Minimal theory of massive gravity Minimal theory of massive gravity - PowerPoint PPT Presentation

Minimal theory of massive gravity 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]

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]

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

Breaking Lorentz invariance ● Usual 4D vielbein approach A B g μ ν =η A B e μ e 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 i e N j

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

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

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

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

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

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

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

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 ∼ ˙ C i ∼ ˙ R 0 ≈ 0, R i ≈ 0,

Hamiltonian of the theory ● Define then the 4 constraints (2 only are new) C 0 ≡{ R 0 , H 1 }+ ∂ R 0 C i ≡{ R i , H 1 } ∂ t , Therefore new Hamiltonian 3 x [− N R 0 − N i R i + m 2 M H 1 +α MN P H = ∫ d [ MN ] +β MN Y [ MN ] +λ C 0 +λ i C i ] ● Dof: (9x2 – 1 – 3 – 3 – 3 – 4)/2 = 2

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

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

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

Two branches ● Two branches solutions exist X = X ± =− c 2 ± √ c 2 2 − c 1 c 3 ● Self accelerating branch c 1 X is constant ● Normal branch ˙ X + N H X − M H = 0 ● For both branches λ = 0

Friedmann evolution ● Friedmann equation 2 M P 2 2 = m 2 H 2 + c 1 X 3 )+ρ ( c 4 + 3 c 3 X + 3 c 2 X 3 M P 2 ● Same background evolution of dRGT ● Self accelerating branch: effective cosmological constant ● No extra constraint: C 0 reduces to Bianchi identity

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 ] , 2 ˙ 2 S = M P 2 2 −(∂ h λ ) h λ 4 x N a 2 ∑ λ=+ ,x ∫ d 2 h λ −μ 2 N a 2 m 2 X [ ( c 2 X + c 3 )+( c 1 X + c 2 ) M N ] μ 2 = 1

Stability of background ● Only two degrees of freedom ● The 2 dof are tensor ● Stable for µ 2 > 0 ● Therefore it is possible to have FLRW (even de Sitter)

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]

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 − 5 Hz Gws from binaries [Finn, Sutton: PRD 2002] μ< 10

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]