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Slow roll in simple non-canonical inflation Gabriela Barenboim∗ Departament de F´ısica Teòrica, Universitat de València, Carrer Dr. Moliner 50, E-46100 Burjassot (València), Spain William H. Kinney† Dept. of Physics, University at Buffalo, the State University of New York, Buffalo, NY 14260-1500 (Dated: February 3, 2008) Weconsiderinflationusingaclassofnon-canonicalLagrangiansforwhichthemodificationtothe kinetictermdependsonthefield,butnotitsderivatives. WegeneralizethestandardHubbleslowroll expansiontothenon-canonicalcaseandderiveexpressionsforobservablesintermsofthegeneralized slow roll parameters. We apply the general results to the illustrative case of “Slinky” inflation, which has a simple, exactly solvable, non-canonical representation. However, when transformed 7 intoacanonical basis,Slinkyinflation consistsofafieldoscillating onamulti-valuedpotential. We 0 calculate the power spectrum of curvature perturbations for Slinky inflation directly in the non- 0 2 canonical basis, and show that the spectrum is approximately a power law on large scales, with a “blue” power spectrum. On small scales, the power spectrum exhibits strong oscillatory behavior. n This is an example of a model in which the widely used solution of Garriga and Mukhanov gives a thewrong answer for the power spectrum. J 6 1 I. INTRODUCTION “k-inflation” [5, 6]. Many particular examples of non- canonical inflation arise in the context of string-inspired 2 inflation models such as the KKLMMT scenario [7] or v There is no doubt at present that modern cosmology DBIinflation[8,9]. Inthispaperweconsiderarestricted 3 leans ona consistenttheoreticalframeworkwhichagrees 4 quantitatively withdata. Basedongeneralrelativityand class of k-inflation models for which the modification to 3 the scalar kinetic term depends on the field but not on the BigBang theory,itcandescribe with amazingpreci- 1 its derivatives. This might at first appear to be a trivial sion the evolution of the Universe from the first fraction 0 modification, since in such a case one can always trans- 7 of a second forward. Nevertheless, such an impressive form the non-canonical scalar into a canonical one via a 0 framework falls short of explaining the flatness and ho- fieldredefinition. However,itisreasonabletoexpectthat / mogeneity of space, let alone the origin of matter and h some models which appear complex or highly contrived structures we observe in the universe today. As a result, p when viewed in a canonical basis may have a simple de- - no self-respecting theory of the Universe can be consid- o ered complete without a solution to these problems, the scription in an equivalent non-canonical representation. r Therefore,itisusefultodevelopasetoftoolsforanalyz- t most successful of which is inflation [1]. s ingslowrollinflationdirectlyintermsofanon-canonical a However, inflation is far from rising to the level of a field, without resorting to field redefinitions. : theory. Inflation is just a general term for models of the v very early universe which involve a period of exponen- i X tial expansion, blowing up an extremely small region to r one equivalent to the current horizon size in a fraction a of a second. In fact there are a bewildering variety of different models to realize inflation. In most of them however,inflationisimplementedthroughasinglescalar The paper is organized as follows: In Section II, we fieldwhoseequationsofmotionarearesolvedwithinthe generalize the standard slow roll expansion to the case slow roll approximation [2, 3, 4]. of simple non-canonical Lagrangians, and derive