Testing the Gaussianity and Statistical Isotropy of the UniverseView this Special Issue
Review Article | Open Access
Non-Gaussianity from Particle Production during Inflation
In a variety of models the motion of the inflaton may trigger the production of some non-inflaton particles during inflation, for example via parametric resonance or a phase transition. Such models have attracted interest recently for a variety of reasons, including the possibility of slowing the motion of the inflaton on a steep potential. In this review we show that interactions between the produced particles and the inflaton condensate can lead to a qualitatively new mechanism for generating cosmological fluctuations from inflation. We illustrate this effect using a simple prototype model for the interaction between the inflaton, , and iso-inflaton, . Such interactions are quite natural in a variety of inflation models from supersymmetry and string theory. Using both lattice field theory and analytical calculations, we study the production of particles and their subsequent rescatterings off the condensate , which generates bremsstrahlung radiation of light inflaton fluctuations . This mechanism leads to observable features in the primordial power spectrum. We derive observational constraints on such features and discuss their implications for popular models of inflation. Inflationary particle production also leads to a very novel kind of nongaussian signature which may be observable in future missions.
In recent years the inflationary paradigm has become a cornerstone of modern cosmology. In the simplest scenario the observed cosmological perturbations are seeded by the quantum vacuum fluctuations of the inflaton field [1–5]. This mechanism predicts a nearly scale invariant spectrum of adiabatic primordial fluctuations, consistent with recent observational data . In addition to this standard mechanism, there are also several alternatives for generating cosmological perturbations from inflation; examples include modulated fluctuations [7–10] and the curvaton mechanism . These various scenarios all lead to similar predictions for the power spectrum. On the other hand, nongaussian statistics (such as the bispectrum) provide a powerful tool to observationally discriminate between different mechanisms for generating the curvature perturbation. In this paper, which is based on [12–15], we will present a qualitatively new mechanism for generating cosmological perturbations during inflation. We discuss in detail the predictions of this new scenario for both the spectrum and nongaussianity of the primordial curvature fluctuations, showing how this new mechanism may be observationally distinguished from previous approaches.
1.1. Non-Gaussianity from Inflation
The possibility to discriminate between various inflationary scenarios has led to a recent surge of interest in computing and measuring nongaussian statistics. Although single field, slow roll models are known to produce negligible nongaussianity [16–18], there are now a variety of scenarios available in the literature which may predict an observable signature. Departures from gaussianity are often parametrized in the following form: where is the primordial curvature perturbation, is a Gaussian random field, and characterizes the degree of nongaussianity. The ansatz (1) is known as the “local” form of nongaussianity.
Although the local ansatz (1) has received significant attention, it is certainly not the only well-motivated model for a nongaussian curvature perturbation. For example, the nongaussian part of need not be correlated with the gaussian part. Consider a primordial curvature perturbation of the form where is some nonlinear (not necessarily quadratic) function, and is a gaussian field which is uncorrelated with . Both (1) and (2) are local in position space; however, these two types of nongaussianity will have very different observational implications. The uncorrelated ansatz (2) for the primordial curvature perturbation can arise, for example, in models with preheating into light fields [19–21]. (See also [22, 23] for more discussion of nongaussianity from preheating and [24, 25] for another model where nongaussianity is generated at the end of inflation.)
A useful quantity to consider is the bispectrum, , which is the 3-point correlation function of the Fourier transform of the primordial curvature perturbation where . The delta function appearing in (3) reflects translational invariance and ensures that depends on three momenta which form a triangle: . Rotational invariance implies that is symmetric in its arguments.
If we assume the ansatz (1) for the primordial curvature perturbation, then has a very particular dependence on momenta; it peaks in the squeezed limit where one of the wavenumbers is much smaller than the remaining two (e.g., ). Such a bispectrum is referred to as having a squeezed shape. However, other shapes of bispectrum are worth considering. A bispectrum is referred to as “equilateral” if it peaks when and “flattened” if it peaks when one of the wavenumbers is half the size of the remaining two (e.g., ).
Without assuming any specific form for the primordial curvature perturbation, such as (1) or (2), one may characterize an arbitrary bispectrum (3) by specifying its shape, running, and size . As discussed above, the shape refers to the configuration of triangle on which is maximal (squeeze, equilateral, or flattened). The running of the bispectrum refers to how the magnitude of depends on the overall size of the triangle. For example, in the case of scale invariant fluctuations, the bispectrum must scale as . Finally, the overall size of the bispectrum is often quantified by evaluating the magnitude of on some fixed equilateral triangle. However, the skewness of the probability density function (defined later) might provide a better measure of the size of nongaussianity.
Different types of nongaussian signatures are correlated with properties of the underlying inflation model. Let us first consider some examples with small running (by “small” here we refer to any model where the running of the bispectrum is proportional to slow-variation parameters or arises due to loop effects. This does not necessarily mean that such running cannot lead to interesting observational signatures, see; [27–30]).(1)A large bispectrum of local shape, along with iso-curvature effects, is associated with models where multiple fields are light (or otherwise dynamically important) during inflation. Examples include the curvaton mechanism [31–34] or models with turning points along the inflationary trajectory [35–39]. The observational bound on local type nongaussianity, coming from the WMAP7  data, is  at 95% confidence level. When combined with large scale structure (LSS) data the bound becomes somewhat stronger: . (2)A large local bispectrum without any isocurvature fluctuations can only be produced by nonlocal inflation models [42–44]. For any single-field inflation model described by a local low-energy effective field theory, the results of  imply that the ratio of the 3-point correlation function to the square of the 2-point function must be of order of the spectral tilt, in the squeezed limit. Hence, it has been argued that a large squeezed bispectrum must be associated with the presence of multiple light degrees of freedom and hence iso-curvature effects. However, in [42–44] it was shown that single field nonlocal inflation models can produce a large squeezed bispectrum in the regime where the underlying scale of nonlocality is much larger than the Hubble scale during inflation. Such constructions evade the no-go theorem of  precisely because they violate the usual assumption of cluster decomposition. Moreover, models of this type are not subsumed by the general analysis of  since nonlocal field theories with infinitely many derivatives cannot be obtained in the regime of low-energy effective field theory. It is nevertheless sensible to study such constructions since they may be derived from ultra-violet (UV) complete frameworks, such as string field theory or -adic string theory. See [47–49] for details concerning the underlying consistency of nonlocal field theories, and see  for a succinct review of nonlocal cosmology. (3)A large equilateral bispectrum is typically associated with a small sound speed for the inflaton perturbations , such as in Dirac-Born-Infeld (DBI) inflation models [51, 52]. However, such a signature may also be obtained in multifield gelaton  or trapped inflation  models. The observational bound on equilateral type nongaussianity is at 95% confidence level . (4)A large flattened bispectrum is associated with nonvacuum initial conditions [26, 56–58]. (To our knowledge there is no explicit computation of the observational bound on flattened nongaussianity. In , a template (the enfolded model) was proposed. The analysis of  is sufficiently general to study this shape; however, they do not explicitly place bounds on but instead constrain an alternative shape (the orthogonal model) which is a superposition of flattened and equilateral shapes.)
