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Unified Model for Inflation and the Dark Side of the Universe Gabriel Zsembinszki GrupdeFísicaTeòricaandInstitutdeFísicad’AltesEnergies 7 UniversitatAutònomadeBarcelona 0 08193Bellaterra,Barcelona,Spain 0 2 Abstract. n We presenta modelwith a complexanda realscalarfieldsanda potentialwhosesymmetryis a J explicitlybrokenbyPlanck-scalephysics.Forexponentiallysmallbreaking,themodelaccountsfor theperiodofinflationintheearlyuniverseandfortheperiodofaccelerationofthelateuniverseor 2 forthedarkmatter,dependingonthesmallnessoftheexplicitbreaking. 1 Keywords: inflation,darkenergy,darkmatter 1 PACS: 98.80.Cq,95.36.+x,95.35.+d v 9 6 INTRODUCTION 3 1 0 The Standard Model (SM) of particle physics based on the gauge group SU(3) 7 × 0 SU(2) U(1)is considered to be a successful model, able to accommodate all existing × / empirical data with high accuracy. Nevertheless, there are many deep questions for h p which the SM is unable to give the right answer, such that many physicists believe that - itisnot theultimatetheoryofnature. In anyextensionoftheSM, theideaofsupposing o r new additional symmetries is quite justified, taking into account that there are known t s symmetries that at low energies are broken, but at higher energies are restored. If we a : assume that global symmetries are valid at high energies, we should expect that they v are only approximate, since Planck-scale physics breaks them explicitly [1, 2]. Even i X with an extremely small breaking, very interesting effects may appear. As discussed in r [3, 4], when aglobalsymmetryis spontaneouslybroken and inadditionthere isa small a explicit breaking, the corresponding pseudo-Golstone boson (PGB) can play a role in cosmology.Thefocusin[3]wastoshowthatthePGBcouldbeadarkmatterconstituent candidate, whereas in [4] it might play the role of a quintessence field responsible for thepresent acceleration oftheuniverse. In the present contribution we will relate the period of very early acceleration of the universe (inflation) either with the present period of acceleration, or with the mysteri- ous dark matter, depending on the smallness of the effects of Planck-scale physics in breaking global symmetries. Direct or indirect observational evidence for the existence ofdarkenergyanddarkmattertogetherwiththeneedforinflationcomemainlyfromsu- pernovaeoftypeIaasstandardcandles[5],cosmicmicrowavebackgroundanisotropies [6], galaxy counts [7] and others [8]. The physics behind inflation, dark matter or dark energy may be completely unrelated, but it is an appealing possibility that they have a commonorigin.Anideaforthiskindofunificationis"quintessentialinflation",thathas beenforwardedbyFriemanandRosenfeld[9].Theirframeworkisanaxionfieldmodel wherethereisaglobalU(1) symmetry,whichisspontaneouslybrokenatahighscale PQ and explicitlybroken byinstantoneffects at thelowenergy QCD scale. Thereal part of thefield is able to inflate in the early universewhilethe axion boson could be responsi- bleforthedark energy period.Theauthorsof[9]comparetheirmodelofquintessential inflation with other models of inflation and/or dark energy. Here, in the framework of a global symmetry with Planck-scale explicit breaking, we offer an explicit scenario of quintessential inflation. As an alternative, we also consider the possibility that, in the same framework, the axion boson is a dark matter constituent. We may have one alter- nativeortheotherdependingon themagnitudeoftheexplicitsymmetrybreaking. THE MODEL In our model, we have a complex field Y that is charged under a certain global U(1) symmetryand apotentialthat containsthefollowingU(1)-symmetricterm 1 V (Y )= l [ Y 2 v2]2 (1) 1 4 | | − where l is a coupling constant and v is the energy scale of the spontaneous symmetry breaking(SSB). WithoutknowingthedetailsofhowPlanck-scalephysicsbreaksourU(1)symmetry, weintroducethemostsimpleeffectiveU(1)-breakingterm 1 V (Y )= g Y n Y e id +Y ⋆eid (2) non−sym − MPn−3| | (cid:16) − (cid:17) with an integern>3. We base our model on the idea that the coupling g is expected to be very small [10]. If g is of order 10 30 then we will see that