A graceful entrance to braneworld inflation James E. Lidsey and David J. Mulryne Astronomy Unit, School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, U.K. Positively-curved,oscillatory universeshaverecentlybeenshowntohaveimportantconsequences for the pre-inflationary dynamics of the early universe. In particular, they may allow a self- interacting scalar field to climb up its potential during a very large number of these cycles. The cycles are naturally broken when the potential reaches a critical value and the universe begins to inflate, thereby providing a ‘graceful entrance’ to early universe inflation. We study the dynamics ofthisbehaviourwithinthecontextofbraneworldscenarioswhichexhibitabouncefromacollaps- ing phase to an expanding one. The dynamics can be understood by studying a general class of braneworldmodelsthataresourcedbyascalarfieldwithaconstantpotential. Withinthiscontext, we determine the conditions a given model must satisfy for a graceful entrance to be possible in 6 principle. We consider the bouncing braneworld model proposed by Shtanov and Sahni and show 0 that it exhibits the features needed to realise a graceful entrance to inflation for a wide region of 0 parameter space. 2 n a I. INTRODUCTION ing monotonically with time. In principle, therefore, the J field may gradually roll up its potential (assuming im- 7 The inflationary scenario is presently the favoured plicitly that it is evolving along a regionof the potential 2 model for large-scalestructure formation in the universe that is increasing in magnitude). As a result, the poten- [1, 2, 3, 4, 5, 6]. (For reviews, see [7, 8, 9, 10]). Infla- tialenergywill graduallyincreaseduring eachsuccessive 1 tion arises whenever the universe undergoes a phase of cycle, with the net result that it becomes progressively v accelerated expansion. Within the context of relativistic harder to satisfy the strong energy condition on scales 3 0 cosmology,a necessaryandsufficientconditionfor accel- above a∗. Eventually, therefore, a cycle will be reached 2 eration is a violation of the strong energy condition, i.e., when the strong energy condition becomes violated dur- 1 ρ+3p< 0. This is equivalent to the condition γ < 2/3, ingtheexpansionaryphaseofthecycle(a>a∗),andthis 0 where γ (ρ +p)/ρ defines the equation of state pa- will initiate a phase of inflation. This implies that slow- 6 rameter. ≡In the simplest versions of the scenario, such roll inflation may be possible even if the field is initially 0 a condition is realised by a scalar ‘inflaton’ field that is located in a region of the potential that would not gen- / h rollingsufficiently slowlydownits self-interactionpoten- erate accelerated expansion, such as a minimum. This -t tial, φ˙2 2/3 for a > a∗. Since these are rather weak verse. This question was recently examined within the conditions, it is important to investigate whether simi- r a context of Loop Quantum Cosmology (LQC), which is lar effects are possible in other cosmological scenarios, the application of Loop Quantum Gravity to symmetric such as the braneworld paradigm. This is the purpose states. (For reviews, see [12, 13, 14]). In this scenario, of the present paper. The braneworld scenario is moti- semi-classical modifications to the standard Friedmann vated by string/M-theory [20, 21, 22] and has attracted equations cause a minimally coupled scalar field to be- considerable attention in recent years. (See [23, 24, 25] haveasaneffectivephantomfluidbelowacriticalvalueof for reviews). Our observable four-dimensional universe the scale factor, a∗, with an effective equation of state, is interpreted as a co-dimension one brane propagating γ < 0 [15, 16, 17, 18, 19]. This causes a collapsing, in a five- (or higher-) dimensional ‘bulk’ space. A num- eff isotropic, spatially closed Friedmann-Robertson-Walker ber of bouncing braneworldmodels have been developed (FRW) universe to undergo a non-singular bounce into to date [26, 27, 28, 29, 30, 31]. In the model proposed an expansionary phase. On the other hand, if the field by Shtanov and Sahni (S-S), for example, the extra bulk satisfiesthestrongenergyconditionabovea∗(whereclas- dimensionis