Localized Features in Non-Gaussianity from Heavy Physics
aa r X i v : . [ a s t r o - ph . C O ] J un Prepared for submission to JCAP
Localized Features in Non-Gaussianityfrom Heavy Physics
Ryo Saito, Yu-ichi Takamizu Yukawa Institute for Theoretical Physics, Kyoto University,Kitashirakawa Oiwake-Cho, Sakyo-ku, Kyoto 606-8502, Japan
Abstract.
We discuss the possibility that we could obtain some hints of the heavy physicsduring inflation by analyzing local features of the primordial bispectrum. A heavy scalarfield could leave large signatures in the primordial spectra through the parametric resonancebetween its background oscillation and the fluctuations. Since the duration of the heavy-mode oscillations is finite, the effect of the resonance is localized in momentum space. In thispaper, we show that the bispectrum is amplified when such a resonance occurs, and that thepeak amplitude of the feature can be O (10 − ), or as large as O (10 ) depending on the typeof interactions. In particular, the resonance can give large contributions in finitely squeezedconfigurations, while the bispectrum cannot be large in the exact squeezed limit. We alsofind that there is a relation between the scales at which the features appear in the bispectrumand the power spectrum, and that the feature in the bispectrum can be much larger thanthat in the power spectrum. If correlated features are observed at characteristic scales in theprimordial spectra, it will indicate the presence of heavy degrees of freedom. By analyzingthese features, we may be able to obtain some information on the physics behind inflation. ontents Primordial non-Gaussianity is a powerful probe to discriminate inflationary models [1]. In thesimplest setup, i.e. single-field slow-roll inflation models with a canonical kinetic term andBunch-Davis initial conditions, the primordial fluctuations are predicted to be approximatelyGaussian-distributed [2]. More generally, for the models of so-called single-clock inflation,where there is only one relevant degree of freedom, it is known that there is a consistencyrelation relating the bispectrum in the squeezed limit to the tilt of the power spectrum [3–8].Thus, a non-Gaussian signal would enable us to narrow down the possible models of inflation.To predict the expected non-Gaussian signal from a given model correctly, we needaccurate knowledge of the inflationary dynamics as well as the Lagrangian. For example, evenin the “single-field” inflation models, where there is a single light scalar field, the consistencyrelation is not always satisfied. It has been argued in Ref. [9] that when the backgroundevolution is provided by a non-attractor solution, the consistency relation is violated whilemaintaining a scale-invariant power spectrum. To probe the model of inflation, we need tounderstand well what can happen under a given model.In standard single-field slow-roll inflation models, non-Gaussian signals in the bispec-trum can be classified into three types: local type, equilateral type, and orthogonal type [10].The recent Planck results have revealed that all these types of non-Gaussianity are consistentwith zero and found no deviations from the prediction of the single-field slow-roll inflationmodels [11]. However, the bispectrum contains a large number of degrees of freedom andtherefore could have much more information. For example, features localized at a specificscale can be induced in the bispectrum by some temporal events like slow-roll violation orparticle production during inflation [12–22]. Localized features can be used to probe theinflationary dynamics.In general, there could be many scalar degrees of freedom other than the inflaton, evenin single-field inflation models. In a model embedded in supergravity or string theory, forexample, such degrees of freedom may appear as moduli fields, Kaluza-Klein modes, or the– 1 –calar supersymmetric partner of inflaton. Usually, the scalar fields are very heavy, m ≫ H ,so that they are assumed to be stuck in their minima, and the model is treated effectively asa single-field model [23]. Recently, however, it has been pointed out that a heavy scalar fieldis not necessarily stuck in its potential minimum during inflation [24–35]. It can be displacedfrom its minimum due to the centrifugal force generated by a turn in the inflaton trajectory.When the turn is very sharp, even oscillations in the heavy direction can be excited [34, 35].Other possibilities are that oscillations can be excited when the heavy scalar field becomesmomentarily light/tachyonic during inflation or that they can be excited at the beginningof inflation [36], which is natural in the case that inflation occurs after tunneling from aneighboring minimum [37–42], for example.In the previous paper [43], we discussed the possibility that excited oscillation of heavymodes can leave non-negligible signatures in the power spectrum through derivative cou-plings, without spoiling inflation (see also Refs. [44] for models with only gravitationalcouplings). We saw that the primordial fluctuations can be enhanced deep in the horizon, k/a ∼ m ≫ H , by the parametric resonance with the excited oscillations. In this paper,we estimate the resonant feature in the bispectrum within the same setup. This gives anexample where resonant non-Gaussianity [45–49] is produced in a realistic model. Since theduration of heavy-mode oscillations is finite, the effect of the resonance is localized in mo-mentum space. We will show that a large feature could be induced in the bispectrum evenwhen the feature in the power spectrum is too small to be observed. We also investigate thebehavior of the bispectrum in the squeezed limit. As in the case that