A Radiative corrections

SU/ITP-16/12, UTTG-09-16

Productive Interactions:

[.3cm] heavy particles and non-Gaussianity

Raphael Flauger, Mehrdad Mirbabayi, Leonardo Senatore, and Eva Silverstein,


Department of Physics, The University of Texas at Austin, Austin, TX, 78712, USA

Institute for Advanced Study, Princeton, NJ 08540, USA

Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA

SLAC National Accelerator Laboratory, 2575 Sand Hill Rd., Menlo Park, CA 94025, USA

Kavli Institute for Particle Astrophysics and Cosmology, Stanford, CA 94305, USA

Abstract
  We analyze the shape and amplitude of oscillatory features in the primordial power spectrum and non-Gaussianity induced by periodic production of heavy degrees of freedom coupled to the inflaton . We find that non-adiabatic production of particles can contribute effects which are detectable or constrainable using cosmological data even if their time-dependent masses are always heavier than the scale , much larger than the Hubble scale. This provides a new role for UV completion, consistent with the criteria from effective field theory for when heavy fields cannot be integrated out. This analysis is motivated in part by the structure of axion monodromy, and leads to an additional oscillatory signature in a subset of its parameter space. At the level of a quantum field theory model that we analyze in detail, the effect arises consistently with radiative stability for an interesting window of couplings up to of order . The amplitude of the bispectrum and higher-point functions can be larger than that for Resonant Non-Gaussianity, and its signal/noise may be comparable to that of the corresponding oscillations in the power spectrum (and even somewhat larger within a controlled regime of parameters). Its shape is distinct from previously analyzed templates, but was partly motivated by the oscillatory equilateral searches performed recently by the Planck collaboration. We also make some general comments about the challenges involved in making a systematic study of primordial non-Gaussianity.  

1 Introduction

The observation of the primordial seeds for structure provides a fertile testing ground for theories of the dynamics that generates them. The theoretical and observational study of the primordial power spectrum and non-Gaussianity is a mature field, with substantial progress recently due to Planck [1] and future possibilities in large-scale structure. But even in the CMB this study is not complete; in fact it is not known if there is a systematic way to complete it.

In principle, there is an infinite space of possibilities, in practice the only useful searches involve -spectra which depend on a limited number of parameters. Such templates can be derived from sufficiently well-defined theories of the primordial perturbations. Distinct analyses are required in order to test shapes which do not overlap strongly, in the precise sense developed in [2]. Moreover, a physical mechanism which generates large non-Gaussianity may generically also affect the power spectrum, so such searches are only well-motivated if the signal/noise in the higher-point functions is competitive with the leading corrections to the power spectrum.

In this work, we will present a new class of shapes motivated by a very basic theoretical possibility: non-derivative couplings of the inflaton to additional heavy fields. In the presence of a discrete shift symmetry such couplings can be significant without spoiling inflation and, as we will see, can lead to non-adiabatic production of very heavy fields, sourcing detectable corrections to the scalar perturbations – including non-Gaussianity – in an interesting range of parameters.

Such couplings occur in axion monodromy inflation [10, 11, 13, 12] at the single-light field level, and were investigated in [5, 6] as a source of tensor emission during inflation (see e.g. [7, 8] for another secondary source of tensor modes). The scalar contribution was a limiting factor on this effect. Here we analyze this contribution in detail, focusing on the regime where the scalar emission is subdominant to the nearly Gaussian vacuum fluctuations, but can be detectable in the power spectrum and non-Gaussianity. In the regime we consider, the particle production does not backreact on the inflationary dynamics, in contrast to [14], where the perturbations were studied in a continuum approximation in the strongly back-reacting regime.

In broader terms, we will show how microscopic subhorizon physics during inflation can be relevant for the superhorizon predictions when the inflaton is coupled to additional fields, even very heavy ones.

