# Correlation functions in stochastic inflation

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## Abstract

Combining the stochastic and \(\delta N\) formalisms, we derive non-perturbative analytical expressions for all correlation functions of scalar perturbations in single-field, slow-roll inflation. The standard, classical formulas are recovered as saddle-point limits of the full results. This yields a classicality criterion that shows that stochastic effects are small only if the potential is sub-Planckian and not too flat. The saddle-point approximation also provides an expansion scheme for calculating stochastic corrections to observable quantities perturbatively in this regime. In the opposite regime, we show that a strong suppression in the power spectrum is generically obtained, and we comment on the physical implications of this effect.

## 1 Introduction

Inflation is one of the leading paradigms describing the physical conditions that prevailed in the very early Universe [1, 2, 3, 4, 5, 6]. It is a phase of accelerated expansion that solves the puzzles of the standard hot Big Bang model, and it provides a causal mechanism for generating scalar [7, 8, 9, 10, 11] and tensor [12] inhomogeneous perturbations on cosmological scales. These inhomogeneities result from the parametric amplification of the vacuum quantum fluctuations of the gravitational and matter fields during the accelerated expansion.

^{1}It consists of an effective theory for the long-wavelength parts of the quantum fields, which are “coarse grained” at a fixed physical scale (i.e. non-expanding), somewhat larger than the Hubble radius during the whole inflationary period.

^{2}The non-commutative parts of this coarse grained field \(\varphi \) are small, and at this scale, short-wavelength quantum fluctuations have negligible non-commutative parts too. In this framework, they act as a classical noise on the dynamics of the super-Hubble scales, and \(\varphi \) can thus be described by a stochastic classical theory, following the Langevin equation

*a*is the scale factor. The Hubble parameter \(H\equiv \mathrm {d}a/(a \mathrm {d}t)\) is related to the potential

*V*via the slow-roll Friedmann equation \(H^2\simeq V/(3M_\mathrm{Pl}^2)\), where \(M_\mathrm{Pl}\) is the reduced Planck mass. The dynamics of \(\varphi \) is then driven by two terms. The first one, proportional to \(V^\prime \) (where a prime denotes a derivative with respect to the inflaton field), is the classical drift. The second one involves a white Gaussian noise, \(\xi \), and renders the dynamics stochastic. It is such that \(\left\langle \xi \left( N\right) \right\rangle =0\) and \(\left\langle \xi \left( N\right) \xi \left( N^\prime \right) \right\rangle =\delta \left( N-N^\prime \right) \).

The stochastic formalism thus accounts for the quantum modification of the super-Hubble scales dynamics. It allows us to calculate quantum corrections on background quantities beyond the one-loop approximation for the inflaton scalar field \(\phi \) (in fact, beyond any finite number of inflaton loops) and to calculate such quantities as e.g. the probability distribution and any moments of the number of inflationary \(e\)-folds in a given point. In turn, cosmological perturbations are affected too, and a natural question to address within the stochastic framework is therefore how quantum effects modify inflationary observable predictions. This is the main motivation of the present work.

Stochastic inflation is a powerful tool for calculating correlation functions of quantum fields during inflation. In Refs. [25, 26, 27, 28, 29], it is shown that standard results of quantum field theory (QFT) are recovered by the stochastic formalism for test scalar fields on fixed inflationary backgrounds, for any finite number of scalar loops and potentially beyond. This result is even extended to scalar electrodynamics during inflation in Refs. [30, 31] and to derivative interactions and constrained fields in Ref. [32]. In Ref. [26], fluctuations of a non-test inflaton field have been studied, too. In this last case, the calculation is performed at linear order in the noise, that is, assuming that the distribution of the coarse grained field remains peaked around its classical value \(\phi _\mathrm {c}\), where \(\phi _\mathrm {c}\) is the solution of Eq. (1.1) without the noise term. However, it may happen that the quantum kicks dominate over the classical drift and provide the main contribution to the inflationary dynamics in some flat parts of the potential. It is therefore legitimate to wonder what observable imprints could be left in such cases. In order to deal with observable quantities, the goal of this paper is therefore to calculate the correlation functions of inflationary perturbations in full generality, taking backreaction of created inflaton fluctuations on its background value into account, starting from Eq. (1.1) and without relying on a perturbative expansion in the noise.^{3}

This work is organized as follows. In Sect. 2, we first discuss the issue of the time variable choice in the Langevin equation (we further elaborate on this aspect in Appendix A). This allows us to set a few notations, and to already argue why some of the effects later obtained (but not all) are Planck suppressed. In Sect. 3, we turn to the calculation of the correlation functions of primordial cosmological perturbations, without assuming them to be small. We first review different methods that have been used in the literature, and motivate our choice of combining the stochastic and \(\delta N\) formalisms. We then settle our computational strategy and proceed with the calculation itself. Results are presented in Sect. 4; see in particular Eqs. (4.1) and (4.6). We show that the standard formulas are recovered in a “classical” limit that we carefully define, and discuss the regimes where they are not valid. Finally, in Sect. 5, we summarize our main results and conclude.

## 2 Time variable issue

*H*is dependent on the full coarse grained field and is therefore a stochastic quantity.