the consistency relationis satisfied, it can be shown that the bispectrum in the squeezed limit cannot be large unlessthe modification to the power spectrum is large. However, we will also show that, for finitelysqueezed configurations, the bispectrum can be greatly enhanced in comparison to the usualsingle-field inflation model (a model with a single light scalar field). Since we cannot ob-serve the exact squeezed limit in actual observations, the enhancement could be practicallyimportant, as recently pointed out in Ref. [50], where the resonant non-Gaussianity froma modulated potential is considered as a concrete example. The detection of such featureswould therefore indicates the presence of heavy degrees of freedom during inflation, and byanalyzing them we may hope to improve our understanding of the physics behind inflation.The organization of this paper is as follows. In §
2, we briefly review the model pre-sented in Ref. [43] to realize an efficient enhancement of the fluctuations in the inflaton fieldand estimate the feature induced in the power spectrum. In §
3, we discuss the resonantenhancement of the bispectrum. Finally, we provide a summary of this paper in § First, we briefly review the model presented in Ref. [43]. We consider a model with a heavyscalar field with mass m ≫ H , χ ( ≡ φ (2) ), which derivatively couples to the inflaton field, φ ( ≡ φ (1) ), as S m ≡ − Z d x √− gP ( X IJ , φ K ) , ( I, J, K = 1 ,
2) (2.1)= − Z d x √− g (cid:20)
12 ( ∂φ ) + V ( φ ) + 12 ( ∂χ ) + m χ + K n + K d + O (cid:0) Xφ / Λ n , X / Λ d (cid:1)(cid:21) , (2.2)– 2 –ith X IJ ≡ − ∂ µ φ I ∂ µ φ J /
2. Here, we have assumed that the derivative couplings at theleading order are given by, K n ≡ λ n n χ ( ∂φ ) , (2.3)and K d ≡ λ d d ( ∂χ ) ( ∂φ ) + λ d d ( ∂χ · ∂φ ) , (2.4)with the dimensional parameters Λ n , Λ d and the dimensionless parameters λ n , λ di ( i = 1 , K n and K d provide the most generalcouplings between the inflaton field and the heavy scalar field at the leading order in 1 / Λ n and 1 / Λ d in a model with the parity symmetry, φ → − φ , and the shift symmetry, φ → φ + c .The approximate shift symmetry is usually assumed to ensure the flatness of the inflatonpotential. In addition, we have assumed the parity symmetry to forbid the kinetic mixing,which obscures the difference between the light and heavy fields. Thus, we first focus on thecouplings K n and K d , though brief comments are presented in § n and Λ d are also expected to appear. To ensure that contributions fromthese terms can be treated perturbatively, we assume here that the background fields satisfythe following conditions, χ ≪ Λ n , ˙ φ, ˙ χ ≪ Λ d . (2.5)We have also suppressed the terms ( ∂φ ) and ( ∂χ ) in Eq. (2.2), because they have littleeffects on our analysis for the background evolution and the power spectrum. The higher-order terms and the self interaction ( ∂φ ) can be important for the bispectrum. Features inthe bispectrum induced by these interactions are discussed in § K n and K d , such as DBI inflation [53]. In thesecases, the conditions (2.5) are not mandatory.Since χ can generally decay into other particles, we assume that χ decays with a rateΓ, which satisfies H ≪ Γ ≪ m . To ensure the slow-roll inflation, we further make twoassumptions. First, the inflaton potential is sufficiently flat, ǫ V ≪ , | η V | ≪ , (2.6)where ǫ V ≡ M p (cid:18) V ′ V (cid:19) , η V ≡ M p V ′′ V , (2.7)are the slow-roll parameters. Here, M p = 2 . × GeV is the reduced Planck mass. Second,the heavy scalar field χ is subdominant, f χ ≪ , (2.8)where f χ ≡ ρ χ ρ ≃ ˙ χ + m χ M p H , (2.9)is the fraction of its energy density to the total one.– 3 – igure 1 . The fluctuations in the inflaton field can be amplified through the parametric resonancewith the background oscillation of the heavy scalar field (2.13). As depicted by the thick line, a modeis redshifted by the cosmic expansion. Since only the modes that have crossed the resonance bandduring the oscillation are amplified, the features in the primordial spectra are localized in momentumspace. Provided that the conditions (2.5), (2.6), and (2.8) are satisfied, the background evolu-tion of the inflation field is given by the slow-roll solution, π φ ( t ) ≃ − V ′ H , (2.10)where π φ is the conjugate momentum for the inflaton field, π φ ≡ ˆ z φ ˙ φ, (2.11)with z φ ≡ a (cid:20) λ n χ Λ n + ( λ d + λ d ) ˙ χ d (cid:21) , (2.12)and ˆ z φ ≡ z φ /a . The heavy scalar field oscillates with a frequency of the mass scale m , χ ( t ) ≃ χ e − Γ t cos( mt ) θ ( t ) , (2.13)with θ ( t ) being the Heaviside function, where we have assumed that the oscillation is excitedinstantaneously at t = 0. The fluctuations are enhanced through a resonance with thebackground oscillation of the heavy scalar field (2.13). We showed the basic picture of theresonant enhancement in Fig. 1.For a later convenience, we introduce parameters which represent the magnitude of thederivative couplings, q n ≡ λ n χ Λ n , (2.14) q d ≡ λ d m χ d , q d ≡ λ d m χ d , (2.15)which are related to the q -parameters introduced in Ref. [43]. We will also use q to representthe order of the q -parameters above. Note that the q -parameters are less than unity under– 4 –he conditions (2.5). They can be expressed in terms of the energy fraction of the heavyscalar field, f χ , as q n ≃ √ λ n (cid:18) E inf m (cid:19) (cid:18) E inf Λ n (cid:19) f χ,t =0 , (2.16)for the couplings K n and q d ≃ λ d (cid:18) E inf Λ d (cid:19) f χ,t =0 , q d ≃ λ d (cid:18) E inf Λ d (cid:19) f χ,t =0 , (2.17)for the couplings K d , where E inf is the energy scale of the inflation, E inf ≡ (3 M p H ) ≃ V . Before investigating the features in the bispectrum, we first estimate the effect