Correlation Functionsin Stochastic Inflation

Correlation Functions
in Stochastic Inflation

[    [
today
Abstract

Combining the stochastic and 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.

a]Vincent Vennin b,c]and Alexei A. Starobinsky


Correlation Functions

in Stochastic Inflation

  • Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth, PO1 3FX, United Kingdom

  • L. D. Landau Institute for Theoretical Physics RAS, Moscow 119334, Russian Federation

  • Department of Physics and Astronomy, Institute for Theoretical Physics, Utrecht University, 3508 TD Utrecht, The Netherlands

E-mail: vincent.vennin@port.ac.uk, alstar@landau.ac.ru

ArXiv ePrint: 1506.04732

 

 

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.

The transition from these quantum fluctuations to classical but stochastic density perturbations [13, 14, 15, 16] gives rise to the stochastic inflation formalism [17, 18, 19, 20, 21, 22, 23, 24, 25].111This formalism was, in fact, first used in LABEL:Starobinsky:1982ee at the level of the Langevin equation, from which results lying beyond the one-loop approximation for the inflaton field were obtained. 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.222More precisely, the coarse grained part of the field consists of the modes for which . Here, is a cutoff parameter satisfying [25] , where is the first slow-roll parameter. Under this condition, the physical results are independent of . The non-commutative parts of this coarse grained field 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 can thus be described by a stochastic classical theory, following the Langevin equation

(1.1)

This equation is valid at leading order in slow roll. Time is labeled by the number of -folds , where is the scale factor. The Hubble parameter is related to the potential via the slow-roll Friedmann equation , where is the reduced Planck mass. The dynamics of is then driven by two terms. The first one, proportional to (where a prime denotes a derivative with respect to the inflaton field), is the classical drift. The second one involves a white Gaussian noise, , and renders the dynamics stochastic. It is such that and .

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 (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 -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 LABEL:Tsamis:2005hd. In LABEL:Finelli:2008zg, 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 , where 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.333In this connection, the approach of LABEL:Enqvist:2008kt 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 formalism (which, in fact, may be called formalism) that does not use an expansion in and in the metric perturbation (in fact, these two quantities are not small in the so called regime of eternal inflation).

This work is organized as follows. In section 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 section 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 formalisms. We then settle our computational strategy and proceed with the calculation itself. Results are presented in section 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 section 5, we summarize our main results and conclude.

2 Time Variable Issue

Because of the Friedmann equation, the Hubble parameter appearing in Eq. (1.1) is sourced by the inflaton field itself, through the slow-roll function . At leading order in the noise, one simply has , which is a classical quantity. Beyond the leading order, however, is dependent on the full coarse grained field and is therefore a stochastic quantity.444Hereafter, by “stochastic quantities”, we simply refer to realization dependent quantities, as opposed to quantities that are fixed for all realizations. This has two consequences. The first one is that starting from a classical time label, any other time variable defined through or 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 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- 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

(2.1)

which we will make use of extensively in the following. Before turning to the calculation of the correlation functions, in this section, we show on a simple example why different time labels in the Langevin equation typically yield results that differ by corrections.

First, let us recast the stochastic process (1.1) through a Fokker-Planck equation, which governs the time evolution of the probability density that at time . In the Itô interpretation555More generally, the last term in Eq. (2.3) can be written in the form (2.2) with , where corresponds to the Itô interpretation and to the Stratonovich one [34]. However, analysis shows that keeping terms explicitly depending on 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. [17, 35, 36], it reads666Note also that we never use the “volume weighted” variant of Eq. (2.3) proposed as an alternative in LABEL:Linde:1993xx since then the resulting distribution is not normalizable: its integral over is time- or -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.

(2.3)

Now let us compare this equation with the one that would have been obtained if the Langevin equation had been written in terms of cosmic time . Performing the simple change of time variable in Eq. (1.1), this is given by

(2.4)

Here we use the notation to stress the fact that, a priori, does not describe the same stochastic process as . The Fokker-Planck equation corresponding to Eq. (2.4) is given by

(2.5)

If is taken to be a function of time only, independent of , the factors can be taken out of the derivatives with respect to 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 , and that they describe the same stochastic process. On the contrary, if explicitly depends on , this is obviously no longer the case and .

