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Single-field inflation, anomalous enhancement of superhorizon fluctuations, and non-Gaussianity in primordial black hole formation

We show a text-book potential for single-field inflation, namely, the Coleman-Weinberg model can induce double inflation and formation of primordial black holes (PBHs), because fluctuations that leave the horizon near the end of first inflation are anomalously enhanced at the onset of second inflation when the time-dependent mode turns to a growing mode rather than a decaying mode. The mass of PBHs produced in this mechanism lies in several discrete ranges depending on the model parameters. We also calculate the effects of non-Gaussian statistics due to higher-order interactions on the abundance of PBHs, which turns out to be small.


I. INTRODUCTION
The primordial perturbation generated in the inflationary epoch [1,2] is believed to be the origin of large-scale structure observed in the universe today. We have accurate information on the primordial perturbation through the observation of anisotropy of the cosmic microwave background(CMB) with the help of cosmological perturbation theory [3].
However, the CMB observation provides us with the information on the perturbations on a limited range of scales. Therefore, we cannot say, a priori, anything on the primordial perturbation at the smaller scales from CMB data. If the perturbation at these scales is order unity, primordial black holes can be produced when the scale of overdensed region crosses the horizon [4]. The typical mass of these black holes is given by horizon mass at horizon crossing 1 : where M G = (8πG) −1/2 = 2.4 × 10 18 GeV is the reduced Planck mass 2 . Though PBH evaporates through the Hawking radiation process [6], those with mass greater than 10 15 g can remain until the present time [7,8]. These PBHs can be an origin of intermediate mass black holes [9] or dark matter in the Universe [7,10].
It is difficult to produce appreciable numbers of PBHs in the simple single-field slow-roll inflation models which predict nearly scale-invariant power spectrum of curvature perturbations because the amplitude of fluctuations on small scales cannot be much different from that normalized by CMB observation on large scales. One exception among others [11,12] is the chaotic new inflation model [13] where double inflation is realized with a single field.
In Ref. [13], however, calculation of curvature fluctuation was done using the slow-roll formula in which only the time-independent mode has been taken into account. In the present paper, we solve the evolution equation of each Fourier mode of fluctuations properly and find an anomalous growth due to the temporal deviation from slow-roll evolution between two inflationary stages. This provides the first realistic example of the anomalous growth of perturbation in the superhorizon regime discussed in Ref. [14]. As a result we obtain a 1 Strictly speaking, the mass of black hole depends on the density perturbations δ and black holes with a small fraction of horizon mass are also produced [5]. Since the contributions of these black holes are small compared to the black holes with horizon mass, we ignore this dependence for simplicity. 2 We use units for which c = 1. power spectrum highly peaked on some scales depending on the values of the model parameters and PBH formation is more easily realized than concluded in Ref. [13]. We search for the values of parameters with which appreciable numbers of PBHs are produced under the observational constrains [15]. We also analyze the effects of non-Gaussianity generated in this model on the abundance of PBHs.
The organization of this paper is as follows. In section II, we introduce the chaotic new inflation model and explain the background evolution in this model. In section III, a mechanism of the enhancement of the perturbation is explained and the power spectrum in the chaotic new inflation model is given. In section IV, we give an expression of PBH abundance resulted from a peaked power spectrum. In section V, we estimate non-Gaussian correction to PBH abundance. In section VI, we calculate the PBH abundance with various values of parameters and give a relation between mass and PBH abundance. Section VII contains conclusion of this paper. In this paper, we use curvature perturbation in the comoving gauge ζ as a degree of freedom of scalar perturbation.

II. CHAOTIC NEW INFLATION MODEL
We consider a single-field inflation model with the Coleman-Weinberg potential [16] Historically, this potential was first used to realize new inflation [17], but phase space consideration has led to the conclusion that chaotic inflation [18] is much more likely to occur [19]. So we start with a large field value.
In this model, inflation can occur twice [20]. First, chaotic inflation occurs. After chaotic inflation, inflaton oscillates between the two minima of the potential. If the parameter v is In describing the evolution of the inflaton during inflation, it is convenient to introduce the following Hubble slow-roll parameters: where dots denote differentiation with respect to the cosmic time t. We show the evolution of these slow-roll parameters in Fig. 2. The values of the model parameters are the same as those employed in Fig. 1. We can see that slow-roll conditions are not satisfied while the inflaton is oscillating between the two minima. As we see in the next section, the existence of this period is important for enhancement of curvature perturbation.

