{'fraction_non_alphanumeric': 0.040155308061714516, 'fraction_numerical': 0.015122100745887402, 'mean_word_length': 4.0165726977618315, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Waterfall field in hybrid inflation and curvature perturbation

We study carefully the contribution of the waterfall field to the curvature perturbation at the end of hybrid inflation. In particular we clarify the parameter dependence analytically under reasonable assumptions on the model parameters. After calculating the mode function of the waterfall field, we use the delta N formalism and confirm the previously obtained result that the power spectrum is very blue with the index 4 and is absolutely negligible on large scales. However, we also find that the resulting curvature perturbation is highly non-Gaussian and hence we calculate the bispectrum. We find that the bispectrum is at leading order independent of momentum and exhibits its peak at the equilateral limit, though it is unobservably small on large scales. We also present the one-point probability distribution function of the curvature perturbation.


I. INTRODUCTION
Currently, primordial inflation [1] is supposed to be the leading candidate to provide the necessary conditions for the successful big bang cosmology [2]. The simplest model of inflation driven by only a single inflaton field is consistent with most recent observations [3].
It is however expected that, in the context of theories beyond the standard model of particle physics e.g. supersymmetry, there is a number of multiple scalar fields which may contribute to the inflationary dynamics [4]. Furthermore, we may be able to observationally detect deviations from the predictions of single field models in the near future and to discuss interesting phenomenology, such as isocurvature perturbations and non-Gaussianity.
Hybrid inflation [5] is an interesting realization with two field contents, the usual inflaton field φ which drives slow-roll inflation and the waterfall field χ which terminates inflation by triggering an instability, a "waterfall" phase transition. Previously, it has been assumed that χ becomes momentarily massless only at the time of waterfall and very heavy otherwise, and thus does not contribute to the curvature perturbation R c on large scales: only the quantum fluctuations of φ contributes to R c and we can follow the well-known calculations of single field case, with the energy density of the universe being dominated by a non-zero vacuum energy.
This naive picture has been receiving a renewed interest [6][7][8] with the common qualitative results that the power spectrum of the curvature perturbation induced by the waterfall field is very blue and extremely small on large scales 1 . However, quantitatively it is not clear if they all agree or not. In particular, in Ref. [8] the δN formalism, which takes account of fluctuations only on super-horizon scales by construction, was employed to derive the power spectrum, but the approach there was not quantitative enough and hence the dependence on the model parameters was not explicitly presented.
In this note, we provide another complementary view. We adopt a few reasonable assumptions on the model parameters and solve the mode functions of χ in terms of the number of e-folds analytically. Then using the δN formalism [10] we calculate the corresponding R c induced by χ explicitly.
The result is consistent with the above references, i.e. the contribution of χ to the large scale curvature perturbation is totally negligible. We also clarify the model parameter dependence on the spectrum of the curvature perturbation. Furthermore, we calculate the corresponding bispectrum, which shows its peak at the equilateral limit. We also compute explicitly the one-point probability distribution function which clearly shows the highly non-Gaussian nature of the curvature perturbation.
The outline of this note is as follows. In Section II, we find the mode function solution of the waterfall field χ valid both on super-horizon and sub-horizon scales. In Section III, we calculate the corresponding curvature perturbation R c induced by χ using the δN formalism.
In Section IV, we present the power spectrum and bispectrum of R c . In Section V, we show the explicit form of the one-point probability distribution function of R c and discuss relates issues. We conclude in Section VI. In Appendices, we discuss some technical details. In Appendix A, to check the consistency of the δN formalism with the standard perturbation theory, we give an estimation of the curvature perturbation by using the linear perturbation equation for R c . We find a good agreement with our result based on the δN formalism. In Appendix B we reconsider the splitting of the super-and sub-horizon modes and compute the average over the horizon scales. The results agree with the formulae we use in the main text.

