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arXiv:1301.4993v3 [astro-ph.CO] 16 Dec 2013 h olaeDsrmd [ mode Boulware-Deser the hc a t w yaisadcnrl [ roll can Φ, and field, realize scalar dynamics to a own how to its of has idea the which One promote to time. is in this change must mass aster,adti osrit hl tn ogrfor- longer cannot self-inflation no that it variable implies solutions, while FRW the flat constraint, bids in this slicings). constraint and theory, analogous other mass an in appears found There are has [ solutions slicing theory flat self-accelerating the dRGT which in identity, Pure evolution Bianchi FRW standard the forbids from model. stemming this constraint, in a diffi- achieve is to cult massive of implementation inflation-like consistent value small a measurements. day attains present ends mass with self-inflation VEV, graviton smaller the a to and rolls a Φ Then, the times, has and constant. late massive, Hubble Φ at large very times a is with graviton early self-inflates the universe the at that of that so mass VEV, imagine the large can sets We then Φ graviton. of (VEV) value expectation larger much [ be today cannot scale Hubble mass know the we graviton than Yet, current inflation. the during scale that be Hubble to the have the order would mass of use graviton inflation To the for “self-inflation”), gravity universe. (i.e. massive early of the solution in self-accelerating inflation drive might to universe late-time the acceleration. in cosmic interest for great account of Since is graviton. [ solution the natural a technically of is mass graviton light the a order having of is factor Hubble admits in- [ gravity of massive solutions dRGT questions self-accelerating Pure addressing cosmology. of in possibility terest the for lowing [ ered nitrcigter famsiegaio,fe of free graviton, massive a of theory interacting An hs o efiflto ob osbe h graviton the possible, be to self-inflation for Thus, nti etr ewl e ht npatc,sc an such practice, in that, see will we letter, this In phenomenon similar a whether is question natural A 2 , 3 NRDCINADOUTLINE AND INTRODUCTION tedG hoy e [ see theory, dRGT (the ] fsltoswihd o peri dRGT. in appear not do identical so which are FRW solutions that Open of solutions exist. self-accelerating not of galileons do branch co of solutions a FRW in coupling flat covariant gravity, form the dRGT brake” pure is “big extension exis the second can w of The dr solutions Furthermore, singularities such be time. while can useful future that, cosmologically self-inflation a find whether for we ask sustained and We theory, field. this of scalar mass a graviton of fixed value the where gravity, massive mass variable esuytebcgon omlg ftoetnin fdG m dRGT of extensions two of cosmology background the study We etrfrPril omlg,Dprmn fPyisadAs and Physics of Department Cosmology, Particle for Center nvriyo enyvna hldlha enyvna1 Pennsylvania Philadelphia, Pennsylvania, of University 5 1 – ,hsrcnl endiscov- been recently has ], 11 14 utHnebclr ae tks n akTrodden Mark and Stokes, James Hinterbichler, Kurt omlge fetne asv gravity massive extended of Cosmologies ,i hc h eSitter de the which in ], ]. 4 o eiw,al- review), a for ] 9 12 , , 17 Dtd ue1,2021) 15, June (Dated: 13 .The ]. 