Aspects of multimetric gravity
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1 Aspects of multimetric gravity Manuel Hohmann Teoreetilise füüsika labor Füüsikainstituut Tartu Ülikool 4. september 2012 Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
2 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Post-Newtonian consistency 5 Gravitational waves 6 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
3 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Post-Newtonian consistency 5 Gravitational waves 6 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
4 Einstein gravity Gravity is described by metric tensor g ab. Einstein-Hilbert action: S G = 1 ωr. 2 Volume form ω. Scalar curvature R. Minimally coupled matter action: S M = ωl M. Einstein equations: R ab 1 2 Rg ab = T ab. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
5 Application to the universe 4.6% visible matter. [Komatsu et al. 09] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
6 Application to the universe 4.6% visible matter. [Komatsu et al. 09] 22.8% dark matter. Galaxy rotation curves. [de Blok, Bosma 02] Anomalous light deflection. [Wambsganss 98] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
7 Application to the universe 4.6% visible matter. [Komatsu et al. 09] 22.8% dark matter. Galaxy rotation curves. [de Blok, Bosma 02] Anomalous light deflection. [Wambsganss 98] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
8 Application to the universe 4.6% visible matter. [Komatsu et al. 09] 22.8% dark matter. Galaxy rotation curves. [de Blok, Bosma 02] Anomalous light deflection. [Wambsganss 98] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
9 Application to the universe 4.6% visible matter. [Komatsu et al. 09] 22.8% dark matter. Galaxy rotation curves. [de Blok, Bosma 02] Anomalous light deflection. [Wambsganss 98] 72.6% dark energy. Accelerating expansion. [Riess et al. 98; Perlmutter et al. 98] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
10 Application to the universe 4.6% visible matter. [Komatsu et al. 09] 22.8% dark matter. Galaxy rotation curves. [de Blok, Bosma 02] Anomalous light deflection. [Wambsganss 98] 72.6% dark energy. Accelerating expansion. [Riess et al. 98; Perlmutter et al. 98] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
11 Application to the universe 4.6% visible matter. [Komatsu et al. 09] 22.8% dark matter. Galaxy rotation curves. [de Blok, Bosma 02] Anomalous light deflection. [Wambsganss 98] 72.6% dark energy. Accelerating expansion. [Riess et al. 98; Perlmutter et al. 98] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
12 Application to the universe 4.6% visible matter. [Komatsu et al. 09] 22.8% dark matter? Galaxy rotation curves. [de Blok, Bosma 02] Anomalous light deflection. [Wambsganss 98] 72.6% dark energy? Accelerating expansion. [Riess et al. 98; Perlmutter et al. 98] Problem: What are dark matter and dark energy? Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
13 Explanations for the dark universe Particle physics: Dark matter: [Bertone, Hooper, Silk 05] Weakly interacting massive particles (WIMPs). [Ellis et al. 84] Axions. [Preskill, Wise, Wilczek 83] Massive compact halo objects (MACHOs). [Paczynski 86] Dark energy: [Copeland, Sami, Tsujikawa 06] Quintessence. [Peebles, Ratra 88] K-essense. [Chiba, Okabe, Yamaguchi 00; Armendariz-Picon, Mukhanov, Steinhardt 01] Chaplygin gas. [Kamenshchik, Moschella, Pasquier 01] Gravity: Modified Newtonian dynamics (MOND). [Milgrom 83] Tensor-vector-scalar theories. [Bekenstein 04] Curvature corrections. [Schuller, Wohlfarth 05; Sotiriou, Faraoni 05] Dvali-Gabadadze-Porrati (DGP) model. [Dvali, Gabadadze, Porrati 00, Lue 06] Non-symmetric gravity. [Moffat 95] Area metric gravity. [Punzi, Schuller, Wohlfarth 07] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
14 Explanations for the dark universe Particle physics: Dark matter: [Bertone, Hooper, Silk 05] Weakly interacting massive particles (WIMPs). [Ellis et al. 84] Axions. [Preskill, Wise, Wilczek 83] Massive compact halo objects (MACHOs). [Paczynski 86] Dark energy: [Copeland, Sami, Tsujikawa 06] Quintessence. [Peebles, Ratra 88] K-essense. [Chiba, Okabe, Yamaguchi 00; Armendariz-Picon, Mukhanov, Steinhardt 01] Chaplygin gas. [Kamenshchik, Moschella, Pasquier 01] Gravity: Modified Newtonian dynamics (MOND). [Milgrom 83] Tensor-vector-scalar theories. [Bekenstein 04] Curvature corrections. [Schuller, Wohlfarth 05; Sotiriou, Faraoni 05] Dvali-Gabadadze-Porrati (DGP) model. [Dvali, Gabadadze, Porrati 00, Lue 06] Non-symmetric gravity. [Moffat 95] Area metric gravity. [Punzi, Schuller, Wohlfarth 07] New idea: repulsive gravity negative mass! Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
15 Mass in Newtonian gravity Three types of mass! [Bondi 57] Active gravitational mass m a - source of gravity: φ = G N m a r. Passive gravitational mass m p - reaction on gravity: F = m p φ. Inertial mass m i - relates force to acceleration: F = m i a. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
