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Trans-Planckian Physics from a Nonlinear Dispersion Relation S.E. Jorás Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil G. Marozzi Dipartimento di Fisica, Università degli Studi di Bologna and INFN, via Irnerio 46, I-40126 Bologna, Italy We study a particular nonlinear dispersion relation ωp(kp) — a series expansion in the physical wavenumber kp — for modeling first-order corrections in the equation of motion of a test scalar field in a de Sitter spacetime from trans-Planckian physics in cosmology. Using both a numerical approachandasemianalyticalone,weshowthattheWKBapproximationpreviouslyadoptedinthe literatureshouldbeusedwithcaution,sinceitholdsonlywhenthecomovingwavenumberk aH. ≫ 9 We determine the amplitude and behavior of the corrections on the power spectrum for this test 0 field. Furthermore,weconsideralso amore realistic model ofinflation, thepower-law model, using 0 only a numerical approach to determine thecorrections on thepower spectrum. 2 n I. INTRODUCTION ical perturbations and then we restrict ourselves to the a J case of a test scalar field in a de Sitter spacetime. We also introduce the WKB approximation in this section. 6 Inflationarymodelsprovideanswerstomanyproblems 1 In Sec. III we adopt a particular way to approach TPP: in standard big bang cosmology, in particular the origin a modification of the equation of motion due to a non- of density fluctuations and the spectrum of cosmic mi- ] linear dispersion relation. The initial conditions for the c crowavebackgroundanisotropies. Thebasisofthewhole test scalar field are set in Sec. IV. In the following sec- q mechanismisthestretchingofquantumfluctuationsgen- - tions we present a numerical calculation, a 3-piecewise r erated at sub-Hubble scales due to the exponential ex- approximation(adoptedinRef.[4])andasemianalytical g pansion of the spacetime during inflation. This model, [ approach to solve the problem. In Sec. VIII we inves- however, has a serious “problem”: if we consider a plain tigate a realistic model of inflation, namely power-law 2 scalar-field-driven inflationary model — say, chaotic in- inflation, using the same nonlinear dispersion relation. v flation — the period of inflation lasts so long that the We then conclude in Sec. IX. 2 wavelengths of the fluctuations which at present corre- 6 spond to cosmological scales were sub-Planckian at the 2 begining of the inflationary phase. Therefore, the evolu- 1 II. STANDARD APPROACH tion of fluctuations at such scales is supposed to follow . 8 different rules from those provided by the standard the- 0 ory of cosmological perturbations — which is based on The spatially flat Friedmann-Robertson-Walker 8 quantum field theory and general relativity. The set of (FRW) metric with first-order cosmological fluctuations :0 rules expected to hold above this energy scale is the so- is given by (ds2 =gµνdxµdxν): v called trans-Planckianphysics (TPP from now on). The Xi TPP could lead to deviations from standard predictions g00 = a2(−1−2α) onthecosmicmicrowavebackgroundradiation(CMBR), a2 r g = (β +B ) a which probes the scales we mentioned before. The ques- 0i − 2 ,i i tion to be asked is “whether the predictions of the stan- g = a2[δ (1 2ψ)+D E+ ij ij ij dard cosmology are insensitive to effects of TPP”. This − +(χ +χ +h )/2] . (1) is the precise statement of the trans-Planckian problem j,i i,j ij [1, 2]. with D = ∂ ∂ 1/3 2δ , considering the conformal ij i j ij Inthepresentpaperweadopt,tomimicTPP,aseries- − ∇ timeη. Tofirst-orderscalar,vectorandtensorperturba- expansionexpressionforthenonlineardispersionrelation tions evolve independently. Vector perturbations can be ω (Eq.(19) below), firstsuggestedby [3], to modify the p omittedbecausetheydieawaykinematically. Thetensor equation of motion of a test scalar field in a de Sitter perturbationh hasonlytwophysicaldegreesoffreedom ij spacetime. We argue that, in this framework, the WKB (polarization states) h and h : + approximation is valid only for k aH (or k k/a ∗ p ≫ ≡ ≫ H with kp the physical wavenumber and H the Hubble hij =h+e+ij +h e∗ij (2) factor). As a consequence, the perturbative approach ∗ based on the WKB approximation, used in [4] to tackle where e+ and e are the polarizationtensors having the this issue, should be used with caution. ij ∗ij following properties in the Fourier space: The outline ofthe paperis the following. InSec.II we present the standard approach for first-order cosmolog- e =e , kie =0, ei =0 (3) ij ji ij i 2 e+e+ij =2, e e ij =2, e+e ij =0 (4) which becomes for a de Sitter spacetime ij ∗ij ∗ ij ∗ On expanding in Fourier modes we can define 2 Ω2(η)=k2 . (14) − η2 h (t,~x)= 1 1 d3kµ (η)ei~k~x, (5) WecannotethatEq.