) Ex: Modified Newtonian dynamics, MOND. M. Milgrom, Astr. J. 270, In this talk, we will explore a third alternative

4 Other observational clues for Dark Matter Galaxies in clusters are moving too fast and should fly apart if mass is what is seen.

5 Other observational clues for Dark Matter Gravitational lensing Angular power spectrum of Cosmic Microwave Background's anisotropy. Structure formations: need seeds for baby galaxies. Primordial nucleosynthesis: 4 He abundance most of the universe mass is non-baryonic....

6 Dark Energy Measurements of far away supernovae indicate that the universe expansion is accelerating (contrarily to what we expect from a matter (normal & dark) dominated universe). Matter is repelled by some force (Dark Energy).

7 Dark Matter and Dark Energy Dark matter and dark energy are not believed to be related. However, their contributions to the expansion is similar at present times. Coincidence or sign of a relation? ( cosmic coincidence problem )

8 Gravity self-interaction as a possible alternate solution Gravitons couple to each other: G G is very small: GMp 2 = with Mp the proton mass (or size -1 ). Usually, such self-interaction effects are negligible. But Gravity always attracts gravity effects add up and for massive enough systems, self-interaction effects may become important. Ex: Magnitude of the gravity field GM/sizesystem. Typical galaxy: M ~10 68 GeV & size~10 35 GeV -1 so GM/sizegalaxy ~10-3.

9 Quantitative estimate of gravity self-interaction We start from General Relativity, i.e the Einstein-Hilbert Lagrangian: 1 LEH= -g g μν Ricci 16πG Rμν tensor Metric det gμν tensor Galaxies have weak gravity fields with stars moving at non-relativistic speeds.! weak field approximation: we can expand LEH in term of spin-2 gravity field ψμν by developing gμν around the Minkowsky metric: gμν~ημν+kψμν+... : LEH= ψ ψ + kψ ψ ψ + k 2 ψ 2 ψ ψ kψμνt μν

10 Quantitative estimate of gravity self-interaction We start from General Relativity, i.e the Einstein-Hilbert Lagrangian: 1 Ricci LEH= -g g 16πG μν Rμν tensor Metric det gμν tensor Galaxies have weak gravity fields with stars moving at non-relativistic speeds.! weak field approximation: we can expand LEH in term of spin-2 gravity field ψμν by developing gμν around the Minkowsky metric: gμν~ημν+kψμν+... : LEH= ψ ψ + kψ ψ ψ + k 2 ψ 2 ψ ψ kψμνt μν Short hand for sum of possible Lorentz-invariant terms of form ψ ψ. Explicitly given by the Fierz-Pauli Lagrangian: ψ ψ ½ λ ψμν λψ μν μ - ½ λψμ λ ν ψν - λ ψλν μψ μν - ν λ ψλ μ ψμν

11 Quantitative estimate of gravity self-interaction We start from General Relativity, i.e the Einstein-Hilbert Lagrangian: 1 Ricci LEH= -g g 16πG μν Rμν tensor Metric det gμν tensor Galaxies have weak gravity fields with stars moving at non-relativistic speeds.! weak field approximation: we can expand LEH in term of spin-2 gravity field ψμν by developing gμν around the Minkowsky metric: gμν~ημν+kψμν+... : LEH= ψ ψ + kψ ψ ψ + k 2 ψ 2 ψ ψ kψμνt μν Coupling constant: k 2 G

12 Quantitative estimate of gravity self-interaction We start from General Relativity, i.e the Einstein-Hilbert Lagrangian: 1 Ricci LEH= -g g 16πG μν Rμν tensor Metric det gμν tensor Galaxies have weak gravity fields with stars moving at non-relativistic speeds.! weak field approximation: we can expand LEH in term of spin-2 gravity field ψμν by developing gμν around the Minkowsky metric: gμν~ημν+kψμν+... : LEH= ψ ψ + kψ ψ ψ + k 2 ψ 2 ψ ψ kψμνt μν Source term (T μν is the energy-momentum tensor). Since we are interested only in a stationary field, we will ignore kψμνt μν.

