Backreaction and Stochastic Effects in Single Field Inflation

Backreaction and Stochastic Effects in Single Field Inflation

Laurence Perreault Levasseur lp375@cam.ac.uk DAMTP, University of Cambridge, Cambridge, CB3 0WA, United Kingdom Kavli Institute for Particle Astrophysics and Cosmology (KIPAC), Menlo Park, CA 94025, USA    Evan McDonough evanmc@physics.mcgill.ca Department of Physics, McGill University, Montréal, QC H3A 2T8, Canada
Abstract

The formalism of stochastic inflation is a powerful tool for analyzing the backreaction of cosmological perturbations, and making precise predictions for inflationary observables. We demonstrate this with the simple model of inflation, wherein we obtain an effective field theory for IR modes of the inflaton, which remains coupled to UV modes through a classical noise. We compute slow-roll corrections to the evolution of UV modes (i.e. quantum fluctuations), and track this effect from the UV theory to the IR theory, where it manifests as a correction to the classical noise. We compute the stochastic correction to the spectral index of primordial perturbations, finding a small effect, and discuss models in which this effect can become large. We extend our analysis to tensor modes, and demonstrate that the stochastic approach allows us to recover the standard tensor tilt , plus corrections.

pacs:
98.80.Cq, 98.80.Qc, 98.70.Vc, 05.10.Gg

I Introduction

Inflation has been tremendously successful in explaining the physics of the very early Universe. It was the first compelling cosmological model to provide a causal mechanism for generating fluctuations on cosmological scales, and it predicted that their spectrum should be almost scale invariant, with small deviations from scale invariance that can be traced back to the precise microphysics of inflation Mukhanov and Chibisov (1981, 1982); Bardeen et al. (1983). These predictions provide a way of connecting theoretical physics to observational cosmology; this has been a very fruitful venture, as has lead to particle-physics based models of inflation Lyth and Riotto (1999), inflation in supergravity Kallosh et al. (2014), and string inflation Baumann and McAllister (2014), to name a few. There is still much to be learned from the CMB, and if the large tensor-to-scalar ratio of Ade et al. (2014) is a hint of good things to come, then the CMB may yet give us an unprecedented opportunity to test models of inflation and quantum gravity.

With the ever-increasing precision of experiments probing the CMB, for example et al. (Planck Collaboration) (2013), it becomes imperative to develop self-consistent methods of calculation for inflationary predictions. The formalism of stochastic inflation is a promising avenue in this direction. It allows for the constant renormalization of background dynamics and in this way circumvents one of the main difficulties of traditional methods: backreaction Mukhanov et al. (1997); Brandenberger (2002); Geshnizjani and Brandenberger (2005); Martineau and Brandenberger (2005); Kolb et al. (2010); Levasseur et al. (2010). This is achieved by separating the dynamics of long, classical wavelengths from short, quantum fluctuation-dominated wavelengths, and studying the interplay of the two sectors. The stochastic formalism then allows for the resummation of corrections to the background dynamics as modes of fluctuations are stretched from the quantum regime into the coarse-grained effective theory.

The resulting theory describes the effective classical dynamics of a large-scale gravitational system, in the presence of a ‘bath’ where all the quantum fluctuations are collected in a classical noise term, through a set of Langevin equations. As required by the fluctuation-dissipation theorem, this noise term comes hand in hand with a dissipation term, which in turn allows for irreversibility and approach to equilibrium. The effective theory therefore belongs to a new class of non-Hamiltonian theories Burgess et al. (2014), which have not been studied much so far in the context of cosmology. (However, see Berera et al. (2009) and references therein in the context of warm inflation.)

Stochastic inflation has a long history. Originally proposed by Starobinsky Starobinsky (1984, 1986), stochastic inflation as studied in the early work of Vilenkin (1983a, b); Bardeen and Bublik (1987); Rey (1987); Goncharov et al. (1987); Sasaki et al. (1988); Nakao et al. (1988); Nambu and Sasaki (1989); Nambu (1989); Salopek and Bond (1990, 1991); Habib (1992); Linde et al. (1994); Starobinsky and Yokoyama (1994) was a simple way to include quantum effects into inflation. The idea was this: quantum fluctuations are generated deep inside the horizon and, at zeroth order in slow-roll, evolve as quantum fields on a fixed de Sitter background. The quantum modes grow and exit the horizon. Doing so, due to their random phase, they provide a kick of a random amplitude to the long-wavelength physics. It follows that the quantum modes act as a source for the classical background, and the physics of this source is probabilistic in nature. More precisely, stochastic inflation provided an ‘educated guess’ that this source should be white noise. The physics of slow-roll inflation can then be studied as per the usual treatment, with the noise included as a source in the equation of motion for the classical (long-wavelength) field.