^{4}This has two consequences. The first one is that starting from a classical time label, any other time variable defined through

*a*or

*H*is a stochastic quantity, and cannot be used to label the Langevin equation, otherwise one would describe a physically different process. The time label must therefore be carefully specified. The second one is that, since

*H*is related to the curvature of space-time, its stochasticity has to do with the one of space-time itself. We are thus a priori describing effective quantum gravitational effects, corresponding to the gravitational interactions and self-interactions of the inflaton field. The corresponding corrections should therefore remain small as long as the energy density of the inflaton field is small compared to the Planck scale. For this reason, it is convenient to define the dimensionless potential

*N*. In the Itô interpretation

^{5}[17, 34, 35], it reads

^{6}

*t*. Performing the simple change of time variable \(\mathrm {d}N=H\mathrm {d}t\) in Eq. (1.1), this is given by

*H*is taken to be a function of time only, independent of \(\varphi \), the

*H*factors can be taken out of the derivatives with respect to \(\phi \) in Eqs. (2.3) and (2.5). In this case, it is straightforward to see that these two are perfectly equivalent through the change of time variable \(\mathrm {d}N=H\mathrm {d}t\), and that they describe the same stochastic process. On the contrary, if

*H*explicitly depends on \(\varphi \), this is obviously no longer the case and \(P\ne \tilde{P}\).

*J*. This current thus needs to be independent of \(\phi \) for a stationary distribution. In most interesting situations, it is actually 0. This is notably the case when the allowed values for \(\phi \) are unbounded. For example, if \(V(\phi )\) is defined up to \(\phi =\infty \), the normalization condition \(\int P_{\mathrm {stat}}(\phi )\mathrm {d}\phi =1\) requires that \(P_{\mathrm {stat}}(\phi )\) decreases at infinity strictly faster than \(\vert \phi \vert ^{-1}\). In this case, both \(P_{\mathrm {stat}}(\phi )\) and \(\partial P_{\mathrm {stat}}(\phi )/\partial \phi \) vanish at infinity. From Eq. (2.6),

*J*vanishes at infinity also, hence everywhere. This yields a simple differential equation to solve for \(P_{\mathrm {stat}}(\phi )\), and one obtains

At this point, we are left with the issue of identifying the right time variable to work with. Actually, one can explicitly show [26, 27, 36] that *N* is the correct answer, and that it is the only time variable that allows the stochastic formalism to reproduce a number of results from QFT on curved space-times. We leave this discussion to Appendix A, where we elaborate on existing results and show why, since we deal with metric perturbations, we must work with *N*.

## 3 Method

Let us now review how correlation functions of curvature fluctuations can be calculated in stochastic inflation, and see which approach is best suited to the issue we are interested in.

*k*(at which the power spectrum is calculated) exits the Hubble radius. If one plugs the expression (A.23) obtained in Appendix A for \(\langle \delta {\phi ^{(1)}}^2 \rangle \) using

*N*as the time variable into Eq. (3.1), one obtains

*t*, whereas, as already said, the number of \(e\)-folds

*N*must be used instead. This has important consequences. Indeed, if one makes use of cosmic time

*t*and plugs the associated expression (A.29) for the quadratic moment of \(\delta \tilde{\phi }^{(1)}\) into Eq. (3.1), one obtains

*N*as the time variable.

Another strategy is followed in Refs. [42, 43, 44], where methods of statistical physics, such as replica field theory, are employed in a stochastic inflationary context. However, only the case of a free test field evolving in a de Sitter or power-law background is investigated, while we need to go beyond the fixed background assumption in order to study the effects of the explicit \(H(\varphi )\) dependence. This is why we cannot directly make use of this computational scheme in the present work.

Finally, in Refs. [45, 46, 47], the \(\delta N\) formalism is used to relate the curvature perturbations to the number of \(e\)-folds statistics. This is this last route that we chose to follow here, since it does not rely on any perturbative expansion scheme, and since it does not prevent us from implementing the explicit \(H(\varphi )\) dependence. In Ref. [47], numerical solutions are obtained for quadratic and hybrid potentials. In the present work, we derive fully analytical and non-perturbative results that apply to any single-field potential, and which do not require a numerical solution of the Langevin equation. As a by-product, this allows us to prove, for the first time, that the standard results are always recovered in the classical limit, for any potential.

### 3.1 The \(\delta N\) formalism

The \(\delta N\) formalism [9, 48, 49, 50, 51, 52] is very well suited to addressing the calculation of correlation functions in stochastic inflation, since it relates the statistical properties of curvature perturbations to the distribution of the number of *e*-folds among a family of homogeneous universes. Let us first recall where this correspondence comes from and, as an example, how the scalar power spectrum is usually calculated in the associated formalism.

*t*slices of space-time have uniform energy density, and that fixed

*x*worldlines be comoving. When doing so, and including scalar perturbations only, the perturbed metric in this gauge (which coincides in the super-Hubble regime with the synchronous gauge supplemented by some additional conditions fixing it uniquely) becomes [9, 53, 54] \(\mathrm {d}s^2=-\mathrm {d}t^2+ a^2(t) e^{2\zeta (\varvec{x})}\delta _{ij}\mathrm {d}x^i \mathrm {d}x^j\), up to small terms proportional to gradients of \(\zeta \). Here, \(\zeta \) is the adiabatic (curvature) perturbation, which is time-independent in single-field inflation once the decaying mode can be neglected. The omission of tensor perturbations is justified by the fact that their amplitude is suppressed compared to the scalar ones by the small slow-roll parameter \(\epsilon _1\). This allows us to define a local scale factor \(\tilde{a}(t,\varvec{x})=a(t)e^{\zeta (\varvec{x})}\). Starting from an initial flat slice of space-time at time \(t_\mathrm {in}\), the amount of expansion \(N(t,\varvec{x})\equiv \ln \left[ \tilde{a}(t,\varvec{x})/a(t_\mathrm {in})\right] \) to a final slice of uniform energy density is then related to the curvature perturbation through

