Ghost-free bigravity theories and their cosmological solutions

Transcription

1 Ghost-free bigravity theories and their cosmological solutions Mikhail S. Volkov LMPT, University of Tours FRANCE M.S.V., JHEP 1201 (2012) 035; Phys.Rev. D86 (2012) ; D86 (2012) Kei-ichi Maeda, M.S.V., arxiv: M.S.V., thematic review for Class.Quan.Grav. IHES, 14 March 2013 Ghost-free bigravity theories and their cosmological solutions p. 1

2 Theories with massive gravitons Deformations of GR that explain the observed universe acceleration, m 1/(cosm. horizon size). Problems: do not reduce to GR in the weak field when m 0 (VdVZ discontinuity), have a ghost, no uniqueness. Remedies exist for the VdVZ problem Vainstein mechanism. Very recently a class of ghost-free models has been discovered. We wish to study cosmologies in these models. Ghost-free bigravity theories and their cosmological solutions p. 2

3 Contents Massive gravity in D=4 Ghost-free theories Proportional backgrounds FLRW cosmologies with non-bidiagonal metrics FLRW cosmologies with bidiagonal metrics Anisotropic cosmologies Ghost-free bigravity theories and their cosmological solutions p. 3

4 I. Massive gravity in D=4 Ghost-free bigravity theories and their cosmological solutions p. 4

5 Non-linear Pauli-Fierz D manifold with two metrics g µν (x) and f µν (x) = η AB µ X A (x) ν X B (x) nd the action S = 1 κ 2 ( 1 2 R+m2 L int ) gd 4 x+s [m] here L int is a scalar function of H α β = gασ f σβ δ α β L int = 1 8 ((Hα α) 2 H α β Hβ α)+o((h α β )3 ) heory is not unique, but has a unique weak field limit. nitary gauge, f µν = η µν, X µ = x µ. Ghost-free bigravity theories and their cosmological solutions p. 5

6 EOM ith ianchi identities G µν = m 2 T µν +κ 2 T [m] µν T µν = 2 L int g µν g µνl int. µ T µν = =6=2+4 propagating DOF. the unitary gauge, f µν = η µν, and for weak fields, µν = η µν +h µν, EOM reduce to Ghost-free bigravity theories and their cosmological solutions p. 6

7 Pauli-Fierz equations 1 2 { h µν +...} = 1 2 m2 (h µν hη µν )+κ 2 T µν [m] applying µ 4 constraints µ h µν ν h = 0 tracing fifth constraint 3m 2 h = 2κ 2 T [m] 10-5=5 DOF = 2s+1 polarizations of massive graviton. or generic g µν no scalar constraint 6-th propagating tate, has negative norm: Boulware-Deser ghost / 72/. VdVZ : m 0, h (2) µν +... = 2κ 2 T [m] µν, h (0) = κ 2 T [m] ainstein: h (0) is strongly coupled for r < r V = (r g /m 4 ) 1/5 Ghost-free bigravity theories and their cosmological solutions p. 7

8 II. Ghost free theories Ghost-free bigravity theories and their cosmological solutions p. 8

9 The drgt massive gravity γ µ ν = g µα f αν γ µ σγ σ ν = g µα f αν A are eigenvalues of γ µ ν. The action is S = 1 8πG ( 1 2 R+m2 U ) gd 4 x ith = b 0 +b 1 λ A +b 2 λ A λ B +b 3 λ A λ B λ C +b 4 λ 0 λ 1 λ 2 λ 3 A A<B A<B<C here is scalar constraint. If only b 1 0 (b 1 = 1) then G µ ν +m 2 (γ µ ν δ µ νγ) = 0 Ghost-free bigravity theories and their cosmological solutions p. 9