(12)isthesameequationofmotion λ a(η)(2π)3/2 tk · as that of a tensor perturbation (6). In this simple case, Z Eq. (12) can be exactly solved. with λ=+/ and where µ will satisfy [5]: ∗ tk The two-point correlation function is given by µ′t′k+ k2− aa′′ µtk =0 (6) 0φ(η,~x)φ(η,~x+~r)0 = +∞ dksinkrPφ(k) (15) (cid:18) (cid:19) h | | i k kr Z0 The scalar sector, in the case of a plain scalar field and the power spectrum is φ(η,~x) = φ(η) +ϕ(t,~x) driven inflationary model, can bereducedtothestudyofasinglegauge-invariantscalar k3 µ 2 k variable defined by Pφ(k)= 2π2 a . (16) (cid:12) (cid:12) µ φ 1 (cid:12) (cid:12) Q= s =ϕ+ ′ ψ+ 2E (7) For superhorizon modes (k a(cid:12)H)(cid:12)the spectrum is time ≪ a 6∇ independent and scale invariant, as one can see from the H(cid:18) (cid:19) exact solution in the limit η 0 . where = a′. This so-called Mukhanov variable [6] Equation(12)canalsobei→nter−pretedasaSchrödinger H a obeys the following equation of motion in Fourier space: equation for a stationary wave function with energy E ω2 = k2 in an effective potential V (η) a /a z ≡ eff ≡ ′′ µ + k2 ′′ µ =0 (8) whichisafunctionofthe“spatial”variableη. Wemight ′s′k − z sk be tempted to solve this equation using WKB approxi- (cid:18) (cid:19) mation, just as it is usually done in basic quantum me- with z = aφ′. Considering an adiabatic vacuum state chanics (QM) [7]. In this approximation the stationary asinitialconHditionforthoseperturbations,allthestatis- solution of Eq. (12) is given by tical properties are characterized by the two-point correlation function, namely by the power spectrum. The c η + µ (η) = exp +i Ω(η )dη + dimensionless power spectrum for scalar and tensor cos- k Ω1/2 ′ ′ mological fluctuations are given, respectively, by (cid:20) Z (cid:21) c η + − exp i Ω(η′)dη′ (17) k3 µ 2 2k3 µ 2 Ω1/2 − P = sk , P = tk . (9) (cid:20) Z (cid:21) Q 2π2 z h π2 a as long as the WKB parameter W is much smaller than (cid:12) (cid:12) (cid:12) (cid:12) Their dependence (cid:12)on th(cid:12)e mode k is defin(cid:12)ed by(cid:12) the spec- 1: (cid:12) (cid:12) (cid:12) (cid:12) tral index in the following way: 2 1 3 Ω 1Ω ′ ′′ dlnP dlnP W 1 . (18) ns−1≡ dlnkQ , nt ≡ dlnkh (10) ≡(cid:12)(cid:12)Ω2 "4(cid:18)Ω(cid:19) − 2 Ω #(cid:12)(cid:12)≪ (cid:12) (cid:12) (cid:12) (cid:12) evaluatedatascalek aHwhenthemodeisoutsidethe Figure 1(cid:12)shows the behavior of(cid:12)W as a function of ≪ horizon. For ns = 1 one has a scale-invariant spectrum the conformal time η in a de Sitter case. According to for the gauge-invariantcosmologicalscalar fluctuation. this plot, the WKB approximation holds for η Letusnowrestrictourselvestothecaseofatestscalar (k aH, subhorizon scales) but not for values c→los−e∞to ≫ field in a de Sitter spacetime where a(η)= 1/(Hη). In zero(k aH,superhorizonscales),whereitequals0.125 this case, cosmologicalfluctuations vanish id−enticaly. (i.e, not≪much smaller than 1) [8]. It fails exactly where If we expand the test scalar field φ(η,~x) in Fourier it is supposed to: near the classical turning point (η tp modes √2/k) and where the effective potential is too stee≡p − φ(η,~x)= a(1η)(2π1)3/2 d3kµk(η)ei~k·~x , (11) (sηolu→tio0n−)o.fFEoqrs.th(1e2s,u1b4h)oisriazopnlasncaelwesa,vwehinerceoEnf≫ormVaefflt,itmhee Z with comoving frequency Ω k, Eq.(17), as expected. ≃ then the equation of motion for each mode is given by µ′k′+Ω2(η)µk =0 (12) III. TRANS-PLANCKIAN BEHAVIOR: NON-LINEAR DISPERSION RELATION with a A modification of the linear dispersion relation (d.r. Ω2(η) k2 ′′ (13) from now on) was proposed by Unruh [9] for describing ≡ − a 3 1 0.8 0.6 ωp 0.4 0.2 0 0 -9 -6 η -3 0 0 0.5 kp3 1 k 1.5 kp2 2 kp1 2.5 p FIG. 1: Total energy (full straight horizontal line), effective FIG. 2: Nonlinear dispersion relation as a function of the potential (dashed blue curve), and WKB parameter (full red physical wavenumber kp for a case with three different solu- curve) as a function of the conformal time η for a linear dis- tions (kp1, kp2 and kp3) of the turning-point equation ωp2 = persion relation. We have used k = Mpl and η is in units of 2H2. H =0.5Mpl and kp is in units of Mpl. 