*N*is to be evaluated in unperturbed universes from an initial epoch when the inflaton field has an assigned value \(\phi \) to a final epoch when the energy density has an assigned value \(\rho \). Since the observed curvature perturbations are almost Gaussian, at leading order in perturbation theory, one has

*H*, on super-Hubble scales, the latter is given by [61] \(\mathcal {P}_{\delta \phi }(k)\simeq H^2(k)/4\pi ^2\), where

*H*(

*k*) means

*H*evaluated at the time when the

*k*mode crosses the Hubble radius, i.e. when \(aH=k\). Together with Eq. (3.7), one therefore obtains

*e*-fold, the classical drift of the inflaton field is of the order \(\Delta \phi _\mathrm {cl}=V^\prime /(3H^2)=\sqrt{2\epsilon _1}M_\mathrm{Pl}\), while the quantum kick is of the order \(\Delta \phi _{\mathrm {qu}}=H/(2\pi )\). This allows us to define a rough “classicality” criterion \(\Delta \phi _{\mathrm {qu}}/\Delta \phi _\mathrm {cl}\) that assesses the amplitude of the stochastic corrections to the classical trajectory. Making use of Eq. (3.8), this ratio can be expressed as

*H*can only decrease) and one is therefore ensured that the stochastic corrections to the inflaton trajectory remain small.

However, they are two caveats to this line of reasoning. The first one is that, as we will show below, \(\Delta \phi _{\mathrm {qu}}/\Delta \phi _\mathrm {cl}\) is not the correct way to assess the importance of stochastic effects and one should use instead another classicality criterion that we will derive. The second one is that, in some situations, \(\epsilon _1\) becomes tiny or even vanishes in some transient phase between the Hubble exit time of the observed modes and the end of inflation. This is the case, for example, when the potential has a flat inflection point, such as in MSSM inflation [62, 63, 64] or as in punctuated inflation [65, 66]. Another situation of interest is when inflation does not have a graceful exit but ends due to tachyonic instability involving an auxiliary field, like in hybrid inflation [67, 68], or by brane annihilation in string-theoretical setups [69, 70]. In such cases, \(\epsilon _1\) can decrease and the last \(e\)-folds of inflation may be dominated by the quantum noise. It is therefore important to study the dispersion \(\delta N\) arising not only from \(\delta \phi _*\) but from the complete subsequent stochastic history of the coarse grained field.

Note also that in these expressions, \(\zeta \) need not be small as was shown in Refs. [9, 51, 71] [note, however, that \(\zeta \) is defined up to a constant due to an arbitrary possible rescaling of *a*(*t*)], thus, \(\delta N\) need not be small, too. As follows from the quasi-isotropic (separate universe) approach, the condition for inflation to proceed is only that \(H\ll M_\mathrm{Pl}\). On the other hand, if \(P_{\zeta }(k)\sim H/(M_\mathrm{Pl}\sqrt{\epsilon _1})\) exceeds unity (the so called regime of “eternal inflation”), then the Universe loses its local homogeneity and isotropy after the end of inflation, but not immediately. This occurs much later than the comoving scale *a*(*t*) / *k* at which this inhomogeneity occurs crosses the Hubble radius \(H^{-1}\) second time. Thus, in the scope of the inflationary scenario \(P_{\zeta }\) may well exceed unity at scales much exceeding the present Hubble radius. The stochastic inflation approach provides us with a possibility to obtain quantitatively correct results in this non-linear regime, too.

### 3.2 Computational program

*k*, let \(\phi _*(k)\) be the mean value of the coarse grained field when

*k*crosses the Hubble radius. If inflation terminates at \(\phi _\mathrm {end}\), let \(\mathcal {N}(k)\) denote the number of \(e\)-folds realized between \(\phi _*(k)\) and \(\phi _\mathrm {end}\). Obviously, \(\mathcal {N}\) is a stochastic quantity, and we can define its variance

The computational program we must follow is now clear. For a given mode *k*, we first calculate \(\phi _*(k)\) (this sets the location of the observational window). We then consider stochastic realizations of Eq. (1.1) that satisfy \(\varphi =\phi _*(k)\) at some initial time,^{7} and denote by \(\mathcal {N}\) the number of \(e\)-folds that is realized before reaching \(\phi _\mathrm {end}\). Among these realizations, we calculate the first moments of this stochastic quantity, \(\langle \mathcal {N} \rangle \), \(\langle \mathcal {N}^2 \rangle \), \(\langle \mathcal {N}^3 \rangle \), etc. We finally apply relations such as Eqs. (3.12) and (3.14) to obtain the power spectrum, the non-Gaussianity local parameter, or any higher order correlation function.