13 Quantitative estimate of gravity self-interaction We start from General Relativity, i.e the Einstein-Hilbert Lagrangian: 1 Ricci LEH= -g g 16πG μν Rμν tensor Metric det gμν tensor Galaxies have weak gravity fields with stars moving at non-relativistic speeds.! weak field approximation: we can expand LEH in term of spin-2 gravity field ψμν by developing gμν around the Minkowsky metric: gμν~ημν+kψμν+... : LEH= ψ ψ + kψ ψ ψ + k 2 ψ 2 ψ ψ kψμνt μν All this is not new: is a know polynomial form of LEH

14 We modify L into a form appropriate for numerical simulations. S= d 4 x L S: action (used for path integrals computed on a lattice). S= d 4 x ψ ψ + kψ ψ ψ + k 2 ψ 2 ψ ψ kψμνt μν Integrating by part: ψ n 1 ψ ψ ψ n+1 2 ψ n+1 The Euler-Lagrange equation of motion + Fierz-Pauli term yield: 2 ψ μν =-k 2 (T μν -½η μν Tr(T)) The weak field approximation implies that T 00 dominates over other T μν components 2 ψ 00 dominates over the other 2 ψ μν. The harmonic gauge condition simplifies the Fierz-Pauli term to ¼ λ ψ 00 λψ00 We can consider only the ψ 00 component for the first term in L, i.e. a scalar field. Easy for numerical simul.

15 We modify L into a form appropriate for numerical simulations. S= d 4 x L S: action (used for path integrals computed on a lattice). S= d 4 x ψ ψ + kψ ψ ψ + k 2 ψ 2 ψ ψ kψμνt μν Integrating by part: ψ n 1 ψ ψ ψ n+1 2 ψ n+1 The Euler-Lagrange equation of motion + Fierz-Pauli term yield: 2 ψ μν =-k 2 (T μν -½η μν Tr(T)) The weak field approximation implies that T 00 dominates over other T μν components 2 ψ 00 dominates over the other 2 ψ μν. The harmonic gauge condition simplifies the Fierz-Pauli term to ¼ λ ψ 00 λψ00 We can consider only the ψ 00 component for the first term in L, i.e. a scalar field. Easy for numerical simul. Higher ψ n+1 2 ψ terms proceed similarly since they all contain 2 ψ. All in all: S d 4 x φ φ + kφ φ φ + k 2 φ2 φ φ with φ=ψ 00 a scalar field.

16 I went through this to show that our L originates from General Relativity. However, there is a more intuitive way to form L : Gravity: spin 2 field Too hard to do numerical simulations. Spin 1 (QED, QCD). Numerical simulation possible (e.g. lattice QCD) but such force doesn't always attract. Spin 0: Always attract, as gravity. By by taking a spin 0, we lose the gravity-light coupling (no black hole, no positive 1919 eclipse measurement by Eddington,...) but this is not important in our context.

17 I went through this to show that 0ur L originates from General Relativity. However, there is a more intuitive way to form L : Needed Feynman graphs are:... K 3! S = d 4 x φ φ + φ φ φ + φ 2 φ φ K 2 4!

18 I went through this to show that 0ur L originates from General Relativity. However, there is a more intuitive way to form L : Needed Feynman graphs are:... K 3! S = d 4 x φ φ + φ φ φ + φ 2 φ φ We could have also φ 2 φ and/or φ 3 associated with but: φ 2 μφ is not Lorentz-invariant and has the wrong dimension. φ 3 has the wrong dimension: [S]=0, [x]=gev -1, [ ]=GeV [φ]=gev k 2 G so [k]=gev -1 and only φ φ φ has the right dimension: [d 4 x k φ φ φ] = 0 = [S] [d 4 x k φ 3 ] = GeV -2 [S] [d 4 x k φ 2 φ] = GeV -1 [S] Same argument for φ 2 φ φ. K 2 4!

