Massive gravitons in arbitrary spacetimes

Transcript

1 Massive gravitons in arbitrary spacetimes Mikhail S. Volkov LMPT, University of Tours, FRANCE Orsay, Modern aspects of Gravity and Cosmology, 8-th November 2017 C.Mazuet and M.S.V. arxiv: Mikhail S. Volkov Massive gravitons in arbitrary spacetimes

2 Linear 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 which gives η µν g µν, µ µ 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 g µν = η µν + h µν E µν 2δG µν + M 2 (h µν λh η µν ) = 0, or explicitly 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. One has µ δg µν 0 4 vector constraints C ν µ E µν = M 2 ( µ h µν λ ν h) = 0, and also C 5 = ν C ν + M 2 η µν E µν = M 2 (1 λ) h + M 4 (1 4λ) h = 0, which becomes constraint if λ = 1.

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 = ν C ν + M 2 g µν E µν = 3M 4 h = 0 but ONLY in vacuum or in Einstein space, for R µν = Λg µν. Otherwise 6 DoF. What to do?

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. Perturbing g µν g µν + δg µν then 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 µ σ in terms of δg µσ is very complicated /Deffayet et al./

8 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

9 Equations in the generic case γ µν is algebraically expressed in terms of background G µν and g µν. 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 )

10 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,k=0 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?

11 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

12 Two special models

13 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.

14 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 ) 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 being the FP mass) model I: M µν = γ µα X α ν g µν γ αβ X αβ, ( γ µν = R µν + M 2 R ) g µν ; 6 model II: M µν = Xµ α γ αν + X γ µν, ( γ µν = R µν M 2 + R ) g µν. 2

15 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 )

16 Constraints

17 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.

18 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 } Next C 5 ρ C ρ E µ µ = 3 2 M2 X 1 2 G µν X µν + ρ I ρ = 0 the number of DoF is 10 5 = 5.

19 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.

20 Einstein space background

21 Einstein spaces, massless limit If R µν = Λg µν γ µν g µν X µν = X νµ the equations reduce to 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. If M H = 0 then equations admit the (diff) gauge symmetry X µν X µν + (µ ξ ν) which reduces the number of DoF: 10 2 = massless limit.

22 Einstein spaces, partially massless limit For M H 0 the divergence of the equations yields four constraints µ X µν = ν X, equations reduce to X µν + µ ν X 2R µανβ X αβ + ΛXg µν + MH(X 2 µν Xg µν ) = 0. Tracing this yields (2Λ 3MH)X 2 = 0 X = = 5 DoF. If M 2 H = 2Λ/3 M2 PM gauge symmetry X µν X µν + µ ν Ω + (Λ/3)g µν Ω there remain only 4 = DoF = partially massless case.

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 auxiliary amplitudes 4 auxiliary amplitudes are expressed in terms of one single S S+ = m2 H ɛ S +, m 2 H S + = 2 m 2 H S + + = 1 Ha 2 Ṡ with H = ȧ/a while (Ṡ + a 2 HS + + ), + 2Hm4 H P2 Ṡ + m 6 H P2 S m 4 H (2P2 + 3m 2 H )S /a2 2H 2 [3m 4 H + 2P2 (2m 2 H ɛ)], S = A Ṡ + B S where A, B are complicated functions of the background scale factor a. The amplitude S fulfils one single master equation following from

31 Scalar sector master equation where the kinetic term K = I (0) = (KṠ 2 US 2 ) a 3 dt 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 U/K M 2 eff as k 0 U/K c 2 (k 2 /a 2 ) as k where c is the sound speed. There is only one DoF in the scalar sector (!!!) In the Einstein space one has m H = M H and the scalar mode does not propagate if either M H = 0 (massless limit) or if M 2 H = 2H2 (partially massless limit). In the generic case one has m H const. and it always propagates.

32 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)

33 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 ).

34 Stability of the system Everything is stable if the background density is small, ρ M 2 M 2 Pl. Model II is stable for any ρ if w = p/ρ < 2/5 stable during inflation. Model I is stable during the 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.

35 Backreaction

36 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, where the energy-momentum tensor T µν is rather complicated. The only solution in the homogeneous and isotropic sector is found in model II and only for M 2 < 0: de Sitter with Λ = 3M 2 > 0 Massive gravitons in our model cannot mimic dark energy.

37 Other applications Holographic superconductors... (?) Black hole solutions? Boson star solutions?

38 Spontaneous hair on black holes

39 Hairy black holes Einstein-Yang-Mills (+Higgs, dilaton) theory, Einstein-Skyrme (XX-th century) Scalars violating the energy conditions (phantom fields, negative potentials) (XXI-st century) Non-minimally coupled scalars (Horndeski) (XXI-st century) Non-linear massive gravity (XXI-st century) Spinning scalar clouds (XXI-st century) /M.S.V. arxiv: /

40 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 black hole is surrounded by mirror walls, the field will be trapped inside the walls but its amplitude will grow black hole bomb /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 /Damour, Deruelle, Ruffini 1976/. Such modes will be amplified but also absorbed by the black hole.

41 Black hole hair via superradiance If the amplification and absorption rates of massive modes are equal, this will lead to non-trivial stationary field clouds around spinning black holes. This suggests that spinning black holes may support massive hair and moreover, spontaneously grow it. First confirmation of this scenario scalar Kerr clouds = spinning black holes with massive complex scalar field /Herdeiro, Radu, 2014/. Next spinning black holes with massive complex vector field /Herdeiro, Radu, Runarsson 2016/.

42 Black hole hair via superradiance First confirmation of the spontaneous growth phenomenon growth of massive complex vector field /East, Pretorius 2017/. 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.

43 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.