We will exhibit two novel effects:

Even fields that are never lighter than the scale in slow-roll inflation1 can make a measurable difference to the primordial -spectra. From the perspective of the effective field theory of inflaton fluctuations [9, 23, 24], the dynamical scales that are visible are the expansion rate and the Fourier frequency of the time dependent mass and couplings. We will see that fields comparable to these scales can modify the primordial -spectra in such a way that is not captured by any effective single field description. There has been much interesting previous work on the potentially observable effects of thermally produced massive particles in the expanding background, such as [16], and the effects of heavier fields on the underlying inflationary dynamics (e.g. [17]). The latter explains inflationary plateaus,2 but does not provide a distinctive signature. The former leads to effects suppressed exponentially in for mass . As we will see, the present mechanism will generate a less suppressed effect of heavy fields, with the number density of the of heavy particles and hence the amplitude governed by the exponential factor

(1.1)

where is the lightest value of the time dependent mass in question, and a coupling which can be order 1. Since this has much greater amplitude for a given mass than the effects arising purely from vacuum fluctuations. This result is pictorially represented in Fig. 1.

For most oscillation frequencies of interest, the amplitude of the resulting oscillatory non-Gaussianity can be parametrically larger than that of previously studied resonant non-Gaussianity [23][24], derived from a slow roll potential with a small sinusoidal term. In fact, for rare events (when the factor (1.1) becomes small), in the regime where the coupling is not too small, the present mechanism produces highly non-Gaussian perturbations with signal/noise easily competitive with that of the oscillatory features in the power spectrum. This provides theoretical motivation for a joint analysis of such templates in the power spectrum and bispectrum, analogous to [15]. The contrast between the present effect and resonant non-Gaussianity is expressed in the simple formulas (4.11) and (4.12) below. In fact, we find that the signal/noise in the primordial point correlators can grow with for a range of ; in this regime it would be interesting to determine the optimal search strategy.

Figure 1: Pictorial representation of our findings: in an inflationary theory with an approximate continuous shift symmetry for the inflaton, only particles that are not much heavier than the Hubble scale are relevant for the dynamics of the fluctuations. However, as we will see, if the continuous shift symmetry is broken, e.g. to a discrete shift symmetry, heavier particles can become relevant as depicted on the right. In the scenarios studied in this work, the new scale is set by . The basic estimate suggests observational sensitivity to these massive particles, which we confirm in a detailed analysis.

In §2 we will review the setup and radiative stability of the model. In §3, we will calculate the correlation functions of scalar perturbations that result from particle production. We discuss the constraints on the model parameters enforced to stay within the regime of validity of our approximations in §4. The phenomenology of the model and the templates for analysis are laid out in §5. Finally, in §6 we will make some general remarks about how our results compare to previous mechanisms generating significant non-Gaussianity, and comment on the interplay between the time-dependent couplings in the EFT of perturbations, non-adiabatic effects, and data searches before concluding in §7.

2 Setup and vacuum loop corrections

We will be interested in the coupling of one or more heavy fields to the inflaton , leading to a field-dependent mass

(2.1)

which implies a time-dependent mass for as rolls during inflation. If this time-dependence is sufficiently rapid, it leads to non-adiabatic production of particles. The produced particles then source inflaton fluctuations as their mass changes in time. We will find that even fields which are never lighter than can contribute measurably to perturbations in some regime of parameters and hence cannot be integrated out.

There are many ways such couplings can appear – in general for each such field there could be an arbitrary mass function. We will develop this in some generality, but ultimately focus on couplings respecting an approximate discrete shift symmetry, weakly broken by the slow roll potential, as motivated by axions. Although that narrows down the possibilities, postulating this symmetry does not suffice to determine the observables, as we will see explicitly, and more theoretical input is required. A mass function that is disordered as discussed recently in [28] is another interesting limit.

The structure of axion monodromy in string theory motivates the discrete shift symmetry, and entails further sectors of fields and couplings. In that theory, there are two types of heavy sectors with masses that arise from the same basic structure and are specific enough to derive concrete oscillatory -spectrum shapes and amplitudes.