1/Mpl. 2 high-energyPhysicsintheblack-holeradiationemission. He was inspired by sound waves, for which a linear d.r. ceases to be valid when the wavelength gets closer to orsmallerthanthe lattice spacing. JacobsonandCorley [10](seealso[2])alsoproposednonlineartermsinthed.r. that could be justified by the inclusion of higher-order 1 derivatives in the Lagrangian. Other changes can also be introducedbyarguingthat the spacetime symmetries migh not survive at high energies [11]. Following this approach to mimic TPP one writes the comoving frequency ω as ω = a(η)ω (k ) — where ω p p p is the physical frequency — and assumes that ω is a p nonlinear function of kp which differs from the standard 0 -9 -6 -3 0 η (linear) one only for physical wavelengths closer to or smallerthanthePlanckscale. Notethatthisreplacement would be innocuous if the dispersion relation was linear FIG. 3: Total energy (full straight horizontal line), effective in k . potential (dashed blu curve), and WKB parameter (full red p curve) as a function of the conformal time η for the nonlin- In this paper we focus on the d.r. ear dispersion relation Eq. (19). We have used α¯ = 0.1325, ω2(k )=k2 αk4+βk6 , (19) β¯ = 0.004375 and H = 0.5Mpl so that there are 3 turning p p p− p p points(indicated bythevertical lines). The WKBcurvewas multiplied by 2 10−2 so its behavior in the region close to with α > 0 and β > 0, proposed in Ref. [3]. See Fig. 2 the turning poin·ts can be clearly seen. Note, however, that for a sketch of this function. The above expression can theWKBapproximation is valid only in theregion η − →−∞ beseenasamereseriesexpansion,butitisalsofoundin sinceW(η 0 )=0.125. Wehaveusedk=Mpl andη isin solid state physics and describes the rotons [12]. There unitsof 1/M→pl. has been suggestions [13] for including a 3-order term in the nonlinear d.r. above, coming from an effective-field- theoryapproach. Sincesuchodd-powertermsviolateCP, where α¯ αH2 and β¯ βH4 are dimensionless quan- ≡ ≡ we do not consider them here. tities. As in the linear case, the WKB approximation is FollowingthediscussioninSectionII,weplotinFig.3 valid only when η . →−∞ theanalogousquantitiestoW (18),thetotalenergyE To compare our results with Ref. [4] we consider only k2, and the effective potential, the latter being given b≡y a particular range of values for α and β, necessary for a positivenonlineardispersionrelationwith threerealdis- 2 tinct solutions for the classical turning-points equation V (η) (α¯k4)η2 (β¯k6)η4+ , (20) eff ≡ − η2 ωp2 =2H2 (see,also,[14]). Therequesttohavethreereal 4 turning-points and a nonlinear ω that differs from the To proceed with either a numerical or an analytical ap- p standard linear one only for physical wavelengths closer proachtofindthesolutionofEq.(24)wehavetofindthe to or smaller than the Planck scale has as consequence initial conditions associated to our nonlinear dispersion that H should be comparable with M . In fact, every- relation in a de Sitter spacetime. We consider a initial pl where in the paper we consider H =0.5M . time η and a fixed value of k (which should be much pl i Defining z 3β¯/α¯2 one obtains the following ranges larger than a(η )H). i ≡ for z and α¯: Inthiscase,forthisfixedvaluek aH,thecomoving frequency ω becomes nearly equal ≫to √βk3/a2 and the 3/4<z <1 , g(z)<α¯ <f(z) (21) WKB vacuum is given by with 1 1 1 3z µ (η) = g(z) 1 (1 z)3/2 (22) wkb −√2β1/4k3/2Hη × ≡ 3z2 2 − − − (cid:20) (cid:21) H2 exp iβ1/2k3 (η3 η3) , (26) 1 3z × − 3 − i f(z) 1+(1 z)3/2 (23) (cid:20) (cid:21) ≡ 3z2 2 − − (cid:20) (cid:21) its derivative by wherez >3/4isthephysicalconditiontohaveapositive nonlinear d.r.. The problem of finding the power spec- 1 1 1 µ (η) = +i β1/4k3/2Hη trum from this particular nonlinear dispersion relation ′wkb √2β1/4k3/2Hη2 √2 × has been tackled in Ref. [4]. In that paper the authors (cid:26) (cid:27) H2 introducedanapproximatepiecewise formofthe nonlin- exp iβ1/2k3 (η3 η3) , (27) ear dispersion relation (see Sec.VI) and apply the WKB × − 3 − i (cid:20) (cid:21) approximation for k such that ω2 > 2H2. They argue p p that the amplitude of the effects of the nonlinearity is and our initial conditions by proportional to ∆ (k k )/k where k and k p1 p2 p1 p1 p2 ≡ − 1 1 are, respectively, the first and the second turning points µ (η ) = (28) wkb i from the right of Fig. 2. Still according to Ref. [4], ∆ −√2β1/4k3/2Hηi measuresthe“time”spentinthe regionwherethe WKB 1 1 