of the reso-nance on the power spectrum. To obtain an analytic expression, here, we use the perturbativemethod assuming that the q -parameters are sufficiently small.To eliminate the gauge degrees of freedom in the metric fluctuations, we use the flatgauge, where the spatial metric becomes a δ ij . In this gauge, we can make some simplifi-cations in estimating the power spectrum. First, the metric fluctuations can be neglectedbecause the resonance takes place deep in the horizon. Moreover, we can also neglect thefluctuations in the heavy scalar field, δχ . This is because δχ oscillates with a frequency ω > m and therefore does not satisfy the resonance condition. In addition, the mass termsuppresses the amplitude of δχ at k/a ∼ m compared with that of the inflaton field.Neglecting the fluctuations in the metric and the heavy scalar field, the second-orderaction for the fluctuations in the inflaton field, ϕ , can be written as, S ≃ Z d t d x z φ (cid:2) ˙ ϕ − c s ( ∇ ϕ ) /a (cid:3) , (2.18)where c s = 1 − λ d ˙ χ d + O (cid:18) ˙ χ Λ d (cid:19) , (2.19)is the speed of sound for the fluctuations in the inflaton field. Note that we can neglect termslike ˙ φ / Λ d in the action (2.18), which could be induced by the self interaction ( ∂φ ) / Λ d ,because they are higher order in the q -parameters.Introducing the canonically normalized variable v ≡ z φ ϕ , the action (2.18) can berewritten as, S = Z d t d x (cid:20) ˙ v − (cid:18) c s ∇ a − ¨ z φ z φ (cid:19) v (cid:21) , (2.20)and then the full Hamiltonian as, H (2) = Z d x (cid:20) ˙ v + (cid:18) c s ∇ a − ¨ z φ z φ (cid:19) v (cid:21) . (2.21)– 5 –ubstituting the background solution (2.13), the quantities c s and ¨ z φ /z φ can be estimatedas, c s ≃ q d e − t cos(2 mt ) , (2.22)¨ z φ z φ ≃ m (cid:20) q n e − Γ t cos( mt ) + ( q d + q d ) e − t cos(2 mt ) + O (cid:18) Hm (cid:19)(cid:21) . (2.23)In the following analysis, we neglect the O ( H/m ) terms in Eq. (2.23) assuming that q islarger than H/m . Taking the free Hamiltonian as the terms independent of the derivativecouplings in Eq. (2.21), the interaction Hamiltonian is given by, H (2) I ≡ m π ) Z d k C (2) k v k v − k , (2.24)where we have moved to Fourier space, v ( x ) = 1(2 π ) Z d k v k e i k · x . (2.25)Here, the coefficient C (2) k is given in terms of the q -parameters as, C (2) k ≡ C (2; n ) k e − Γ t cos( mt ) + C (2; d ) e − t cos(2 mt ) , (2.26)where C (2; n ) k ≡ − q n , (2.27) C (2; d ) k ≡ q d " (cid:18) kam (cid:19) − − q d . (2.28)Hence, using the in-in formalism, the correction to the power spectrum can be estimated as,∆ h ϕ k ( t ) ϕ k ′ ( t ) i ≃ i Z t d t ′ h [ H (2) I ( t ′ ) , ϕ k ( t ) ϕ k ′ ( t )] i , (2.29)at the leading order in the q -parameters, where the fields in LHS are operators in the Heisen-berg picture while, in RHS, they represent operators in the interaction picture. For brevity,we denote them in the same way since it will be clear which they represent from the context.In terms of the dimensionless power spectrum, P ϕ , defined through, h ϕ k ( t ) ϕ k ′ ( t ) i ≡ (2 π ) δ ( k + k ′ ) 2 π P ϕ k , (2.30)the correction is expressed as, ∆ P ϕ ( k ) / P ϕ = I (2) k , (2.31) Note that infinitely many terms would appear in the interaction Hamiltonian if we define the free Hamilto-nian as terms independent of the derivative couplings without introducing the canonically normalized variable v . – 6 –here I (2) k ≡ − m Re (cid:20) i Z t d t ′ C (2) k u k ( t ′ ) (cid:21) . (2.32)Here, the estimation time t should be taken sufficiently after the decay of the oscillation.The function u k is the mode function for v k , which is defined through, v k ( t ) = u k ( t ) a k + u ∗ k ( t ) a − k † , (2.33)for the creation/annihilation operators. Though the state could be excited depending on theexcitation mechanism of the oscillation, we simply assume that the state was in the vacuumstate at the excitation of the oscillation. Even if the state was excited, the resonance is notspoiled except for some specific excited states. Assuming that the state is not excited, themode function is provided by, u k ( t ) = r a k (cid:18) − ikτ (cid:19) e − ikτ , (2.34)where τ is the conformal time. Because the resonance occurs deep in the horizon, we canapproximate the mode functions in the integral (2.32) as, u k ( t ) ≃ r a k e − ikτ . (2.35)Hence, I (2) k can be approximated as, I (2) k ≃ − Z z d z ′ (cid:18) kam (cid:19) − C (2) k sin(2 kτ ′ ) , ( z ≡ mt ) . (2.36)The resonance occurs between the factor sin(2 kτ ′ ) and the oscillatory components, cos( mt ′ )and cos(2 mt ′ ), in C (2) k (see Eq. (2.26)). If the negative energy mode exists, it would introducean additional oscillatory component in Eq. (2.36) with a constant phase shift. However, thisadditional component does not spoil the resonance unless the amplitude of the negativeenergy mode and the phase shift have specific values for the scales of the resonance.Here, we evaluate I (2) k for the couplings K n . Replacing the parameters ( q n , m/ , Γ / q d , m, Γ), we can obtain a similar result for the couplings K d . Using the product-to-sumidentities for the trigonometric functions, we have two oscillatory functions in Eq. (2.36).The integration has a resonant contribution from one of the two in the interval around thestationary point of the phase function, θ (2; n ) ≡ mt − kτ . Taking the derivative of the phasefunction θ (2; n ) , we can find that the resonance occurs at time t ∗ where, ka ( t ∗ ) = m . (2.37)Hence, the resonance occurs ∆ N ≡ ln( m/ H ) e-folds before the horizon crossing of themodes. Solving the above equation, the value of z at the resonance is estimated to be z ∗ ≃ mH ln (cid:18) ka m (cid:19) , (2.38)– 7 –here a is the scale