Massive Gravity Cosmology Moriond.Key

Cosmological Solutions of Massive Gravity and its Extensions

Kurt Hinterbichler (Perimeter Institute)

KH: Rev. Mod. Phys. 84, 671–710 (review)

arXiv:1402.5424 Melinda Andrews, KH, James Stokes, Mark Trodden arXiv:1306.5743 Melinda Andrews, KH, James Stokes, Mark Trodden arXiv:1304.0449 A. Emir Gumrukcuoglu, KH, Chunshan Lin, Shinji Mukohyama, Mark Trodden arXiv:1301.4993 KH, James Stokes, Mark Trodden

Moriond Cosmology, March 25, 2016 Motivation: The cosmological constant problem

1 1 122 Rµ⌫ Rgµ⌫ + ⇤gµ⌫ = 2 Tµ⌫ 2 10 2 MP MP really small

⇤ = ⇤ + ⇤ + ⇤ + observed UV known physics phase transitions ···

• The CC is small (not natural)

• A small value does not stay small under deformations of the physics (not technically natural) Motivation: theory vs. observation Observational side: the coming years and decades will bring ever more precise large scale structure data

···

Square Kilometer WFIRST Array SDSS

Euclid Dark Energy Survey

LSST DESI

Theoretical side: a distinct lack of good ideas to test against �CDM Possible solutions

1) Naturalness fails (landscape? no deeper explanation?) conservative 2) The light degrees of freedom are not what we think they are

3) The rules of eﬀective quantum ﬁeld theory are not what we think they are on large scales radical (See e.g. Federico Piazza’s introduction) New degrees of freedom

• GR is the unique theory of an interacting massless helicity-2 at low energies → to modify gravity is to change the degrees of freedom

• Locality, Lorentz-Invariance → degrees of freedom are classiﬁed by mass and spin/helicity

First thought: make the graviton massive IR modiﬁcation scale

M 1 mr V (r) 2 e ,m H ⇠ MP r ⇠

Extra DOF: 5 massive spin states as opposed to 2 helicity states

• If it fails → learn something about why the graviton is massless Massive graviton: linear theory

Massive spin 2 particle: 5 degrees of freedom (as opposed to 2 for massless helicity 2)

Fierz-Pauli action: Fierz, Pauli (1939)

1 µ⌫ ⌫ µ µ⌫ 1 1 2 µ⌫ 2 1 µ⌫ = @hµ⌫ @ h + @µh⌫@ h @µh @⌫ h + @h@ h m (hµ⌫ h h )+ hµ⌫ T L 2 2 2 MP

Einstein-Hilbert (massless) part. Mass term breaks gauge symmetry. Gauge symmetry: hµ⌫ = @µ⇠⌫ + @⌫ ⇠µ Fierz-Pauli tuning ensures 5 D.O.F. Helicity (Stükelberg) analysis

Introduce Stükelberg ﬁelds: h h + @ A + @ A +2@ @ µ⌫ ! µ⌫ µ ⌫ ⌫ µ µ ⌫

h helicity 2 2 DOF µ⌫ ⇠ ± hµ⌫ 8 Aµ helicity 1 2 DOF relativistic limit m 0 ⇠ ± 5 DOF ! > <> helicity 0 1 DOF ⇠ > :>

This is the vDVZ discontinuity: scalar ﬁfth force

1 ˆ ˆµ⌫ ˆ µ ˆ 1 µ⌫ 1 ˆ m=0(h0) Fµ⌫ F 3 @µ@ + hµ0 ⌫ T + T L 2 MP MP

helicity-2 helicity-1 helicity-0 where the interaction potential U is the most general one that reduces to Fierz-Pauli at linear order,

U(g(0),h)=U (g(0),h)+U (g(0),h)+U (g(0),h)+U (g(0),h)+ , (6.35) 2 3 4 5 ···

U (g(0),h)= h2 [h]2 , (6.36) 2 (0) 3 2 3 U3(g ,h)=+ C⇥1 h + C2 h [h]+C3 [h] , (6.37) (0) 4 3 2 2 2 2 4 U4(g ,h)=+D1 h ⇥ + D2 h ⇥ [h]+D3 h + D4 h [h] + D5 [h] , (6.38) (0) 5 4 3 2 3 2 2 2 U5(g ,h)=+F1 h ⇥+ F2 h ⇥[h]+F3 h ⇥[h] + F4 h⇥ h + F5 h [h] 2 3 5 +F6 h ⇥ [h] +F7⇥[h] , ⇥ ⇥ ⇥ ⇥ (6.39) . . ⇥

(0),µ⇤ (0)µ⇤ The square bracket indicates a trace, with indices raised with g , i.e. [h]=g hµ⇤, 2 (0)µ (0)⇥⇤ [h ]=g h⇥g h⇤µ, etc. The coecients C1,C2, etc. are generic coecients. Note that (0) the coecients in Un(g ,h)forn>Dare redundant by one, because there is a combination of the various contractions, the characteristic polynomial TD(h) (see Appendix A), which Interaction termsLn (0) vanishes identically. Thus one of the coecients in Un(g ,h)forn>D(or any one linear combination) can be set to zero. Effective field theory philosophy: write every possible term parametrizedIf we like, weby canarbitrary re-organize coefficients the terms in the potential by raising and lowering with the full metric gµ⇤ rather than the absolute metric g(0)µ⇤,

M 2 1 1 1 S =P d4x dD(px gR(⌅) gRp ) g ⌅m2Vg(g,m h2)V ,(g, h) , (6.40) 2 22 4 4 Z ⇧  ⇤ ⌅ where V (g, h)=V (g, h)+V (g, h)+V (g, h)+V (g, h)+ , (6.41) 2 3 4 5 ···

V (g, h)= h2 h 2, (6.42) 2 ⇥ ⇤⇥ ⇤ V (g, h)=+c h3 + c h2 h + c h 3, (6.43) 3 1⇥ ⇤ 2⇥ ⇤⇥ ⇤ 3⇥ ⇤ V (g, h)=+d h4 + d h3 h + d h2 2 + d h2 h 2 + d h 4, (6.44) 4 1⇥ ⇤ 2⇥ ⇤⇥ ⇤ 3⇥ ⇤ 4⇥ ⇤⇥ ⇤ 5⇥ ⇤ V (g, h)=+f h5 + f h4 h + f h3 h 2 + f h3 h2 + f h2 2 h 5 1⇥ ⇤ 2⇥ ⇤⇥ ⇤ 3⇥ ⇤⇥ ⇤ 4⇥ ⇤⇥ ⇤ 5⇥ ⇤ ⇥ ⇤ 2 3 5 +f6 h h + f7 h , (6.45) . ⇥ ⇤⇥ ⇤ ⇥ ⇤ .

55 Arkani-Hamed, Georgi and Schwartz (2003) The effective field theory Creminelli, Nicolis, Pappuchi, Trincherini (2005) de Rham, Gabadadze (2010)

After replacement hµ⌫ hµ⌫ + @µA⌫ + @⌫ Aµ +2@µ@⌫ + there are interaction terms: ! ··· 2 2 n n 2 n 4 nh 2nA 3n n n 2 n m M h h (@A) A (@ ) ⇤ hˆ h (@Aˆ) A (@ ˆ) P ⇠

1 1/ 3n +2nA + nh 4 Various strong coupling scales: ⇤ = M m , = P n + n + n 2 A h

The smallest scale is carried by a cubic scalar interaction: Energy scale

MP 2 ˆ 3 (@ ) 4 1/5 5 , ⇤5 =(MP m ) ⇠ ⇤5 ⇤5 This is the (UV) strong coupling scale of the theory m

• Scalar self-interactions display the Boulware-Deser ghost Boulware, Deser (1972) Deffayet, Rombouts (2005) The Λ3 effective theory de Rham, Gabadadze, Tolley (2011)

Key insight: Choose the interactions to cancel bad operators and make the cutoﬀ as high as possible.