expres- While it is clearly possible to generate a variety of sions for the scalar and tensor power spectra generated by a non-canonical inflaton. In Section III we introduce interesting inflationary models from the simplest La- grangians, i.e. those involving a minimally coupled in- theexampleof“Slinky”inflation[10,11],whichisanex- flaton with a canonical kinetic term, it is a limited ap- actlysolvablenon-canonicalinflationmodelwithahighly proach. Modelswithnon-canonicalkinetictermsareone nontrivial canonical equivalent which consists of a field possible way to move beyond the simplest scalar field oscillating on a multi-valued potential. In Section IV we Lagrangians. Non-canonical inflation is certainly not a calculate the curvature power spectrum for Slinky infla- new idea, having first been studied in a general sense as tion, which displays both power-law and oscillatory regions. One very interesting feature of Slinky inflation is that the solution of Garriga and Mukhanov [6], widely used in stringy model building, can be seen to give the ∗Electronicaddress: [email protected] wronganswer. SectionVpresentsdiscussionandconclu- †Electronicaddress: whkinney@buffalo.edu sions. 2 II. INFLATION FROM A NON-CANONICAL TogetherwithEq. (3),thisformsacompletesetofequa- LAGRANGIAN tions for the evolution of the universe. It is convenient to re-write the equations (3) and (9) In this section, we derive general expressions for infla- in the useful Hamilton-Jacobi form [12, 13, 14, 15]. Dif- tionary parameters from a Lagrangianof the form ferentiating Eq. (9) with respect to time gives = 1F (θ)gµν∂µθ∂νθ V (θ). (1) 2HH′(θ)θ˙ = 8π 1F′(θ)θ˙3+F (θ)θ˙θ¨+V′(θ)θ˙ L 2 − 3m2 2 Pl (cid:20) (cid:21) A general k-inflation model may also include a function = 8π F(θ)H(θ)θ˙, (10) of the field derivatives ∂ θ in the kinetic term, but we −m2 µ Pl concentrateonthissimplercaseandshowthatitcanex- hibitsurprisinglycomplexbehavior. TheEuler-Lagrange where we have used the equation of motion (3). Using equationforanarbitrarymetricg correspondingtothis the FriedmannEquation(9) wethen havethe Hamilton- µν Lagrangianis: Jacobi Equations 1 4π ∂ √ gF (θ)gµν∂ θ +V′(θ) H′(θ) = F(θ)θ˙ √ g ν − µ −m2 −12−gµν∂µ(cid:2)θ∂νθF′(θ)=0. (cid:3) (2) 8π V (θ) = H2(Pθl) 1 m2Pl H′(θ) 2 (.11) 3m2Pl " − 12πF(θ)(cid:18)H(θ)(cid:19) # Specializing to the case of a Friedmann-Robertson- Walkermetric,gµν =diag 1, a2(t), a2(t), a2(t) and These equation are completely equivalent to the origi- − − − the homogeneous mode of the scalar field θ =0 results nal equations (3) and (9), but use the field value as a (cid:2) ∇ (cid:3) in the equation of motion clock instead of the coordinate time, which is consistent as long as the field evolution is monotonic. We can then 1 1 θ¨+3Hθ˙+ V′(θ)+ θ˙2F′(θ) =0, (3) express the second equation above in terms of the slow F (θ) 2 (cid:20) (cid:21) roll parameter ǫ, defined as where H =(a˙/a) is the Hubble parameter. We can sim- m2 1 H′(θ) 2 ilarly expressthe stress-energyofthe non-canonicalfield ǫ(θ) Pl , (12) as ≡ 4π F (θ) H(θ) (cid:18) (cid:19)(cid:18) (cid:19) Tµν =F (θ)∂µθ∂νθ gµν, (4) so that −L which for the homogeneous mode reduces to 8π 1 V (θ)=H2(θ) 1 ǫ(θ) . (13) 3m2 − 3 T00 = 1 1F(θ)θ˙2+V (θ) Pl (cid:20) (cid:21) a2 2 This is exactly the same form as the Hamilton-Jacobi (cid:20) (cid:21) Tij = 1 1F(θ)θ˙2 V (θ) . (5) equationforacanonicalscalarfield;allofthedependence a2 2 − on the non-canonical function F (θ) has been absorbed (cid:20) (cid:21) into the expression (12) for the slow-roll parameter ǫ. Wecanthereforeexpresstheenergydensityandpressure Wecanverifythatǫhasitsusualinterpretationinterms of