Non-Gaussianity and finite length inflation
aa r X i v : . [ a s t r o - ph . C O ] A p r Preprint typeset in JHEP style - HYPER VERSION
November 1, 2018
Non-Gaussianity and finite length inflation
Shiro Hirai
Department of Digital Games, Osaka Electro-Communication University1130-70 Kiyotaki, Shijonawate, Osaka 575-0063, JapanE-mail: [email protected]
Tomoyuki Takami
Department of Digital Games, Osaka Electro-Communication University1130-70 Kiyotaki, Shijonawate, Osaka 575-0063, JapanE-mail: [email protected]
Abstract:
In the present paper, certain inflation models are shown to have large non-Gaussianity in special cases. Namely, finite length inflation models with an effective higherderivative interaction, in which slow-roll inflation is adopted as inflation and a scalar-matter-dominated period or power inflation is adopted as pre-inflation, are considered.Using Holman and Tolley’s formula of the nonlinearity parameter f flattenedNL , we calculate thevalue of f flattenedNL . A large value of f flattenedNL ( f flattenedNL > e -folds and f NL has strongdependence on the length of inflation. Interestingly, this length is similar to that for thecase in which the suppression of the CMB angular power spectrum of l = 2 was derivedusing the inflation models described in our previous papers. Keywords: . ... ontents
1. Introduction 22. Scalar perturbations 33. Calculation of the nonlinearity parameter 54. Discussion 7 – 1 – . Introduction
Non-Gaussianity of primordial perturbations is one of the most interesting problems impliedby the WMAP data [1, 2]. The observational limits on the nonlinearity parameter fromWMAP seven-year data [2] are − < f localNL <
74 (95% CL) , − < f equilNL <
266 (95%CL) and − < f orthogNL < l = 2 as indicated by Wilkinson Microwave Anisotropy Probe (WMAP)data [1] may be explained to a certain extent by the finite length of inflation for an inflationof 50-60 e -folds [9]. Of course, there are many attempts [10] to derive this suppression.Based on the physical conditions before inflation, we have shown that the initial state ofscalar perturbations in inflation is not simply the Bunch-Davies state, but rather a moregeneral state (a squeezed state), where a scalar-matter-dominated period, a radiation-dominated period, or another inflation is considered as pre-inflation, and the general initialstates in inflation were calculated analytically. In the present paper, we demonstrate anew property of the proposed inflation model. Using Holman and Talley’s formula for thenonlinearity parameter f flattenedNL , we calculate the value of f flattenedNL for the case in which theproposed finite inflation models have effective higher-derivative interactions, where slow-roll inflation is adopted as inflation and a scalar-matter-dominated period or power-lawinflation period is adopted as pre-inflation. The obtained results are very interesting.– 2 – . Scalar perturbations We consider curvature perturbations in inflation and a scalar-matter-dominated epoch. Thebackground spectrum considered is a spatially flat Friedman-Robertson-Walker (FRW)universe described by metric perturbations. The line element for the background andperturbations is generally expressed as [11] ds = a ( η ) { (1 + 2 A ) dη − ∂ i Bdx i dη − [(1 − δ ij + 2 ∂ i ∂ j E + h ij ] dx i dx j } , (2.1)where η is the conformal time, the functions A , B , Ψ, and E represent the scalar pertur-bations, and h ij represents tensor perturbations. The density perturbation in terms of theintrinsic curvature perturbation of comoving hypersurfaces is given by R = − Ψ − ( H/ ˙ φ ) δφ ,where φ is the inflaton field, δφ is the fluctuation of the inflaton field, H is the Hubbleexpansion parameter, and R is the curvature perturbation. Overdots represent derivativeswith respect to time t , and primes represent derivatives with respect to the conformal time η . Introducing