19 I went through this to show that 0ur L originates from General Relativity. However, there is a more intuitive way to form L : Needed Feynman graphs are:... K 3! S = d 4 x φ φ + φ φ φ + φ 2 φ φ K 2 4! We have this form for the action, with the form of the generic term φ n φ φ imposed by the dimension of k.

20 I went through this to show that 0ur L originates from General Relativity. However, there is a more intuitive way to form L : Needed Feynman graphs are:... K 3! S = d 4 x φ φ + φ φ φ + φ 2 φ φ We consider one global field on the lattice. Neglecting nonlinear effects, we have: k 2 = m 16πG K 2 4! 16πG m=1 m=1 16πG 16πG 2(16πG) m=2 m=2 m=2 m=2

21 We use this scalar L and Feynman path integral formalism on a lattice to compute the 2-point green function. 2-point green function: probability for a force carrier to go from one point to another.! gravity potential.

22 Keeping only φ φ in L (no field self-interaction), we obtain: Checks of lattice calculations V 1/r: Newton potential.

23 Checks of lattice calculations Keeping only φ φ in L and adding a mass term mφ 2 φ 2, we obtain: V e -m φr/ r: Yukawa potentials. Simulation makes sense (for the φ.φ part)

24 With k 10-3 the full L gives: Results of lattice calculations i.e V= a r + br

25 Application to galaxies Is for a 2-point system. It implies the collapse: Force~constant For an homogeneous spherical distribution, there is no preferred collapse direction There should be no field self-interaction effects. We have F~constant spreads out over the sphere surface, i.e.!! F~constant /(4/3 πr 2 ) V~1/r as expected.

26 Application to galaxies Is for a 2-point system. It implies the collapse: Force~constant For an homogeneous spherical distribution, there is no preferred collapse direction There should be no field self-interaction effects. We have F~constant spreads out over the sphere surface, i.e.!! F~constant /(4/3 πr 2 ) V~1/r as expected. For an homogeneous cylindrical distribution, only the field lines outside of the disk collapse. We have F~constant/(2πr) V~log(r). Adding back the unaffected part from φ φ in L, we get: a r b 2π V= + log(r) with a & b given by the numerical computation. We use this potential for spiral galaxies. Taking ρ= e -r/r 0 as the 2πr0 2 profile density of a galaxy of mass M, we can compute rotation curves. M

27 Rotation curves of spiral galaxies Typical spiral galaxies Full calculation. No field self-interaction (Newton). data. Dwarf spiral galaxies

28 The Tully-Fisher relation Empirical Observation that the luminosity (i.e visible mass) and rotation speed of spiral galaxies are correlated: Important relation used to estimate a galaxy absolute luminosity (thus their distances to earth). The relation is not explained by the dark matter models. It helped to selected between various ways to modify gravity! The Tully-Fisher relation is assumed in MOND model.

29 The Tully-Fisher relation Empirically: log(m)= log(v)+ε ( =3.9±0.2, ε~1.5) In our framework: a r b -GMmb Force from potential V= + log(r) is: Ur (for large distances) Equilibrium between centripetal and gravity forces for a body of mass m yields: -mv 2 Ur -GMmb = Ur v 2 = 2 2 ar 2 ar 2log(V) = log(b) + log( ) r GMb 2 a with a: Coef. of Newton potential. b: string tension. a is a function of GM. Expanding: b( GM )=b0 + b1 GM + GM 2 a

30 The Tully-Fisher relation Empirically: log(m)= log(v)+ε ( =3.9±0.2, ε~1.5) In our framework: b -GMmb Force from potential V= + log(r) is: Ur (for large distances) Equilibrium between centripetal and gravity forces for a body of mass m yields: -mv 2 Ur -GMmb = Ur v 2 = 2log(V) = log(b) + log( ) GM 2 a a~gm so is independent of M. a r 2 r 2 ar 2 ar GMb 2 a with a: Coef. of Newton potential b: string tension b is a function of GM. Expanding: b( GM )=b0 + b1 GM +b2gm +... GM 2 a b(0)=0 so b0=0 (no field line collapse without graviton-graviton coupling) GM GM for galaxies