### 3.3 First passage time analysis

Because any part of the potential can a priori be explored, here we consider two possible ending points, \(\phi _1\) and \(\phi _2\), located on each side of \(\phi _*\). If the potential is, say, of the hilltop type (left panel), \(\phi _1\) and \(\phi _2\) can be taken at the two values where inflation has a graceful exit, on each side of the maximum of the potential. If, on the other hand, a flat potential extends up to \(\phi =\infty \) (right panel), one of these points, say \(\phi _2\), can be taken where *V* becomes super-Planckian and inhomogeneities prevent inflation from occurring. In such cases, the precise value of \(\phi _2\) plays a negligible role, as we will show in Sect. 3.3.1. Let \(\mathcal {N}\) be the number of \(e\)-folds realized during this process.

*f*of \(\varphi \). The Taylor expansion of such a function at second order is given by \(f\left( \varphi +\mathrm {d}\varphi \right) =f\left( \varphi \right) +f^{\prime }\left( \varphi \right) \mathrm {d}\varphi +f^{\prime \prime }\left( \varphi \right) /2\,\mathrm {d}\varphi ^2+\mathcal {O}\left( \mathrm {d}\varphi ^3\right) \). Now, if \(\varphi \) is a realization of the stochastic process under study, \(\mathrm {d}\varphi \) is given by Eq. (1.1) and at first order in \(\mathrm {d}N\), one obtains

#### 3.3.1 Ending point probability

As a first warm-up, let us calculate the probability \(p_1\) that the inflaton field first reaches the ending point located at \(\phi _1\) [i.e. \(\phi \left( \mathcal {N}\right) =\phi _1\)] or, equivalently, the probability \(p_2=1-p_1\) that the inflaton field first reaches the ending point located at \(\phi _2\) [i.e. \(\phi \left( \mathcal {N}\right) =\phi _2\)]. This will also allow us to determine when the ending point located at \(\phi _2\) plays a negligible role.

*h*and \(\psi \) are linearly related, see Eq. (3.17), the same equation is satisfied by \(\psi \). When averaged over all realizations,

^{8}its right hand side vanishes. One then obtains \(\left\langle \psi \left[ \varphi \left( \mathcal {N}\right) \right] \right\rangle =\psi \left( \phi _*\right) \), which is the probability \(p_1\) one is seeking for. All one needs to do is therefore to solve Eq. (3.18) to obtain \(h\left( \varphi \right) \), to plug the obtained expression into Eq. (3.17) to derive \(\psi (\varphi )\), and finally to evaluate this function at \(\phi _*\). A formal solution to Eq. (3.18) is given by \(h\left( \varphi \right) =A\int _{B}^{\varphi }\exp \left[ -1/v\left( x\right) \right] \mathrm {d}x\), where

*A*and

*B*are two integration constants that play no role, since they cancel out when calculating \(\psi \) thanks to Eq. (3.17). Indeed, the latter gives rise to

^{9}

A few remarks are in order about this result. First, one can check that, since \(\phi _*\) lies between \(\phi _1\) and \(\phi _2\), the probability (3.20) is ensured to be between 0 and 1. Second, one can also verify that when \(\phi _*=\phi _1\), \(p_1=1\), and when \(\phi _*=\phi _2\), \(p_1=0\), as one would expect. Third, in the case depicted in the right panel of Fig. 1, in the limit where \(\phi _2\rightarrow \infty \), one is sure to first reach the ending point located at \(\phi _1\), that is, \(p_2=\int _{\phi _1}^{\phi _*}e^{-1/v}/\int _{\phi _1}^{\phi _2}e^{-1/v}=0\). Indeed, the numerator of the expression for \(p_2\) is finite, since a bounded function is integrated over a bounded interval. If the potential is maximal at \(\phi _2\), and if it is monotonous over an interval of the type \([\phi _{0},\phi _{2}[\), its denominator is on the contrary larger than the integral of a function bounded from below by a strictly positive number, over an unbounded interval \([\phi _{0},\phi _{\infty }[\). This is why it diverges, and why \(p_2\) vanishes. This means that if \(\phi _2\) is sufficiently large, its precise value plays no role, since inflation always terminates at \(\phi _1=\phi _\mathrm {end}\).

#### 3.3.2 Mean number of *e*-folds

^{10}

^{11}but one can be more specific. First of all, as can be seen in Fig. 2, \(\bar{\phi }\) must be such that, when

*f*is evaluated at \(\phi _2\), the integration domain of Eq. (3.24) possesses a positive part and a negative part, which are able to compensate for each other. This implies that \(\bar{\phi }\) must lie between \(\phi _1\) and \(\phi _2\). A second generic condition comes from splitting the

*x*-integral in Eq. (3.24) into \(\int _{\phi _1}^{\varphi }\mathrm {d}x = \int _{\phi _1}^{\phi _2} \mathrm {d}x+ \int _{\phi _2}^{\varphi }\mathrm {d}x\). The first integral vanishes because \(f(\phi _2)=0\), which means that in order for