the gauge-invariant potential u ≡ a ( η )( δφ + ( ˙ φ/H )Ψ) allows the action forscalar perturbations to be written as [12] S = 12 Z dηd x { ( ∂u∂η ) − c s ( ∇ u ) + Z ′′ Z u } , (2.2)where c s is the velocity of sound, and in inflation Z = a ˙ φ/H , and u = − Z R . The field u ( η, x ) is expressed using annihilation and creation operators as follows: u ( η, x ) = 1(2 π ) / Z d k { u k ( η ) a k + u ∗ k ( η ) a †− k } e − i kx . (2.3)The field equation for u k ( η ) is derived as d u k dη + ( c s k − Z d Zdη ) u k = 0 , (2.4)where c s = 1 is assumed in inflation. The solution of u k satisfies the normalization condition u k du ∗ k /dη − u ∗ k du k /dη = i .First, slow-roll inflation is considered. The slow-roll parameters are defined as [13,14]: ǫ = 3 ˙ φ φ V ) − = 2 M ( H ′ ( φ ) H ( φ ) ) , (2.5) δ = 2 M H ′′ ( φ ) H ( φ ) , (2.6) ξ = 4 M H ′ ( φ ) H ′′′ ( φ )( H ( φ )) . (2.7)The quantity V ( φ ) is the inflation potential, and M P is the reduced Plank mass. Otherslow-roll parameters ( ǫ V , η V , ξ V ) can be written in terms of the slow-roll parameters ǫ , δ ,and ξ for first-order slow roll, i.e., ǫ = ǫ V , δ = η V − ǫ V , and ξ = ξ V − ǫ V η V + 3 ǫ V , where– 3 – V = M / V ′ /V ) , η V = M ( V ′′ /V ), and ξ V = M ( V ′ V ′′′ /V ). Using the slow-rollparameters, ( d Z/dη ) /Z is written exactly as1 Z d Zdη = 2 a H (1 + ǫ − δ + ǫ − ǫδ + δ ξ , (2.8)and the scale factor is written as a ( η ) = − ((1 − ǫ ) ηH ) − . Here, the slow-roll parametersare assumed to satisfy ǫ < δ <
1, and ξ <
1. As only the leading-order terms of ǫ and δ are adopted, ǫ and δ may be considered to be constant, allowing the scale factor to bewritten as a ( η ) ≈ ( − η ) − − ǫ [14]. Equation (2.4) can be rewritten as d u k dη + ( k − ǫ − δη ) u k = 0 . (2.9)The solution to Eq. (2.9) is written as [13] f I k ( η ) = √ π e i ( ν +1 / π/ ( − η ) / H (1) ν ( − kη ) , (2.10)where ν = 3 / ǫ − δ , and H (1) ν is the Hankel function of the first kind of order ν . Themode functions u k ( η ) of a general initial state in inflation are written as u k ( η ) = c f I k ( η ) + c f I* k ( η ) , (2.11)where the coefficients c and c obey the relation | c | − | c | = 1. The important pointhere is that the coefficients c and c do not change during inflation. In ordinary cases, thefield u k ( η ) is considered to be in the Bunch-Davies state, i.e., c = 1 and c = 0, becauseas η → −∞ , the field u k ( η ) must approach plane waves ( e − ikη / √ k ). Second, in the caseof power-law inflation, where a ( t ) ∝ t q , a similar method can be used, and the solution isobtained as Eq.(2.10) with ν = 3 / / ( q − u k can be written in a form similar to Eq. (2.4) with a value of c s = 1and with Z ∝ a p ( η )[ H − H ′ ] / / H , (where H = a ′ p /a p ). The solution to Eq. (2.4) is thenwritten as f S k = (1 − i/ ( kη )exp[ − ikη ]) / √ k .– 4 – . Calculation of the nonlinearity parameter Here, an inflation model is considered. Since we consider slow-roll inflation to have a finitelength, we assume a pre-inflation period to be a scalar-matter-dominated period in whichthe scalar field is the inflaton field, or is power-law inflation, i.e., double inflation. A simplecosmological model is assumed, as defined byPre-inflation: a p ( η ) = b ( − η − η j ) r , (3.1)Slow-roll inflation: a I ( η ) = b ( − η ) − − ǫ , (3.2)where η j = − ( r ǫ + 1) η , b = ( − − ǫr ) r ( − η ) − − ǫ − r b . (3.3)The scale factor a I ( η ) represents slow-roll inflation. Here, de-Sitter inflation ( ǫ = 0) is notconsidered. Slow-roll inflation is assumed to begin at η = η . In pre-inflation, for the caseof r = 2, the scale factor a p ( η ) indicates that pre-inflation is a scalar-matter-dominatedperiod, and, for the case of r = − q/ ( q − a p ( t ) ∝ t q .Using above the pre-inflation model, the initial state of inflation given by Eq. (2.11)will be fixed as follows. The coefficients c and c are fixed using