16 Mass in Newtonian gravity Three types of mass! [Bondi 57] Active gravitational mass m a - source of gravity: φ = G N m a r. Passive gravitational mass m p - reaction on gravity: F = m p φ. Inertial mass m i - relates force to acceleration: F = m i a. Theory relates the different types of mass: Momentum conservation: m a m p. Weak equivalence principle: m p m i. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
17 Mass in Newtonian gravity Three types of mass! [Bondi 57] Active gravitational mass m a - source of gravity: φ = G N m a r. Passive gravitational mass m p - reaction on gravity: F = m p φ. Inertial mass m i - relates force to acceleration: F = m i a. Theory relates the different types of mass: Momentum conservation: m a m p. Weak equivalence principle: m p m i. m a m p m i experimentally verified. Gravity is always attractive. Convention: unit ratios and signs such that m a = m p = m i > 0. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
18 Mass in Newtonian gravity Three types of mass! [Bondi 57] Active gravitational mass m a - source of gravity: φ = G N m a r. Passive gravitational mass m p - reaction on gravity: F = m p φ. Inertial mass m i - relates force to acceleration: F = m i a. Theory relates the different types of mass: Momentum conservation: m a m p. Weak equivalence principle: m p m i. m a m p m i experimentally verified. Gravity is always attractive. Convention: unit ratios and signs such that m a = m p = m i > 0. Observations exist for visible mass only. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
19 Dark universe from negative mass Idea for dark universe: standard model with m a = m p = m i < 0. Both copies couple only through gravity dark. Preserves momentum conservation. Breaks weak equivalence principle only for cross-interaction. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
20 Dark universe from negative mass Idea for dark universe: standard model with m a = m p = m i < 0. Both copies couple only through gravity dark. Preserves momentum conservation. Breaks weak equivalence principle only for cross-interaction. Explanation of dark matter. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
21 Dark universe from negative mass Idea for dark universe: standard model with m a = m p = m i < 0. Both copies couple only through gravity dark. Preserves momentum conservation. Breaks weak equivalence principle only for cross-interaction. Explanation of dark matter. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
22 Dark universe from negative mass Idea for dark universe: standard model with m a = m p = m i < 0. Both copies couple only through gravity dark. Preserves momentum conservation. Breaks weak equivalence principle only for cross-interaction. Explanation of dark matter. Explanation of dark energy. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
23 Dark universe from negative mass Idea for dark universe: standard model with m a = m p = m i < 0. Both copies couple only through gravity dark. Preserves momentum conservation. Breaks weak equivalence principle only for cross-interaction. Explanation of dark matter. Explanation of dark energy. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
24 Dark universe from negative mass Idea for dark universe: standard model with m a = m p = m i < 0. Both copies couple only through gravity dark. Preserves momentum conservation. Breaks weak equivalence principle only for cross-interaction. Explanation of dark matter. Explanation of dark energy. Advantage: Dark copy Ψ of well-known standard model Ψ + : No new parameters. No unknown masses. No unknown couplings. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
25 Repulsive Einstein gravity Positive and negative test masses follow different trajectories. Two types of test mass trajectories two types of observers. Observer trajectories are autoparallels of two connections ±. Observers attach parallely transported frames to their curves. Frames are orthonormalized using two metric tensors g ± ab. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
26 Repulsive Einstein gravity Positive and negative test masses follow different trajectories. Two types of test mass trajectories two types of observers. Observer trajectories are autoparallels of two connections ±. Observers attach parallely transported frames to their curves. Frames are orthonormalized using two metric tensors g ± ab. No-go theorem forbids bimetric repulsive gravity. [MH, M. Wohlfarth 09] Solution: N 3 metrics gab I and standard model copies ΨI. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
27 Action and equations of motion N metric tensors and N standard model copies. Matter action: sum of standard model actions. Gravitational action: S G [g 1,..., g N ] = 1 d 4 x N g 0 (x + yδ IJ )g Iij Rij J + F(S IJ ). 2 I,J=1 Symmetric volume form g 0 = (g 1 g 2... g N ) 1/N. F(S IJ ) quadratic in connection difference tensors S IJ = Γ I Γ J. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