II. MODE FUNCTION SOLUTION OF WATERFALL FIELD
Before we begin explicit computations, first of all we make the physical picture clear. Our purpose is to calculate the contribution of the waterfall field χ to the curvature perturbation R c . This is only possible when χ becomes dynamically relevant. While χ is well anchored at its minimum during the phase of slow-roll inflation and hence does not participate in the inflationary dynamics, it controls the physical processes from the moment of waterfall till the end of inflation. Thus, in the context of the δN formalism, if we can find the evolution of χ during this phase as a function of the number of e-folds N, it amounts to finding R c by the geometrical identity R c = δN. Therefore, our aim in this section is to calculate χ = χ(N) starting from the moment of waterfall. We will directly use this result to calculate R c in the next section.
We consider the potential of the two fields, the inflaton φ and the waterfall field χ, as We note that during the most period of inflation of our interest, it is assumed that the vacuum energy V 0 = M 4 /(4λ) dominates so that the Hubble parameter is effectively a constant, H = H 0 . This is a good approximation even after the waterfall phase transition until the last moment of inflation. The slow-roll and the waterfall conditions are respectively.
The equations of motion are given byφ where the spatial gradient term for φ is neglected as usual. Note that before waterfall, is well anchored at its minimum χ = 0 so it is itself the same as its fluctuation, χ = δχ. Thus we may regard (2.5) as the equation for δχ 2 , which arises from the vacuum fluctuations. Then after the waterfall transition, δχ becomes unstable and δχ 2 starts to grow rapidly, and inflation ends when the inflaton starts to roll fast, which happens when the term g 2 δχ 2 exceeds m 2 in (2.4). Here we adopt the mean field approximation, i.e. we replace g 2 δχ 2 by its expectation value g 2 δχ 2 , which should be valid for the motion of the homogeneous inflaton field φ. We also assume that the nonlinear term λδχ 2 in (2.5) can be neglected until the end of inflation. That is, we assume At the end of calculation, we must check if this condition is satisfied for the range of the parameters of our interest.
2 Note that during inflation δρ φ ∼ δφ while δρ χ ∼ δχ 2 , and thus the metric fluctuations are relatively second order with respect to δχ and does not appear in the equation of motion for δχ. This situation is closely analogous to the case of false vacuum inflation [11]. The correlation functions also show similar momentum dependence to those produced during false vacuum inflation [11,12] We can rewrite (2.4) and (2.5) in a more convenient form by using the number of e-folds as the time variable, dN = Hdt. Denoting the derivative with respect to N by a prime, we where n = N − N c is the number of e-folds measured relative to the time of the waterfall transition, and we have introduced the parameter r by We note that we can write the scale factor a and the conformal time η = −1/(aH) using n as a =a c e n = k c H 0 e n , (2.11) respectively, where a c = a(N c ) and k c = a c H 0 .
Inserting the background solution (2.9) for φ into (2.8) and neglecting the nonlinear term in accordance with the assumption (2.6), we obtain the equation for δχ in the Fourier space,

13)
A. High frequency limit k/a → ∞: WKB solution In the high frequency limit, we can solve (2.13) in terms of the WKB approximation. In this limit the proper asymptotic behavior of the positive frequency function is given by (2.14) The WKB solution that has this asymptotic behavior is readily obtained as where for convenience we have definedβ bỹ The above WKB solution is valid for any k at sufficiently early times, −n ≫ 1.
B. Low frequency limit k/a → 0: Hankel function solution In the large scale limit k → 0, (2.13) becomes Then, the solution is easily found to be to the Hankel function solution at some time well before the waterfall, n < 0 and |n| ≫ 1.
In the limit k/k c → 0, the WKB solution (2.15) becomes where we have assumed e −2rn ≫ 1.
As for the Hankel function solution, the argument is very large in the limit −n ≫ 1, √ βe −rn /r ≈ νe −rn ≫ ν. Thus using the asymptotic form of the Hankel function, we find that (2.18) becomes ν gives the correct phase factor dependence of (2.20) and thus we have c 2 = 0 and Thus, the long wavelength positive frequency function is given by Here let us evaluate the mode function at the moment of waterfall n = 0. At n = 0, remembering that β ≫ 1, the Hankel function takes the form with ν ≈ √ β/r. That is, the index and the argument of the Hankel function are the same.
In this case, the Hankel function solution takes the form Then, denoting by a subscript L the long wavelength modes which are on super-horizon scales at n = 0, we can write where the numerical factor reads 2 √ π/ 3 2/3 Γ(2/3) ≈ 1.25854, and we have defined 3 α ≡ 2rβ . (2.28) As we will see in the next section, we must require α ≫ 1. The above result (2.27) implies that all the super-horizon modes have the same amplitude at the moment of waterfall. The the moment of waterfall will be taken as the "initial" time to estimate the contribution of δχ to the curvature perturbation R c .
For the modes that are still on sub-horizon scales at the time of waterfall, k ≫ k c , the WKB solution is valid until n = 0. Denoting them by a subscript S, (2.15) readily gives so that at the moment of waterfall This is the "initial" amplitude of the sub-horizon modes.
Before we move on, we mention that the initial amplitudes of large scale limit (2.27) and that of small scale limit (2.30) do not match at k = k c if we extrapolate from both sides, but are different by a suppression factor α −1/3 . This indicates that in the intermediate regime around k = k c these two extreme values are deviating from the limiting values and smoothly connected [8]. In particular, this implies that the sub-horizon modes with k k c have slightly different initial amplitudes from (2.30). However this will not affect our subsequent discussions because of the phase volume ∼ k 3 that gives rise to a sharp peak in the spectrum at k ≈ αk c ≫ k c , as we will see below. Hence we just use (2.27) for the initial amplitude of the large scale modes with k < k c and (2.30) for that of the small scale modes with k > k c .