Stochastic inflation was put on a more solid footing by Morikawa (1990) and Hosoya et al. (1989), where the equations of motion for stochastic inflation were derived from a path integral Calzetta and Hu (1994, 1995); Calzetta et al. (1997); Matacz (1997); Calzetta and Hu (1999); Matarrese et al. (2004); Franco and Calzetta (2010). Given these equations of motion, the vast majority of modern applications of stochastic inflation take the same approach as Starobinsky: calculate the variance of quantum modes in a pure de Sitter background, include this as white noise in the Klein-Gordon equation for long-wavelength modes, and study slow-roll inflation in the presence of this white noise (see, e.g., Afshordi and Brandenberger (2001); Geshnizjani and Afshordi (2005a); Tsamis and Woodard (2005a); Martin and Musso (2006); Kunze (2006); Finelli et al. (2009, 2010); Clesse (2011); Martin and Vennin (2012); Weenink and Prokopec (2011)). However, this method misses a key element of the physics: as pointed out in Boyanovsky et al. (1995) the short-wavelength and long-wavelength physics are coupled. Namely, the quantum modes do not evolve on a pure de Sitter background, but rather on a background that is both slow-roll and stochastically corrected. In terms of the path integral, the coupling of the two sectors (long-wavelength, or ‘coarse-grained’ fields, and short-wavelength, or ‘bath’ fields) manifests itself as loop diagrams calculated in the Schwinger-Keldysh ‘in-in’ formalism of quantum field theory, which has become widely applied in cosmology since Weinberg (2005); Maldacena (2003), after the early work of Calzetta and Hu (1987); Jordan (1986) (however, see Altland and Simons (2010); Calzetta and Hu (2008) and references within for an introduction in the context of out-of-equilibrium QFT and open systems). This approach was developed in Perreault Levasseur (2013), where it was dubbed the ‘recursive formalism of stochastic inflation’.

Cosmological perturbations have also been studied in the context of stochastic inflation, see for example Kunze (2006) and Fujita et al. (2013, 2014). We will use a method inspired from the approach used in Kunze (2006), with some modifications that will be discussed in section VI. An alternative, and relatively recent, proposal Fujita et al. (2013, 2014) is to apply the formalism to stochastic inflation. This makes intuitive sense: the formalism can be qualitatively understood as a ‘separate universe approach’, and one would not expect a local noise to invalidate this approach. This approached will also be touched upon in section VI.

In Perreault Levasseur et al. (2013), the recursive formalism was applied to hybrid inflation Linde (1994a). In this scenario, the spectral index is strongly dependent on the duration of the ‘waterfall phase’ of inflation Clesse and Rocher (2009), where the field dominating the energy density of the Universe during inflation becomes tachyonic and ‘waterfalls’ down the side of the potential. This generates a red tilt, provided that the waterfall phase lasts for a suitable number of -folds. It was found in Perreault Levasseur et al. (2013) that the recursive corrections caused the tilt of the inflaton perturbations to become bluer in the valley, while also causing the waterfall phase to end earlier than otherwise expected, making a red tilt much more difficult, if not impossible, to achieve.

In the present paper we have more modest goals, that is, to study recursive stochastic effects in single field inflation, both analytically and numerically, in particular the simple model Linde (1983); Linde et al. (1994); Linde (1994b), away from the regime of eternal chaotic inflation Linde (1986a, b); Guth (2007).111 See also Li and Wang (2007); Kohli and Haslam (2014); Feng et al. (2010); Qiu and Saridakis (2012) and references therein for existing studies of stochastic eternal inflation. We find that the recursive approach gives corrections to quantum modes that could not be deduced from naively including slow-roll effects alone. We then study the effect of this on long-wavelength perturbations, and calculate the power spectrum of primordial perturbations. We extend this approach to include tensor perturbations, and discuss the effect of couplings to heavy fields.

The outline is as follows: in section II we outline the usual approach to stochastic inflation and review the recursive formalism. In section III we calculate the classical noise induced by quantum fluctuations on a classical background which is zeroth order in slow-roll. In section IV we study the effect of this noise by computing the stochastic (and slow-roll) corrected classical background, and continue in section V to compute the backreaction on the quantum modes. We then use this in section VI to compute the backreaction on IR modes and the spectrum of curvature perturbations. We extend this to a class of simple multifield models in section VII, and to tensor modes in section VIII. We conclude and discuss our results in section IX.

Ii Stochastic Inflation: Basic Setup and Recursive Strategy

Let us first consider the action of a single scalar field in a fixed background. The matter part of the action is given by:

(1)

which leads to the equation of motion

(2)
(3)

In the present paper, we will more specifically be interested in the chaotic potential

(4)

Moreover, to ensure that we remain away from the eternal inflation regime throughout our analysis, we impose the condition throughout this paper Guth (2007), where is the initial value of the inflaton at the beginning of inflation.