31 The Tully-Fisher relation Empirically: log(m)= log(v)+ε ( =3.9±0.2, ε~1.5) In our framework: b -GMmb Force from potential V= + log(r) is: Ur (for large distances) Equilibrium between centripetal and gravity forces for a body of mass m yields: -mv 2 Ur -GMmb = Ur v 2 = 2log(V) = log(b) + log( ) GM 2 a a~gm so is independent of M. GMb 2 a with a: Coef. of Newton potential b: string tension b is a function of GM. Expanding: b( GM )=b0 + b1 GM +b2gm +... So b~b1 GM log(b)~½log(m)+log(b1 G ), and the equilibrium condition yields: log(m) = 4 log(v) + c The Tully-Fisher relation is explained and the proportionality coefficient quantitatively given. a r 2 r 2 ar GM 2 a 2 ar

32 Galaxy clusters Our calculations expressed from the dark matter standpoint, indicate that clusters contain typically 94% of dark mass, to compare with the 80-95% observed. In 2006, direct evidence of dark matter was found in the Bullet cluster.!! Direct evidence: observations compatible with dark matter scenario but not with MOND.!! Our approach is also naturally compatible with the Bullet cluster observations.

33 Possible link to Dark Energy Energy conservation:! Increase in galaxy binding decrease of outside potential energy.!!!!!!! equivalently! Field lines are partially confined in systems (galaxies or clusters). Two clusters or two spiral galaxies should interact less than expected using a Newtonian potential:!! F true = F expected D, where D is due to the depletion of field lines outside of the system. -D would act like a (time and space -dependent) negative force. It may be relevant to dark energy (but F true always greater than 0: no net repulsion). If true, this would also naturally explain the cosmic coincidence problem :!!!Dark Matter and Dark energy are actually related.

34 Gravity is not the only force that self-interacts. Quantum Chromodynamique (QCD), the theory of strong nuclear force is the quintessencial self-interacting theory. It s worthwhile to explore possible parallels between gravity and the strong force. Reminder on Strong force: It binds quarks together into hadrons (particles held together by the strong force, e.g. a nucleon). The vectors of the strong force (the gluons) strongly interact with each other. Origin of quark confinement. Residual effects bind nucleons together into the nuclei of atoms.

35 Theoretical parallels between gravity and QCD gravity Gravitons couple to each other: (m graviton =0 but they have energy & momentum 0 and gravity couples to that) A gauge theory of gravity would be a Yang-Mills (i.e. non-abelian) theory. G QCD Gluons couple to each other: s The gauge theory of strong force is the prototype of a Yang-Mills theory.

36 Theoretical parallels between gravity and QCD gravity Gravitons couple to each other: (m graviton =0 but they have energy & momentum 0 and gravity couples to that) A gauge theory of gravity would be a Yang-Mills (i.e. non-abelian) theory. QCD Gluons couple to each other: Differences between gravity and QCD G is very small (GMp 2 = ) a priori, effects of self-interaction are small. Graviton spin: 2 (gravity always attracts gravity effects add up) G s The gauge theory of strong force is the prototype of a Yang-Mills theory. s is large: ~1 Gluon spin: 1 (QCD attracts or repulses, as for QED)

37 Theoretical parallels between gravity and QCD gravity Gravitons couple to each other: (m graviton =0 but they have energy & momentum 0 and gravity couples to that) A gauge theory of gravity would be a Yang-Mills (i.e. non-abelian) theory. QCD Gluons couple to each other: Differences between gravity and QCD G is very small (GMp 2 = ) a priori, effects of self-interaction are small. Graviton spin: 2 (gravity always attracts gravity effects add up) G s The gauge theory of strong force is the prototype of a Yang-Mills theory. s is large: ~1 Gluon spin: 1 (QCD attracts or repulses, as for QED) As already discussed, the two differences balance each other for massive enough systems and gravity self-interaction effects similar to the ones seen in QCD may happen.