This can be better illustrated by calculating the stationary distributions associated with these processes. Let denote a stationary probability distribution for the stochastic process (1.1), or equivalently, (2.3). By definition, , hence

(2.6)

which defines the probability current . This current thus needs to be independent of for a stationary distribution. In most interesting situations, it is actually . This is notably the case when the allowed values for are unbounded. For example, if is defined up to , the normalization condition requires that decreases at infinity strictly faster than . In this case, both and vanish at infinity. From Eq. (2.6), vanishes at infinity also, hence everywhere. This yields a simple differential equation to solve for , and one obtains

(2.7)

Here, an overall integration constant, which makes the distribution normalized, , is omitted. Similarly, Eq. (2.5) can be written as , and requiring that the current vanishes gives rise to a differential equation for the stationary distribution , which can easily be solved. One obtains

(2.8)

The two distributions are close, and the effects coming from the dependence are small, only in the regions of the potential where .

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, 37] that 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 .

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.

The problem can first be treated at linear order [38, 39, 40] by expanding the coarse grained field about its classical counterpart , . Here, recall that is the solution of the Langevin equation (1.1) without the noise term. The quadratic moment of can be calculated as in appendix A.2, see Eq. (A.23). It corresponds to the integrated power spectrum of the field fluctuations on super-Hubble scales, and can therefore be related [40] to the power spectrum of curvature perturbations thanks to the relation

(3.1)

In this expression, the right hand side needs to be evaluated when the scale associated with the wavenumber (at which the power spectrum is calculated) exits the Hubble radius. If one plugs the expression (A.23) obtained in appendix A for using as the time variable into Eq. (3.1), one obtains

(3.2)

As before, needs to be evaluated when the scale associated with the wavenumber at which the power spectrum is calculated exits the Hubble radius. The quantity is the first slow-roll parameter. At leading order in slow roll, it verifies . The above expression exactly matches the standard result [41, 42]. In order to get the first corrections to this standard result, one thus needs to go to higher orders in . Actually, one can show that no contributions arise at next-to-leading order, and that one needs to go at least to next-to-next-to-leading order. This renders the calculation technically difficult. This is why we will prefer to make use of non-perturbative techniques. In passing, let us stress that in LABEL:Kunze:2006tu, the Langevin equation is written and solved with , whereas, as already said, the number of -folds  must be used instead. This has important consequences. Indeed, if one makes use of cosmic time and plugs the associated expression (A.29) for the quadratic moment of into Eq. (3.1), one obtains

(3.3)

Here, we have adopted the same notation as in section 2 where a tilde recalls that not the same quantity is worked out and is not . This result matches Eq. (2.11) of LABEL:Kunze:2006tu. However, in this work, it is concluded that, because of the second term in the braces of Eq. (3.3), which is always negative, the amplitude of the power spectrum in the stochastic approach is in general reduced with respect to the standard result. One can see that such a statement is incorrect, since the extra term in Eq. (3.3) is simply due to not working with the correct time variable. This is why, if such an approach were to be followed and extended to higher orders, it would again be crucial to work with as the time variable.

Another strategy is followed in Refs. [43, 44, 45], 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 dependence. This is why we cannot directly make use of this computational scheme in the present work.

Finally, in Refs. [33, 46, 47], the formalism is used to relate the curvature perturbations to the number of -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 dependence. In LABEL:Fujita:2014tja, 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 Formalism

The 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 -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.