E. Evolution of the relevant modes after waterfall
Having found the "initial" amplitudes of both large scale and small scale modes, now we can calculate the subsequent evolution of the modes until the end of inflation.
Let us first consider the large scale modes. The solution is given by (2.24) and is valid for n > 0 as well. Then, using the asymptotic form of the large ν = √ β/r we can find [13] H (1) This is obtained with rn ≪ 1, and is thus valid for n ≪ 1/r. For any sensible model of hybrid inflation r ≪ 1, while the number of e-folds after waterfall until the end of inflation, n f , is O(1) or at most a few. Hence this asymptotic form is valid until the end of inflation.
Then, plugging (2.31) into (2.24), we can find that after waterfall the mode function on super-horizon scales evolves as 4 where the initial amplitude of the large scale modes δχ L (n = 0) is given by (2.27). As the logarithmic term indicates, (2.32) does not hold precisely at n = 0 but is valid for, as mentioned above, some time after waterfall till the end of inflation. As we will evaluate δχ at the end of inflation n f = O(1), we can justifiably use (2.32) to calculate the curvature perturbation.
Let us now turn to the small scale modes. An important point to calculate the evolution of sub-horizon modes is that the end of inflation is determined by the quanta of χ which become tachyonic right after waterfall [7], and affect the effective mass of φ in the form g 2 δχ 2 . The modes which become tachyonic satisfy, by definition, (k/k c ) 2 < |β| in (2.13).
Assuming n = O(1), we have |β| ≈ 2βrn ∼ α 2 . Hence we find that the modes with become tachyonic. Thus α must be much greater than unity in order to have an effective tachyonic instability.
To summarize, the small scale modes of our interest, which contribute to the tachyonic instability and control the end of inflation, are those in the interval Since (k/k c ) 2 can be neglected in comparison withβ at leading order approximation, they satisfy the same equation as the equation for the large scale modes, (2.17). Hence the evolution of these modes at n > 0 is the same as that given by (2.32). That is, where we have set the overall coefficient as

III. CURVATURE PERTURBATION INDUCED BY WATERFALL FIELD
In this section, we calculate the curvature perturbation R c by using (2.32) and (2.35) in the context of the δN formalism. In the δN formalism the spacetime geometry is spatially smoothly varying over super-horizon scales while each Hubble horizon size region is regarded as a homogeneous and isotropic universe. Hence we first need to smooth over the horizon where δχ 2 L (0) and hence δχ 2 (n) is spatially varying on super-horizon scales. Note that we have omitted the logarithmic dependence term on n in the exponent, which is subdominant when we evaluate at n = n f = O(1). We have also subtracted the contribution from the modes with k > αk since they remain stable and behave in the same way as the flat Minkowski vacuum modes, in accordance with the regularization we adopted, i.e. δχ 2 (n) = 0 at n < 0.
With α ≫ 1, from the initial amplitudes (2.27) and (2.30) we can see that the contribution of sub-horizon modes is much bigger than the one from super-horizon modes if the average is taken. At the end of inflation we have δχ 2 (n f ) = m 2 /g 2 , so that using which follows from (2.30), we have we find Now let us rephrase the above discussion in a form more convenient for the δN formalism.
With coordinate dependence explicit, δχ 2 given by (3.1) is recast as As mentioned in the first paragraph of this section, since the smoothing is done over the horizon scales H −1 0 , there remains no spatial coordinate dependence in δχ 2 S . Meanwhile, we do have a spatial coordinate dependence for the modes with wavelengths longer than , which is what we should take care of in the context of the δN formalism. Neglecting −3n in the exponential for simplicity since α ≫ 1, splitting n =n + δn and expanding in terms of δn, (3.1) is written as where δχ 2 S ≫ δχ 2 L as discussed above. Now we evaluate δn at a later time, say, at the end of inflation n = n f . Here it is important to note that the end of inflation is controlled by the value of δχ 2 at each spatial point, namely, Analogous to the case when the value of the inflaton field determines the end of inflation hypersurface, this condition determines the end of inflation hypersurface on which the energy density is uniform (at leading order approximation where the contribution of the inflaton to the energy density is negligible). Then, using (3.2) and (3.4), we find where we have truncated at linear order in δn. Inverting this relation, we can write the curvature perturbation generated between the moment of phase transition and the end of inflation as .