If we relax the assumption that the bispectrum is close to scale invariant, then a much richer variety of nongaussian signatures is possible. For example, in models with sharp steps in the inflaton potential [59, 60] the bispectrum is large only for triangles with a particular characteristic size. We will refer to such a signature as a localized nongaussian feature. Localized nongaussianities are not well constrained by current observation but may be observable in future missions.
Given the significant role that nongaussianity may play in discriminating between different models of the early universe, it is of crucial importance to explore and classify all possible consistent signatures for the bispectrum and other nongaussian statistics. Indeed, in this paper we will describe a new kind of signature—uncorrelated nongaussian features—which is predicted in a variety of simple and well-motivated models of inflation, but which has nevertheless been overlooked in previous literature.
1.2. Inflationary Particle Production
Recently, a new mechanism for generating cosmological perturbations during inflation was proposed . This new mechanism, dubbed infra-red (IR) cascading, is qualitatively different from previous proposals (such as the curvaton or modulated fluctuations) in that it does not rely on the quantum vacuum fluctuations of some light scalar fields during inflation. Rather, the scenario involves the production of massive iso-curvature particles during inflation. These subsequently rescatter off the slow-roll condensate to generate bremsstrahlung radiation of light inflaton fluctuations (which induce curvature perturbations and temperature anisotropies in the usual manner). IR cascading can also be distinguished from previous mechanisms from the observational perspective: this new mechanism leads to novel features in both the spectrum and bispectrum.
In principle, IR cascading may occur in any model where non-inflaton (iso-curvature) particles are produced during inflation. Models of this type have attracted considerable interest recently; examples have been studied where particle production occurs via parametric resonance [12, 13, 54, 61–66], as a result of a phase transition [19, 20, 67–74] or otherwise . Recent interest in inflationary particle production has been stimulated by various considerations.(1)Particle production arises naturally in a number of microscopically realistic models of inflation, including examples from string theory  and supersymmetric (SUSY) field theory . In particular, inflationary particle production is a generic feature of open string inflation models , such as brane/axion monodromy [77–79]. (2)The energetic cost of producing particles during inflation has a dissipative effect on the dynamics of the inflaton. Particle production may therefore slow the motion of the inflaton, even on a steep potential. This gives rise to a new inflationary mechanism, called trapped inflation [54, 80, 81], which may circumvent some of the fine-tuning problems associated with standard slow-roll inflation. See  for an explicit string theory realization of trapped inflation and  for a generalization to higher-dimensional moduli spaces and enhanced symmetry loci. The idea of using dissipative dynamics to slow the motion of the inflaton is qualitatively similar to warm inflation  and also to the variant of natural inflation [83, 84] proposed recently by Anber and Sorbo . (3)Observable features in the primordial power spectrum, generated by particle production and IR cascading, offer a novel example of the non-decoupling of high scale physics in the Cosmic Microwave Background (CMB) [61, 85–87]. In the most interesting examples, the produced particles are extremely massive for (almost) the entire history of the universe; however, their effect cannot be integrated out due to the nonadiabatic time dependence of the iso-inflaton mode functions during particle production. In  particle production during large field inflation was proposed as a possible probe of Planck-scale physics.
In this paper we study in detail the impact of particle production and IR cascading on the observable primordial curvature perturbations. In order to illustrate the basic physics we focus on a very simple and general prototype model where the inflaton, , and iso-inflaton, , fields interact via the coupling We expect, however, that our results will generalize in a straightforward way to more complicated models, such as fermion iso-inflaton fields or gauged interactions, wherein the physics of particle production and rescattering is essentially the same. Our result may also have implications for inflationary phase transitions, because spinodal decomposition can be interpreted as a kind of particle production, and similar bilinear interactions will induce rescattering effects.
Scalar field interactions of type (4) have also been studied recently in connection with nonequilibrium Quantum Field Theory (QFT) [88–90], in particular with applications to the theory of preheating after inflation [91–95] and also moduli trapping [80, 81] at enhanced symmetry points. Although our focus is on particle production during inflation (as opposed to during preheating, after inflation) some of our results nevertheless have implications for preheating, moduli trapping, and also non-equilibrium QFT more generally. For example, in  analytical and numerical studies of rescattering and IR cascading during inflation made it possible to observe, for the first time, the dynamical approach to the turbulent scaling regime that was discovered in [96, 97].
Particle production during inflation in model (4) leads to observable features in the primordial power spectrum, . A number of recent studies have found evidence for localized features in that are incompatible with the simplest power-law model [62, 73, 98–110]. Although these observed features may simply be statistical anomalies (see, e.g., ), there remains the tantalizing possibility that they represent some new physics beyond the simplest slow-roll model. Upcoming polarization data may play an important role in distinguishing these possibilities . In the meantime, it is interesting to determine the extent to which such features may be explained by a simple and well-motivated model such as (4). Moreover, because (4) is a complete microscopic model (as opposed to a phenomenological modification of the power spectrum) it is possible to predict a host of correlated observables, such as features in the scalar bispectrum and tensor power spectrum. Hence, it should be possible to robustly rule out (or confirm) the possibility that some massive iso-curvature particles were produced during inflation.
If detected, features from particle production and IR cascading will provide a rare and powerful new window into the microphysics driving inflation. This scenario opens up the possibility of learning some details about how the inflaton couples to other particles in nature, as opposed to simply reconstructing the inflaton potential along the slow-roll trajectory. Moreover, due to the non-decoupling discussed above, features from particle production and IR cascading may probe new (beyond the standard model) physics at extraordinarily high energy scales.
The outline of this paper is as follows. In Section 2 we provide a brief, qualitative overview of the dynamics of particle production and IR cascading in model (4). In Section 3 we study in detail this same dynamics using fully nonlinear lattice field theory simulations. In Section 4 we provide an analytical theory of particle production and IR cascading in an expanding universe. A complimentary analytical analysis, using second-order cosmological perturbation theory, is provided in Section 5. In Section 6 we consider the observational constraints on inflationary particle production using a variety of data sets. In Section 7 we provide several explicit microscopic realizations of our scenario and study the implications of our observational constraints on models of string theory inflation, in particular brane monodromy. In Section 8 we quantify and characterize the nongaussianity generated by particle production and IR cascading. Finally, in Section 9, we conclude and discuss possible future directions.
2. Overview and Summary of the Mechanism
In this section we provide a brief overview of the dynamics of particle production and IR cascading in model (4) and also summarize the resulting observational signatures. In the remainder of this paper we will flesh out the details of this mechanism with analytical and numerical calculations.
We consider the following model: where is the Ricci curvature constructed from the metric , is the inflaton field, and is the iso-inflaton. As usual, we assume a flat FRW space-time with scale factor and employ the reduced Planck mass . We leave the potential driving inflation unspecified for assuming that it is sufficiently flat in the usual sense, that is, , where are the usual slow-roll parameters.
Note that one might wish to supplement (5) by its supersymmetric completion in order to protect the flatness of the inflaton potential from large radiative corrections coming from loops of the field. We expect that our results will carry over in a straightforward way to SUSY models and also to more complicated scenarios such as higher spin iso-inflaton fields and (possibly) inflationary phase transitions.
The coupling in (5) is introduced to ensure that the iso-inflaton field can become instantaneously massless at some point along the inflaton trajectory (which we assume occurs during the observable range of -foldings of inflation). At this moment particles will be produced by quantum effects.