β1/4k3/2 approximationis not satisfied,butit actuallyis simply a µ′wkb(ηi) = √2β1/4k3/2Hη2 +i √2 Hηi (29) measure of the distance between the two largest turning i points—k andk inFig.2. ItisclearlyseeninFig.3 p1 p2 Tobemoreaccuratetheaboveequationsaremeaning- that ∆ is much smaller that such time interval. ful only when We show below the full result (i.e, the exact solution withinthethree-piecewiseapproximation)andtwoother k4 k6 2 k6 differentapproaches—onenumericalandtheothersemi- Ω2(η) k2 α +β β (30) analytical — for evaluating the nonlinear effects on the ≡ − a2 a4 − η2 ≃ a4 power spectrum. Butbeforeproceedingtoprovingthe claimsabove,we so one obtains the following constraints on β: shall first determine the initial conditions we assume. a6 a4 a2 β 2H2 , β , β α . (31) ≫ k6 ≫ k4 ≫ k2 IV. THE EQUATION OF MOTION: INITIAL CONDITIONS The WKB solution is an exact one at the infinite past andthusthechoiceoftheadiabaticvacuumissomewhat The equation of motion for the mode function of a “natural”. Wealsorecallthatallvacuaprescriptionsare test scalar field, following the approach just introduced equivalent up to zero-order when the WKB approxima- to consider the TPP effects, is given by tion holds [15]. The choice of a different set of initial conditions at a a µ′k′+ a2ωp2− a′′ µk =0 (24) given η = η1, though, can always be seen as the out- (cid:18) (cid:19) come froma particular (trans-Planckian)evolution from a different (and perhaps more consensual) set of initial with ω given by Eq.(19). p conditions defined at η = η < η . In other words, the For k aH the WKB approximation is valid (see 0 1 Fig. 3), t≫he term a2ω2 = ω2 dominates and the mode choice of initial conditions is equivalent to the choice of p the physics that takes place before the moment when function is given,chosingthe adiabatic vacuum(see [4]), they are set. Nevertheless, that choice does not replace by: the discussion on the physics that takes place after that 1 η moment, while energies above the Planck scale are still µ (k,η)= exp i ω(k,τ)dτ . (25) wkb 2ω(k,η) − at play. (cid:20) Zηi (cid:21) p 5 12 z=0.775 10 8 1 Cαβ 6 ωp z=0.950 4 2 0 -2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 ∆ k p FIG.4: Correction Cαβ onthepowerspectrum,with z vary- FIG.5: Three-piecewiselinearapproximationtothenonlinear ing from 0.775 up to 0.950 with step 0.025, as a function of d.r. as suggested in Ref. [4]. The horizontal line is √2H. ∆ from a full numerical calculation using the nonlinear d.r. Regions 1,2,3 are defined by the vertical lines and labeled (19). from right to left. kp is in unitsof Mpl. V. NUMERICAL APPROACH thisnonlineard.r. yieldsnocorrectionwhatsoevertothe power spectrum if the parameters happen to be around Inthissectionwefindthecorrectiontothepowerspec- those values. trum,forourtestscalarfieldinadeSitterspacetimewith a nonlinear d.r., using a fully numerical approach with initial conditions given by Eqs. (28,29). This has been VI. LINEAR APPROXIMATION done using a C code and the GSL library [16], where we have set H = 0.5M and considered a fixed value pl of k a(η )H, stopping the evolution at a time η for Since the WKB factor(18) is notmuchsmallerthan 1 i f which≫k a(η )H. Wehaveverifiedthattheresultisin- except for k aH, one cannot use the WKB expression f ≪ ≫ dependentfromthevalueofk,namelythat,asexpected, (17)forthesolutionof(24)(exceptinthe forementioned the power spectrum is still scale independent. region, of course). Nevertheless, there is indeed an ex- We write the power spectrum as act solution of Eq. (24) if we approximate the nonlinear dispersionrelationbyastraightlines,asdoneinRef.[4]. 2 H If we write ω (k )=A k +B for each of the regions P (k)= [1+C ] . (32) i p i p i φ 2π αβ i=1,2,3 (see Fig. 5), the solutions are (cid:18) (cid:19) The function C represents the correction due to the αβ iB 9 i nonlinear d.r., which obviously depends on the parame- µ (η) = c W , γ ,2iA kη + i i M i i ters α and β. " H r4 − # In Fig. 4 we show the correctionC for different val- αβ iB 9 i ues of z (going from z =0.775 to 0.950 with step 0.025) + diWW , γi,2iAikη (33) in function of ∆. We can clearly see that ∆ is not the " H r4 − # onlyparameterthatplaysaroleinthe calculationofthe nonlinear effects on the power spectrum. Indeed, effec- where γi (Bi/H)2 and WM,W(, , ) are Whittaker ≡ · · · tive potentials with the same ∆ have different heights, functions [17]. We pick Ai and Bi such that the