Massive gravitons in arbitrary spacetimes

Transcription

1 Massive gravitons in arbitrary spacetimes Mikhail S. Volkov LMPT, University of Tours, FRANCE Kyoto, YITP, Gravity and Cosmology Workshop, 6-th February 2018 C.Mazuet and M.S.V., Phys.Rev. D96, (2017) arxiv: Mikhail S. Volkov Massive gravitons in arbitrary spacetimes

2 Massive gravitons in curved space Cosmology Black holes

3 Massive fields in curved space Spin 0. One has in Minkowski space η µν µ ν Φ = M 2 Φ To pass to curved space one replaces η µν g µν, µ µ which gives g µν µ ν Φ = M 2 Φ Similarly for spins 1/2 (Dirac), 1 (Proca), 3/2 (Rarita-Schwinger). The procedure fails for the massive spin 2.

4 Massive spin 2 in flat space Fierz-Pauli equations E µν σ µ h σν + σ ν h σµ h µν µ ν h + η µν ( h α β h αβ ) + M 2 (h µν λh η µν ) = 0 which imply 4 vector constraints C ν µ E µν = M 2 ( µ h µν λ ν h) = 0, and also C 5 = ( µ ν + M 2 η µν )E µν = M 2 (1 λ) h + M 4 (1 4λ) h = 0 which becomes constraint if λ = 1, C 5 = 3M 2 h = 0.

5 Fierz-Pauli: λ = 1 ( + M 2 )h µν = 0, µ h µν = 0, h = 0, 10 5 = 5 propagating DoF. For λ 1 there are 6 DoF. Passing to curved space via η µν g µν and µ µ yields E µν σ µ h σν + σ ν h σµ h µν µ ν h + g µν ( h α β h αβ ) + M 2 (h µν h g µν ) = 0. This implies the 5 constraints C ν µ E µν = M 2 ( µ h µν ν h) = 0, C 5 = ( µ ν + M 2 g µν )E µν = 3M 4 h = 0 ONLY in Einstein spaces, if R µν = Λg µν. For R µν Λg µν there are 5+1 DoF ghost is present.

6 Linear theory from the nonlinear one

7 Ghost-free massive gravity Let g µν and f µν be the physical and reference metrics and γ µ σγ σ ν = g µσ f σν, γ µν = g µσ γ σ ν, [γ] = γ σ σ. The equations are /drgt, 2010/ E µν G µν (g) + β 0 g µν + β 1 ([γ] g µν γ µν ) + β 2 γ ( [γ 1 ] γµν 1 γµν 2 ) + β3 γ γµν 1 = 0.

8 Ghost-free massive gravity Let g µν and f µν be the physical and reference metrics and γ µ σγ σ ν = g µσ f σν, γ µν = g µσ γ σ ν, [γ] = γ σ σ. The equations are /drgt, 2010/ E µν G µν (g) + β 0 g µν + β 1 ([γ] g µν γ µν ) + β 2 γ ( [γ 1 ] γµν 1 γµν 2 ) + β3 γ γµν 1 = 0. Perturbing g µν g µν + δg µν yields E µν E µν + δe µν with δe µν = δg µν + β 0 δg µν + β 1 ([δγ] g µν + [γ] δg µν δγ µν ) +... where δγ µ σγ σ ν + γ µ σδγ σ ν = δg µσ f σν δγγ + γδγ = δg 1 f. Solution for δγ in terms of δg is very complicated /Deffayet et al./

9 Ghost-free massive gravity in tetrad formalism Introducing two tetrads e a µ and φ a µ such that g µν = η ab e a µe b ν, f µν = η ab φ a µφ b ν, one has and the equations γ a b = φa σe σ b, γ ab = η ac γ c b = γ ba E ab G ab + β 0 η ab + β 1 ([γ] η ab γ ab ) + β 2 γ ( [γ 1 ] γ 1 ab ) γ 2 ab + β3 γ γ 1 ab = 0. The idea is to linearize with respect to tetrad perturbations e a µ e a µ + δe a µ with δe a µ = X a b eb µ and then project to e µ a and express everything in terms of X µν = η ab e a µδe b ν δg µν = X µν + X νµ