The starting point of stochastic inflation is to split the field into long-wavelength modes (c for classical), and short-wavelength modes (q for quantum). Note that both and are quantum fields in nature; technically corresponds to a quantum averaged field, coarse grained on a radius of constant physical size. We choose this coarse-graining scale to be the scale at which quantum fields undergo squeezing, i.e. the Hubble scale., at which point the commutators of the fields and their derivatives scale as and are therefore exponentially suppressed (see Calzetta and Hu (1987); Habib (1992); Albrecht et al. (1994); Calzetta and Hu (1995); Hu and Sinha (1995); Polarski and Starobinsky (1996); Calzetta et al. (1997); Matacz (1997); Kiefer et al. (1998, 2007a, 2007b); Franco and Calzetta (2010); Weenink and Prokopec (2011); Burgess et al. (2014) and references therein concerning the topics of quantum average versus classical fields, decoherence, and the conditions required for classicalization).

The splitting into and is defined by

(5)
(6)

where the is the mode expansion of the quantum fields in terms of creation and annihilation operators, and is a time-dependent window function. The window function acts to project onto only the modes with a comoving wavenumber somewhat larger than the physical Hubble scale. To be precise, we take the filtering scale to be large enough to make sure that all modes that are part of have undergone squeezing and classicalization. To see this splitting at the level of the equation of motion, we can Taylor expand the equation of motion222In a dynamical spacetime depends on the full quantum field , and hence the extension of this heuristic argument to realistic inflationary setups is slightly more involved, although conceptually the derivation is identical. about ,

(7)

where our perturbation variable has been chosen to be the number of quantum fields (which can be seen from the path integral formulation to be equivalent to counting powers of in a Schwinger-Keldysh loop expansion).

Given that the coarse-graining radius is chosen to correspond to the classicalization radius, the quantum-averaged field corresponds to an effective classical field which we call , endowed with a probability density function (PDF). This effective classical field allows us to treat collectively all realizations of the universe with consistent histories. Its PDF gives different probabilistic weights to classical realizations coming from different sets of random phases of the mode functions as they successively cross the Hubble radius and freeze. The PDF allows for a notion of ensemble average, which is equal to the quantum expectation value provided the ergodic hypothesis is satisfied. This point is further clarified in Tsamis and Woodard (2005b) and discussed in more details in Perreault Levasseur (2013).

It follows that alongside the system , , {coarse-grained quantum field , small scale quantum fluctuations}, we can write a corresponding classical, probabilistic system consisting of and a set of classical Gaussian noises modeling the effects of the incoming modes of joining the coarse-grained theory. Using the definition (6) to rewrite the in square brackets in terms of their linear mode expansion, as well as the fact that the linearized mode functions satisfy their linearized equation of motion, equation (7) can be rewritten as

(8)

where the only surviving terms in the square brackets (i.e. the ones containing at least one time derivative acting on the time-dependent window function ) have been defined as the classical noise. A simple calculation, again using only equation (6), reveals that the noise terms are drawn from a random Gaussian probability distribution given by

(9)

and we have defined, letting , the matrix to have components given by

(10)

with

(11)

where in the above equation is larger than the coarse-graining scale. The terms written in parenthesis on the right-hand side (r.h.s.) of (8), which are the only ones still containing quantum fields, can also be rewritten in terms of classical noise terms,333In fact, equation (8) is inconsistent unless this is done, since as written they are a quantum contribution to a classical equation of motion. as was done in Perreault Levasseur (2013). However, for the case of a quadratic potential, which we shall consider here, these higher-order terms vanish and hence will not contribute to equation (8). To emphasize the split between quantum and classical: the modes are quantum, while the noises are classical, as the noise terms appearing in equation (8) are evaluated at the moment the modes exit the horizon and give a ‘kick’ to the IR (classical) theory.

The variance of and can be read directly from this definition, by equating ensemble averages and quantum expectation values under the ergodicity assumption. To solve for the stochastic background, it is necessary to solve simultaneously for the linear mode function of the bath field, which satisfies the equation:

(12)

where the wavenumber is larger than the coarse-graining scale, i.e. for wavelengths smaller than the coarse-graining radius. In what follows, we will be interested in solving the classical system perturbatively and will not look any further at the quantum averaged field . We therefore drop the tilde for the sake of simplicity, and from now on by we mean the classical, stochastic coarse-grained field.

This is not an easy system to solve: the coarse-grained field , which obeys (8), depends on the amplitude and statistics of the noise terms , which are given in terms of the mode functions of the quantum field, . These mode functions in turn depend on a specific realization of the background in which they evolve, through their self-energy, which acts as a -dependent mass term.

A word on the precise structure of the perturbative expansion: we will solve the system perturbatively in the number of quantum fields, , and in the slow-roll parameter . For example, the solution for at order corresponds to a constant background field with no stochastic corrections. A calculation of the quantum mode function can be done in this background, which is now at order . An equivalent counting is in powers of , which counts loops in the Schwinger-Keldysh formalism, i.e. corresponds to . This was shown in Perreault Levasseur (2013). Given a quantum mode valid at order , the variance of the noise can consistently be computed at order , and hence the PDF of IR modes can also be computed at order . In particular, the variance of IR modes (which encodes the spectral tilt), is valid at order . We will use this notation extensively in this paper.