38 Key features of Strong Force Facts Quarks and gluons are confined inside hadrons Explanations α s Large & gluons are color-charged field-lines collapse into string-like flux-tubes. Evidences from lattice QCD (action density, G. Bali et al):

39 Key features of Strong Force Facts Quarks and gluons are confined inside hadrons Explanations α s Large & gluons are color-charged field-lines collapse into string-like flux-tubes. q q No strong force at large distances (but for residual effects: Yukawa potential, etc...) All field lines are inside the hadron:

40 Key features of Strong Force Facts Quarks and gluons are confined inside hadrons Explanations α s Large & gluons are color-charged field-lines collapse into string-like flux-tubes. q q No strong force at large distances (but for residual effects: Yukawa potential, etc...) Hadrons lie on Regge trajectories All field lines are inside the hadron: The more a hadron spins, the larger the string tension (i.e the binding energy, i.e. the mass) to compensate for the centrifugal force. Ang. Mom. M 2 +cste QCD potential: V= + r (inside the nucleon)

41 Empirical parallels between cosmology and Hadronic physics Cosmology Galaxies (or clusters of galaxies) have a larger mass than the sum of their known constituents.!!! Alternatively Gravity is stronger than we think for these systems. Tully-Fisher relation: log(m)= log(v)+ ( =3.9±0.2, ~1.5) (M galaxy visible mass, v rotation speed) Unexplained with dark matter. Assumed by MOND.! Negative pressure pervades the universe and repels galaxies from each other. Alternatively Total force is smaller than we think at very large distances.

42 Empirical parallels between cosmology and Hadronic physics Cosmology Galaxies (or clusters of galaxies) have a larger mass than the sum of their known constituents.!!! Alternatively Gravity is stronger than we think for these systems. Tully-Fisher relation: log(m)= log(v)+ ( =3.9±0.2, ~1.5) (M galaxy visible mass, v rotation speed) Unexplained with dark matter. Assumed by MOND.! Negative pressure pervades the universe and repels galaxies from each other. Alternatively Total force is smaller than we think at very large distances. Hadronic physics 2 quarks ~10 MeV, Pion mass 140 MeV 3 quarks ~15 MeV, Nucleon: 938 MeV For non-relativistic quarks, this extra mass comes from large binding energy: VQCD= + r.

43 Empirical parallels between cosmology and Hadronic physics Cosmology Galaxies (or clusters of galaxies) have a larger mass than the sum of their known constituents.!!! Alternatively Gravity is stronger than we think for these systems. Hadronic physics 2 quarks ~10 MeV, Pion mass 140 MeV 3 quarks ~15 MeV, Nucleon: 938 MeV For non-relativistic quarks, this extra mass comes from large binding energy: VQCD= + r. Tully-Fisher relation: log(m)= log(v)+ ( =3.9±0.2, ~1.5) (M galaxy visible mass, v rotation speed) Unexplained with dark matter. Assumed by MOND.! Regge trajectories:! log(m)=c log( J )+b (c=0.5) (M, hadron mass, J angular momentum) Negative pressure pervades the universe and repels galaxies from each other. Alternatively Total force is smaller than we think at very large distances.

44 Empirical parallels between cosmology and Hadronic physics Cosmology Galaxies (or clusters of galaxies) have a larger mass than the sum of their known constituents.!!! Alternatively Gravity is stronger than we think for these systems. Hadronic physics 2 quarks ~10 MeV, Pion mass 140 MeV 3 quarks ~15 MeV, Nucleon: 938 MeV For non-relativistic quarks, this extra mass comes from large binding energy: VQCD= + r. Tully-Fisher relation: log(m)= log(v)+ ( =3.9±0.2, ~1.5) (M galaxy visible mass, v rotation speed) Unexplained with dark matter. Assumed by MOND.! Negative pressure pervades the universe and repels galaxies from each other. Alternatively Total force is smaller than we think at very large distances. Regge trajectories:! log(m)=c log( J )+b (c=0.5) (M, hadron mass, J angular momentum) No strong force between hadrons (except for residual effects).

