Cosmological perturbations in nonlinear massive gravity A. Emir Gümrükçüoğlu 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
Why massive gravity? 1 Is there a massive gravity theory which reduces smoothly to GR in the massless limit? Are the predictions of GR stable against small graviton mass? 2 Galactic curves, supernovae new types of (dark) matter 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) van Dam, Veltman 70 Discontinuity with GR in the limit m g 0 ( 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)
Nonlinear massive gravity de Rham, Gabadadze, Tolley 10 Gauge invariant, nonlinear mass term: S m [g µν, f µν ] = Mp 2 mg 2 d 4 x g (L 2 + α 3 L 3 + α 4 L 4 ) L 2 = 1 ( [K] 2 [K 2 ] ), L 3 = 1 ( [K] 3 3[K][K 2 ] + 2[K 3 ] ), 2 6 L 4 = 1 ( [K] 4 6[K] 2 [K 2 ] + 3[K 2 ] 2 + 8[K][K 3 ] 6[K 4 ] ), 24 K µ ν δ µ ν ( g 1 f ) µ ν, [ ] Tr( ), f µν η ab µ φ a ν φ b f µν : fiducial metric; φ a : Stückelberg fields. < φ a > breaks the general coordinate invariance. Unitary gauge: φ a = δ a µx 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
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 ϕ a = δ a µx µ in the unitary gauge. Metric ansatz: g µν dx µ dx ν = N(t) 2 dt 2 + a(t) 2 Ω ij dx i dx j Equations of motion for φ a 3 branches of solutions Branch I : ȧ/n = α/n = Trivial, evolution determined by f µν. Branches II ± : Two cosmological branches α(t) = X ± a(t), with X ± 1 + 2α 3 + α 4 ± 1 + α 3 + α3 2 α 4 = constant α 3 + α 4 Ωij ({ϕ k })=δ ij + K δ il δ jm ϕl ϕ m 1 K δ lm ϕ l ϕ m
Background equations of motion H ȧ a N Dynamics of Branch II ±, with generic (conserved) matter source: 3 H 2 + 3 K a 2 = Λ ±+ 1 m 2 g Λ ± (α 3 + α 4 ) 2 MPl 2 [ (1 + α 3 ) ρ, 2Ḣ N + 2 K a 2 = 1 (2 + α 3 + 2 α 2 3 3 α 4 ) ± 2 (ρ+p), MPl 2 ) ] 3/2 (1 + α 3 + α3 2 α 4 For Minkowski f µν, only K < 0 solutions exists. For ds fiducial, flat/open/closed FRW are allowed. Chunshan Lin s talk
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, Stückelberg fields and matter fields: ] g 00 = N 2 (t) [1 + 2φ], g 0i = N(t)a(t)β i, g ij = a 2 (t) [Ω ij (x k ) + h ij ϕ a = x a + π a + 1 2 πb b π a + O(ɛ 3 ), σ I = σ (0) I + δσ I Matter sector: a set of independent degrees of freedom {σ I }.
Gauge invariant variables Scalar-vector-tensor decomposition: β i = D i β + S i, π i = D i π + π T i, h ij = 2ψΩ ij + ( D i D j 1 3 Ω ij ) E + 1 2 (D if j + D j F i ) + γ ij } Di Ω ij, Ω ij D i D j D i S i = D i π T i = D i F i = 0 D i γ ij = γ i i = 0 Gauge invariant variables without Stückelberg fields: Q I δσ I L Z σ (0) I, Φ φ 1 t(nz 0 ), N Originate from g µν and matter fields δσ I Ψ ψ ȧ 1 E, a 6 B i S i a 2N Ḟi, However, we have 4 more degrees of freedom: Z 0 a β + a2 Ė N 2N 2 Z i 1 2 Ωij (D j E + F j ) Under x µ x µ + ξ µ : Z µ Z µ + ξ µ ψ π ψ 1 3 π ȧ a π0, E π E 2 π, Fi π F i 2 πi T Associated with Stückelberg fields
Quadratic action S (2) mass = M 2 p After using background constraint for Stückelberg fields: S (2) = S (2) EH + S(2) matter + S (2) Λ ± }{{} depend only on Q I, Φ, Ψ, B i, γ ij + (2) S mass }{{} S (2) mass=s (2) mass S (2) Λ ± The first part is equivalent to GR + Λ ±+ Matter fields σ I. The additional term: d 4 x N a 3 [ Ω MGW 2 The only common variable is γ ij. E π, ψ π, F π i 3(ψ π ) 2 1 12 E π ( + 3K )E π + 1 16 F i π( + 2K )F π i 1 8 γij γ ij ] have no kinetic term! We treat them as nondynamical. 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: S mass (2) = M2 p d 4 x N a 3 Ω MGW 2 γ ij γ ij 8
Tensor modes Assuming no tensor contribution from matter sector, S (2) tensor = M2 Pl 8 d 4 x N a 3 Ω [ 1 N 2 γij γ ij + 1 ] a 2 γij ( 2K )γ ij MGW 2 γ ij γ ij, The mass function MGW 2 is time dependent: MGW 2 ±(r 1)mg 2 X± 2 1 + α 3 + α3 2 α 4 ( ) Time dependence provided by r na = 1 H Nα X ±, H ȧ, H H f Na f α nα Stability is determined by the sign of (r 1)m 2 g. Fiducial metric f µν Evolution of r. eg.1: Minkowski fiducial r ȧ eg.2: ds fiducial r ȧ/a
Possible signals? For MGW 2 > 0, the spectrum of stochastic GW will undergo a suppression (w.r.t GR) when (k/a) 2 MGW 2. For M GW O(H 0 ), the suppression may be observed. 4 3 2 1 1 2 3 2 4 6 8 Example for M 2 GW = constant. Assumed initial scale invariance. Small scales: Same as GR signal. Larges scales: Suppression. Frequency dominated by M 2 GW at large scales. Cutoff: k/k eq = (M GW /2H eq) 1/3 (Here:.02) Work in progress, with S. Kuroyanagi, C. Lin, S. Mukohyama, N. Tanahashi
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?