Let us first consider the homogeneous dynamics of the inflaton field, . Near the point we can generically expand where , and we have arbitrarily set the origin of time so that corresponds to the moment when . (We are, of course, assuming that .) The interaction (4) induces an effective (time varying) mass for the particles of the form where we have defined the characteristic scale It is straightforward to verify that the simple expression (9) will be a good approximation for which, in most models, will be true for the entire observable 60 -foldings of inflation.
Note that, without needing to specify the background inflationary potential , we can write the ratio as where is the usual amplitude of the vacuum fluctuations from inflation. In this work we assume that which is easily satisfied for reasonable values of the coupling . In particular, for we have .
The scenario we have in mind is the following. Inflation starts at some field value and the inflaton rolls toward the point . Initially, the iso-inflaton field is extremely massive and hence it stays pinned in the vacuum, , and does not contribute to superhorizon curvature fluctuations. Eventually, at , the inflaton rolls through the point where and particles are produced. To describe this burst of particle production one must solve for the following equation for the particle mode functions in an expanding universe: Equations of this type are well-studied in the context of preheating after inflation  and moduli trapping . The initial conditions for (12) should be chosen to ensure that the q-number field is in the adiabatic vacuum in the asymptotic past (see Sections 3 and 4 for more details). In the regime particle production is fast compared to the expansion time and one can solve (12) very accurately for the occupation number of the created particles Very quickly after the moment , within a time , these produced particles become nonrelativistic (), and their number density starts to dilute as .
Following the initial burst of particle production there are two distinct physical effects which take place. First, the energetic cost of producing the gas of massive out-of-equilibrium particles drains energy from the inflaton condensate, forcing to drop abruptly. This velocity dip is the result of the backreaction of the produced fluctuations on homogeneous condensate . The second physical effect is that the produced massive particles rescatter off the condensate via the diagram in Figure 1 and emit bremsstrahlung radiation of light inflaton fluctuations (particles).
Backreaction and rescattering leave distinct imprints in the observable cosmological perturbations. Let us first discuss the impact of backreaction. In Figure 2 we plot the velocity dip resulting from the backreaction of the produced particle on the homogeneous inflaton condensate . From this figure we see that the quantity becomes large in the dip. This violation of slow roll is a transient effect; at late times the produced particles become extremely massive and their number density dilutes as .
One can understand the temporary slowing down of the inflaton from an analytical perspective. Backreaction is taken into account using the mean-field equation where the vacuum average is computed following [80, 92] In (14) we have implicitly assumed that the usual Coleman-Weinberg corrections to the inflaton potential have already been absorbed into , hence the vacuum average should include only the effects of nonadiabatic particle production. (Here is the total number density of produced particles, and the factor reflects the usual volume dilution of non-relativistic matter.) In Figure 2 we have plotted the solution of (14) along with the exact result obtained from lattice field theory simulations, illustrating the accuracy of this simple treatment.
Using the mean-field approach, one finds that the transient violation of slow roll leads to a “ringing pattern” (damped oscillations) in the power spectrum of inflaton fluctuations . This ringing pattern is localized around wavenumbers which left the horizon at the moment when particle production occurred. The effect is very much analogous to Fresnel diffraction at a sharp edge.
The second physical effect, rescattering, was considered for the first time in the context of inflationary particle production in . Figure 1 illustrates the dominant process: bremsstrahlung emission of long-wavelength fluctuations from rescattering of the produced particles off the condensate. The time scale for such processes is set by the microscopic scale, , and is thus very short compared to the expansion time, . Moreover, the production of inflaton fluctuations deep in the infrared (IR) is extremely energetically inexpensive, since the inflaton is very nearly massless. The combination of the short time scale for rescattering and the energetic cheapness of radiating IR leads to a rapid build-up of power in long wavelength inflaton modes: IR cascading. This effect leads to a bump-like feature in the power spectrum of inflaton fluctuations, very different from the ringing pattern associated with backreaction. The bump-like feature from rescattering dominates over the ringing pattern from backreaction for all values of parameters.
In  model (5) was studied using lattice field theory simulations, without neglecting any physical processes (that is to say that full nonlinear structure of the theory, including backreaction and rescattering effects, was accounted for consistently). However, this same dynamics can be understood analytically by solving the equation for the inflaton fluctuations in the approximation that all interactions are neglected, except for the diagram in Figure 1. The appropriate equation is See  for a detailed analytical theory. The solution of (16) may be split into two parts: the solution of the homogeneous equation and the particular solution which is due to the source term. The former simply corresponds to the usual scale invariant quantum vacuum fluctuations from inflation. The particular solution, on the other hand, corresponds to inflaton fluctuations generated by rescattering. The abrupt growth of inhomogeneities at sources the particular solution and generates inflaton fluctuations which subsequently cross the horizon and freeze in.
As mentioned earlier, rescattering generates a bump-like contribution to the primordial power spectrum of the curvature perturbations. To good approximation this may be described by a simple semi analytic fitting function where the first term corresponds to the usual vacuum fluctuations from inflation (with amplitude and spectral index ) while the second term corresponds to the bump-like feature from particle production and IR cascading. The amplitude of this feature () depends on while the location () depends on .
In  the simple fitting function (17) was used to place observational constraints on inflationary particle production using a variety of cosmological data sets. Current data are consistent with rather large spectral distortions of the type (17). Features as large as of the usual scale-invariant fluctuations from inflation are allowed, in the case that falls within the range of scales relevant for CMB experiments. Such a feature corresponds to a realistic coupling . Even larger values of are allowed if the feature is localized on smaller scales. In Figure 3 we have illustrated the primordial power spectrum in model (5) for a representative choice of parameters. We also plot the CMB angular Temperature-Temperature (TT) power spectrum for the same parameters.
The prototype model (5) may be realized microscopically in a variety of different particle physics frameworks. In particular, particle production is a rather generic feature of open string inflation models  where the inflaton, , has a geometrical interpretation as the position of some mobile D-brane. In this context the iso-inflaton, , corresponds to a low-lying open string excitation which is stretched between the mobile inflationary brane and any other (spectator) branes which inhabit the compactification volume. If the inflationary and spectator branes become coincident during inflation, then the symmetry of the system is enhanced  and some low-lying stretched string states will become instantaneously massless, mimicking interaction (4) (see also ). An explicit realization of this scenario is provided by brane/axion monodromy models [77–79]. Our observational constraints on inflationary particle production may be used to place bounds on parameters of the underlying string model .
The bump-like feature in , illustrated in Figure 3, must be associated with a nongaussian feature in the bispectrum [12, 14]. Indeed, it is evident already from inspection of (16) that the inflaton fluctuations generated by rescattering are significantly nongaussian; the particular solution of (16) is bi-linear in the gaussian field . The nongaussian signature from IR cascading is rather novel. The nongaussian part of is uncorrelated with the gaussian part. Moreover, the bispectrum is very far from scale invariant; it peaks strongly for triangles with a characteristic size , corresponding to the location of the bump in the power spectrum (17). The shape of the bispectrum therefore depends sensitively on the size of the triangle and is not well described by any of the templates that have been proposed in the literature to date.
