Concistency of Massive Gravity LAVINIA HEISENBERG

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1 Universite de Gene ve, Gene ve Case Western Reserve University, Cleveland September 28th, University of Chicago in collaboration with C.de Rham, G.Gabadadze, D.Pirtskhalava

2 What is Dark Energy? 3 options? Cosmological Constant cosmological constant problem? Dark Energy (Why don t we see them? Similar fine-tuning problem?) Dark Gravity (Is there any viable model?) massive gravity? fine-tuning problem?

3 Quantum corrections t Hooft s naturalness argument any physical parameter c i at any energy scale E can remain small if the limit c i 0 increases the symmetry of the system Example: electron mass m e electroweak scale, BUT the electron mass is technically natural quantum corrections only give rise to δm e m e m e 0 implies an additional chiral symmetry representing the conservation of left- and right-handed leptons So in the massless limit, the electron mass receives no quantum corrections CC: there is no symmetry recovered in the limit Λ 0 and any massive particle of mass M contributes to the vacuum energy proportional to M 4.

4 Tuning versus Fine-tuning in massive gravity Tuning the mass needs to be tuned m H 0, same tuning as Cosmological Constant Λ M 4 p H2 0 M 2 p m2 M 2 p Fine-tuning t Hooft s naturalness argument applies in the massless limit, the graviton mass receives no quantum corrections since we recover a symmetry in the m 0 limit in the full theory the quantum correction give rise to counterterms which are proportional to the mass itself δm m 2

5 Decoupling limit (DL) of drgt theory DL (M p, m 0 with Λ 3 3 = m2 M p const ) gives the following scalar-tensor interactions L = 1 2 hµν Eµν αβ h αβ + h µν 3 a n X µν (n) [Φ] n=1 Λ 3(n 1) 3 where h µν =helicity-2, φ =helicity-0 field and a 1 = 1 2 and a 2,3 are two arbitrary constants and X (1,2,3) µν denote X (1) µν [Φ] = ε µ αρσ ε ν β ρσ Φ αβ, X (2) µν [Φ] = ε µ αργ ε ν βσ γ Φ αβ Φ ρσ, X (3) µν [Φ] = ε µ αργ ε ν βσδ Φ αβ Φ ρσ Φ γδ with Φ αβ = α β φ. The structure of the interactions are very similar to the Galileon interactions L Gal = c n φε µν ε αβ Φ µα Φ νβ Φ Φ.

6 Similarities and differences to Galileons Common IR modification of gravity as due to a light scalar field with non-linear derivative interactions respects the symmetry φ(x) φ(x) + c + b µ x µ Second order equations of motion, containing at most two time derivatives non-renormalization theorem applies Different undiagonazable interaction + a3 h µν X Λ 6 µν (3) 3 important for the self-accelerating solution extra coupling µ φ ν φt µν important for the degravitating solution only 2 free-parameters

7 Non-renormalization theorem in Galileons Galileon interactions L Gal = c n φε µν ε αβ Φ µα Φ νβ Φ Φ counter terms arising in the 1-loop effective action: come with at least one extra derivative as compared to the original interactions don t take Galileon form at all Galileon coupling constants are technically natural tunes to any value and remain radiatively stable

8 Non-renormalization theorem in Galileons Any external particle comes with at least two derivatives applied on it in the 1-PI action. Example: consider the following vertex: V = φε µν ρσε αβρσ Φ µα Φ νβ

9 Non-renormalization theorem in Galileons this vertex give a contribution to the transition amplitude d 4 k im i (2π) 4 G kg k+p ε αργδ ε βσ γδ k αk β (p + k) ρ (p + k) σ with the Feynman propagator G k = i k 2 iɛ. terms linear in momentum p and independent of it, cancel due to antisymmetric structure of the vertex only term which can be contracted with the antisymmetric tensor is p ρ p σ im iε αργδ ε βσ d 4 γδ p k ρp σ (2π) 4 G kg k+p k α k β any loop that involves a vertex V = φε µν ρσε αβρσ Φ µα Φ νβ will lead to at least two derivatives on the external leg

10 Non-renormalization theorem in the DL The same non-renormalization theorem applies in the decoupling limit of massive gravity: The only difference is that we now have the helicity-2 field appearing in the interactions. V = h µν X (2) µν [Φ] h µν ε αργ µ εν βσ γφ αβ Φ ρσ

11 Quantum corrections beyond decoupling limit interactions beyond the decoupling limit are symbolically of the form L = ( ) f n,l h n h( 2 π) l, M p n 1, l 0 Λ 3(l 1) 3 they are all suppressed by an integer power of M 1 p compared to the vertices arising in the decoupling limit non-renormalization theorem no longer applies since there is no quantum correction in the decoupling limit, graviton mass itself cannot receive quantum corrections which are larger than ( ) n Λ3 m 2 with n 1. δm 2 = Λ 2 3 M p

12 Cosmology in the decoupling limit The Lagrangian in the decoupling limit L = hµν Eµν αβ h αβ + h µν a n n=1 Λ 3(n 1) 3 X (n) µν [Φ] + 1 M p h µν T µν Self-accelerating solution T µν = 0 H 0 Degravitating solution T µν 0 H = 0 Gravitons form a condensate whose energy density sources self-acceleration Gravitons form a condensate whose energy density compensates the cosmological constant

13 Introduction Quantum corrections DL of drgt non-renormalization Cosmology Self-accelerating solution a2 + H 2 = m2 2a2 q 2 + 2a3 q 3 q and q = 3a 3 (2a22 +3a3 )1/2 3 2a3 stability stable self-accelerating solution: 2a2 a2 a2 < 0 and 3 2 < a3 < 2 2 (3) hµν Xµν plays a crucial role for the stability (a3 = 0 ghost) kinetic term of the perturbation of the helicity-0 mode survives no strong coupling issues no quadratic mixing term between perturbations of the helicity-2 and helicity-0 cosm. evolution very similar to ΛCDM

14 Degravitating solution degravitating solution: high pass filter modifying the effect of long wavelength sources such as a CC vacuum energy gravitates very weakly H = 0 g µν = η µν a 1 q + a 2 q 2 + a 3 q 3 = λ Λ 3 3 Mp as long as the parameter a 3 is present, this equation has always at least one real root this static solution is stable for any region of the parameter space for which 2(a 1 + 2a 2 q + 3a 3 q 2 ) 0 and real

15 Summary technical naturalness of massive gravity there are no quantum corrections in the m 0 limit beyond this limit the quantum corrections are proportional to the mass itself δm 2 = m 2 cosmology in massive gravity stable selfaccelerating solutions in the decoupling limit degravitating solutions