Minimal theory of massive gravity

Transcription

1 Minimal theory of massive gravity Antonio De Felice Yukawa Institute for Theoretical Physics, YITP, Kyoto U. 2-nd APCTP-TUS Workshop Tokyo, Aug 4, 2015 [with prof. Mukohyama]

2 Introduction drgt theory: deep insight into massive gravity [de Rham, Gabadadze, Tolley: PRL 2011] Difficult to find viable phenomenology No BD ghost [Boulware, Deser: PRD 1972] But other ghost are present in simple/crucial backgrounds [ADF, Gumrukcuoglu, Mukohyama: PRL 2012]

3 Motivation Key idea Is it possible to make the drgt idea work? The theory needs to be changed Bigravity, quasidilaton, [de Rham etal: IJMP 14; D Amico etal: PRD 13; ADF, Mukohyama: PLB 14] What if we make the theory simpler? Remove unwanted degrees of freedom Massive gravity with less than 5 dof: need to break LI

4 Breaking Lorentz invariance Usual 4D vielbein approach g μ ν =η A B e A μ e B ν Invariant under a vielbein local Lorentz transf: e A μ Λ A C e C μ Split 4D into 1+ 3, and remove local Lorentz transformation: this fixes a preferred frame Introduce the following variables N, N i, e I j Define 3D metric γ i j =δ M N e M ie N j

5 Initial variables We have 9 variables, e I j, 3 shifts: N i, lapse N Define N i =γ i j N j, N I =e I j N j Build up ADM 4D vielbein No boost: 13 vars instead of 16 (general 4D vielbein) e A μ=( N 0 T N I e I j) Metric in ADM form: g μ ν =η A B e A μ e B ν

6 Fiducial variables Along the same lines fiducial variables M, M i E I j 3D fiducial metric ~ γi j =δ M N E M i E N j ADM 4D unboosted fiducial vielbein E A μ=( M M I E I j) 0 T, M I =E I ~ j γ j k M k Unitary gauge: fixed-dynamics external fields

7 Precursor Lagrangian EH term for physical metric L EH = g R(g), M P 2 =2 Mass term: Total L 0 = m2 24 ϵ ABCD ϵ αβ γδ E A α E B β E C γ E D δ L 1 = m2 6 ϵ ABCD ϵ αβ γ δ E A α E B β E C γ e D δ L 2 = m2 4 ϵ ABCD ϵ αβ γδ E A α E B βe C γ e D δ L 3 = m2 6 ϵ ABCD ϵ αβ γδ E A α e B βe C γ e D δ L 4 = m2 24 ϵ ABCD ϵ αβ γδ e A α e B βe C γ e D δ L TOT =L EH + i=0 Same in form as drgt but asymmetrical ADM vielbein 4 c i L i

8 Precursor Hamiltonian Consider 3D vielbein as fundamental variables Physical lapse and shift as Lagrange multipliers Canonical momentum of 3D vielbein π jk =Π j I δ I J e J k, e J k e I k=δ J I Symmetry P [M N ] =e M j Π j I δ I N e N j Π j I δ I M 0 Hamiltonian linear in lapse and shift Mass term does not include shift variables

9 Precursor Hamiltonian/constraints Primary constraints: H (1) = d 3 x [ N R 0 N i R i +m 2 M H 1 +α MN P [ MN ] ] Time derivative of constraints ~ C τ, τ=1,2, Y [MN ] =δ M L E L j e N j δ N L E L j e M j, E L j E M j=δ L M Precursor Hamiltonian H (2) pre = d 3 x[ N R 0 N i R i +m 2 M H 1 +α M N P [ M N ] + ~ λ τ C ~ τ +β M N Y [M N ] ]

10 Degrees of freedom for precursor theory Phase space variables: 2x9 = 18 psv: e I j, Π j I All constraints are independent and second-class R 0, R i, P [ MN ], Y [ MN ], Number of dof: ( )/2 = 3 Reason: vielbein in ADM form (breaking LI) in mass term ~ C τ Still we try to find a theory with 2 dof (τ=1,2)

