Multimetric cosmology and structure formation
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1 Multimetric cosmology and structure formation Manuel Hohmann Teoreetilise füüsika labor Füüsikainstituut Tartu Ülikool 3. oktoober 2012 Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
2 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
3 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
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 3. oktoober / 39
5 Application to the universe 4.6% visible matter. [Komatsu et al. 09] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 c 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 3. oktoober / 39
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 c 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 J,K =1 + terms linear in I S JK + terms quadratic in S IJ. c JK g Jij R K ij + Repulsive Newtonian limit for N 3. [MH, M. Wohlfarth 10] N c IJ R J ab Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39 J=1
29 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
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 3. oktoober / 39
31 Simple cosmological model Early universe: radiation; late universe: dust. Copernican principle: common evolution for all matter sectors. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
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 3. oktoober / 39
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 for repulsive Newtonian limit: (2 N)T ab = R ab 1 2 Rg ab. Negative effective gravitational constant for early / late universe. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
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 3. oktoober / 39
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 3. oktoober / 39
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 min (cosh 2 3 ρ = (N 2)amin 2 ( cosh )) 2 3ω+1 η, ( 3ω )) 6ω+6 3ω+1 η. Positive minimal radius a min (Big Bounce). [MH, M. Wohlfarth 10] Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
37 Cosmological evolution Ρ Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
38 Cosmological parameters Friedmann equation: (2 N)Ω M + Ω K = 1. Matter density: Ω M = ρ ( ) 0 3ω + 1 3H0 2 sinh 2 η 0. 2 Curvature parameter: Ω K = k a 2 0 H2 0 Fitting of supernova data: [Amanullah et al. 10] m = 1 ȧ 2 (t 0 ) z Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
39 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
40 Ingredients Metrics g I ab = g0 ab + hi ab with g 0 = dt dt + a 2 (t)γ αβ dx α dx β and a(t) determined by cosmology. Scale for structure formation curvature radius of the universe: Cubic volume 0 x α l. Approximate γ αβ by δ αβ. Periodic boundary conditions. Matter content: n point masses M for each sector. Model for dust matter: p = 0. Matter density: ρ = Mn (al) 3. Large mean distance al/ 3 Nn 2GM. Small peculiar velocities v α Ii = aẋ α Ii 1. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
41 Local dynamics Masses of type I follow geodesics of their metric g I ab : ẍii α = αh00 I 2a 2 Antisymmetric Poisson equation: 2ȧ aẋ α Ii. N h00 I = 2 (2δ IJ 1)Φ J. J=1 Individual Newtonian potentials Φ I (t, x): Φ I (t, x) = M n 1 a(t) d( x, x Ii (t)). i=1 Periodic distance function d( x, x ): d( x, x ) = min x x + l k. k Z 3 Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
42 Implementation N = 4 matter types. n = point masses for each matter type calculation steps. Simulation written in C. Calculation using 3.0 GHz Intel Core 2 Duo E8400 CPU. 7.5 days computation time. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
43 Evolution - all matter types (N = 4, n = 16384) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
44 Final state - all matter types (N = 4, n = 16384) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
45 Final state - only visible matter (N = 4, n = 16384) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
46 Intermediate result Different matter types separate. Formation of clusters. Seemingly empty voids contain invisible matter types. Structures are very coarse. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
47 Intermediate result Different matter types separate. Formation of clusters. Seemingly empty voids contain invisible matter types. Structures are very coarse. Increase number of point masses. N = 4 matter types. n = point masses for each matter type calculation steps. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
48 Intermediate result Different matter types separate. Formation of clusters. Seemingly empty voids contain invisible matter types. Structures are very coarse. Increase number of point masses. N = 4 matter types. n = point masses for each matter type calculation steps. Higher computation power required. Simulation written in CUDA. Calculation using NVidia Tesla C2075 GPU. 2 months computation time. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
49 Evolution - all matter types (N = 4, n = ) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
50 Final state - all matter types (N = 4, n = ) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
51 Evolution - only visible matter (N = 4, n = ) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
52 Final state - only visible matter (N = 4, n = ) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
53 Intermediate result Structures are still very coarse. Voids are not very empty. Violent dynamics. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
54 Intermediate result Structures are still very coarse. Voids are not very empty. Violent dynamics. Increase number of matter types. N = 16 matter types. n = point masses for each matter type calculation steps. Simulation written in CUDA. Calculation using NVidia Tesla C2075 GPU. 3.5 months computation time. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
55 Evolution - all matter types (N = 16, n = 65536) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
56 Final state - all matter types (N = 16, n = 65536) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
57 Evolution - only visible matter (N = 16, n = 65536) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
58 Final state - only visible matter (N = 16, n = 65536) Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
59 Results Finer structures. Filaments between galactic clusters. Large voids free of visible matter. Voids contain clusters of repulsively interacting matter. Possible explanation for local velocity anomaly? [Tully 07] Important contribution to weak lensing. Strong negative gravitational lenses? Calculate gravitational lensing from simulation data! Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
60 Results Finer structures. Filaments between galactic clusters. Large voids free of visible matter. Voids contain clusters of repulsively interacting matter. Possible explanation for local velocity anomaly? [Tully 07] Important contribution to weak lensing. Strong negative gravitational lenses? Calculate gravitational lensing from simulation data! Further increase number of point masses used in the simulation? Problem: Currently used algorithm scales as O(n 2 ). Use different algorithm! Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
61 Future work Adapt GADGED-2 code [Springel 05] to multimetric gravity. TreeSPH algorithm: Gravitational forces from hierarchical multipole expansion. Gas dynamics from smoothed particle hydrodynamics (SPH). Better scaling behavior O(n log n). Usable on multicore PCs & clusters. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
62 Outline 1 Introduction 2 Multimetric cosmology 3 Simulation of structure formation 4 Conclusion Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
63 Summary Idea: Repulsive gravity might explain dark matter & dark energy. Multimetric repulsive gravity with N 3 by explicit construction. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
64 Summary Idea: Repulsive gravity might explain dark matter & dark energy. Multimetric repulsive gravity with N 3 by explicit construction. Cosmology: Big Bounce cosmology. Accelerating expansion becomes small at late times. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
65 Summary Idea: Repulsive gravity might explain dark matter & dark energy. Multimetric repulsive gravity with N 3 by explicit construction. Cosmology: Big Bounce cosmology. Accelerating expansion becomes small at late times. Structure formation: Result highly depends on number N of matter types. Formation of galactic clusters. Voids contain repulsively interacting, invisible matter. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
66 Outlook Further cosmological calculations: Analyze stability of cosmological solutions. Apply cosmological perturbation theory. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
67 Outlook Further cosmological calculations: Analyze stability of cosmological solutions. Apply cosmological perturbation theory. Improved simulations of structure formation: Use GADGED-2 code for multimetric structure formation. Include thermodynamics into simulation. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
68 Outlook Further cosmological calculations: Analyze stability of cosmological solutions. Apply cosmological perturbation theory. Improved simulations of structure formation: Use GADGED-2 code for multimetric structure formation. Include thermodynamics into simulation. Connection to observations: Test multimetric cosmology against CMB fluctuations. Peculiar motion of galaxies due to repulsive matter? Repulsive matter distribution in voids from weak lensing? Search for negative gravitational lenses. Manuel Hohmann (Tartu Ülikool) Multimetric gravity 3. oktoober / 39
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