10 Constraints g µν = η AB e µ A eν B, f µν = η AB ω A µω B ν, γ µ ν = e µ A ωa ν µν G µν +m 2 (γ µν g µν γ) = 0 γ [µν] = 0 ˆγ 2 = ĝ 1 ˆf E E µ µ = R 3m 2 γ = 0. 4 : 0 = C µ ν E µ ν = m 2 ( ν γ µ ν µ γ); 1 : 0 = C = σ ((γ 1 ) σ µc µ )+ m2 2 E = ( = m 2 { σ (γ 1 ) σ µ( ν γ µ ν µ γ) } R ) 2 3m2 2 γ umino 70/ Ghost-free bigravity theories and their cosmological solutions p. 10

11 The ghost-free bigravity = 1 2κ 2 g R gd 4 x+ 1 2κ 2 f R fd 4 x m2 κ 2 U gd 4 x + S m [g,g-matter]+s m [f,f-matter], κ g = κcosη, κ f = κsinη, γ µ ν = g µα f αν b k U k = b 0 +b 1 λ A +b 2 λ A λ B +b 3 λ A λ B λ C +b 4 λ 0 λ 1 λ 2 λ 3 U = k A A<B A<B<C g f, λ A 1 λ A, k b k U k g k b 4 k U k f lat space is the solution and m is the FP mass if only 0 = 4c 3 +c 4 6, b 1 = 3 3c 3 c 4, b 2 = 2c 3 +c 4 1, 3 = (c 3 +c 4 ), b 4 = c 4. /Hassan,Rosen 2011/ Ghost-free bigravity theories and their cosmological solutions p. 11

12 Field equations G ρ λ = m2 cos 2 ηt ρ λ +T[m]ρ λ, Gρ λ = m2 sin 2 ηt ρ λ +T [m]ρ λ, T ρ λ = τ ρ λ δρ λ U, T ρ λ = g f τ ρ λ, τ ρ λ = {b 1 U 0 +b 2 U 1 +b 3 U 2 +b 4 U 3 }γ µ ν {b 2 U 0 +b 3 U 1 +b 4 U 2 }(γ 2 ) µ ν + {b 3 U 0 +b 4 U 1 }(γ 3 ) µ ν b 4 U 0 (γ 4 ) µ ν Massive gravity for η 0 if f µν becomes flat. g µν = f µν = η µν g µν = η µν +δg µν, f µν = η µν +δf µν, h (mass) µν = cosηδg µν +sinηδf µν, h (0) µν = cosηδf µν sinηδg µν Ghost-free bigravity theories and their cosmological solutions p. 12

13 III. Proportional backgrounds Ghost-free bigravity theories and their cosmological solutions p. 13

14 f µν = C 2 g µν γ µ ν = Cδ µ ν G ρ λ +Λ g(c)δ ρ λ = T[m]ρ λ, Gρ λ +Λ f(c)δ ρ λ = T [m]ρ λ. Λ g (C) = cos 2 η ( b 0 +3b 1 C +3b 2 C 2 +b 3 C 3), Λ f (C) = sin2 η C 3 ( b1 +3b 2 C +3b 3 C 2 +b 4 C 3). ν µ = G ν µ/c 2 Λ f = Λ g /C 2, T [m]µ ν = T [m]µ ν /C 2 (fine tuning) f Λ g /C 2 = 0 4 solutions for C,Λ. 3 = 1,c 4 = 0.3,η = 1, Λ g = {10.126; 0; 0.509; 4.505}. = 1 Λ g = 0 GR. Λ g > 0 self acceleration. o massive gravity limit. Ghost-free bigravity theories and their cosmological solutions p. 14

15 IV. FLRW cosmologies with non-bidiagonal metrics (de Sitter sector) Ghost-free bigravity theories and their cosmological solutions p. 15

16 Spherical symmetry ds 2 g = Q 2 dt 2 +N 2 dr 2 +R 2 dω 2 ds 2 f = (aqdt+cndr) 2 +(cqdt bndr) 2 +u 2 R 2 dω 2,,N,R,a,b,c,u depend on t,r, a cn/q 0 0 γ µ ν = g µα cq/n b 0 0 f αν = 0 0 u u. T 0 r = cn Q [(c 3 +c 4 )u 2 +2(1 2c 3 c 4 )u+3c 3 +c 4 3]. Ghost-free bigravity theories and their cosmological solutions p. 16