posi- depths and steepnesses, all of which influence effects on tionsofthemaximumandminimumofthepiecewised.r. thepowerspectrum. Evenforsmall∆,thecorrectioncan coincide with those of the full nonlinear case and that be large depending on the values of α and β. Therefore, the firstandthe second“turningpoints”kp1 andkp2 are it can hardly be considered a perturbation. the same as well (see Fig. 5). In this way we have the ¿From Fig. 4, we can note that for the smaller value same ∆ of the exact nonlinear d.r.. Of course, we pick of z one obtains a minimum for Cαβ at ∆ > 0 (which A3 = 1 and B3 = 0, corresponding to the linear d.r. in corresponds to a finite value of β for a fixed z). Besides the long-wavelength(small kp) limit. we have no correction (C = 0) for particular values The coefficients c , d are given by matching the αβ 1 1 of z and ∆. This means that we could still have no asymptotic behavior of the previous equation to the ini- correction in spite of finite α and β. In other words, tial conditions yielded by the WKB approximation (see 6 below). The former is given by [17] 12 µ1(η) ≈ e−iA1kη(2iA1kη)+iBH1 c1Γ(Γ1(1++ν2+ν)κ) +d1 + 10 z=0.775 (cid:20) 2 (cid:21) + e+iA1kη(2iA1kη)−iBH1 c1Γ(Γ1(1++ν2ν)κ) (34) 8 (cid:20) 2 − (cid:21) 6 β α where ν 9/4 γ and κ 2B /H. Comparing C ≡ − 1 ≡ − 1 4 Eq. (34) to the expected WKB solution p z=0.950 1 η 2 µ (η) = exp i ω(η)dτ wkb √2ω − (cid:20) Zηi (cid:21) 0 1 η iBH1 = e−iA1k(η−ηi) (35) -2 √2A1k (cid:18)ηi(cid:19) 0 0.1 0.2 0.3 0∆.4 0.5 0.6 0.7 0.8 we get FIG.6: Correction tothespectrumusingthe3-piecewiseap- c =0 proximation,withz varyingfrom 0.775upto0.950withstep 1 (36) d = 1 (2iA kη ) iB1/H. 0.025, as a function of ∆. (cid:26) 1 √2A1k 1 i − Note that the expression for d reduces to the expected 1 Suchacompactnotationallowusto write intermsof form when A1 =1 and B1 =0, which correspond to the C3 as usual d.r.. C1 asTisheusaumaplllyituddoneeofinthQeMgr.oSwiinncgemwoedkencoawn btheecaexlcaucltatseod- C3 =M−41·M3·M−21·M1·C1 (41) The coefficient of the growing mode is given by d — lution in each region (Eq. (33)), all we have to do is to 3 the second component of — since it is the coefficient match them and their derivatives at the boundaries. In 3 C of the divergent Whittaker function when η 0 : matrix notation, at the boundary between regions 1 and − → 2, we write: iB √1 4γ i W 3, − 3,2iA k2η − (42) W 3 WM(1)(η12) WW(1)(η12) c1 (cid:18)−2H 2 (cid:19)→ kη = (37) while W 0 at the same limit. Therefore, M    → WM′(1)(η12) WW′(1)(η12) d1 i  W(2)(η ) W(2)(η )  c  µ3(η)→ −kηd3 (43) M 12 W 12 2 = as η 0 . We can thus write the power spectrum as    − WM′(2)(η12) WW′(2)(η12) d2 → k3 µ 2    P (k) = 3 where the superscripts (1,2) indicate the value of the φ 2π2 a sautbtshcartipbtoiuinndEaqry.((3s3e)e,Fη1ig2.≡5)−aknpd12(/′)(Hinkd)icisattehsedvearliuveatoifvηe ≃ H2π22k(cid:12)(cid:12)(cid:12)|d3|(cid:12)(cid:12)(cid:12)2, (44) with respect to η. We will write the above equation in a in the limit η 0 (k aH). This is, as expected, more compact form as → − ≪ scale invariant. = (38) In Fig. 6 we have plotted the correction of the spec- 1 1 2 2 M ·C M ·C trum, due to this three-piecewise approximation, as a with anobviousnotation. Analogously,we canwrite the function of ∆. The reason for such large values is that matching at the boundary of regions 2 and 3 as the nonlinear d.r. canbe qualitatively different from the linear one (ω k) at large η even for small ∆. W(2)(η ) W(2)(η ) c ∼ | | M 23 W 23 2 As it is clear, comparing Fig. 6 to Fig. 4, the re- = (39) sults obtained with this three-piecewise approximation    WM′(2)(η23) WW′(2)(η23) d2 areprettydifferentfromthenumericaloneobtainedwith  W(3)(η ) W(3)(η )  c  the exact nonlinear d.r.. M 23 W 23 3 =    WM′(3)(η23) WW′(3)(η23) d3 VII. SEMIANALYTICAL APPROACH    and as In this section we use k as the independent variable. p = . (40) Since it is proportional to η (k = k/a = ηHk), there 3 2 4 3 p M ·C M ·C − 7 is no particular advantage to choose either way, but this 10 variable is more “physical” and thus one can rely on her/his physical intuition. For a linear d.r., Eq. (12) is then written as 8 d2µ 1 2 k + µ =0. (45) dk2 H2 − k2 k 6 p (cid:18) p(cid:19) E Using a nonlinear d.r. amounts to the substitution 4 1 1 ωp2(kp) (46) H2 → H2 k2 p 2 in the previous equation. Although there is no exact solution of Eq. (45) with the substitution (46) when one 0 0 0.5 1 k 1.5 2 2.5 3 uses the d.r. (19), we