the resulting PGB is a − darkmattercandidate, whileforg-valuesoforder10 119 itwillbeaquintessencefield. − Thecomplexscalarfield Y maybewrittenin theform Y =f eiq /v. (3) Ourbasicideaisthattheradialpartf ofthefieldY isresponsibleforinflation,whereas the angular part q can play either the role of the present dominating dark energy of the universe, or of the dark matter, depending on the values of g parameter that appears in (2). In order for f to inflate, one has to introduce a new real field c that assists f to inflate. The c field is supposed to be massive and neutral under U(1). In the process of SSB at temperatures T v in the early universe, the scalar field f develops in time, startingfromf =0andgoi∼ngtovaluesdifferentfromzero,asininvertedhybridinflation [11,12]models.Weshallfollowref.[12]andcouplec toY witha Y Y c 2 term.More ∗ − specificallyweintroducethefollowingcontributiontothepotential 1 a 2 Y 2c 2 2 V2(Y ,c )= 2m2c c 2+(cid:18)L 2− |4L |2 (cid:19) (4) where a is a coupling and L and mc are mass scales. The interaction between the two fieldswillgivetheneededbehavioroftherealpartofY togiveinflation.Suchmodelsof inflationarerealizedinsupersymmetry,usingagloballysupersymmetricscalarpotential [12]. Tosummarize,ourmodelhasacomplexfieldY andarealfieldc withatotalpotential V(Y ,c )=V (Y ,c )+V (Y )+C (5) sym non sym − whereC is a constant that sets the minimumof the effective potential to zero. The non- symmetricpart isgivenby(2), whereas thesymmetricpart is thesumof(1)and (4), V (Y ,c )=V (Y )+V (Y ,c ) (6) sym 1 2 Inflation Let us study, firstly, the conditions to be imposed on our model to describe the inflationary stage of expansion of the primordial Universe. In order to do this, we will only work with the symmetric part of the effective potential, which dominates over the non-symmetricpartat early times,and aftermakingthereplacement(3) weobtain 1 a 4f 4c 4 1 Vsym(f ,c )=L 4+2 m2c −a 2f 2 c 2+ 16L 4 +4l (f 2−v2)2, (7) (cid:16) (cid:17) Here, f is the inflaton field and c is the field that plays the role of an auxiliary field, which ends the inflationary regime through a "waterfall" mechanism. We note that the f 4c 4 term in Eq.(7) does not play an important role during inflation, but only after it ends,and itsetsthepositionoftheglobalminimumofV (f ,c ). sym From (7) we notice that the field c has an effectivemass givenby M2 =m2 a 2f 2, c c − so that for f <f = mc , the only minimum ofV (f ,c ) is at c =0. The curvature of c a sym theeffective potentialin the c direction is positive,whilein the f direction is negative. Because we expect that after theSSB, f is close to the origin and displaced from it due to quantum fluctuations, it will roll down away from the origin, while c will stay at its minimum c =0 until the curvature in c direction changes sign. That happens when f >f and c becomesunstableand startstoroll downits potential. c Theconditionstobeimposedonourmodelare thefollowing: • Thevacuumenergy termin (7)shoulddominateovertheothers:L 4 > 1l v4 4 • The absolute mass squared of the inflaton should be much less than the c -mass squared, mf2 = l v2 m2c , which fixes the initial conditions for the fields: c is | | ≪ initially constrained at the stable minimum c =0, and f may slowly roll from its initialpositionf 0 ≃ • Slow-rollconditionsinf -direction,whicharegivenbythefollowingrequirements: e MP2 Vs′ym 2 1, h MP2Vs′y′m 1, where a prime means derivative with res≡pe1c6tp to(cid:16)Vfsym(cid:17) ≪ | | ≡ (cid:12)(cid:12)8p Vsym(cid:12)(cid:12) ≪ (cid:12) (cid:12) • Swuhfefirceiefnt numf b(etr o)f=e-ffolmdsarokfsitnhfleaetniodno:fNsl(ofw)-r=ollRtitennfldaHti(otn)dt = M8pP2 Rffend VVss′yymmdf end end c ≡ • Fastrollofc fieldattheendofinflation: D Mc2 H2,where D Mc2 istheabsolute variationofthec -masssquaredinaHubb|letim|e≫H,aroundth|epoin|twheref f c ≃ • Fast roll of f after c settles down to the minimum. This is possible because the potentialhasanon-vanishingfirstderivativeatthatpointwhichforcesf tooscillate around the minimum of the potential, with a frequency w which we want to be greaterthan theHubbleparameterH:w >H. Fromthelastconditionweobtainan upperlimitfortheSSB scalev