Starting from the unperturbed flat Friedmann-Lemaître-Robertson-Walker (FLRW) line element, , deviations from homogeneity and isotropy can be included in a more general, perturbed metric, which contains some gauge redundancy. A specific gauge choice consists in requiring that fixed slices of space-time have uniform energy density, and that fixed 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] , up to small terms proportional to gradients of . Here, 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 . This allows us to define a local scale factor . Starting from an initial flat slice of space-time at time , the amount of expansion to a final slice of uniform energy density is then related to the curvature perturbation through

(3.4)

where is the unperturbed amount of expansion. From this, an important simplification arises on large scales where anisotropy and spatial gradients can be neglected, and the local density, expansion rate, etc., obey the same evolution equations as a homogeneous FLRW universe. Thus we can use the homogeneous FLRW solutions to describe the local evolution, which is known as the “quasi-isotropic” [55, 56, 57, 58] or “separate universe” approach [59, 60, 51]. It implies that is the amount of expansion in unperturbed, homogeneous universes, so that can be calculated from the knowledge of the evolution of a family of such universes. Written in terms of the inflaton field , consisting of an unperturbed, homogeneous piece and of a perturbation originating from quantum fluctuations, Eq. (3.4) gives rise to

(3.5)

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

(3.6)

Here, is usually evaluated with the slow-roll, classical formula

(3.7)

Once is decomposed into Fourier components, , the power spectrum is defined from the quantum expectation value . It can be expressed in terms of the power spectrum of (defined by similar relations) thanks to Eq. (3.6). For quasi-de Sitter inflation, and when the curvature of the inflaton potential is much smaller than , on super-Hubble scales, the latter is given by [61] , where means evaluated at the time when the mode crosses the Hubble radius, i.e. when . Together with Eq. (3.7), one therefore obtains

(3.8)

which is the same as Eq. (3.2) and which matches the standard result [41, 42].

A fundamental remark is that in the above usual calculation, the quasi-isotropic (separate universe) approximation is assorted with the assumption that on super-Hubble scales, the evolution of the inflaton field is governed by its classical equation of motion (3.7). The stochastic dispersion in the number of -folds thus only comes from the field dispersion at Hubble crossing . In most cases, this is a good approximation for the following reason. From the Langevin equation (1.1), one can see that during the typical time scale of one -fold, the classical drift of the inflaton field is of the order , while the quantum kick is of the order . This allows us to define a rough “classicality” criterion that assesses the amplitude of the stochastic corrections to the classical trajectory. Making use of Eqs. (3.8), this ratio can be expressed as

(3.9)

which is valid for single-field slow-roll models of inflation with canonical kinetic terms. Since for the modes observed in the Cosmic Microwave Background (CMB), stochastic effects are already small when these modes cross the Hubble radius. If one further assumes that monotonously grows toward during the last stages of inflation, decreases (since 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, 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, 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, can decrease and the last -folds of inflation may be dominated by the quantum noise. It is therefore important to study the dispersion arising not only from but from the complete subsequent stochastic history of the coarse grained field.

Note also that in these expressions, need not be small as was shown in Refs. [9, 51, 71] [note, however, that is defined up to a constant due to an arbitrary possible rescaling of ], thus, need not be small, too. As follows from the quasi-isotropic (separate universe) approach, the condition for inflation to proceed is only that . On the other hand, if 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 at which this inhomogeneity occurs crosses the Hubble radius second time. Thus, in the scope of the inflationary scenario 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 Programme

This is why we now generalize this approach to a fully stochastic framework. For a given wavenumber , let be the mean value of the coarse grained field when crosses the Hubble radius. If inflation terminates at , let denote the number of -folds realized between and . Obviously, is a stochastic quantity, and we can define its variance

(3.10)

It is related with the curvature perturbation of Eq. (3.6) in the following manner. Since is computed between two fixed points and , it receives an integrated contribution from all the modes crossing the Hubble radius between these two points, and one has

(3.11)

Here we have used the relation , where stand for slow-roll corrections that we do not need to take into account at leading order in slow roll. One then has

(3.12)

In the same manner, the third moment of the number of -folds distribution,

(3.13)

receives a double integrated contribution from the local bispectrum , and one has . The local parameter, measuring the ratio between the bispectrum and the power spectrum squared, is then given by

(3.14)

where is a conventional historical factor. Analogously, the trispectrum is related to the third derivative of with respect to , and so on and so forth.