III. ENHANCEMENT OF CURVATURE PERTURBATION
Curvature perturbation is enhanced in the models where slow-roll conditions are temporarily broken as stated in Ref. [14]. In this section, we briefly describe the mechanism of the enhancement and give a power spectrum of curvature perturbation in the chaotic new inflation model.

A. Evolution of curvature perturbation
Curvature perturbation in the comoving gauge ζ, in terms of which the amplitude of perturbation in the intrinsic spatial curvature of the comoving slicing R c is written as evolves according to an equation where N is the number of e-folds and ζ k is the Fourier transform of ζ: In the slow-roll inflation regime, the coefficient of the second term in Eq. (6) is positive and therefore the solutions of Eq. (6) in the long-wavelength regime, where the last term is negligible, are a constant mode and a decaying mode. The time derivative dζ k /dN, which corresponds to the decaying mode, diminishes in proportional to a −2 . If we neglect the last term in Eq. (6), dζ k /dN diminishes as a −3 . However, the last term decreases as (k/aH) 2 ζ k ∝ a −2 . As a result the second term in Eq. (6) soon becomes comparable to the last term in Eq. (6), and dζ k /dN diminishes as a −2 . Therefore ζ k soon becomes constant after horizon crossing and the power spectrum of curvature perturbation is given by the squared amplitude of the vacuum fluctuation 3 at the time the mode crossed the horizon: The power spectrum given by Eq. (9) is well approximated by the form P ζ (k) ∝ k n−1 where n − 1 = −2ǫ − η, namely, a nearly scale-invariant spectrum with small value of slow-roll parameters. Since the observed amplitude is small and single-field inflation models generically gives spectral index n < 1 [21], this power spectrum leads to a small number of primordial black holes. On the other hand, the coefficient of the second term in Eq. (6) can be negative in the model where slow-roll conditions are temporarily broken such as the chaotic new inflation model, in which we find ǫ ≃ 3 and η < 0 near the end of the oscillatory phase so that 3 − ǫ + η < 0. In this case, Eq. (6) has a growing mode solution instead of the decaying mode solution outside the horizon. Therefore ζ k grows even after horizon crossing and its amplitude is enhanced. This enhancement of the perturbation enables even a single-field inflation model to produce a large number of PBHs. 3 In subhorizon scales, ζ k is given by the quantum fluctuation in the vacuum state. Therefore, ζ k π ζ k has the minimum value 1 under the uncertainty relation ζ k π ζ k ≥ 1. Here, π ζ k is the momentum conjugate to ζ k and expressed as 2a 3 ǫM 2 Gζ * k . Then, approximatingζ k ∼ (k/a)ζ k , we obtain the amplitude (8).

FIG. 3: Power spectrum of curvature perturbation (solid line). This spectrum is calculated under
. We show also a power spectrum estimated by using the formula (9), which is used for a slow-roll inflation model (dashed line).
In Fig. 3, it is observed that the power spectrum deviates from the one estimated by using the slow-roll formula (9) and the enhancement of the perturbation has occurred. The power spectrum has a peak with amplitude ∼ 6.

