3 January 2002

Physics Letters B 524 (2002) 15–20www.elsevier.com/locate/npe

Curvature perturbation at the local extremumof the inflaton potential

Jun’ichi Yokoyama, Shogo InoueDepartment of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan

Received 13 April 2001; received in revised form 1 November 2001; accepted 16 November 2001Editor: J. Frieman

Abstract

The spectrum of curvature perturbation generated during inflation is studied in the case the inflation-driving scalar field(inflaton)φ crosses over its potential extremum. It is shown that the nondecaying mode of perturbation has a finite value and aproper formula is given. The result is also extended to more general cases whereφ is nonnegligible. 2002 Elsevier ScienceB.V. All rights reserved.

Inflation in the early universe [1] provides a mech-anism to generate primordial density fluctuations outof quantum fluctuations of the inflation-driving scalarfield, inflaton φ [2], in addition to explain the large-scale homogeneity and isotropy. In the standard infla-tion models such as new [3] and chaotic [4] inflation,inflation is driven by the potential energy of the in-flaton,V [φ], as it slowly rolls the potential hill andthey predict formation of adiabatic fluctuations witha nearly scale-invariant spectrum. More specifically,the amplitude of the curvature perturbation on comov-ing hypersurfaces,Rc , on comoving scaler = 2π/kis given by the formula [5,6],

(1)Rc(r)∼= H 2

2π |φ|∣∣∣∣tk

(2)∼= 3H 3

2π |V ′[φ]|∣∣∣∣tk

,

E-mail address: yokoyama@vega.ess.sci.osaka-u.ac.jp(J. Yokoyama).

where a dot denotes time derivative andH is theHubble parameter. Here the right-hand-sides shouldbe evaluated att = tk when the relevant scalekleft the Hubble radius during inflation. In the secondapproximate equality (2), we have used the equationof motion with the slow-roll approximation,

(3)3Hφ + V ′[φ] = 0.

The reason why (1) gives an almost scale-invariantspectrum is that bothH and φ change very slowlyduring slow-roll inflation.

Recently, however, new classes of inflation mod-els have been proposed in whichφ is not necessar-ily slowly rolling during the entire period of inflation[7–9]. Wheneverφ is nonnegligible in the full equa-tion of motion of the homogeneous part ofφ,

(4)φ + 3Hφ + V ′[φ] = 0,

the slow-roll approximation (3) fails. There are twotypical cases where this happens. One is the caseφ vanishes. The proper analysis of the amplitude ofdensity fluctuation in this case was recently done in[10], where it was found that the correct amplitude is

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)01369-7

16 J. Yokoyama, S. Inoue / Physics Letters B 524 (2002) 15–20

not given by the formula (1), which is divergent, butby (2). It has been further argued recently [11] that theformula (2) may continue to be a useful approximationeven when evolution ofφ is not described by a slow-roll solution. The other is the caseV ′[φ] vanishes,namely, when the inflaton crosses over an extremum ofthe potential. In this case, (2) is divergent and wouldnot be applicable. The purpose of the present paperis first to analyze density fluctuation in this situationand then extend it to the generation of perturbationsin more general nonslow-roll inflation models whosepotential satisfies several conditions indicated belowto allow our analytic treatment.

First, as a specific model we consider the caseφ overshoots the local maximum of the potential atφ = 0 during the new inflation stage in the chaotic newinflation model [7]. More specifically, we approximatethe potential near the extremum as

(5)V [φ] = V0 − 1

2m2φ2, m2 > 0,

and consider the caseφ crosses overφ = 0 fromthe positive side to the negative one with a smallkinetic energy so that the cosmic expansion rateH ispractically constant in the relevant period. We assumeH 2 � m2 andV0 � m2φ2. Then the solution of (4)reads:

φ(t)

(6)= φ∗[

1

λ+ − λ−eλ+(t−t∗) − 1

λ+ − λ−eλ−(t−t∗)

],

φ(t)

(7)= φ∗[

λ+λ+ − λ−

eλ+(t−t∗) − λ−λ+ − λ−

eλ−(t−t∗)],

(8)λ± = −3

2H ±

√9

4H 2 +m2,

where we have setφ(t∗) = 0 andφ(t∗) = φ∗. Usingλ+ ∼= m2/(3H) andλ− ∼= −3H(1 + m2/(9H 2)), wefind