1/3 Cutoff raised to: ⇤ M m2 3 ⇠ P LongitudinalThe scalar mode self-interactions is described in (by9.27 Galileon) are given interactions: by the following four lagrangians, Energy scale 1 2 = (⇥) , (⇧µ⌫ @µ@⌫ ) M L2 2 ⌘ P 1 = (⇥)2[] , L3 2 1 = (⇥)2 []2 [2] , L4 2 ⇤3 1 = (⇥)2 []3 3[][⇥2] + 2[3] . m(9.28) L5 2 ⇥ • EquationsThese of are motion known are as the secondgalileon orderterms (no [135 ghost)] (see also Section II of [136] for a summary of the galileons). They share two special properties: their equations of motion are purely secondµ • Wess-Zumino terms for shifts of the field and its derivative (x) (x)+c + cµx order (despite the appearance of higher derivative terms in the lagrangians),! andGoon, they KH, Joyce, are Trodden (2012) • Not renormalized at any loop (no quantum corrections in the decoupling limit) Luty, Porrati, Rattazziµ (2003) invariant up to a total derivative under the galilean symmetry (8.8), (x) (x)+c + bµx . ⇥ Nicolis, Rattazzi (2004) As shown in [135], the terms (9.28) are the only polynomial terms in four dimensionsKH, Trodden, with Wesely (2010) • Side theseeffect: properties. full theory has no Boulware-Deser ghost (propagates 5 DOF non- linearly) The galileon was first discovered in studies of the DGP brane world modelHassan, Rosen [35] (2011) (which KH, Rosen (2012) we will explore in more detail in Section 10.2), for which the cubic galileon,+ many others, was following found L3 to describe the leading interactions of the brane bending mode [144, 145]. The rest of the galileons were then discovered in [135], by abstracting the properties of the cubic term away from DGP. They have some other very interesting properties, such as a non-renormalization theorem (see e.g. Section VI of [136]), and a connection to the Lovelock invariants through brane embedding [146]. Due to these unexpected and interesting properties, they have since taken on a life of their own. They have been generalized in many directions [147, 148, 149, 150, 151, 152, 153], and are the subject of much recent activity (see for instance the > 100 papers citing [135]). The fact that the equations are second order ensures that, unlike (8.10), no extra degrees of freedom propagate. In fact, as pointed out in [34], the properties (A.17) of the

tensors Xµ guarantee that there are no ghosts in the lagrangian (9.10) of the decoupling limit theory.20 By going through a hamiltonian analysis similar to that of Section 2.1,we

can see that h00 and h0i remain Lagrange multipliers enforcing ﬁrst class constraints (as they should since the lagrangian (9.10) is gauge invariant. In addition, the equations of motion

20 This is contrary to [100], which claims that a ghost is still present at quartic order. As remarked however in [34], they arrive at the incorrect decoupling limit lagrangian, which can be traced to a minus sign mistake in their Equation 5, which should be as in (9.4).

91 Arkani-Hamed, Georgi and Schwartz (2003) The Λ3 effective theory Creminelli, Nicolis, Pappuchi, Trincherini (2005) de Rham, Gabadadze (2010)

2 (@) ⇤ 2 1/3 3 , ⇤3 = MP m ⇠ ⇤3

• Scalar self-interactions responsible for Vainshtein radius:

Source of mass M

r 2 rSchwarzschild M/MP M 1/3 1 rYukawa 1/m ⇠ r ⇠ ⇤ ⇠ M (m2M )1/3 ✓ P ◆ P The Λ3 theory resummed: dRGT gravity de Rham, Gabadadze, Tolley (2011)

A B Vierbein formulation: gµ⌫ = eµ e⌫ ⌘AB KH, Rachel Rosen (arXiv:1203.5783)

D 2 MP D 2 A1 An An+1 An d x e R[e] m an ✏A A e e 1 1 2 | | 1··· D ^ ···^ ^ ^ ···^ n Z X Z

Mass terms are simply all possible ways of wedging a vierbein and background vierbein:

✏ eA1 eA2 eA3 eA4 A1A2A3A4 ^ ^ ^ ✏ eA1 eA2 eA3 1A4 A1A2A3A4 ^ ^ ^ ✏ eA1 eA2 1A3 1A4 A1A2A3A4 ^ ^ ^ ✏ eA1 1A2 1A3 1A4 A1A2A3A4 ^ ^ ^ ✏ 1A1 1A2 1A3 1A4 A1A2A3A4 ^ ^ ^ Some features of the theory Theoretical: • Consistent, ghost free eﬀective ﬁeld theory propagating a single massive graviton • Issues with superluminal propagation, breakdown of initial value formulation, associated with the scalar mode when going beyond its regime of validity (needs a UV completion) Burrage, de Rham, Heisenberg, Andrew J. Tolley (2011) Deser, Waldron (2012) Brito, Terrana, Johnson, Cardoso (2014) CC problem:

• Exist self accelerating solutions, in the absence of a CC (acceleration is caused by

New CC graviton mass m ∼ H) de Rham, Gabadadze, Heisenberg, Pirtskhalava (2010) 8 Gumrukcuoglu, Lin, Mukohyama (2011) problem > < • A small graviton mass is protected from large quantum corrections (diﬀ invariance restored as m → 0) de Rham, Heisenberg, Ribeiro (2013) > Dvali, Gabadadze, Shifman (2002) : Arkani-Hamed, Dimopolous, Dvali Gabadadze (2002) Old CC 8 Dvali, Hoffman, Khoury (2007) > • Screening of a large CC (hard to realize non-linearly) problem <> > :> Phenomenology/new signals:

• Vainshtein mechanism hides 5-th force from experiments. Vainshtein (1972) • Residual screening eﬀects might be observable Expectation for how cosmology should work

non-linear, looks like GR linear, 5th-force Exponentially suppressed force Source of mass M

M 1/3 1 r rYukawa 1/m ⇤ ⇠ M (m2M )1/3 ✓ P ◆ P ⇠

consider matter of constant density ρ in a sphere of radius R,

1/3 ⇢ 2 2 r R, ⇢c MP m ⇤ ⇠ ⇢ ⇠ ✓ c ◆ R ⇢ Expectation for how cosmology should work

Early times: Domain of size ~1/m

1 ⇢ ⇢c,H 1/m ⌧ Hubble patch ~1/H << 1/m ⇢ 1/3 1 1 r ⇤ ⇠ ⇢c H H ✓ ◆ Vainshtein radius >> 1/H

Hubble patch is well within its Vainshtein radius → dynamics described by GR

Very late times: Hubble patch ~1/H >> 1/m

1 Vainshtein radius << 1/H ⇢ ⇢ ,H 1/m ⌧ c Domains of size ~1/m ⇢ 1/3 1 1 r ⇤ ⇠ ⇢ H ⌧ H ✓ c ◆ Hubble patch is larger than its Vainshtein radius and is unscreened → dynamics diﬀerent from GR Massive gravity cosmology Massive gravity cosmology Kurt Hinterbichler April 11, 2013 Kurt Hinterbichler

April1Massivegravity 11, 2013

Our starting point is the massive gravity action 1Massivegravity S = SEH + Smass + SM , (1.1) Our starting point is the massive gravity action where S = S + S + S , 1 (1.1) EH mass MS = M 2 d4x p gR , EH 2 P where Z 2 2 4 Smass = MPm d x p g ( 2 + ↵3 3 + ↵4 4) , (1.2) 1 2 4 L L L SEH = MP d x p gR , Z 2 4 Z SM = d x M (g, ) . (1.3) L S = M 2m2 d4x p g ( + ↵ + ↵Z ) , (1.2) mass P L2 3L3 4L4 (TheZ ↵’s here di↵er by a factor: ↵standard = 8↵here.) The mass term consists of the S = d4x ghost-free(g, ) . combinations, (1.3) M LM Z 1 2 2 (The ↵’s here di↵er by a factor: ↵ = 8↵2 =.) The[ mass] [ term] , consists of the standard L here 2 K K ghost-free combinations, 1 = [ ]3 3[ ][ 2]+2[ 3] , L3 6 K K K K 1 2 2 2 = [ ] [ ] , 1 4 2 2 2 2 3 4 L 2 K K 4 = [ ] 6[ ] [ ]+3[ ] +8[ ][ ] 6[ ] , (1.4) L 24 K K K K K K K 1 3 2 3 3 = [ ] 3[ ][ ]+2[µ ] µ, µ L 6 K KwhereK K⌫ K= ⌫ pg g¯⌫ andg ¯µ⌫ is the background (non-dynamical) metric which 1 = [ ]4 6[we]2[ take2]+3[ to be2 Minkowski.]2 +8[ ][ 3] 6[ 4] , (1.4) L4 24 K K K K K K K