IV. ABUNDANCE OF PRIMORDIAL BLACK HOLES
In this section, we give an expression of the PBH abundance resulting from a strongly peaked power spectrum.
We estimate the PBH abundance based on the Press-Schechter method [22,23]. In this method, a PBH with mass greater than M is formed when the perturbation which is smoothed on scale R M corresponding to M exceeds the threshold ζ th . The smoothed perturbation ζ R M is defined by where W (x/R) is a window function. The fraction of the energy density of the Universe collapsing into PBHs with mass M < M BH < M + ∆M at the time they form is given by where the prefactor 2 is due to Press-Schechter's prescription and P R M is the probability Therefore, we omit the argument x in the following. For the moment, we assume ζ k to be Gaussian distributed, which is the case to the lowest order of perturbation, and consider non-Gaussian correction in the next section to test the validity of this assumption. Then where W (kR) is the volume-normalized Fourier transform of the window function W (x/R).
We estimate the PBH abundance, (11), resulting from a strongly peaked power spectrum at the scale k −1 = k −1 peak . Since the power spectrum is strongly peaked we can approximate it to be δ-function in Eq. (12). This approximation is adequate, for P R M is exponentially sensitive to σ R M and a small difference in σ R M leads to a large difference in P R M so that only the value of σ 2 R at the peak is important. Under this approximation and using a top-hat function as W (kR), we obtain dσ 2 R M /dM ∼ P ζ (k peak )δ(M − M peak ) where M peak is the mass corresponding to the scale of the peak. Therefore, only PBHs whose mass is M peak form. In this approximation, its abundance is given by where β is independent of ∆M under the current approximation. In the peaked spectrum, the amplitude at the scale of the peak can be large while the amplitude at the observed scale is consistent with the observational value. Therefore, a large number of PBHs are produced from strongly peaked power spectrum.

V. NON-GAUSSIAN CORRECTION TO PBH ABUNDANCE
The amplitude of perturbation producing PBHs is so large that non-Gaussinity of curvature perturbation due to higher-order effects can be important. In the case of slow-roll inflation, this effect on the formation of PBHs has been studied in Ref. [24] and shown to be negligibly small contrary to the intuitive expectation. However, in models where slow-roll conditions are temporarily violated such as the chaotic new inflation model, large non-Gaussinity arises [25] and can modify the PBH abundance obtained above. In this section, we investigate this possibility.

A. Three-point correlation functions
Here in order to estimate the effects of the deviation from Gaussian due to the higherorder interaction in the action, we first evaluate three-point correlation functions.
We sketch the derivation of the three-point correlation functions [26]. The three-point functions are estimated by perturbative expansion with respect to the interaction. For the estimation of the three-point correlation functions, the cubic terms of ζ 4 in the action give relevant interactions, which consist of where ψ is defined by From these terms, the interaction Hamiltonian up to the third order reads where π ζ is the momentum conjugate to ζ and the variables with subscript I denote variables in the interaction picture. Coefficients H (i) (i = 1, 2, 3, 4) are given by where we have defined for simplicity of expression. For an equilateral triangle, k 1 = k 2 = k 3 = k, the values of these quantities arek = k aH , cos θ = − 1 2 , z = 1.
In calculating the three-point correlation functions, the in-in formalism [27] is used since we want to calculate an expectation value with respect to the vacuum state at t → −∞: where ǫ 0 is a positive infinitesimal constant 5 . To leading order of H I , the three-point corre- 5 In an abuse of language, we use the same symbol ζ k for denoting a quantized variable as a classical one. lation function is given by where the t ′ integration contour is deformed so that both bra and ket are projected on to the vacuum state at t → −∞. According to the behavior of ζ kI , the integrals in Eq. (23) can be splitted into three parts, an integral over the region inside the horizon, the region around horizon crossing and the region outside the horizon. In the former two parts the deviation from the slow-roll inflation models is small, since the third term in Eq. (6) is dominant in these parts. The deviation arises in the last part, because ζ k grows outside the horizon in the chaotic new inflation model and not in the slow-roll inflation models. In the slow-roll inflation models, the contribution from the part outside the horizon is negligible since each term in the interaction (16) includesζ I or k/aH which have small values outside the horizon and, furthermore, the commutator of ζ k 's or its time derivatives vanish as a −ν (ν ≥ 2) 6 .
The three-point correlation functions are suppressed by the slow-roll parameters estimated at horizon crossing [26]. In contrast, in the chaotic new inflation model,ζ I has a non-negligible value outside the horizon and the integrals over the region outside the horizon contributes to the three-point correlation functions. The reason for obtaining large non-Gaussianity is different from that in Ref. [25]. In the chaotic new inflation model it is the growth of the perturbation outside the horizon, while in Ref. [25] it is the characteristic behavior of the slow roll parameters near or inside the horizon.