17 No radial flux [(c 3 +c 4 )u 2 +2(1 2c 3 c 4 )u+3c 3 +c 4 3] = 0 ( ) 1 u = 2c 3 +c 4 1± 1 c 3 +c 4 +c 2 3 c 3 +c 4 T 0 0 = T r r = const. ) µ T µ ν T r r T θ θ = (c 3u u c 3 +2)((u a)(u b)+c 2 ) = 0 ssuming the latter is true, T µ ν = const δ µ ν, T µ ν = const δ µ ν Ghost-free bigravity theories and their cosmological solutions p. 17

18 Equations (A) G µ ν +Λ g δ µ ν = T [m]µ ν (B) G µ ν +Λ f δ µ ν = T [m]µ ν (C) (c 3 u u c 3 +2)[(u a)(u b)+c 2 ] = 0 Λ g = m 2 cos 2 η(1 u)(c 3 u u c 3 +3) > 0, Λ f = m 2 sin 2 η u 1 u 2 (c 3 u c 3 +2) < 0 T [m]µ ν = diag[ ρ(t),p(t),p(t),p(t)], T [m]µ ν = 0 hamseddine, M.S.V, 11/, /d Amico et al. 11/, /Kobayashi et al. 12/, ratia, Hu, Wyman 12/, /M.S.V. 12/ Ghost-free bigravity theories and their cosmological solutions p. 18

19 (A)+ (B) ds 2 g = dt 2 +a 2 (t) ( ) dr 2 1 kr 2 +r2 dω 2, ds 2 f = (U)dT 2 + du2 (U) +U2 dω 2. here ȧ 2 a2 3 (Λ+ρ) = k, (U) = 1 Λ f 3 U2 ow to relate T,U to t,r? One had ds 2 f = (aqdt+cndr)2 +(cqdt bndr) 2 +u 2 R 2 dω 2. Ghost-free bigravity theories and their cosmological solutions p. 19

20 (C) s 2 f = (ω0 ) 2 +(ω 1 ) 2 +u 2 R 2 dω 2 = (ϑ 0 ) 2 +(ϑ 1 ) 2 +U 2 dω 2. hile 0 = dt, θ 1 = du, ω 0 = a(adt+cdr), ω 1 = a(bdr cdt), = ur = ura(t), while θ 0,θ 1 are boost-related to ω 0,ω 1 ω 0 = θ 0 secα+θ 1 tanα, ω 1 = θ 1 secα+θ 0 tanα ( ) quating coefficients in front of dt, dr in ( ) gives a,b,c,α in rms of U,. Inserting to (u a)(u b)+c 2 = 0 gives Ghost-free bigravity theories and their cosmological solutions p. 20

21 Consistency a 1 kr 2 ( UT TU ) u 2 a 2 A+ A +ua = 0 ( ) ± = a( T ± U)+ 1 kr 2 (U ± T ), = 1 Λ f 3 U2, = ura. Exact solutions of ( ) are found in the massive ravity limit, when η = Λ f = 0, = 1, t ( ) dt u 2 k = 0 : T(t,r) = C ȧ + 4C +Cr2 a, t = ±1 : T(t,r) = (C 2 +ku 2 )(ȧ +k)dt+ca 2 1 kr 2 T(t, r), U(t, r) are found, consistency is fulfilled. Ghost-free bigravity theories and their cosmological solutions p. 21