(3.9)
This explicitly shows that the spectrum of R c is determined by the spectrum of δχ 2 L . From the result obtained previously in Sec. II C, the mode function is k-independent for k < k c . This implies that the power spectrum of δχ L is white: P δχ L (k) is constant (in the conventional terminology used in cosmology, it is blue with the spectral index of 4: P δχ L (k) ≡ k 3 /(2π 2 )P δχ L (k) ∝ k n−1 with n = 4. See (5.4) for example). Assuming that the spectrum has a ultraviolet cutoff at k = k c , this implies that δχ 2 L also has the same white spectrum, since the convolution of two white spectra is white. Thus apart from the amplitude which we will calculate below, we can already conclude that P R ∝ k 3 , so that the spectral index is strongly blue with n Rc = 4, indicating that the curvature perturbation is strongly suppressed on large scales.
Before we move to the computation of the power spectrum, let us also observe that R c seems to be always negative. This can be also read from (3.8): although δχ L may be positive or negative, it appears in the form of a square in (3.8). So irrespective of the sign of δχ L its contribution is always positive. Meanwhile, the left hand side of (3.8) is a constant. Thus, to compensate the positive contribution of δχ 2 L to make the left hand side a constant, δn is always negative. Also we note that the average value of R c is not zero, . (3.10) We will consider these issues a little further later.
Finally, before closing this section, let us discuss constraints on the model parameters.
First we consider the condition that comes from the fact that the initial value of δχ 2 must be smaller than the final value of it. From (3.2) and (3.7), we find On the other hand, for this hybrid inflation model to be viable, the amplitude of the curvature perturbation due to the inflaton field φ must not exceed the observed value, P (φ) R 10 −9 , hence g 2 (2π) 2 10 −9 r 2 β . Comparing with (3.14), we see that this condition is also well satisfied for typical values of the model parameters.

A. Power spectrum
In this section, we drop the subscript L from δχ L for notational simplicity. Since R c = 0, it is more relevant to consider R c − R c rather than R c itself given by (3.9). Nevertheless, the difference becomes irrelevant in the Fourier space as long as we focus on a finite wavenumber.
We will discuss this point in the next section.
Moving to the Fourier space, we can write so that the power spectrum is written as where we have used (3.2) in the last equality.
Before waterfall, δχ is purely quantum and it can be expressed in terms of the creation and annihilation operators a † k and a k as where a † k and a k satisfy the canonical commutation relations otherwise zero, and the mode function χ k follows the same equation as that of δχ. Since the Fourier component of δχ 2 is written as a convolution we have to correlate four creation and annihilation operators with different momenta, To calculate the above, we should note that what we are interested in are connected graphs, correlating different (δχ 2 ) k 's. Thus the meaningful contractions are δχ 2 k δχ 2 q = (δχ p δχ k−p ) (δχ l δχ q−l ) + (δχ p δχ k−p ) (δχ l δχ q−l ) , (4.7) while the remaining possible contractions are within the same (δχ 2 ) k 's and hence are irrelevant. Then, we can easily find Thus, eliminating one of the momenta using the delta functions, and using the remaining delta function δ (3) (k + q) to replace q with −k, we find However, from (2.27), we have already seen that the super-horizon mode χ k is independent of k, and thus can be pulled out of the integral. Hence, we only have to integrate over the relevant super-horizon scale momentum, for which the upper limit is k = k c . 5 Therefore, using (2.27), we finally obtain (4.10) Since this expression has, as it should, the correct delta function dependence, we can readily extract the power spectrum P R . Noting (4.2) we find where the numerical coefficient reads , the maximum amplitude is found at k = k c as P R ∼ α −22/3 which is already much smaller than unity for α ≫ 1. For larger scales, it is exponentially suppressed and thus becomes absolutely negligible: for example, for a scale that exited the horizon at 50 e-folds before waterfall, it is suppressed by a factor (e −50 ) 3 ≈ 10 −65 . As already discussed in the previous section, setting P R ∝ k n R −1 , the spectrum is very blue with the index n R = 4.