*f*to be symmetrical in \(\phi _1\leftrightarrow \phi _2\), \(\bar{\phi }(\phi _1,\phi _2)\) must satisfy this symmetry too, that is to say, \(\bar{\phi }\left( \phi _1,\phi _2\right) =\bar{\phi }\left( \phi _2,\phi _1\right) \). Third, in the case where the potential is symmetric about a local maximum \(\phi _\mathrm {max}\) close to which inflation proceeds, the integrand in Eq. (3.24) is symmetric with respect to the first bisector in Fig. 2. The two green triangles must therefore have the same surface, which readily leads to \(\bar{\phi }=\phi _\mathrm {max}\). Fourth, finally, in the case displayed in the right panel of Fig. 1, if \(\phi _2\) is sufficiently large, we have established in Sect. 3.3.1 that \(p_2=0\) and the quantity we compute is the mean number of \(e\)-folds between \(\phi _*\) and \(\phi _1=\phi _\mathrm {end}\). For explicitness, let us assume that \(v^\prime >0\) (the same line of arguments applies in the case \(v^\prime <0\)). Inflation proceeds at \(\phi <\phi _2\). In the domain of negative contribution in Fig. 2, the argument of the exponential in Eq. (3.24) is positive. As a consequence, if \(\bar{\phi }\) is finite and \(\phi _2\rightarrow \infty \), the negative contribution to the integral is infinite while the positive one remains finite, which is impossible. In order to avoid this, one must then have \(\bar{\phi } = \phi _2\). In practice, almost all cases boil down to one of the two previous ones and \(\bar{\phi }\) is specified accordingly. Combining Eqs. (3.23) and (3.24), one finally has

^{12}

This quantity is plotted for large and small field potentials in Fig. 3, where it is compared with the results of a numerical integration of the Langevin equation (1.1) for a large number of realizations over which the mean value of \(\mathcal {N}\) is computed. One can check that the agreement is excellent.

*Classical limit*Let us now verify that the above formula boils down to the classical result (3.7) in some “classical limit”. This can be done by performing a saddle-point approximation of the integrals appearing in Eq. (3.25). Let us first work out the

*y*-integral, that is to say, \(\int ^{\bar{\phi }}_x\mathrm {d}y/v(y)\exp [1/v(y)]\). Since the integrand varies exponentially with the potential, the strategy is to evaluate it close to its maximum, i.e. where the potential is minimum. The potential being maximal at \(\bar{\phi }\) in most cases (see the discussion above), the integrand is clearly maximal

^{13}at

*x*. Taylor expanding 1 /

*v*at first order around

*x*, \(1/v(y)\simeq 1/v(x)-v^\prime (x)/v^2(x)(y-x)\), one obtains, after integrating by parts,

^{14}\(\int ^{\bar{\phi }}_x\mathrm {d}y/v(y)\exp \left[ 1/v(y)\right] \simeq v(x)/v^\prime (x)\exp \left[ 1/v(x)\right] \). Plugging back this expression into Eq. (3.25), one finally obtains

*v*can be trusted as long as the difference between 1 /

*v*(

*x*) and 1 /

*v*(

*y*) is not too large, say \(\vert 1/v(y)-1/v(x)\vert < R\), where

*R*is some small number. If one uses the Taylor expansion of 1 /

*v*at first order, this means that \(\vert y-x\vert <Rv^2/v'\). Requiring that the second order term of the Taylor expansion is small at the boundary of this domain yields the condition \(\vert 2v-v^{\prime \prime }v^2/{v^\prime }^2\vert \ll 1\). For this reason, we define the classicality criterion

*v*at second order. One obtains

*e*-folds, while when \(\epsilon _1\) decreases as inflation proceeds, the correction is negative and the stochastic effects tend to decrease the number of

*e*-folds, at least at linear order.

#### 3.3.3 Number of *e*-folds variance

*e*-folds, defined in Eq. (3.10). If one squares Eq. (3.22) and takes the stochastic average of it, one obtains

^{15}

*f*is the function defined in Eq. (3.24). When the Itô lemma (3.16) is applied to \(g\left( \phi \right) \), if one further sets \(g\left( \phi _1\right) =g\left( \phi _2\right) =0\), one obtains

*Classical limit*As was done for the mean number of \(e\)-folds in Sect. 3.3.2, let us derive the classical limit of Eq. (3.33). Obviously, in the classical setup the trajectories are not stochastic and \(\delta \mathcal {N}^2=0\), and what we are interested in here is the non-vanishing leading order contribution to \(\delta \mathcal {N}^2\) in the limit \(\eta _\mathrm {cl}\ll 1\). As before, the

*y*-integral can be worked out with a saddle-point approximation, and one obtains

^{16}\(\int ^{\bar{\phi }_2}_x\mathrm {d}y {f^\prime }^2\left( y\right) \exp \left[ 1/v(y)\right] \simeq v^4\left( x\right) /{v^\prime }^3\left( x\right) \exp \left[ 1/v\left( x\right) \right] /M_\mathrm{Pl}^4\). Plugging back this expression into Eq. (3.33), one obtains

#### 3.3.4 Number of *e*-folds skewness and higher moments

## 4 Results

We are now in a position where we can combine the intermediary results of the previous sections to give explicit, non-perturbative and fully generic expressions for the first correlation functions of curvature perturbations in stochastic inflation. We first derive the relevant formulas and their classical limits, before commenting on their physical implications in Sect. 4.3.

### 4.1 Power spectrum

*k*such that when it crosses the Hubble radius, the mean inflaton field value is \(\phi _*\). This formula provides, for the first time, a complete expression of the curvature perturbations power spectrum calculated in stochastic inflation. It is plotted for large and small field potentials in Fig. 4.