The magnitude of this new kind of nongaussianity may be quite large. To quantify the effect it is useful to introduce the probability density function (PDF), , which is the probability that the curvature perturbation has a fluctuation of size . If we define the central moments of the PDF as then a useful measure of nongaussianity is the dimensionless skewness of the PDF, defined by where the subscript indicates that only the connected part of the correlator should be included. The skewness encodes information about the bispectrum integrated over all size and shape configurations and thus provides a meaningful single number to compare the nongaussianity of inflation models which may have very different shapes or running .
If we choose (which is compatible with observation for all values of ), then model (5) produces the same value of as a local model (1) with . This large value suggests that nongaussianity from particle production during inflation may be observable in future missions.
Depending on model parameters, the nongaussian features predicted by model (5) may lead to a rich variety of observable consequences for the CMB or Large Scale Structure (LSS). The phenomenology of this model is quite different from other constructions that have been proposed to obtain large nongaussianity from inflation. However, the underlying microscopic description (5) is extremely simple and, indeed, rather generic from the low-energy perspective. Explicit realizations of interaction (4) have been obtained from string theory and SUSY. Moreover, in order to obtain an observable signature it was not necessary to fine-tune the inflationary trajectory or appeal to re-summation of an infinite series of high-dimension operators.
3. Numerical Study of Rescattering and IR Cascading
3.1. HLattice Simulations
In this section we study numerically the creation of fluctuations by rescattering of the produced particles off the condensate in model (5). To this end, we have written a new lattice field theory code, HLattice , for simulating the interactions of scalar fields in a cosmological setting. HLattice can be used to simulate the dynamics of any number of interacting scalar fields with arbitrary scalar potential and metric on field space . We solve the Klein-Gordon equations for the scalar field dynamics in an expanding FRW space-time and also solve the Friedmann equation self-consistently for the scale factor, . Since the production of long wavelength modes is so energetically inexpensive, a major requirement for successfully capturing this effect is respecting energy conservation to very high accuracy. HLattice conserves energy with an accuracy of order ~, as compared to , which has been obtained using previous codes such as DEFROST  or LATTICEASY . A minimum accuracy of order is required for the problem at hand.
The box size of our simulations corresponds to a comoving scale which is initially times the horizon size , while . We run our simulations for roughly 3 -foldings from the initial moment when the particles are produced although a single -folding would have been sufficient to capture the effect. For the sake of illustration, we have chosen the standard chaotic inflation potential with for our numerical analysis. However, our results do not depend sensitively on the choice of background inflation model. (The model independence of our result arises simply because all the dynamics of rescattering and IR cascading occurs within a single -folding from the moment when . Over such a short time it will always be a good approximation to expand . Hence the dependence on the background dynamics arises only through which is determined by the Hubble scale and the observed amplitude of curvature perturbations. This claim of model independence is born out by explicit analytical calculations in the next section.) We have considered both and and also three different values of the coupling constant: . As expected, the coupling determines the magnitude of the effect while simply shifts the location of the power spectrum feature. For this choice of inflationary potential, the choice corresponds to putting the feature on scale slightly smaller than today’s horizon. On the other hand, corresponds to placing the feature on scales much smaller than those probed by the CMB (we considered this case in order to be able to directly contrast our results with ).
In order to capture the quantum production of particles using classical lattice simulations, we start our numerical evolution very shortly after particle production has occurred, when the modes are nearly adiabatic, but before any significant inflaton fluctuations have been produced. In practice, this corresponds to initializing the simulation at a time . The initial conditions for the modes are given by the usual Bogoliubov computation [80, 92]. These are chosen to reproduce the occupation number , while ensuring that the source term for the fluctuations is turned on smoothly at the initial time. As long as the initial conditions are chosen appropriately, our results are not sensitive to the choice of .
At the initial time, the occupation numbers in the inflaton and iso-inflaton fluctuations are small. However, very quickly the massive particles are diluted away by the expansion of the universe, and the occupation number of the produced IR fluctuations grows large compared to unity. Thus, classical lattice field theory simulations are sufficient to capture the late-time dynamics. (In the next section we will provide a quantum mechanical treatment of the dynamics of particle production and IR cascading during inflation, which will serve as an a posteriori justification for our classical lattice calculation.)
Our approach is very similar to the methodology that has been employed successfully in studies of preheating after inflation for many years [114, 115]. In that case the initial fluctuations of the fields are chosen to reproduce the exact behaviour of the quantum correlation functions. The occupation numbers of the fields are small at the initial time. However, these grow rapidly as a result of the preheating instability, and classical simulations are sufficient to capture the late-time dynamics.
3.2. Numerical Results
We have studied the fully nonlinear dynamics of particle production and the subsequent interactions of the produced with the inflaton field in model (5), as described above. We are interested in the power spectrum of the inflaton fluctuations This contains a contribution coming from the usual quantum vacuum fluctuations from inflation that is close to the usual power-law form on large scales. Such a contribution would be present even in the absence of particle production and is not particularly interesting for us. In order to isolate the effects of rescattering we have subtracted off this component in Figures 4, 5, and 6. In all cases we have normalized to the amplitude of the usual vacuum fluctuations from inflation, .
Figure 4 shows time evolution of the power in the inflaton fluctuations generated by rescattering, for three different time steps early in the evolution. This figure illustrates how multiple rescatterings lead to a dynamical cascading of power into the IR. To illustrate the magnitude of this effect, the horizontal yellow line corresponds to the amplitude of the usual vacuum fluctuations from inflation. For , the fluctuations from rescattering come to dominate over the vacuum fluctuations within a single -folding. In Figure 5 we illustrate how the magnitude of the spectral distortion depends on the coupling, . (The apparent change in the location of the feature for different values of arises because we are plotting the power spectrum as a function of and depends on .)
At late times, the IR portion of the power spectrum illustrated in Figure 4 will remain fixed since the modes associated with these scales have gone outside the horizon and become frozen. On the other hand, the UV portion of this curve corresponds to modes that are still inside the horizon, hence we expect and the UV tail of the power spectrum should damp as , due to the Hubble expansion. We observe precisely this behaviour in our lattice field theory simulations, and this is illustrated in Figure 6, which displays the dynamics of IR cascading over a much longer time scale.
Within a few -foldings from the time of particle production, the entire bump-like feature from IR cascading becomes frozen outside the horizon. At this point the fluctuations have become classical, large-scale adiabatic density perturbations and are observable in the present epoch (presuming that occurs during the observable range of -foldings). In Figure 3 we have illustrated this bump-like feature in both the primordial power spectrum and angular TT spectrum, for a representative choice of parameters.
3.3. Backreaction Effects
As discussed previously, the production of fluctuations at backreacts on the homogeneous causing a transient violation of slow roll. We can study this backreaction numerically, by averaging the inhomogeneous field over the simulation box. The result is plotted in Figure 2. We have also plotted the analytical solution of the mean field (14), showing that this agrees with the exact numerical result.
The dynamics illustrated in Figure 2 is easy to understand physically. The production of particles at drains kinetic energy from the condensate and hence must decrease abruptly. However, within a few -foldings of the moment , the produced iso-inflaton particles become extremely massive and are diluted by the expansion as . At late times the inflaton velocity must tend to the slow-roll value. Notice that the velocity including backreaction effects is not changed significantly, as compared to the usual slow-roll result. This illustrates the energetic cheapness of particle production and IR cascading in model (5).