45 Intriguing correspondance between key facts of hadronic physics and observations involving dark matter and dark energy. Might be due to the similarities between gravity theory and QCD.

46 Caveats (and justifications) Approximations to go from LEH to scalar L. (Those were justified.) Lattice approximations, e.g. finite grid high energy cut off. (High energy modes should not be important for our stationary, classical, weak field.) Ignored the source term in L. Should be fine for stationary systems. (Same is done in Lattice QCD simplest Calculations: done in gluonic sector only). Guess for the coupling k. Neglects non-linear effects while those are important.(but they can only make k larger.) Simple models for galaxies and clusters. Particular choice of boundary conditions: φ=0 on boundaries. Typical circular boundary conditions would not work for us. (We tested these conditions with k=0 and recovered a Newton or Yukawa potential). Many Approximations, but they are justified.

47 Caveats (and justifications) Approximations to go from LEH to scalar L. (Those were justified.) Lattice approximations, e.g. finite grid high energy cut off. (High energy modes should not be important for our stationary, classical, weak field.) Ignored the source term in L. Should be fine for stationary systems. (Same is done in Lattice QCD simplest Calculations: done in gluonic sector only). Guess for the coupling k. Neglects non-linear effects while those are important.(but they can only make k larger.) Simple models for galaxies and clusters. Particular choice of boundary conditions: φ=0 on boundaries. Typical circular boundary conditions would not work for us. (We tested these conditions with k=0 and recovered a Newton or Yukawa potential). Many Approximations, but they are justified. Main message: We may have effects, known and seen QCD, that can explain cosmological observations without dark matter and possibly dark energy. An approximative estimate indicates quantitatively that these effects should be relevant.

48 Summary Gravity self-interaction suggests a mechanism to explain naturally galaxy rotation curves and cluster dynamics. A calculation established within a weak field approximation agrees well with observations involving dark matter, without requiring any arbitrary parameters or exotic particles. The Tully-Fisher relation arises naturally. Suggests that dark energy could partly be a consequence of energy conservation between the increased binding energy of the system and the outside potential energy. All these effects have parallels in hadronic physics, maybe because the respective theories (gravity and QCD) have strong similarities.

49 Back-up slides

50 Bullet cluster In 2006, direct evidence of dark matter was found in the Bullet Cluster: The bullet cluster is formed of two colliding clusters. Gravitational lensing and x-ray imaging of the gas show that the locations of the two mass maximums are offset from the two maximums of the gas density. Galaxies and whatever appears as dark mass are unaffected by the collision. Gas (which dominates the cluster visible mass) was affected and stayed in the collision region.

51 Bullet cluster This is compatible with: Galaxies: small cross section since galaxy sizes cluster size. Dark matter: small cross section since weakly interacting particles. Gas: large cross section since interacts electromagnetically and size~cluster size. Since gas dominates the visible mass of a cluster, the observation is seen as the first proof for dark matter. But it can be viewed as the fact that MOND (or any modification of gravity that scales with visible mass) fails in this particular case.

52 Bullet cluster In our framework: Field self-interaction effects are associated with geometrically asymmetric distributions: : no effects; :some effects; :large effects. Field self-interaction effects ( dark matter ) should be less suppressed for the homogeneous gas. The Field self-interaction effects are associated with the galaxies (increases of stars binding inside galaxies AND of galaxies interaction).!! We should observe that most of the dark matter follows the galaxy system.

53 Particular choice of boundary conditions: φ=0 on boundaries. Typical circular boundary conditions would not work for us: Assuming a perfectly linear potential is created by :

54 Dark Energy F true = F expected D, where D is due to the depletion of field lines outside of the system. If one Includes D in the universe evolution equation: dl*h0: attenuation redshift z: ~distance PRELIMINARY

55 Dark Energy (residual) F true = F expected D, where D is due to the depletion of field lines outside of the system. If one Includes D in the universe evolution equation: PRELIMINARY dl*h0: attenuation redshift z: ~distance