*Classical limit*Before commenting further on the physical implications of the above result, let us make sure that in the classical limit, \(\eta _\mathrm {cl}\ll 1\), the standard formula is recovered. Combining Eqs. (3.12), (3.26), and (3.34), one has

### 4.2 Non-Gaussianity and higher moments

Obviously, one can go on and calculate any higher order correlation function with Eq. (3.40). However, with the power spectrum and non-Gaussianity local parameter at hand, we already are in a position where we can draw important physical conclusions.

### 4.3 Discussion

A first important consequence of Eqs. (4.1) and (4.6) is the correctness of their classical limits. They show the validity of our computational program for calculating correlation functions in general. This may be particularly useful for investigating other cases than single-field slow-roll inflation, especially when the standard procedure is difficult to follow. Indeed, our method can easily be numerically implemented, and it could then be applied to more complicated scenarios such as multi-field inflation where it has been shown [77] that the \(\delta N\) formalism retains reliable, modified kinetic terms where the stochastic inflation formalism has been generalized [78, 79, 80, 81]. In particular, it is well suited to situations where stochastic effects dominate the inflationary dynamics in some parts of the potential [47, 68, 82] and where one must take the stochastic effects into account.

However, even if the stochastic effects within the CMB observable window need to be small, let us stress that the location of the observable window along the inflationary potential can be largely affected. This notably happens when the potential has a flat region between the location where the observed modes exit the Hubble radius and the end of inflation, as is the case e.g. in hybrid inflation or in potentials with flat inflection points.

Another important point to note is that, contrary to what one may have expected, the corrections we obtained are not controlled by the ratio \(\Delta \phi _{\mathrm {qu}}/\Delta \phi _\mathrm {cl}\) extensively used in the literature, but by the classicality criterion \(\eta _\mathrm {cl}\) derived in Eq. (3.27). This has two main consequences.

*v*, which means that it is Planck suppressed.

^{17}This makes sense, since, as noted in Sect. 2, some of the corrections we obtained physically correspond to the self-interactions and gravitational interactions of the inflaton field.

^{18}This is why it can be useful to compare our results with loop calculations performed in the literature by means of other techniques. In particular, the self-loop correction to the power spectrum is derived in Ref. [86], and graviton loop corrections are obtained in Ref. [87] (for a nice review, see also Ref. [88]). A diagrammatic approach based on the \(\delta N\) formalism is also presented in Ref. [89] where the power spectrum and the bispectrum are calculated up to two loops. In all these cases, the obtained corrections are of the form \(\mathcal {P}_\zeta ^{1\mathrm {-loop}}=\mathcal {P}_\zeta ^\mathrm {tree}(1+\alpha \mathcal {P}_\zeta ^\mathrm {tree}\epsilon ^2 N)\). Here, \(\alpha \) is a numerical factor of order one that depends on the kind of loops one considers, and \(\epsilon ^2\) stands for second order combinations of slow-roll parameters. When the number of \(e\)-folds

*N*is of the order \(1/\epsilon \), this is exactly the kind of leading corrections we obtained. This feature is therefore somewhat generic. Obviously, it remains to understand which loops exactly our approach allows one to calculate, and how our results relate to the above mentioned ones. We leave it for future work.

Classicality criterion \(\eta _\mathrm {cl}\) defined in Eq. (3.27) for a few types of inflationary potentials. Expect for “large field”, the expression given for \(\eta _\mathrm {cl}\) is valid close to the flat point of the potential

Potential type | \({v(\phi )}\) | \({\eta _\mathrm {cl}}\) |
---|---|---|

Large field | \(\propto \phi ^p\) | \(\left( 1+\frac{1}{p}\right) v\) |

Hilltop | \( v_0\left[ 1-\left( \frac{\phi }{\mu }\right) ^p\right] \) | \(\frac{v_0}{p}\left( \frac{\mu }{\phi }\right) ^p\) |

Polynomial plateau | \( v_\infty \left[ 1-\left( \frac{\phi }{\mu }\right) ^{-p}\right] \) | \(\frac{v_\infty }{p}\left( \frac{\phi }{\mu }\right) ^p\) |

Exponential plateau | \( v_\infty \left[ 1-\alpha \exp \left( -\frac{\phi }{\mu }\right) \right] \) | \(\frac{v_\infty }{\alpha }\exp \left( \frac{\phi }{\mu }\right) \) |

Inflection point | \(v_0\frac{n\left( n-1\right) }{\left( n-1\right) ^2}\left[ \left( \frac{\phi }{\phi _0}\right) ^2-\frac{4}{n}\left( \frac{\phi }{\phi _0}\right) ^n+\frac{1}{n-1}\left( \frac{\phi }{\phi _0}\right) ^{2n-2}\right] \) | \(\frac{v_0}{n\left( n-1\right) }\left| \frac{\phi }{\phi _0}-1\right| ^{-3}\) |