The computational programme we must follow is now clear. For a given mode , we first calculate (this sets the location of the observational window). We then consider stochastic realizations of Eq. (1.1) that satisfy at some initial time,777This calculation therefore relies on a specific choice of initial (in fact, pre-inflationary) conditions, since all trajectories emerge from at initial time. In principle, other choices could be made, even if most physical quantities (in particular, perturbations during the last -folds) do not depend on them. and denote by the number of -folds that is realized before reaching . Among these realizations, we calculate the first moments of this stochastic quantity, , , , 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

In what follows, this calculation is performed using the techniques developed in “first passage time analysis” [72, 73], which was applied to stochastic inflation in LABEL:Starobinsky:1986fx. We consider the situation sketched in Fig. 1, where the inflaton is initially located at and evolves in some potential under Eq. (1.1).

Figure 1: Sketch of the dynamics solved in section 3. The inflaton is initially located at and evolves along the potential under the stochastic Langevin equation (1.1), until it reaches one of the two ending values or . The left panel is an example where inflation always terminates by slow-roll violation, while the right panel stands for a situation where one of the ending points, , corresponds to where above which inhomogeneities prevent inflation from occurring.

Because any part of the potential can a priori be explored, here we consider two possible ending points, and , located on each side of . If the potential is, say, of the hilltop type (left panel), and 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 (right panel), one of these points, say , can be taken where becomes super-Planckian and inhomogeneities prevent inflation from occurring. In such cases, the precise value of plays a negligible role, as we will show in section 3.3.1. Let be the number of -folds realized during this process.

Before proceeding with the calculation of the moments, a first useful result to establish is the Itô lemma, which is a relation verified by any smooth function of . The Taylor expansion of such a function at second order is given by . Now, if is a realization of the stochastic process under study, is given by Eq. (1.1) and at first order in , one obtains

(3.15)

Integrating this relation between where and where or , one gets the Itô lemma [74]

which we will repeatedly make use of in the following.

3.3.1 Ending Point Probability

As a first warm-up, let us calculate the probability that the inflaton field first reaches the ending point located at [i.e. ], or, equivalently the probability that the inflaton field first reaches the ending point located at [i.e. ]. This will also allow us to determine when the ending point located at plays a negligible role.

First of all, let be a function of the coarse grained field that can be expressed as

(3.17)

where will be specified later. By construction, one has and . This implies that the mean value of evaluated at is given by . The idea is then to find an function that makes easy the evaluation of the left hand side of the previous relation, so that we can deduce . In order to do so, let us apply the Itô lemma (LABEL:eq:ito) to . If one requires that the integral of the second line of Eq. (LABEL:eq:ito) vanishes, that is,

(3.18)

one obtains

(3.19)

Because and are linearly related, see Eq. (3.17), the same equation is satisfied by . When averaged over all realizations,888The fact that the averaged integral in the right hand side of Eq. (3.19) vanishes is non-trivial since both the integrand and the upper bound are stochastic quantities, but this can be shown rigorously (see e.g. p. 12 of LABEL:Gihman:1972). its right hand side vanishes. One then obtains , which is the probability one is seeking for. All one needs to do is therefore to solve Eq. (3.18) to obtain , to plug the obtained expression into Eq. (3.17) to derive , and finally to evaluate this function at . A formal solution to Eq. (3.18) is given by , where and are two integration constants that play no role, since they cancel out when calculating thanks to Eq. (3.17). Indeed, the latter gives rise to

(3.20)

and a symmetric expression for .999This is in agreement with Eq. (29) of LABEL:Starobinsky:1986fx, derived in the case where is constant, hence , where and lie at correspondingly, and where the initial condition for Eq. (2.3) is chosen to be .

A few remarks are in order about this result. First, one can check that, since lies between and , the probability (3.20) is ensured to be between and . Second, one can also verify that when , , and when , , as one would expect. Third, in the case depicted in the right panel of Fig. 1, in the limit where , one is sure to first reach the ending point located at , that is, . Indeed, the numerator of the expression for is finite, since a bounded function is integrated over a bounded interval. If the potential is maximal at , and if it is monotonous over an interval of the type , 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 . This is why it diverges, and why vanishes. This means that if is sufficiently large, its precise value plays no role, since inflation always terminates at .