µ µ µ where K⌫ = ⌫ pg g¯⌫ andg ¯µ⌫1.1is the Spatially background flat (non-dynamical) metric which we take to be Minkowski. ConsiderCosmological first the case solutions of spatially flat in FRW, the full theory ds2 = N(t)2dt2 + a(t)2dx2. (1.5) 1.1 Spatially flat LookThis for spatially- is the time-diflat ↵FRWinvariant solutions: ansatz. Unitary gauge corresponds to taking f(t)=t, Consider first the case of spatiallya We flatb take FRW, the Stückelberg degrees of freedom to be so that @µ @⌫ ⌘ab = ⌘µ⌫. This is the time-di2 ↵ invariant2 2 ansatz.2 2 UnitaryThis is the gauge time-dii i corresponds↵ invariant0 ansatz.˙ to Unitary taking gaugef(t)= correspondst, to taking f(t)=t, The mini-superspace action is (for expandinga = bx , universe = f(a,t ˙).f 0), (1.6) This is the time-di↵ invarianta ansatz.ds b = UnitaryN(t) dt gauge+ a(t) correspondsdx . so that @ toµ taking@⌫ ⌘ab =f(⌘tµ)=⌫. (1.5)t, so that @µ @⌫ ⌘ab= ⌘µ⌫. a b The mini-superspace action is (for expanding universea, ˙ f˙ 0), so that @µ @⌫ ⌘ab = ⌘µ⌫. 2 ˙ We take the StückelbergThe mini-superspace degrees of freedom action to be is (for expanding2 4 a universe˙ a a, ˙ f 0), Mini-superspace action: S =3M ˙d x 1 2 (1.7) The mini-superspace action is (for expandingR universePa, ˙ f 0), 2 4 a˙ a N SR =3M d x (1.7) i i 0 Z a˙ 2a P N = x , a˙ 2a= f(2t). 4 (1.6) Z  2 4 SR =3MP d2x 2 4 ˙ 2 2 4 (1.7)˙ SR =3MP d x Smass+=3MPm Nd x NF(a)Smass(1.7)fG=3(MaP)m. +d xmatterNF(a) fG(a(1.8)) . (1.8) N Z  L Z  1 Z Z h i 2 2 where4 h i 2 2 Smass4 =3M m ˙ d x NF(a) fG˙ (a) . (1.8) Smasswhere=3MPm d x NF(a)P fG(a) . (1.8) ↵3 2 ↵4 3 Z Z h↵ F (a)=a(a 1)(2i a 1)↵ + (a 1) (4a 1) + (a 1) (1.9) F (a)=ha(a 1)(2a 1)i + 3 (a 1)2(4a 1) + 4 (a3 1)3 (1.9)3 where where 2 2 ↵4 3 3 G(a)=a (a 1) + ↵3a3(a 1)+ (a 1) . (1.10) 3 2 ↵ 2 ↵4 3↵ G(a↵)=3 a (a 2 1) + ↵ a3(↵a4 1) 2+3 (a 1) . 4 3 (1.10) F (a)=a(a 1)(2Fa(a)=1) +a(a (a 1)(21)a(4a1) +1)3 + (Varyinga(a 1)1) w.r.t.(4a3 f,1)N +(1.9)and a,andfixingthegauge(a 1) (1.9)N =1,wegetrespectively Friedmann equations: 3 (adding3 3 back in the matter) 3 2 2 2 ↵4 3 2 ↵4 3 G(a)=a (a Varying1)G +(↵a3)=a( w.r.t.a a (1)a f+,1)N +(anda↵ a1)(aa,andfixingthegauge. 1) + (a 1) .(1.10)N =1,wegetrespectively(1.10) 3 3 d (adding back in the matter) 3 G(a)=0 (1.11) 2 2 F (a, f) dt Varying w.r.t. f, N and a,andfixingthegauge3MP HN =1,wegetrespectively+ m 3 = ⇢ Varying w.r.t. f, N and a,andfixingthegaugea N =1,wegetrespectivelyF (a) (adding back in the matter)  d 3M 2 H2 + m2 = ⇢, (1.12) (adding back in the matter) G(a)=0P a3 (1.11) dt  2 d 2 2 2ä m d M H + + dF 0(a) fG˙ 0(a) = p, (1.13) G(a)=0P (1.11)a a2 dt 2G(a2)=02 F (a)  a =0 (1.11) 3MdtP H + m = ⇢, ⇣ ⌘ (1.12) along with mattera3 conservation⇢ ˙ +3Hdt(⇢ + p)=0. 2 2 2 F (a)  The gauge symmetry of time re-parametrization invariance in the mini-superspace 3MP H + m 3 =2 ⇢, 2 action22F acting(a) on a, N and(1.12)f becomes a2 3M2 H2ä + mm = ⇢, (1.12)  M H P+ +D’Amico, Fde3 0 Rham,(a) Dubovsky,fG˙ 0( Gabadadze,a) = p, (1.13) No non-trivial flat FRW Psolutions! 2 a d 2  a a Pirtskhalava, Tolley (2011a =) ⇠a,˙ N = (⇠N) , f = ⇠f.˙ (1.14) 2 2 2ä m  ˙ ⇣ ⌘ dt MP H + + F 0(a) fG0(a) 2 = p, (1.13) along witha mattera2 2 conservation2 2ä m⇢ ˙ +3H(⇢ + p)=0. M H + + TheF 0( Noethera) fG˙ identity0(a) is then= p, (1.13)  The gauge symmetryP⇣ ofa timea⌘2 re-parametrization invariance in the mini-superspace along with matter conservation⇢ ˙ +3H(⇢+ p)=0. d action acting on a, N and f becomes⇣ ⌘ Lf˙ N L +˙a L =0. (1.15) The gauge symmetryalong with of time matter re-parametrization conservation⇢ ˙ invariance+3H(⇢ + inp the)=0. mini-superspacef dt N a action acting on a, N and f becomes The gauge symmetry of time re-parametrizationThe equationd invariance (1.11)tellsusthat in˙ thea mini-superspace= const. Thus there are no expanding flat a = ⇠a,˙ FRWN = solutions(⇠N in) massive, f gravity.= ⇠f. (1.14) action acting on a, N andd f becomes dt a = ⇠a,˙ N = (⇠N) , f = ⇠f.˙ (1.14) The Noether identitydt is then 1.2d Spatial curvature a = ⇠a,˙ N = (⇠N) , f = ⇠f.˙ (1.14) The Noether identity is then Wedt taked possibility of open spatial sections (k<0). The metric is f˙ N +˙a =0. (1.15) L L L 2 2 2 2 i j The Noether identityd is then f dt N a ds = N (t)dt + a (t)⌦ijdx dx (1.16) Lf˙ N L +˙a L =0. (1.15) f dt N a where The equation (1.11)tellsusthatd a = const. Thus there are nok expanding flat Lf˙ N L +˙a L =0. ⌦ = + x x(1.15). (1.17) The equation (1.11)tellsusthatFRW solutionsa in= massiveconst. Thus gravity. there are no expanding flat ij ij 1 kr2 i j f dt N a FRW solutions in massive gravity. The equation (1.11)tellsusthata = const. Thus there are no expanding2 flat 1.2 Spatial curvature 1.2 SpatialFRW curvature solutions in massive gravity. We take possibility of open spatial sections (k<0). The metric is We take possibility of open spatial sections (k<0). The metric is 1.2 Spatial curvature 2 2 2 2 i j ds = N (t)dt + a (t)⌦ijdx dx (1.16) 2 2 2 2 i j ds = N (t)dt + a (t)⌦ijdx dx (1.16) We takewhere possibility of open spatial sections (k<0). The metric is k where 2 2 2 2 i j dsk = N ⌦(tij)d=t +ij +a (t)⌦ijdxxidxxj. (1.16)(1.17) ⌦ = + xx . 