timelike andthis results in modifications to sicaldynamics is recovered),sucha universe willeventu- the effective four-dimensional Friedmann equation that ally recollapse. The overall result, therefore, is that the induce a non-singular bounce [26]. universe oscillates between cycles of contraction and ex- The paper is organized as follows. In Section II, we pansion [16]. outline howa phasespace analysisfor afield witha con- If the field’s kinetic energy dominates its potential en- stant potential can yield valuable insight into the cos- ergy during these cycles, it will effectively behave as a mic dynamics that leads to a graceful entrance for infla- massless field with its value either increasing or decreas- tion. We then proceed in Section III to determine the 2 necessary and sufficient conditions for graceful entrance when the Friedmann equation has an arbitrary depen- denceontheenergydensityofthe universe. Weconsider the S-S braneworld scenario as an explicit example in Sections IV–VI and determine the regions of parameter spacewhere agracefulentranceis possible. We conclude with a discussion in Section VII. II. THE PHASE SPACE DESCRIPTION OF GRACEFUL ENTRANCE TO INFLATION Theoscillatorydynamicsofisotropicuniversessourced by a scalar field with a constant potential proves very useful when determining the necessary conditions for a graceful entrance to inflation for a given cosmological model. In general, the dynamics is determined by three evolution equations for the scale factor, the Hubble parameter and the velocity of the scalar field, respectively. In addition, there is the Friedmann equation, which represents a constraint. This implies that the dynamics can be expressed as a two-dimensional system of equations and presented on a two-dimensional phase space. In the LQC scenarios considered previously, it proved convenienttoparametrizethephasespaceintermsofthe scale factor and Hubble parameter [18]. The qualitative natureofthephasespacethatcanresultinagracefulen- trance to inflation is illustrated in Fig. 1. For a positive potentialtakingvaluesinthe range0 Vcrit. In the jectorieswhichpassabovethesaddlepointrepresentini- former case, the important features for a graceful entrance tiallycollapsinguniverseswhichevolvethroughabounce areanareaofphasespaceinwhichoscillations takeplace,as into an eternal (de Sitter) inflationary era. This is illus- indicated by the region enclosed by the dotted line, and tra- trated qualitatively in the top panel of Fig. 1. jectories which evolve into an inflationary phase. The centre equilibrium point about which oscillations occur is marked If the value of the potential is increased, the effect on with a circle, theseparatrix is represented by thedotted line the phase space is such that the centre and saddle point andthesaddleequilibriumpointoccursattheself-intersection arepositionedclosertooneanother. Atthecriticalvalue of the separatrix. In the lower panel (V > Vcrit), there ex- V = V , the points merge and disappear. The phase crit ist nocentreor saddle equilibrium pointsand all trajectories space for V > V therefore contains no equilibrium crit evolveinto an inflationary phase. points and all trajectories represent initially contract- ing universes which bounce and subsequently inflate, as shown in the lower panel of Fig. 1. This simplifiedphasespacestructureenablesus to de- stillbe representedasatrajectoryin the phasespaceas- velop a graceful entrance mechanism for more realistic sociated with an instantaneous value of the potential, as cosmologies, where the potential is field-dependent. Let illustrated in the top panel of Fig. 1. Over a large num- us assume that the scalar field is initially located in a ber of cycles, however, this ‘instantaneous phase space’ region of the potential such that V(φ) < V and that will become significantly modified. Indeed, if the poten- crit the universe is undergoing oscillations about the centre tial energy increases monotonically as the field evolves, equilibrium point. The dynamics in this regime will be the region of phase space in which oscillations are pos- dominated by the field’s kinetic energy and we will fur- sible will become progressively smaller. Since the oscil- thersupposethatthefield’svelocityissuchthatitmoves latory region disappears completely at V = V , this crit up the potential. If the change in the magnitude of the corresponds to the largest value that the field’s poten- potential is negligible over a given cycle, this cycle can tialenergycanacquirebeforetheoscillationsarebroken. 