(a) particle sectors with monodromy structure

There is a part of the spectrum which undergoes monodromy in analogy to the potential energy, with a different sector reaching a minimal mass or tension each time the field traverses an underlying period in the axion field space. If these degrees of freedom are particles (as opposed to strings), we have

(2.2)

where is the periodic axion field as a function of the canonically normalized inflaton field . In the absence of drift [26], with an axion decay constant , leading to the last expression in (2.2) with .

As just mentioned, depending on the microphysical details the produced sources may be extended strings rather than particles. Non-adiabatic production of strings has some very interesting subtleties and potentially distinguishing features [5][27]. To be specific, we will focus on the particle production case in this paper.

(b) sinusoidally modulated masses

Additional fields coupled to the inflaton generically have masses modulated by the periodic term in the potential. For example, as the axion traverses its underlying period, massive fields, such as moduli or Kaluza-Klein modes, will undergo periodic modulation of their masses. This motivates a mass-squared of the form

(2.3)

and requires

(2.4)

Near a point of minimal mass, where the argument of the cosine is , this behaves as

(2.5)

In the regime we will be interested in, the form (2.5) governs the physics near a particle production point as we will explain shortly. Although this regime is similar to case (a), the sinusoidal behavior of the resulting source of perturbations will lead to distinct results for the scalar perturbations in case (b). In both cases the angular frequency of production events is given by

(2.6)

2.1 EFT perspective and dependence on high energy scales

Before proceeding to analyze this mechanism in detail, let us address how the need to incorporate the heavy fields arises in the effecitve field theory context [9]. One of the lessons of the present work will be that the precision of current data can require inclusion of very heavy fields which one cannot trivially integrate out. If the EFT Lagrangian has some time dependent couplings and in the action (see equation (6.1) below), and if their Fourier transforms have support at frequency of order , any additional particles that exist with mass of order will be produced.

Before moving to the present work, let us first note that this fact is already familiar in standard inflationary scenarios where the time dependence induced by the Hubble scale in the metric induces fluctuations in particles of mass . As particles get parametrically heavier than , one might naively imagine reducing to a single-field EFT by integrating out such particles. But the resulting EFT would only include all terms in an analytic expansion in ; non-analytic effects in this parameter, for example those which scale as , would be missed. Depending on the precision of the experiment, these non-perturbatively small effects may not be negligible.

Exactly the same considerations apply to all time-dependent couplings in the Lagrangian for the EFT of inflationary perturbations (6.1). If the EFT Lagrangian contains functions of time with support at frequencies of order the mass , a naive single-clock theory would miss effects that scale as .

It is very possible for the effective theory of perturbations to have functions of time that have support at frequencies of order : even in slow-roll inflation, time-translations are broken at the scale . Let us apply this, for example, to slow roll inflation in the presence of a particle whose mass depends on the inflaton as . If we were to integrate out this particle, we would get contributions to the low energy theory proportional to whose Fourier transform scales as (taking ). This has support up to frequencies of order .

As we will see in detail in this paper, this implies that particles with mass , and even slightly larger than this, can be produced at a detectable level. In fact, a cosmological experiment that measures cosmological modes, has a relative precision of about ; for an experiment such as Planck, this is about . This allows for the exponent of the exponential suppression to be large (with the details depending on additional power law factors as we will see in this work).

Notice that there are no surprises from the EFT point of view: in all EFT’s, the number of relevant degrees of freedom should be declared a priori. If the time-dependent couplings have a Fourier transform with support at frequencies of order , insisting on a single-clock description of inflation amounts to assuming that no additional particles are present with mass lighter than (and even a bit larger than because we can afford for some exponential suppression). This depends on the full model of inflation. As such, this provides a new role for UV completion that goes beyond simply deriving the light field spectrum from it and controlling Planck-suppressed corrections to the inflationary dynamics.