the matching condition inwhich the mode function and first η -derivative of the mode function are continuous at thetransition time η = η . ( η is the time at which slow-roll inflation begins.) For simplicitypre-inflation states are assumed to be the Bunch-Davies vacuum. The coefficients c and c can be calculated analytically in the case of the scalar-matter-dominated period: c = − i z / r π e i (( − δ − ǫ ) π/ − z/ (1+ ǫ )) n z ( − − iz − ǫ ) H (2) ν ( z )+(4 z + (3 − δ + 3 ǫ )(1 + ǫ + 2 iz )) H (2) ν ( z ) o , (3.4) c = − i z / r π e − i (( − δ − ǫ ) π/ z/ (1+ ǫ )) n z ( − − iz − ǫ ) H (1) ν ( z )+(4 z + (3 − δ + 3 ǫ )(1 + ǫ + 2 iz )) H (1) ν ( z ) o , (3.5)and in the case of the double inflation model: c = − π p q ( q − ǫ ) e iπ (6+2 / ( q − − δ +4 ǫ ) / {− qzH (1)5 / / ( q − ( zz ) H (2) ν ( z ) − H (1)3 / / ( q − ( zz )( − qzH (2) ν ( z ) + (1 + q ( − δ − ǫ ) + ǫ ) H (2) ν ( z ) } (3.6) c = − π p q ( q − ǫ ) e iπ ( − δ + q (1+ δ − ǫ )+2 ǫ ) / q − {− qzH (1)5 / / ( q − ( zz ) H (1) ν ( z ) − H (1)3 / / ( q − ( zz )((1 + q ( δ − − ǫ ) + ǫ ) H (1) ν ( z ) − qzH (1) ν ( z ) } (3.7)– 5 –here ν = 3 / ǫ − δ , ν = 5 / ǫ − δ , z = − kη and zz = qz/ (( q − ǫ )).The initial states of inflation can be written in terms of the slow-roll parameters, the starttime of slow-roll inflation η , and the double inflation parameter q . Here, three slow-rollinflation models are adopted: the new inflation model with the potential term given by V ( φ ) = λ ν (1 − φ/ν ) p ), where p = 3 , ν ≈ M p , the chaotic inflation model withthe potential term given by V ( φ ) = M φ/m ) a , where a = 2 , , m ≈ M p and the hybridmodel V ( φ ) = α (( ν − σ ) + m / φ + g φ σ ) ≃ α ( ν + m / φ ), where ν ≈ − M p and m ≈ × − M p [16]. Using the normalization value from the WMAP five-year data,we obtain the values of the spectral index and the slow-roll parameters, such asNew inflation: n s = 0 . ǫ = 1 . × − , δ = − . n s = 0 . ǫ = 0 . δ = 0 . φ model: n s = 0 . ǫ = 0 . δ = 0 . φ model: n s = 0 . ǫ = 0 . δ = 0 . φ model: n s = 0 . ǫ = 0 . δ = 0 . f flattenedNL . Holman andTolley [3] showed that if the effective action for the inflaton contains the higher-derivativeinteraction [17] L = √− g λ M (( ∇ φ ) ) , which is derived, for example, from k-inflation orDBI inflation, and the initial state of inflaton is not the Bunch-Davies vacuum, then theenhanced non-Gaussianity is derived as follows: f flattenedNL ≈ ˙ φ M | c | ( ka ( η ) H ) = 2 ǫM H z | c | , (3.8)where M is the cutoff scale, which is the limit of effective theory, and we assum M ≈ k/a ( η )where η is the beginning time of slow-roll inflation, and z = − kη . The present treatmentconsiders the effect of the length of inflation, where z = 1 indicates that inflation startsat the time when the present-day size perturbation k = 0 . / Mpc) exceeds the Hubbleradius in inflation (i.e., inflation of close to 60 e -folds). Using the values of the aboveparameters we can calculate the values of | c | , | c | , and f flattenedNL in terms of z (= − kη ).The values of | c | change only slightly among the models, but vary with the value of z , as0.0063 for z = 8, 0.004 for z = 10, and 0.001 for z = 20, and | c | ∼ = 1. From all of themodels except for the φ model, similar values of f flattenedNL are calculated, i.e., f flattenedNL ≈ z = 8, and f flattenedNL ≈