3 At this point the trajectory in the phase space rapidly and Eqs. (1) and (2) fully determine the cosmic dynam- evolvesinto the inflationaryregime correspondingto the ics. Differentiating Eq. (1) with respect to cosmic time lower panel of Fig. 1. Hence, the value V sets the implies that crit energy scale at the onset of inflation. Once inflation begins, the field’s kinetic energy will rapidly tend to zero H˙ = 3γeffH2+ 1 3γeff 1 , (4) asthe fieldslowsdown,reachesapointofmaximumdis- − 2 − 2 a2 (cid:18) (cid:19) placement and rolls back down the potential. Slow-roll inflationwillthenariseifthe potentialsatisfiestheusual and Eqs. (3) and (4) then describe a closed dynamical slow-rollconstraints. system, where the Friedmann equation (1) represents a Although the dynamics may be further complicated constraint. The equilibrium points of this system arise bytheintroductionofadditionalmattersources[19],the whenever a˙ =a¨=0, which implies that above description outlines how a field can evolve up its 2 self-interactionpotentialwhiletheuniverseundergoesos- γ (a )= , H(a )=0 . (5) eff eq eq cillations. This forms the basis for a graceful entrance 3 mechanism that naturally generates the conditions for The stability of the system (3)-(4) can be determined slow-roll inflation. A phase plane analysis is important by linearising about the equilibrium points and evaluat- since it highlights the relevance of the centre and saddle ing the corresponding eigenvalues. It is straightforward equilibriumpoints. Inparticular,asufficientbutnotnec- to show that the eigenvalues are given by essaryconditionfor cyclicbehaviouris the existenceofa centreequilibriumpoint,while asaddlepointisrequired 3 1 dγ toseparatethoseregionsofphasespacewherecyclicand λ2 = eff . (6) −2 a2dlna non-cyclic behaviour takes place. Furthermore, the dis- (cid:20) (cid:21)eq appearance of the saddle point at a critical value of the Ingeneral,therefore,anecessaryandsufficientcondition potential is key to the graceful entrance mechanism. for the equilibrium point to be a centre is InthefollowingSection,therefore,wewillconsiderthe conditions for the existence of saddle and centre equilib- 2 dγ rium points in cosmologiesdescribed by a set of general- λ2 <0 , H =0 , γeff = , eff >0 , 3 dlna ized Friedmann equations. (7) whereas the corresponding condition to be a saddle is III. GENERAL CONDITIONS FOR CENTRE λ2 >0 , H =0 , γ = 2 , dγeff <0 . AND SADDLE EQUILIBRIUM POINTS eff 3 dlna (8) It is interesting to note that under the assumptions we A. Relativistic Cosmology have made, centre and saddle points are the only types of equilibrium points permitted. We wish to study classes of cosmological models that are motivated by the braneworld paradigm and to determine whether such models display the general char- B. Braneworld Scenarios acteristics required for a graceful entrance to inflation. However,we will first consider the relativistic cosmology Thefour-dimensionalcosmologicaldynamicsforawide basedonclassicalEinsteingravity,sincethissystempro- classofpositively-curvedFRWbraneworldscenarioscan vides a suitable framework for considering more general bemodeledintermsofageneralizedFriedmannequation models. Specifically, we will consider a positively-curved of the form FRWcosmologysourcedbyaneffectiveperfectfluidwith an energy density and pressure related by an arbitrary 8πℓ2 1 equationofstate,p =[γ 1]ρ ,whereitisassumed H2 = PlρL2(ρ)+f(a) , (9) eff eff− eff 3 − a2 implicitlythattheequationofstateparameterisaknown function of the scale factor, γeff =γeff(a). where ρ is the total energy density of the matter con- The Friedmann and fluid equations for this model are fined to the brane. The function L(ρ) is assumed to be given by positive-definite and parametrizes the departure of the model from the standard relativistic