Iii First Step of the Recursion: Stochastic Noise

To make progress with the system (8) and (12) while maintaining the consistency of the solution, and in order to capture the fact that the quantum modes sit in a stochastic background, we use the recursive method from Perreault Levasseur (2013). At step 0 of the method, we start by approximating the background to zeroth order in slow-roll (i.e. pure de Sitter space). Step one of the method is to find the amplitude of the noise in this zeroth-order background.

More precisely, we want to calculate the amplitude of the classical noise arising from quantum fluctuations evolving in such a background and joining the coarse-grained theory. That is, we want to calculate to order by solving equation (12) and then use equation (10) to find the statistical properties of the noise terms. Fortunately, it is possible to show (after simple algebra) that is always suppressed by some power of the slow-roll parameters, since it is proportional to at least one time derivative of the quantum field mode function , and hence, at this order in perturbation theory, it is sufficient to calculate alone.

We start by making an explicit choice of the window function

(13)

where parametrizes how long after their Hubble crossing modes can be considered classicalized, and so acts as an ‘ignorance parameter’ (however, see Casini et al. (1999); Winitzki and Vilenkin (2000); Liguori et al. (2004) for discussions of subtleties concerning this choice and explorations of different possibilities).

Using this, and changing time variables to the number of -folds (for reasons that will be discussed in sections IV and VI), the variance can be computed to the required order in slow-roll at this stage of the recursive method

(14)

where we have used the PDF (9), leaving the mode function unspecified in (11), to explicitly calculate the variance of the noise , keeping only terms of order (where we define the first slow-roll by ). Here the modes need to be evaluated at the time when they join the coarse-grained scales, which can be evaluated using the usual expression for mode functions in the de Sitter background,

(15)

and it follows that the mean and variance of the noise are given by

(16)

The variance is constant and proportional to , and hence acts as white Gaussian noise444This will not remain true at higher orders in the recursive method: the noise will become colored due to interactions of the bath and the system, as was discussed in Starobinsky and Yokoyama (1994). with zero mean. The noise variance is local in time, and although it might appear to be nonlocal in space, the factor in fact acts as a theta function at the coarse-graining radius, being one within the coarse-graining length, and zero outside. This ensures that the noise is only () correlated within each coarse-graining region, but is not correlated between different regions. Equivalently, this can be stated by saying the nonlocalities are only within the coarse-graining scale, and so the coarse-grained theory remains local.

Iv Step Two: Stochastically-Corrected Coarse-Grained Theory

iv.1 Analytic solution

In the previous subsection we assumed a classical nondynamical background and used it to calculate the noise to order . Using this, we can calculate the corrected classical background at order . To do this, we solve equation (8), which is a Langevin equation for at this order 555Recall, as mentioned above equation (13), that the noise terms and are higher order in slow-roll.:

(17)

In the above, all quantities are valid to zeroth order in slow-roll. In particular, as should be explicit from the previous section, the variance of the noise term is valid to leading order in but to zeroth order in slow-roll. Consistent perturbation theory then requires that the Hubble parameter appearing in (17) also be evaluated at zeroth order in slow-roll, and hence is simply constant.

Solving (17) gives a solution valid at . At this stage in the recursive method, we are considering quasi-de Sitter space rather than a nondynamical de Sitter spacetime. Gauge fixing therefore becomes necessary and we choose to work with gauge-invariant variables. In the stochastic formalism, this can be achieved by using the number of -folds elapsed since the beginning of inflation, , as the time variable. Recalling that we are working with the potential

(18)

we get

(19)

The advantage of using as the time variable is that linear order perturbations of the resulting stochastic process then coincide with the Mukhanov gauge-independent variable, as shown in Finelli et al. (2009). This is because in terms of the number of -folds, Taylor expanding to linear order the full fields equations of motion yields the gauge-fixed perturbation equations. This will be discussed further in section VI.

The solution to equation (19) can easily be written in terms of an integral equation:

(20)

where we have made the rescaling , and is therefore a regular Brownian motion with unit variance. From this, along with equation (16) for the statistics of the noise, we see that the incoming quantum modes leave the mean (i.e. the first-order moment) of the effective classical background unaffected and only modify higher moments of the background PDF.

Other derived stochastic quantities can also be calculated from the coarse-grained field at this order. For example, the slow-roll parameter will now have a stochastic piece (the same is true of the time-dependent Hubble parameter ), which can be expressed as:

(21)

This can be rewritten as,

(22)

where we have separated the first slow-roll parameter into a classical piece and a stochastic piece .

It is important to note that (20) is not the most general expression to characterize solutions to (19), since each individual solution is only one realization the stochastic process , as is made explicit by the presence of the Wiener process in the expression. Alternatively, we can solve for the probability density function , which gives the probability of a field configuration over the whole length of inflation, using a Fokker-Planck equation. Another, perhaps simpler, option is to solve (19) numerically, by solving many different realizations and from there inferring the shape of the underlying PDF using Bayes’ theorem. This can be done by maximizing the likelihood on the - space (provided we assume the PDF is Gaussian) or, in the absence of Gaussianity, by finding the 68% confidence levels. In the following, this is the strategy we will adopt.

iv.2 Numerical solution to the coarse-grained theory

In terms of the rescaled noise , the stochastic differential equation (SDE) to solve is:

(23)
(24)

In order to discretize the SDE, we need to discretize the time delta function in (24):

(25)

Here, is the integration time step used in the numerical solver. This simple SDE can be solved using the Euler method to integrate:

(26)

where the are independent random numbers drawn from a random normal distribution with standard deviation .