φ(t)∼= φ∗3H

[e(m

2/(3H2))H(t−t∗)

− e−3H(1+m2/(9H2))(t−t∗)]

(9)

= φ∗3H

[e(m

2/(3H2))H(t−t∗)

− e−(m2/(3H2))H(t−t∗)(a(t∗)a(t)

)3],

φ(t)∼= φ∗[m2

9H 2e(m

2/(3H2))H(t−t∗)

+(

1− m2

9H 2

)e−3H(1+(m2/(9H2))(t−t∗)

]

= φ∗[m2

9H 2e(m

2/(3H2))H(t−t∗)

+(

1− m2

9H 2

)e−(m2/(3H2))H(t−t∗)

(a(t∗)a(t)

)3]

(10)≡ φs (t)+ φr (t),

with

φs(t) ≡ φ∗m2

9H 2e(m

2/(3H2))H(t−t∗),

(11)

φr (t) ≡ φ∗(

1− m2

9H 2

)e−(m2/(3H2))H(t−t∗)

(a(t∗)a(t)

)3

,

near the local maximum of the potential. Herea(t) ∝eHt denotes the scale factor. As is seen here,φ canbe decomposed into a slowly varying modeφs (t) anda rapidly changing decaying modeφr(t) which isapproximately proportional toa−3(t) ∝ e−3Ht .

We now incorporate linear metric perturbations tothe spatially flat Robertson–Walker background in thelongitudinal gauge,

ds2 = −[1+ 2Ψ (x, t)

]dt2

(12)+ a2(t)[1+ 2Φ(x, t)

]dx2,

where we use the notation of [12] for the gauge-invariant perturbation variables [13]. Hereafter, allperturbation variables represent Fourier modes like

(13)Φk =∫

d3x

(2π)3/2Φ(x, t)eikx,

and we omit the suffixk. We use the followingcombination of gauge-invariant variables [5,6,10,14,15]:

(14)Y = X − φ

HΦ, where

(15)X = δφ − a

kφσg,

is the gauge-invariant scalar field fluctuation withσgbeing the shear of each constant time slice. The lattervanishes on Newtonian slice including the longitudinalgauge.

J. Yokoyama, S. Inoue / Physics Letters B 524 (2002) 15–20 17

The above quantityY is related to the gauge-invariant variableRc as

(16)Rc = Φ − aH

kv = Φ + 2

3

Φ +H−1Φ

1+w= −H

φY,

in the present situation where the universe is domi-nated by the scalar field. Herev is a gauge-invariantvelocity perturbation [12] andw denotes the ratio ofpressure to energy density.

From the perturbed Einstein equations, we obtainthe following equations:

(17)Xφ − φX + φ2Φ = 2

κ2

k2

a2Φ,

(18)Φ +HΦ = −κ2

2φX,

whereκ2 = 8πG. From these equations and (14), theequation of motion ofY reads:

(19)Y + 3HY +[(

k

a

)2

+M2Yeff

]Y = 0,

with

M2Yeff ≡ V ′′(φ)+ 3κ2φ2

(20)− κ4

2H 2 φ4 + 2κ2 φ

HV ′(φ).

This equation has the following exact solution in thelong-wavelength limitk → 0 [14,16]:

(21)Y (t)= c1(k)Y1(t)+ c2(k)k3Y2(t),

(22)Y1(t)= φ

H,

(23)Y2(t)= φ

H

t∫ti

H 2

a3φ2dt,

wherec1(k) andc2(k) are integration constants to bedetermined by quantum fluctuations generated duringinflation, andti is some initial time which may bechosen arbitrarily because its effect can be absorbedby redefiningc1(k). We chooseti so thatY2 consistsonly of the decaying mode. Despite its appearance,the decaying modeY2(t) is regular even at the epochsφ = 0 [14]. In the second term of the right-hand-sideof (21), an extra factork3 has been introduced so thatthe scale factora(t) appears in the rescaling-invariantform of k/a(t), and c1(k) and c2(k) have the samedimension [10].

