Claudia de Rham Dec. 18 th 2013

Why Modify Gravity in the IR ? Late time acceleration & CC problem First signs of the breakdown of GR on cosmological scales ?

MG from Extra Dimensions Best way to obtain a sensible theory of Modified Gravity is to start with General Relativity 5d Gravity Soft massive Gravity (DGP) Hard massive Gravity Ghost-free mGR Galileons

Infinite extra dimension Confine Gravity on a brane DGP model 5d Gravity 4d gravity Dvali Gabadadze Porrati, 2000

Infinite extra dimension Confine Gravity on a brane DGP model 5d Gravity Finite-size extra dimension KK or deconstruction Hard massive Gravity (or multi-gravity) 4d gravity 5d Gravity

MG from Extra Dimensions In both cases In both cases (DGP and hard mass gravity), 5 dof from massless spin-2 in 5d 5 dof for massive spin-2 in 4d (no BD ghost) ensures

MG from Extra Dimensions In both cases In both cases (DGP and hard mass gravity), 5 dof from massless spin-2 in 5d 5 dof for massive spin-2 in 4d (no BD ghost) In both cases In both cases, the scalar dof behaves as a Galileon scalar field (in some limit) Screened via Vainshtein mechanism Screened via Vainshtein mechanism ensures

Deconstructing Gravity Start with 5d gravity Start with 5d gravity Discretize it along the extra dimension (after gauge fixing) y Deconstruction: Arkani-Hamed, Cohen & Georgi, PRL 86, 4757 (2001) 5d Gravity

Deconstructing Gravity Start with 5d gravity Start with 5d gravity Discretize it along the extra dimension (after gauge fixing) y 5d Gravity

Deconstructing Gravity Start with 5d gravity Start with 5d gravity Discretize it along the extra dimension (after gauge fixing) y 5d Gravity

Ghost-free Massive Gravity 4d Ghost-free Deconstructing 5d GR naturally leads to a 4d Ghost-free multi-gravity theory

Ghost-free Massive Gravity 4d Ghost-free Deconstructing 5d GR naturally leads to a 4d Ghost-free multi-gravity theory CdR, Matas & Tolley,

Ghost-free Massive Gravity Deconstructing 5d GR in a gauge-fixed way (5d lapse=1) leads to Ghost-free massive gravity or multi-gravity Truncated theory is consistent Truncated theory is consistent Deconstruction is EQUIVALENT to KK (after non-linear field redefinition) Deconstruction is EQUIVALENT to KK (after non-linear field redefinition) CdR, Matas & Tolley, mass

Ghost-free Massive Gravity Structure of mass term is essential to avoid BD ghost Boulware & Deser, PRD 6, 3368 (1972) CdR & Gabadadze, PRD 82, (2010) CdR, Gabadadze & Tolley, PRL 106, (2011)

Ghost-free Massive Gravity Structure of mass term is essential to avoid BD ghost Helicity-0 mode appears as a Galileon (in some limit) Boulware & Deser, PRD 6, 3368 (1972) CdR & Gabadadze, PRD 82, (2010) CdR, Gabadadze & Tolley, PRL 106, (2011) After Stückelbergisation

Ghost-free Massive Gravity Structure of mass term is essential to avoid BD ghost Helicity-0 mode appears as a Galileon (in some limit) Boulware & Deser, PRD 6, 3368 (1972) CdR & Gabadadze, PRD 82, (2010) CdR, Gabadadze & Tolley, PRL 106, (2011) After Stückelbergisation

Open Questions in MG Viable Cosmological Solutions & Phenomenology Quantum Corrections - is the small mass technically natural ? - how does the special structure of the potential gets affected by QC ? Superluminalities Strong Coupling

Tuning of the Mass We typically consider the graviton mass to be same tuning as for the CC problem !