v<M . (8) P Dark matter As stated above, our idea is that the PGB q that appears after the SSB of U(1) can play theroleofquintessenceorof dark matter, depending on thevaluesof g-parameter. Let us start investigating the case where q describes dark matter. For a detailed study we send the reader to our work [3]. Here, we will just highlight the main features and conclusionsofourstudyin [3]. Duetothesmallexplicitbreaking oftheU(1)symmetry,q getsamass n 1 v m2 =2g − M2 (9) q (cid:18)M (cid:19) P P which depends on the two free parameters v and g. In what follows, we fix the value of n=4 exceptifexplicitlymentioned. For q to be a dark matter candidate, it should satisfy the following astrophysical and cosmologicalconstraints: • It shouldbestable,witha lifetimet q >t0, wheret0 isthelifetimeoftheuniverse • Itsdensityshouldbecomparableto thedark matterdensityW q W DM 0.25 ∼ ∼ • Because it can be produced in stars, it should not allow for too much energy loss and rapidcoolingofstars • Even if it is stable, q can be decaying in the present and thus contribute to the diffusephotonbackgroundoftheuniverse,whichisboundedexperimentally. In order to calculate the density of produced q -particles we took into account the dif- ferent production mechanisms: thermal production in the hot plasma, and non-thermal productionby q -field oscillationsand from the decay of cosmicstringsproduced in the SSB. A detailed study [3] showed that for v < 7.2 1012 GeV, there is thermal pro- duction of q particles, and the number density produ×ced is given by n 0.12T3. The th numberdensity produced by themisalignmentmechanism is nosc ≃ 21mq≃v2 and by cos- mic strings decay is n v2/t . Also, we have to take into account that non-thermal str str produced q may finally t≈hermalize, depending on the values of g and v. Astrophysical constraintsplacealimitonv, butnot ong v>3.3 109GeV. (10) × The combinations of astrophysical and cosmological constraints lead to the following valuesfor vandg forq tobeadark mattercandidate v 1011GeV, g 10 30. (11) − ∼ ∼ Asafinalcomment,wementionthatonecouldobtainvaluesofordertheelectriccharge forg,ifoneputsn=7, withalln<7 prohibitedforsomeunknownreason. Dark energy Let us find now the values for v and g in order for q to be a quintessence field responsible for the present acceleration of the universe. There are two conditions it shouldsatisfy: • The field q should be displaced from the minimum of the potential Vnon sym(q ), and we suppose that its value is of order v; it will only start to fall tow−ards the minimumin thefuture mq <3H0 (12) • The energy density of the q field, r 0, should be comparable to the present critical densityr , ifwewant q to explainall ofthedark energy contentoftheuniverse. c 0 r q r c (13) ∼ 0 Intheaboveequation(12),H istheHubbleconstant.Takingintoaccounttheexpression 0 n 1 forthemassofq , Eq. (9), mq =√2g(cid:16)MvP(cid:17) −2 MP,condition(12)becomes v n 1 9H2 g − < 0 . (14) (cid:18)M (cid:19) 2M2 P P The energy density of the q field is givenby the valueof the non-symmetricpart of the effective potential, V (f ,q ), with the assumption that the present values of both non sym fieldsare oforderv − n 1 v r q ≃Vnon−sym(v,v)=g(cid:18)MP(cid:19) − MP2v2. (15) Introducing (15) into (13) and remembering that the present critical energy density r = 3H02MP2, wehavethat c0 8p v n 1 3H2 g − 0 . (16) (cid:18)M (cid:19) ≃ 8p v2 P Combining(14)and (16)weobtaina constrainton v 1 v> M . (17) P 6 This is the restriction to be imposed on v in order for q to be the field describing dark energy. Notice that it is independent of n. It is also interesting to obtain the restriction onthecouplingg, whichcan bedoneifweintroduce(17)into(16)giving 3 6n+1 H2 g< × 0 . (18) 8p M2 P ReplacingthevalueforH 10 42 GeVandtakingthesmallestvaluen=4,weobtain 0 − ∼ thelimit g<10 119. (19) − CONCLUSIONS We havepresented a model that is able to explain inflation and dark energy, or inflation and dark matter. Although it is possible that there is no connection between them, the idea of unifying such important ingredients of cosmology into the same model is exciting. Our model contains two scalar field: one, Y , which is complex and charged under a certain global U(1) symmetry, and another one, c , which is real and neutral under U(1). The real part of Y is supposed to give inflation by coupling to the real field c . The imaginary part of Y can be either a dark matter candidate, or a quintessence field responsible for the recent acceleration of the universe. We suppose that we have a U(1)-symmetric potential to which we add a small term which explicitly breaks the symmetry due to Planck-scale physics. Our conclusion is that the explicit breaking has to be exponentially suppressed. In fact, this is suggested by quantitative studies on the breakingofglobalsymmetriesbygravitationaleffects[10].Ifthesuppressionparameter g is of order 10 30 and v 1011 GeV, the PGB that appears after the SSB ofU(1) is a − dark matter candidate. Fo∼r a much stronger suppression g 10 119 and a higher SSB − ∼ scalev M , thePGB isacandidateto thedark energy oftheuniverse. P ∼ Previous work on explicit breaking of global symmetries can also be found in [13], and related to Planck-scale breaking, in [14]. Cosmological consequences of some classesofPGBs are discussedin[15]. ACKNOWLEDGMENTS I thank Eduard Massó for all the advices and the good things he taught me during last years ofcollaboration.Thiswork was supportedby DURSI undergrant 2003FI00138. REFERENCES 1. SeeforinstanceT.Banks,Physicalia12,19(1990),andreferencestherein. 2. S.B.GiddingsandA.Strominger,Nucl.Phys.B307,854(1988). S.R.Coleman,Nucl.Phys.B310,643(1988). G.Gilbert,Nucl.Phys.B328,159(1989). 3. E.Masso,F.RotaandG.Zsembinszki,Phys.Rev.D70,115009(2004)[arXiv:hep-ph/0404289]. 4. E.MassoandG.Zsembinszki,“UnifiedmodelforinflationanddarkenergywithPlanck-scaleJCAP 0602,012(2006)[arXiv:astro-ph/0602166]. 5. S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517, 565 (1999) [arXiv:astro-ph/9812133]. A.G.Riessetal.[SupernovaSearchTeamCollaboration],Astron.J.116,1009(1998)[arXiv:astro- ph/9805201]. P.Astieretal.,Astron.Astrophys.447,31(2006)[arXiv:astro-ph/0510447]. 6. D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 148, 175 (2003) [arXiv:astro- ph/0302209]. A. Balbi et al., Astrophys. J. 545, L1 (2000) [Erratum-ibid. 558, L145 (2001)] [arXiv:astro- ph/0005124]. C.Pryke,N.W.Halverson,E.M.Leitch,J.Kovac,J.E.Carlstrom,W.L.HolzapfelandM.Dragovan, Astrophys.J.568,46(2002)[arXiv:astro-ph/0104490]. C. B. Netterfield et al. [Boomerang Collaboration], Astrophys. J. 571 (2002) 604 [arXiv:astro- ph/0104460]. J.L.Sieversetal.,Astrophys.J.591,599(2003)[arXiv:astro-ph/0205387]. A. Benoit et al. [the Archeops Collaboration], Astron. Astrophys. 399, L25 (2003) [arXiv:astro- ph/0210306]. C.J.MacTavishetal.,arXiv:astro-ph/0507503. A.G.Sanchezetal.,Mon.Not.Roy.Astron.Soc.366,189(2006)[arXiv:astro-ph/0507583]. M.Tegmarketal.[SDSSCollaboration],Phys.Rev.D69,103501(2004)[arXiv:astro-ph/0310723]. 7. L.Verdeetal.,Mon.Not.Roy.Astron.Soc.335(2002)432[arXiv:astro-ph/0112161]. M.S.Turner,TheAstrophysicalJournal,576:L101-L104,2002. A.LewisandS.Bridle,Phys.Rev.D66,103511(2002)[arXiv:astro-ph/0205436]. X. m. Wang, M. Tegmark and M. Zaldarriaga, Phys. Rev. D 65, 123001 (2002) [arXiv:astro- ph/0105091]. 8. J.Garcia-Bellido,arXiv:astro-ph/0502139. 9. R.RosenfeldandJ.A.Frieman,JCAP0509,003(2005)[arXiv:astro-ph/0504191]. 10. R. Kallosh, A. D. Linde, D. A. Linde and L. Susskind, Phys. Rev. D 52, 912 (1995) [arXiv:hep- th/9502069]. 11. B.A.OvrutandP.J.Steinhardt,Phys.Rev.Lett.53,732(1984). 12. D.H.LythandE.D.Stewart,Phys.Rev.D54,7186(1996)[arXiv:hep-ph/9606412]. 13. C.T.HillandG.G.Ross,Phys.Lett.B203,125(1988). C.T.HillandG.G.Ross,Nucl.Phys.B311,253(1988). 14. M.Lusignoli,A.MasieroandM.Roncadelli,Phys.Lett.B252,247(1990). S.Ghigna,M.LusignoliandM.Roncadelli,Phys.Lett.B283,278(1992). D.Grasso,M.LusignoliandM.Roncadelli,Phys.Lett.B288,140(1992). 15. C.T.Hill,D.N.SchrammandJ.N.Fry,CommentsNucl.Part.Phys.19,25(1989). A.K.Gupta,C.T.Hill,R.HolmanandE.W.Kolb,Phys.Rev.D45,441(1992). J.A.Frieman,C.T.HillandR.Watkins,Phys.Rev.D46,1226(1992). J. A. Frieman, C. T. Hill, A. Stebbins and I. Waga, Phys. Rev. Lett. 75, 2077 (1995) [arXiv:astro- ph/9505060].

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