Using (11), we find

Y1(t) = φs(t)

H+ φr (t)

H

(24)

= φ∗H

[m2

9H 2e(m2/(3H2))H(t−t∗)

+(

1− m2

9H 2

)e−(m2/(3H2))H(t−t∗)

×(a(t∗)a(t)

)3],

Y2(t) = − 1

3φs(t)a3(t)

(1+ 2m2

9H 2

)−1

= − 3H 2

m2φ∗

(1+ 2m2

9H 2

)−1

e−(m2/(3H2))H(t−t∗)

(25)× 1

a3(t),

so that the solution in the long wavelength limit isgiven by

Y (t)= c1(k)

(φs (t)

H+ φr (t)

H

)

(26)− c2(k)

H 3

3φs(t)

(1+ 2m2

9H 2

)−1(k

a(t)H

)3

.

We can also incorporate the finite wave numbercorrection using the iterative expression [10,14] as

Y = c1(k)Y1 + c2(k)k3Y2

(27)+ k2Y1

∫aY2Y dt − k2Y2

∫aY1Y dt.

In the present case, we find the leading finite-wavenumber correction is proportional to(k/a(t))2. As aresult, we obtain

Y (t)= c1(k)φs(t)

H

+ c1(k)φs(t)

2H

(1− 2m2

3H 2

)(k

a(t)H

)2

+ c1(k)φr (t)

H

− c2(k)H 3

3m2φs (t)

(1− 2m2

9H 2

)(k

a(t)H

)3

(28)+ · · · ,to the lowest-order inm2/H 2.

18 J. Yokoyama, S. Inoue / Physics Letters B 524 (2002) 15–20

Next we consider evolution ofY in its short-wavelength regime in order to set the initial conditionout of quantum fluctuations. Since we are primarilyinterested ink-mode that leaves the horizon when theinflaton crosses over its potential extremum att = t∗,we only need to consider the evolution ofY whenm2|t− t∗|/3H � 1 holds during inflation. Thus, in thisregime (19), becomes

(29)Y + 3HY +(k

a

)2

Y = 0.

Under the condition|H/H 2| � 1, the solution of thisequation, satisfying the normalization condition

(30)Y Y ∗ − Y Y ∗ = i

a3,

is approximately given by

(31)

Y = iH√2k3

[αk(1+ ikη)e−ikη − βk(1− ikη)eikη

],

where η = −1/(Ha) is a conformal time, andαk

andβk are constants which satisfy|αk|2 − |βk|2 = 1.We shall choose(αk , βk) = (1,0) so that the vacuumreduces to the one in Minkowski space–time at theshort-wavelength limit (−kη → ∞). We thereforeobtain

Y = iH√2k3

(1+ ikη)e−ikη

(32)

= iH√2k3

[1+ 1

2

(k

a(t)H

)2

+ i

3

(k

a(t)H

)3

+ · · ·],

where the latter expansion is useful afterk-mode hasgone out of the horizon in the periodt < t∗ +3H 2/m2.

From (32) and (28), we find that under the conditionm2 � H 2 we can match the first and the secondterms of (28) with those of (32) and the third andthe fourth terms of (28) with the third term of (32).The former constitutes the growing (or more properly,nondecreasing) mode and the latter corresponds to thedecaying mode. As a result, we find

c1(k)= iH 2

√2k3 φs(tk)

(33)= 9iH 4

√2k3m2φ∗

e−(m2/(3H2))H(tk−t∗),

c2(k)= φs(tk)√2k3H 2

(1+ 2m2

9H 2

)[1

+ i27H 2

m2

(1− m2

9H 2

)e−(2m2/(3H2))H(tk−t∗)

(34)×(a(t∗)H

k

)3].

This means that the nondecaying mode of the cur-vature perturbation on the comoving horizon scale att = t∗, r∗ is given by,

(35)Rc(r∗)=[∣∣c1(k)

∣∣2 4πk3

(2π)3

]1/2

= 9H 4

2πm2|φ∗| .

Apparently it is enhanced by a factor 9H 2/m2

compared with (1). This does not mean, however,that we find a severe deviation from a scale-invariantspectrum. On a more general comoving scaler =r(tk) = 2π/k, leaving the Hubble radius during newinflation, we find from the first equality of (33):

(36)Rc(r)= H 2

2π |φs(tk)|,

which is practically scale-invariant form2 � H 2.The deviation from the scale-invariant spectrum ismanifest only in the decaying modec2(k), as is seenin the second term of (34).

