Aspects of massive gravity Sébastien Renaux-Petel CNRS - IAP Paris Rencontres de Moriond, 22.03.2015
Motivations for modifying General Relativity in the infrared Present acceleration of the Universe: {z } Dark Energy or Infra-Red modification of GR Way out of the (old) cosmological constant problem? Better understand GR
New degrees of freedom General Relativity: the only theory of an interactive massless helicity-2 field Weinberg, QFT Modifying GR means adding new degrees of freedom (dof) Examples : - scalar tensor: GR + explicit scalar field - f(r): GR + a scalar field in disguise. Identifying the new dof may be non-trivial. Today: example of massive gravity
Massive gravity One interesting IR modification: giving a mass to the graviton (of order the present Hubble scale) Motivation: weakens gravity on large scales and possibly degravitates the cosmological constant Massive gravity: a theory of an interactive massive spin-2 field. What is it? V e mr Several interesting pathologies to cure/phenomena with possible applications in other areas: strong classical nonlinearities, ghost instability, low cut-off EFT, IR/UV interplay, screening of long-range scalars... Reviews: Hinterbichler 1105.3735 de Rham 1401.4173 r obs bare
Complex history 1939 Fierz-Pauli 1970 van Dam, Veltman, Zakharov 1972 Vainshtein 1972 Boulware, Deser Linear ghost-free MG Discontinuity......resolved by non-linearities? Ghost at non-linear level 2003 Arkani-Hamed, Georgi, Schwartz Stückelberg method 2009 Babichev, Deffayet, Ziour Vainshtein mechanism works! 2010 de Rham, Gabadadze, Tolley Non-linear ghost-free MG
Ghost-free massive and bimetric gravity L MG = M 2 Pl 2 g R + m 2 K 2 K 2 µ + 3 (K 3 +...)+ 4 (K 4 +...) K µ = µ gµ de Rham, Gabadadze, 10 de Rham, Gabadadze, Tolley, 10 In 4d (d dim), there is a 2-parameter family (d-2) of ghost-free theories of Lorentz-invariant massive gravity Absence of Boulware-Deser ghost has been proved exactly in many different languages de Rham, Gabadadze, Tolley, 10, 11, Hassan, Rosen, 11 Mirbabay, 11, Hassan, Schmidt-May, von Strauss, 12 Deffayet, Mourad, Zahariade, 12 And around any reference metric, even dynamical: bigravity µ! f µ Hassan, Rosen, Schmidt-May, 11 Hassan, Rosen, 11 Reformulation in terms of vielbeins and extension to multi-metric Hinterbichler, Rosen, 12, Noller et al 14 See talk by Blanchet
Degrees of freedom in massive gravity Mass term breaks diffeomorphism invariance of GR 2! 5 degrees of freedom General Relativity 1 massless spin-2: Massive Gravity 1 massive spin-2: - 2 helicity-2-2 helicity-2-2 helicity-1-1 helicity-0 The details are really only well known for a Minkowski reference metric, and around Minkowski space
Stückelberg trick Central idea: gauge-invariance is a redundancy of description. So any theory can be made gauge-invariant, with any gauge-invariance we like. We restore gauge invariance with the replacement: gauge-invariant under Vector example: L = 1 4 F µ F µ 1 2 m2 A µ A µ + A µ J µ L = 1 4 F µ F µ 1 2 m2 (A µ + @ µ ) 2 + A µ J µ @ µ J µ A µ = @ µ, = breaks the U(1) gauge invariance A µ! A µ + @ µ The original theory is simply a gauge-fixed version of the gaugeinvariant theory. Same physical content and same predictions. Stückelberg, 1938
Stückelberg for massive gravity Gauge-invariance of GR: full diffeomorphism invariance: f µ (x)! @g @x µ @g @x f (g(x)) We restore gauge-invariance in MG with the introduction of 4 scalar fields and the replacement: Arkani-Hamed et al, 03 Reference metric f µ (x)! f µ (x) =f (Y (x)) @ µ Y @ Y {z } f µ g µ fµ transforms like a metric tensor under diff transforms like a scalar under diff Original Lagrangian recovered in the unitary gauge Y = x
Stückelberg for massive gravity Expansion Y = x A with reference metric f µ = µ Metric fluctuation: h µ g µ µ! H µ = h µ + @ µ A + @ A µ @ µ A @ A Additional replacement: A µ! A µ + @ µ H µ = h µ + @ µ A + @ A µ +2@ µ @ @ µ A @ A @ µ A @ @ @ µ @ @ A @ µ @ @ @