Second, \(\eta _\mathrm {cl}\) contains \(1/v'^2\) terms. This means that, even if *v* needs to be very small,^{19} if the potential is sufficiently flat, \(\eta _\mathrm {cl}\) may be large. In Table 1, we have summarized the shape of \(\eta _\mathrm {cl}\) for different prototypical inflationary potentials. For large field potentials, \(\eta _\mathrm {cl}\) is directly proportional to *v*. This is why, in the left panels of Figs. 3 and 4, departure of the stochastic results from the standard formulas occur only when \(v\gg 1\), in a regime where our calculation cannot be trusted anyway. However, for potentials with flat points, different results are obtained. If the flat point is of the hilltop type, \(\eta _\mathrm {cl}\) diverges at the maximum of the potential. This is why, in the right panels of Figs. 3 and 4, even if *v* saturates to a small maximal value, the stochastic result differs from the classical one close to the maximum of the potential. However, this happens many \(e\)-folds before the scales probed in the CMB cross the Hubble radius, that is to say, at extremely large, non-observable scales. The same conclusion holds for plateau potentials (either of the polynomial or exponential type) where stochastic effects lead to non-trivial modifications in far, non-observable regions of the plateau. On the other hand, if the potential has a flat inflection point, \(\eta _\mathrm {cl}\) can be large at intermediate wavelengths, too small to lie in the CMB observable window but still of astrophysical interest. This could have important consequences in possible non-linear effects at those small scales, such as the formation of primordial black holes (PBHs) [90]. In such models, the production of PBHs is calculated making use of the standard classical formulas for the amount of scalar perturbations. However, we have shown that in such regimes, stochastic effects largely modify its value. An important question is therefore how this changes the production of PBHs in these models. In particular, it is interesting to notice that if the potential is concave (\(v^{\prime \prime } < 0\)), which is the case favored by observations [91, 92, 93], the leading correction in Eq. (4.4) is an enhancement of the power spectrum amplitude. However, as can be seen in the right panel of Fig. 4, as soon as one leaves the perturbative regime, this is replaced by the opposite trend: at the flat point, the classical result accounts for a diverging power spectrum while the stochastic effects make it finite. If generic, this effect may be important for the calculation of PBHs formation, and we plan to address this issue in a future publication.

## 5 Conclusion

Let us now summarize our main results. Making use of the \(\delta N\) formalism, we have shown how curvature perturbations can be related to fluctuations in the realized amount of inflationary \(e\)-folds in stochastic inflation trajectories. We have then applied “first passage time analysis” techniques to derive all the statistical moments of the number of *e*-folds, hence all scalar correlation functions in stochastic inflation.

We have shown that the standard results can be recovered as saddle-point limits of the full expressions. The situation is therefore analogous to, e.g., path integral calculations. A new simple classicality criterion has been derived, which should replace the common estimate based on the ratio between the mean quantum kick and the classical drift during one *e*-fold. It shows that quantum corrections to inflationary observables are Planck suppressed in general (that is to say, they are proportional to \(V/M_\mathrm{Pl}^4\)), but can be large if the potential is flat enough, even at sub-Planckian scales. For simple inflationary models where \(\vert \mathrm {d}V/\mathrm {d}\phi \vert /V\) increases monotonously as inflation proceeds, the corresponding effects play a non-trivial role only at extremely large, non-observable scales. However, models containing a flat point in the potential between the Hubble exit location of the modes currently observed in the CMB and the end of inflation behave differently. First, the stochastic effects change the mean total number of inflationary \(e\)-folds and can therefore largely modify the location of the observational window along the inflationary potential. Second, the amount of scalar perturbations produced around the flat point is strongly modified by stochastic effects. This may be crucially important for a number of non-linear effects computed at these small scales, such as the formation of PBHs, or non-Gaussianity.

Together with the case of tensor perturbations, which we have not addressed in this paper, we plan to study these issues in future publications.

## Footnotes

- 1.
This formalism was, in fact, first used in Ref. [9] at the level of the Langevin equation, from which results lying beyond the one-loop approximation for the inflaton field were obtained.

- 2.
More precisely, the coarse grained part of the field consists of the modes

*k*for which \(k\lesssim \sigma a H\). Here, \(\sigma \) is a cutoff parameter satisfying [25] \(e^{-1/(3\epsilon _1)}\ll \sigma \ll 1\), where \(\epsilon _1\) is the first slow-roll parameter. Under this condition, the physical results are independent of \(\sigma \). - 3.
In this connection, the approach of Ref. [45] is close to ours. However, we use a different form of the Fokker–Planck equation, a different initial condition for the inflaton probability distribution, and a different form of the \(\delta N\) formalism (which, in fact, may be called

*N*formalism) that does not use an expansion in \(\delta N\) and in the metric perturbation \(\zeta \) (in fact, these two quantities are not small in the so called regime of eternal inflation). - 4.
Hereafter, by “stochastic quantities”, we simply refer to realization dependent quantities, as opposed to quantities that are fixed for all realizations.

- 5.More generally, the last term in Eq. (2.3) can be written in the formwith \(0\le \alpha \le 1\), where \(\alpha =0\) corresponds to the Itô interpretation and \(\alpha =1/2\) to the Stratonovich one [33]. However, analysis shows that keeping terms explicitly depending on \(\alpha \) exceeds the accuracy of the stochastic approach in its leading approximation (1.1). In particular, corrections to the noise term due to self-interactions of small-scale fluctuations (if they exist) are at least of the same order or even larger.$$\begin{aligned} \frac{\partial }{\partial \phi } \left\{ \frac{H^{2\alpha }}{8\pi ^2}\frac{\partial }{\partial \phi }\left[ H^{2(1-\alpha )}P\left( \phi ,N\right) \right] \right\} \end{aligned}$$(2.3)
- 6.
Note also that we never use the “volume weighted” variant of Eq. (2.3) proposed as an alternative in Ref. [24], since then the resulting distribution is not normalizable: its integral over \(\mathrm {d}\phi \) is time- or