Thus we can obtain the correct expression for thenondecreasing mode of the curvature perturbationif we replaceφ in (1) with its nondecaying modeonly. This was also the case whenφ vanishes dur-ing inflation which was studied in [10]. Figure 1 de-picts the actual spectrum of curvature perturbation to-gether with the conventional slow-roll formula (1) forH 2/m2 = 30. Using (1) would seriously underesti-mate the actual amplitude in the regimeφ is not slowlyrolling.

Next, we consider the applicability of this approachto a more general potentialV [φ]. As it is clear fromthe above analysis, our method is applicable if thesolution of φ is separable to a sum of a slowlyvarying mode and a decaying mode approximatelyproportional toa−3(t), at least for a few expansiontime scales after thek-mode under considerationleaves the Hubble radius att = tk . Let us expandV [φ]aroundφ = φ(tk) ≡ φk up to second order:

V [φ] = V [φk] + V ′[φk](φ − φk)

J. Yokoyama, S. Inoue / Physics Letters B 524 (2002) 15–20 19

Fig. 1. Actual spectrum of the curvature perturbation (solid line) andthe result of slow-roll formula (broken line) forH2/m2 = 30.

+ 1

2V ′′[φk](φ − φk)

2 + · · ·

(37)≡ Vk + V ′

k(φ − φk)+ 1

2V ′′k (φ − φk)

2 + · · · .

Then a solution of the equation of motion,

φ + 3Hkφ + V ′k + V ′′

k (φ − φk)= 0,

is given by a linear combination ofeΛ+(t−tk) andeΛ−(t−tk), whereΛ± are solutions ofΛ2 + 3HkΛ +V ′′k = 0. Here a subscriptk denotes the value of each

quantity at the timetk .In order that the solution ofφ is separable to

a slowly changing mode and a decaying mode, werequire|V ′′

k | � H 2k . Then we find, to the lowest order

in |V ′′k |/H 2

k :

φ(t)=[V ′′k

9H 2k

φk +(

1− V ′′k

9H 2k

)V ′k

3Hk

]

× e(V ′′k /

(3H2

k

))Hk(t−tk)

+(

1− V ′′k

9H 2k

)(φk − V ′

k

3Hk

)

(38)× e−3(1+(

V ′′k /

(9H2

k

)))Hk(t−tk).

The first and the second terms of the above solutioncorrespond toφs(t) and φr (t) in (10) of the previous

analysis, respectively. If we apply the same matchingmethod as before, we obtain

(39)Rc(r)= H 2k

2π

∣∣∣∣ V ′′k

9H 2k

φk +(

1− V ′′k

9H 2k

)V ′k

3Hk

∣∣∣∣−1

,

instead of (36) for the nondecaying mode of thecurvature perturbation on the comoving horizon scaleat t = tk . In order that (39) gives the correct expressionfor the curvature perturbation, the expansion (37)should remain valid at least for a few expansion timescales afterk-mode has crossed the horizon att = tk .That is, we require

(40)∣∣V ′

k

∣∣∣∣φk∣∣H−1k < Vk,

∣∣V ′′k

∣∣(φkH−1k

)2<Vk.

Since thek-mode leaves the horizon during inflation,we haveφ2

k < Vk. Then together with the assumption|V ′′

k | � H 2k , the second inequality is trivially satisfied.

Furthermore, sinceV ′k/Hk is of the same order ofφ in

the slow-roll case, we find|V ′k||φk|H−1

k � φ2k < Vk is

also automatically satisfied. Thus the only nontrivialcondition for our analysis to be valid is|V ′′

k | � H 2k

apart from the condition for accelerated expansionφ2k < Vk.In summary, we have analytically studied genera-

tion of curvature perturbation in the case the slow-rollequation of motion does not hold during inflation, bymatching the quantum fluctuation (32) with the long-wave exact solution (26). For this matching to be pos-sible, it is essential that the solution ofφ can be de-scribed by a sum of a slowly changing mode and adecaying mode approximately proportional toa−3(t),which requires|V ′′

k | � H 2k . Models with more com-

plicated potentials that violate this condition must bestudied numerically [11,17].

Acknowledgements

We are grateful to M. Sasaki and O. Seto for use-ful communications. The work of J.Y. was supportedin part by the Monbukagakusho Grant-in-Aid for Sci-entific Research “Priority Area: Supersymmetry andUnified Theory of Elementary Particles (No. 707).”

20 J. Yokoyama, S. Inoue / Physics Letters B 524 (2002) 15–20

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