After simulating a large number of realizations of this coarse-grained background, the underlying PDF of the random variable can be reconstructed. This can be done by assuming an underlying Gaussian PDF and sampling the likelihood of the - space parametrizing the possible Gaussians to find the maximum likelihood. This means, at each time step, and for every plausible value of and , we apply Bayes theorem to find the probability that values from all realizations of are drawn from the Gaussian defined by a given choice of and .

Figure 1: Left panel: Top: mean trajectory of the coarse grained inflaton . Note that the large discrepancy of compared to the standard slow-roll value is simply due to the fact that at this level in the recursion, we solved the EoM for keeping , a constant. This means that by the end of inflation, we can expect corrections to our coarse-grained solution as large as . Middle: 1 deviation from the mean trajectory, i.e. standard deviation of maximum likelihood. Bottom: percentage error at 1. Right panel: The reconstructed (un-normalized) PDF of the random variable for a few fixed -folds during inflation, and assuming inflation lasted 60 -folds. These values were inferred from 250 realizations, with , , and . All fields shown are in units of .

The resulting PDF for a few fixed -folds over the course of inflation are shown for the potential in Figure 1. One can see that the variance of the long-wavelength field is initially zero, as one would expect from the fact that the PDF was initialized as a delta function in probability space. Its variance then grows as more modes exit the horizon and join the coarse-grained theory, which can be seen in the middle panel on the left-hand side (l.h.s.), as well as the plot on the r.h.s. The fractional variance of the field also grows during this time period, as shown in the bottom panel of the l.h.s.

The absolute variance does not grow indefinitely. As can be seen from both the l.h.s. and r.h.s. panels, the variance saturates during the last 10 -folds of inflation, approaching a maximal value that can easily be estimated from (19):

(27)

Since the field is massive, this is as one should expect: quantum fluctuations becoming classical push the field fluctuations to roll up their potential, but the shape of the potential tends to make the field roll back down to its minimum. These two competing effects eventually reach an equilibrium point, which can be calculated in standard perturbative analysis to coincide with (27).

As a final comment, note that, (27) being a constant, the power spectrum of curvature perturbations that we obtain at this stage in the recursive method is exactly flat. That is to say, the maximal equilibrium value that the field fluctuations reach is constant with time, which is consistent with a constant push from the incoming quantum modes. If the spectrum were tilted, this would correspond to kicks with time-dependent amplitude, and this would in turn modify the quasiequilibrium position for , making it time dependent. This is what we will observe in the next level of recursion.

We can apply the same method to infer the underlying PDF of the slow-roll parameter , displayed in Figure 2, which exhibits a qualitative behavior similar to . These graphs depict an interesting perspective: the super-Hubble classical theory we obtain is a ‘fuzzy’ one, in the sense that the classical parameters have an inherent uncertainty stemming from the constant incoming quantum modes. Therefore, on the large scales of the coarse-grained theory (by which we mean on scales of many Hubble volumes), the value of varies from point to point with a standard deviation shown in Figure 2.

Furthermore, even at a single point the value of the classical parameters, such as , are constantly fluctuating. In particular, this means that, when one averages over macroscopic timescales, there is a minimum possible value for the slow-roll parameter . Indeed, even in the limit where (or equivalently ) starts out very large, in such a way that as defined in (22) tends to 0, the contribution from will always remain finite. This is true in general for any model of inflation: the root-mean-square of always provides a minimal value of the first slow-roll parameter, regardless of how small it is engineered to be classically. It is worth stressing how the picture that we obtain differs from the standard one: the super-Hubble theory is now fundamentally probabilistic, and each realization of the quantum modes in the bath sees one of this ensemble of fluctuating field trajectories as a background.

Figure 2: The reconstructed (un-normalized) probability density function of the slow-roll parameter (the subscript 1 denotes first slow-roll parameter) for a few fixed -folds during the last 60 -folds of inflation. These values were inferred from 250 realizations, with , .

V Step Three: Quantum Fields Evolving on a Stochastic Background

v.1 Analytic solution

Now that a solution to (8) valid to has been found, we can go back to the bath (i.e. short-wavelength) sector of the theory and solve the linearized mode function of the quantum field to . This will allow us to find the noise variance to leading order in slow-roll.

To do this, we treat the twice-corrected quantum modes, which we denote , as perturbations about a fixed background field . It is important to note that this procedure requires a careful treatment of metric perturbations, the necessity of which was realized in Geshnizjani and Afshordi (2005b), where the authors considered the backreaction of cosmological perturbations (i.e. the effect of second-order perturbations on the background) in the presence of stochastic effects. However they did not consider the backreaction of the shifted background on the noise itself, which is precisely what we are interested in here.