The transient violation of slow roll illustrated in Figure 2 is expected to induce a ringing pattern in the vacuum fluctuations from inflation . This effect is accounted for automatically in our HLattice simulations. However, we would like to disentangle the effect of backreaction on the cosmological fluctuations from the effect of rescattering. This will be useful in order to compare the relative importance of different physical processes and also to guide our analytical efforts in the next section. To this end, we consider the evolution of the curvature perturbation on comoving hypersurfaces, . In linear theory the equation for the Fourier modes is well known Here the prime denotes derivatives with respect to conformal time and . Equation (21) is only strictly valid in the absence of entropy perturbations. However, in our case the field is extremely massive for nearly the entire duration of inflation, hence one may expect that direct iso-curvature contributions to are small. We have solved (21) numerically. In order to take backreaction effects into account we compute the dynamics of by averaging over our HLattice simulation box. Next, we solve (21) given this background evolution and compute the power spectrum The result is very close to the usual power-law form , with small superposed oscillations resulting from the transient violation of slow roll; see Figure 7. In order to make the ringing pattern more visible, we have subtracted off the usual (nearly) scale-invariant result which would be obtained in the absence of particle production. For comparison, we also plot the bump-like feature from rescattering and IR cascading. This latter contribution was obtained using the results for from the previous subsection and the naive formula (so that ).
From Figure 7 we see that IR cascading has a much more significant impact on the observable curvature fluctuations than does backreaction. Indeed, for the transient violation of slow roll yields an order correction to the vacuum fluctuations while the correction from IR cascading is of order . This dominance is generic for all values of the coupling. Thus, in developing an analytical theory of particle production during inflation, it is a very good approximation to completely ignore backreaction effects.
4. Analytical Formalism
In the last section, we have studied particle production, rescattering, and IR cascading using nonlinear lattice field theory simulations. In this section we will develop a detailed analytical theory, in order to understand those results from a physical perspective. These results were first presented in . We consider, again, model (5). The equations of motion that we wish to solve are where is the covariant d'Alembertian. It will be useful to work with conformal time , related to cosmic time via . In terms of conformal time the metric takes the form We denote derivatives with respect to cosmic time as and with respect to conformal time as . The Hubble parameter has conformal time analogue . For an inflationary (quasi-de Sitter) phase () one has to leading order in the slow roll parameter .
As discussed in Section 2, the motion of the homogeneous inflaton leads to the production of a gas of particles at the moment when . The first step in our analytical computation is to describe this burst of particle production in an expanding universe. Following the initial burst, both backreaction and rescattering effects take place. Our formalism will focus on the latter effect, which is much more important, and we provide only a cursory treatment of backreaction.
4.1. Particle Production in an Expanding Universe
The first step in our scenario is the quantum mechanical production of particles due to the motion of . To understand this effect we must solve the equation for the fluctuations in the rolling inflaton background. Approximating (24) gives where . We remind the reader that for reasonable values of the coupling; see (11).
The flat space analogue of (27) is very well understood from studies of broad-band parametric resonance during preheating  and also moduli trapping at enhanced symmetry points . One does not expect this treatment to differ significantly in our case since both the time scale for particle production and the characteristic wavelength of the produced fluctuations are small compared to the Hubble scale: . Hence, we expect that the occupation number of produced particles will not differ significantly from the flat-space result (13), at least on scales . Furthermore, notice that the field is extremely massive for most of the inflation Since , it follows that , except in a tiny interval which amounts to roughly -foldings for . Therefore, we do not expect any significant fluctuations of to be produced on superhorizon scales .
Let us now consider the solutions of (27). We work with conformal time and write the Fourier transform of the quantum field as Note the explicit factor of in (29) which is introduced to give a canonical kinetic term. The q-number valued Fourier transform can be written as where the annihilation/creation operators satisfy the usual commutation relation and the c-number valued mode functions obey the following oscillator-like equation: The time-dependent frequency is where is the time-dependent effective mass of the particles, and is the usual cosmic time variable. We have arbitrarily set the origin of conformal time so that corresponds to the moment when .
In Figure 8(a) we have plotted a representative solution of (32) in order to illustrate the qualitative behaviour of the modes . In Figure 8(b) we plot the occupation number of particles with momentum , defined as the energy of the mode divided by the energy of each particle. Explicitly, we define where the term comes from extracting the zero-point energy of the linear harmonic oscillator (see  for a review). From Figure 8(a) we see that, near the massless point , the fluctuations get a “kick”, and from Figure 8(b) we see that the occupation number jumps abruptly at this same moment.
Let us now try to understand analytically the behaviour of the solutions of (32). At early times , the frequency varies adiabatically In this in-going adiabatic regime the modes are not excited and the solution of (32) is well described by the adiabatic solution where We have normalized (37) to be pure positive frequency so that the state of the iso-inflaton field at early times corresponds to the adiabatic vacuum with no particles. (Inserting (37) into (35) one finds for the adiabatic solution, as expected.)
The adiabatic solution (37) ceases to be a good approximation very close to the moment when , that is at times . In this regime the adiabaticity condition (36) is violated for modes with wavenumber and particles within this momentum band are produced. During the non-adiabatic regime we can still represent the solutions of (32) in terms of the functions as This expression affords a solution of (32) provided the time-dependent Bogoliubov coefficients obey the following set of coupled equations: The Bogoliubov coefficients are normalized as , and the assumption that no particles are present in the asymptotic past (this assumption is justified since any initial excitation of would have been damped out exponentially fast by the expansion of the universe) fixes the initial conditions , for . This is known as the adiabatic initial condition.
From the structure of (39) it is clear that violation of condition (36) near leads to a rapid growth in the coefficient. The time variation of can be interpreted as a corresponding growth in the occupation number of the particles
At late times () adiabaticity is restored and the growth of must saturate. By inspection of (39) we can see that the Bogoliubov coefficients must tend to constant values in the out-going adiabatic regime. Therefore, within less than an -folding from the moment of particle production the solution of (32) can be represented as a simple superposition of positive frequency modes and negative frequency modes. Our goal now is to derive an analytical expression for the modes which is valid in this out-going adiabatic region.
Let us first study the adiabatic solution . If we focus on the interesting region of phase space, , then the adiabatic solution (37) is very well approximated by where is defined by (34). It is interesting to note that (41) is identical to the analogous flat-space result , except for the factor of . Taking into account also the explicit factor of in our definition of the Fourier transform (29), we recover the expected large-scale behaviour for a massive field in de Sitter space, that is, . This dependence on the scale factor is easy to understand physically, it simply reflects the volume dilution of nonrelativistic particles: .
Next, we seek an expression for the Bogoliubov coefficients , in the out-going adiabatic regime . From (39) it is clear that the value of the Bogoliubov coefficients at late times can depend only on dynamics during the interval where adiabaticity condition (36) is violated. This interval is tiny compared to the expansion time, and we are justified in treating as a constant during this phase. Hence, it follows that the flat space computation of the Bogoliubov coefficients [80, 92] must apply, at least for scales . To a very good approximation we therefore have the well-known result in the out-going adiabatic regime. Equation (43) gives the usual expression (13) for the co-moving occupation number of particles produced by a singe burst of broad-band parametric resonance:
Finally, we arrive at an expression for the out-going adiabatic modes which is accurate for interesting scales . Putting together results (41) and (38) along with well-known expressions (42) and (43) we arrive at valid for . Equation (45) is the main result of this subsection. We will now justify that this expression is quite sufficient for our purposes.