behaviour, H2 ρ. H2 = 8πℓ2Plρ 1 , (1) The function f(a) is a function of the scale factor a∝nd, 3 eff − a2 ina braneworldcontext, usuallyparametrizesthe effects ρ˙ = 3Hγ ρ , (2) of a bulk black hole on the four-dimensional dynamics eff eff eff − [32,33,34]. Ifthematterfieldsareconfinedtothebrane, where they satisfy the standard conservation equation a˙ =Ha , (3) ρ˙ = 3Hγρ , (10) − 4 where the equation of state parameter is defined by p = and substituting Eq. (14) into Eq. (12) implies that the (γ 1)ρ and is once more to be viewed implicitly as a effective equation of state parameter is given by − known function of the scale factor. γρ(1 ρ/σ)a4+m/2πℓ2 forTmheofsyEsqtse.m(1()9-)(-2()10b)ycdaenfibneinegxapnreesffseedctiinvetheenesrtgaynddaernd- γeff = ρ(1−−ρ/2σ)a4+3m/8πℓP2Pll . (15) sity [35] Since we are interested in whether a graceful entrance toinflationcanoccurinthismodel,wewilldeterminethe 3 ρ =ρL2+ f(a) . (11) equilibrium points that arise when the matter confined eff 8πℓ2 Pl to the brane corresponds to a scalar field that is rolling along a constant potential, V. This is equivalent to a DifferentiatingEq. (11)andcomparingwithEq. (2)then matter source comprised of a massless scalar field and impliesthattheeffectiveequationofstateparameterhas a cosmological constant. The equation of state for the the form system is given by γeff =(cid:20)γ(cid:18)ρL2+ρ2d(dLρ2)(cid:19)− 8π1ℓ2Plddlnfa(cid:21) γ =2(cid:18)1− Vρ(cid:19) . (16) −1 3 ρL2+ f . (12) In general, the equilibrium points for this system will × 8πℓ2 arise whenever Eqs. (7) or (8) are satisfied. It follows (cid:18) Pl (cid:19) from Eqs. (13) and (15) that such points occur when In principle, therefore, if the conservation equation (10) can be solved for a given equation of state, γ(a), 8πℓ2Pl ρ ρ2 + m = 1 , (17) the effective equationof state (12) will be a knownfunc- 3 − 2σ a4 a2 (cid:18) (cid:19) tion of the scale factor once the functional forms of L(ρ) ρ2 2 ρ2 m 1 and f(a) have been specified for a particular braneworld γ ρ ρ = . (18) − σ − 3 − 2σ −4πℓ2 a4 model. Differentiation will then yield the necessary in- (cid:18) (cid:19) (cid:18) (cid:19) Pl formation required in order to determine the existence For finite values of the scale factor, Eq. (17) may be (or not) of the centre and saddle equilibrium points, as simplified after substitution of Eq. (18): summarized in Eqs. (7) and (8). 1 4 2ρ ρ In the following Section, we will employ these results =4πℓ2 ρ +γ 1 , (19) within the context of a specific braneworld model. a2 Pl 3 − 3σ σ − (cid:20) (cid:16) (cid:17)(cid:21) and Eq. (19) may then be employed to express Eq. (17) in the form of a quartic equation in the energy density: IV. SHTANOV-SAHNI BRANEWORLD 2 2ρ2 1 V 8πℓ2 m eq + ρ +V a cToh-deimShetnasnioonv-oSnaehnbira(Sn-eSw)ibthraanenweograltdivsecetneanrsiioonemσ˜biendas Pl " 3σ −(cid:18)3 σ(cid:19) eq # five-dimensionalEinsteinspacesourcedbyapositivecos- 5ρ2eq 2 V + + ρ V =0 . (20) eq mologicalconstant, where the fifth dimension is timelike − 6σ 3 σ − (cid:18) (cid:19) [26, 36]. The effective Friedmann equation on the brane The solutions to Eq. (20) yield the values of the energy is given by [26, 36, 37] density at the equilibrium points and the corresponding 8πℓ2 ρ2 m 1 value of the scale factor can then be deduced directly H2 = Pl ρ + , (13) from Eq. (19). For physical solutions, one must ensure 3 − 2σ a4 − a2 (cid:20) (cid:21) that the energy density and scale factor are positive and real at each equilibrium point. where σ σ˜. If the bulk space is not conformally ≡ − Thedarkradiationonthebranecansignificantlyinflu- flat, the projection of the five-dimensional Weyl tensor encethedynamicsofthesystem. Inviewofthis,wecon- induces an effective ‘dark radiation’ term on the brane, sider separately the cases where this radiation is present parametrized by the constant m [32, 33, 34]. A confor- or absent in the following Sections. mally flat bulk corresponds to m = 0. Under quite general conditions, a negative quadratic dependence on the energy density in the Friedmann equation (13) implies V. NO DARK RADIATION that a collapsing brane world-volume is able to undergo a non-singular bounce [26]. If no dark