still can get a fair analytical ap- * k p proximation by writing FIG. 7: Effective potential Veff(kp) (thick black line) and its Σ2(kp) ≡ H12ωp2k(2kp) − k22 aΣp2p(kropx)imanadtioEnqss(.th(4in8)c,ofloorreαd¯ l=ine0s).1g3i2v5e,nβ¯by=Ve0ff.0(k0p4)37=5Ean−d p p H =0.5Mpl, as a function of kp (in Mpl units). The straight = Hβ2kp4− Hα2kp2− k22 + H12 hoforki∗zo(nsteaeltleinxetifsorEd≡efi1n/itHio2n.)T. heverticallinemarksthevalue p Σ2(k ) β k4 α k2+ 1 c ,k k 1 p ≡ H2 p− H2 p H2 − p ≥ ∗ (47) where Ht (k ) HeunT(A,0,B,+iζ) and ≈ Σ2(k ) α k2+ 1 2 d ,k k 1 p ≡  2 p ≡−H2 p H2 − kp2 − p ≤ ∗ Ht2(kp) ≡ HeunT(A,0,B,−iζ) are triconfluent Heun functions [18], with wherec, d and the matching point k are defined by requiring that ∗ 3 2/3 α2 A 1+cH2+ , (51) Σ21(k )=Σ22(k )=Σ2(k ) ≡ (cid:18)2H2√β(cid:19) (cid:18)− 4β(cid:19) ∗ ∗ ∗ 1/3 , (48) 3  ddΣk21 = ddΣk22 B ≡ (cid:18)2H2β2(cid:19) α (52) k∗ k∗ (cid:12) (cid:12) 1/3 which correspond t(cid:12)(cid:12)o k = (H(cid:12)(cid:12)2/β)1/6, d = (β/H2)1/3 ζ ≡ 3√2β1/4H kp. (53) andc=2(β/H2)1/3. T∗heapproximatedexpr−essionshave (cid:16) (cid:17) We can constrain the coefficients (c ,d ) in Eq. (50) by the same limiting behaviors (both at k 0 and k 1 1 p p → → comparing the asymptotic behavior of the above equa- ) than the originalone. That feature providesanother ∞ tion to the expected WKB one, Eq. (26). The asymp- piece of information on the dependence of the outcome totic limit of interest here is taken along a Stokes line on the details of the nonlinear d.r. for intermediate k , p (arg(iζ)=π/2)ofthe triconfluentHeunfunction, which asopposedtoits small-andlarge-scalelimits. SeeFig.7 means that there are two equally dominant terms in the for a qualitative comparison. asymptotic expansion: The calculation itself is carried out by the usual approach in QM, as in the previous section. The require- HeunT(A,0,B,iζ) ζ 1[a+bexp( iζ3)] (54) − ments that the “wave function” and its first derivative ∼ − are continuous at k are easily accomplished since the as ζ + , where a and b are constants. Since neither solutions of the equa∗tion of motion (45,46) with the ap- → ∞ one is dominant over the other, both must be taken into proximations(47)are knownanalytically. For k>k we account. The forementioned comparison yields find: ∗ d b c a=0 d2fd1k(2kp) −Σ21(kp)f1(kp)=0, (49) (cid:26) −1c1−b+1d1a=C (55) p whereC i/[(3H√2β1/4)1/3√k]. Thereareonly2equa- ≡ tionsfor4unknowns,whichshouldbeexpectedfromthe i α √β f (k ) = c exp + k + k3 Ht (k )+ lack of a dominant behavior in the asymptotic expan- 1 p 1 H −2√β p 3 p 1 p (cid:20) (cid:18) (cid:19)(cid:21) sion along a Stokes line, as mentioned above. One could i α √β choose whichever 2 of the above parameters (say, a and +d exp k + k3 Ht (k ) (50) 1 −H −2√β p 3 p 2 p b) to be specified by comparing the final result to the (cid:20) (cid:18) (cid:19)(cid:21) 8 outcome of the numerical calculation using the approxi- g2(k ) h2(k ) c ∗ ∗ 2 mated effective potential. Formally, that determination = (64) is supposed to be done at every value of α and β, which  dg2(kp) dh2(kp) d  would render our semianalytical approach useless. Nev-  dkp kp=k∗ dkp kp=k∗  2 (cid:16) (cid:17) (cid:16) (cid:17)   ertheless,wehavefoundthatwritingbandd intermsof 1 where aandc yieldsaqualitativelygoodbehaviorfordifferent 1 values of z, i.e, for different pairs α,β when compared +i α √β { } g (k ) exp − k + k3 Ht (k )(65) to the numerical calculation (see Fig. 8 below). Such 1 p ≡ H 2√β p 3 p 1 p procedure yields (cid:20) (cid:18) (cid:19)(cid:21) i α √β b = c1a (56) h1(kp) ≡ exp −H 2−√βkp+ 3 kp3 Ht2(kp)(66) d (cid:20) (cid:18) (cid:19)(cid:21) 1 1 3 √αk2 C C2+4a2c2 g (k ) W D, , p (67) d1 = − ± 2a 1 (57) 2 p ≡ kp M 4 H ! p The sign in the above equation was again numerically p1 3 √αkp2 h (k ) W D, , (68) determined to be the lower one (see below). Following 2 p ≡ kp W 4 H ! this line of reasoning,we used a=0.1 and c = 1.8 for 1 all z. − which can be wpritten in a more compact form, as in the The last step of the semianalytical procedure is the previous section, as evolution in the second region, when k <k : ∗ = . (69) 1 1 2 2 d2f (k ) M ·C M ·C 2 p Σ2(k )f (k )=0, (58) dk2 − 2 p 1 p As before, one can invert such equation and write p whose exact solution is C2 =M−21·M1·C1, (70) f (k ) = c 1 W D,3,√αkp2 + which shows once more that d2 is a linear combination 2 p 2 kp M 4 H ! of c1 and d1. Since we have fixed c1 from the beginning, we are