*N*-dependent. Thus, it leads to probability non-conservation. Neither is it justified from the physical point of view, since it is based on the assumption that all Hubble physical volumes (“observers”) emerging from the expansion of a previous inflationary patch are clones of each other, while they are strongly correlated. - 7.
This calculation therefore relies on a specific choice of initial (in fact, pre-inflationary) conditions, since all trajectories emerge from \(\phi _*\) at initial time. In principle, other choices could be made, even if most physical quantities (in particular, perturbations during the last 60

*e*-folds) do not depend on them. - 8.
- 9.
This is in agreement with Eq. (29) of Ref. [17], derived in the case where

*H*is constant, hence \(v^{-1}\approx v_*^{-1} - (v-v_*)\,v_*^{-2}\), where \(\phi _2\) and \(\phi _1\) lie at \(\pm \infty \) correspondingly, and where the initial condition for Eq. (2.3) is chosen to be \(P(\phi ,0)=\delta (\phi - \phi _*)\). - 10.
Here again, since both the integrand and the upper bound are stochastic quantities, it is non-trivial that the integral in the right hand side of Eq. (3.22) vanishes when averaged, but it can be shown rigorously.

- 11.Alternatively, one can write Eq. (3.24) in the explicit form [17]where \(p_1\) is given by Eq. (3.20) and, in the configuration of Fig. 1, \(\theta (x-x_*)=1\) when \(x>x_*\) and 0 otherwise.$$\begin{aligned} f\left( \varphi \right) = \int _{\phi _1}^{\phi _2} \frac{\mathrm {d}y}{M_\mathrm{Pl}} \int _y^{\phi _2} \frac{\mathrm {d}x}{M_\mathrm{Pl}}\frac{1}{v(y)}\exp \left[ \frac{1}{v(y)}- \frac{1}{v(x)}\right] \left[ \theta (x-x_*) - p_1\right] , \end{aligned}$$
- 12.
This is again in agreement with Eq. (35) of Ref. [17] if

*H*is constant and \(\phi _*=\bar{\phi }=0\), while \(\phi _\mathrm {end}=\infty \). - 13.
Strictly speaking, this is only true if the potential is a monotonous function of the field, but this is most often the case in the part of the potential that is relevant to the inflationary phase.

- 14.
Since \(v(\bar{\phi })\gg v(x)\) and if

*v*is monotonous, one can also show that \(\exp \left[ -v^\prime (x)/v^2(x)(\bar{\phi }-x)\right] \) is exponentially vanishing and this term can be neglected. - 15.
This is again a non-trivial result, since both the integrand and the upper bound of the integral appearing in Eq. (3.22) are stochastic quantities, but, as before, it can be shown in a rigorous way.

- 16.
- 17.
This remark also sheds some new light on the old debate [83, 84, 85] whether quantum gravitational corrections should affect inflationary predictions through powers of \(\phi /M_\mathrm{Pl}\) or \(V/M_\mathrm{Pl}^4\). This analysis reveals \(V/M_\mathrm{Pl}^4\) corrections only, regardless of the value of \(\phi /M_\mathrm{Pl}\).

- 18.
For this reason, one may think that performing the calculation in Fourier space as we did does not allow us to properly account for self-interaction effects and that a real space calculation should be carried out instead. However, since the separate Universe approximation is exponentially well verified on large scales, this is not the case. Making use of the same formalism as in Ref. [25], we have indeed explicitly checked that performing the calculation in real space leads to the same results as the ones presented here.

- 19.
Since

*v*can only decrease, \(v<10^{-10}\) for all observable modes. - 20.
In this discussion, vector and tensor perturbations are irrelevant, which is why they are not taken into account.

- 21.
In spite of the complexity of the field equations at second order, see e.g. Ref. [97], in the long-wavelength limit, it is sufficient [59] to use the local conservation of energy-momentum to establish Eqs. (A.6 and (A.7). Because this is not the main subject of this discussion, the corresponding calculations are not reproduced here but they can be found in Refs. [59, 95].

- 22.
The 1 / 2 factor comes from the relation \(\int _{x_1}^{x_2}f(x)\delta (x-x_2)\mathrm {d}x=f(x_2)/2\), which applies when the Dirac function is centered at a boundary of the integral.

- 23.
One can show [100] that the potential associated with power-law inflation, for which \(a(t)\propto t^p\), is given by \(V(\phi )\propto e^{-\sqrt{2/p}\phi /M_\mathrm{Pl}}\). Since \(H^2=V(\phi )/(3M_\mathrm{Pl}^2)\) at leading order in slow roll, one obtains the given \(H(\phi )\) profile.

- 24.
- 25.
- 26.
As shown in Sect. 3.1, this difference is crucial, since it leads to an incorrect result for the power spectrum of scalar perturbations.

- 27.

## Notes

### Acknowledgments

It is a pleasure to thank Eiichiro Komatsu, Jérôme Martin, Gianmassimo Tasinato, David Wands and Jun’ichi Yokoyama for very useful comments and enjoyable discussions. VV’s work is supported by STFC grant ST/L005573/1. A.S. was partially supported by the RFBR Grant No. 14-02-00894. His visit to the Utrecht University was supported by the Delta ITP Grant BN.000396.1. He thanks Profs. S. Vandoren and T. Prokopec and Dr. J. van Zee for hospitality during the visit.

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