the field θ as: of the equation of state by writing 1 ρ = 2F (θ)θ˙2+V (θ) ρ+3p = 2F(θ)θ˙2 2V (θ) p = 1F (θ)θ˙2 V (θ), (6) = 3m2PlH−2(θ)[1 ǫ(θ)]. (14) 2 − − 4π − so that the speed of sound in the scalar field fluid is [6] The equationofstate of the non-canonicalfield θ is then ∂p/∂θ˙ c2 =1. (7) 1 S ≡ ∂ρ/∂θ˙ p=ρ 1− 3ǫ(θ) , (15) (cid:20) (cid:21) Fromtheseexpressionsitisstraightforwardtoshowthat and the condition for inflation p < ρ/3 is then, as for the continuity equation − a canonical field, ǫ < 1. The scale factor evolves as a ρ˙+3H(ρ+p)=0 (8) e−N, where the number of e-folds N is given by ∝ is equivalent to the equation of motion (3) for the scalar H 2√π F (θ) field. The Einstein Field Equation for the Hubble pa- N = Hdt= dθ = dθ. (16) rameter is then −Z −Z θ˙ mPl Z s ǫ(θ) H2 = 8π 1F(θ)θ˙2+V (θ) . (9) NotethatN isdefinedtobethenumberofe-foldsbefore 3m2 2 the end of inflation, so that it is large at early times and Pl (cid:20) (cid:21) 3 decreases as inflation progresses. It is straightforwardto This calculation may appear to be trivial, since for F verify that the lowest-orderflow relation dependent only on θ and not its derivatives ∂ θ, we are µ free to transform the problem into a canonical basis 1 dH ǫ= (17) H dN φ= √Fdθ (26) holds in the non-canonical case. We can define higher- Z order slow roll parameters in terms of the flow relations and then use the usual canonical expressions for calcu- [16], lating the slow roll parameters and the inflationary ob- ′ servables. In the next section, we consider the case of 1 dǫ m2 H′(θ)/ F (θ) “Slinky” inflation [10, 11], for which the transformation η =ǫ+ = Pl , (18) 2ǫ dN 4π (cid:16) F (θp)H(θ) (cid:17) (26) is multi-valued, and therefore the associatedcanon- (cid:18) (cid:19) ical field evolution is nontrivial. p dη ξ2 = ǫη+ dN III. SLINKY INFLATION ′ ′ H′/√F H′/√F /√F = m4Pl (cid:16) (cid:17)(cid:20)(cid:16) (cid:17) (cid:21) , (19) A “Slinky” is a large spring with a very weak spring 16π2 √FH2 constant which “walks” down a staircase, producing in time-lapse a pattern resembling the potential describing and similarly for higher-order parameters. In fact it is the Slinky model, i.e. clear that any expression involving a differential of a canonicalfielddφcanbe translatedto the non-canonical 3 case by dφ √Fdθ. Especially important is the ampli- V(θ) = V0cos2θexp (2θ sin2θ) , (27) → b − tude of the curvature power spectrum, (cid:20) (cid:21) where V is the dark energy density observed today and 0 δN δN 2√π F P1/2 = δφ δθ = δθ. (20) whose kinetic function is chosen to be R δφ → δθ m ǫ Plr 3m2 Since the amplitude of the fluctuation in a canonical F(θ) = Pl sin2θ. (28) πb2 scalar is given by the Hubble parameter, whereb is anarbitraryrealparameterwhichgovernsthe H δφ= , (21) number of steps of the potential. Figure (1) shows the 2π Slinky potential. thefluctuationamplitudeofthenon-canonicalfieldθwill Thisapparentlyharmlesscombinationofpotentialand be given by kinetic terms produces not only anunusual cosmological history but also breaks down the possibility of switching H betweenthecanonicalandnon-canonicaldescriptions,as δθ = , (22) 2π√F the tool to do so, F(θ), periodically goes to zero, which results in divergences in the slow-roll parameters. The and the curvature power spectrum is given by the usual model is characterized by a periodic equation of state, expression in terms of ǫ, w(a)= cos2θ(a). (29) P1/2 = H . (23) − R m √πǫ Pl In this way, the same field can serve the role both of the inflaton in the early universe and the quintessence Using field, which drives the accelerated expansion in the cur- d 1 d rent universe. The model