Cosmological perturbations in nonlinear massive gravity A. Emir G - PowerPoint PPT Presentation

Cosmological perturbations in nonlinear massive gravity A. Emir G¨ umr¨ ukc ¸ ¨ uo˘ glu IPMU, University of Tokyo AEG, C. Lin, S. Mukohyama, JCAP 11 (2011) 030 [arXiv:1109.3845] AEG, C. Lin, S. Mukohyama, To appear in JCAP [arXiv:1111.4107] Asia Pacific School/Workshop on Cosmology and Gravitation YITP , March 2, 2012 uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity

Why massive gravity? Is there a massive gravity theory which reduces smoothly 1 to GR in the massless limit? Are the predictions of GR stable against small graviton mass? Galactic curves, supernovae ⇒ new types of (dark) matter 2 and energy. Alternative approach: can these components be associated with the gravity sector, by large distance modifications of GR? Massive extension of GR? Linear mass terms ( Fierz, Pauli ’39 ) ⇒ Discontinuity with GR in the limit m g → 0 ( van Dam, Veltman ’70 ) Zakharov ’70 Nonlinear effects can recover continuity ( Vainshtein ’72 ) Nonlinear extensions have generically an additional ghost degree. ( Boulware, Deser ’72 ) ( See G. Gabadadze’s lectures on Sat and Sun ) uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity

Nonlinear massive gravity de Rham, Gabadadze, Tolley ’10 Gauge invariant, nonlinear mass term: � d 4 x √− g ( L 2 + α 3 L 3 + α 4 L 4 ) S m [ g µν , f µν ] = M 2 p m 2 g 1 L 3 = 1 [ K ] 2 − [ K 2 ] [ K ] 3 − 3 [ K ][ K 2 ] + 2 [ K 3 ] � � � � L 2 = , , 2 6 1 [ K ] 4 − 6 [ K ] 2 [ K 2 ] + 3 [ K 2 ] 2 + 8 [ K ][ K 3 ] − 6 [ K 4 ] � � L 4 = , 24 � µ �� f µν ≡ η ab ∂ µ φ a ∂ ν φ b g − 1 f K µ ν ≡ δ µ ν − ν , [ · · · ] ≡ Tr ( · · · ) , f µν : fiducial metric; φ a : St¨ uckelberg fields. < φ a > breaks the general coordinate invariance. Unitary gauge: φ a = δ a µ x µ , f µν = η µν . By construction, free of BD ghost in the decoupling limit. For generic f µν , free of BD ghost away from the decoupling limit. Hassan, Rosen ’11 uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity

Cosmological backgrounds AEG, Lin, Mukohyama ’11 (a) f µν with FRW symmetry ⇒ cosmological solutions f µν = − n 2 ( ϕ 0 ) ∂ µ ϕ 0 ∂ ν ϕ 0 + α 2 ( ϕ 0 )Ω ij ( ϕ k ) ∂ µ ϕ i ∂ ν ϕ j ❑ ❆ K δ il δ jm ϕ l ϕ m ❆ Ω ij ( { ϕ k } )= δ ij + ϕ a = δ a µ x µ in the unitary gauge. 1 − K δ lm ϕ l ϕ m g µν dx µ dx ν = − N ( t ) 2 dt 2 + a ( t ) 2 Ω ij dx i dx j Metric ansatz: Equations of motion for φ a ⇒ 3 branches of solutions ˙ Branch I : a / N = ˙ α/ n = ⇒ Trivial, evolution determined by f µν . Branches II ± : Two cosmological branches α ( t ) = X ± a ( t ) , � 1 + α 3 + α 2 1 + 2 α 3 + α 4 ± 3 − α 4 with X ± ≡ = constant α 3 + α 4 uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity

Background equations of motion Dynamics of Branch II ± , with generic (conserved) matter source: ˙ a H ≡ P a N P q − 2 ˙ 3 H 2 + 3 K a 2 = Λ ± + 1 N + 2 K H 1 ρ , a 2 = ( ρ + P ) , M 2 M 2 Pl Pl � � 3 / 2 � m 2 � � � g 2 + α 3 + 2 α 2 1 + α 3 + α 2 Λ ± ≡ − ( 1 + α 3 ) 3 − 3 α 4 ± 2 3 − α 4 ( α 3 + α 4 ) 2 Chunshan Lin’s For Minkowski f µν , only K < 0 solutions exists. ← − talk For dS fiducial, flat/open/closed FRW are allowed. uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity

Perturbing the solution AEG, Lin, Mukohyama ’11 (b) Lack of BD ghost does not guarantee stability. e.g. Higuchi’s ghost ( Higuchi ’87 ) Scalar sector may include additional couplings, giving rise to potential conflict with observations. Can we distinguish massive gravity from other models of dark energy/modified gravity? Introducing perturbations f µν does not depend on physical metric. FRW symmetry is preserved even when φ a are perturbed. Perturbations in the metric, St¨ uckelberg fields and matter fields: � � g 00 = − N 2 ( t ) [ 1 + 2 φ ] , g ij = a 2 ( t ) Ω ij ( x k ) + h ij g 0 i = N ( t ) a ( t ) β i , ϕ a = x a + π a + 1 2 π b ∂ b π a + O ( ǫ 3 ) , σ I = σ ( 0 ) + δσ I I Matter sector: a set of independent degrees of freedom { σ I } . uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity

Gauge invariant variables Scalar-vector-tensor decomposition: � D i ← Ω ij , △ ≡ Ω ij D i D j π i = D i π + π T β i = D i β + S i , i , D i S i = D i π T i = D i F i = 0 � � D i D j − 1 E + 1 h ij = 2 ψ Ω ij + 3 Ω ij △ 2 ( D i F j + D j F i ) + γ ij D i γ ij = γ i i = 0 Gauge invariant variables without St¨ uckelberg fields:   δσ I − L Z σ ( 0 ) Z 0 ≡ − a a 2 2 N 2 ˙ , Q I ≡ N β + E I Originate from g µν   φ − 1 N ∂ t ( NZ 0 ) , Z i ≡ 1 and matter fields δσ I Φ ≡ 2 Ω ij ( D j E + F j ) ❍ ❥ ❍     a Z 0 − 1 ψ − ˙ a Ψ 6 △ E ,  Under x µ → x µ + ξ µ :  ≡   Z µ → Z µ + ξ µ 2 N ˙ a B i S i − F i , ≡ Associated with However, we have 4 more degrees of freedom: St¨ uckelberg fields ✟ ✟ ✙ 3 △ π − ˙ ψ π ≡ ψ − 1 a a π 0 , E π ≡ E − 2 π , F π ≡ F i − 2 π T i i uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity

Quadratic action After using background constraint for St¨ uckelberg fields: S ( 2 ) = S ( 2 ) EH + S ( 2 ) matter + S ( 2 ) S ( 2 ) ˜ + Λ ± mass ���� � �� � S ( 2 ) ˜ mass = S ( 2 ) mass − S ( 2 ) depend only on Q I , Φ , Ψ , B i , γ ij Λ ± The first part is equivalent to GR + Λ ± + Matter fields σ I . The additional term: d 4 x N a 3 √ � � 3 ( ψ π ) 2 − 1 12 E π △ ( △ + 3 K ) E π + 1 i − 1 � S ( 2 ) ˜ mass = M 2 Ω M 2 16 F i π ( △ + 2 K ) F π 8 γ ij γ ij p GW The only common variable is γ ij . E π , ψ π , F π have no kinetic term! We treat them as nondynamical. i Scalar and vector sector ⇒ same dynamics as GR, with additional cosmological constant Λ ± and same matter content. The only modification at linear order is in the tensor sector: � mass = − M 2 d 4 x N a 3 √ ˜ p S ( 2 ) Ω M 2 GW γ ij γ ij 8 uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity

Tensor modes Assuming no tensor contribution from matter sector, � 1 � � tensor = M 2 d 4 x N a 3 √ γ ij + 1 γ ij ˙ S ( 2 ) a 2 γ ij ( △ − 2 K ) γ ij − M 2 GW γ ij γ ij Pl Ω N 2 ˙ , 8 The mass function M 2 GW is time dependent: � M 2 GW ≡ ± ( r − 1 ) m 2 g X 2 1 + α 3 + α 2 3 − α 4 ± � � ˙ na 1 H a α ˙ Time dependence provided by r ≡ N α = H f , H ≡ Na , H f ≡ X ± n α Stability is determined by the sign of ( r − 1 ) m 2 g . Fiducial metric f µν → Evolution of r . eg.1: Minkowski fiducial ⇒ r ∝ ˙ a eg.2: dS fiducial ⇒ r ∝ ˙ a / a uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity

Possible signals? For M 2 GW > 0, the spectrum of stochastic GW will undergo a suppression (w.r.t GR) when ( k / a ) 2 � M 2 GW . For M GW ∼ O ( H 0 ) , the suppression may be observed. P Log 10 Example for M 2 P GR k GW = constant . 3 Log 10 Assumed initial scale invariance. � 4 � 3 � 2 � 1 1 2 k eq Small scales: Same as GR signal. � 2 Larges scales: Suppression. � 4 Frequency dominated by M 2 GW at � 6 large scales. Cutoff: k / k eq = ( M GW / 2 H eq ) 1 / 3 � 8 (Here: ∼ . 02) � 10 Work in progress , with S. Kuroyanagi, C. Lin, S. Mukohyama, N. Tanahashi uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity

Summary/Discussion Gauge invariant study of perturbations of self-accelerating cosmological solutions in potentially ghost-free nonlinear massive gravity. Dynamics of scalar and vector modes are same as in GR, at the level of quadratic action. ⇒ No stability issues in scalar/vector sectors. Tensor sector acquires a time dependent mass. ⇒ Modification of stochastic GW spectrum, CMB B–mode polarization at large scales. Expected 5 degrees for massive spin 2 ⇒ Only 2 degrees (2 GW polarizations). Cancellation of kinetic terms at quadratic level. Possible connection with the cosmological branch of solutions? Strong coupling vs Nondynamical? ⇒ Need to go beyond perturbation theory. Radiative stability? ⇒ First step: strong coupling scale in the cosmological branch? uo˘ A. Emir G¨ umr¨ ukc ¸ ¨ glu APS2012 Cosmological perturbations in nonlinear massive gravity