1 kr2 (1.17) ij ij 1 kr2 i j where k 2 2 ⌦ = + x x . (1.17) ij ij 1 kr2 i j 2 This is the time-di↵ invariant ansatz. Unitary gauge corresponds to taking f(t)=t, a b so that @µ @⌫ ⌘ab = ⌘µ⌫. The mini-superspace action is (for expanding universea, ˙ f˙ 0), a˙ 2a S =3M 2 d4x (1.7) R P N Z  S =3M 2m2 d4x NF(a) fG˙ (a) . (1.8) mass P Z h i where This is the time-di↵ invariant ansatz. Unitary gauge corresponds to taking f(t)=t, so that @ a@ b⌘ = ⌘ . µ ⌫ ab ↵ µ⌫ ↵ F (a)=a(a The1)(2 mini-superspacea 1) + 3 ( actiona 1) is2 (for(4a expanding1) + universe4 (a 1)a, ˙ 3f˙ 0), (1.9) 3 3 a˙ 2a 2 S2 =3↵M4 2 d4x 3 (1.7) G(a)=a (a 1) + ↵3a(a 1)R + P(a 1) . N (1.10) 3 Z  S =3M 2m2 d4x NF(a) fG˙ (a) . (1.8) Varying w.r.t. f, N and a,andfixingthegaugemass P N =1,wegetrespectively Z h i (adding back in the matter)where ↵ ↵ F (a)=d a(a 1)(2a 1) + 3 (a 1)2(4a 1) + 4 (a 1)3 (1.9) G(a)=0 3 3 (1.11) ↵ G(a)=dt a2(a 1) + ↵ a(a 1)2 + 4 (a 1)3. (1.10) 3 3 Varying2 w.r.t. 2 f, N 2andF (aa),andfixingthegaugeN =1,wegetrespectively 3MP H + m = ⇢, (1.12) (adding back in the matter) a3  2 d 2 2 2ä m G(a)=0 (1.11) M H + + F 0(a) fG˙ 0(a) = p, (1.13) P a a2 dt  ⇣ ⌘F (a) along with matter conservation⇢ ˙ +3H(⇢ + p)=0.3M 2 H2 + m2 = ⇢, (1.12) P a3 The gauge symmetry of time re-parametrization invariance in the mini-superspace 2 2 2 2ä m action acting on a, N and f becomes M H + + F 0(a) fG˙ 0(a) = p, (1.13) P a a2 Take the Stückelbergd fields to⇣ be ⌘ alonga = with⇠a,˙ matterN conservation= (⇠N)⇢ ˙,+3Hf(⇢=+ ⇠p)=0.f.˙ (1.14) The gauge symmetry of time re-parametrization invariance in the mini-superspace dt 0 = f(t)p1 kr2 (1.18) action acting on a, N and f becomes The Noether identity is then i p i d= kf(t)x . (1.19) a = ⇠a,˙ N = (⇠N) , f = ⇠f.˙ (1.14) d dt a b This isL thef˙ time-diN ↵L invariant+˙a L =0 ansatz.. The fiducial metric(1.15) is then @µ @⌫ ⌘ab = The Noetherf ˙2 identitydt2 isN then ka diag f , ( k)f ij + 1 kr2 xixj . Unitary gauge corresponds to taking f(t)= d The equation (1.11)tellsusthatt, so⇣Take that @ thea Stückelberg@a =b⌘ const.=Ldiagf˙ fieldsThusN to1,⌘ thereL( be+˙k)at2 areL =0 no+. expandingk x x , which flat (1.15) is Minkowski in µ ⌫ ⇥ ab f dt ⇤N a ij 1 kr2 i j FRW solutions in massiveMilne gravity. coordinates. 0 ⇥ 2 ⇤ TheInTake mini-superspace,equation the ( Stückelberg1.11)tellsusthat the action fieldsa = becomes toconst.= bef(tThus)p1 therekr are no expanding flat (1.18) FRW solutions in massive gravity. i p i =0 kf(t2)x . 2 (1.19) 1.2 Spatial curvature 2 =4f(t)a˙pa1 kr (1.18) Take the Stückelberg fields to be SR =3MP d x +kNa Gumrukcuoglu,(1.20) Lin, Mukohyama (2011) 1.2 Spatial curvatureFRW solutions:i openN i slicing a b We take possibility of openThis spatial is the sections time-di↵ (kMinkowski space (in (there are twoC branches of solution here,C since we must solve a quadratic equation fora a whicha ˙ pF (a,k =0.ThesehaveaR) a(t)=p kt, whichMilne is just coordinates). Minkowski space (in C± Looking at (1.25), theref). Then are plugging twomodification branches back into of (1.26 solutions.a3), weis obtain a constant, First a modified are depending the Friedmann solutions only equation. for on ↵ The2, ↵3, so when this constant @G MilneF (a, aR coordinates).) 2 2 Then there are the solutions for which @a =0.Solvingthisalgebraicallyforf, modification is negative,C± is a constant, we have depending self-accelerationLooking only onat↵ (1.25 with, ↵ ,), so⇢ there when thisM arem constant two. branches of solutions. First are the solutions for whicha ˙ p k =0.Thesehaveaa(3t)=p kt, which is just Minkowski2 3 self space@G P (in Then there are the solutions2 2 for which ⇠ =0.Solvingthisalgebraicallyforwe find f = aR for some constantf, depending only on the coecients ↵2, ↵3 is negative, we have self-acceleration which with ⇢selfa ˙ MpP m k. =0.Thesehave@a a(t)=p kt, whichC± is just MinkowskiC± space (in Milne coordinates). ⇠ (there are two branches of solution here, since we must solve a quadratic equation for we find f =@G aR for some constant depending only on the coecients ↵2, ↵3 ± Milne coordinates). 3 ± f Then there are the solutions for which @a C=0.Solvingthisalgebraicallyfor3 C f, ). Then plugging back into (1.26), we obtain a modified Friedmann equation. The (there are two branches of solution here, since we must solve a@G quadraticF (a, aR equation) for Then there are the solutions for which =0.SolvingthisalgebraicallyforC± f, we find f = aR for some constant depending only on the coecients ↵2, ↵3 modification@a a3 is a constant, depending only on ↵2, ↵3, so when this constant ± f). Then± plugging back into (1.26), we obtain a modified Friedmann equation. The 2 2 C C we find f = aR for some constant is negative,depending we have only self-acceleration on the coe withcients⇢self↵2,M↵3 m . (there are two branches of solution here, since weF (a, mustaR) solve a quadraticC± equation for C± ⇠ P C± modification a3 (thereis a are constant, two branches depending of solution only on here,↵2, since↵3, so we when must this solve constant a quadratic equation for f). Then plugging back into (1.26), we obtain a modified Friedmann equation. The 2 2 F (a, aR) is negative, we havef). self-acceleration Then plugging with back⇢ intoself (1.26M ),m we. obtain a modified Friedmann equation.3 The C± P modification a3 is a constant, depending only on ↵2, ↵3, soF when(a, aR this) constant⇠ 2modification2 C3± is a constant, depending only on ↵2, ↵3, so when this constant is negative, we have self-acceleration with ⇢self M m . a ⇠ Pis negative, we have self-acceleration with ⇢ M 2 m2. 3 self ⇠ P 3 3 Take the Stückelberg fields to be