radiation is present (m = 0), the quartic Comparison of the Friedmann equations (9) and (13) equation (20) reduces to the quadratic constraint implies immediately that ρ m 10ρ2eq 8 4V L2 = 1 , f(a)= , (14) + ρeq+4V =0 , (21) − 2σ a4 3σ − 3 σ (cid:18) (cid:19) (cid:16) (cid:17) 5 and this equation can be solved in terms of the field’s at the saddle. kinetic energy: It follows from Eqs. (27) and (28) that the centre and saddleequilibriumpointsmovetowardseachotherasthe 4σ 4V 2 9V2 18Vσ+4σ2 1/2 value of the potential is increased, eventually merging φ˙2 = − ± − . (22) eq 5 and disappearing at V = Vcrit. These are precisely the (cid:2) (cid:3) requirementsforagracefulentrancetoinflationthatwere It follows that there can be at most two static (physi- outlined in Section II. cal) solutions to the Friedmann equations (9) and (10), We now proceed to illustrate the cosmic dynamics depending on the magnitude of the potential. If V < 0, in the phase space. The dynamics is clearly two- there is only one solution to Eq. (21) with φ˙2 > 0, im- dimensional due to the Friedmann constraint equation plying there is only one equilibrium point in the phase (13). In principle, therefore, the phase space could be space. For V = 0, there is one solution with φ˙2 > 0 represented in two dimensions by introducing appropri- and a second, but physically uninteresting, point where ate variables. However, this may be non-trivial in prac- thefield’skineticenergyvanishesandthescalefactordi- tice,sincemorethanonesetofvariablesmayberequired. verges. Forapositivepotential,onthe otherhand,there (This would then imply that more than one phase plane are two real roots to Eq. (21) if the condition diagram would be needed in order to present the full dynamics). In the context of the present discussion, 9V2 18Vσ+4σ2 >0 , (23) it proves convenient to numerically integrate the field − equations and then view the evolution of the universe is satisfied. The values of the potential which bound in the three-dimensional space spanned by a,H,φ˙ . the region of parameter space in which there are no real { } Since the Friedmann constraint defines a surface in this solutions are therefore given by phase space, the evolution of the universe can then be parametrized in terms of a trajectory on this surface. √5 V = 1 σ . (24) In the following, we will refer to such a surface as the ± 3 ! Friedmannsurface. Identifying a coordinate systemthat coverstheFriedmannsurfaceisthenequivalenttofinding We find that for 0 Vcrit, where the equilibrium points have from condition (6) after substituting Eqs. (15), (17) and effectively merged. Alltrajectoriesnow evolvethrougha (21) and we find that the eigenvalues are given by bounce into an ever-expanding inflationary period of de Sitter expansionwith H constant, φ˙ 0 anda . → → →∞ 1 Comparison between the phase spaces of Fig. 1 and λ2 =4πℓ2 φ˙2 4 5φ˙2+4V . (26) Pl − σ those of Figs. 2 and 3 confirms that a graceful entrance (cid:20) (cid:16) (cid:17)(cid:21) to inflation, as outlined in section II, can occur in this Thisimpliesthatagivenpointwillcorrespondtoacentre model. if φ˙2 > 4(σ V), otherwise it will represent a saddle. 5 − Consequently, for physical roots to Eq. (21), the field’s kinetic energy has a value VI. EFFECTS OF DARK RADIATION 4V +4σ+2√9V2 18Vσ+4σ2 φ˙2 = − − , (27) In the case where dark radiation plays a dynamical eq 5 role in the Friedmann equation (13), the full quartic ex- atthecentreequilibriumpoint,whereasittakesthevalue pression given by Eq. (20) must be solved to determine the nature of the equilibrium points. One must also en- 4V +4σ 2√9V2 18Vσ+σ2 sure that the scale factor and kinetic energy of the field φ˙2eq = − − 5 − , (28) take positive and real values at these points. The con- 6 ∞ ∞ . φ . φ −∞ ∞ 0 H −∞ ∞ H FIG.2: Thetoppanelrepresentsthephasespaceinthevari- ables H,φ˙,a , when m = 0, σ = 0.05 and 0 < V < Vcrit, FTIhGer.e3a:reAnsoffiorniFteige.qu2i,libbruitumnowpoifnortsainpothteenptihaalsVe s>pacVecriitn. { } whereVcrit isdefinedinEq. (25). Numericalvaluesaregiven this region of parameter space. in Planckunits. Thesurface definedbytheFriedmannequa- tion(13),withρ=φ˙2/2+V,isalsoshownandallphasespace trajectories