Tuning of the Mass We typically consider the graviton mass to be same tuning as for the CC problem ! However the graviton mass is stable against quantum corrections Technically natural tuning, unlike the tuning of the CC to zero

Non-renormalization In the DL, we recover a Galileon theory which enjoys a non-renormalization theorem graviton mass does not get renormalized in the DL Beyond the DL, the graviton mass gets renormalized by Technically natural tuning, unlike the tuning of the CC to zero CdR, Gabadadze, Heisenberg & Pirtskhalava,

Open Questions in MG Viable Cosmological Solutions & Phenomenology Quantum Corrections - is the small mass technically natural ? - how does the special structure of the potential gets affected by QC ? Superluminalities Strong Coupling See CdR, Heisenberg & Ribeiro,

Galileons in trouble (?) Perturbations about a background

Galileons in trouble (?) Change the kinetic structure !!! Fluctuations see a different effective metric Fluctuations can propagate faster than light ! Adams, Arkani-Hamed, Dubovsky, Nicolis & Rattazzi, JHEP 0610 (2006) 014 Hinterbichler, Nicolis & Porrati, JHEP 0909, 089 (2009) Burrage, CdR, Heisenberg & Tolley, JCAP 1207, 004 (2012) Deser & Waldron, PRL 110, (2013)

Galileons in trouble (?) Strong coupling scale gets redressed: At the surface of the Earth (only from the effect of the Earth)

Galileons in trouble (?) Strong coupling scale gets redressed: After that ? … break tree-level unitarity At the surface of the Earth (only from the effect of the Earth)

Galileons in trouble (?) Strong Coupling Issues & Superluminalities Ubiquitous to Galileon theories Strong Coupling Issues & Superluminalities Ubiquitous to Galileon theories & to massive gravity (?) This was the state of Affairs …till the end of 2012

2012 Breakthrough July: Higgs Boson discovered August: Curiosity Lands on Mars Summer: Quantum Teleportation Distance Record Broken of more than 50 miles October:Earth’s Exoplanet Twin discovered orbiting Alpha Centauri B December 31 st 20:55:59 GMT: …

2012 Breakthrough July: Higgs Boson discovered August: Curiosity Lands on Mars Summer: Quantum Teleportation Distance Record Broken of more than 50 miles October:Earth’s Exoplanet Twin discovered orbiting Alpha Centauri B December 31 st 20:55:59 GMT:

Missed the final breakthrough because… December 31 st 2012, 20:55:59 GMT

Dual Galileon In 3d

Dual Galileon Curtright & Fairlie, CdR, Fasiello & Tolley, in preparation In 3d

Dual Galileon Curtright & Fairlie, CdR, Fasiello & Tolley, in preparation In 3d

Dual Galileon Large x no Vainshtein Small x Vainshtein Large Small Vainshtein region in original theory maps to weak region in dual theory.

Dual Galileon Canonical normalization of 

Dual Galileon Canonical normalization of  Strong coupling Weak coupling

Strong Coupling in the Dual theory The dual theory is under control as long as Back to the original theory, this implies For the Earth CdR, Fasiello & Tolley, in preparation

Other eg. of duality In 4d

Other eg. of duality In 4d

Other eg. of duality The original theory admits superluminal propagation in the vacuum ! (ie. no matter or other sources)

Other eg. of duality The original theory admits superluminal propagation in the vacuum ! (ie. no matter or other sources) plane wave solutions

Other eg. of duality The original theory admits superluminal propagation in the vacuum ! (ie. no matter or other sources) plane wave solutions Speed of fluctuations: Superluminal propagation for

Other eg. of duality The original theory admits superluminal propagation Yet the dual theory is just free ! Trivially causal, unitary, UV complete,…

Group vs Front velocity In the original theory, What is SL is the GROUP velocity What counts for causality is the FRONT velocity here

Group vs Front velocity In the original theory, What is SL is the GROUP velocity What counts for causality is the FRONT velocity Dualityfront velocity is LUMINAL Duality implies that the front velocity is LUMINAL No acausality

Massive Gravity is a specific framework to study IR modifications of Gravity It could play a role for - the late-time acceleration of the Universe - the cosmological constant problem We now have the theoretical formalism to describe a stable theory of massive gravity It behaves a scalar-tensor Galileon theory in some limit, hand in hand with a Vainshtein mechanism Outlook

Superluminalities & Strong Issues Galileon theories are plagued with both Superluminalities & Strong Issues These issues could be tackled by a field redefinition DUAL GALILEON ? Massive Gravity requires tuning Massive Gravity requires tuning The graviton mass is technically natural At 1-loop the potential does not get strongly distabilized At higher loops, it still need to be explored… Outlook