0 = f(t)p1 kr2 (1.18) i = p kf(t)xi. (1.19) a b This is the time-di↵ invariant ansatz. The ﬁducial metric is then @µ @⌫ ⌘ab = ˙2 2 k diag f , ( k)f ij + 1 kr2 xixj . Unitary gauge corresponds to taking f(t)= a b 2 k t, so⇣ that @µ @⌫⇥⌘ab = diag 1⇤,⌘( k)t ij + 1 kr2 xixj , which is Minkowski in Milne coordinates. ⇥ ⇤ In mini-superspace, the action becomes

a˙ 2a S =3M 2 d4x + kNa (1.20) R P N Z  S =3M 2m2 d4x NF fG˙ , (1.21) mass P Z h i where ↵ ↵ F (a, f)=a(a p kf)(2a p kf)+ 3 (a p kf)2(4a p kf)+ 4 (a p kf)3 3 3 (1.22) ↵ G(a, f)=a2(a p kf)+↵ a(a p kf)2 + 4 (a p kf)3. (1.23) 3 3

Varying w.r.t. f and fixing the gauge N =1,wegetTake the Stückelberg fields to be

0 = f(t)p1 kr2 (1.18) @F @G +˙a =0. i = p(1.24)kf(t)xi. (1.19) @f @a a b @F p @G This is the time-di↵ invariant ansatz. The fiducial metric is then @µ @⌫ ⌘ab = Using the relation @f +Self-acceleratingk @a =0,thiscanbewrittenas solutions˙2 2 k diag f , ( k)f ij + 1 kr2 xixj . Unitary gauge corresponds to taking f(t)= a b 2 k t, so⇣ that @µ @⌫⇥⌘ab = diag 1⇤,⌘( k)t ij + 1 kr2 xixj , which is Minkowski in @G Milne coordinates. a˙ p k =0. (1.25)⇥ ⇤ @a In mini-superspace, the action becomes ⇣ ⌘ a˙ 2a S =3M 2 d4x + kNa (1.20) Varying w.r.t. N and fixing the gauge N = 1, we get (adding back in theR matter)P N Z  p Trivial Minkowksi solution2 2 4 a˙ = k Smass =3M m d x NF fG˙ , (1.21) 2 2 k 2 F (a, f) P 3M H + + m = ⇢. (1.26)Z Two branches of solutions: P a2 a3 h i 8  where ↵ ↵ > @G F (a, f)=a(a p kf)(2a p kf)+ 3 (a p kf)2(4a p kf)+ 4 (a p kf)3 Looking at (1.25), there<> are two=0 branches of solutions.Algebraic First are equation the solutions for 3f for 3 p @a p (1.22) whicha ˙ k =0.Thesehavea(t)= kt, which is just Minkowski space (in ↵ G(a, f)=a2(a p kf)+↵ a(a p kf)2 + 4 (a p kf)3. (1.23) Milne coordinates). > 3 3 > @G Then there are the: solutions for which @a =0.SolvingthisalgebraicallyforVarying w.r.t. f and fixing the gauge N f=1,weget, we find f = aR for some constant depending only on the coecients@F ↵2,@↵G3 Self-accelerating branchC± solutions: C± +˙a =0. (1.24) (there are two branches of solutionf here,= sinceap we mustk solve a quadratic equation@f @ fora ± C @F p @G f). Then plugging back into (1.26), we obtain a modifiedUsing the Friedmann relation @f + equation.k @a =0,thiscanbewrittenas The F (a, aR) modification C3± is a constant, depending only on ↵2, ↵3, so when this constant@G a p Dimensionless, depends on parameters2 2 ↵3, ↵4 a˙ k =0. (1.25) is negative, we have self-acceleration with ⇢self MP m . @a ⇠ ⇣ ⌘ Varying w.r.t. N and fixing the gauge N = 1, we get (adding back in the matter) 3 k F (a, f) p 3M 2 H2 + + m2 = ⇢. (1.26) 2 2 k 2 F (a, a k) P 2 3 ± a a 3MP H + 2 + m C 3 = ⇢  a a Looking at (1.25), there are two branches of solutions. First are the solutions for  whicha ˙ p k =0.Thesehavea(t)=p kt, which is just Minkowski space (in Milne coordinates). Then there are the solutions for which @G =0.Solvingthisalgebraicallyforf, Constant, independent of a @a we find f = aR for some constant depending only on the coecients ↵2, ↵3 ± ± (there are twoC branches of solution here,C since we must solve a quadratic equation for 2f). Then2 plugging back into (1.26), we obtain a modified Friedmann equation. The Looks like a dark energy component with: ⇢self MP m F (a, aR) modification C± is a constant, depending only on ↵ , ↵ , so when this constant ⇠ a3 2 3 is negative, we have self-acceleration with ⇢ M 2 m2. self ⇠ P

3 4

B. Self-accelerating background solutions

We now discuss solutions, starting with the St¨uckelberg constraint (10). Integrating this equation gives 1 X (1 X) 3 3(X 1)↵ +(X 1)2↵ ⇠ X3 = constant . (15) 3 4 a4 ⇥ n o ⇥ ⇤ 4 In an expanding universe, the right hand side of the above equation decays as a . Thus after a suciently long time, X saturates to a constant value XSA, corresponding to a zero of the left hand side of (15). These constant X solutions lead to an e↵ective energy density which acts like a cosmological constant. As pointed out in [23], there are four such solutions for which X is constant. Of these, X =0implies , and as in [23], we drop this solution !1 to avoid strong coupling.2 What remains are the three solutions to the cubic equation 3

2 3 (1 X) 3 3(X 1)↵3 +(X 1) ↵4 ⇠ X =0. (17) X=XSA ⇥ ⇤ For these solutions, we write the Friedmann equation (11) as ! 3 H2 = ⇤ + ⇤ , (18) 2 SA ⇣ ⌘ where the e↵ective cosmological constant from the mass term is

⇤ m2 (X 1) 3X +6+(X 4)(X 1)↵ +(X 1)2↵ . (19) SA ⌘ g SA SA SA SA 3 SA 4 The Friedmann equation (18) also provides⇥ a condition on the parameter !; on the self-accelerating⇤ solutions, one needs to have ! < 6 in order to keep the left hand side of the Friedmann equation (18) positive. This ensures that when ordinary matter is added to the right hand side, we will have standard cosmology during matter domination. (Although we do not include matter fields, the sign of the matter energy density can be determined by replacing the 2 bare ⇤ with ⇢/Mp .) Finally, on the self-accelerating solutions, with constant H specified by (18), the equation of motion for fixes the ratio r = a/N. From Eq.(13), we obtain

! H2 (X 1) r =1+ SA . (20) SA m2 X2 [↵ (X 1)2 2(X 1) ⇠ X2 ] g SA 3 SA SA SA

Here, to simplify the expression we have used the St¨uckelberg equation (15) to eliminate ↵4.

C. Perturbations

To ﬁnd the action for quadraticSelf-accelerating perturbations, we expand solutions: the physical perturbations metric in small ﬂuctuations gµ⌫ around a (0) solution gµLinear⌫ , Perturbations around self-accelerating solutions Gumrukcuoglu, Lin, Mukohyama (2012)

(0) gµ⌫ = gµ⌫ + gµ⌫ , (21) SVT decomposition g 00 ⇠ 2 T As we discuss at the end of AppendixgA30i ,theremainingsolutionsalsoleadtostrongcouplinginthevectorandscalarsectors,whenBi + iB ⇠ =0andtheparameters↵ and ↵ are such⇠ that Xr 0. 3 4 TT T T 3 For the choice ⇠ =0,thesystemsimpliﬁesasoneofthesolutionstoEq.(g h + ' + E +17)becomesE + X =1,whiletheremainingtwoareE ij ⇠ ij ij ri j rj i rSAirj 2 3 ↵3 +2↵4 9 ↵ 12 ↵4 We expect 5 degrees of freedom: ± 3 X XSA = . (16) ± ⌘ ⇠=0 2 ↵q4 In this special setting, the solution X =1isuninteresting:thee ↵ective cosmological constant1 transverse-traceless from the mass term istensor zero and in the presenttransverse-traceless scenario, the background tensor becomes equivalent to a de Sitter universe driven by a (bare) cosmological constant 6 ⇤/(6 !). 1 transverse vector However,2 in transverse the presence vectors of matter ﬁelds- and no bare cosmological constant,- 1 scalar this solution= asymptotically1 transverse approachesvector a Minkowski background and is unstable [23]. 2 scalars 4 scalars 1 scalar

lapse and shift appear dRGT ghost-free constraint without time derivatives

1 d4xa3p⌦ (0)˙2 + + (0)v ˙ 2 + + h˙ 2 + h ~ 2 2k hij M 2 h2 L2 ⇠ ··· i ··· ij a2 ij r GW ij Z  ⇣ ⌘ Tensor modes get a mass m2 Scalar and vector kinetic terms vanish ⇠ g What to do?

Look for non-isotropic and/or non-homogeneous solutions:

• Homogeneous, non-isotropic solutions can be found (Bianchi type) De Felice, Gumrukcuoglu, Lin, Mukohyama (2012) Mourad, Steer (2013-14) • non-isotropy suppressed by graviton mass, only relevant at late times • Some of these solutions have stable ﬂuctuations

• Non-homogeneous, isotropic solutions can also be found Koyama, Niz, Tasinato (2011) Gratia, Hu, Wyman(2012)

Decoupling limit: self-accelerating solutions exist and appear well-behaved (See Douglas Spolyar’s talk)

does not match onto the full open self-accelerating solution, rather it matches onto a self-accelerating non- homogeneous/isotropic solutions

Domains of size ~1/m Full cosmology is diﬃcult to study What to do?