lie on this surface. The compactified coordinates x arctan(H), y arctan(lnφ˙) and z arctan(lna) have √m 1 2 4πℓ2 σ 1 be≡en employed in o≡rder to show the entir≡e phase space. The − a2 − 2√m = 3Pl − 4m . (29) lower panel represents thetwo-dimensional projection of this (cid:18) (cid:19) space onto the plane spanned by the variables H,φ˙ . The { } centre equilibrium point is denoted by the solid circle and Provided V < σ, the hyperboloid’s topology depends the separatrix is represented by the dotted line. The saddle only the relative values of the dark radiation parameter, pointoccurs atthepointwheretheseparatrix self-intersects. m, and the brane tension, σ. Indeed, there is a criti- Theaxeshavebeencompactifiedusingthecoordinatechange cal value of m at which the topology changes and this is x=arctan(H) and y=arctan(lnφ˙). given by 3 1 straint(20)canbesolvedanalyticallyusingthestandard m = . (30) techniques for quartic equations, but since the result- crit 16πℓ2 σ Pl ing expressions for the roots are very lengthy, we will not present them here. However, further insight may be Consequently, the surface defined by Eq. (29) consists gained by investigating the topology of the Friedmann of two disconnected pieces if m < m , whereas it is crit surface. This may be achieved by rewriting the Fried- a single surface for m > m . It is possible that one crit mann constraint equation (13) in the form of a hyper- of the two surfaces may correspond to a region where boloid: φ˙2 < 0, in which case it would be unphysical. However, 2 the qualitative dynamics of the universe will clearly be 8πℓ2 φ˙2 √2σ V H2+ Pl radicallydifferentaboveandbelowthecriticalvaluemcrit 3 2√2σ − 2 − √2σ!! andwethereforeproceedtoconsidereachregimeinturn. 7 0.15 spacecanberepresentedasasingletwo-dimensionalplot by employing cylindrical polar coordinates. These are shown in the lower panels of the figures. The dynamics in the upper sectors of the Friedmann surfaceinFigs. 5and6isqualitativelysimilartothecase 0.1 . where no dark radiation is present (m = 0). The lower φ2 sector of the Friedmann surface contains the new centre equilibrium point that arises when m =0. This point is 6 real for all values of the potential. Hence, as shown in 0.05 Figs. 5 and 6, the universe remains trapped in an indef- inite cycle of expansion and contraction if it is initially located in the lower half of the Friedmann surface. This behaviour can be understood since a sufficiently large (positive) cosmological constant introduces a large neg- 0 −0.05 0 Vcrit2 V 0.1 0.15 ative effective cosmological constant in the Friedmann equation (13) as a consequence of the quadratic term in FIG. 4: Illustrating the position and nature of the equilib- the energy density. This will prevent the universe from rium points as a function of the field’s kinetic and potential entering a phase of accelerated inflationary expansion. energies when 0 < m < mcrit. A dashed line corresponds to Indeed, although is is not relevant for the graceful en- a saddle point and a solid line to a centre point. The critical trance mechanism,it canbe shownthat if V exceedsyet value Vcrit2 represents the magnitude of the potential when anothercriticalvalue,theupperregionoftheFriedmann one of the centre points and the saddle disappear from the surface is no longer physical since the field’s kinetic en- phase space. The numerical values chosen for the plot were ergy becomes negative. In effect, therefore, the size of σ = 0.05 and m = 1.0 and the axes are measured in Planck the universe is bounded from above. units. Bycomparingthequalitativedynamicsoftheuniverse whennodarkradiationispresent(Figs. 2and3)tothat A. mmcrit dark radiation parameter and the brane tension. Eq. (19)wasalsoemployedtoverifythatthesolutionstoEq. We may adopt a similar approach for m>m . Fig. crit (20)correspondtophysicalvaluesofthescalefactor,and 7illustratesthepositionandnatureofthephysicallyrele- their stability was determined from Eq. (6). vantequilibriumpointsasafunctionofthefield’skinetic TheresultsareshowninFig. 