B. Correction to PBH abundance from three-point correlation functions
In the following, we give an expression of the PBH abundance resulting from the perturbation with the three-point correlation functions.
The probability distribution P R M can be expressed as 6 If ζ kI (t) is constant ζ kI (t) = ζ const kI , the three-point functions vanishes because a term ζ k 1 I (t)ζ k 2 I (t)ζ k 3 I (t)ζ * k1I (t ′ )ζ * k2I (t ′ )ζ * k3I (t ′ ) becomes a real number |ζ cosnt k1I ζ const k2I ζ const k3I | 2 in this case. Then the leading term of the commutator of ζ k 's have a decaying mode in ζ k 's, and decreases as a −2 . With time derivatives of ζ k 's, the commutators decreases more rapidly.
where Φ is defined by Φ can be expanded by cumulants of ζ R M , ζ n R M c : The cumulants of ζ R M can be expressed by the connected part of the correlation functions of ζ k . With the formula (26), we can estimate the correction to P R M from higher-order correlation functions.
The correction from the three-point correlation functions can be calculated, retaining terms up to m = 3 in Eq. (26). In the case | ζ 3 R M c | is much smaller than σ 3 R M , we can give a concrete expression of the corrected PBH abundance. In this case, we can expand Φ with respect to J which is defined by and get J can be expressed by the three-point correlation functions of ζ k as Substituting the probability distribution P R M given by Eq. (28) to Eq. (11), we get the PBH abundance including the correction from the three-point correlation functions.
Because of homogeneity of the universe, the three-point correlation functions of ζ k can be written as The isotropy of the universe guarantees that A(k 1 , k 2 , k 3 ) is in fact a function of "shape" and "size" of triangle spanned by k i (i = 1, 2, 3) such as the quantities (21) and independent of direction of the triangle. In the chaotic new inflation model, ζ k grows most significantly at k = k peak and A(k 1 , k 2 , k 3 ) has the largest value at k 1 = k 2 = k 3 = k peak . Therefore, as before, we can take approximation d ζ 3 R M c /dM ∼ 4πk 6 peak A peak δ(M − M peak ), where A peak represents A(k 1 , k 2 , k 3 ) estimated at k 1 = k 2 = k 3 = k peak . Under this approximation, the PBH abundance is given by where J peak ≡ 2π 3 The PBH abundance with some values of J peak is shown in Fig. 4. For the values of the parameters (λ, v) = (5.4 × 10 −14 , 0.355139M G ) with which the amplitude at the peak has the values for producing relevant number of PBHs, ∼ 6.2 × 10 −3 , the value of A peak is estimated to be On the other hand, the value obtained by estimating the contribution from the region around horizon crossing as done in slow-roll inflation models is calculated to be [26] where the value of P ζ is estimated at horizon crossing, ∼ 10 −11 (see Fig. 3). f NL is an estimator usually used for parameterizing the size of non-Gaussianity observed in the CMB [26] and has the value of the order of the slow-roll parameters estimated at horizon crossing, ∼ 10 −1 (see Fig. 2).
We get larger three-point correlation functions than those without the enhancement of the perturbation. However, the parameter J peak which is an estimator for the correction to the PBH abundance from the three-point correlation functions is estimated to be This value is too small to modify the PBH abundance relevantly. Though large three-point correlation functions are obtained, since a denominator of the estimator is also large, only a small correction is obtained.
The correction to the PBH abundance from N-point correlation functions appears with a factor P −N/2 ζ . Therefore the corrections are expected to be small as the correction from three point functions are. We can therefore use the expression (13) for the PBH abundance safely.