10 Equations in the generic case with the kinetic term E µν µν + M µν = 0 µν = 1 2 σ µ (X σν + X νσ ) σ ν (X σµ + X µσ ) 1 2 (X µν + X νµ ) µ ν X ( ) + g µν X α β X αβ + R αβ X αβ R σ µ X σν R σ ν X σµ and the mass term M µν = β 1 ( γ σ µx σν g µν γ αβ X αβ ) + β 2 { γ α µγ β ν X αβ (γ 2 ) α µ X αν + γ µν γ αβ X αβ +[γ] γ α β X αν + ((γ 2 ) αβ X αβ [γ] γ αβ X αβ ) g µν } + β 3 γ ( X µσ (γ 1 ) σ ν [X ](γ 1 ) µν )

11 Equations for γ µν Background equations G µν + β 0 g µν + β 1 ([γ] g µν γ µν ) +β 2 γ ( [γ 1 ] γµν 1 γµν 2 ) + β3 γ γµν 1 = 0 can be viewed as cubic algebraic equations for γ µν. For any g µν the solution is γ µν (g) = n=0 k=0 3 b nk (β A ) R n (R k ) µν, There are special values of β A for which the sum is finite. How many propagating DoF are there?

12 Constraints There are 16 equations E µν µν + M µν = 0 for 16 components of X µν. The imply the following 11 conditions: [µν] = 0 M [µν] = 0 6 algebraic constraints C ν = µ E µν = 0 4 vector constraints C 5 = µ ((γ 1 ) µν C ν ) + β 1 2 E α α + β 2 γ µν E µν ( ) γ + β 3 (γ 1 ) 0α (γ 1 ) 0β (γ 1 ) 00 (γ 1 ) αβ g 00 ( E αβ 1 2 g αβ(e σ σ 1 )) g 00 E 00 ) The number of DoF is = 5. = 0 scalar constraint

13 Two special models

14 Models I and II Background equations G µν + β 0 g µν + β 1 ([γ] g µν γ µν ) +β 2 γ ( [γ 1 ] γµν 1 γµν 2 ) + β3 γ γµν 1 = 0 are non-linear in γ µν. There are two exceptional cases: Model I: β 2 = β 3 = 0, G µν + β 0 g µν + β 1 ([γ] g µν γ µν ), which can be resolved with respect to γ µν ; Model II: β 1 = β 2 = 0, G µν + β 0 g µν + β 3 γ γ 1 µν = 0, which can be resolved with respect to γ γ 1 µν.

15 Equations for the two special models with the kinetic term E µν µν + M µν = 0 µν = 1 2 σ µ (X σν + X νσ ) σ ν (X σµ + X µσ ) and the mass term 1 2 (X µν + X νµ ) µ ν X ( ) + g µν X α β X αβ + R αβ X αβ R σ µ X σν R σ ν X σµ model I: M µν = γ µα X α ν g µν γ αβ X αβ, ( γ µν = R µν + M 2 R ) g µν ; M 2 = β 0 /3 6 model II: M µν = Xµ α γ αν + X γ µν, ( γ µν = R µν M 2 + R ) g µν, M 2 = β 0. 2

16 Action I = 1 2 X νµ E µν g d 4 x L g d 4 x (order of indices!) with L = L (2) + L (0) where L (2) = 1 4 σ X µν µ X νσ α X µν α X µν α X β X αβ 1 8 αx α X with X µν = X µν + X νµ and X = X α α. One has in model I L (0) = 1 2 X µν R σ µx σν (M2 R 6 )(X µνx νµ X 2 ) and in model II L (0) = 1 2 X µν R σ µx σν 1 2 X µν R σ νx σµ 1 2 X µν X να R α µ + XR µν X µν (M2 + R 2 )(X µνx νµ X 2 )

17 Constraints

18 Algebraic constraints E µν µν + M µν = 0 are 16 equations for 16 components of X µν. One has µν = νµ hence one should have M [µν] = 0 which yields 6 algebraic conditions Model I: γ µα X α ν = γ να X α µ Model II: Xµ α γ αν = Xν α γ αµ which reduce the number of independent components of X µν to 10.