11 Introducing further constraints We need to introduce extra constraints Without overkilling the modes Without killing interesting backgrounds Notice that on the constraint surface H pre H 1 m 2 d 3 x M H 1

12 Constraints Reconsider the time-evolution of the primary constraints {R 0, H pre } 0, {R i, H pre } 0, But, as seen before, they introduce only 2 secondary constraints Then we constrain the model by imposing all 4 derivatives of primary constraints to vanish C 0 Ṙ 0 0, C i Ṙ i 0,

13 Hamiltonian of the theory Define then the 4 constraints (2 only are new) C 0 {R 0, H 1 }+ R 0, C t i {R i, H 1 } Therefore new Hamiltonian H = d 3 x[ N R 0 N i R i +m 2 M H 1 +α MN P [MN ] +β MN Y [ MN ] +λ C 0 +λ i C i ] Dof: (9x )/2 = 2

14 Building blocks We have R 0 =R 0 GR m 2 H 0, R 0 GR = γ R[ γ] 1 γ ( γ nl γ mk 1 2 γ nm γ kl ) πnm π kl, R i =R i GR =2 γ ik D j π kj, H 0 = ~ γ(c 1 +c 2 Y I I )+ γ(c 3 X I I +c 4 ), X I J =e I l E J l, X I L Y L J =δ I J, H 1 = ~ γ[ c 1Y I I + c )] 2 2 (Y I I Y J J Y J I Y I J +c 3 γ

15 Consequences The theory is now given 14 second-class independent constraints The theory is defined via the Hamiltonian (breaking Lorentz invariance) Possible to define a Lagrangian

16 Cosmology Consider a time-dependent diagonal M (t ), E I j= ~ a (t) δ I j Symmetry of also symmetric Y I J =δ I M E M l e J l e J l On the background N (t ), e I j=a(t ) δ I j The constraints C i 0 are trivially satisfied on FLRW The constraint C 0 0 equivalent to Bianchi identity (c 3 +2 c 2 X+c 1 X 2 ) ( Ẋ +N H X M H )=0, X = ~ a /a, H =ȧ/(n a)

17 Two branches Two branches solutions exist Self accelerating branch X is constant X= X ± = c 2± c 2 2 c 1 c 3 c 1 Normal branch Ẋ + N H X M H =0 For both branches

18 Friedmann evolution Friedmann equation 3 M 2 P H 2 = m2 2 M P 2 (c 4 +3c 3 X +3 c 2 X 2 +c 1 X 3 )+ρ Same background evolution of drgt Self accelerating branch: effective cosmological constant No extra constraint: C0 reduces to Bianchi identity

19 Stability of the background In drgt 3 out of 5 dof are non-dynamical Ghosts are present (not BD ghost) In this theory only 2 dof exist 3[ 2 ḣλ S= M 2 P d 4 x N a 2 λ=+, x N ( h λ) 2 2 a 2 ] μ 2 2 h, λ μ 2 = 1 2 m2 X [ (c 2 X +c 3 )+(c 1 X +c 2 ) M N ]

20 Stability of background Only two degrees of freedom The 2 dof are tensor Stable for Therefore it is possible to have FLRW (even de Sitter)

21 Phenomenology Only tensor modes propagate (besides matter field) No extra scalar/vector mode arises from gravity No need of screening any extra force No need of Vainshtein mechanism [Vainshtein: PLB 1972] No Higuchi ghost will be present (only tensor modes) [Higuchi: NPB 1987]

22 Constraints The self-accelerating branch induces an effective cosmological constant For the normal branch (for non-trivial dynamics of M, ~ a ) the background is non-trivial but no scalar dof is present Constraint coming from modification of emission rate of Gws from binaries μ<10 5 Hz [Finn, Sutton: PRD 2002]

23 Conclusions Massive gravity extensions Reducing dof to only 2 Only tensors modes remain FLRW becomes stable Phenomenology simplifies and constraints get weaker Gws are massive: phenomenology (sharp peak in GW spectrum) [Gumrukcuoglu, Kuroyanagi, Lin, Mukohyama, Tanahashi: CQG 2012]