Minimal Theory of Massive gravity
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1 Minimal Theory of Massive gravity Shinji Mukohyama (YITP Kyoto) Ref. arxiv: & with Antonio DeFelice Based also on other works with Antonio DeFelice, Garrett Goon, Emir Gumrukcuoglu, Lavinia Heisenberg, Kurt Hinterbichler, David Langlois, Chunshan Lin, Ryo Namba, Atsushi Naruko, Takahiro Tanaka, Norihiro Tanahashi, Mark Trodden
2 Massive gravity: history Simple question: Can graviton have mass? May lead to acceleration without dark energy Yes? No?
3 Massive gravity: history Simple question: Can graviton have mass? May lead to acceleration without dark energy Yes? No?
4 Massive gravity: history Simple question: Can graviton have mass? May lead to acceleration without dark energy Yes? No? Fierz-Pauli theory (1939) Unique linear theory without instabilities (ghosts) van Dam-Veltman- Zhakharov discontinuity (1970) Massless limit General Relativity
5 Massive gravity: history Simple question: Can graviton have mass? May lead to acceleration without dark energy Yes? No?
6 Massive gravity: history Simple question: Can graviton have mass? May lead to acceleration without dark energy Yes? No? Vainshtein mechanism (1972) Nonlinearity Massless limit = General Relativity Fierz-Pauli theory (1939) Unique linear theory without instabilities (ghosts) Boulware-Deser ghost (1972) 6 th d.o.f.@nonlinear level Instability (ghost) van Dam-Veltman- Zhakharov discontinuity (1970) Massless limit General Relativity
7 Nonlinear massive gravity de Rham, Gabadadze 2010 de Rham, Gabadadze & Tolley 2010 First example of fully nonlinear massive gravity without BD ghost since 1972! Purely classical (but technically natural) Properties of 5 d.o.f. depend on background 4 scalar fields f a (a=0,1,2,3) Poincare symmetry in the field space: fiducial metric Pullback of Minkowski metric in field space to spacetime
8 Systematic resummation de Rham, Gabadadze & Tolley 2010 K No helicity-0 ghost, i.e. no BD ghost, in decoupling limit No BD ghost away from decoupling limit (Hassan&Rosen)
9 Massive gravity: history Simple question: Can graviton have mass? May lead to acceleration without dark energy Yes? No?
10 Cosmological solutions in nonlinear massive gravity Good? Bad?
11 Cosmological solutions in nonlinear massive gravity Good? Bad? Consistent Theory found in 2010 but No Viable Cosmology?
12 Open FLRW solutions Gumrukcuoglu, Lin, Mukohyama, arxiv: [hep-th] f mu spontaneously breaks diffeo. Both g mu and f mu must respect FLRW symmetry Need FLRW coordinates of Minkowski f mu No closed FLRW chart Open FLRW ansatz
13 Open FLRW solutions Gumrukcuoglu, Lin, Mukohyama, arxiv: [hep-th] EOM for f a (a=0,1,2,3) The first sol implies g mu is Minkowski we consider other solutions Latter solutions do not exist if K=0 Metric EOM self-acceleration
14 Self-acceleration X 0 0 X 0 0 0
15 Cosmological solutions in nonlinear massive gravity Good? Bad? Open universes with selfacceleration GLM (2011a) D Amico, et.al. (2011) Non-existence of flat FLRW (homogeneous isotropic) universe! GLM = Gumrukcuoglu-Lin-Mukohyama
16 Cosmological solutions in nonlinear massive gravity Good? Bad? GLM = Gumrukcuoglu-Lin-Mukohyama
17 Summary of Introduction + a Nonlinear massive gravity free from BD ghost FLRW background No closed/flat universe Open universes with self-acceleration! More general fiducial metric f mu closed/flat/open FLRW universes allowed Friedmann eq does not depend on f mu Cosmological linear perturbations Scalar/vector sectors same as in GR Tensor sector time-dependent mass
18 Nonlinear instability DeFelice, Gumrukcuoglu, Mukohyama, arxiv: [hep-th] de Sitter or FLRW fiducial metric Pure gravity + bare cc FLRW sol = de Sitter Bianchi I universe with axisymmetry + linear perturbation (without decoupling limit) Small anisotropy expansion of Bianchi I + linear perturbation nonlinear perturbation around flat FLRW Odd-sector: 1 healthy mode + 1 healthy or ghosty mode Even-sector: 2 healthy modes + 1 ghosty mode This is not BD ghost nor Higuchi ghost.