22 Properties of the solutions g-metric is FLRW with open, closed or flat sections. Matter-dominated at early times, Λ-dominated at late time self-acceleration. Without matter de Sitter or static Schwarzschild-de Sitter /Isham,Story 78/ f-metric is AdS. When η 0, Λ f sin 2 η 0 f µν is flat, massive gravity is recovered. In the massive gravity limit this gives the complete FLRW solution. In the literature is called inhomogeneous, since the fluctuations are expected to be non-flrw, although this effect should be suppressed by smallness of m. Ghost-free bigravity theories and their cosmological solutions p. 22

23 V. FLRW cosmologies with diagonal metrics (Vainstein sector) M.S.V. 11 M. von Strauss et al. 11 M. Cristosomi et al. 11 Ghost-free bigravity theories and their cosmological solutions p. 23

24 Diagonal metrics ( ) dr ds 2 g = dt 2 +e 2Ω 2 1 kr 2 +r2 dω 2, k = 0,±1 ( ) dr ds 2 f = A 2 dt 2 +e 2W 2 1 kr 2 +r2 dω 2. quations (here Λ g (ξ),λ f (ξ) are polynomials in ξ = e W Ω ) 2 = Λ g(ξ)+ρ g 3 k 4 e 2Ω, Ẇ 2 A 2 = Λ f(ξ)+ρ f 3 k 4 e 2W, ( ) nd the conservation condition [ (e W ) A ( e Ω ) ]( b1 +2b 2 ξ +b 3 ξ 2) = 0. Ghost-free bigravity theories and their cosmological solutions p. 24

25 Generic solutions [ (e W ) A ( e Ω ) ] = 0 Ẇξ = ΩA e difference of 2 ODE s ( ) gives algebraic relation Λ g (ξ)+ρ g (Ω) = ξ 2 (Λ f (ξ)+ρ f (Ω,ξ)) ξ = ξ(ω). ( ) jecting ξ(ω) back to ( ) gives the Friedmann equation ȧ 2 +U(a) = k here a = 2e Ω and U = (a 2 /3)(Λ g +ρ g ). here are several roots of ( ) several types of U(a). Ghost-free bigravity theories and their cosmological solutions p. 25

26 Physical and exotic cosmologies 1 5 k= IV III III III k=0 k=1 A IIa B IIb I IIc U(a)/m C D V k=-1 k=0 k= a a physical: ρ m 2 T 0 0 for small a, ρ m2 T 0 0 for large a exotic: ρ m 2 T 0 0 for any a. f µν is not flat for η 0 no massive gravity limit Ghost-free bigravity theories and their cosmological solutions p. 26

27 Special solutions = e W Ω is constant and satisfies b 1 +2b 2 ξ +b 3 ξ 2 = 0, = 2e Ω fulfills ȧ 2 (a 2 /3)(Λ g (ξ)+ρ g ) = k cosmology with constant Λ g (ξ), also A 2 = f 00 = (Λ g +ρ g )a 2 3k (Λ f +ρ f )a 2 3k/ξ 2 or k = 1 (open universe) admits the massive gravity limit,λ f,ρ f 0, f µν η µν the only truly homogeneous and otropic massive gravity cosmology, although unstable at on-linear level. /Mikohyama et al./ Ghost-free bigravity theories and their cosmological solutions p. 27

28 VI. Anisotropic cosmologies with diagonal metrics Kei-ichi Maeda, M.S.V. arxiv: Ghost-free bigravity theories and their cosmological solutions p. 28

29 Bianchi class A types s 2 g = α(t) 2 dt 2 +h ab (t)ω a ω b, ds 2 f = A2 (t)dt 2 +H ab (t)ω a ω b. [e a,e b ] = C c ab e c, C c ab = ncd ǫ dab, n ab = diag[n (1),n (2),n (3) ] I II VI 0 VII 0 VIII IX n (1) n (2) n (3) h ab,h ab are diagonal G 0 r = G 0 r = 0 no radial fluxes. h ab = diag[α 2 1,α 2 2,α 2 3], H ab = diag[a 2 1,A 2 2,A 2 3]. Ghost-free bigravity theories and their cosmological solutions p. 29