factor at the onset of the oscillation. Then, the integral can be evaluatedas, I (2) k ≃ C (2; n ) k, ∗ Im I ( n )2 k , (2.39)where I ( n )2 k ≡ Z z d z ′ e − Γ m z ′ + i ( z ′ − kτ ′ ) , (2.40)and C (2; n ) k, ∗ is the coefficient function (2.27) at the resonance, t = t ∗ . The duration of theresonance can be roughly estimated by ∆ z ≡ m/ p ¨ θ , which is given by∆ z (2; n ) = p m/H, (2.41)for the couplings K n . Approximating the oscillating factor e i ( z ′ − kτ ′ ) in Eq. (2.40) by atop-hat function, then, we can roughly estimate the contribution to I ( n )2 k from the resonancethrough the couplings K n as, I ( n )2 k ≃ C (2; n ) k, ∗ sin θ (2; n ) ∗ Z max(0 ,z ∗ + √ mH )max(0 ,z ∗ − √ mH ) d z ′ e − Γ m z ′ (2.42) ≡ C (2; n ) k, ∗ sin θ (2; n ) ∗ I ( n )2 k, approx (2.43)= (cid:18) ln (cid:16) ka m (cid:17) < − q Hm (cid:19) , − m Γ (cid:20) − (cid:16) ka m (cid:17) − Γ H e − Γ √ Hm (cid:21) sin θ (2; n ) ∗ q n (cid:18) − q Hm < ln (cid:16) ka m (cid:17) < q Hm (cid:19) , − m Γ (cid:16) ka m (cid:17) − Γ H sinh (cid:16) Γ √ Hm (cid:17) sin θ (2; n ) ∗ q n (cid:18) ln (cid:16) ka m (cid:17) > q Hm (cid:19) , (2.44)where θ (2; n ) ∗ is the phase function at the resonance, θ (2; n ) ∗ ≃ mH (cid:20) ln (cid:18) ka m (cid:19) + 1 (cid:21) . (2.45)Hence, the power spectrum has a peak at k p /a ≃ me √ H/m / P ( n ) ϕ ( k p ) / P ϕ ∼ m Γ (cid:16) − e − √ Hm (cid:17) q n . (2.46)In particular, when the decay rate is small, Γ ≪ √ Hm , the peak amplitude becomes,∆ P ( n ) ϕ ( k p ) / P ϕ ∼ q n r mH , (2.47)which is of the order of 2 q n ∆ z (2; n ) .In Fig. 2, we showed the scaling function I ( n ) k (Eq. (2.40)). The function I ( n ) k has a peakat k/a m ∼
1. As can be seen in the figure, the approximate function I ( n ) k, approx introduced inEq. (2.44) gives a reasonable fit of I ( n ) k . Note that we have used 2 k for the argument of I ( n ) k for later convenience. The factor 2 k in Eq. (2.40)corresponds to k + k ′ if we don’t take into account the delta function in Eq. (2.30). Similarly, the scalingfunction I ( n ) K with K = k + k + k appears for the bispectrum. – 8 – igure 2 . The scaling function I ( n ) k (Eq. (2.40)), which determines the scale dependence of theresonant enhancement. The thick line represents the approximate function, I ( n ) k, approx . The functionhas a peak at k/ ( a m ) ≃ e √ H/m . In the plot, we have set Γ = 5 H and m = 10 H . Once the correlation function for the inflaton field are obtained, we can calculate thosefor the comoving curvature fluctuations ζ by using the gauge transformation. After theoscillation has damped out, the gauge transformation is given as usual, ζ = H ˙ φ ϕ, (2.48)at the linear order. At the non-linear level, in general, we can obtain the correlation functionsof ζ by using δN -formalism [55–58] from those of ϕ on superhorizon scales. Since ζ isproportional to ϕ at the linear order, the correction to the power spectrum for ζ , ∆ P ζ / P ζ ,is provided also by Eq. (2.31):∆ P ζ ( k ) / P ζ = I (2) k ≃ C (2; n ) k, ∗ sin θ (2; n ) ∗ I ( n )2 k, approx . (2.49)Hence, the peak amplitude is given by Eq. (2.46) (or Eq. (2.47)):∆ P ( n ) ζ ( k p ) / P ζ ∼ m Γ (cid:16) − e − √ Hm (cid:17) q n (2.50) ∼ q n r mH (for Γ ≪ √ Hm ) . (2.51)We close this section with a comment on the perturbativity. Even when the q -parametersare smaller than unity, the higher-order terms in the expansion with respect to the interac-tions can be comparable to the leading term when the duration of the resonance ∆ t issufficiently long. This is because the resonance coherently accumulates the effects of theinteraction. Then, the order parameter of the expansion is given by H (2) I ∆ t , which could belarge even when q is very small. The perturbative method is valid only when the correctionto the power spectrum (2.47) is less than unity. When this condition is not satisfied, weshould solve the full equation of motion numerically to get the mode function as done in Ref.[43]. Note that the notation R is also used to denote the comoving curvature fluctuations. The variable ζ herecorresponds to R in Ref. [54], for example. – 9 –ther than the resonance, the derivative couplings could induce the loop corrections tothe power spectrum and the Green function. To ensure that the corrections are suppressed,the model (2.2) should have a cutoff at the energy scale Λ cut ≡ min(2 π Λ n , (2 π ) / Λ d ). Hence, to perform the calculation perturbatively at the resonance scale, k/a ∼ m , the massscale should not exceed the cutoff scale, m < Λ cut . Next, we investigate the effect of the resonance on the bispectrum, h ζ k ( t ) ζ k ( t ) ζ k ( t ) i ≡ (2 π ) δ ( k + k + k ) P ζ B ζ k k k . (3.1)After showing which cubic interactions are important for the resonance, we estimate thecorrection to the bispectrum perturbatively by using the in-in formalism.We use the dimensionless quantity B ζ to represent the amplitude of the bispectrum,which is made dimensionless by the factor k k k as in Ref. [59]. Note that this normalizationfactor differs from that in the definition of the local-type f NL . In addition, we use theuncorrected power spectrum P ζ to define the amplitude B ζ . Here, we discuss which cubic interactions play important roles in the resonance. First,they should obviously contain oscillatory components. Secondly, the interactions with morederivatives become important for the resonance. This is because the resonance occurs deepin the horizon, k/a ∼ m ≫ H , where the fluctuations oscillate with high frequencies. Notethat the fluctuations in the heavy scalar field can be neglected at the leading order as inthe case of the power spectrum. Hence, we do not need to consider the mixing between theinflaton field and the heavy