also assumes that the infla- = , (24) dlnk ǫ 1dN ton/quintessence field has weak perturbative couplings − with matter and radiation. This introduces two addi- we see that the usual expression for the scalar spectral tional degrees of freedom, so that Slinky has only three index applies in the non-canonical case,1 adjustable parameters, which are chosen such that the Ω /Ω and Ω /Ω come out to their measured values dlnP r Λ m Λ R n 1 =2η 4ǫ. (25) at a = 1 in a flat Universe. The temperature history − ≡ dlnk − of the model is quite nonstandard as well, in that the coupling between radiation and the non canonical scalar field force them track each other. Here we ignore the 1 Anequivalentexpressionforthespectralindexintermsofalter- late-time behaviorof the Slinky field and concentrateon natelydefinedslowrollparameterswasderivedinRef. [17]. the period of inflation in the early universe. 4 Despite the apparently complicated form of the po- From Eqs. (31) and (35), we also have an exact expres- tential (27), the Hamilton-Jacobi Equations (11) can be sion for ǫ in terms of the canonical field φ: solved exactly: 2 φ 3 ǫ(φ)=3 1 , (37) H(θ)=H0exp 2b (2θ−sin2θ) " −(cid:18)φ0(cid:19) # (cid:20)(cid:18) (cid:19) (cid:21) b θ˙ = H(θ). (30) and inflation occurs for (φ/φ )2 >2/3, or −2 0 We then have exact expressions for the slow roll param- φ 2 2 eters, > . (38) m πb2 (cid:18) Pl(cid:19) ǫ(θ) =3sin2θ The canonical expression for ǫ (37) can be recognized as b η(θ) =3sin2θ+ . (31) aspecialcaseofthenon-slowrollsolutionderivedinRef. 2tanθ [18], The equation of state is oscillatory, with ǫ varying from ǫ = 3 (p = ρ) to ǫ = 0 (p = ρ) periodically while the V (φ) − ǫ 3 1 , (39) field value θ and the Hubble parameter H both decrease ≃ − V (φ ) (cid:20) 0 (cid:21) monotonically. Thisbehaviorisnotpossibleforacanon- ical field, since ǫ = 0 implies φ˙ = 0 for a canonical field where φ is a stationary value of the field, φ˙ = 0. Near 0 φ. Taking the interval around θ = 0 for definiteness, the stationary value, the potential is approximately inflation occurs for ǫ<1, or V (φ) 1 φ 2 1 1 1 ǫ= . (40) −sin−1 √3 <θ <sin−1 √3 . (32) V (φ0) ≃ − 3 (cid:18)φ0(cid:19) (cid:18) (cid:19) (cid:18) (cid:19) SoweexpectthatthecanonicalequivalentofSlinkyevo- Thenumberofe-foldsofinflation(16)duringthisperiod lutionis qualitativelythat ofafield evolvingnear aclas- is sical turning point on a potential that is approximately 2 θi 4 1 2.46 quadratic in the vicinity of the turning point. This ex- N = dθ = sin−1 = , (33) plains the apparent divergence in the power spectrum, b b √3 b Zθf (cid:18) (cid:19) sincethestandardslowrollexpressionsbreakdownfora so that N >60 requires b<0.04.2 Note that the Slinky fieldevolvingneara turningpoint[19]. However,we can be more precise than this. Since the expresion for ǫ (37) model is strongly non-slow roll near θ = 0, π, 2π,..., ± ± is exact, we can derive a corresponding exact expression since the parameter η(θ) diverges at these points even for the potential [18], as the equation of state approaches the de Sitter limit, ǫ=0. The expression(23)formallydivergesintheǫ=0 limit as well, but (as we show below) this divergence is 4√π φ 1 V (φ)=V (φ )exp ǫ(φ)dφ 1 ǫ(φ) . 