28 Action and equations of motion N metric tensors and N standard model copies. Matter action: sum of standard model actions. Gravitational action: S G [g 1,..., g N ] = 1 d 4 x N g 0 (x + yδ IJ )g Iij Rij J + F(S IJ ). 2 I,J=1 Symmetric volume form g 0 = (g 1 g 2... g N ) 1/N. F(S IJ ) quadratic in connection difference tensors S IJ = Γ I Γ J. Equations of motion: T I ab = g0 g I 1 2N gi ab N (x + yδ JK )g Jij R K ij + J,K =1 + terms linear in I S JK + terms quadratic in S IJ. Repulsive Newtonian limit for N 3. [MH, M. Wohlfarth 10] N (x + yδ IJ )R J ab Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32 J=1
29 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Post-Newtonian consistency 5 Gravitational waves 6 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
30 Cosmological symmetry Standard cosmology: Robertson Walker metrics g I = n 2 I (t)dt dt + a2 I (t)γ αβdx α dx β. Lapse functions n I. Scale factors a I. Spatial metric γ αβ of constant curvature k { 1, 0, 1} and Riemann tensor R(γ) αβγδ = 2kγ α[γ γ δ]β. Perfect fluid matter: Normalization: g I ab uia u Ib = 1. T I ab = (ρ I + p I )u Ia u Ib + p I g Iab. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
31 Simple cosmological model Early universe: radiation; late universe: dust. Copernican principle: common evolution for all matter sectors. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
32 Simple cosmological model Early universe: radiation; late universe: dust. Copernican principle: common evolution for all matter sectors. Single effective energy-momentum tensor Tab I = T ab. Single effective metric gab I = g ab. Common scale factors a I = a and lapse functions n I = n. Rescale cosmological time to set n 1. Ricci tensors Rab I = R ab become equal. Connection differences S IJ i jk = 0 vanish. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
33 Simple cosmological model Early universe: radiation; late universe: dust. Copernican principle: common evolution for all matter sectors. Single effective energy-momentum tensor Tab I = T ab. Single effective metric gab I = g ab. Common scale factors a I = a and lapse functions n I = n. Rescale cosmological time to set n 1. Ricci tensors Rab I = R ab become equal. Connection differences S IJ i jk = 0 vanish. Equations of motion simplify: (2 N)T ab = R ab 1 2 Rg ab. Negative effective gravitational constant for early / late universe. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
34 Cosmological equations of motion Insert Robertson Walker metric into equations of motion: ρ = 3 (ȧ2 2 N p = 1 2 N ), a 2 + k a 2 ( 2ä a + ȧ2 a 2 + k a 2 ). Positive matter density ρ > 0 requires k = 1 and ȧ 2 < 1. No solutions for k = 0 or k = 1. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
35 Cosmological equations of motion Insert Robertson Walker metric into equations of motion: ρ = 3 (ȧ2 2 N p = 1 2 N ), a 2 + k a 2 ( 2ä a + ȧ2 a 2 + k a 2 ). Positive matter density ρ > 0 requires k = 1 and ȧ 2 < 1. No solutions for k = 0 or k = 1. Acceleration equation: ä a = N 2 (ρ + 3p). 6 Acceleration must be positive for standard model matter. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
36 Explicit solution Equation of state: p = ωρ; dust: ω = 0, radiation: ω = 1/3. General solution using conformal time η defined by dt = a dη: ( 3ω + 1 a = a 0 (cosh 2 3 ρ = (N 2)a0 2 ( cosh )) 2 3ω+1 (η η 0 ), ( 3ω )) 6ω+6 3ω+1 (η η 0 ). Positive minimal radius a 0. [MH, M. Wohlfarth 10] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
37 Cosmological evolution a Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32 0
38 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Post-Newtonian consistency 5 Gravitational waves 6 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
39 Structure formation Formation of galactic structures not fully understood: Missing dwarf problem. [Moore et al. 99] Core-cusp-problem. [Dubinski, Carlberg 91; Navarro et al. 96] Structure formation in multimetric gravity: Perturbation of cosmological background solution. Model dust matter by point particles. Interaction between point particles given by Newtonian limit. Implementation: Large particle number requires high computing power. Use GPU computing! Results: Galactic clusters and filament-like structures. Seemingly empty voids contain invisible matter. Repulsive gravity effects from galactic voids. Negative gravitational lenses in galactic voids? Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
40 Structure formation - all matter types Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
41 Structure formation - all matter types Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
42 Structure formation - only visible matter Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
43 Structure formation - only visible matter Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
44 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Post-Newtonian consistency 5 Gravitational waves 6 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
45 Parametrized post-newtonian formalism Obtain fingerprint of single-metric gravity theories. [Thorne, Will 71; Will 93] 10 parameters, constrained by solar system experiments. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