B. Bispectrum
Having found the curvature perturbation and the solution of the mode function, it is now straightforward to calculate the three-point correlation function. We can start from the 5 Note, however, that mathematically there seems no apparent reason to set the upper limit of the integral at k = k c . It seems reasonable to extend the range of integration into sub-horizon scales up to an arbitrary ultraviolet cutoff at k = k UV with k UV ≫ k c . If proceeding with the sub-horizon mode function solution (2.30), one finds that the squared mode function |χ k | 2 is suppressed by a factor of k c /k relative to super-horizon modes. But this suppression factor is not strong enough to make the integral independent of the ultraviolet cutoff. Since the integrand |χ p | 2 χ |k−p| 2 is proportional to p −2 , the integral will be dominated by the contribution from the ultraviolet cutoff, leading to the result in proportional to k UV . Of course there is a natural choice for the cutoff in the present case; k UV = αk c , up to which the modes become tachyonic, as advocated in Ref. [6,14]. If we are to take this choice, then the resulting amplitude of curvature perturbations will be substantially enhanced, though the qualitative result will not change. Nevertheless, this strong dependence of super-horizon fluctuations on the ultraviolet cutoff deep inside the horizon looks physically strange because it seems to imply the violation of causality. In fact if this were indeed the case, then we would have a first example in which the δN formalism fails even for the curvature perturbation on super-horizon scales. This may be originated from our assumption of the knowledge of the entire universe beyond the horizon scale in the Fourier transformation. We discuss this point of maintaining causality regarding the horizon scale patches in the inflating universe in Appendix B, justifying (3.1) and (4.9) which are the very foundation of our computation of the correlation functions. definition (4.12) As before, we are interested in the connected graphs. This means we only take contractions between those coming from different (δχ) k 's. It is immediately seen that there (4.13) Each of these terms exactly corresponds to the term with δ (3) (k 1 + k 2 + k 3 ). Indeed, we find these 8 contractions give (4.14) Again noting that |χ k | 2 is independent of momentum, and has a cut-off at k = k c , we obtain Comparing this expression with the definition of the bispectrum, we find where the numerical coefficient reads 16(2π) 7 / {3 5 [Γ(2/3)] 6 } ≈ 4128.89. To leading order, the bispectrum has no momentum dependence, and thus the dimensionless shape function (k 1 k 2 k 3 ) 2 B R (k 1 , k 2 , k 3 ) exhibits its maximum amplitude at the equilateral limit k 1 = k 2 = k 3 . This is anticipated, since the curvature perturbation produced by the waterfall field is intrinsically highly non-Gaussian. Note, however, that this bispectrum is completely unobservable on large scales: in the equilateral limit, multiplying k 6 , we see for example that it is exponentially suppressed by a factor of (e −50 ) 6 ≈ 10 −130 for a scale that exited the horizon at 50 e-folds before the waterfall. Thus this bispectrum is totally hopeless to be detected on large scales.