For modes deep in the UV, , our expression (45), is not accurate. (Expression (41) for the adiabatic modes is not valid at high momenta where .) However, such high momentum particles are not produced, condition (36) is always satisfied for . Note that the absence of particle production deep in the UV is built into our expression (45): as this function tends to the vacuum solution .
Our expression (45) is also not valid deep in the IR, for modes . To justify this neglect requires somewhat more care. Notice that, even very far from the massless point, , long wavelength modes should not be thought of as particle like. The large-scale mode functions are not oscillatory but rather damp exponentially fast as . Hence, even if we started with some super-horizon fluctuations of at the beginning of inflation, these would be suppressed by an exponentially small factor before the time when particle production occurs. Any super-horizon fluctuation generated near would need to be exponentially huge to overcome this damping. However, resonant particle production during inflation does not lead to exponential growth of mode functions. (In this regard our scenario is very different from preheating at the end of inflation. In the latter case the inflaton passes many times through the massless point , and there are, correspondingly, many bursts of particle production. After many oscillations of the inflaton field, the particle occupation numbers build up to become exponentially large, and, averaged over many oscillations of the background, the mode functions grow exponentially. However, in our case there is only a single burst of particle production at . The resulting occupation number (13) is always less than unity, and the solutions of (32) never display exponential growth.)
To verify explicitly that there is no significant effect for super-horizon fluctuations let us consider solving (27), neglecting gradient terms. The equation we wish to solve, then, is (For simplicity we take for this paragraph; however, this has no effect on our results.) The solution of this equation may be written in terms of parabolic cylinder functions as For our purposes the precise values of the coefficients , are not important. Rather, it suffices to note that for function (47) behaves as This explicit large-scale asymptotics confirms our previous claims that the super-horizon fluctuations of damp to zero exponentially fast, as . As discussed previously, this damping is easy to understand in terms of the volume dilution of non-relativistic particles. We can also understand the power-law damping that appears in (48) from a physical perspective. The properly normalized modes behave as while on large scales we have . Hence, the late-time damping factor which appears in (48) reflects the fact that the particles become ever more massive as rolls away from the point .
Finally, it is straightforward to see that function (47) does not display any exponential growth near . Hence, we conclude that there is no significant generation of super-horizon fluctuations due to particle production. (This is strictly true only in the linearized theory. It is possible that particles are generated by nonlinear effects such as rescattering. However, even such second-order fluctuations will be extremely massive compared to the Hubble scale and must therefore suffer exponential damping on large scales.)
In this subsection we have seen that the quantum production of particles in an expanding universe proceeds very much as it does in flat space. This is reasonable since particle production occurs on a time scale short compared to the expansion time and involves modes which are inside the horizon at the time of production.
4.2. Inflaton Fluctuations
In Section 4.1 we studied the quantum production of particles which occurs when rolls past the massless point . Subsequently, there are two distinct physical processes which take place: backreaction and rescattering. As we have argued in Section 3, the former effect has a negligible impact of the observable spectrum of cosmological perturbations. Hence, we will not study this effect in any detail (see [61, 63, 64] for analytical calculations). Instead we provide a cursory treatment of backreaction in Appendix A, in order to clear up some common misconceptions.
In this subsection we study the rescattering of produced particle off the inflaton condensate. The dominant process to consider is the diagram illustrated in Figure 1, corresponding to bremsstrahlung emission of fluctuations (particles) in the background of the external field. (There is also a subdominant process of the type which is phase-space suppressed.) Taking into account only the rescattering diagram illustrated in Figure 1 is equivalent to solving the following equation for the q-number inflaton fluctuation: where we have introduced the notation for the inflaton effective mass. (Note that we are not assuming a background potential of the form , only that in the vicinity of the point .)
Equation (49) may be derived by noting that (5) gives an interaction of the form between the inflaton and iso-inflaton, in the background of the external field . Equivalently, one may construct this equation by a straightforward iterative solution of (23).
We work in conformal time and define the q-number Fourier transform of the inflaton fluctuation analogously to (29): (To avoid potential confusion we again draw the attention of the reader to the explicit factor in our convention for the Fourier transform.) The equation of motion (49) now takes the form The solution of (51) consists of two parts: the solution of the homogeneous equation and the particular solution which is due to the source. The former corresponds, physically, to the usual vacuum fluctuations from inflation. On the other hand, the particular solution corresponds physically to the secondary inflaton modes which are generated by rescattering.
4.3. Homogeneous Solution and Green Function
We consider first the homogeneous solution of (51). Since the homogeneous solution is a Gaussian field, we may expand the q-number Fourier transform in terms of annihilation/creation operators , and c-number mode functions as Here the inflaton annihilation/creation operators , obey and commute with the annihilation/creation operators of the -field:
Using (26) and (7) it is straightforward to see that the homogeneous inflaton mode functions obey the following equation: where we have defined The properly normalized mode function solutions are well known and may be written in terms of the Hankel function of the first kind as This solution corresponds to the usual quantum vacuum fluctuations of the inflaton field during inflation.
In passing, let us compute the power spectrum of the quantum vacuum fluctuations from inflation. Using the solutions (57) we have on large scales . The explicit factor of in (58) appears to cancel the in our definition of the Fourier transform (50). The spectral index is using (56).
4.4. Particular Solution: Rescattering Effects
We now consider the particular solution of (51). This is readily constructed using the Green function (60) as Notice that this particular solution is statistically independent of the homogeneous solution (52). In other words, the particular solution can be expanded in terms of the annihilation/creation operators associated with the field whereas the homogeneous solution is written in terms of the annihilation/creation operators associated with the inflaton vacuum fluctuations. These two sets of operators commute with one another.
We will ultimately be interested in computing the -point correlation functions of the particular solution (62). For example, carefully carrying out the Wick contractions, the connected contribution to the 2-point function is The power spectrum of fluctuations generated by rescattering is then defined in terms of the 2-point function in the usual manner (The explicit factor of in definition (63) appears to cancel the factor of in our convention for Fourier transforms (50).)
The total power spectrum is simply the sum of the contribution from the vacuum fluctuations (58) and the contribution from rescattering (63): There are no cross-terms, owing to the fac that and commute.
We now wish to evaluate the 2-point correlator (62). In principle, this is straightforward: first substitute result (45) for the modes and result (60) for the Green function into (62), next evaluate the integrals. However, there is a subtlety. The resulting power spectrum is formally infinite. Moreover, the 2-point correlation function (62) receives contributions from two distinct effects. There is a contribution from particle production, which we are interested in. However, there is also a contribution coming from quantum vacuum fluctuations of the field interacting nonlinearly with the inflaton. The latter contribution would be present even in the absence of particle production, when , .
In order to isolate the effects of particle production on the inflaton fluctuations, we would like to subtract off the contribution to the 2-point correlation function (62) which is coming from the quantum vacuum fluctuations of . This subtraction also has the effect of rendering the power spectrum (63) finite, since it extracts the usual UV divergent contribution associated with the Minkowski-space vacuum fluctuations.
As a step towards renormalizing the 2-point correlation function of inflaton fluctuations from rescattering (62), let us first consider the simpler problem of renormalizing the 2-point function of the gaussian field . We defined the renormalized 2-point function in momentum space as follows: In (65) the quantity is the contribution which would be present even in the absence of particle production, computed by simply taking solution (38) with , . Explicitly, we have where are the adiabatic solutions (37).
To see the impact of this subtraction, let us consider the renormalized variance for the iso-inflaton field, . Employing prescription (65) we have where is the contribution from the Coleman-Weinberg potential. This proves that our prescription reproduces the scheme advocated in . The renormalized variance (67) is finite and may be computed explicitly using our solutions (45). We find where is the total co-moving number density of produced particles. Result (68) was employed in  to quantify the effect of backreaction on the inflaton condensate in the mean field treatment (14). Hence, the renormalization scheme (65) was implicit in that calculation also.
At the level of the 2-point function, our renormalization scheme is tantamount to assuming that Coleman-Weinberg corrections are already absorbed into the definition of the inflaton potential, . In general, such corrections might steepen and spoil slow-roll inflation. Here, we assume that this problem has already been dealt with, either by fine-tuning the bare inflaton potential or else by including extended SUSY (which can minimize dangerous corrections). See also  for a related discussion.
Having established a scheme for renormalizing the 2-point function of the gaussian field , it is now straightforward to consider higher-order correlation functions. We simply rewrite the 4-point function as a product of 2-point functions using Wick's theorem. Next, each Wick contraction is renormalized as (65). Applying this prescription to (62) amounts to where are the adiabatic solutions defined in (37).
4.6. Power Spectrum
We are now in a position to compute the renormalized power spectrum of inflation fluctuations generated by rescattering, . We renormalize the 2-point correlator of the inflaton fluctuations generated by rescatter according to (70) and extract the power spectrum by comparison to (63). We have relegated the technical details to Appendix B and here we simply state the final result: where the functions , are the curved space generalization of the characteristic integrals defined in . Explicitly we have The characteristic integral can be evaluated analytically; however, the resulting expression is not particularly enlightening. Evaluation of the integral requires numerical methods. More details are in Appendix B. Equation (71) is the main result of this section.
4.7. Comparison to Lattice Field Theory Simulations
In Section 3 the results of our analytical formalism were plotted alongside the output of fully nonlinear HLattice simulations. It is evident from Figures 4, 5, and 6 that the agreement between these approaches is extremely good, even very late into the evolution and in the regime . The consistency of perturbative quantum field theory analytics and nonlinear classical lattice simulations provides a highly nontrivial check on our calculation.
4.8. The Bispectrum
So far, we have shown how to compute analytically the power spectrum generated by particle production, rescattering, and IR cascading during inflation in model (5). We found that IR cascading leads to a bump-like contribution to the primordial power spectrum of the inflaton fluctuations. However, this same dynamics must also have a nontrivial impact on nongaussian statistics, such as the bispectrum. Indeed, it is already evident from our previous analysis that the inflaton fluctuations generated by rescattering may be significantly nongaussian. From expression (62) we see that the particular solution (due to rescattering) which is bi-linear is the gaussian field .
We define the bispectrum of the inflaton field fluctuations in terms of the three-point correlation function as The factor appears in (73) to cancel the explicit factors of in our convention (50) for the Fourier transform. It is well known that the nongaussianity associated with the usual quantum vacuum fluctuations of the inflaton is negligible [16–18]; therefore, when evaluating the bispectrum (73) we consider only the particular solution (62) which is due to rescattering. Carefully carrying out the Wick contractions, we find the following result for the renormalized 3-point function: where the modes are defined by (38) and are the adiabatic solutions (37). On the last line of (74) we have labeled schematically terms which are identical to the preceding three lines, only with and interchanged. One may verify that this expression is symmetric under interchange of the momenta by changing dummy variables of integration.
It is now straightforward (but tedious) to plug expressions (41) and (45) into (74) and evaluate the integrals. This computation is tractable analytically because the time and phase-space integrals decouple. The bispectrum is then extracted by comparison to (73). This computation is carried out in detail in  where we have shown that peaks only over triangles with a characteristic size, corresponding to the location of the bump in the power spectrum. This result is easy to understand on physical grounds, all the dynamics of rescattering and IR cascading take place over a very short time near the moment . Hence, the effect of this dynamics on the primordial fluctuations must be limited to scales leaving the horizon near the time when particle production occurs.
We will provide a cursory discussion of the bispectrum in Section 8 when we discuss nongaussianity from particle production during inflation.
4.9. Inclusion of a Bare Iso-Inflaton Mass
In passing, it may be interesting to consider how the analysis of this section is modified in the case that our prototype action (5) is supplemented by a bare mass term for the iso-inflaton field of the form . Thus, in place of (5) suppose that we consider the model Now the particles do not become massless at the point , but rather the effective mass-squared reaches a minimum value (which we assume to be positive). Such a correction may arise due to a variety of effects and will reduce the impact of particle production and IR cascading on the observable cosmological fluctuations.
Let us briefly consider how the additional bare mass term in (75) alters the dynamics of particle production. The iso-inflaton fluctuations now obey the equation rather than (27). This equation was solved in  in the regime where particle production is fast compared to the expansion time. (In the opposite regime, which corresponds to a fine-tuned coupling , the iso-inflaton will be light for a significant fraction of inflation. In that case the theory (5) must be considered as a multifield inflation model and one can no longer consistently assume . In other words, relaxing the assumption of fast particle production significantly changes the scenario under consideration, and we do not pursue this possibility any further.) The occupation number of produced particles is which differs from our previous result (13) by the suppression factor Therefore, the effect of the inclusion of a bare mass for the iso-inflaton is to suppress the number density of produced particles by an amount . This suppression reflects the reduced phase space of produced particles: the adiabaticity condition is violated only for modes with .
The reduction of translates into a suppression for the -point correlation functions of the iso-inflaton. For example, the renormalized variance is suppressed by a factor of . The power spectrum of inflaton fluctuations generated by rescattering, , is proportional to the 4-point correlator of and hence picks up a suppression factor of order . Similarly, the bispectrum is proportional to the 6-point correlator of and must be reduced by a factor of order . The condition is equivalent to and ensures that the addition of a bare iso-inflaton mass will have a negligible impact on any observable.
For models obtained from string theory or supergravity (SUGRA), it is natural to expect of the order of the Hubble scale during inflation [116–118]. In the context of SUGRA, the finite energy density driving inflation breaks SUSY and induces soft scalar potentials with curvature of order . In the case of string theory, many scalars are conformally coupled to gravity  through an interaction of the form where the Ricci scalar is during inflation. More generally, any nonminimal coupling between gravity and the iso-inflaton will induce a contribution of order to the effective mass of , as long as . In all models where condition (80) is satisfied for reasonable values of the coupling , see; (11). Thus, we expect that corrections of the form will not alter our results in a wide variety of well-motivated models.
5. Cosmological Perturbation Theory
In Section 4 we developed an analytical theory of particle production and IR cascading during inflation which is in very good agreement with nonlinear lattice field theory simulations. However, this formalism suffers from a neglect of metric perturbations, and, consequently, we were unable to rigorously discuss the gauge invariant curvature perturbation . (This variable is related to the quantity , defined in Section 3.3 as on large scales.) Hence, the reader may be concerned about gauge ambiguities in our results. In this section we address such concerns, showing that metric perturbations may be incorporated in a straightforward manner and that their consistent inclusion does not change our results in any significant way. We will do so by showing explicitly that, with appropriate choice of gauge, (49) and (27) for the fluctuations of the inflaton and iso-inflaton still hold, to first approximation. We will also go beyond our previous analysis by explicitly showing that in this same gauge the spectrum of the curvature fluctuations, , is trivially related to the spectrum of inflaton fluctuations, .