19 Differential constraints, model I with ( γ µν = R µν + M 2 R ) g µν 6 one obtains the four vector constraints C ρ (γ 1 ) ρν µ E µν = σ X σρ ρ X + I ρ = 0 with I ρ = (γ 1 ) ρν { X αβ ( α G βν ν γ αβ ) + µ γ µα X α ν } There is also a scalar constraint C 5 ( ρ (γ 1 ) ρν µ + 12 ) g µν E µν = 3 2 M2 X 1 2 G µν X µν + ρ I ρ = 0 the number of DoF is 10 5 = 5.

20 Differential constraints, model II With one has ( γ µν = R µν M 2 + R ) g µν 2 C ρ γ ρν µ E µν = Σ ρναβ ν X αβ = 0 with Σ ρναβ γ ρν γ αβ γ ρβ γ να and C 5 ρ C ρ + 1 2g 00 Σ00α β ( 2E β α δα β (E σ σ 1 ) g 00 E 00 ) = 0. This does not contain the second time derivative constraint.

21 Einstein space background

22 Einstein spaces, massless limit R µν = Λg µν γ µν g µν X µν = X νµ everything reduces to the standard Higuchi equations µν + MH(X 2 µν Xg µν ) = 0 where the Higuchi mass I: MH 2 = Λ/3 + M 2, II: MH 2 = Λ + M 2. Massless limit: M H = 0 X µν X µν + (µ ξ ν) = 2 DOF Partially massless limit: M 2 H = 2Λ 3 X µν X µν +( µ ν + Λ 3 g µν)ω = 4 DOF None of these limits exists for R µν Λg µν.

23 Short summary Six algebraic conditions and five differential constraints C ρ = 0 and C 5 = 0 reduce the number of independent components of X µν from 16 to 5. This matches the number of polarizations of massive particles of spin 2. When restricted to Einstein spaces, the theory reproduces the standard description of massive gravitons. Unless in Einstein spaces, no massless (or partially massless) limit. For any value of the FP mass M the number of DoF on generic background is 5.

24 Cosmological background

25 FLRW cosmology Line element g µν dx µ dx ν = dt 2 + a 2 (t)dx 2 where a(t) fulfills the Einstein equations 3 ȧ2 a 2 = ρ M 2 Pl ρ, 2 ä a + ȧ2 a 2 = p M 2 Pl p. Here M Pl is the Planck mass and ρ, p are the energy density and pressure of the background matter.

26 Fourier decomposition X µν (t, x) = a 2 (t) k X µν (t, k)e ikx where the Fourier amplitude splits into the sum of the tensor, vector, and scalar harmonics, X µν (t, k) = X µν (2) + X µν (1) + X µν (0) The spatial part of the background Ricci tensor R ik δ ik hence X ik = X ki X µν has only 13 independent components. Assuming the spatial momentum k to be directed along the third axis, k = (0, 0, k), the harmonics are

27 Tensor, vector, scalar harmonics X µν (2) = 0 D + D 0 0 D D + 0, X (1) X (0) µν = µν = S iks + 0 S S 0 iks S k2 S 0 W + + W + 0 W+ 0 0 ikv + W 0 0 ikv, 0 ikv + ikv 0, where D ±, V ±, S, W ± ±, S ± ± are 13 functions of time. The equations split into three independent groups one for the tensor modes X µν (2), one for vector modes X µν (1), and one for scalar modes X µν (0).

28 Tensor sector The effective action is I (2) = (KḊ 2 ± UD 2 ±) a 3 dt with where K = 1, U = M 2 eff + k2 /a 2 I : M 2 eff = M ρ, m2 H = M 2 eff, II : M 2 eff = M 2 p, m 2 H = M 2 + ρ m H reduces to the Higuchi mass in the Einstein space limit.

29 Vector sector 4 auxiliary amplitudes are expressed in terms of two V ± W + ± = P 2 m 2 H V ± m 4 H + P2 (m 2 H ɛ/2), W ± = P2 [m 2 H ɛ] V ± m 4 H + P2 (m 2 H ɛ/2), (with ɛ = ρ + p) and the effective action I (1) = (K V ± 2 UV±) 2 a 3 dt with K = U = M 2 eff k2 k 2 m 4 H m 4 H + (k2 /a 2 )(m 2 H ɛ/2), In Einstein spaces one has m H = M H (Higuchi mass), vector modes do not propagate if M H = 0 (massless limit). Otherwise m H const. they always propagate.