scalar field.Taking into account the above points, we can pick up the relevant interactions as (seeAppendix A), H (3) I = a M p r ǫ V Z d x (cid:26) − ˆ z φ (cid:20) (3 − c s ) ˙ ϕ + c s a ( ∇ ϕ ) (cid:21) ϕ + (cid:2) ˆ z φ + 2( q d + q d ) e − t sin ( mt ) (cid:3) ∇ i ( ∇ − ˙ ϕ ) ∇ i ϕ ˙ ϕ (cid:27) . (3.2)Note that we need to include the gravitational couplings to obtain the interactions (3.2)because the couplings K n and K d do not contain cubic interactions of φ such as ( ∂χ · ∂φ )( ∂φ ) due to the parity symmetry. This is why the interactions (3.2) are suppressed by the Planckscale, M p . However, direct cubic interactions exist if the self interaction ( ∂φ ) or higher-orderinteractions are introduced, though we have neglected them in the previous section becausethey have little effects on the background evolution and the power spectrum. They are alsoexpected to arise if we relax the requirement of the parity symmetry. We will discuss thesepossibilities in § Here, we have assumed that the coupling constants of the derivative couplings, λ , are of the order of unity. – 10 –n terms of the canonically normalized variable v ≡ z φ ϕ , the interaction Hamiltoniancan be written as, H (3) I = 1 a M p (2 π ) r ǫ V Z d k d k d k δ ( k + k + k ) h m C (3 O ) k k k v k v k v k + mC (3 I ) k k k v k ( v k v k ) · + C (3 II ) k k k v k ˙ v k ˙ v k i . (3.3)Here, each coefficients are decomposed into the K n - and K d -coupling parts as, C (3 i ) k k k ≡ C (3 i ; n ) k k k e − Γ t cos( mt ) + C (3 i ; d ) k k k e − t cos(2 mt ) , ( i = O, II ) , (3.4)and C (3 I ) k k k ≡ C (3 I ; n ) k k k e − Γ t sin( mt ) + C (3 I ; d ) k k k e − t sin(2 mt ) , (3.5)where C (3 O ; n ) k k k ≡ − q n k · k ( am ) , (3.6) C (3 I ; n ) k k k ≡ − q n (cid:18) − k · k k (cid:19) , (3.7) C (3 II ; n ) k k k ≡ q n (cid:18) − k · k k (cid:19) , (3.8)and C (3 O ; d ) k k k ≡ q d + 3 q d k · k ( am ) , (3.9) C (3 I ; d ) k k k ≡ − q d + q d (cid:18) − k · k k (cid:19) , (3.10) C (3 II ; d ) k k k ≡ − q d (cid:18)
12 + 3 k · k k (cid:19) + 3 q d (cid:18) − k · k k (cid:19) . (3.11)Here, we have extracted only oscillatory components from the interactions (3.2) and neglectedthe higher-order terms in the q -parameters and H/m . Provided the interaction Hamiltonian (3.3), we can estimate the correction to the bispectrumby using the in-in formalism, h ϕ k ( t ) ϕ k ( t ) ϕ k ( t ) i ≃ i Z t d t ′ h [ H (3) I ( t ′ ) , ϕ k ( t ) ϕ k ( t ) ϕ k ( t )] i , (3.12)at the leading order in the q -parameters. To see the contributions from the resonance, it issufficient to consider the linear gauge transformation (2.48), ζ = Hϕ/ ˙ φ . Then, in terms ofthe dimensionless bispectrum (3.1), the correction can be estimated as,∆ B ζ ≃ ¯ ǫ V I (3) k k k . (3.13)– 11 –ere, I (3) k k k ≡ Re (cid:20) mH Z z d z ′ C (3) k k k e − iKτ ′ (cid:21) + (5 perms) , ( K ≡ k + k + k ) , (3.14)where the coefficient function C (3) k k k is defined as, C (3) k k k ≡ C (3 O ) k k k − i (cid:18) k + k am (cid:19) C (3 I ) k k k − (cid:18) k am (cid:19) (cid:18) k am (cid:19) C (3 II ) k k k . (3.15)We have also introduced the averaged slow-roll parameter ¯ ǫ V by,¯ ǫ V ≡ √ ǫ V, ∗ ǫ V,c , (3.16)where ǫ V, ∗ and ǫ V,c are evaluated at the resonance and the horizon crossing, respectively.In deriving Eq. (3.14), we have used the subhorizon approximation for all mode functions, u k i ( t ′ ) ( i = 1 , , k i /a ∗ H >
1. This approximation is not valid when we consider the squeezed limit, whereone of the modes can be outside of the horizon when the oscillation is excited. We considerthis limit separately in the last part of this subsection.Since the integral has a similar form as Eq. (2.36), we can estimate it as in the previoussection. The oscillatory components, e imt ′ and e imt ′ , in C (3) k k k (see Eqs. (3.4) and (3.5))resonate with the factor from the mode functions, e − iKτ ′ . The resonance occurs in a similarway as in the power spectrum, where the oscillatory components in the interactions resonatewith the factor from the mode functions e − ikτ ′ . Here, we evaluate I (3) k k k for the couplings K n . Replacing the parameters ( q n , m/ , Γ /
2) by ( q d , m, Γ), we can obtain a similar result forthe couplings K d . The phase function θ (3; n ) ≡ mt − Kτ has a stationary point at time t ∗ where Ka ( t ∗ ) = m. (3.17)Then, using the function (2.40) with K in the argument, the integral can be evaluated as, I (3) k k k ≃ Re I ( n ) K (cid:20) C (3 O ) k k k , ∗ − k + k K C (3 I ) k k k , ∗ − (cid:18) k K (cid:19) (cid:18) k K (cid:19) C (3 II ) k k k , ∗ (cid:21) mH , (3.18)where the coefficient functions are evaluated at the resonance, t = t ∗ . The duration of theresonance is given by Eq. (2.41). Note that the bispectrum has contributions from thesubhorizon regime [45–49] in contrast to the cases without a resonance. Then, I (3) k k k canbe approximated as, I (3; n ) k k k ≃ cos θ (3; n ) ∗ I ( n ) K, approx (cid:20) C (3 O ) k k k , ∗ − k + k K C (3 I ) k k k , ∗ − (cid:18) k K (cid:19) (cid:18) k K (cid:19) C (3 II ) k k k , ∗ (cid:21) mH +(5 perms) , (3.19)and then the correction to the bispectrum (3.13) can be roughly estimated as,∆ B ( n ) ζ ≃ ¯ ǫ V (cid:16) mH (cid:17) q n cos θ (3; n ) ∗ I ( n ) K, approx S k , k , k , (3.20)– 12 – igure 3 . The resonant enhancement of the bispectrum through the couplings K n . In the left panel,we showed the shape function (3.21). Along the solid line, the bispectrum scales as shown in the rightpanel. The lines indicate the same in Fig. 2. The bispectrum has large values along the dashed line, K ≡ k + k + k = 2 k p , where k p is the peak scale of the feature in the power spectrum. Here, wehave set Γ = 5 H , m = 10 H , q n = 0 .
1, and ¯ ǫ V = 0 .