0 nonphysical. The physical meaning of the divergence in "mPl Zφ0 #(cid:18) − 3 (cid:19) the parameters can be understood by transforming to a p (41) canonical basis. We can transform to a canonical field φ Itis straightforwardto use this expressionto recoverthe by taking originalexpressionfortheSlinkypotentialbytransform- ing the integral into the non-canonical basis: 3m2 dφ=√Fdθ = Pl sinθdθ, (34) πb2 4√π φ 4√3π θ r ǫ(φ)dφ = φ sin2θdθ 0 so that mPl Zφ0 mPl Z0 p 3 = (2θ sin2θ), (42) φ= φ cosθ, (35) 0 b − − (cid:18) (cid:19) with so that we recover exactly the Slinky potential (27). We 3 can also solve the canonical equivalent of the integral φ0 =mPl πb2. (36) (42), r 4√π φ φ x ǫ(φ)dφ=4√3π 0 1 x2dx m m − Pl Zφ0 (cid:18) Pl(cid:19)Z1 2 Slinky inflation naturally allows for a large amount of entropy p 2 p 3 φ φ φ productionafterinflation,sothatthenumberofe-foldsrequired = 1 +sin−1 ttohasnol6v0e.tFhoerhthoreizpounrpporsoebsleomfthciasnpainpeprr,inwceipwleillbaessmumucehscsmaleaslleorf (cid:18)2b(cid:19)(cid:18)φ0(cid:19)s −(cid:18)φ0(cid:19) (cid:18)φ0(cid:19) ordertheCMBquadrupoleexitedthehorizonaround N =60. f1(φ).  (43) ≡ 5 The corresponding canonical potential is then Log(V(θ )) 2 φ V (φ)=V exp[f (φ)], (44) 1 0 1 φ (cid:18) 0(cid:19) where f (φ) is defined by Eq. (43). Here we take the 1 sign convention that √ǫ has the same sign as H′(φ), m H′(φ) √ǫ=+ Pl , (45) 2√π H(φ) θ so that √ǫ > 0 implies φ˙ < 0, and the solution (44) 0 Π 2Π 3Π therefore represents the potential for a field rolling from φ=+φ0toφ= φ0,where φ0areturningpointsinthe FIG.1: ThepotentialforSlinkyinflationintermsofthenon- − ± field evolution. We can similarly solve for the evolution canonical fieldθ. Inflation takesplace nearthe“flat”regions of the field from φ = φ to φ = +φ . In this case, for around θ=0,±π,.... 0 0 φ˙ >0, we must take √−ǫ<0, and 4√π φ φ −1 Log(V (φ )) ǫ(φ)dφ=4√3π 0 1 x2dx m m − Pl Z−φ0 (cid:18) Pl(cid:19)Zx p p 2 3 φ φ φ = 1 cos−1 +π −(cid:18)2b(cid:19)(cid:18)φ0(cid:19)s −(cid:18)φ0(cid:19) − (cid:18)φ0(cid:19)  f2(φ).  (46) ≡ The corresponding potential is: φ 2 V (φ)=V ( φ ) exp[f (φ)]. (47) 2 1 0 2 − φ (cid:18) 0(cid:19) Note that this potential is not the same as the potential −φ φ 0 (44): the field rolls to the left along one potential, but 0 thenrollsbacktotherightalongadifferentpotential! We havechosenthe normalizationofEq. (47)toensurecon- FIG. 2: The canonical equivalent of the Slinky potential. In tinuityattheturningpoint,V ( φ )=V ( φ ). Mono- contrast to the monotonic non-canonical field θ, the corre- 2 0 1 0 tonic evolutionofthe non-canon−icalscalarθ−corresponds sponding canonical field φ oscillates on a multi-valued po- to oscillationsofthe canonicalscalarφ onamulti-valued tential, with inflation happening near the stationary points where φ˙ =0. potential, with each oscillation rolling down a succes- sively lower “ramp”, as shown in Fig. 2. The Hamilton- Jacobiequation,validonlyformonotonicfieldevolution, This isthe solutionofGarrigaandMukhanov[6]. Inthe can nonetheless be exactly solved for the canonical field limitofslowroll,the powerspectrumis approximatelya in a piecewise fashion. powerlaw,withspectralindexgivenintermsoftheslow “Slinky”inflationprovidesanexampleofasimplenon- roll parameters by canonicalinflationmodelwithahighlynon-trivialcanon- icalcounterpart,consistingofascalarfieldoscillatingon b a multi-valued potential. In the next section, we discuss n 1=2η 4ǫ= 6sin2θ+ . (49) − − − 2tanθ thegenerationofdensityperturbationsinSlinkymodels. For this expression to be valid, ǫ and η must be much smaller than unity. From Eq. (31) this results in the IV. THE CURVATURE PERTURBATION IN conditions SLINKY INFLATION b 1 θ , (50) 2 ≪ ≪ √3 In the standard slow roll approximation,we can write the power spectrum of curvature perturbations by eval- both of which can be satisfied in the limit b 1. For uating the mode amplitude at horizon crossing, ≪ θ 1, ≪ H2 b P = . (48) R m2 πǫ n 1 , (51) Pl (cid:12)k=aH − ≃ 2θ (cid:12) (cid:12) (cid:12) 6 so that in slow roll, with θ b/2, we expect a nearly Defining ≫ power-law