30 3-curvature andu [α 1,α 2,α 3 ] = e Ω [A 1,A 2,A 3 ] = e W [ e β ++ 3β,e β + ] 3β,e 2β + [ e B ++ 3B,e B + ] 3B,e 2B + (3) R = 2n(1) n (3) α e 4Ω{ n (1) α 2 1 n (2) α 2 2 +n (3) α 2 3 } U g = b 0 e 3Ω +b 3 e 3W + b 1 e W+2Ω( e 2(B + β + ) +2e B + β + cosh[ ) 3(B β )] + b 2 e 2W+Ω( e 2(B + β + ) +2e (B + β + ) cosh[ ) 3(B β )]. (3) R : α a A a, U f : b k b k+1, U g = αu g +AU f. Ghost-free bigravity theories and their cosmological solutions p. 30

31 Equations ( ) 2 ( 3Ω Ω e = e β ) 2 3Ω + +( e β ) 2 [ 3Ω + 1 α α α 6 ( ) [ 3Ω Ω e = 1 ( ) U cos 2 η α 6 Ω +3αU g e β ) ( ) 3Ω ± = 1 2cos 2 ηu αe 3Ω(3) R, α 12 β ± ) e 3W Ẇ 2 ( = e ) 2 3W B + +( e ) 2 [ 3W B + 1 A A A 6 ) e 3W Ẇ [ = 1 ( ) U sin 2 η A 6 W +3AU f ) ( ) 3W Ḃ± e = 1 2sin 2 ηu Ae 3W(3) R. A 12 B ± 2cos 2 ηe 3Ω U g e 6Ω(3) R +2e 6Ω ρ g ], 2αe 3Ω(3) R +3αe 3Ω (ρ g P g ) 2sin 2 ηe 3W U f e 6W(3) R +2e 6W ρ f ], 2Ae 3W(3) R +3Ae 3W ( ρ f P f ) ], ], α ( Ẇ ) W + B + + B U g = A B + B ( Ω ) Ω + β + + β U f. β + β Ghost-free bigravity theories and their cosmological solutions p. 31

32 f µν = C 2 g µν equal anisotropies s 2 g = dt 2 +e 2Ω( e 2β ++ 3β dx 2 1 +e 2β + 3β dx 2 2 +e 4β + dx 2 3 Bianchi I, equations ) ( e 3Ω Ω) 2 = σ 2 + +σ (Λ g +ρ g )e 6Ω, e 3Ω β± = σ ± here Λ g = cos 2 η ( b 0 +3b 1 C +3b 2 C 2 +b 3 C 3) 3H 2, b 0 +3b 1 C +3b 2 C 2 +b 3 C 3) = tan2 η C ( b1 +3b 2 C +3b 3 C 2 +b 4 C 3) Ω = Ht+O(e 3Ht ), β ± = B ± = β ± ( )+O(e 3Ht ). Ghost-free bigravity theories and their cosmological solutions p. 32

33 Dynamical system formulation ith ẏ N = F N (α,a,y M ), y 0 = e Ω, y 1 = e β +, y 2 = e 3β, y 3 = e W, y 4 = e B +, y 5 = e 3B, y 6 = e3ω α Ω, y 7 = e3ω α β +, y 8 = e3ω α β, y 9 = e3w A Ẇ, y 10 = e3w A B +, y 11 = e3w A B. lus three constraints C 1 (y N ) = 0, C 2 (y N ) = 0, C 3 (y N ) = 0. Ghost-free bigravity theories and their cosmological solutions p. 33

34 Constraints C 1 = 11 N=0 C 1 y N F N C 2 = 11 N=0 C 2 y N F N C 3 0 C 3 = 0 C 1,C 2 propagate. Does C 3 propagate itself? C 3 = 11 N=0 C 3 y N F N = αx α (y M )+AX A (y M ) 0 condition of propagation of all constraints A = X α X A α it is enough to impose the constraints only at t = t 0. Ghost-free bigravity theories and their cosmological solutions p. 34