01. If other interactions exist, larger enhancementscan be realized ( § where S k , k , k ≡ q − n (cid:20) C (3 O ) k k k , ∗ − k + k K C (3 I ) k k k , ∗ − (cid:18) k K (cid:19) (cid:18) k K (cid:19) C (3 II ) k k k , ∗ (cid:21) + (5 perms) , (3.21)for the couplings K n . Therefore, the amplitude of the correction to the bispectrum can bewritten in terms of that to the power spectrum (2.50) as,∆ B ( n ) ζ ∼ ¯ ǫ V (cid:16) mH (cid:17) ∆ P ( n ) ζ P ζ S k , k , k . (3.22)As is clear from the expression (3.22), the correction to the bispectrum can be much largerthan that to the power spectrum thanks to the large factor m/H . In Fig. 3, we showed theshape and scaling of the correction to the bispectrum. The bispectrum has large values for K = 2 k p , where k p is the peak scale of the feature in the power spectrum, k p /a m ≃ e √ H/m / Equilateral configurations ( k ≃ k ≃ k ≃ k p / B ( n ) ζ, equilateral ≃ ǫ V (cid:16) mH (cid:17) ∆ P ( n ) ζ P ζ (3.23) ≃ ǫ V q n (cid:16) mH (cid:17) (for Γ ≪ √ Hm ) , (3.24)– 13 –t the equilateral point k ≃ k ≃ k ≃ k p /
3, while for the equilateral-type f NL , B ζ, equilateral = 910 f equilNL . (3.25)- Squeezed configurations ( k ≪ k ≃ k ≃ k p )In the finitely squeezed configurations at the specific scale, k ≪ k ≃ k ≃ k p but k > a ∗ H ≃ ( H/m ) K , we obtain a non-vanishing contribution,∆ B ( n ) ζ, not so squeezed ≃ ǫ V (cid:16) mH (cid:17) ∆ P ( n ) ζ P ζ (3.26) ≃ ǫ V q n (cid:16) mH (cid:17) (for Γ ≪ √ Hm ) , (3.27)while for the local-type f NL , B ζ, local = 310 f localNL (cid:18) k p k (cid:19) . (3.28)In terms of the bispectrum, h ζ k ( t ) ζ k ( t ) ζ k ( t ) i (2 H/m ) k p ≪ k ≪ k p , k ≃ k ≃ k p ≃ (2 π ) δ ( k + k + k ) P ζ ( k )∆ P ( n ) ζ ( k ) (cid:18) k k p (cid:19) (cid:20) ǫ V (cid:16) mH (cid:17)(cid:21) , (3.29)where P ζ is the power spectrum for ζ , P ζ ≡ π P ζ /k . On the other hand, in the standardsingle-field slow-roll inflation models, the bispectrum for the squeezed configurations is of theorder of the slow-roll parameters: h ζ k ( t ) ζ k ( t ) ζ k ( t ) i k ≪ k ≃ k ≃ k p = − (2 π ) δ ( k + k + k ) P ζ ( k ) P ζ ( k ) " n s − O (cid:18) k k p (cid:19) , (3.30)where n s is the spectral index of the power spectrum. Hence, the excitation of a heavy scalarfield can give a non-negligible effect on the bispectrum for the configurations, (2 H/m ) k p ≪ k ≪ k p , k ≃ k ≃ k p , as recently pointed out in Ref. [50]. If the peak scale k p is not so large,the scales k ≪ (2 H/m ) k p could be in the superhorizon regime at the present time. In thatcase, these contributions could provide practically the most important one in observationsof the bispectrum in the squeezed limit. Note that the contribution (3.29) corresponds tothe O ( k /k p ) correction in Eq. (3.30). This correction can be large in the resonant casesbecause it appears in the combination ( k τ ∗ ) ≃ ( k / k p ) ( m/H ) , which is not necessarilysmall even for the squeezed configurations k ≪ k ≃ k .As mentioned in the text below Eq. (3.14), our subhorizon approximation is not validin the squeezed limit where k ≪ (2 H/m ) k p , k ≃ k ≃ k p . In this case, the mode function– 14 – k and its time derivative ˙ u k should be replaced by those with the extra factors i/kτ ′ and1 / ( kτ ′ ) , respectively. Hence, the contributions to the squeezed limit can be estimated as,∆ B ( n ) ζ, squeezed ∼ ¯ ǫ V (cid:18) k p k (cid:19) ∆ P ( n ) ζ P ζ (3.31) ∼ ¯ ǫ V q n (cid:18) k p k (cid:19) (cid:16) mH (cid:17) (for Γ ≪ √ Hm ) , (3.32)and the bispectrum behaves as, h ζ k ( t ) ζ k ( t ) ζ k ( t ) i k ≪ k ≃ k ≃ k p ∼ (2 π ) δ ( k + k + k ) P ζ ( k )∆ P ( n ) ζ ( k ) ǫ V . (3.33)Therefore, the bispectrum scales as in the usual case (Eq. (3.30)) and cannot become largeunless the modification in the power spectrum is large.As discussed in the previous section, we need to require the conditions ∆ P ( n ) ζ / P ζ < m < π Λ n for the perturbativity. The correction has the maximum value whenthese conditions are saturated. In this case, both energy scales m and 2 π Λ n are given by O (10 )( f χ,t =0 / ¯ ǫ V ) / H . Then, our perturbative calculation is reliable if∆ B ( n ) ζ ∼ ¯ ǫ V (cid:16) mH (cid:17) ∆ P ( n ) ζ P ζ < O (10 ) × (¯ ǫ V f χ,t =0 ) . (3.34)Here, we have used the observed value for the amplitude of the power spectrum, P ζ = O (10 − )[60, 61]. Similarly, in the case of the couplings K d , we can find the upper limit for theperturbativity as, ∆ B ( d ) ζ ∼ ¯ ǫ V (cid:16) mH (cid:17) ∆ P ( d ) ζ P ζ < O (10 ) × (¯ ǫ V f χ,t =0 ) . (3.35)The energy scales m and (2 π ) / Λ d are given by O (10 )( f χ,t =0 / ¯ ǫ V ) / H when the inequalityis saturated.Therefore, the correction can be O (10 − ), which is much larger than that expected inthe standard single-field slow-roll inflation models, even when the correction to the powerspectrum is small. More larger non-Gaussianity can be induced when the correction to thepower spectrum is measurably large. As in the case of the power spectrum, the higher-order terms could be comparable to theleading term even when the q -parameters are small. Here, we discuss the validity of theperturbation for the cubic interactions (3.2).First, we estimate the magnitude of the interaction Hamiltonian (3.2) in the resonanceregime from the dimensional argument. The interaction Hamiltonian (3.2) can be estimatedas, H (3) I ∼ aq √ ǫ V (2 π ) M p k ϕ k , (3.36)– 15 –here we have assumed that the wavenumbers have the same order k . The fluctuations ϕ k can be estimated as, ϕ k ∼ a √ k (cid:18) πk (cid:19) . (3.37)Here, we have replaced the creation/annihilation operators by (2 π/k ) , taking into accounttheir commutation relation [ a k , a k ′ † ] = (2 π ) δ ( k − k ′ ). As mentioned in the previous sec-tion, the order parameter of the expansion is H (3) I ∆ t , where ∆ t ∼ / √ mH . Using Eqs.(3.36) and (3.37), it can be estimated as, H (3) I ∆ t ∼ (2 π ) − q √ ǫ V (cid:18) HM p (cid:19) (cid:16) mH (cid:17) (3.38) ∼ (cid:18) P ζ π (cid:19) ∆ B ζ , (3.39)at the resonance, k/a ∼ m . Therefore, the higher-order terms are subdominant for the cubicinteractions as long as ∆ B ζ < O (10 ).The quadratic interaction Hamiltonian (2.24) also gives corrections to Eq. (3.12). In asimilar way as above, the order parameter of the corrections is estimated to be, H (2) I ∆ t ∼ q r mH . (3.40)Hence, we can use the perturbation safely when the feature in the power spectrum is negligiblysmall, ∆ P ζ / P ζ ∼ q p m/H <