power spectrum, with a “blue” tilt (i.e. n > 1). Since θ˙ < 0, shorter-wavelength modes cross outside x 2θ, (60) the horizon at smaller values of θ, so that we expect the ≡ b spectrum to run such that the blue tilt will be steeper we can write the mode equation near the singular point on small scales than on large scales, i.e. a “positive” θ 1 as approximately running. ≪ However, slow roll inevitably breaks down as θ 0, 3 b2 andthesecondslowrollparameterηbecomessingula→r,as u′k′(x)−u′k(x)+ y02e2x+ x + 4 −2 uk(x)=0. (61) does the horizon-crossing expression for the power spec- (cid:20) (cid:21) trum (48). The singularity corresponds to the turning ThevariablexcanbeseenfromEq. (33)tobeameasure point of the equivalent canonical field φ. It is straight- ofthe number of e-folds ofinflation, andis here evolving forwardtoseethattheequationforquantumfluctuations frompositivetonegative. Thiscanbesolvedanalytically inthe inflatonisalsosingular. Intermsofthe conformal for b 1 and for modes much larger than the horizon, time τ, the equation for the gauge-invariant Mukhanov y y≪ex 0 0 variable u is [20] ≃ → k xe−x d2uk + k2 1d2z u =0, (52) uk = e−x[e3x 3xEi(3x)] . (62) dτ2 − zdτ2 k (cid:26) − (cid:27) (cid:18) (cid:19) Both solutions are regular at the singularity x = 0, but where the second has a singular derivative. Since, for x 1, ≪ z 2√π aφ˙ =2√π a√Fθ˙ . (53) z xe−x, (63) ≡ H ! H ! ∝ the first solution in Eq. (61) represents the constant Using Eq. (30), we can write z as: mode of the power spectrum on superhorizon scales, z = √3mPla0e−2θ/bsinθ, (54) uk const., y 0. (64) − z → → where a is defined as the arbitrary constant of propor- (cid:12) (cid:12) tionality0in the evolution of the scale factor The second solut(cid:12)(cid:12)ion(cid:12)(cid:12)is a decaying mode. Contact can also be made with the slow roll limit when a(θ)=a e−2θ/b. (55) 0 3 b2 . (65) We can relate the conformal time to the field value by x ≪ 4 2 where Eq. (61) has solution dθ =(aH)dτ, (56) b u ex/2H (y ex)=√yH (y), (66) k ν o ν ∝ and the mode equation (52) can be written exactly in terms of the field θ as [10]: where we have applied the Bunch-Davies boundary condition u e−ikτ when k aH, and k b2 d2u b du ∝ ≫ 4 dθ2k + 2 3sin2θ−1 dθk+ 1 3 b2 (cid:18) (cid:19) (cid:18) (cid:19) ν = 9 b2 1 . (67) k 2 1 2 (cid:0)b2+6cos2θ(cid:1) 6bcos3θ u 2p − ≃ 2(cid:18) − 18(cid:19) k "(cid:18)aH(cid:19) − 4(cid:18) − − sinθ (cid:19)# Thissolutionisvalidbothinsideandoutsidethehorizon, =0. (57) as long as slow roll is a good approximation. In the long wavelength limit, Note that the last term is singular at θ =0. We can express the wavelengthof the mode relative to the horizon u 2 P k3 k y3−2ν, (68) size as R ∝ z ∝ (cid:12) (cid:12) y k =y exp 1 θ 3sin2θ , (58) and the scalar spectral in(cid:12)(cid:12)dex(cid:12)(cid:12)is given by 0 ≡ aH −b − 2 (cid:20) (cid:18) (cid:19)(cid:21) b2 n=4 2ν 1+ , (69) where the constant of proportionality y is evaluated at 0 − ≃ 6 θ =0, so that in the extreme slow roll limit, we expect to gen- k erate an almost exactly scale-invariant power spectrum, y . (59) 0 ≡ aH with a very tiny blue tilt. (cid:12)θ=0 (cid:12) (cid:12) (cid:12) 7 For interesting values of the model parameters, the which is straightforwardto evaluate using the identity slow roll limit is strongly violated, and we resort to nu- dy(x) mericallysolvingthe exactmodeevolutionequation(57) =y(x)[1 ǫ(x)], (77) mode-by-modeink. Forthispurpose,itisusefultowrite dx − Eq. (57) as: so that u′′(x)+ 3sin2(bx/2) 1 u′(x)+ x − kτ = y(x)dx. (78) y02Y2(x(cid:0))− 32cos(bx)+ 3(cid:1)2bcsoins3(b(xbx//22)) + b42 − 21 u(x) − Zxf (cid:20)=0, (cid:21) (70) We