46 Parametrized post-newtonian formalism Obtain fingerprint of single-metric gravity theories. [Thorne, Will 71; Will 93] 10 parameters, constrained by solar system experiments. Extension to multimetric gravity theories. [MH, M. Wohlfarth 10] Additional 14 unobserved parameters. 8 parameters can be obtained from linearized field equations. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
47 Parametrized post-newtonian formalism Obtain fingerprint of single-metric gravity theories. [Thorne, Will 71; Will 93] 10 parameters, constrained by solar system experiments. Extension to multimetric gravity theories. [MH, M. Wohlfarth 10] Additional 14 unobserved parameters. 8 parameters can be obtained from linearized field equations. Example: multimetric action can be chosen such that α + = 1, θ + = 0: standard PPN gauge choice. γ + = 1, σ + + = 2: experimental consistency. α = 1: repulsive Newtonian limit. γ = 1, θ = 0, σ + = 2: additional dark PPN parameters. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
48 Multimetric example theory Gravitational action with parameters x, y, u, v, w, r, s [MH 12]: S G = 1 16π d 4 x g 0 [x N g I ij R J ij + I,J=1 + v S I k S I k ij + w S I k im S I m jk + g I kl g I mn N g (yr I ij I ij + u S I S i I j I=1 ( r S I m ik S I n jl + s S I m ij S I n kl) )]. Restriction of input parameters by PPN consistency: y = 1 2 N Nx, v = 6 N Nx + 2u, 4 2N w = 6 N 1 + Nx 3u, r = 4 2N 2 N + Nx u. PPN consistent theory with parameters x, u, s. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
49 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Post-Newtonian consistency 5 Gravitational waves 6 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
50 Linearized multimetric gravity Perturbation ansatz: g I = η + h I. Most general linearized vacuum field equations: P p (a h b)p +Q h ab +R a b h +M p q h pq η ab +N hη ab = 0 Parameter matrices P, Q, R, M, N. Diagonalize parameter matrices. 10 parameters P 1, P 0, Q 1, Q 0, R 1, R 0, M 1, M 0, N 1, N 0. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
51 Linearized multimetric gravity Perturbation ansatz: g I = η + h I. Most general linearized vacuum field equations: P p (a h b)p +Q h ab +R a b h +M p q h pq η ab +N hη ab = 0 Parameter matrices P, Q, R, M, N. Diagonalize parameter matrices. 10 parameters P 1, P 0, Q 1, Q 0, R 1, R 0, M 1, M 0, N 1, N 0. Calculate wave-like solutions of vacuum field equations. Gravitational waves propagate at the speed of light. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
52 Polarizations and E(2) class Polarizations classified by reps. of E(2). [Eardley, Lee, Lightman et al. 73] E(2) class depends on parameters P i, R i, M i. 0 P =0 =0 P + 2R 0 0 M =0 N 2 N 3 III 5 II 6 2 tensors +1 scalar +2 vectors +1 scalar Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
53 PPN consistent multimetric example Parameter values [MH 12]: P 1 = 2Q 1 = 2R 1 = 2M 1 = 2N 1 = 1 2 N, P 0 = 6 N 4 2N 2Nx + 2u 2s, N 0 = 4 N 8 4N + Nx + u s, 2 Q 0 = 1 4, R 0 = M 0 = 6 N 8 4N + Nx u + s. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
54 PPN consistent multimetric example Parameter values [MH 12]: P 1 = 2Q 1 = 2R 1 = 2M 1 = 2N 1 = 1 2 N, P 0 = 6 N 4 2N 2Nx + 2u 2s, N 0 = 4 N 8 4N + Nx + u s, 2 Q 0 = 1 4, R 0 = M 0 = 6 N 8 4N + Nx u + s. Dependent only on single parameter p := Nx u + s. Generic case: E(2) class for h 1 : N 2 E(2) class for h i : N 2 Effective E(2) class: N 2 Special case p = 6 N 8 4N : E(2) class for h 1 : N 2 E(2) class for h i : II 6 Effective E(2) class: II 6 Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
55 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Post-Newtonian consistency 5 Gravitational waves 6 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
56 Summary Idea: Repulsive gravity might explain dark matter & dark energy. Multimetric repulsive gravity with N 3 by explicit construction. Cosmology features late-time acceleration and big bounce. Structure formation features clusters and voids. Repulsive gravity is consistent with solar system experiments. Gravitational waves are null. E(2) class can be one of N 2, N 3, III 5, II 6. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
57 Outlook Work in progress: Emission of gravitational waves from binary systems. Post-Newtonian approximation of axially symmetric solutions. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
58 Outlook Work in progress: Emission of gravitational waves from binary systems. Post-Newtonian approximation of axially symmetric solutions. Future work: Remaining PPN parameters from full multimetric PPN formalism. Restrict multimetric gravity theories by additional PPN bounds. Further construction principles, e.g., higher symmetries. Construct further exact solutions. Stabillity of cosmological solutions. Obtain restrictions from cosmological perturbation theory. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 4. september / 32
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