1, as discussed in the previous section. When this condition isnot satisfied, we should solve the full equation of motion to get the mode function. In thatcase, the mode function contains the negative energy mode and the bispectrum has additionalresonant contributions [47]. We can also estimate the cutoff scale Λ cut discussed in the lastpart of § t = 2 π/m . In the previous subsections, we have seen that the bispectrum can be affected much by theresonance through the couplings K n and K d at the specific scale. However, as can be seen inEq. (3.2), their effects are suppressed by the Planck scale. This is because cubic interactionsof the inflaton field are not contained in the couplings K n and K d . Then, they are inducedthrough the gravitational couplings. Direct cubic interactions appear if there is the selfinteraction of the inflaton field, λ s d ( ∂φ ) . (3.41)They are also induced by the interactions that are higher order in Λ d , λ h d ( ∂χ · ∂φ ) ( ∂φ ) , (3.42)or violate the parity symmetry φ → − φ , λ p d ( ∂χ · ∂φ )( ∂φ ) . (3.43)– 16 –ere, we estimate the magnitude of their effects on the bispectrum. The effects of theinteractions (3.41), (3.42), and (3.43) on the background evolution and the power spectrumcan be analyzed in a similar way as the couplings K d ; The interactions (3.41) and (3.42)have little effects because they are higher order in the q -parameters, while the interaction(3.43) could have an effect as large as the couplings K d with the q -parameter of the order of q ( ǫ V f − χ,t =0 ) . In the parity-vaiolating case, the kinetic mixing term ( ∂χ · ∂φ ) could appear.We assume that this term is so suppressed that the background evolution and the powerspectrum are not affected much.The interaction (3.41) induce the interaction Hamiltonian as, a λ s ˙ φ Λ d ! Z d x ˙ ϕ ( ∂ϕ ) . (3.44)Oscillatory components are induced in this Hamiltonian at the higher order in the q -parameters(see Eqs. (2.10) and (2.11)).On the other hand, the interactions (3.42) and (3.43) induce the interaction Hamiltonianas, a λ h ˙ χ ˙ φ d ! Z d x ˙ ϕ (cid:20) ϕ − a ( ∇ ϕ ) (cid:21) , (3.45)and a (cid:18) λ p ˙ χ d (cid:19) Z d x ˙ ϕ ( ∂ϕ ) , (3.46)respectively. Using the dimensional argument in the previous subsection, their effects on thebispectrum can be estimated to be,∆ B ( s ) ζ ∼ ∆ B ( h ) ζ ∼ q ( ǫ V f − χ,t =0 ) (cid:16) mH (cid:17) ∼ ( ǫ V f − χ,t =0 ) (cid:16) mH (cid:17) (cid:18) ∆ P ζ P ζ (cid:19) , (3.47)∆ B ( p ) ζ ∼ q ( ǫ V f − χ,t =0 ) (cid:16) mH (cid:17) ∼ ( ǫ V f − χ,t =0 ) (cid:16) mH (cid:17) ∆ P ζ P ζ , (3.48)at the peak scale. Therefore, the corrections to the bispectrum through the interactions(3.41), (3.42), and (3.43) are enhanced respectively by the factors ( m/H ) / and ( m/H ) ,which can be much larger than the factor m/H for the couplings K n and K d . Since all modefunctions appear with a derivative, these contributions scale as k − for finitely squeezedconfigurations where the subhorizon approximation (2.35) is appropriate. In the squeezedlimit, on the other hand, we cannot use the subhorizon approximation and the bispectrumscales as k − as in the usual cases.If we require the conditions ∆ P ( d ) ζ / P ζ < m < (2 π ) / Λ d for the perturbativity,they should be bounded from above as,∆ B ( s ) ζ , ∆ B ( h ) ζ < O (10 ) × ( ǫ V f − χ,t =0 ) , (3.49)∆ B ( p ) ζ < O (10 ) × ( ǫ V f − χ,t =0 ) − . (3.50)Therefore, the interactions (3.42) and (3.43) could induce large features in the bispectrumeven when the features in the power spectrum are too small to be detected. Note that the– 17 –cale of the resonance induced by the interaction (3.43) is different from that for the couplings K d ; The resonance occurs at K/a ∼ m for the interaction (3.43) while K/a ∼ m for thecouplings K d . If we consider much higher-order interactions, the scales of the resonanceappear at the integral multiples of the mass scale. Though they are suppressed by the q -parameters, they could still induce non-negligible features in the bispectrum. Non-Gaussianity could contain various information on the physics behind inflation. In thispaper, we have discussed the possibility that we could obtain hints on the heavy physics dur-ing inflation by analyzing local features in the primordial bispectrum. A heavy scalar fieldcan leave non-negligible signatures in the primordial spectra through the parametric reso-nance between its background oscillation and the fluctuations in the inflaton field. We haveestimated the contributions from the resonance perturbatively by picking up the interactionsrelevant to it. The bispectrum is amplified at specific configurations, and its amplitude canbe O (10 − ), or as large as O (10 ) depending on the type of interactions within the param-eter region where the perturbative expansion is applicable. In particular, the resonance cangive large contributions in finitely squeezed configurations, while the bispectrum cannot belarge in the squeezed limit unless the modification to the power spectrum is large as in thecase that the consistency relation is satisfied. Since we cannot observe the exact squeezedlimit, these contributions could practically be most important in actual observations of thebispectrum