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"Massive Gravity"

Claudia de Rham

	Abstract
1	Introduction
2	Massive and Interacting Fields
	2.1	Proca field
	2.2	Spin-2 field
	2.3	From linearized diffeomorphism to full diffeomorphism invariance
	2.4	Non-linear Stückelberg decomposition
	2.5	Boulware–Deser ghost
I	Massive Gravity from Extra Dimensions
3	Higher-Dimensional Scenarios
4	The Dvali–Gabadadze–Porrati Model
	4.1	Gravity induced on a brane
	4.2	Brane-bending mode
	4.3	Phenomenology of DGP
	4.4	Self-acceleration branch
	4.5	Degravitation
5	Deconstruction
	5.1	Formalism
	5.2	Ghost-free massive gravity
	5.3	Multi-gravity
	5.4	Bi-gravity
	5.5	Coupling to matter
	5.6	No new kinetic interactions
II	Ghost-free Massive Gravity
6	Massive, Bi- and Multi-Gravity Formulation: A Summary
7	Evading the BD Ghost in Massive Gravity
	7.1	ADM formulation
	7.2	Absence of ghost in the Stückelberg language
	7.3	Absence of ghost in the vielbein formulation
	7.4	Absence of ghosts in multi-gravity
8	Decoupling Limits
	8.1	Scaling versus decoupling
	8.2	Massive gravity as a decoupling limit of bi-gravity
	8.3	Decoupling limit of massive gravity
	8.4	$Λ3$ -decoupling limit of bi-gravity
9	Extensions of Ghost-free Massive Gravity
	9.1	Mass-varying
	9.2	Quasi-dilaton
	9.3	Partially massless
10	Massive Gravity Field Theory
	10.1	Vainshtein mechanism
	10.2	Validity of the EFT
	10.3	Non-renormalization
	10.4	Quantum corrections beyond the decoupling limit
	10.5	Strong coupling scale vs cutoff
	10.6	Superluminalities and (a)causality
	10.7	Galileon duality
III	Phenomenological Aspects of Ghost-free Massive Gravity
11	Phenomenology
	11.1	Gravitational waves
	11.2	Solar system
	11.3	Lensing
	11.4	Pulsars
	11.5	Black holes
12	Cosmology
	12.1	Cosmology in the decoupling limit
	12.2	FLRW solutions in the full theory
	12.3	Inhomogenous/anisotropic cosmological solutions
	12.4	Massive gravity on FLRW and bi-gravity
	12.5	Other proposals for cosmological solutions
IV	Other Theories of Massive Gravity
13	New Massive Gravity
	13.1	Formulation
	13.2	Absence of Boulware–Deser ghost
	13.3	Decoupling limit of new massive gravity
	13.4	Connection with bi-gravity
	13.5	3D massive gravity extensions
	13.6	Other 3D theories
	13.7	Black holes and other exact solutions
	13.8	New massive gravity holography
	13.9	Zwei-dreibein gravity
14	Lorentz-Violating Massive Gravity
	14.1	SO(3)-invariant mass terms
	14.2	Phase $m1 = 0$
	14.3	General massive gravity ( $m0 = 0$ )
15	Non-local massive gravity
16	Outlook
	Acknowledgments
	References
	Footnotes
	Figures

List of Figures

	Figure 1: Spectral representation of different models. (a) DGP (b) higher-dimensional cascading gravity and (c) multi-gravity. Bi-gravity is the special case of multi-gravity with one massless mode and one massive mode. Massive gravity is the special case where only one massive mode couples to the rest of the standard model and the other modes decouple. (a) and (b) are models of soft massive gravity where the graviton mass can be thought of as a resonance.
	Figure 2: Codimension-2 brane with positive (resp. negative) tension brane leading to a positive (resp. negative) deficit angle in the two extra dimensions.
	Figure 3: Seven-dimensional cascading scenario and solution for one the metric potential $Φ$ on the $(5 + 1)$ -dimensional brane in a 7-dimensional cascading gravity scenario with tension on the $(3 + 1)$ -dimensional brane located at $y = z = 0$ , in the case where $M 4∕M 3= M 5∕M 4 = m 6 5 7 6 7$ . $y$ and $z$ represent the two extra dimensions on the $(5 + 1 )$ -dimensional brane. Image reproduced with permission from [149], copyright APS.
	Figure 4: Degrees of freedom for massive gravity on a maximally symmetric reference metric. The only theoretically allowed regions are the upper left green region and the line $m = 0$ corresponding to GR.
	Figure 5: Difference between phase, group, signal and front velocities. At $t = δt$ , the phase and group velocities are represented on the left and given respectively by $v = δx ∕δt phase P$ and $vgroup = δxG ∕δt$ (in the limit $δt → 0$ .) The signal and front velocity represented on the right are given by $vsignal = δxS ∕δt$ (where $δxS$ is the point where at least half the intensity of the original signal is reached.) The front velocity is given by $vfront = δxF∕δt$ .
	Figure 6: Polarizations of gravitational waves in general relativity and potential additional polarizations in modified gravity.
	Figure 7: Alternative ways in deriving the cosmology in massive gravity.