35 Strategy t the initial moment t = 0 the universe is an anisotropic eformation of a finite size FLRW. One chooses Ω(0) = 0 e initial universe size e Ω 1 in 1/m units. The initial nisotropies β ±, B ±, β ±, B ± The f-sector is empty, f = 0. The g-sector contains radiation + dust, ρ g = 0.25 e 4Ω e 3Ω he dimensionful energy m 2 M 2 pl ρ g (ev) 4, assuming at m ev. or all Bianchi types, the solutions rapidly approach a state ith a constant expansion rate and constant and non-zero nisotropies = late time attractor. Ghost-free bigravity theories and their cosmological solutions p. 35

36 Expansion rate and anisotropies 0.02 VIII Bianchi IX 0.01 B -- VI 0 II 0 β + I,VII 0 β IX B Ht Ht or Bianchi I one can scale away the constant values of ± = B ±, but not for other Bianchi types universe enerically approaches an anisotropic state, although it xpands with a constant rate. Ghost-free bigravity theories and their cosmological solutions p. 36

37 f-metric and shears 0.06 ξ=exp(w-ω) VI A 0.02 VIII Ht Ht oth A and e W Ω approach the same value f µν = C 2 g µν. ight: Σ = β2 + + β 2 / Ω, the relative contribution of shears the total energy. If only one or two Hubble times have lapsed since the acceleration started, then Σ is not small. Ghost-free bigravity theories and their cosmological solutions p. 37

38 Late time anisotropies 0.8 Bianchi IX Ht he rescaled shears e 3Ω/2 β± and e 3Ω/2 B ± oscillate with onstant amplitudes. Linearizing around f µν = C 2 g µν, β ± B ± e 3Ht/2 cos(hωt) with ω = ω(c,b k,η,h) he shear energy β2 + + β 2 e 3Ω 1/a 3 falls off as the nergy of a non-relativistic (dark?) matter. In GR 1/a 6. Ghost-free bigravity theories and their cosmological solutions p. 38

39 Near singularity behaviour 1 exp(ω) a a Bianchi IX Bianchi I exp(w) -0.5 A a Ht Ht hen continued to the past, the solutions show a ingularity where both e Ω and e W vanish. For Bianchi IX nisotropies start fluctuating near singularity. Ghost-free bigravity theories and their cosmological solutions p. 39

40 Empty Bainchi IX 1 empty Bianchi IX 0.5 small anisotropy exp(ω) large anisotropy Ht ρ g = ρ f = 0 and anisotropies vanish de Sitter with a ounce in the past, 2He Ω = cosh(t t 0 ). or small anisotropies it is still a bounce, while for larger nisotropies a singularity appears. Ghost-free bigravity theories and their cosmological solutions p. 40

41 Empty Bainchi IX chaos 10 0 ln(α 1 ) ln(α 2 ) ln(α 3 ) empty Bianchi IX Ω ear singularity a sequence of Kasner-like periods with α a t p a with p 1 +p 2 +p 3 = p 2 1 +p 2 2 +p 2 3 = 1. atter cannot change this, as ρ grows slower than shears, 1/a 6 shear energy = β β 2 1/a 3 Ghost-free bigravity theories and their cosmological solutions p. 41

42 Conclusions types of FLRW self-accelerating cosmologies in bigravity: ] f µν = C 2 g µν, require source fine-tuning, ρ f = ρ g /C 2 ] bidiagonal, approach [a] at late times when ρ f = ρ g 0 ] non-bidiagonal, admit the limit of flat f-metric nisotropic cosmologies approach anisotropic versions of ]. In GR shear energy 1/a 6, while in bigravity it is 1/a 3, which could perhaps mimic dark matter. The ianchi IX bigravity cosmology is chaotic near singularity. is unclear if there exist non-bidiagonal anisotropic osmologies. Ghost-free bigravity theories and their cosmological solutions p. 42