in that limit. In the analysis, we have assumed that oscillations of a heavy scalarfield are excited without introducing other effects. This assumption may be too idealisticwhen we consider a specific excitation mechanism of the oscillations. In general, there mightbe slow-roll violations [29], excitations from the vacuum, or mixing of the light and heavyfields through non-derivative couplings [24–35]. We will explore their effects on the resonancein future work.We have also found that there is a relation between the scales at which the featuresappear in the bispectrum and the power spectrum, and that the feature in the bispectrum canbe much larger than that in the power spectrum. Moreover, if we consider a specific excitationmechanism like a turn in the inflaton trajectory, other features could be also induced in theprimordial spectra around the horizon scale at the excitation time. In particular, as discussedin Ref. [62], correlated features in the bispectrum and the power spectrum are induced bya turn in the inflaton trajectory. Since the resonance scale is comparable to the mass scaleat the excitation time, we could determine the mass of the heavy scalar field if both of thesefeatures are detected. Though we have not discussed the observability of localized features inthe bispectrum, it will be discussed in the upcoming paper by one of the present author [63].If the correlated features are observed at the characteristic scales in the primordial spectra,it will indicate the presence of heavy degrees of freedom. By analyzing them, we could obtainsome information on the physics behind inflation. Acknowledgement
We thank J. Yokoyama for initial collaboration and J. R. White for reading a part of themanuscript. The work is supported by a Grant-in-Aid through JSPS (No. 23-3430 and No.24-2236). – 18 –
Interactions Relevant to the Resonance
Here, we discuss which cubic interactions are most relevant to the resonance. The cubicinteractions for the action (2.1) have been obtained in Refs. [64, 65]. They can be separatedinto the gravitational and matter parts as, H (3) g = − a M p H Z d x (cid:20) α + 2 a H α ∇ β + 1 a H ( ∇ β ∇ β − ∇ i ∇ j β ∇ i ∇ j β ) (cid:21) α, (A.1) H (3) m = − a Z d x (cid:20) P IJ ( X IJ + αX IJ ) + 12 P IJ,KL (2 X IJ + αX IJ ) X KL + P IJ,K ( X IJ + αX IJ ) ϕ K + 12 P I,J αϕ I ϕ J + 16 P IJ,KL,MN X IJ X KL X MN + 12 P IJ,KL,M X IJ X KL ϕ M + 12 P IJ,K,L X IJ ϕ K ϕ L + 16 P I,J,K ϕ I ϕ J ϕ K (cid:21) . (A.2)Here, α and β are respectively the perturbations in the lapse and shift functions at the firstorder, which are given by solving the constraint equations as, α = π I M p H ϕ I , (A.3) ∇ βa H = − α + 12 M p H h P I ϕ I − P IJ,K ˙ φ I ˙ φ J ϕ K + ( P IJ + P IJ,LM ˙ φ L ˙ φ M )( ˙ φ I ˙ φ J α − ˙ φ I ˙ ϕ J ) i . (A.4)The function X IJn ( n = 1 , ,
3) represents the perturbations in X IJ ≡ − ( ∂ µ φ I )( ∂ µ φ J ) / n -th order. They can be read from X IJ = 12(1 + α ) X n =0 V IJn − a ∇ i ϕ I ∇ i ϕ J , (A.5)where V IJ = ˙ φ I ˙ φ J , (A.6) V IJ = 2 ˙ φ ( I ˙ ϕ J ) , (A.7) V IJ = ˙ ϕ I ˙ ϕ J − a ∇ i β ˙ φ ( I ∇ i ϕ J ) , (A.8) V IJ = − a ∇ i β ˙ ϕ ( I ∇ i ϕ J ) , (A.9) V IJ = 1 a ( ∇ i β ∇ i ϕ I )( ∇ i β ∇ i ϕ J ) . (A.10)Here, () in the superscript represents the symmetrization.Since the resonance occurs deep in the horizon, the interactions with more derivativesare more relevant to the resonant enhancement of the bispectrum. Taking the terms withmore derivatives, the lapse and shift functions are estimated to be, α ≃ − r ǫ V ϕM p , (A.11) ∇ βa H ≃ r ǫ V ϕM p H ˆ z φ + 2( q d + q d ) e − t sin ( mt )ˆ z φ , (A.12)– 19 –t the leading order in the slow-roll parameter ǫ V and the fraction f χ . Though the energyfraction of the heavy scalar field f χ contains an oscillatory component, f χ ⊃ (cid:18) Γ m (cid:19) f χ,t =0 e − t sin(2 mt ) , (A.13)we have neglected it because the decay rate Γ is assumed to be much smaller than the massscale m , and then it cannot lead to a large enhancement.Counting the number of derivatives on ϕ , we can find the ( ∂ϕ ) -type interactions in theterms 16 P IJ,KL,MN X IJ X KL X MN , P IJ,KL X IJ X KL . (A.14)However, these terms are absent unless we include the interaction ( ∂φ ) , or those that arehigher order in Λ d or violate the parity symmetry φ → − φ . In § ∂ϕ ) -type interactions.The next candidates are provided by the ϕ ( ∂ϕ ) -type interactions. They can be foundin the terms 1 a H ( ∇ β ∇ β − ∇ i ∇ j β ∇ i ∇ j β ) α, P IJ,K X IJ ϕ K , (A.15)12 P IJ,KL (2 X IJ + αX IJ ) X KL , P IJ ( X IJ + αX IJ ) . (A.16)The first term is higher order in the slow-roll parameter ǫ V and the second term is absentunless parity-violating interactions like χφ ( ∂φ ) are introduced. Hence, the last two terms(A.16) provide the cubic interactions that are most relevant to the resonance when only thecouplings K n and K d exist. In terms of z φ and c s , these terms can be explicitly written as, H (3)relevant = z φ Z d x (cid:26) α (cid:20) (3 − c s ) ˙ ϕ + c s a ( ∇ ϕ ) (cid:21) + 1 a ∇ i β ∇ i ϕ ˙ ϕ (cid:27) (A.17) ≃ a M p r ǫ V Z d x (cid:26) − ˆ z φ (cid:20) (3 − c s ) ˙ ϕ + c s a ( ∇ ϕ ) (cid:21) ϕ + (cid:2) ˆ z φ + 2( q d + q d ) e − t sin ( mt ) (cid:3) ∇ i ( ∇ − ˙ ϕ ) ∇ i ϕ ˙ ϕ (cid:27) . (A.18) References [1] N. Bartolo, E. Komatsu, S. Matarrese and A. Riotto, Phys. Rept. , 103 (2004)[astro-ph/0406398].[2] J. M. Maldacena, JHEP , 013 (2003) [astro-ph/0210603].[3] P. Creminelli and M. Zaldarriaga, JCAP , 006 (2004) [astro-ph/0407059].[4] C. Cheung, A. L. Fitzpatrick, J. Kaplan and L. Senatore, JCAP , 021 (2008)[arXiv:0709.0295 [hep-th]].[5] P. Creminelli, G. D’Amico, M. Musso and J. Norena, JCAP , 038 (2011) [arXiv:1106.1462[astro-ph.CO]]. In the case that Γ ≪ H , the suppression factor should be replaced by H/m . – 20 –
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