One might wonder how effects that are exponentially suppressed as can dominate over the ones that come from integrating out these particles. As we will see in our detailed analysis below, this can be understood as follows. Integrating out heavy particles in slow roll inflation, for example, will induce operators such as . This operator induces a non-Gaussianity with signal to noise ratio (as can be read off from the ratio of the three point interaction Lagrangian to the kinetic term, ) [19]. This is if . The non-Gaussianity from particle production works differently. As we will review in detail below (see e.g. [20, 22]), it induces a number of particles in an Hubble patch, , (which is the relevant quantity) that scales as , where we have just kept the most important parametric dependence. The non-Gaussianity of this distribution is controlled by , which can be order one for . This non-Gaussianity will then be transferred to the inflaton through the relevant couplings. The prefactor is a very large number, allowing for some exponential suppression to be present, while preserving the dominance of the effect.

In short, the observable fluctuations during inflation can be sensitive to scales that are as much as a few orders of magnitude higher than . If happens to be sufficiently high, this corresponds to scales close to the GUT scale or so, which makes the possibility even more interesting.

2.2 Vacuum loop corrections

The coupling of the fields to the inflaton generates two types of corrections to the dynamics of , which can roughly be characterized as those coming from loops (vacuum fluctuations) and those coming from production; of course in general there is a combination of the two. The latter effects are the main subject of this paper. The former must be taken into account as well. Their size depends on microscopic details such as the level of broken supersymmetry in the sector. The contribution from vacuum fluctuations generates various periodic terms in the effective action. This may be the leading such contribution, or it may be subdominant, depending on parameters.

In general, before computing the effects of particle production, we should ensure that the system we are considering is controlled against radiative corrections. The condition for radiative stability is important to our assessment of the non-Gaussianity: it restricts the strength of the coupling constant in the theory, and hence leads to some constraint on the strength of nonlinear interactions visible in the perturbations. In this regard, it is worth emphasizing that with microscopic supersymmetry, the contributions of bosons and fermions can (partially) cancel each other in the corrections to the effective action, whereas they arise additively in the non-adiabatic effects that we are concerned with in this paper. Therefore, we analyze radiative corrections in appendix A assuming some degree of supersymmetry. We will find that for the corrections to the slow-roll potential to be subdominant we need

(2.7)

and for the higher derivative corrections not to induce large non-Gaussianity (as in DBI inflation [37])

(2.8)

Another way to see the existence of a regime of sub-dominance of the non-Gaussianity induced by power law corrections to effective action was described above in §2.1.

3 Power spectrum and non-Gaussianity from particle production

In this section, we will derive the shape and amplitude of the contribution to the power spectrum and bispectrum (as well as higher-point correlators) from repeated particle production events. We will first give a detailed description of the non-adiabatic production and evolution of the heavy particles (sources) in §3.1. In §3.2, we will derive the spectrum of classical scalar emission by these sources.3 In §3.3, we include the effect of interaction with vacuum scalar fluctuations. Finally, in §3.4 we consider interference terms between two branches of the wave-function: one without particle production and one in which a pair of particles are produced and subsequently annihilate into quanta.

3.1 Source dynamics

We will work in a regime where the timescale associated with each production event is shorter than half the time period separating the events. Each production event is then well modeled by a time-dependent mass of the form

(3.1)

where or in our two specific cases described above. Here labels the event, and in the regime the timescale on which the production occurs is

(3.2)

This follows from maximizing as a function of , giving . Particle production including cosmological applications has been discussed extensively in the literature; see for example [20][21].

Before proceeding further, let us check that in case (b) we can indeed obtain an inequality allowing us to model the production event using (3.1). This requires

(3.3)

where we used equations (2.6) and (3.2). This is consistent with the basic requirement (2.4) above in case (b), since there is a window

(3.4)

To fix our conventions and notation, the inflationary metric is approximately de Sitter

(3.5)

with conformal time coordinate . Let us denote the comoving momenta by and physical momenta by , which are given at the time of the -th production event by

(3.6)

Starting from the vacuum, evolving through the window of times where the particles reach their minimal mass generates a squeezed state

(3.7)

where is a normalization factor, and the Bogoliubov coefficients satisfy

(3.8)

This leads to a source for scalar (and tensor) perturbations which is essentially a step function times a more slowly varying source

(3.9)

where

(3.10)

is the average number density of particles produced in each event. This density dilutes with the expansion of the universe, but every a new generation of sources is produced. The physical momentum scale of the production events can be seen to be of order from (3.8).