30 Scalar sector I (0) = (KṠ 2 US 2 ) a 3 dt where the kinetic term K = 3k 4 m 4 H (m2 H 2H2 ) (m 2 H 2H2 )[9m 4 H + 6(k2 /a 2 )(2m 2 H ɛ)] + 4(k4 /a 4 )(m 2 H ɛ) and the potential (c being the sound speed) U/K M 2 eff as k 0 U/K c 2 (k 2 /a 2 ) as k There is only one DoF in the scalar sector (!!!) In Einstein spaces one has m H = M H and the scalar mode does not propagate if M H = 0 (massless limit) or if M 2 H = 2H2 (PM limit). In the generic case one has m H const. and it always propagates.

31 No ghost conditions lim K > 0 k with K (2) = 1, K (1) = K (0) = k 2 m 4 H m 4 H + (k2 /a 2 )(m 2 H ɛ/2), 3k 4 m 4 H (m2 H 2H2 ) (m 2 H 2H2 )[9m 4 H + 6(k2 /a 2 )(2m 2 H ɛ)] + 4(k4 /a 4 )(m 2 H ɛ)

32 No tachyon conditions c 2 > 0 with c 2 (2) = 1, c 2 (1) = M2 eff m 4 H (m 2 H ɛ/2), c 2 (0) = (m2 H ɛ)[m4 H + (2H2 4M 2 eff ɛ)m2 H + 4H2 M 2 eff ] 3m 4 H (2H2 m 2 H ).

33 Stability of the system Everything is stable if the background density is small, ρ M 2 M 2 Pl. Model II is stable during inflation. Model I is stable during inflation if the Hubble rate is not very high, H < M. Both models are always stable after inflation if M GeV. Both models are stable now if M 10 3 ev. Assuming that X µν couples only to gravity and hence massive gravitons do not have other decay channels, it follows that they could be a part of Dark Matter (DM) at present.

34 Backreaction

35 Self-coupled system I = 1 2 (M 2 Pl R + X νµ E µν ) g d 4 x. Varying this with respect to the X µν and g µν yields M 2 Pl G µν = T µν, E µν = 0. The only solution in the homogeneous and isotropic sector is de Sitter with Λ = 3M 2 > 0, hence for M 2 < 0. Massive gravitons in our model cannot mimic dark energy.

36 Black hole hair via superradiance

37 Superradiance Incident waves with ω < mω H are amplified by a spinning black hole /Zel dovich 1971/, /Starobinsky 1972/, /Bardeen, Press, Teukolsky 1972/ If the field has a mass µ then its modes with ω < µ cannot escape to infinity and will stay close to the black hole. Such modes will be amplified but also absorbed by the black hole. /Damour, Deruelle, Ruffini 1976/. It follows that massive hair should grow spontaneously on black holes

38 Black hole hair via superradiance First confirmation of this scenario scalar Kerr clouds = stationary spinning black holes with massive complex scalar field /Herdeiro, Radu, 2014/. Next spinning black holes with massive complex vector field /Herdeiro, Radu, Runarsson 2016/. First confirmation of the spontaneous growth phenomenon growth of massive complex vector field /East, Pretorius 2017/.

39 Black hole hair via superradiance As the supperadiance rate increases with spin, the vector massive hair grows faster than the scalar one easier to simulate. However, the tensor hair should grow still faster. This suggest there should be spinning black holes with complex massive graviton hair. Complexification replacing in the action I = 1 2 X νµ E µν X νµ E µν + X νµ Ē µν (M 2 Pl R + X νµ E µν ) g d 4 x.

40 Summary of results The consistent theory of massive gravitons in arbitrary spacetimes presented in the form simple enough for practical applications. The theory is described by a non-symmetric rank-2 tensor whose equations of motion imply six algebraic and five differential constraints reducing the number of independent components to five. The theory reproduces the standard description of massive gravitons in Einstein spaces. In generic spacetimes it does not show the massless limit and always propagates five degrees of freedom, even for the vanishing mass parameter. The explicit solution for a homogeneous and isotropic cosmological background shows that the gravitons are stable, hence they may be a part of Dark Matter. An interesting open issue possible existence of stationary black holes with massive graviton hair.