3.3.2 Mean Number of -folds

Figure 2: Integration domain of Eq. (3.24) when evaluated at , in the case (the opposite case proceeds the same way). The discrete parameter is integrated between and , while varies between and . The resulting integration domain is displayed in green. When , one has and one integrates a positive contribution to the mean number of -folds. Conversely, when , one has and one integrates a negative contribution. This is necessary in order for the overall integral to vanish. This is why must lie between and .

Let us now turn to the calculation of the mean number of -folds . As above, we want to make use of the Itô lemma (LABEL:eq:ito). To do so, let us define as the solution of the differential equation

(3.21)

with boundary conditions . Such a solution will be explicitly calculated in due time. For now, it is interesting to notice that when this is plugged into the Itô lemma (LABEL:eq:ito), the first term of the left hand side, , vanishes, and the second integrand of the right hand side is . Thus, the Itô equation can be rewritten as

(3.22)

By averaging over realizations, one obtains101010Here 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.

(3.23)

What one needs to do is therefore to solve the deterministic differential equation (3.21) with the associated boundary conditions, and to evaluate the solution at . One obtains

(3.24)

where is an integration constant set to satisfy the condition . There is no generic expression for it,111111Alternatively, one can write Eq. (3.24) in the explicit form [17] where is given by Eq. (3.20) and, in the configuration of Fig. 1, when and otherwise. but one can be more specific. First of all, as can be seen in Fig. 2, must be such that, when is evaluated at , 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 must lie between and . A second generic condition comes from splitting the -integral in Eq. (3.24) into . The first integral vanishes because , which means that in order for to be symmetrical in , must satisfy this symmetry too, that is to say, . Third, in the case where the potential is symmetric about a local maximum 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 . Fourth, finally, in the case displayed in the right panel of Fig. 1, if is sufficiently large, we have established in section 3.3.1 that and the quantity we compute is the mean number of -folds between and . For explicitness, let us assume that (the same line of arguments applies in the case ). Inflation proceeds at . In the domain of negative contribution in Fig. 2, the argument of the exponential in Eq. (3.24) is positive. As a consequence, if is finite and , 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 . In practice, almost all cases boil down to one of the two previous ones and is specified accordingly. Combining Eqs. (3.23) and (3.24), one finally has121212This is again in agreement with Eq. (35) of LABEL:Starobinsky:1986fx if is constant and , while .

(3.25)

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 is computed. One can check that the agreement is excellent.

Figure 3: Mean number of -folds  realized in the large field (left panel) and small field (where , right panel) potentials, as a function of . The location refers to the value of for which the classical number of -folds  and is where . In both panels, the overall mass scale in the potential is set to the value that fits the observed amplitude of the power spectrum , when the latter is calculated with the classical formula (3.8), -folds before the end of inflation. The green line corresponds to the analytical exact result (3.25), and the red circles are provided by a numerical integration of the Langevin equation (1.1) for a large number of realizations over which the mean value of is computed. The orange dashed line corresponds to the classical limit (3.26). The top axes display and the classicality criterion . The yellow shaded area stands for , where inhomogeneities are expected to prevent inflation from occurring and our calculation cannot be trusted anymore.
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 -integral, that is to say, . 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 in most cases (see the discussion above), the integrand is clearly maximal131313Strictly 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. at . Taylor expanding at first order around , , one obtains, after integrating by parts,141414Since and if is monotonous, one can also show that is exponentially vanishing and this term can be neglected. . Plugging back this expression into Eq. (3.25), one finally obtains

(3.26)

which exactly matches the classical result (3.7). The classical trajectory thus appears as a saddle-point limit of the mean stochastic trajectory, analogously to what happens e.g. in the context of path integral calculations.