VI. PARAMETER SEARCH
In §III, we have shown the curvature perturbation ζ is enhanced in the chaotic new inflation model. In this section, we calculate the PBH abundance using Eq. (13) with various values of the parameter v and search the parameter with which relevant number of PBHs can be produced. For each v, the parameter λ is fixed by the power spectrum of ζ observed by WMAP, P ζ = 2 × 10 −9 at k = 0.002/Mpc. In the linear perturbation theory, the power spectrum P ζ (k) scales as 7 Since the horizon mass at the matter-radiation equality is ∼ 10 17 M ⊙ , PBHs that can be the origin of intermediate mass black holes or dark matter are produced in the radiationdominated epoch. In the radiation-dominated epoch, the threshold value of the density perturbation δ th is given by 1/3 [4,7]. Then, we find the corresponding value of the curvature perturbation ζ th = 0.75 with the help of the formula in the linear perturbation theory, δ k=aH = 4ζ k=aH /9 [23]. A similar value has been obtained by numerical calculation in Ref. [28]. Equation (1) where the variables with subscript " * " denote those estimated at arbitrary time in the radiation-dominated epoch. We assume the period of reheating is negligible and have chosen the time at which inflation finished as time at which the variables with subscript " * " are estimated. We write these variables with subscript "end". We where R end ≡ 1/(a end H end ), which is the comoving scale crossing the horizon at the end of the entire inflation. The PBH abundance β is plotted as a function of the mass corresponding to the scale of the peak M peak in Figs. 5 and 6 with the observational constraints. The constraint on PBH abundance with mass above 10 15 g is obtained from the condition that the abundance of PBHs is less than that of matter today. The other constraints are obtained from the consistency with nucleosynthesis and γ-ray observation. In Fig. 6, we can observe that PBHs with mass 6×10 20 g can constitute a large part of matter in the universe. Though PBHs with some range of mass are observationally excluded to be the dominant component of dark matter [29], there are no strong constraints in mass range 10 20 g −10 26 g (10 −13 M ⊙ − 10 −7 M ⊙ ). The mass 6 × 10 20 g are in this range. Therefore, we can obtain PBHs that can be an origin of dark matter in the chaotic new inflation model. waves generated from scalar perturbations through non-linear couplings [30]. The scale of the peak which gives mass M = 1.4 × 10 13 g corresponds to GW frequency 8 ∼ 100 Hz, 8 The power spectrum of the induced GW have the peak at k peak / √ 3π [30]. and power spectrum for the induced GW is estimated to be 9 ∼ P 2 ζ ∼ 10 −4 . Therefore, the induced GW can be detected by detectors such as GEO600 [ In both parameter regions, however, the values of spectral index and tensor-to-scalar ratio are not in the preferred region of the observational data. This is because, in the chaotic new inflation model, the observed perturbations are produced at the chaotic inflationary epoch with a potential close to the quartic one, which is not preferred by the observational data. In this paper we adopted the Coleman-Weinberg potential as a simple model to drive inflationary dynamics without specifying particle physics background. In reality, we expect supergravity corrections to the scalar potential is important, especially for ϕ > M G . If such corrections modifies the potential in an appropriate way, this scenario could be a feasible one. Another way to rescue it is to adopt a smaller self coupling λ using the scaling law (36), so that it predicts smaller amplitudes of both scalar and tensor fluctuations on large scales observed by CMB etc., assuming that the observed scalar perturbations on these scales were created through a different mechanism, say, curvaton [36], or modulated rehating [37]. Since the peak we have found is so prominent that one could produce observable amount of PBHs even in such a case with smaller λ.

Acknowledgement
This work was partially supported by JSPS Grant-in-Aid for Scientific Research Nos. 16340076 and 19340054.  9 In Ref. [30], the energy density of the induced GW is calculated to be Ω GW ∼ 10 −17 A 4 where A is the amplitude at the scale of the peak relative to the observed amplitude. At M = 1.4 × 10 13 g, A is estimated to be ∼ 10 3 , and Ω GW ∼ 10 −5 . This value is comparable to the nucleosynthesis bound Ω GW ∼ 10 −5 [31]. However, the amplitude of the scalar perturbations is so large that the effects neglected in Ref. [30] such as backreactions to the scalar perturbations from the tensor perturbations should be properly taking into account.