9 ,such ], (the ] oe oasaa edΦ, pro- field is scalar squared a mass to graviton moted the which in theory dRGT c osttegaio as hc sfie.S eare we So equations.) exploring fixed. cosmological just is are basic but the which here, mass, self-inflation in graviton interested the not set to act rnhcnit fnvlsltoswihaentfudin found not gravity. are massive which pure second solutions novel The of self- theory. consists dRGT the branch pure of to of consists identical solutions branch are accelerating first that and The solutions exist branches. self-accelerating can two solutions in For however, come they solutions. ansatz, dRGT FRW FRW pure flat open in an forbids as constraint that, neg- this find and We theory, zero of curvature. case spatial the ative in We solutions dRGT. possible pure leads the in scalar than discuss the constraint of complicated more presence a the to that find and theory, this in Ulk h aibems theory, mass dimen- variable spatial the additional (Unlike an freedom sion. in of bending degree brane scalar new describes a has theory This uaiiso h bgbae type. brake” “big sin- curvature the future of exhibit gularities may cosmological theory non-inflationary this to that solutions time. show of we length addition, relevant cosmologically In a for sustained be aienetnino asv rvt nrdcdi [ in introduced gravity covariant massive the of consider extension we galileon first), the from independently esatwt aibems asv rvt.Ti is This gravity. massive mass variable with start We edrv h akrudcsooia qain for equations cosmological background the derive We ntescn afo hslte wihcnb read be can (which letter this of half second the In otoeo RT n e eodbranch second new a and dRGT, of those to AIBEMS ASV GRAVITY MASSIVE MASS VARIABLE eosrt htteegnrlyexist generally there that demonstrate e omsiegaiy efidta,a in as that, find We gravity. massive to fdG srpae yteexpectation the by replaced is dRGT of mlgclsltost hs theories. these to solutions smological o hr eid hycno be cannot they period, short a for t vnb h efaclrtdbranch self-accelerated the by iven uin oeit–te oss of consist they – exist do lutions S = siegaiy h rtis first The gravity. assive 14 USA 9104, S EH tronomy, + S mass + S Φ , π eede not does here π which , 15 (1) ]. 2 where Inserting (6) and (9) into the , we obtain the mini-superspace action 1 2 4 SEH = M d x √ g R , (2) 2 P Z − 2 2 4 2 a˙ a Smass = M d x √ g Φ ( 2 + α3 3 + α4 4) , (3) SEH =3M dt , (10) P Z − L L L P Z − N  1 4 2 2 ˙ SΦ = d x √ g g(Φ)(∂Φ) + V (Φ) . (4) Smass =3MP dt Φ NF (a) fG(a) , (11) − Z − 2  Z h − i 3 1 −1 2 Here α3, α4 are the two free parameters of dRGT theory. SΦ = dta N g(Φ)Φ˙ NV (Φ) . (12) We have allowed for an arbitrary kinetic function g(Φ) Z 2 −  and potential V (Φ), so that there is no loss of generality where in the scalar sector. The mass term consists of the - free combinations [3], F (a)= a(a 1)(2a 1) − − 1 α3 2 α4 3 = [ ]2 [ 2] , + (a 1) (4a 1)+ (a 1) , (13) L2 2 K − K 3 − − 3 −  2 2 α4 3 1 3 2 3 G(a)= a (a 1)+ α3a(a 1) + (a 1) . (14) 3 = [ ] 3[ ][ ] + 2[ ] , − − 3 − L 6 K − K K K  1 4 2 2 2 2 3 4 This mini-superspace action is invariant under time 4 = [ ] 6[ ] [ ] + 3[ ] + 8[ ][ ] 6[ ] , L 24 K − K K K K K − K reparametrizations, under which f transforms like a  (5) scalar. There are four equations of motion, obtained by vary- where Kµ = δµ √gµση , η is the non-dynamical ν ν − σν µν ing with respect to F,N, Φ and a. As in GR, the Noether fiducial metric which we have taken to be Minkowski, identity for time reparametrization invariance tells us and the square brackets are traces. To work in the gauge that the acceleration equation obtained by varying with invariant formalism, we introduce four St¨uckelberg fields respect to a is a consequence of the other equations, so φa through the replacement η ∂ φa∂ φbη . µν → µ µ ab we may ignore it. After deriving the equations, we will Variable mass massive gravity was first considered in fix the gauge N = 1 (this cannot be done directly in the [9], and further studied in [17–20] (see also [16] for a more action without losing equations). symmetric scalar extension of dRGT). dRGT gravity has Varying with respect to f we obtain the constraint been demonstrated to be ghost-free through a variety of pointed out in [18], different approaches [21–25], and the introduction of the scalar field does not introduce any new Boulware-Deser Φ= C , (15) like ghost degrees of freedom into the system [17]. G(a) For cosmological applications we take a Friedmann, Robertson-Walker (FRW) ansatz for the metric, so that where is an arbitrary integration constant. (Note that the analogousC equation in the fixed mass theory implies ds2 = N 2(t)dt2 + a2(t)Ω dxidxj , (6) − ij that a = constant, so there are no evolving flat FRW solutions in that case [9].) Varying with respect to N, where we obtain the Friedmann equation, κ i j Ωij = δij + 2 x x (7) 1 κr 2 2 ΦF (a) 1 ˙ 2 − 3MP H + 3 = g(Φ)Φ + V, (16) is the line element for a maximally symmetric 3-space of  a  2 2 2 2 2 curvature κ and r = x + y + z . We also take the and varying with respect to Φ we obtain the scalar field assumptions of homogeneity and isotropy for the scalar equation field, ¨ ˙ 1 ′ ˙ 2 ′ Φ=Φ(t). (8) g(Φ) Φ+3HΦ + g (Φ)Φ + V (Φ) h i 2 Consider first the case of flat Euclidean sections (κ = 2 F (a) ˙G(a) =3MP 3 f 3 . (17) 0). We work in the gauge invariant formulation, and the  a − a  Stueckelberg degrees of freedom take the ansatz [9, 10]. Rather than solving the coupled second-order Einstein- φi = xi, φ0 = f(t), (9) scalar equations of motion, one can instead reduce the system to a single first-order Friedmann equation. The where f(t), like a(t), is a monotonically increasing func- relation (15) can be used to eliminate Φ and its first tion of t. derivative from (16), which then becomes a first-order 3 differential equation in a which determines the scale fac- self-inflation should end, and the graviton mass should tor, become small at late times. Thus, assume we have an inflationary solution a C 2 F (a) Ht ∼ V 3M 3 e , with H constant. The scale factor is growing 2 G(a) − P C a G(a) ∼ H =   ′ 2 2 . (18) exponentially, so we Taylor expand the entire right hand 2 1 2 C G (a) a 3MP 2 g G(a) G(a)4 side of (18) for large a, as − C   V ′(0) Once we have solved for the scale factor, the scalar Φ 2 (6+4α3 + α4) 2 V (0) MP − 1 1 is determined from (15) and the Stueckelberg field f is H = 2 +   3 + 4 . 3M C 3+3α3 + α4 a O a  1 P determined by solving (17) .   (20) We see that the dependence on all of the massive grav- Singularities ity modifications redshifts away exponentially, at least as fast as a−3, and we are left with inflation driven only by the value of the potential at Φ = 0. (In particular, con- One feature of this model that has not been noticed tributions sensitive to the function g only start to enter previously is that it allows for the possibility of curvature at (1/a7).) singularities at finite values of a. These can happen when SaidO another way, we only have self-inflation if the the evolution attempts to pass through values of a for φF (a) quantity 3 in (16) is approximately constant when which the denominator of the right hand side of (18) a a eHt. But the constraint equation (15) makes this im- goes to zero. ∼ 1 3 possible: we see from (15) that Φ 3 , since G(a) a When this happens a is finite, but the Hubble param- a 3 ∼ ΦF (a) ∼ eter, and hencea ˙, is blowing up. The for large a. Since F (a) a , the quantity a3 behaves like Φ a−3, so it goes∼ to zero exponentially fast and is also blowing up, so this is a genuine curvature singu- ∼ ∼ larity; a “big brake” where