Bose enhancement and backreaction

In certain situations we should consider the effect of previously produced particles on a given event. So far we discussed particle creation from the vacuum, but the calculation easily generalizes. If there are many production events per Hubble time, i.e. , and if the massive particles of interest are bosons there will be an enhancement in their production. To apply the flat space analysis we restrict attention to a subset of the events occurring in one Hubble time. The number of produced particles in the presence of an existing number excited is modified to

(3.11)

Using a continuum approximation4 when there are many events with frequency and ignoring the redshift of momenta we obtain

(3.12)

In one Hubble time this gives

(3.13)

In order to match current limits on oscillatory features in the primordial power spectrum [4], we will be interested in a regime with sufficiently large that , and ranging up to of order . The effect of the previously produced particles therefore only becomes marginally important for the highest frequencies.

Before moving to the perturbations, we should note the conditions for our produced particles not to strongly affect the background evolution of the inflaton . We can estimate the back reaction of effective scalar potential contributed by the -dependent energy density in as . First, we must keep its effect subdominant to the original slow roll background evolution by imposing

(3.14)

In addition, we will impose

(3.15)

where in the last step we used that our background is slow-roll inflation. This prevents the production of the particles from draining significant kinetic energy from the inflaton, we are working far from the regime of [14]. We will verify that these conditions are satisfied below in our parameter window of interest after deriving the perturbations.

Action for the fluctuations and their mode functions

We will be interested in repeated production events whose distribution in time will determine the scale-dependence of our perturbations. A discrete shift symmetry, as arises in axion monodromy inflation, will lead to shapes respecting a discrete version of scale-invariance, i.e. a symmetry under for integer . The shapes will exhibit residual oscillatory features which we will compute in detail.

Working with the conformal time coordinate , and decomposing , the action is

where

(3.17)

describes interactions higher order in which descend from the -dependent mass term.

The -point functions can be computed by standard in-in perturbation theory

(3.18)

where the interaction picture fields and are evolved with the quadratic Hamiltonian, including the time-dependent mass-squared term for the particles, obtained from the background homogeneous evolution of the inflaton. To evaluate (3.18) we need the mode functions for the scalar fluctuations and fields.

mode function
We start by expanding the interaction picture field in terms of lowering and raising operators and

(3.19)

which satisfy , as well as , and we defined the shorthand notation

(3.20)

Considering the leading de Sitter expansion with approximately constant , the properly normalized mode solution is

(3.21)

These mode functions satisfy

(3.22)

ensuring canonical commutation relations for and its canonical momentum . At early times this becomes

(3.23)

mode function
Similarly for a given sector of particles, we have the mode expansion

(3.24)

where the mode function is a solution of the free equation of motion including the effects of the time-dependent mass, and . This encodes the evolution of the operator via the free Hamiltonian, appropriate for the interaction picture. In case (b), with sinusoidal , inside the horizon this is a Mathieu function. But there is a simple WKB approximation valid in our regime: between bursts of particle production, the solution is a linear combination of adiabatic modes which we can write as

(3.25)

with normalization , where is the timescale of the production event (3.2). We can consider a mode solution which is pure positive frequency initially, i.e. take . After the first event, a nontrivial contribution is generated. The full set of within the Minkowski regime (when Hubble dilution is negligible) can be understood in a simple way from the analogue Schrodinger problem solved by the mode solution

(3.26)

with effective potential .

In our specific case (a), at each time , there is a different sector that reaches its minimal mass, so we have one production event per sector. In case (b), we have a single sector with an oscillating mass leading to repeated particle production events for this sector. The analysis leading to (3.13) suggests that for our purposes, even in this latter case we can treat the events as independent, with the correlator being a sum over their contributions. In our case (a), for each sector of particles this is simply scattering off of an inverse Harmonic oscillator potential, with the reflection coefficient of order (see for example the appendix of [5] for a derivation). In our case (b), this is scattering off a sinusoidal potential, which behaves as a sequence of inverse Harmonic oscillators near its maxima. As mentioned above, the full solution for this is given by Mathieu functions (one-dimensional Bloch waves in a sinusoidal potential).