19 Cosmological solutions in nonlinear massive gravity Good? Bad? More general fiducial metric f mu closed/flat/open FLRW universes allowed GLM (2011b) Open universes with selfacceleration GLM (2011a) NEW Nonlinear instability of FLRW solutions DGM (2012) D Amico, et.al. (2011) Non-existence of flat FLRW (homogeneous isotropic) universe! GLM = Gumrukcuoglu-Lin-Mukohyama DGM = DeFelice-Gumrukcuoglu-Mukohyama
20 New backgrounds or Extended theories New nonlinear instability [DeFelice, Gumrukcuoglu, Mukohyama 2012] (i) new backgrounds, or (ii) extended theories (i) Anisotropic FLRW (Gumrukcuoglu, Lin, Mukohyama 2012): physical metric is isotropic but fiducial metric is anisotropic (ii) Extended quasidilaton (De Felice&Mukohyama 2013), Bimetric theory (Hassan, Rosen 2011; DeFelice, Nakamura, Tanaka 2013; DeFelice, Gumrukcuoglu, Mukohyama, Tanahashi, Tanaka 2014), Rotationinvariant theory (Rubakov 2004; Dubovsky 2004; Blas, Comelli, Pilo 2009; Comelli, Nesti, Pilo 2012; Langlois, Mukohyama, Namba, Naruko 2014), Composite metric (de Rham, Heisenberg, Ribeiro 2014; Gumrukcuoglu, Heisenberg, Mukohyama 2014, 2015), New quasidilaton (Mukohyama 2014), They provide stable cosmology.
21 New class of cosmological solution Gumrukcuoglu, Lin, Mukohyama, arxiv: [hep-th] + De Felice, arxiv: [hep-th] Healthy regions with (relatively) large anisotropy Are there attractors in healthy region? Classification of fixed points Local stability analysis Global stability analysis At attractors, physical metric is isotropic but fiducial metric is anisotropic. Anisotropic FLRW universe! statistical anisotropy expected (suppressed by small m g2 )
22 New class of cosmological solution Gumrukcuoglu, Lin, Mukohyama, arxiv: [hep-th] + De Felice, arxiv: [hep-th] Anisotropy in Expansion Anisotropy in fiducial metric
23 Cosmological solutions in nonlinear massive gravity Good? Bad? GLM = Gumrukcuoglu-Lin-Mukohyama DGM = DeFelice-Gumrukcuoglu-Mukohyama
24 New backgrounds or Extended theories New nonlinear instability [DeFelice, Gumrukcuoglu, Mukohyama 2012] (i) new backgrounds, or (ii) extended theories (i) Anisotropic FLRW (Gumrukcuoglu, Lin, Mukohyama 2012): physical metric is isotropic but fiducial metric is anisotropic (ii) Extended quasidilaton (De Felice&Mukohyama 2013), Bimetric theory (Hassan, Rosen 2011; DeFelice, Nakamura, Tanaka 2013; DeFelice, Gumrukcuoglu, Mukohyama, Tanahashi, Tanaka 2014), Rotationinvariant theory (Rubakov 2004; Dubovsky 2004; Blas, Comelli, Pilo 2009; Comelli, Nesti, Pilo 2012; Langlois, Mukohyama, Namba, Naruko 2014), Composite metric (de Rham, Heisenberg, Ribeiro 2014; Gumrukcuoglu, Heisenberg, Mukohyama 2014, 2015), New quasidilaton (Mukohyama 2014; De Felice, Gumrukcuoglu, Heisenberg, Mukohyama, Tanahashi 2016), They provide stable cosmology.
25 Cosmological solutions in nonlinear massive gravity Good? Bad? GLM = Gumrukcuoglu-Lin-Mukohyama DGM = DeFelice-Gumrukcuoglu-Mukohyama
26 More recent development Minimal Theory of Massive Gravity 1. Fix local Lorentz to realize ADM vielbein in drgt 2. Switch to Hamiltonian 3. Add 2 additional constraints De Felice & Mukohyama, arxiv: physical dof only = massive gravitational waves exactly same FLRW background as in drgt no BD ghost, no Higuchi ghost, no nonlinear ghost Three steps to the Minimal Theory (It is easy to go back to Lagrangian after 3.)
27 Constraint Cosmology of MTMG Self-accelerating branch eff from graviton mass term (even with c 4 =0) Scalar/vector parts are the same as CDM Time-dependent mass for gravity waves
28 Self-acceleration X 0 0 X 0 0 0
29 Constraint Cosmology of MTMG Normal branch Dark component with w -1 without extra dof Scalar part recovers GR in UV (L m -1 ) but deviates from GR in IR (L m -1 ) Vector part is the same as GR Time-dependent mass for gravity waves
30 Cosmological solutions in nonlinear massive gravity Good? Bad? GLM = Gumrukcuoglu-Lin-Mukohyama DGM = DeFelice-Gumrukcuoglu-Mukohyama DGHM = DeFelice-Gumrukcuoglu-Heisenberg-Mukohyama
31 Summary Nonlinear massive gravity free from BD ghost FLRW background No closed/flat universe Open universes with self-acceleration! More general fiducial metric f mu closed/flat/open FLRW universes allowed Friedmann eq does not depend on f mu Cosmological linear perturbations Scalar/vector sectors same as in GR Tensor sector time-dependent mass All homogeneous and isotropic FLRW solutions in the original drgt theory have infinitely strong coupling and ghost instability Stable cosmology realized in (i) new class of cosmological solution or (ii) extended theories Minimal theory of massive gravity with 2dof provides a nonlinear completion of drgt self-accelerating cosmology
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