evaluate this integral numerically for each mode. To relate the choice of y to a physical scale in the universe 0 where today,itisusefultocalculateN ,definedasthenumber H y(x) x 3sin(bx) of e-folds before the end of inflation at which a given Y(x)= =exp + . (71) mode crossed outside the horizon. The current CMB y −2 2b 0 (cid:20) (cid:21) quadrupole then corresponds to N of around 60. This H Expressed in this form, the choice of wavenumber k is is straightforward,since mappedentirelytothechoiceofdimensionlessparameter a(x ) y0 as defined by Eq. (59). We set the boundary condi- eNH f e(xH−xf), (79) tion on the mode at shortwavelengthusing the slow roll ≡ a(xH) ≃ expression. where x horizon crossing for a particular mode, H k1/2u = √π√ kτH ( kτ). (72) y(xH)=y0exH =1. We then have k ν 2 − − y e−xH, (80) 0 This boundary condition corresponds to the Bunch- ≃ Davies vacuum in the ultraviolet limit, and will be ac- and the number of e-folds is curate as long as a given mode spends a sufficiently long N (y ) ln(y ) x . (81) time inside the horizon during slow roll evolution. Since H 0 0 f ≃− − the durationofinflationis finite, thisapproximationwill Figure 3 shows the power spectrum as a function of y0 breakdownformodeswhichhavewavelengthscloseto,or and N for b = 0.025, which gives a total amount of H larger than, the horizon size at the begining of inflation. inflation of around a hundred e-folds. The power spec- For these modes, the initial condition is set by the pre- trumhasqualitativelydifferentbehaviorsformodeswith inflationaryevolution.3 Thiswillmodifythepowerspec- y <1, which exit the horizonprior to the singularity at 0 trum only at the largest scales, which we neglect in this θ = 0, and those with y > 1, which exit after. This 0 analysis and simply use the boundary condition (72) for results in an approximately power-law behavior on large all modes. Each mode is integrated from k/(aH) = 100 scales,andanoscillatorypowerspectrumonsmallscales. to the end of inflation. In terms of the variable x, this Notethattheslowrollexpressionforthepowerspectrum, corresponds to evolution from (48)isapoorfittotheexactpowerspectrum. Theshape x =ln100/y , (73) of the power spectrum is qualitatively similar for other i 0 choices of the parameter b, with power-law behavior at to large scales giving way to oscillations at small scales. If 2 1 we make b smaller, the fit to the slow roll expression at xf = sin−1 . (74) large scales improves significantly. Figure 4 shows the −b √3 (cid:18) (cid:19) power spectrum for b = 0.01. Slow roll is a good fit In order to correctly set the boundary condition (72) on to the exact result at very large scales, with N > 125, the mode function, we must relate the conformal time τ whicharefaroutsidethe currentcosmologicalhorizonat to the field value x. In the slow roll limit, we have N =60. Inanycase,theWMAP3datastronglydisfavor modelswithabluespectrumandnegligibletensormodes y(x) kτ = . (75) [22, 23, 24, 25, 26], and perturbations from a single pe- − 1 ǫ riod of Slinky Inflation prove to be a poor fit to the real − In the non-slow roll case, this generalizes to the exact world. expression dy dy(x) dx V. CONCLUSIONS d( kτ)= = , (76) − 1 ǫ dx 1 ǫ(x) − − We have considered inflation driven by a scalar field with a Lagrangianof the form 3 Foradiscussionoftheeffectofthepre-inflationaryvacuumina 1 = F(θ)gµν∂ θ∂ θ V (θ). (82) generalsetting,seeRef. [21]andreferencestherein. L 2 µ ν − 8 and ′ m2 H′(θ)/ F(θ) η = Pl . (84) 4π (cid:16) F(θp)H(θ) (cid:17) In terms of these parampeters, the expressions for the scalar power spectrum and the associated spectral index take on