V. DISTRIBUTION OF CURVATURE PERTURBATION
In this section, we consider the one-point probability distribution function of R c . Basically, we can guess the form of the probability distribution function. At leading order R c is proportional to the square of δχ L which is very close to Gaussian. Thus, the probability distribution of R c ∼ δχ 2 L is expected to be very close to the chi-squared distribution. The fully nonlinear distribution function can be obtained from (3.1). By setting n = n f + δn and the left hand side of it to be m 2 /g 2 , and regarding δn = R c as a function of δχ L , the distribution function P of R c is given as Here, we already know that P χ (δχ L ) is a Gaussian distribution with zero mean, i.e.
and the variance σ 2 δχ L ≡ δχ 2 L − δχ L 2 , with δχ L = 0, is given by where the power spectrum P δχ L (k) of the fluctuations δχ L can be found from (2.27) as Noting that the large scale modes has a cut-off at k = k c , we obtain Now, we evaluate (3.1) at n = n f + δn = n f + R c to write This equation can be easily solved for δχ L as a function of R c , Thus from (5.1) we can immediately find the probability distribution of R c as This is a fairly complex probability distribution function, and is very different from the Gaussian one. We plot it in Fig. 1.
Having the distribution function of R c at hand, let us consider the mean value R c . We can formally write it as where R c is now regarded as a function of δχ L . Although we cannot invert (5.7) to find R c (δχ L ) exactly, we can obtain an approximate expression by assuming |R c | ≪ 1 as where we have expanded in the limit δχ 2 L ≪ α 2 H 2 0 /(8π 2 ). It is trivial to find that for a Gaussian distribution P(x), and thus the average value of R c is found, using (5.5), as As we have mentioned before, the relevant curvature perturbation is not R c itself but R c − R c . Therefore, although R c is always negative, the true fluctuations from the mean value can become positive. Nevertheless, since the mean value R c turns out to be very small, there is no chance to have a large positive fluctuation: the fluctuation is bounded from above as Recalling that R c is negative of the gravitational potential, we can see that the curvature perturbation induced by the waterfall field repels matter around rather than attract. This implies that there would be no primordial black hole formation even on scales as small as the Hubble horizon scale, but rather bubbles of void may appear. But this is a highly qualitative argument and we need more explicit calculations, which we do not pursue in this note.

VI. CONCLUSION
In this paper, Here to check if our result based on the δN formalism is consistent with the standard perturbation theory, we evaluate the curvature perturbation by using the linear perturbation equation for R c . See also Ref. [6] for this approach.
In linear theory, on super-horizon scales, it is known that the curvature perturbation on comoving slices R c satisfiesṘ where δP c is the pressure perturbation on comoving slices. The comoving slice is defined by In the present case, this means where we have used the fact that χ = δχ L and P δχ L (k) ∝ k 3 , namely the fact that on super-horizon scales the contribution from the waterfall field to δT 0 i is negligible compared to that from the inflaton field. That is, on super-horizon scales, the comoving slices are defined solely in terms of the inflaton as those on which the inflaton field is homogeneous.
Therefore the contribution to the pressure perturbation δP c comes totally from the waterfall field χ, As for ρ + P , we have Hence the time dependent ratio of δχ ′ S 2 to φ ′ 2 is written as Using (3.7) and (A6), at the end of inflation we have Therefore, for n f 1, δχ ′ With the above result in mind, we rewrite (A1) as Since δχ 2 L (n)/ δχ 2 S (n) is time-independent, we can just replace it by that evaluated at n = 0. Then (A11) can be expressed as The last factor on the right hand side is negligible for n < n eq and approximately equal to one for n eq < n < n f . Therefore, with the initial condition that R c (0) = 0, it can be easily integrated to give With the identification that ∆n ≈ 1/(2αn 1/2 f ) as evaluated in (A10), this agrees with our result using the δN formalism 6 . Therefore, in stead of (B2), it is more reasonable to divide the field in such a way that φ S (x) and φ S (y) will not be correlated if |x − y| > L H . To incorporate this prescription, we proceed as follows. We introduce two boxes of different size, a very large box L 3 where L = NL H with N ≫ 1 being a very large integer and the horizon size box L 3 H . The large box would correspond to the present horizon size of the universe.
We define χ S (x) for each horizon size box as where n = (n 1 , n 2 , n 3 ) (n i are integers), x i is the center of i-th box, and θ (i) (x) = 1 if x is in the i-th horizon size region and zero otherwise. The long wavelength part is defined by Thus we have the decomposition, This guarantees that there is no correlation of between two short wavelength modes that belong to two different horizon size regions: for |x − y| > L H , Now we take the square of φ(x) and average over the horizon scale. We obtain It is reasonable to assume that | φ (i) (k)| 2 is independent of the region (i). Hence we may set . Then since i θ (i) (x) = 1, we obtain This agrees with (3.1).
Also, if we consider the sum on the short wavelength modes, which appears in (4.9), it is apparent that this is non-vanishing only for |k| ≥ 2π/L H , because there exists no sum for |k| < 2π/L H by definition. This means there will be no contribution from the short wavelength modes to the curvature perturbation on super-horizon scales.