Other options: look for normal solutions in the next-simplest theories

• Extended massive gravity

• Quasi-dilaton theory

• Massive gravity coupled to galileons

• Bi-metric theory

• … 11

In the following, for clarity, we will use the deﬁnitions a˙ ↵ ↵˙ n H ,X , H¯ ,r , (64) ⌘ Na ⌘ a ⌘ n ↵ ⌘ NX

and we will omit the dependence of the functions mg, ↵3, ↵4 and V on the field value . (We also caution the reader that the above definitions of X and r are di↵erent than the ones in the quasi-dilaton theory, which we introduced in Section II.) 10 The equation of motion for the temporal Stückelberg field f is

1 S 1 d 1 2 of the kinetic matrix one can show that there are two ghost= modesm in2a the3 (X scalar1) 1 sector,(X provided1) ↵ + ( thatX 1)H2 ↵,determined 3M 2Nn f N dt g 3 3 4 p f=t  by the Friedmann equation, is an increasing function of ⇤. n o + a3 Hm¯ 2 X 3 X(2 + r)+(X 1)((1 + 2r)X 3)↵ (X 1)2(rX 1)↵ =0. Thus, the form of the dispersion relations for the perturbations dependsg on and receives order one correction 3 due 4 to UV e↵ects, but the presence of the ghost seems to be a robust feature. h i (65)

The Friedmann equation is obtained by varying the action with respect to N,

III.1 VARYINGS MASS1 THEORY˙ 2 =3H2 ⇤ + V m2 (XD’Amico,1) de3( XRham,2) Dubovsky, + (X Gabadadze,4)(X 1) ↵ +(X 1)2↵ =0, Mass-varyingM 2a3 N massive M 2 2 N gravity2 g 3 4 p f=t p ✓ ◆ Pirtskhalava, Tolley (2011) ⇥ ⇤ (66) We now turn to the varying mass theory, obtained by introducing a scalar into dRGTHuang, theoryPiao, Zhou and (2011 allowing) the and by taking a variation with respect to a, we obtain the dynamical equation which, after forming a linear combination graviton mass to be a function of this scalar. This theory was first considered in [10], and further studied in [29]. Allow graviton mass/couplingswith (to66 ),depend can be expressed on a new as scalar field: The action is ˙ 2 1 S 1 S 2 H ˙ 2 2 2 = + m (r 1) X 3 2X +(X 3)(X 1)↵3 +(X 1) ↵4 =0. M 3M 2Na2 a M 2a3 N N M 2 N 2 1g 4 p p p 2f=t p f=t p ⌫ S = d x g R[g] 2⇤ +2mg()[ 2 + ↵3() 3 + ↵4() 4] @µ@ ⇥ V () . (56) ⇤ (67) ( 2 L L L 2 ) Z h Finally, the equation of motion for is i We have further generalized to allow the dRGT parameters ↵ and ↵ to depend on the scalar . The part of the 1 S 1 3d ˙ 4 ˙ 12 = +3H + V 0 actionThere which providesare now mass 6 degrees to the gravitonof freedom: takesa3N 5 the graviton same formN +dt 1 as scalarN in Eq.(2),N but here the building block tensor is f=t ✓ ◆ K the same as in dRGT theory [2], i.e. in terms of which Eqs. (65)-(68) can be re-written, respectively, as 2 2 2 Mp mg (X 1) ↵30 (4 X(1 + 3 r)) + ↵40 (X 1) (rX 1) µ ( µ µ 1 There now exists flat FRW self-accelerating⇢˙m ⌫ = ⌫ solutions:g g¯ . (57) +3KH (⇢m +pm)= Q,⌫ 2 mg0 3(X(r + 1) 2) N ⇣ ⌘ + (X(1 + 3r) 4)↵3 +(X 1)(rX 1)↵4 =0, (68) p mg " X 1 #) One of our goals is to compare our results2 with the1 analogous analysis of perturbations in the dRGT theory. Since 3 H = ⇤ + 2 (⇢ + ⇢m) , the original theory does not allow flatwhere solutions a prime for denotesM ap Minkowski di↵erentiation reference with respect metric, to it. is necessary to adopt a more It is convenient to cast these equations into a more familiar perfect fluid-like form, by defining the following general form. We therefore extend theH˙ fiducial1 metric to be an arbitrary spatially flat homogeneous and isotropic quantities metric, = Eff2ective[(⇢ +darkp)+( energy:⇢m + pm)] , N 2 Mp 2 2 2 1 ⇢m M m (X 1) 3(X 2) + (X 4)(X 1)↵3 +(X 1) ↵4 , X 0 ⇢˙ µ ⌫ p g 2 2 2 i j g¯µ⌫ dx dx⌘ = n(t)dt +↵(t)ijdx dx. , ⌘ a !(58) +3H (⇢2 +2 p)=Q. 2 2 (70) 2 pm M m 6 3X(⇥r + 2) + X (1 + 2r) (X 1)(4 X(3r + 2) +⇤ rX )↵3 (X 1) (rX 1)↵4 , N ⌘ p g 2 2 ⇥˙ 2 ⇤ From here on, we will use these forms of theQ cosmologicalMp mg ( backgroundX 1) ↵30 (4 equations.X(1Gumrukcuoglu, + 3 r)) + ↵40 (KH,X Lin,1) Mukohyama, (rX 1) For some of these solutions,A. perturbations Cosmological⌘ Nare Background all stable.( Equations Trodden (2013)

2 m0 3(X(r + 1) 2) + g (X(1 + 3r) 4)↵ +(X 1)(rX 1)↵ , We first study the cosmological background equations (see [29–34] for morem on backgroundX 1 cosmological solutions3 4 B. Tensor perturbations g " #) to mass-varying massive gravity.) For the physical˙ 2 background metric, we adopt the flat FRW ansatz ⇢ + V, ⌘ 2 N 2 We proceed with the perturbation theoryµ in⌫ the˙ 2 same2 manner2 as2 describedi j in Section II C. We start with tensor gµ⌫ dxp dx = N(V,t) dt + a(t) ijdx dx . (59) (69) perturbations around the background (59), ⌘ 2 N2 To write the mini-superspace action, we introduce time reparametrization via a Stückelberg field f(t), by replacing 2 TT the fiducial metric with gij = a hij , (71)

i TT TT i µ ⌫ 2 2 2 2 i j where @ hij = hi = 0. g¯ dx dx = n(f(t)) f 0(t) dt + ↵(f(t)) dx dx . (60) µ⌫ ij The action for the tensor perturbations reads Unitary gauge corresponds to the choice f(t)=t. M 2 1 k2 The mini-superspace actionS = isp d3ka3 Ndt h˙ TT 2 + M 2 hTT 2 , (72) 8 N 2 | ij | a2 GW | ij | 2 ✓ ✓ ◆ ◆ S 2 Za˙ a 2 3 1 1 2 2 = dt 3M + m NF fn˙ (f)G + a N ˙ NV() NM ⇤ , (61) where the mass termV is P N g 2 P Z ⇢  ⇣ ⌘  2 where V is the comoving2 volume(r and1) X 2 ⇢m 1 2 X ⇢m + pm MGW = 2 mg (X 1) 2 + 2 . (73) (X 1) ↵ Mp r 1 X ↵1 Mp F a(a ↵(f))(2a ↵(f)) + 3 (a ↵(f))2✓(4a ↵(f)) + 4◆(a ↵(f))3, (62) ⌘ 3 3 To obtain this, we have used the background acceleration↵ equation (67). In the case of self accelerating solutions G a2(a ↵(f)) + ↵ a(a ↵(f))2 + 4 (a ↵(f))3. (63) [11, 15] of the standard dRGT⌘ theory, i.e. when3 ˙ = 0 and ⇢ 3 = p = const., the last term in (73) drops out of the m m calculation, and the mass reduces to the one found in [15]. We also stress that MGW here is di↵erent than the one deﬁned for the quasi-dilaton theory in Section II.

C. Vector perturbations

Next we consider transverse vector perturbations to the metric

a2 g = NaBT , g = (@ ET + @ ET ) , (74) 0i i ij 2 i j j i

i T i T where @ Bi = @ Ei = 0. The ﬁeld Bi appears without time derivatives and may be eliminated as an auxiliary ﬁeld using its own equation of motion,

˙ T T a Ei Bi = . (75) 2 a2 N 2 1 k2 M 2 (r2 1) (⇢m + pm) p h i Once this solution is inserted back into the action, what remains is an action for one dynamical vector

M 2 k2 M 2 S = p d3ka3 Ndt TV E˙ T 2 GW ET 2 , (76) 8 N 2 | i | 2 | i | Z ✓ ◆ Galileons coupled to massive gravity

There is a way to couple galileons to massive gravity: Gabadadze, KH, Khoury, Pirtskhalava, Trodden (2012)

M P p gR + M 2 m2p g e ( g1g¯)+ M 2m2p g¯ + L ⇠ 2 P n ··· n X p ⇥ ⇤ ↵ g¯µ⌫ = @µX @⌫ X ⌘↵ + @µ⇡@⌫ ⇡

X↵ = x↵ @↵ Galileon shift symmetries are preserved, and no ghostly degrees of freedom are introduced

Cosmological solutions are similar to those of pure massive gravity: Andrews, KH, Stokes, Trodden (2013)