4form=1andσ =0.05, and potential energies for the specific choices m = 1.4 where the locations of the saddle and centre equilibrium and σ = 0.05. The qualitative nature of this plot re- pointsforgivenfieldenergiesarerepresentedbythesolid mains unaltered for all m>mcrit. It follows from Fig. 7 anddashedlines,respectively. Thequalitativebehaviour that there is only a single saddle equilibrium point when is the same for all m < m . There is a critical value the magnitude of the potential is below a critical value crit of the potential, Vcrit2, such that there are three real Vcrit3. Thereisthenafinite rangeofvaluesofthepoten- roots to the quartic equation (20) when 0 < V < Vcrit2. tial, Vcrit3 V Vcrit4, for which there are no physical ≤ ≤ These correspond to three distinct equilibrium points, equilibrium points. For V > Vcrit4, on the other hand, two of which are centres whereas the third is a saddle thereexistsbothacentreandasaddleequilibriumpoint, point. At V = V , however, the saddle and one of andatstillhighervaluesofthepotential,thesaddlepoint crit2 the centre points merge and, for V >V , these points disappears once more. However, the equilibrium points crit2 correspond to unphysical roots. Consequently, there is which occur above Vcrit4 are not relevant to the graceful only one equilibrium point for V > V and this is a entrance scenario. crit2 centre. Figs. 8 and 9 illustrate the Friedmann surface and Thequalitativeevolutionoftheuniverseintheregimes and the corresponding phase trajectories for V < V crit3 V < V and V > V is shown in Figs. 5 and 6, and V < V < V , respectively. In Fig. 8, the crit2 crit2 crit3 crit4 respectively, where the trajectories have been calculated saddlepointrepresentsadividebetweencyclicandinfla- bynumericallyintegratingthefieldequationsforthespe- tionary behaviour, although it should be noted that the cific choices m = 1 and σ = 0.05. The topology of the cyclic dynamics does not occur around a centre equilib- Friedmann surface in the upper panels of these figures, rium point in this case. For V >V , the saddle point crit3 together with Eq. (29), implies that the entire phase disappears and all trajectories eventually evolve into an 8 ∞ ∞ a a 0 −π π 0 θ −π θ π FIG.5: Thetoppanelrepresentsthephasespaceinthevari- FIG. 6: As in Fig. 5, but now for a potential V > Vcrit2. ables H,φ˙,a , when m=1.0, σ =0.05 and 0mcrit. Numerical values chosen for the parameters are m=1.4 and pointoccursatthepointwheretheseparatrixintersectswith σ=0.05 in Planck units. itself. inflationary region, as shown in Fig. 9. As was the case 9 ∞ ∞ a a 0 0 −π θ π −π θ π FIG.8: ThetopandbottompanelsareasdescribedforFig. FIG.9: ThetopandbottompanelsareasdescribedforFig. 5, but now for m > mcrit and 0 < V < Vcrit3. Numerical 5, but now for m>mcrit and Vcrit3 m is more difficult to answer. the region of parameter space relevant to inflation, or in crit The key features of a cyclic region and a saddle point in aregiondisconnectedfromit,suchasthelowersectionof thephasespace,whichdisappearatacriticalvalueofthe the Friedmannsurface. Inthe presentcase,however,the potential,areindeedpresent. However,thereisafurther two regions of the previously disjointed Friedmann sur- complication. For m < m , the cyclic dynamics is al- face are connected. In effect, therefore, the turn around crit ways dominated by the field’s kinetic energy, but this is in the field may occur too soon for inflation to occur if no longer the case when m>m . m>m . crit crit FurtherinsightmaybegainedfromFigs. 8and9. The Morespecifically,let us assumeas beforethat wehave shaded gray areas (red in the online versions) represent a realistic inflationary potential, with a magnitude ini- the regions of phase space where the field’s kinetic en- tially in the range 0 φ˙2,andconsequently drivesa phaseof inflationary expansion. It follows that the critical value V sets the energy scale for the onset of inflation. crit Acknowledgments Wehavedevelopedaframeworkforinvestigatingagen- eral class of braneworld models in this context that are characterizedbyaFriedmannequationwithanarbitrary DJM is supported by PPARC. We thank R. Tavakol dependence on the energy density. Such a class of mod- for numerous discussions.

A graceful entrance to braneworld inflation

Similar A graceful entrance to braneworld inflation