Stückelberg for massive gravity Decoupling limit: concentrates on the new interactions beyond GR at the lowest energy scale In the decoupling limit, the massive gravity Lagrangian is invariant under h µ = @ µ + @ µ A µ = @ µ = 0 GR-like Maxwell-like In the decoupling limit, h µ,a µ, encode the helicity-2, helicity-1 and scalar dofs of the theory
Massive gravity on de Sitter Reference metric: Minkowski de Sitter. Still a maximally symmetric spacetime (same amount of symmetry). Study the theory around de Sitter spacetime How the ds reference metric affects the helicity-0 mode is well-known at linear order: m 4 ( ) 2 m 2 m 2 (d 2)H 2 ( ) 2 Higuchi, 87, Deser, Waldron, 01 Grisa, Sorbo 09 Higuchi bound: m 2 > (d 2)H 2 The helicity 0-mode disappears completely, at the linear level, for m 2 =(d 2)H 2 (partially massless)
Strategy de Rham, RP, 1206.3482 Embed d-ds into (d+1)-minkowski... ds Y ds 2 d+1 = e 2HY dy 2 + (ds) µ dx µ dx = AB dz A dz B Y =0... and copy the procedure on Minkowski Minkowski Introduction of Stückelberg fields: µ dx µ dx = ( AB dz A dz B ) projected = MN A M B N dz A dz B projected = MN µ M N dx µ dx M = Z M MN N behaves as a scalar field in the decoupling limit and captures the physics of the helicity-0 mode
Decoupling limit Covariantized metric fluctuation in terms of the helicity-0 mode H µ = h µ +2 µ 2 µ with µ = r µ r + H 2 ( ) 2 ( µ µ ) ( µ µ ) + O(H 4 ) Decoupling limit: M Pl!1 m! 0, (m 2 M H! 0 H m fixed (d 2)/2 Pl ) 2/(d+2) fixed Suppress nonlinearities of GR Keep finite the lowest energy interactions of the helicity-0 mode Satisfy the Higuchi bound Study the Partially Massles case
Decoupling limit L (dec) = X n c n ( n,d,h 2 /m 2 )L (n) Gal (@ ) 2 (@ 2 ) n 2 (d+2)/2 n 2 non-diagonalizable + terms mixing h and The kinetic term vanishes for m 2 =(d 2)H 2 (known) Remarkably, all the other interactions vanish simultaneously for 3 = 1 3 d 1 d 2 n = 1 n n 1 for n 4 Helicity-0 mode disappears completely in the DL! Unique candidate theory for partially massless gravity. Prompted lots of studies: the helicity-0 mode reappears at higher energy de Rham et al 13
Another limitation of the decoupling limit, in screening mechanisms Challenge: Modifications of GR on cosmological scales but recovery of GR in the Solar System Modified Gravity Screening mechanisms are necessary, which hide the additional dof. Chameleon, Symmetron, Vainshtein screening etc See talks by Babichev, Minazzoli and Mota General Relativity
Vainshtein mechanism and decoupling limit Most studies of the Vainshtein mechanism consider static and spherically symmetric configurations, and rely on the decoupling limit Particular model: minimal model, with 3 = 1 3, 4 = 1 12 has no interactions in the decoupling limit. No non-linearities, so no Vainshtein mechanism? e.g. Koyama, Niz & Tasinato Not necessarily. Simply decoupling limit is not enough here. Need to study the structure of interactions at higher energies
Vainshtein mechanism and decoupling limit Static and spherically symmetric M Pl Generic M Pl m 2n n+1 RP, 1401.0947 Tower of interactions, of energy n+1 3n+1! M Pl m 2 1/3 n!1 Hard to decipher whether the Vainshtein mechanism is effective = M Pl m 2 1/3 Energy scales of interactions in the minimal model of massive gravity Study of structure of solutions: no recovery of GR Decoupling limit is not enough + Static spherically symmetric configurations can be misleading due to their high degree of symmetry
Conclusion Identifying the new degrees of freedom in modified gravity can be non-trivial Identification of degrees of freedom is not known in massive gravity with a general reference metric and background metric See talk by Bernard Some insights for maximally symmetric reference metrics: (A)dS de Rham, RP Importance of studying screening mechanisms beyond static spherically symmetric configurations (small breaking in the Solar System, and walls an filaments in Large Scale Structure) Time-dependence: Babichev et al 11 Shape-dependence: Bloomfield et al 14, Bloomfield, Burrage & RP 15, to appear