• No ﬂat FRW solutions

• There is an open FRW self-accelerating solution

• Kinetic terms for vector and scalar ﬂuctuations vanish on the self-accelerating solution 2

the details of the matter propagating in the loops. If we wish to restore the kinetic terms at the classical level, one avenue is to move away from homogeneous and isotropic cosmologies [21, 22]. Another is to keep homogeneity and isotropy, but change the theory by adding more degrees of freedom. In this paper, we take the latter approach. We study cosmological perturbations in two extensions of dRGT theory: quasi-dilaton massive gravity, and mass-varying massive gravity. Massive gravity admits an extension to a theory with a global scale symmetry through the inclusion of a specific scalar field, dubbed the quasi dilaton [23]. We examine self-accelerating background solutions to the quasi-dilaton model, and consider the behavior of cosmological perturbations (see also [24]). We find the conditions under which the backgrounds are free of ghosts. We find that there is always a ghost-like instability in the scalar sector, for fluctuations of physical wavelength shorter than the Hubble radius. The other extension we consider is mass-varying massive gravity, the theory obtained by promoting2 the mass to a

the detailsscalar of field the matter [10, 29 propagating]. We study in the the behavior loops. If we of cosmological wish to restore perturbations the kinetic terms in this at the model, classical and level, find one the conditions under avenuewhich is to move the backgrounds away from homogeneous are free of and ghosts. isotropic cosmologies [21, 22]. Another is to keep homogeneity and isotropy, but change the theory by adding more degrees of freedom. In this paper, we take the latter approach. We study cosmological perturbations in two extensions of dRGT theory: quasi-dilaton massive gravity, and mass-varying massive gravity. II. QUASI-DILATON THEORY Massive gravity admits an extension to a theory with a global scale symmetry through the inclusion of a specific scalar field,We dubbed start with the thequasi quasi-dilaton dilaton [23]. Wetheory, examine introduced self-accelerating in [23]. background The action solutions governs toa dynamical the quasi-dilaton metric gµ⌫ and a scalar model,field and consider,1 the behavior of cosmological perturbations (see also [24]). We find the conditions under which the backgrounds are free of ghosts. We find that there is always a ghost-like instability in the scalar sector, for 2 Mp ! fluctuations ofS = physical wavelengthd4x p shorterg R[g than] 2 the⇤ +2 Hubblem2 radius.( + ↵ + ↵ ) @ @⌫ +2m2 ⇠ p ge¯ 4 /Mp . (1) 2 g L2 3 L3 4 L4 M 2 µ g The other extensionZ we consider⇢ is mass-varying massive gravity, the theory obtained byp promoting the mass to a scalar field [10, 29]. We study the behavior of cosmological perturbations in this model, and find the conditions under The part of the action which provides the mass to the graviton is which the backgrounds are free of ghosts.

1 2 2 Quasi-dilaton2 = [ ] massive, gravity L II.2 QUASI-DILATONK K THEORY 1 ⇣ 3 ⇥ ⇤⌘ D’Amico, Gabadadze, Hui, Pirtskhalava (2013) = [ ] 3[ ] 2 +2 3 , L3 6 K K K K Add aWe new start scalar with thein a quasi-dilaton manner fi theory,xed by introduced scaling symmetry: in [23]. The action governs a dynamical metric gµ⌫ and a scalar 1⇣ 4 ⇥2 ⇤ ⇥ ⇤⌘ 2 field ,1 = [ ] 6[ ] 2 +3 2 +8[ ] 3 6 4 , (2) L4 24 K K K K K K K 2 ⇣ ⌘ Mp ⇥ ⇤ ⇥! ⇤ ⇥ ⇤ ⇥ ⇤ Swhere= squared4x bracketsp g R denote[g] 2 ⇤ a+2 trace.m2 ( While+ ↵ these+ ↵ expressions) are@ @ similar⌫ +2 inm form2 ⇠ p toge¯ the4 /M dRGTp . (1) theory, in the case of 2 g L2 3 L3 4 L4 M 2 µ g the quasi-dilatonZ ⇢ theory, the building block tensor is definedp as K The part of the action which provides the mass to the graviton is µ µ = µ e/Mp g 1g¯ , (3) 1 ⌫ ⌫ = [ ]2 2 , K ⌫ L2 2 K K ⇣p ⌘ whereg ¯ is a non-dynamical fiducial metric. The theory is invariant under a global dilation of the space-time µ⌫ 1 ⇣ 3 ⇥ ⇤⌘ There is a global symmetry given= by[ ] scaling3[ ] coordinates2 +2 3 and, shifting the scalar: coordinates, accompaniedL3 6 byK a shift K of K. ThisK symmetry rules out a non-trivial potential for . 1⇣ 4 ⇥2 ⇤2 ⇥ ⇤⌘2 2 3 4 Throughout the analysis,4 = µ we[ ] choose6[µ] the fiducial+3 metric+8[ to] be Minkowski,6 , (2) L 24x K e xK , K K MPKK K ⇣ ⌘ ! ⇥ ⇤ !µ⇥ ⇤⌫ 2⇥ ⇤ ⇥ i ⇤ j where square brackets denote a trace. While these expressionsg¯µ⌫ dx dx are= similardt in+ formijdx to thedx dRGT. theory, in the case of (4) the quasi-dilaton theory, the building block tensor is defined as K • There are flat FRW solutions with self-acceleration. µ µ µ /Mp 1 ⌫ = ⌫ e g g¯ , (3) Gumrukcuoglu, KH, Lin, Mukohyama, TroddenK (2013A.) Background equations⌫ of motion ⇣ ⌘ Gannouji, Hossain, Sami, Saridakis (2013) p whereg ¯µ⌫ is a non-dynamical fiducial metric. The theory is invariant under a global dilation of the space-time coordinates,For accompanied the physical by background a shift of . metric, This symmetry we adopt rules the out flat a non-trivial FRW ansatz potential for . • Scalar perturbations around these solutions are unstable: Throughout the analysis, we choose the fiducial metric to be Minkowski, g dxµ dx⌫ = N(t)2 dt2 + a(t)2 dxi dxj . (5) Gumrukcuoglu, KH, Lin, Mukohyama, Trodden (2013) µ⌫ ij g¯ dxµ dx⌫ = dt2 + dxi dxj . (4) D’Amico, Gabadadze, Hui, Pirtskhalava (2013) µ⌫ ij

• Decoupling1 We thank limit the solutions authors of now [23]forpointingouttheexistenceofthe haveA. Backgroundstable, sub-luminal equations ofperturbations motion⇠ term in the most(including general quasi-dilatonthe vector) action. Gabadadze, Kimura, Pirtskhalava (2014) For the physical background metric, we adopt the ﬂat FRW ansatz

g dxµ dx⌫ = N(t)2 dt2 + a(t)2 dxi dxj . (5) µ⌫ ij

1 We thank the authors of [23]forpointingouttheexistenceofthe⇠ term in the most general quasi-dilaton action. 2

1 2 2 2 = [ ] , ExtendedL II.2 QUASI-DILATONK Quasi-dilaton K THEORY De Felice, Mukohyama (2013) 1 ⇣ 3 ⇥ ⇤⌘ = [ ] 3[ ] 2 +2 3 , L3 6 K K K K CombineWe startgalileon with theand quasi-dilaton quasi-dilaton theory, construction: introduced in [23]. The action governs a dynamical metric gµ⌫ and a scalar 1⇣ 4 ⇥2 ⇤ ⇥ ⇤⌘ 2 field ,1 = [ ] 6[ ] 2 +3 2 +8[ ] 3 6 4 , (2) L4 24 K K K K K K K 2 ⇣ ⌘ Mp ⇥ ⇤ ⇥! ⇤ ⇥ ⇤ ⇥ ⇤ Swhere= squared4x bracketsp g R denote[g] 2 ⇤ a+2 trace.m2 ( While+ ↵ these+ ↵ expressions) are@ @ similar⌫ +2 inm form2 ⇠ p toge¯ the4 /M dRGTp . (1) theory, in the case of 2 g L2 3 L3 4 L4 M 2 µ g the quasi-dilatonZ ⇢ theory, the building block tensor is definedp as K The part of the action which provides the mass to the graviton is µ µ µ /Mp 1 ⌫ = ⌫ e g g¯ , (3) 1 2 2 K ⌫ 2 = [ ] , L 2 K K ⇣p 2 ⌘ whereg ¯ is a non-dynamical fiducial metric. The theory is invariantMP under a global dilation of the space-time µ⌫ 1 ⇣ 3 ⇥ ⇤⌘ g¯µ⌫ g¯µ⌫ + ce @µ@⌫ = [ ] 3[ ] 2 +2 3 , coordinates, accompaniedL3 6 byK a shift K of K. ThisK! symmetry rules out a non-trivial potential for . 1⇣ 4 ⇥2 ⇤ ⇥ ⇤⌘ 2 Throughout the analysis,= we[ ] choose6[ ] the2 fiducial+3 2 metric+8[ to] be3 Minkowski,6 4 , (2) L4 24 K K K K K K K ⇣ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤⌘ where square brackets denote a trace. While these expressionsµ are⌫ similar2 in form toi thej dRGT theory, in the case of • Still flat FRW solutions with self-acceleration.g¯µ⌫ dx dx = dt + ijdx dx . (4) the quasi-dilaton theory, the building block tensor is defined as K Gabadadze, Kimura, Pirtskhalava (2014) µ µ µ /Mp 1 Kahniashvili, Kar, Lavrelashvili, Agarwal, ⌫ = ⌫ e g g¯ , (3) K A. Background equations⌫ of motionHeisenberg, Kosowsky (2015) • All perturbations can now be made stable ⇣p ⌘ Heisenberg, (2015) whereg ¯µ⌫ is a non-dynamical fiducial metric. The theory is invariant under a global dilation of the space-time Motohashi, Hu (2015) coordinates,For accompanied the physical by background a shift of . metric, This symmetry we adopt rules the out flat a non-trivial FRW ansatz potential for . Throughout the analysis, we choose the fiducial metric to be Minkowski, µ ⌫ 2 2 2 i j • First example of a stable self-acceleratinggµ solution⌫ dx dx away= N from(t) dt decoupling+ a(t) ij limitdx dx . (5) g¯ dxµ dx⌫ = dt2 + dxi dxj . (4) µ⌫ ij