the universe comes to some we cannot sustain self-inflation. finite scale factor and gets stuck [26, 27]. Similar types Having encountered an obstacle to the possibility of of singularities also occur in DGP [28]. self-inflation in the flat slicing, we now investigate the For example, Taylor expanding the denominator of possibility in the open slicing (κ< 0). Following [10] we (18) for large a we obtain a critical value of the scale take the Stueckelberg ansatz to be factor at which the Hubble parameter diverges, φ0 = f(t) 1 κr2, φi = √ κf(t)xi. (21) p − − g(0) 1/6 1/3 √ The mini-superspace action then becomes acr = 3 2 C . (19) 2MP  3+3α3 + α4  2 2 a˙ a SEH =3M dt + κNa , (22) P Z − N  Self-Inflation 2 ˙ Smass =3MP dt Φ NF (a,f) fG(a,f) , (23) Z h − i We now consider the possibility that the graviton has 3 1 −1 2 SΦ = dta N g(Φ)Φ˙ NV (Φ) , (24) a large mass in the early universe, through some natu- Z 2 −  ral displacement (and resulting VEV) of the scalar field where from its true minimum near zero. We seek dynamics such that the scalar field slowly rolls down its potential, F (a,f)= a(a √ κf)(2a √ κf) during which time the graviton remains massive, result- α − − − − α + 3 (a √ κf)2(4a √ κf)+ 4 (a √ κf)3, ing in a large self-acceleration of the universe. This self- 3 − − − − 3 − − acceleration comes from the second term on the left-hand (25) side of the Friedmann equation (16). For this to be true G(a,f)= a2(a √ κf)+ α a(a √ κf)2 self-acceleration this term should be much larger than 3 α − − − − both the scalar kinetic energy and potential on the right + 4 (a √ κf)3. (26) hand side, so that the acceleration is primarily driven 3 − − by the modification to gravity and not by the scalar Again, we have time reparametrization invariance so we field. After many e-folds, Φ should roll towards zero, can ignore the acceleration equation, and we will fix the gauge N = 1 after deriving the equations of motion. The constraint equation arising from varying with respect to f is 1 Note that in general the Stueckelberg field cannot be chosen ar- bitrarily as in [18] but is non-trivially constrained by the choice ∂G(a,f) (˙a √ κ)Φ + G(a,f)Φ=0˙ . (27) of mass term, or in this case, kinetic function g(Φ). − − ∂a 4

The Friedmann equation obtained by varying with re- flat bulk, with embedding coordinates XA(x), coupled spect to N is to the induced metric through dRGT interactions. The worldvolume theory is 2 2 κ ΦF (a,f) 1 ˙2 3MP H + 2 + 3 = g(Φ)φ + V (Φ), (28)  a a  2 S[g, g¯]= SR[g]+ Smass[g, g¯]+ Sπ[¯g], (31) and the equation of motion for Φ is where SR and Smass are the usual Einstein-Hilbert and dRGT mass terms except that now 1 g(Φ) Φ+3¨ HΦ˙ + g′(Φ)Φ˙ 2 + V ′(Φ) A B h i 2 g¯µν = ∂µX ∂ν X ηAB (32) F (a,f) G(a,f) =3M 2 f˙ . P  a3 − a3  is the pull-back of the 5D Minkowski metric to the 4D (29) brane. The action Sπ consists of ghost-free scalars con- structed fromg ¯µν and the extrinsic curvature [29]. The In order to obtain inflation driven by the graviton only term which will be relevant for us is the DBI term mass, the term ΦF (a,f)/a3 in the Friedmann equation Ht (28) must be approximately constant when a e . Re- 4 4 Sπ = Λ d x√ g,¯ (33) arranging the constraint equation (27) to isolate∼ Φ gives − Z −

˙ ∂G(a,f) where Λ is some characteristic mass scale. In the flat slic- Φ √ κ a ∂a = H − . (30) ing, the other possible higher order terms in Sπ all vanish Φ −  − a  G(a,f) on the cosmological backgrounds we consider, so we may Now there are three possibilities: the first is that exclude them without loss of generality. They do con- a(t) eHt grows faster than f(t). In this case, the right- tribute in the curved slicing, however the final constraint hand∼ side of (30) approaches a constant at late times, equation (48) and the conclusions drawn from it are not Φ˙ /Φ 3H + (1/a), which tells us that Φ decreases modified in their presence, so we ignore them in what exponentially∼ − atO late times, Φ(t) e−3Ht. This, in turn, follows. The embedding functions are scalar fields; the implies that the self-acceleration∼ quantity ΦF (a,f)/a3 first four become the Stueckelberg fields, and the fifth be- in (28) decreases exponentially like Φ, since F (a,f)/a3 comes an extra physical scalar, the brane bending mode approaches a constant. So once again, we cannot sus- π, tain inflation in this case. The second possibility is that Xµ = φµ, X5 π. (34) f(t) grows faster than a(t) eHt. In this case, the right- ≡ hand side of (30) goes to zero∼ at late times, so Φ becomes 3 Consider first the case of flat Euclidean space (κ = 0). constant. The self-acceleration quantity ΦF (a,f)/a in We take the FRW ansatz (6) and (9), along with ho- (28) now grows without bound as f 3( κ)3/2/a3 at ∼ − mogeneity for the scalar π = π(t). The mini-superspace late times, so again we do not achieve sustained infla- action is tion. Finally, there is the possibility that f(t) e−Ht, growing at the same rate as a(t). This case follows∼ the 2 same pattern as the first possibility – the right-hand side 2 a˙ a SR =3M dt , (35) of (30) approaches a constant at late times, and the self- P Z − N  acceleration quantity decreases exponentially. 2 2 ˙2 2 In summary, flat and open FRW solutions exist in Smass =3M dtm NF (a) G(a) f π˙ , (36) P Z  − q −  mass-varying massive gravity, but the constraint equa- 4 2 2 tion (the one which forbids flat FRW solutions in Sπ = Λ dt f˙ π˙ , (37) pure dRGT theory) does not allow for long-lasting self- − Z q − inflation. Finally, homogeneous and isotropic closed where F (a) and G(a) are as in (13) and (14). FRW solutions are not possible for the same reason they The mini-superspace action is invariant under time are not in dRGT – the fiducial Minkowski metric cannot reparametrizations, under which both f and π transform be foliated by closed slices. like scalars. Thus, as before we may ignore the accelera- tion equation of motion obtained by varying with respect THE DBI MODEL to a, since it is redundant. We fix the gauge N = 1 after deriving the equations. Varying with respect to N we obtain the Friedmann We now turn to the cosmology of the coupled galileon- equation massive gravity model introduced in [15]. One way to construct the theory is to consider a dynamical metric, F (a) H2 + m2 =0. (38) gµν , on the worldvolume of a 3-brane in a five dimensional a3 5

The f and π equations are, respectively

δS δS d f˙ = =  3M 2m2G(a)+Λ4  =0, (39) δπ δf dt P  f˙2 π˙ 2  q −  d 2 2 4 3 π˙  3M m G(a,f)+Λ (√ κf)  =0. − dt P −  f˙2 π˙ 2  q −  δS d π˙ (47) =  3M 2m2G(a)+Λ4  =0. (40) δπ −dt P f˙2 π˙ 2  ˙ δS δS  q −  Taking the combination f δf +π ˙ δπ of the two equa- tions of motion (46) and (47), and using the relation Taking the combination f˙ δS +π ˙ δS of the two equa- δf δπ ∂F (a,f) = √ κ ∂G(a,f) , which can be checked straight- tions of motion (39) and (40) , we arrive at the following ∂f − − ∂a constraint equation, forwardly from the definitions (25) and (26), we arrive at the following constraint equation, d ˙2 2 f π˙ [G(a)]=0. (41) ∂G(a,f) q − dt a˙ f˙2 π˙ 2 √ κf˙ =0. (48) ∂a  q − − −  The square root cannot be zero because it appears in the denominator of (39) and (40), so we must have d There are now two possible branches of solutions. The dt [G(a)] = 0, which implies that a must be constant and first consists of the solutions for which ∂G = 0. Solving thus there are no evolving flat FRW solutions. This is the ∂a this algebraically for f, we find f = ±aR for some con- same phenomenon as in pure dRGT theory [9], and we C stant ± depending only on the coefficients α2, α3 (there see that the DBI extended theory has a similar structure. can beC two branches here, since we must solve a quadratic The inclusion of matter minimally coupled to gµν does equation for f). Then, reinserting this into (45), we ob- not affect this conclusion because it does not contribute tain a modified Friedmann equation. The modification to the equations (39), (40) and hence the constraint (41) F (a,C±aR) 3 is a constant, depending only on α , α , so that is unchanged. a 2 3 when this constant is negative, we have self-acceleration We now turn to the case of nonzero spatial curvature 2 2 with ρself MP m . These solutions are exactly the same κ< 0. Using again the ansatz (21) we obtain self-accelerated∼ solutions that exist for the pure dRGT 2 theory [10]. The solution for π can then be determined 2 a˙ a SR =3M dt + κNa , (42) by solving (47). P Z − N  However, for the theory at hand, there exists a new 2 2 ˙2 2 possibility. This second branch consists of solutions for Smass =3MP dtm NF (a,f) G(a,f) f π˙ , Z  − q −  which (43)

4 3 2 2 a˙ f˙2 π˙ 2 = √ κf.˙ (49) Sπ = Λ dt (√ κf) f˙ π˙ , (44) − Z − q − q − − In the case of the pure dRGT theory where π = 0, this with F (a,f) and G(a,f) as in (25) and (26). branch gives only solutions for which a = √ κt, which is Varying with respect to N and setting N = 1, we ob- just in Milne coordinates.− Here we have tain the Friedmann equation the possibility of non-trivial solutions on this branch. ˙ κ κ F (a,f) Solving (49) forπ ˙ givesπ ˙ = f 1+ a˙ 2 , and substitut- H2 + + m2 =0. (45) ± ˙ a2 a3 ing this into the π equation of motionp (47) we see that f cancels and we are left with an algebraic equation in f, The f and π equations are, respectively namely

δS ∂F (a,f) ∂G(a,f) =3M 2m2 f˙2 π˙ 2 a˙ 2 + κ δf P  ∂f − ∂f −  3M 2m2G(a,f)+Λ4(√ κf)3 = (50) q P − r κ C  − 3Λ4(√ κ)3f 2 f˙2 π˙ 2+ − − q − where is the integration constant from integrating (47). C d f˙ This is a cubic equation which can be solved for f. Elim- +  3M 2m2G(a,f)+Λ4(√ κf)3  =0, dt P − inating f from the Friedmann equation (45) yields a sep-  f˙2 π˙ 2 arable equation of motion for the scale factor which can  q −  (46) have solutions with non-trivial evolution. 6

SUMMARY AND CONCLUSIONS [3] C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Rev. Lett. 106, 231101 (2011) [arXiv:1011.1232 [hep-th]]. [4] K. Hinterbichler, arXiv:1105.3735 [hep-th]. In this letter we have explored the cosmology of two [5] C. de Rham, G. Gabadadze, L. Heisenberg and different extensions of the dRGT massive gravity theory. D. Pirtskhalava, Phys. Rev. D 83, 103516 (2011) The first model is variable mass massive gravity, which [arXiv:1010.1780 [hep-th]]. results from promoting the fixed mass term of the dRGT [6] K. Koyama, G. Niz and G. Tasinato, Phys. Rev. Lett. model to the vacuum expectation value of a dynamical 107, 131101 (2011) [arXiv:1103.4708 [hep-th]]. scalar field, as suggested in [9]. In dRGT theory, there is [7] T. M. Nieuwenhuizen, Phys. Rev. D 84, 024038 (2011) a constraint equation that forbids non-trivial flat FRW [arXiv:1103.5912 [gr-qc]]. solutions, though self-accelerating open solutions exist. [8] A. H. Chamseddine and M. S. Volkov, Phys. Lett. B 704, 652 (2011) [arXiv:1107.5504 [hep-th]]. In the variable mass theory, the form of the constraint [9] G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, is different, and it no longer forbids flat FRW solutions. D. Pirtskhalava, A. J. Tolley, Massive Cosmologies, The constraint, however, makes it difficult to realize the [arXiv:1108.5231]. idea of self-inflation, i.e. using the self-acceleration prop- [10] A. E. Gumrukcuoglu, C. Lin and S. Mukohyama, JCAP erties of massive gravity in the early universe to drive 1111, 030 (2011) [arXiv:1109.3845 [hep-th]]. inflation. Furthermore, we have demonstrated for the [11] L. Berezhiani, G. Chkareuli, C. de Rham, G. Gabadadze first time that a large class of cosmologies within these and A. J. Tolley, Phys. Rev. D 85, 044024 (2012) models exhibit a future curvature singularity of the “big [arXiv:1111.3613 [hep-th]]. [12] C. de Rham, G. Gabadadze, L. Heisenberg and D. Pirt- brake” form. skhalava, arXiv:1212.4128 [hep-th]. The second extension is the coupled galileon-massive [13] N. Arkani-Hamed, H. Georgi and M. D. Schwartz, Annals gravity scalar-tensor theory of [15]. In the flat slicing, Phys. 305, 96 (2003) [hep-th/0210184]. the constraint equation takes a similar form to that of [14] A. S. Goldhaber and M. M. Nieto, Rev. Mod. Phys. 82, the pure dRGT theory, and it similarly rules out non- 939 (2010) [arXiv:0809.1003 [hep-ph]]. trivial FRW solutions. In the open slicing, the theory [15] G. Gabadadze, K. Hinterbichler, J. Khoury, D. Pirt- reproduces the self-accelerating branch discovered in pure skhalava and M. Trodden, arXiv:1208.5773 [hep-th]. [16] G. D’Amico, G. Gabadadze, L. Hui and D. Pirtskhalava, massive gravity, and in addition, provides a new branch arXiv:1206.4253 [hep-th]. of evolving cosmological solutions, the detailed general [17] Q. -G. Huang, Y. -S. Piao and S. -Y. Zhou, properties of which will be the topic of future work. arXiv:1206.5678 [hep-th]. Acknowledgments: We would like to thank Gregory [18] E. N. Saridakis, arXiv:1207.1800 [gr-qc]. Gabadadze, A. Emir G¨umr¨uk¸c¨uo˘glu and Emmanuel Sari- [19] Y. -F. Cai, C. Gao and E. N. Saridakis, arXiv:1207.3786 dakis for helpful comments. MT is supported in part [astro-ph.CO]. [20] D. -J. Wu, Y. -F. Cai and Y. -S. Piao, arXiv:1301.4326 by the US Department of Energy, and JS and MT are [hep-th]. supported in part by NASA ATP grant NNX11AI95G. [21] S. F. Hassan and R. A. Rosen, Phys. Rev. Lett. 108, Research at Perimeter Institute is supported by the Gov- 041101 (2012) [arXiv:1106.3344 [hep-th]]. ernment of Canada through Industry Canada and by the [22] S. F. Hassan and R. A. Rosen, JHEP 1204, 123 (2012) Province of Ontario through the Ministry of Economic [arXiv:1111.2070 [hep-th]]. Development and Innovation. The work of KH was made [23] M. Mirbabayi, Phys. Rev. D 86, 084006 (2012) possible in part through the support of a grant from the [arXiv:1112.1435 [hep-th]]. John Templeton Foundation. The opinions expressed in [24] C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Lett. B 711, 190 (2012) [arXiv:1107.3820 [hep-th]]. this publication are those of the authors and do not nec- [25] K. Hinterbichler and R. A. Rosen, JHEP 1207, 047 essarily reflect the views of the John Templeton Founda- (2012) [arXiv:1203.5783 [hep-th]]. tion. [26] J. D. Barrow, Class. Quant. Grav. 21, L79 (2004) [gr- qc/0403084]. [27] M. P. Dabrowski and A. Balcerzak, gr-qc/0701056. [28] R. Gregory, N. Kaloper, R. C. Myers and A. Padilla, JHEP 0710, 069 (2007) [arXiv:0707.2666 [hep-th]]. [1] D. G. Boulware and S. Deser, Phys. Rev. D 6, 3368 [29] C. de Rham and A. J. Tolley, JCAP 1005, 015 (2010) (1972). [arXiv:1003.5917 [hep-th]]. [2] C. de Rham and G. Gabadadze, Phys. Rev. D 82, 044020 (2010) [arXiv:1007.0443 [hep-th]].