their usual forms in terms of the slow roll parameters, H P1/2 = , (85) R m √πǫ Pl and 0.0001 0.001 0.01 0.1 1 10 dlnP R n 1 =2η 4ǫ. (86) − ≡ dlnk − FIG.3: Thepower spectrum(normalized tounityat y0 =1) for Slinky inflation plotted as a function of ln(y0) ∝ ln(k), Such a field redefinition might appear to be trivial, for b = 0.025. The top axis shows the number of e-folds N since one could just as easily work with the equivalent before the end of inflation. The green (dashed) line is the canonical field φ. However, we describe a simple exam- slowrollsolution(48). (Despiteitssuperficialresemblanceto ple, “Slinky” inflation,in which the transformationfrom the Cℓ spectrum of the Cosmic Microwave Background, this the non-canonical to the canonical basis is multi-valued. plot shows the primordial power spectrum!) In this case, a field evolving monotonically along a potential in a non-canonical representation corresponds in the canonical representation to a field oscillating on a multi-valued potential. The non-canonical representa- tionprovesmuchsimpler,andtheevolutionequationsfor thebackgroundcosmologycanbesolvedexactly,consist- ingofalternatingperiodsofacceleratinganddecelerating expansion, and has been proposed as a unified model of inflation and quintessence [10, 11]. In this paper, we do not consider the late-time behavior of the Slinky field, but instead concentrate on the early inflationary period of the model. We write an exact equation for the evolution of gauge-invariantcurvature perturbations during inflation, which is complicated by the presence of singu- larities at θ = 0, π,..., which in the canonical descrip- ± tion correspond to the extrema of the oscillating field, 0.0001 0.001 0.01 0.1 1 10 where the field velocity φ˙ = 0. We evaluate the mode equationnumerically,andfindthat,formodeswhichexit the horizon prior to the singular point, the power spec- FIG.4: Thepower spectrum(normalized tounityat y0 =1) trum is an approximate power law with a “blue” (i.e. forSlinkyinflationplottedasafunctionofln(y0)∝ln(k),for n>1) power spectrum. Modes with shorter wavelength b=0.01. The top axis shows the numberof e-folds N before which are still inside the horizon at the singular point the end of inflation. The green (dashed) line is the slow roll produce an oscillatory power spectrum. In the slow roll solution (48). limit,thesolutionofGarrigaandMukhanov[6],inwhich the power spectrum is evaluated when a mode exits the horizon,providesagoodfit to the exactpowerspectrum Since the modification to the kinetic term depends on However,thesolutionfailsinthenon-slowrollregime,in the field but not its derivatives, it is possible to write an particular near the singularity. As a whole, the pertur- equivalent canonical Lagrangian via a field redefinition bationsproducedinSlinky inflationarenota goodfitto dφ =√Fdθ. It is then straightforwardto generalize the the currentdata. However,we haveonly consideredper- standardHubbleslowrollexpansiontoderiveexpressions turbations generated during a single epoch of inflation. for the flow parametersǫ, η, and so forth in terms of the Since Slinky evolution is periodic, perturbations may in field θ and the function F (θ), for example principle be processed through multiple periods of infla- tionanddeceleratedexpansionintheearlyuniverse,and m2 1 H′(θ) 2 it is unclear what the effect of such evolution would be ǫ(θ)= Pl , (83) 4π F(θ) H(θ) on the primordial power spectrum. (cid:18) (cid:19)(cid:18) (cid:19) 9 Acknowledgments WHKissupportedinpartbytheNationalScienceFoun- dation under grant NSF-PHY-0456777. 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