1 We thank the authors of [23]forpointingouttheexistenceoftheA. Background equations of motion⇠ term in the most general quasi-dilaton action.

For the physical background metric, we adopt the ﬂat FRW ansatz

g dxµ dx⌫ = N(t)2 dt2 + a(t)2 dxi dxj . (5) µ⌫ ij

1 We thank the authors of [23]forpointingouttheexistenceofthe⇠ term in the most general quasi-dilaton action. the linear cosmological perturbation equations. Here we derive and study the (equivalent) set of equations in general terms and, more importantly, proceed to make the link to observable quantities. Recently, Könnig and Amendola explored large-scale structures in aone-parameterbigravitymodel[59]. Our results are in complete agreement with theirs. the linear cosmological perturbation equations. HereCosmological we derive and study perturbations the (equivalent) in bimetric gravity have also been studied in Refs. [30, 60]. set of equations in general terms and, more importantly,This pro articleceed to is make organized the link as follows. to In section 2 we review the bimetric massive gravity observable quantities. Recently, Könnig and Amendolatheory explored and itslarge-scale cosmological structures self-accelerating in background solutions. We then present the aone-parameterbigravitymodel[59]. Our results arecosmological in complete perturbation agreement with equations theirs. in section 3 and focus on the formation of structures in Cosmological perturbations in bimetric gravity havethe also matter-dominated been studied in Refs. era. [30 The, 60]. observationally relevant part of the matter power spectrum This article is organized as follows. In section 2 weis review at the the subhorizon bimetric massive scales, and gravity in this limit we arrive at a convenient closed-form evolution theory and its cosmological self-accelerating backgroundequationsolutions. in section We then4 that present captures the the modifications to the general relativistic growth rate cosmological perturbation equations in section 3 and focus on the formation of structures in of linear structures, as well as the leading order scale-dependence which modifies the shape the matter-dominated era. The observationally relevant part of the matter power spectrum of the spectrum at near-horizon scales. The full perturbation equations are detailed in is at the subhorizon scales, and in this limit we arrive at a convenient closed-form evolution equation in section 4 that captures the modifications toappendix the generalA,andthegeneralformofthecoe relativistic growth rate fficients in the closed-form equation is given of linear structures, as well as the leading order scale-depin appendixendence whichB.Theresultsareanalyzednumericallyinsection modifies the shape 4,wherewediscussthe of the spectrum at near-horizon scales. The full perturbatigeneralon features equations of the are models detailed and in confront them with the observational data. We conclude appendix A,andthegeneralformofthecoefficientsin in section the closed-form5. equation is given in appendix B.Theresultsareanalyzednumericallyinsection4,wherewediscussthe general features of the models and confront themGhost-free with2Theoryandcosmologicalframework the observational bi-metric data. We conclude gravity in section 5. Hassan, Rosen (2011) Allow the fiducial metric to become2.1 non-minkowski Massive bigravity, or allow the fiducial 2Theoryandcosmologicalframework metric to become dynamical: The ghost-free theory of massive bigravity is defined by the action [22] 2.1 Massive bigravity M 2 M 2 S = g d4x det gR(g) f d4x det fR(f) The ghost-free theory of massive bigravity is defined by the action [22] − 2 − − 2 − ! ! " 4 " M 2 M 2 S = g d4x det gR(g) f d4x det fR(f) + m2M 2 d4x det g β e g−1f − 2 − − 2 − g − n n ! ! n=0 " 4 " ! " # $" % + m2M 2 d4x det g β e g−1f 4 g − n n + d x det g m (g, Φ) , (2.1) n=0 − L ! " # $" % ! 4 " 7 degrees+ ofd xfreedom:det g 2m (forg, Φ massless) , where gravitong and + f5 forare massive spin-2(2.1) tensorgraviton fields with metric properties. They are dynamical, − L µν µν ! " and they define the gravitational sector of the theory. For conciseness we will frequently where gµν and fµν are spin-2 tensor fields with metricrefer properties. to the metrics They as areg dynandamical,f,suppressingtheirindices.Theirkinetictermsaretheusual and they define the gravitational sector of the theory. For conciseness we will frequently Einstein-Hilbert terms, and the fields interact through a potential comprising five terms refer to the metrics as g and f,suppressingtheirindices.Theirkinetictermsaretheusual which are particular functions of the fields g and f (but not their derivatives): Einstein-Hilbert terms, and the fields interact through a potential comprising five terms FRW solutions, come in two branches: von Strauss, Schmidt-May, Enander, Mortsell, Hassan (2011) which are particular functions of the fields g and f (but not their derivatives):e ( ) 1, 0 ≡ Fasiello, Tolley (2012, 2013) Akrami, Koivisto, Sandstad (2013) e0 ( ) 1, e1 ( ) [ ] , ≡ ≡ Solomon, Akrami, Koivisto (2014) • Self-acceleratinge1 ( ) [ ]branch, again has vanishing kinetic terms 1 2 2 ≡ e2 ( ) [ ] De Felice, Gumrukcuoglu,, Mukohyama, Tanahashi, Tanaka (2014) 1 2 2 ≡ 2 − e2 ( ) [ ] , 1 $ 3 & '% 2 3 ≡ 2 − e ( ) [ ] Langlois,3[ Naruko] (2012)+2 , 1 $ % 3 3 & ' 2 3 ≡ 6 Langlois,− Mukohyama, Namba, Naruko (2012) e3 ( ) [ ] 3[ ] +2 , $ % • Normal branch≡ 6 can have− self-accelerating behavior e4 ( ) det ( ) . & ' & ' (2.2) e ( ) det$ ( ) . & ' & '% ≡(2.2) 4 ≡

–3– –3– Cosmology of ghost-free bi-metric gravity See Reviews & refs therein: Bull et. al. arXiv:1512.05356 Schmidt-May, von Strauss:1512.00021 • Self-accelerating solutions can be found, compatible w/ observations at the background level

Inﬁnite branch (ratio of scale factors → inﬁnity at early times) Perturbations: tend to have ghostly instabilities Normal branch has sub-branches 8 > Finite branch (ratio of scale factors → constant at early times) <> Perturbations: no ghosts, everything ok except scalar perturbation (tends to have gradient instabilities at early times) > • Finite branch instabilities:> can be pushed to unobservably early times, beyond the regime of validity of the theory, but then cosmology becomes GR-like Akrami, Hassan, Konnig, Schmidt-May, Solomon (2015)

General trend:

if there is no GR-like limit theory gets ruled out

if there is a GR-like limit gets pushed towards this limit no observable diﬀerence

May be a solution to dark energy problem, but we won’t know if it’s right Summary

• Theories with massive gravitons have a chance of explaining acceleration in a technically natural way, but do not yet solve the old CC problem.

• For pure massive gravity, one must either work with complicated inhomogeneous/anisotropic solutions, or work in certain limits (decoupling limit)

• Alternatively, work with standard solutions in extended theories (extra scalars, bi-metric theories)

• UV completion will eventually be needed.