40 at z = 10. Details are shown in Table 1. With respect tothe other values of z , larger values of f flattenedNL can be derived at smaller z ( z < f flattenedNL can be derived at larger z ( z > f flattenedNL appears to depend strongly on the value of z , which represents the lengthof inflation, and the difference of the value of f flattenedNL among our three slow-roll inflationmodels is not large. Since the z -dependence of f flattenedNL is very steep, any value of f flattenedNL can be derived at some point of z . We next consider the case of double inflation, the valueof f flattenedNL is 100 at 3 < z < < z < z ≈ q -dependence( a ( t ) ∝ t q ), the values of f flattenedNL are similar at very large q but change at q ≈ . Discussion We have derived a new property of the proposed finite inflation model. The possibilityof large non-Gaussianity is demonstrated. The proposed inflation model is a finite lengthinflation model with an effective higher derivative interaction, where slow-roll inflation isadopted as inflation and a scalar-matter-dominated period or power inflation is adopted aspre-inflation. Owing to the existence of pre-inflation, the initial state in inflation is not theBunch-Davies state, but is instead a more general state. The coefficients c and c can beanalytically calculated. Using Holman and Tolleys formula of the nonlinearity parameter f flattenedNL , we calculated the value of f flattenedNL . For the case in which the scalar-matter-dominated period is considered to be pre-inflation, large values of f flattenedNL ( f flattenedNL ≈ < z <
10 in all the models considered herein, and similar results arederived for the case of double inflation at 3 < z <
4. These ranges can be written as 60-63 e -folds. This length is similar to that obtained when the suppression of CMB angular powerspectrum of l = 2 was derived using the inflation models described in previous papers [9],although such spectral suppression is not inconsistent when considering cosmic variance.On the experimental value of f flattenedNL , the orthogonal shape ( f orthogNL ) is peaked both onequilateral-triangle configurations ( f equilNL ) and on flattened-triangle configurations ( f flattenedNL )[18], but we think we need further considerations to derive the constraint of f flattenedNL fromthe constraints of f orthogNL and f equilNL . Therefore, we do not show it here. We assume sucha high-derivative interaction in order to obtain non-linearity and effective interactions forslow-roll interaction. This high-derivative interaction appears to influence the parametersof slow-roll inflation. In order to clarify this problem, we must investigate a concreteinflation model such as k-inflation or DBI inflation. In the future, we would like to applythe proposed method to other inflation models and investigate the dependence of the lengthof inflation on f flattenedNL . Acknowledgments
The authors would like to thank the staff of Osaka Electro-Communication University fortheir valuable discussions. – 7 – eferences [1] Spergel D N et al. 2007
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JCAP – 8 – able 1: Values of f flattenedNL for the case of the matter-dominated period as pre-inflation New inflation Hybrid Chaotic inflation p = 3 p = 4 φ φ φ z = 8 123.8 123.7 122.7 123.8 187.7 126.4 z = 10 40.5 40.4 40.1 40.4 61.3 41.3 z = 20 1.26 1.26 1.25 1.26 1.91 1.28 Table 2:
Values of f flattenedNL in the hybrid inflation for double inflation q = 10 q = 10 q = 10 q = 10 q = 10 z = 3 108.5 109.1 115.3 190.6 1096.5 z = 4 23.4 23.6 25.3 45.1 266.8 z = 5 7.24 7.30 7.93 14.8 88.6 Table 3:
Values of for f flattenedNL the new inflation case of n = 3 and for the new inflation case of n = 4 for double inflation n = 3 q = 10 q = 10 q = 10 q = 10 q = 10 z = 4 254.0 254.1 256.0 275.1 478.5 z = 5 81.8 81.9 82.5 89.1 157.7 z = 6 32.5 32.6 32.8 35.6 63.6 n = 4 q = 10 q = 10 q = 10 q = 10 q = 10 z = 4 194.9 195.1 197.0 216.5 424.2 z = 5 62.8 62.9 63.5 70.2 140.1 z = 6 25.0 25.0 25.3 28.0 56.5– 9 – able 4: Values of f flattenedNL for the Chaotic inflation in the cases φ , φ , and φ for double inflation φ model q = 10 q = 10 q = 10 q = 10 q = 10 z = 3 227.6 228.2 234.7 306.9 1196.5 z = 3 . z = 4 49.8 50.0 51.8 71.1 291.0 φ model q = 10 q = 10 q = 10 q = 10 q = 10 z = 3 181.1 181.5 185.6 242.9 1130.3 z = 3 . z = 4 36.1 36.2 37.5 53.9 274.4 φ model q = 10 q = 10 q = 10 q = 10 q = 10 z = 3 165.5 165.5 165.4 191.2 1061.5 z = 3 . zz