Before we progress further, we must first fix a gauge. The most natural choice is the same gauge as classical perturbations of , that is, the spatially flat gauge. Following the treatment of Gordon et al. (2001), we fix the gauge to the spatially flat gauge, and find the equation of motion for the field perturbations:

(28)

It is important to note that is a corrected version of , as opposed to an additional contribution to the mode function . Hence, we impose the Bunch-Davies initial condition on .

Solving the gauge constraints, transforming to Fourier space, and using the canonically normalized variable , we find the equation of motion:

(29)

where prime denotes a derivative with respect to , the conformal time defined by . The solution to this, at first order in slow-roll, and after matching to Bunch-Davies initial conditions, is given by:

(30)
(31)

where we have ignored the time dependence of coming from since it is suppressed relative to .

Now that we have an expression for the mode function evolution, we can obtain the expression for the noise variance, valid to order . After some computation, see Appendix A, and doing a combined expansion in and , the variance simplifies to (at first order in both and )

(32)

where is defined as

(33)

Here, denotes the Euler-Mascheroni constant. Note that the Hubble parameter appearing in the above two equations stands for the full , i.e. stochastically and slow-roll corrected. Stochastic corrections to the variance of the noise are therefore included in , as per equation (21), as well as in .

Also, note that in order to impose that the variance of the noise is independent of the choice of coarse-graining radius, that is, independent of , we must impose the hierarchy Starobinsky and Yokoyama (1994). This is consistent with the expressions found in Perreault Levasseur et al. (2013), and ensures that the effective theory we obtain through the coarse-graining process is a sensible and physical one.

v.2 Numerical solution to the mode function equation

To proceed with the numerical solution, we first recast equation (28) in a more useful form. This equation can easily be rewritten in terms of the number of -folds :

(34)

where a dot refers to a derivative with respect to cosmic time . Retaining only terms up to leading order in , the classical piece of , and up to , we obtain the equation

(35)

To solve for the PDF of the stochastic linearized quantum mode function corresponding to (34) is very difficult, since (34) is now proportional to the square of the noise666In this case, an analytical solution for the PDF of the mode functions through a Fokker-Planck equation is not possible anymore.. We therefore proceed numerically, using a modified version of the Runge-Kutta method for solving SDEs (which reduces to the improved Euler method in the absence of a stochastic term), as explained in Appendix B.

Figure 3: Left panel: Top: mean trajectory of the real part of the linearized mode function with , freezing at . Middle: 1- deviation form the mean trajectory, i.e. standard deviation of maximum likelihood. Bottom: percentage error at 1. Right: The reconstructed (un-normalized) PDF of the random variable for a few fixed -folds during the last 60 -folds of inflation. These values were inferred from 250 realizations, with , . Fields are shown in units of .
Figure 4: Power spectrum of the noise correlator at step 3 of the recursive method, . Top panel: Average power spectrum, , in units of . The full (blue) line is the corrected power spectrum obtained numerically at step 3, while the red line is the analytic computation. The grey dashed line is the variance obtained at the previous step of the recursion, , and the yellow dashed line is , i.e. the naive correction to the noise to account for the fact that inflation happens is quasi-de Sitter space by simply making the Hubble radius a time-dependent quantity. Middle panel: 1 deviation from the mean trajectory, i.e. standard deviation of maximum likelihood, again in units of . Bottom panel: Percentage error at 1-. These graphs were obtained by simulating 100 realizations, and, assuming a Gaussian underlying PDF, the mean and variances were inferred by maximizing the likelihood of the space. Simulations were performed using , .

In order to solve for each realization of the mode function in a given realization of the background (solved for in step two of the recursive method, see section IV.1), for every realization, the background and the mode function equation must be solved simultaneously (such that each realization of the mode function ‘sees’ the background generated by the right Wiener process ). The result for a fixed mode is displayed in Figure 3.

Figure 3 highlights the generic behavior of a mode of the gauge-invariant Mukhanov variable inside and outside the Hubble horizon. Before horizon crossing, the mode has a constant norm. Therefore, plotted here has an amplitude that decays as . The reason why it appears to be oscillating widely in the top panel of Figure 3 is that only the real part to the mode function is shown, and the real and imaginary parts oscillate at identical speed with a phase shift. Around , the mode plotted crosses the Hubble radius and freezes, and its amplitude remains constant from then on (real and imaginary parts independently).

The absolute variance of the mode also decays as while the mode is inside the horizon, as shown in the middle panel of the l.h.s., as well as the plot on the r.h.s. The fractional variance, displayed in the bottom panel of the l.h.s., diverges each time the real part of the mode crosses zero (which should not be interpreted as a physical effect), However, after the mode has exited the horizon, both the field and the variance approach a constant (this can be seen in the r.h.s. and the middle panel of the l.h.s.), and the fractional variance converges to roughly 0.25%. Note that, for our purpose in the present paper, we only apply the description of as a UV mode up until this mode joins the coarse-grained theory via the noise , which occurs a few -folds after horizon exit to ensure classicalization, around in Figure 3.

