Higuchi VS Vainshtein

Transcription

1 Higuchi VS Vainshtein ArXiv: soon to appear. Matteo Fasiello and Andrew J. Tolley Case Western Reserve University

2 Outline The Higuchi bound The Vainshtein mechanism Higuchi vs Vainshtein drgt setup FRW on FRW Despair not: a quicker route to the resolution

3 The Higuchi bound is a condition that stems from requiring stability from the classical theory of linear Massive Gravity L = L EH + L m = X p T q h 1 2 pt P p qt Q q + p T i PQ q (1) Roughly speaking: stability <==> Q, P positive definite (Higuchi + gradient instability) (2) Essential literature: A. Higuchi Nucl.Phys. B282 (1987) 397 S. Deser, A. Waldron Phys.Lett. B508 (2001) hep-th/ L.Grisa, L.Sorbo Phys.Lett. B686 (2010) arxiv:

4 Let s take a look example: Fierz-Pauli S = S EH m 2 4 Z d 4 x p ḡ (4) h µ h h f µ f f µ f i where: f µ =ḡ µ EH usual tensor decomposition T ij = T Tt ij +2@ (i T t j) ij ˆ@ ij T t + ˆ@ ij T l We are looking at the scalar here, the helicity 0 mode use ADM formalism solve constraint equations, solve for p t,h t canonical transformation: p l! p 0 + h t m 2 2H 2 /4H ; h l! q 0 + h t /2

5 I 0 = p 0 q 0 h 1 2 h 3 2 m 2 12H 2 ip h 12H 2 i 2 m 2 q 0 r 2 + m 2 9H 2 4 q 0 i 2 = m 2 2H 2 Immediately then, stability dictates: 2 > 0 in this setup, the Higuchi bound reads: m 2 > 2H 2

6 a quick, heuristic derivation: Vainshtein radius R µ + m 2 h µ T µ * underlying assumption: f µ 6=ḡ µ h µ 1 R m 2 therefore r<r V R r 2 ; GM r r V = M M 2 P m2 ) R GM r 3 m 2 1/3 r>r V C. Deffayet, G. Dvali, G. Gabadadze, A. Vainshtein hep-th/ Phys.Rev. D65 (2002) G. Chkareuli, D. Pirtskhalava airxiv

7 Inside the Vainshtein radius lies the region where one recovers GR schematically: 3H 2 = +3m 2 (1) one must require m 2 <H 2 Combining Higuchi and Vainshtein then: want our theory to be stable m 2 > 2H 2 GR works all around us m 2 <H 2

8 Clearly, there s a problem... In deriving the Higuchi bound, a number of assumptions were made: Shall we add matter content? Of course: Z S = S EH + S m 2 d 4 x p h 1 i g (4) 2 +V ( ) [Grisa and Sorbo, 2010] m 2 > 2(H 2 + Ḣ) but FP gravity, i.e. ghosts! Shall we use a different reference metric f? Yes, no reason not to. f µ 6=ḡ µ Plan: Study a ghost-free theory of massive gravity with matter content

9 drgt: Ghost-free m.g. theory at fully non-linear level * No Boulware-Deser Ghost, at all orders De Rham, Gabadadze, Tolley Hassan, Rosen * Screening mechanism in the non linear regime that restores continuity with G.R. as m approaches 0 * High enough cutoff so that the theory different regimes can be described S = S EH +2m 2 Z d 4 x p h g " 2 ( p g 1 f)+ 3 " 3 ( p g 1 f)+ 4 " 4 ( p g 1 f)i

10 Our set up * drgt theory of massive gravity with S m 2 =2m 2 Z d 4 x p h g " 2 ( p g 1 f)+ 3 " 3 (..)+ 4 " 4 (..)i " 2 (X) = 1 2 Tr2 [X] Tr[X 2 ] ; " 3 (X) = 1 6 Tr3 [X] 3Tr[X 2 ]Tr[X]+2Tr[X 3 ] " 4 (X) = 1 24 Tr4 [X] 6Tr[X 2 ]Tr 2 [X]+3Tr 2 [X 2 ]+8Tr[X 3 ]Tr[X] 6Tr[X 4 ] ** The reference metric f and g0 need not be the same, parametrize this as: f µ =(1+z)ḡ µ * * in ds

11 Higuchi bound: m 2 (1 z 2z 2 )(m 2 (1 z 2z 2 ) 2H 2 ) > 0 in other words, the Higuchi bound has the generic form m 2 ( m 2 2H 2 ) > 0 m is the dressed mass, we ask m 2 > 0 to avoid instabilities in the vector sector. Two branches of solutions: 0 <H< 3 2 H 0 ; H> 3 2 H 0 ; * 1+z = H/H0 m 2 > 2HH2 0 3H 0 2H. m 2 < 2HH 2 0 2H 3H 0. includes the H0 =H branch new branch apparently, for H>>H0, m 2 /H 2 0 > 1 this is a much weaker Higuchi bound, but Vainshtein will require the opposite inequality to hold, a.k.a. : back to square 1.

12 Add matter: L = L EH + L drgt + Z d 3 x p (4) g V ( ) Background: H 2 = m 2 (z Ḣ = 2 4 z 2 ) V 6 ; m z 2z 2 M +2Mz ; +3H + V 1 =0; V 1 = dv ( ) d ; ż = H (M 1 z); 1+z = b a = H H 0 0; H b ḃ Mb. * fµ = diag M 2 (t), (1 + z(t)) 2 a(t) 2 ; 3 =0= 4.

13 Ḣ drops out of the Higuchi inequality! The bound is independent from the equation of state for matter: m 2 (H) =m 2 H H 0 ( ) 2( ) H H 0 +( ) H2 H 2 0 2H 2. interesting feature, but the problem remains. Could the freedom on the alpha s pay off? It doesn t. Time evolution does not help either. poly (k) 1 (z) poly (k) 2 (z) 1 3 = 1= 4 structure also makes it hard. In this setup there is no regime which is simultaneously observationally acceptable and ghost-free.

14 A quicker method and a resolution of the H-V tension Use the properties of the mini superspace action: ds 2 = 2 dt 2 + b( 2 )d~x 2 B n m 2 M 2 Pl (1, 3, 4 ) S = Z dtna 3h 3M 2 Pl ȧ2 N 2 a 2 k a 2 N 3X n=0 A n b( ) a n 3X n=0 B n b( ) a n (a)i A n (3 n) =B n+1 (n + 1) field redefinition: =1/H 0 ln(a)+ /H 0 ; M = Na 3 ; = a 3 /3

15 fluctuations + diagonalize S (2) = Z dt h M 6M 2 Pl M 2 i 3M 2 Pl M 2! gravity sector has decoupled from helicity zero mode! we read off the Higuchi bound Proceeding analogously for bigravity, when f too is dynamical:! m 2 dressed(h) H 2 + H2 0 M 2 P M 2 f 2H 4 new!

16 What to do now? Fully bi-metric theories Inhomogeneities in the s? work in progress... Reasons to be hopeful: see Gabadadze et al., massive cosmologies.