The effect can be shuffled between the mode solution and the basis of creation and annihilation operators, via the Bogoliubov transformation . The state satisfying is a squeezed state in terms of the Fock space with .

For a given particle production event at time , we generate a squeezed state excited above the vacuum. To compute the contributions to the correlation functions from this event we can work with the Fock space built from . (We will suppress the label in subsequent expressions.) The formula corresponding to (3.18) is given by writing the state in this basis,

(3.27)

with the normalization . In our calculations below, we will find that the leading effects come from saddle points in the integrals over in the interaction Hamiltonian, and that these saddles are at or after the production event, . As a result, we can replace the lower limits of integration with to good approximation. The expansion of in this basis is then simply of the form

(3.28)

The Bogoliubov coefficients describing the particle production now appear in the the state rather than the mode functions. We have included the prefactor appropriate for the de Sitter background. We have written this in a WKB form, as is justified by the massiveness of the particles which leads to small variation .

As we will see, different classes of diagrams dominate in different regimes of parameters. We will first consider the contributions generated by the 3-point vertex , focusing on those which do not involve annihilations of particles. This will generate effects similar to those predicted by the classical model described in the appendix of [6]. The leading such contributions scale like the density of produced particles times other factors determined by a simple stationary phase approximation to the time integrals.5 In the following section, we will consider other diagrams which encode quantum interference and annihilation effects. These include contributions scaling like , but with more rapidly oscillating integrands and hence different prefactors. The two classes of diagrams dominate in different regimes of parameters which we will spell out below. The shape of the non-Gaussianity in the first class we will consider is novel, whereas the second class of diagrams has more similarity to resonant non-Gaussianity [23].

3.2 Sourced contributions at order

We will begin by computing contributions that are similar to those that would arise from a classical source created by the production event. These contributions to the correlators of the scalar fluctuations will be given as in [6] by correlators of the source convolved with the retarded Green’s function. The retarded Green’s function derived from (3.21) is

(3.29)

so that

(3.30)

Here the factor of comes from the source term. (As above, we defined itself in proper units.) We will be interested in late-time observations, related to

(3.31)

Let us unpack the source (3.9) a bit more. We have

(3.32)

where for now we are approximating by its background evolution. Close enough to the event, this is of the form for both cases (a) and (b), but we will require the later evolution of the source. This leads to

(3.33)

where we replaced the from the product of mode functions with . This is often a good approximation, for the following reason. In both of our cases (a) and (b), the frequency can be written as

(3.34)

where as above denotes the minimal mass, either or , and . The term dilutes exponentially, and its initial value is the dominant momentum squared in our particle production process, . This is smaller than in our regime of interest, for which the exponential Bogoliubov coefficient (1.1) is .

The second line in (3.33) indicates the rest of the terms quadratic in raising and lowering operators.6 These terms will lead to interesting interference effects we will include below, in fact starting at order (rather than order as we have here). These contributions have a net oscillation with time . At a given order in , this causes some suppression relative to effects we will find here which in our case (b) will resonate at the scale . For the present section, we will therefore focus on the contributions that arise from the terms. The last factor captures the Hubble dilution of the source particles after their creation.

The next step is to derive the correlators of (3.33), from which using (3.30) we will obtain the desired contribution to the correlators of . These are expectation values in the squeezed state

(3.35)

where is a normalization factor, and

(3.36)

Here is a step function smoothed out over the non-adiabaticity timescale (3.2). For the two point function we find the behavior

(3.37)

Finally we can plug the above into (3.30) to estimate the Gaussian scalar perturbations. In this step, we treat the functions as simply step functions , since the production timescale is much shorter than that of the oscillatory features we are considering. We can view this as a test of the UV sensitivity of this part of the calculation – if the result does not blow up, then evidently the high energy scale scale is not cutting off any divergence. Defining

(3.38)

and using (3.30) and (3.37) we find the power spectrum

(3.39)

Translating to using (plus slow-roll suppressed higher order corrections), and using the standard result

(3.40)

we find

(3.41)

The subscript refers to the particle production contribution. The truly oscillatory contribution is a piece of this as we will describe below.