Repeating this procedure for every mode exiting the coarse-graining radius during the last 60 -folds of inflation in a given realization of the background, we obtain the corrected power spectrum of the stochastic noise. Figure 4 shows the resulting average noise correlator (thick blue, top panel), when averaging over 100 realizations, and the 1- error on this correlator on the middle (absolute) and bottom (fractional) panels. The red dot-dashed line in the top panel, representing the analytical calculation from equation (33), shows very good agreement between our numerical and analytical treatments. The top panel of the figure also shows, for comparison, the result at zeroth order in slow-roll which was obtained in section III and used in section IV, as well as the naive slow-roll correction obtained by taking in the zeroth-order result, as one would obtain by following the procedure of, e.g., Finelli et al. (2009) (yellow line).

Vi Step Four: Corrected Coarse-Grained Theory

vi.1 Overview and numerical approach

We have thus far completed two levels of recursion: 1) for our first ‘guess’, we began with a nondynamical de Sitter background, then calculated the amplitude of the noise generated by quantum modes evolving on such a background in section III; 2) using this noise (valid to leading order in and zeroth order in slow-roll) as a source, we went back to the large scales and solved for the statistics of the coarse-grained classical inflaton, , in section IV. Using this as a background (valid to first order in slow-roll and ) for the short-scale physics, we then evaluated the corrected quantum modes in section V. This then allowed us to find the variance of the noise arising from this bath, , valid to .

Next, we shall use this noise to, once more, come back to the large-scale physics and source the coarse-grained theory. This will allow us to obtain a coarse-grained field valid to . That is, we must now solve:

(36)

where now and is a random Gaussian variable sampled from a distribution with mean 0 and variance given by equation (32).

Recall that, although it should be thought of as a background when discussing the short-scale dynamics of the quantum mode functions inside the bath, the resulting is not homogeneous, i.e. the stochastic contribution to is inherently inhomogeneous. Rather, the PDF for contains all the information about the classicalized field, including perturbations. This is an elegant way to encode a large amount of information; however, we are left with the problem of calculating the standard phenomenological parameters of inflationary cosmology, such as the spectral tilt.

Numerically, however, solving this equation is quite easy. Using a method analogous to what was done in section IV, we solve for each realization of the coarse-grained theory using a realization of the noise output that was used toward the construction of Figure 4. After constructing 100 realizations of , we then use Bayes theorem to infer the two first moments of the underlying PDF (assuming Gaussianity), i.e. its mean and variance as functions of time.

As one should expect with a choice of parameters excluding eternal inflation, this additional step in the recursive method does not give significant corrections to the mean trajectory of the inflaton. Its variance, however, is the quantity capturing the integrated power of the classicalized field fluctuations, and is of great interest to us. This is the quantity presented in Figure 5, where the numerical result is the black solid line.

Figure 5: Total integrated power in, or variance of, the classical long-wavelength fluctuations, , in units of . The full (black) line is the numerical result, while the dashed light- and dark-blue dashed lines correspond to different analytic approximations to the integral in equation (43), see footnote 8 for details. The yellow dashed line corresponds to a noise variance of , i.e. the naive approximation obtained by slow-roll corrected to account for the fact that the bath evolves in a slow-rolling background Finelli et al. (2009). This graph was obtained by simulating 100 realizations, and, assuming a Gaussian underlying PDF, the mean and variances were inferred by maximizing the likelihood of the space. Simulations were performed using , .

vi.2 Inflaton fluctuations beyond leading order

vi.2.1 Perturbation equations for the random variable

As alluded to in section IV, the analysis of fluctuations in stochastic inflation can be done quite simply by using the number of -folds as the time variable. This fact can also be seen by considering the formalism. As an example, let us consider single field inflation, following Sugiyama et al. (2013). The background equation of motion is given by (where is the energy density):

(37)
(38)

The power of the formalism comes from realizing that, in the absence of entropy perturbations, the above equation applies nonperturbatively. This leads to a statement of the ‘separate universe approach’ to perturbations,

(39)

This equation states that the nonperturbative dynamics in one region of spacetime are captured by solving for the background , but given a set of perturbed initial conditions . In fact, the full formalism is much more powerful than this, as it is easily generalizable to a gradient expansion.

This same formalism can (and has, see Fujita et al. (2013, 2014)) be applied to stochastic inflation, since the noise evolves independently in different Hubble patches, and hence does not spoil the ‘separate universe approach’.777To be more precise: fluctuation and dissipation terms that arise at 3rd order are non-local, meaning this approach would need to be modified. However, these non-localities are at most at the coarse-graining radius, and therefore do not spill into neighboring Hubble patches. Furthermore, here we are considering a free scalar field, and are neglecting the coupling between tensor and scalar perturbations. In fact, this locality was shown to be a necessary condition for stochastic inflation in Tolley and Wyman (2008). The applicability of the formalism could in principle be shown more rigorously by expanding the action for the stochastic inflation, which was derived in Perreault Levasseur (2013).