To render the analysis tractable we would like to take full advantage of the results derived in the last section. To do so, we employ the Seery et al. formalism for working directly with the field equations  and make considerable use of results derived by Malik in [120, 121]. (Note that our notations differ somewhat from those employed by Malik. The reader is therefore urged to take care in comparing our formulae.)
We expand the inflaton and iso-inflaton fields up to second order in perturbation theory as The perturbations are defined to average to zero so that and . (The condition is ensured by the fact that for nearly the entire duration of inflation.)
We employ the flat slicing and threading throughout this section. With this gauge choice the perturbed metric takes the form so that spatial hypersurfaces are flat. Note also that in this gauge the field perturbations , coincide with the Sasaki-Mukhanov variables  at both first and second order.
This perturbative approach, of course, neglects the momentary slowdown of the inflaton background due to backreaction. However, we have already shown in Section 3.3 that backreaction has a tiny impact on the observable cosmological perturbations (see also ).
5.1. Gaussian Perturbations
In  Malik has derived closed-form evolution equations for the field perturbations , at both first order () and, second () order in perturbation theory. Let us first study the Gaussian perturbations. The closed-form Klein-Gordon equation for derived in  can be written as Following our previous analysis we expand the first-order perturbation in terms of annihilation/creation operators as where denotes the Hermitian conjugate of the preceding term, and we draw the attention of the reader to the the explicit factor of in our definition of the Fourier transform. Working to leading order in slow-roll parameters we have This equation coincides exactly with (55), and the properly normalized solutions again take the form of (57). The only difference is that the order of the Hankel function, , is now given by rather than by (56). The power spectrum of the Gaussian fluctuations is, again, given by (58). The correction to the order of the Hankel function translates into a correction to the spectral index: instead of (59) we now have which is precisely the standard result .
Thus, as far as the quantum vacuum fluctuations of the inflaton are concerned, the only impact of consistently including metric perturbations is an correction to the spectral index .
Let us now turn our attention to the first-order fluctuations of the iso-inflaton. The closed-form Klein-Gordon equation for derived in  can be written as This coincides exactly with (27), which we have already solved. The fact that linear perturbations of do not couple to the metric fluctuations follows from the condition .
5.2. Non-Gaussian Perturbations
Now let us consider the second-order perturbation equations. The closed-form Klein-Gordon equation for derived in  can be written as As usual, the left-hand side is identical to the first-order equation (83) while the source term is constructed from a bi-linear combination of the first order quantities and . In order to solve (89) we require explicit expressions for the Green function and the source term . The Green function is trivial for the case at hand; it is still given by our previous result (56), provided that one takes into account the fact that the order of the Hankel functions is now given by (86) rather than (56). In other words, the Green function for the non-Gaussian perturbations (89) differs from the result obtained neglecting metric perturbations only by corrections.
Next, we would like to consider the source term, , appearing in (89). Schematically, we can split the source into contributions bi-linear in the Gaussian inflaton fluctuation and contributions bi-linear in the iso-inflaton : The contribution would be present even in the absence of the iso-inflaton. These correspond, physically, to the usual nongaussian corrections to the inflaton vacuum fluctuations coming from self-interactions. This contribution to the source is well studied in the literature and is known to contribute negligibly to the bispectrum . Thus, in what follows, we will ignore .
On the other hand, the contribution appearing in (90) depends only on the iso-inflaton fluctuations . This contribution can be understood, physically, as generating nongaussian inflaton fluctuations by rescattering of the produced particles off the condensate. Hence, the contribution may source large nongaussianity and is most interesting for us. It is straightforward to compute explicitly for our model using the general results of . We find where the upper sign is for and the lower sign is for . Notice that the contributions to on the third and fourth line of (91) contain the inverse spatial Laplacian and are thus nonlocal. These terms all contain at least as many gradients as inverse gradients, and hence the large-scale limit is well defined. In  it is was argued that these terms nearly always contribute negligibly to the curvature perturbation on large scales.
Let us now examine the structure of the iso-inflaton source , (91). The first line of (91) goes like . This coincides exactly with the source term in (49) which was already studied in Section 4. On the other hand, the terms on the second, third and fourth lines of (91) are new. These represent corrections to IR cascading which arise due to the consistent inclusion of metric perturbations. We will now argue that these “extra” terms are negligible as compared to the first line. If we denote the energy density in gaussian iso-inflaton fluctuations as then, by inspection, we see that the first line of (91) is parametrically of order while the remaining terms are of order . Hence, we expect the first term to dominate for the field values which are relevant for IR cascading. This suggests that the dominant contribution to is the term which we have already taken into account in Section 4.
Let us now make this argument more quantitative. Inspection reveals that the only “new” contribution to (91) which has any chance of competing with the “old” term is the one proportional to (the first term on the second line). This new correction has the possibility of becoming significant because it grows after particle production, as rolls away from . This growth, which reflects the fact that the energy density in the particles increases as they become more massive, cannot persist indefinitely. Within a few -foldings of particle production the iso-inflaton source term must behave as , corresponding to the volume dilution of non-relativistic particles. Hence, in order to justify the analysis of Section 4 we must check that the term does not dominate over the term which we have already considered during the relevant time after particle production. It is straightforward to show that where is the number of -foldings elapsed from particle production to the time when IR cascading has completed. Hence, , and we conclude that the second, third, and fourth lines of (91) are (at least) slow-roll suppressed as compared to the first line.
In summary, we have shown that consistent inclusion of metric perturbations yields corrections to the inflaton fluctuations which fall into two classes.(1)Slow-roll suppressed corrections to the inflaton vacuum fluctuations (these amount to changing the definition of in solution (57)). These corrections have two physical effects. First, they yield an correction to the spectral index. Second, they modify the propagator by an correction. (2)Corrections to the source for the nongaussian inflaton perturbation . These corrections are the second, third, and fourth lines of (91)) which, as we have seen, are slow-roll suppressed.
It should be clear that neither of these corrections alters our previous analysis in any significant way.
So far, we have shown that a consistent inclusion of metric perturbations does not significantly alter our previous results for the field perturbations. Specifically, is identical to our previous solution of (27) for the iso-inflaton while coincides with the homogeneous solution of (49), up to slow-roll corrections. At second order in perturbation theory, we have seen that To a leading order in slow roll, , and the first term coincides with our previous result for the particular solution of (49). The terms of order represent nongaussian corrections to the vacuum fluctuations from inflation (coming from self-interactions of and the nonlinearity of gravity). These would be present even in the absence of particle production and are known to have a negligible impact on the spectrum and bispectrum .
We are ultimately interested in the connected -point correlation functions of . For example, the 2-point function gets a contribution of the form which gives the usual nearly scale-invariant large-scale power spectrum from inflation. The cross-term is of order and represents a negligible “loop” correction to the scale-invariant spectrum from inflation. (The cross-term does not involve the iso-inflaton since and are statistically independent.) Finally, there is a contribution which involves terms of order coming from rescattering and terms of order