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]

• 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]

Introduction

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
• Invariant under a vielbein local Lorentz transf:
• Split 4D into 1+ 3, and remove local Lorentz

transformation: this fixes a preferred frame

• Introduce the following variables
• Define 3D metric

gμ ν=ηA Be

A μe B ν

e A

μ→ΛA CeC μ

N , N i,eI

j

γi j=δM N eM

ieN j

Initial variables

• We have 9 variables, , 3 shifts: , lapse
• Define
• Build up ADM 4D vielbein
• No boost: 13 vars instead of 16 (general 4D vielbein)
• Metric in ADM form:

e

A μ=(

N ⃗

T

N I eI

j)

e

I j

N Ni N

i=γ i j N j, N I=e I j N j

gμ ν=ηA Be A

μeB ν

Fiducial variables

• Along the same lines fiducial variables
• 3D fiducial metric
• ADM 4D unboosted fiducial vielbein
• Unitary gauge: fixed-dynamics external fields

M , Mi E

I j

E

A μ=(

M ⃗

T

M I EI

j),

M

I=E I j~

γ

j k Mk

~ γi j=δM N E

M i E N j

Precursor Lagrangian

• EH term for physical metric
• Mass term:
• Total
• Same in form as dRGT but asymmetrical ADM vielbein

ℒ EH=√−g R(g), M P

2=2

ℒ TOT=ℒEH+∑

i=0 4

ci ℒi ℒ 0= m

2

24 ϵABCDϵ

αβ γδ E A α E B β E C γ E D δ

ℒ1=m

2

6 ϵABCDϵ

αβγ δ E A α E B βE C γe D δ

ℒ 2=m

2

4 ϵABCDϵ

αβ γδ E A α E B βe C γe D δ

ℒ3= m

2

6 ϵABCDϵ

αβ γδ E A αe B βe C γe D δ

ℒ 4= m

2

24 ϵABCDϵ

αβ γδe A αe B βe C γe D δ

• Consider 3D vielbein as fundamental variables
• Physical lapse and shift as Lagrange multipliers
• Canonical momentum of 3D vielbein
• Symmetry
• Hamiltonian linear in lapse and shift
• Mass term does not include shift variables

Precursor Hamiltonian

π

jk=Π j I δ I J eJ k, eJ k e I k=δJ I

P

[M N ]=e M jΠ j I δ I N−e N jΠ j I δ I M≈0

Precursor Hamiltonian/constraints

• Primary constraints:
• Time derivative of constraints
• Precursor Hamiltonian

H

(1)=∫d 3 x[−N R0−N i Ri+m 2 M H 1+αMN P [ MN ]]

~ C τ , τ=1,2 , Y

[MN ]=δ M L E L j e N j−δ N L E L je M j ,

E L

j E M j=δL M

H pre

(2)=∫d 3 x[−N R0−N i Ri+m 2 M H 1+αM N P [ M N ]+~

λ

τ ~

C τ+βM N Y

[M N ]]

Degrees of freedom for precursor theory

• Phase space variables: 2x9 = 18 psv:
• All constraints are independent and second-class
• 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

e

I j , Π j I

R0, Ri , P

[ MN ] , Y [ MN ], ~

C τ (τ=1,2)

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≈H1≡m

2∫d 3 x M H 1

Constraints

• Reconsider the time-evolution of the primary constraints
• 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

{R0, H pre}≈0, {Ri, H pre}≈0, C0∼ ˙ R0≈0, Ci∼ ˙ Ri≈0,

Hamiltonian of the theory

• Define then the 4 constraints (2 only are new)

Therefore new Hamiltonian

• Dof: (9x2 – 1 – 3 – 3 – 3 – 4)/2 = 2

C0≡{R0, H1}+ ∂ R0 ∂t , Ci≡{Ri, H1}

H =∫d

3 x[−N R0−N i Ri+m 2 M H 1+αMN P [MN ]+βMN Y [ MN ]+λC 0+λ iCi]

Building blocks

• We have

R0=R0

GR−m 2H 0,

R0

GR=√γ R[γ]− 1

√γ(γnl γmk−1

2 γnm γkl)π

nmπ kl,

Ri=Ri

GR=2γik D jπ kj,

H 0=√~ γ(c1+c2Y I

I)+√γ(c3 X I I+c4),

X I

J=eI l E J l, X I LY L J=δI J ,

H 1=√~ γ[c1Y I

I+ c2

2 (Y I

I Y J J−Y I J Y J I)]+c3 √γ

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
• Symmetry of also symmetric
• On the background
• The constraints are trivially satisfied on FLRW
• The constraint equivalent to Bianchi identity

M (t), E

I j=~

a(t) δ

I j

Y

I J=δ I M EM le J l → e J l

N (t), e

I j=a(t) δ I j

Ci≈0 C0≈0 (c3+2c2 X+c1 X

2) ( ˙

X+N H X−M H )=0, X=~ a /a, H=˙ a/(N a)

Two branches

• Two branches solutions exist
• Self accelerating branch

X is constant

• Normal branch
• For both branches λ = 0

X=X±=−c2±√c2

2−c1c3

c1 ˙ X+N H X−M H=0

Friedmann evolution

• Friedmann equation
• Same background evolution of dRGT
• Self accelerating branch: effective cosmological constant
• No extra constraint: C0 reduces to Bianchi identity

3 M P

2 H 2=m 2 M P 2

2 (c4+3c3 X+3 c2 X

2+c1 X 3)+ρ

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

S= M P

2

2 ∑

λ=+,x∫d 4 x N a 3[

˙ hλ

2

N

2−(∂ hλ) 2

a

2

−μ

2hλ 2],

μ2=1 2 m2 X[(c2 X+c3)+(c1 X+c2) M N ]

Stability of background

• Only two degrees of freedom
• The 2 dof are tensor
• Stable for µ2 > 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 )

the background is non-trivial but no scalar dof is present

• Constraint coming from modification of emission rate of

Gws from binaries [Finn, Sutton: PRD 2002]

M , ~ a μ<10

−5 Hz

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]