not able to find analytically the dependence of + d p1 W D,3,√αkp2 (59) d2 on k and, therefore, to say if the spectrum is scale 2 W kp 4 H ! invariant,asitshouldbe. Nevertheless,weapproachthe sameproblemnumericallyandweverifythattheproblem where D (1 dH2)p/(4H√α). exhibits such invariance. ≡ − Thecoefficientd2 determinestheamplitudeoftheper- In order to be able to measure the acuracy of the turbations since the function WW(kp) is the growing so- semianalytical approach, we have used the approxima- lution: tion (47) and numerically evolved the initial conditions (28,29). Theevolutionwassplitintwopieces: fork >k W (k ) √πH1/4 1 lim W p = (60) and k < k . Such calculation, besides fixing the sign in∗ kp→0 kp 2α1/8Γ(ξ)kp Eq. (57), a∗lso yields the “best” (i.e., more robust with respect to changes in α and β) values of c = 1.8 and whereξ 5 1 dHp2 . Theaboveexpressionallowsus 1 − ≡ 4 − 4−H√α a=0.1. to write th(cid:16)e spectrum a(cid:17)s InFig.8 weshowthe functionCαβ fordifferentvalues of z as a function of ∆, as given by the numerical evolu- k3 f 2 2 tion using the approximatedexpressionsfor the effective P (k) (61) φ ≡ 2π2 a potential and as given by the semianalytical approach (cid:12) (cid:12) k(cid:12)(cid:12)√H(cid:12)(cid:12) with the same approximation. We can see that the cor- ≃ 8πα1(cid:12)/4Γ(cid:12)2(ξ)|d2|2 (62) rectiondependsalsoonz,showingthat∆isnottheonly variable to work with. H 2 The semianalytical solution is not expected to work = 2π (1+Cαβ), (63) well for small β¯, which means, for a fixed z, small ∆. (cid:18) (cid:19) For that, we shouldhavetakeninto accountthe next-to- in the superhorizon limit k H. The coefficient d p 2 leading order terms in the asymptotic expansion of the ≪ is determined by the forementioned procedure, requiring Heun functions, Eq. (54). That would have, however, that the function and its derivative are continuous at introduced new parameters that would have had to be k =k . In matrix notation, it can be written as fixed by comparing to the numerical solutiononce more. ∗ Since we see no advantage in having a large number of g (k ) h (k ) 1 1 c ∗ ∗ 1 such parameters, we did not do so. =  dg1(kp) dh1(kp)  d  We note thatwe reproduceaminimum Cαβ ata finite  dkp kp=k∗ dkp kp=k∗  1 ∆asintheprevioussection. Thisapproximationyieldsa  (cid:16) (cid:17) (cid:16) (cid:17)   9 40 with ηi = ti/(1 p). The equation of motion for the − Mukhanov variable, Eq.(8), becomes (using η as the in- 35 z=0.775 z=0.950 z=0.775 dependent variable) 30 a µ + k2 ′′ µ =0 (76) 25 ′s′k − a sk (cid:18) (cid:19) 20 β andthe canonicallynormalizedsolution,associatedwith α C 15 the adiabatic vaccum for k aH, is given by (see, for z=0.950 ≫ example, [20]) 10 πη 1/2 5 µ = H(1)( kη), (77) k − 4 ν − 0 (cid:16) (cid:17) withν =3/2+1/(p 1). So,inthelong-wavelengthlimit, − -5 we obtain the following scale dependent power spectrum 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ∆ 2p FIG.8: CorrectionCαβ asafunctionof∆whenzvariesfrom PQ(k)= (2π1)2 −η2 p−1 π1Γ2(ν)k−p−21 (78) 0.775to0.950 (asshownbythelabels)forthesemianalytical (cid:18) i(cid:19) approach (dashed bluelines) and for thenumericalevolution with a spectral index n = 1 2 . Using this result, with approximated effective potential (47) (full red lines). s − p 1 namely the dependence on k of th−e power spectrum, the range p<60 is disfavoredat 2σ by the observation(see, correctionC with a goodqualitative behavior as com- for example, [21]). αβ pared to Fig. 4, but a large difference in the magnitude Nowwewanttoseeiftheintroductionoftheparticular of the effect. nonlinear d.r. given in Eq. (19) changes the dependence of the power spectrum on k. Let us set our background: Eq. (76) can be written, using Eq. (75), as VIII. POWER-LAW INFLATION 2p2 p 1 A numerical approach can also be applied to find the µ′k′+(cid:20)k2− (1−−p)2η2(cid:21)µk =0. (79) correction to the power spectrum in a realistic model For p 1, this can be approximated as of inflation for the nonlinear dispersion relation under ≫ consideration. In this sectionwe restrictourselvesto the µ + k2 2a2H2 µ =0, (80) case of power-law inflation [19] where, in proper time, ′k′ − k a(t) tp, with p>1. This expansionis generated by an so, in this limit and(cid:2)following th(cid:3)e same reasoning as be- ∼ exponential potential fore, to obtain a positive nonlinear d.r. that gives three real distinct classical turning-points (ω2 = 2H2) at the p λ initial time [25] we require that V(φ)=V exp (φ φ ) (71) 0 i −M − (cid:20) pl (cid:21) g(z) f(z) 3/4<z <1 , <α< . (81) withV = Mp2lp(3p 1)andλ= 2 1/2. Thescalefactor H(ηi)2 H(ηi)2 0 t2i − p andthe homogeneussolutionof(cid:16)the(cid:17)scalarfieldaregiven with g(z) and f(z) given in Eqs. (22,23). by The new equation of motion will be given by a(t) = tt p, (72) µ′k′+ a2 ka22 −αka44 +βka66 − (21p2−p)p2η12 µk =0 (cid:18) i(cid:19) (cid:20) (cid:18) (cid:19) − (cid:21) (82) t φ(t) = φi+Mpl(2p)1/2log (73) and, repeating the calculation in Sec. IV with the new t i background,one obtains the following initial conditions respectively, and the slow-roll parameters by 1 µ (η ) = (83) wkb i ǫ Mp2l Vφ 2 = 1 , ǫ M2 Vφφ = 2, (74) √2β1/4k3/2 1 ≡ 2 V p 2 ≡ pl V p 1 p 1 β1/4 where Vφ ≡ dV(cid:16)/dφ(cid:17)and so on. In conformal time η, the µ′wkb(ηi) = −√2β1/4k3/2p−1ηi −i √2 k3/2(84) scale factor and the Hubble parameter become We now proceed with the numerical analysis taking p p t = (p 1)/M (and thus η = 1/M ). The restric- a(η)= η 1−p , H(η)= p η p−1 1 (75) tiion to a−pply thpel adiabatic iniitial−conditpilon ( kη 1) ηi −p−1 ηi η − i ≫ (cid:16) (cid:17) (cid:16) (cid:17) 10 -3.805 -2.26 -3.81 -2.28 -2.3 -3.815 -2.32 -3.82 ) ) Q Q -2.34 P P 3k -3.825 3k (n (n -2.36 l l -3.83 -2.38 -3.835 -2.4 -3.84 -2.42 -3.845 -2.44 4.6 4.8 5 5.2 5.4 5.6 5.8 4.6 4.8 5 5.2 5.4 5.6 5.8 ln(k) ln (k) FIG. 9: Logarithm of the power spectrum in a power-law FIG. 10: Logarithm of the power spectrum in a power-law modelofinflation infunctionofln(k)foralineard.r.. k isin modelofinflation usingthenonlineard.r. (12)in functionof unitsof Mpl. ln(k). k is in unitsof Mpl. becomes equivalentto k M ; therefore,weareindeed pl ≫ in the energy scale of TPP. Besides, with those initial conditions and in the limit p 1, we have H(η ) M i pl ≫ ≃ and the new d.r. becomes very different from the linear one only for k = (M ). For the numerical analysis we pl O take z = 0.80, α = f(0.80)+g(0.80) and the limiting value 2H(ηi)2 p = 60, making the analysis for k between 100M and pl 300M and going from η to a time for which k aH. pl i ≪ We give in Fig. 9 the logarithm of the power spectrum calculated for the standard linear case in function ofln(k),whileinFig. 10thelogarithmofthepowerspec- trum calculatedfor the nonlineard.r. case in functionof ln(k)formanydifferentpointsinsidetheaforementioned range of k. As one can see from this plot, the power spectrum for the nonlinear case still has a power-lawde- pendence on k, but this d.r. produces a change both in FIG. 11: Plot of the function 1/(1 + W) in terms of the the normalization factor and in the spectral index when wavenumberk (inunitsofMpl)andtheconformaltimeη us- compared to the linear case. Fitting the data in Fig. 10 ingthesameparametersasinFig.10(seetextfortheprecise we obtain a spectral index smaller than that of the lin- values). One can clearly see that the WKB approximation ear case: 0.857 instead of 0.966. The new spectral index holds (W 1) only for early times. ≪ wouldbe almostthe sameas thatof the linear caseif we have taken p 15. One obtains similar results starting ∼ from different values of p or with different values for the defined in Eq. (18). One can easily see that the WKB parameters α and β. Therefore, the introduction of the approximation does not hold at small absolute values of nonlinear d.r. gives a stronger dependence of the power the conformal time, which stresses our very point[26]. spectrumonk and,asaconsequence,apossibledifferent range of disfavored values for p. A different approach to this problem in a power-law IX. CONCLUSIONS model is shown in [22]. The authors use the minimum- uncertaintyprincipletofixtheinitialconditionsandalso In this paper we have shown that the nonlinear d.r. find corrections to the power spectrum. We also would (19) presents nonlinear results, i.e, which are not shown like tounderline thatour resultsarenotindisagreement in the perturbative approaches used in previous papers with Ref. [23]. Indeed, we consider a regime for which in the subject. In particular, the correction C can αβ H is of the order of M while the results of [23] assume be almost as large as 10 depending on the value of the pl H M . parameters α, β. Therefore, it can hardly be considered pl ≪ UsingthesameparametersasinFig.10,wehaveplot- a perturbation. ted in Fig. 11 the behavior of 1/(1+W), where W is Our results agree, in part, with the literature on the