This calculation also allows us to identify under which conditions the classical limit is recovered. A priori, the Taylor expansion of can be trusted as long as the difference between and is not too large, say , where is some small number. If one uses the Taylor expansion of at first order, this means that . Requiring that the second order term of the Taylor expansion is small at the boundary of this domain yields the condition . For this reason, we define the classicality criterion

(3.27)

This quantity is displayed in the top axes in Fig. 3 and one can check that indeed, the classical trajectory is a good approximation to the mean stochastic one if and only if . In the following, we will see that is the relevant quantity to discuss the strength of the stochastic effects in general and in section 4.3, we will further discuss the physical implications of Eq. (3.27).

For now, and for future use, let us give the first correction to the classical trajectory. This can be obtained going one order higher in the saddle-point approximation, that is to say, using a Taylor expansion of at second order. One obtains

(3.28)

where the dots stand for higher order terms. In the brackets of Eq. (3.28), the two last terms stand for the first stochastic correction and one should not be surprised that, in general, when is small, it is small. It is also interesting to notice that it is directly proportional to . When increases as inflation proceeds, the stochastic leading correction is therefore positive and the stochastic effects tend to increase the realized number of -folds, while when decreases as inflation proceeds, the correction is negative and the stochastic effects tend to decrease the number of -folds, at least at linear order.

3.3.3 Number of -folds Variance

Let us now move on with the calculation of the dispersion in the number of -folds, defined in Eq. (3.10). If one squares Eq. (3.22), and takes the stochastic average of it, one obtains151515This 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.

(3.29)

In order to make use of the Itô lemma, let then be the function defined by

(3.30)

where is the function defined in Eq. (3.24). When the Itô lemma (LABEL:eq:ito) is applied to , if one further sets , one obtains

(3.31)

where the second equality is just a consequence of Eq. (3.29) and where the third equality is just a consequence of Eq. (3.23). Therefore, one just needs to solve Eq. (3.30) with boundary conditions and to evaluate the resulting function at in order to obtain . The differential equation (3.30) can formally be integrated, and one obtains

(3.32)

where is an integration constant that must be chosen in order to have . One can show that it satisfies the four properties listed in section 3.3.2 for and can therefore be specified in the same manner. With , one then has

(3.33)
Classical Limit


As was done for the mean number of -folds in section 3.3.2, let us derive the classical limit of Eq. (3.33). Obviously, in the classical setup the trajectories are not stochastic and , and what we are interested in here is the non-vanishing leading order contribution to in the limit . As before, the -integral can be worked out with a saddle-point approximation, and one obtains161616In the limit, is close to the classical trajectory (3.26) as shown in section 3.3.2, and one can take . . Plugging back this expression into Eq. (3.33), one obtains

(3.34)

Finally, and for future use again, let us give the first correction to this classical limit. Going one order higher in the saddle-point approximation, one obtains

(3.35)

3.3.4 Number of -folds Skewness and Higher Moments

In the same manner, if one denotes the third moment of the distribution of number of -folds by defined in Eq. (3.13), one can show that is the solution of the differential equation that obeys . As before, taking and , one obtains

(3.36)

where can be set as . Similarly to above, making use of Eqs. (3.28) and (3.35), a saddle-point approximation of this integral leads to the classical limit

(3.37)

Let us finally explain how the same procedure can be iterated and higher order moments can be calculated. Let us denote the momentum of the number of -folds distribution by

(3.38)

where, by convention, we set and . As above, one can recursively show that is the solution of the differential equation

(3.39)

satisfying . On then has

(3.40)

When , this yields the variance (3.33); when , the skewness (3.36) is obtained; when , the kurtosis could be derived as well, and so on and so forth for any moment.

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 section 4.3.

4.1 Power Spectrum

Figure 4: Scalar power spectrum for the large field (left panel) and small field (where , right panel) potentials, as a function of . The conventions are the same as in Fig. 3. The green line corresponds to the analytical exact result (4.1), and the orange dashed line to the classical limit (4.3).

Following the programme we settled in section 3.2, if one plugs Eqs. (3.25) and (3.33) into Eq. (3.12), one obtains , that is,

(4.1)

In this expression, stands for the power spectrum calculated at a scale such that when it crosses the Hubble radius, the mean inflaton field value is . 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.

From this, a generic expression for the spectral index can also be given. Since, at leading order in slow roll, , one has