This approach can be applied here as follows. The equation of motion for the classical coarse-grained field is given by equation (36). We can split the field into a homogeneous mode, which is just the expectation value , and an inhomogeneous piece containing all the classical fluctuations.888We will not write a subscript on the inhomogeneous piece of for the sake of simplifying the notation. We therefore expand (36) around using:

(40)

to find the equation for the first-order fluctuations Finelli et al. (2009):

(41)

where, as in equation (36), is defined as including both the slow-roll and stochastic corrections. The above equation can be solved to give the PDF for , from which we would like to extract information about the power spectrum.

Alternatively, the variance of can be calculated from (41) by multiplying both sides by and averaging, without having to solve for the full PDF of the classical field,

(42)

where, apart for the occurrences of in , all other powers of are evaluated at . To arrive at this equation, we have used the relation , which can be deduced by expanding and in terms of their Fourier modes, and enforcing continuity of the (amplitude of the) full field across the horizon . We emphasize that the cumbersome notation is necessary: the slow-roll correction to the variance of the noise was calculated with respect to the background , where as the occurrences of in equation (36) are defined with respect to .

This equation is easily solved in terms of the homogeneous solution for ,

(43)

In the last step, we have assumed that the time evolution of and are the same. To be precise, they differ by terms that are higher in slow-roll than the precision to which is defined.

Substituting the form of in the above equation yields the following solution for the total power in the fluctuations of the coarse-grained inflaton as a function of time:999If one had instead solved the integral exactly, i.e. kept the occurrences of in (43), the integral to solve would have had the form . Solving this integral, (43) would become . The theoretical prediction from this result for is shown in Figure 5 (dark blue dashed line), and is negligibly close to the result from the expression in (44).

(44)

where the right arrow denotes the asymptotic value approached towards the end of inflation (as the field approaches its minimum).

The first equality in the above equation (before the limit is taken) is the analytic result of the recursive formalism, and is shown in Figure 4 (dashed dark-blue line). The pale-blue dashed line shows the first equality in equation (43), before any approximation on is done. The good agreement between this, the final analytical result, and the variance of the classical field obtained numerically (full black line) supports the validity of our approach. For comparison, we have also plotted (dashed yellow line) the obtained total integrated power in classical fluctuations obtained in Finelli et al. (2009, 2010), by using the slow-roll corrected to account for the fact that the bath evolves in a slow-rolling background (by enforcing it by hand). In contrast, the recursive method we apply here self-consistently accounts for this correction, in a natural way.

vi.2.2 Recovering the power spectrum of scalar fluctuations

Clearly, when calculating the power spectrum of scalar perturbations, the often-used procedure of quantizing the fluctuations deep inside the horizon and then evaluating at horizon crossing cannot be applied here, as the quantum fluctuations have been replaced by a classical noise which, at every given time, is only nonzero at the coarse-graining scale. The power spectrum of perturbations can instead be calculated by noting that the variance of fluctuations is the integral of the power spectrum from an IR cutoff to the Hubble horizon, or more precisely, the classicalization radius, which corresponds to the coarse-graining scale. That is,

(45)

which is a standard textbook result (see, for example, Lyth and Liddle (2009)). The power spectrum in the above expression can be written in terms of mode functions as

(46)

where the tilde is to denote that is not a Fourier mode of , but rather the mode function that one would find in the standard procedure of quantizing perturbations and computing the power spectrum.

Another approach consists in using the following trick: we can parametrize a generic power spectrum in terms of a general spectral index , and explicitly compute the integral on the r.h.s. of equation (45). Then, by solving equation (42) for , we can deduce the value of . More explicitly, in standard perturbation theory, a generic power spectrum of field fluctuations far outside the Hubble radius can be written as (see for example Martin (2004); Finelli et al. (2004), where we keep only the terms that depend on ):

(47)

where is a time-dependent amplitude and is the spectral index. Therefore, integrating over all super-Hubble modes as in (45), we obtain

(48)

where a standard calculation would take . Performing the renormalisation of this quantity through adiabatic subtraction as done in Finelli et al. (2004), we can obtain a value for that is independent of the IR cutoff. From there, evaluating this result after a sufficiently long period of inflation (in order for the one-point correlator to saturate to its maximal asymptotic value and for the memory of initial conditions to disappear), the terms in square brackets in equation (48) (plus counter terms) simplify to . Therefore, given , one can solve for the spectral index:

(49)

vi.2.3 Computation of the time-dependent amplitude

In the specific case of inflation, the standard theory of cosmological perturbations gives the following result for the time dependence of mode functions far outside of the Hubble horizon:

(50)

where is the Hankel function of the first kind, and . At small or , i.e. outside the Hubble radius, the mode functions can be asymptotically approximated by:

(51)

This allows us to calculate the r.h.s. of equation (45), which gives the total amount of power in the inflaton fluctuations:

(52)

The arrow in the last line denotes the value at which the correlator saturates towards the end of inflation. From this, we can deduce an explicit form for the time-dependent amplitude of the power spectrum: