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The rise of a tensor instability in Eddington-inspired gravity Celia Escamilla-Rivera Fisika Teorikoaren eta Zientziaren Historia Saila, Zientzia eta Teknologia Fakultatea, Euskal Herriko Unibertsitatea, 644 Posta Kutxatila, 48080, Bilbao, Spain. Ma´ximo Banãdos P. Universidad Católica de Chile, Avenida Vicuna Mackema 4860, Santiago, Chile Pedro G. Ferreira Astrophysics, University of Oxford, DWB, Keble Road, Oxford, OX1 3RH, UK InthisworkanextensiontoEddington’sgravitationalactionisanalyzed. Weconsiderthetensor perturbations of a FLRW space-time in the Eddington regime in where the tensor mode is linearly 3 unstabledeep and theresulting modifications to Einstein regime are quitestrong. 1 0 PACSnumbers: 98.80.−k,04.30.−w,14.70.Kv 2 Keywords: Cosmology,Graviton,GravitationalWaves. n a J Introduction.-Oneofthegreatestjigsawsinthecurrent q dxµdxν = X2dη2+Y2(δ +γ )dxidxj, (3) µν ij ij − 2 physics research is to understand the starting point to 2 general relativity. The Einstein-Hilbert action has been wherea,X andY aresolelyfunctionsofconformaltime, η. Here we shall work in the transverse traceless (TT) usually the initial core of gravitational theories, but an ] gauge, which leaves only the tensor modes in perturba- c alternative idea was proposed by Eddington in Ref. [1], q wheretheaffineconnectionΓµ isthefundamentalfield. tions, i.e ∂ihij =∂iγij =0. After straightforward calcu- - αβ lationsinwherewefoundthebackgroundequationsfrom r Stylishly,thetwosuggestionsareequivalentbyafactorof g Λ. Moreover, Eddington proposal is incomplete because Eq.(1)andthe perturbedfieldequationsusingEq.(2)we [ have that even though the tensor perturbations in both itdoesn’tincludeamattersector. Atthispointanalter- the metric and the auxiliary metrics are multiplied by 1 native theory of gravity was proposed in Ref. [2] where v the matter fields are introduced inside the gravitational differentconformalfactors,they areidenticalinthisthe- 4 ory as we presented in Ref. [6] . Furthermore, even in action in a Palatini form 6 the Einstein regime, where X =Y =a, we find that γ ij 2 2 5 SEBI[g,Γ,Ψ]= d4x gµν +κRµν(Γ) λ√g is non trivial and completely locked to the behaviour of κZ (cid:20)q| |− (cid:21) h , which evolution equation is given by . ij 1 +S [g,Ψ], (1) 0 m Y′ X′ X 2 ′′ ′ 2 13 where κ =8πG, Ψ denotes any additional matter fields. hij +(cid:18)3Y − X (cid:19)hij +(cid:18)Y (cid:19) k hij =0. (4) R is the symmetric Ricci tensor constructed with Γ. : µν v This actioncanreproduceEddington’soriginalactionat We now wish to see how this system evolves in the dif- i X large values of κR and Einstein’s at small values. Addi- ferent regimes. In the Einstein regime we find that the tionally, in this new theory the Eddington regime arise evolutionis indistinguishablefromEinsteingravity,even r a in the very early Universe and it may have led to a min- though the auxiliary metric is perturbed and present. It imum scale factor. Hence, and more significantly, in this is in the Eddington regime that we find novel behaviour regime seems we can preventthe formation of cosmolog- for a two different values of κ: ical singularities. As an enthusiastic proposal there are Case with κ > 0. In Ref. [7] it was shown a set several literatures that proof an interesting physics, for • ofequationsthatresurrectsanotherofEddington’s exampleinRef.[3]wasshowedthe possibilitytotestthe ideas of doing away with beginning and instead Eddington corrections to Newtonian gravity using Solar have the Universe indefinitely loitering in stasis in physicsandinRef.[4]thetestwasemployedaroundcom- thedistantpast. Toproceedwereexpressourback- pact rotating sources. The kindness of this new theory groundquantitiesintermsofconformaltime,ηand gives the possibility to reexpressed as a bigravity theory the evolution equation for the tensor mode is as was demonstrated in Ref. [5]. Instability in the Eddington Born-Infeld Theory.- Our exp(α∆η) exp(α∆η) main point is to consider a perturbed homogeneous and h′′ +2α h′ + k2h =0. (5) ij [1 exp(α∆η)] ij [1 exp(α∆η)] ij isotropic space-time by choosing the following two met- − − rics Clearly we found an instability in the Eddington g dxµdxν = a2dη2+a2(δ +h )dxidxj, (2) regime, in the asymptotic past when ∆η . µν ij ij − →−∞ 2 Case with κ < 0. We found that, if one choses theories in regions of density and curvature and gives us • a closed, positively curved spatial metric, it is a new way of looking at the generation and evolution of possible to construct an oscillating (or Phoenix cosmologicalperturbations. Universe)[7] which undergoes an indefinite number ofcycles. Suchamodelshould,inprincipleallowus Acknowledgments.- C. E-R is grateful to Ma´ximo tostudytheevolutionofperturbationsthroughthe Banãdos and Pedro G. Ferreira for proposing this prob- various cycles and shed light on some of the issues lem and their devoted guidance and hospitality during that have been raised in the study of cyclic cos- the author’s stay at the University of Oxford. Also mologies. As we calculated in our previous work, thankstoR.Jantzen,K.RosquistandR.Ruffinifortheir Ref.[6],theevolutionequationforthetensormode invitation and support in the MG13 meeting and Shinji is given by MukohyamachairmanintheSessionAT4: ModifyGrav- ity. TheauthorissupportedbyPabloGarc´ıaFundation, 2 k2 FUNDEC, Mexico and the Department of Theoretical ′′ ′ h + h + h =0, (6) ij η ij 3β2η2 ij Physics UPV/EHU Research Group 317207ELBE. As in the case of previous case we find an instability this time at the bounce and the solutions blow up rendering such a space-time unstable to tensor [1] A. Eddington, The Mathematical Theory of Relativity, mode perturbations. (CambridgeUniversityPress,Cambridge,England,1924); Conclusions.- In this work we learn that trace- E. Schrodinger, Spacetime Structure (Cambridge Univer- sity Press, Cambridge, England, 1950). less/transverse modes may play an unexpected role and [2] M. Banados, P.G. Ferreira, Phys. Rev. Lett. 105, 011101 should be included if possible in gravitational theories. (2010). [arXiv:1006.1769 [astro-ph.CO]] Indeed by allowing more general perturbations it should [3] J. Casanellas, P. Pani, I. Lopes, V. Cardoso, Astrophys. also be possible trace the effect of nonlinear evolution of J., 745, 15, 2012 [arXiv:1109.0249[astro-ph]] the tensor modes (coupled to radial modes) to search if [4] P.Pani,V.Cardoso andT. Delsate, Phys. Rev. Lett. 107 the singularity canbe stabilized in the nonlinear regime. (2011) 031101 [arXiv:1106.3569 [gr-qc]]. Itisremarkabletomentionedthatexistacoupleofanaly- [5] T. Delsate, J. Steinhoff (2012) [arXiv:1201.4989] sisofperturbationsinRef.[8]wheredidnotfindsuchany [6] C.Escamilla-Rivera,M.BanadosandP.G.Ferreira,Phys. Rev. D 85, 087302 (2012) [arXiv:1204.1691 [gr-qc]]. instability in the scalar sector and this might be a hint [7] J. H. C. Scargill, M. Banados and P. G. Ferreira, that it is the particular form of the Eddington inspired arXiv:1210.1521 [astro-ph.CO]. Born-Infeld theory that gives rise to such interesting be- [8] M. Banados, P. G. Ferreira and C. Skordis, Phys. Rev. D haviour. 79,063511(2009)[arXiv:0811.1272[astro-ph]].T.Clifton, Last but not least, our analysis clearly hints at the P. G. Ferreira, A. Padilla, C. Skordis Phys. Rep. 513 1 possibility that interesting effects might arise in these (2012) [arXiv:1106.2476]