For the N point function, following similar steps, we find a connected contribution

(3.42)

which can similarly be traded for .

In the regime where these contributions dominate, these formulas directly lead to templates for analysis, collected below in section 5. Let us examine their amplitude and shape. One important quantity is the ratio of signal/noise in the three and two point functions, which in the cosmic variance dominated Gaussian approximation is given by

(3.43)

where the prime on expectation values denotes dropping and we evaluated the numerator and denominator at to get a sense of the relative amplitudes; we will discuss the shape in momentum space (including the scale dependence) further below.

To proceed, let us apply this result to a situation with an approximate discrete shift symmetry, with events evenly spaced in proper time , corresponding to conformal times

(3.44)

where depends inversely on the underlying field period . In our case (b), this frequency appears in the cosine term in the potential. In both cases (a) and (b) it describes the frequency of particle production events: events per Hubble time.

The behavior of these -spectra, and their ratios, is somewhat different in our two cases (a) and (b). Case (a) will prove to overlap strongly with existing templates for , whereas case (b) has additional resonances in the time integrals as a result of the oscillating mass and has small overlap with existing templates for . So in much of this work we will focus on the behavior of case (b). But let us evaluate them in turn.

Estimates for the integrals in case (a)

In this case, from (2.2) we have approximately a step function source since

(3.45)

for , equivalently . Thus the integral we need is

(3.46)

The sum in (3.39) and (3.42) is dominated at the horizon crossing time because the summand becomes small if for any of the momenta is much different from 1. When the Green’s function is suppressed as . When we have

(3.47)

This justifies glossing over the short scale details of the production event and approximating the source by a step function. Incorporating this, we estimate the relevant ratio for in the -point function (3.42) as

(3.48)

where

(3.49)

is the number of events per Hubble time and Hubble volume.

Estimates for the integrals for case (b)

We would like to estimate the dominant contributions to the integral over :

(3.50)

From (2.3) we have

(3.51)

For sufficiently small , this reduces to the simpler form , giving the integral

(3.52)

with coefficient

(3.53)

from (2.3).

We will present our final analysis of the parameter windows for our template based on the approximation (3.52) below in §4. We will find there that for the lower end of frequencies of interest, the ratio is not hierarchically suppressed. We should therefore either include the full form (3.62) (along with an extra parameter varying over a small range of values), or determine that the overlap between the two templates is strong enough to justify the simplification to the pure sinusoidal function. To begin, we will analyze the pure sinusoidal form in case (b), and then return to this point. By calculating the Fourier coefficients of the full expression, we find that even for , the simpler expression gives very similar results for the leading Fourier mode contributing to the resonant integral.

Let us now estimate the size of this effect by approximating the integral over and sum over . First, note that the Green’s function (3.31) is of order as , suppressing any contributions outside the horizon. For , the second term in dominates over the first. The dominant contribution to the integral in (3.38) is easily estimated by a stationary phase approximation, taking into account the two sources of oscillation in the integrand in case (b). That is, for the integrand has two oscillating functions: and , which resonate at . Explicitly,

(3.54)

A saddle point integration with leads to

(3.55)

valid for (so that the saddle point is well separated from the endpoint of the integral). The leading contributions to the sum in both numerator and denominator of (3.43) then come from the smallest value of which is consistent with picking up this saddle, i.e. . From (3.44) we note that of order terms in the sum over contribute with approximately the same value of (there are events within a Hubble time ). Altogether this leads to a ratio (3.43) of order

(3.56)

This behavior can be checked numerically. This result also controls at tree level, within a finite range of for which the cosmic variance limited Gaussian approximation applies. We will find in §4 that (3.56) can be somewhat larger than 1 consistently with our conditions for control of the model.

Let us also separately record the amplitude of the correction to the power spectrum and bispectrum. For the power spectrum, from (3.39) we obtain