A Notation and Conventions

UTTG-08-09

TCC-24-09

SU-ITP-09-34

Oscillations in the CMB

[.3cm] from Axion Monodromy Inflation

Raphael Flauger, Liam McAllister, Enrico Pajer, Alexander Westphal, and Gang Xu

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

Department of Physics, Cornell University, Ithaca, NY 14853

Department of Physics, Stanford University, Stanford, CA 94305

We study the CMB observables in axion monodromy inflation. These well-motivated scenarios for inflation in string theory have monomial potentials over super-Planckian field ranges, with superimposed sinusoidal modulations from instanton effects. Such periodic modulations of the potential can drive resonant enhancements of the correlation functions of cosmological perturbations, with characteristic modulations of the amplitude as a function of wavenumber. We give an analytical result for the scalar power spectrum in this class of models, and we determine the limits that present data places on the amplitude and frequency of modulations. Then, incorporating an improved understanding of the realization of axion monodromy inflation in string theory, we perform a careful study of microphysical constraints in this scenario. We find that detectable modulations of the scalar power spectrum are commonplace in well-controlled examples, while resonant contributions to the bispectrum are undetectable in some classes of examples and detectable in others. We conclude that resonant contributions to the spectrum and bispectrum are a characteristic signature of axion monodromy inflation that, in favorable cases, could be detected in near-future experiments.

## 1 Introduction

Inflation [1, 2, 3] is a successful paradigm for describing the early universe, but it is sensitive to the physics of the ultraviolet completion of gravity. This motivates pursuing realizations of inflation in string theory, a candidate theory of quantum gravity. Considerable progress has been made on this problem in recent years, so much so that the most pressing task, particularly in view of upcoming CMB experiments, is to learn how to distinguish various incarnations of inflation in string theory from each other and from related models constructed directly in quantum field theory.

Fortunately, the additional constraints inherent in realizing inflation in an ultraviolet-complete framework can leave imprints in the low-energy Lagrangian, and hence ultimately in the cosmological observables. In favorable cases, a given class of models may make distinctive predictions for a variety of correlated observables, allowing one to exclude this class of models given adequate data.

One decisive observable for probing inflation is the tensor-to-scalar ratio, . A promising class of string inflation models producing a detectable tensor signature are those involving monodromy [4], in which the potential energy is not periodic under transport around an angular direction in the configuration space. The first examples [4] involved monodromy under transport of a wrapped D-brane in a nilmanifold, and a subsequent class of examples invoked monodromy in the direction of a closed string axion [5].

The axion monodromy inflation scenario of [5] is falsifiable on the basis of its tensor signature, . However, primordial tensor perturbations have not been detected at present, while the temperature anisotropies arising from scalar perturbations have been mapped in great detail [6]. One could therefore hope to constrain axion monodromy inflation more effectively by understanding the signatures that it produces in the scalar power spectrum and bispectrum. Characterizing these signatures is the subject of the present paper.

As we shall explain, the potential in axion monodromy inflation is approximately linear, but periodically modulated: each circuit of the loop in configuration space can provide a bump on top of the otherwise linear potential. Modulations of the inflaton potential with suitable frequency and amplitude can yield two striking signatures: periodic undulations in the spectrum of the scalar perturbations, and resonant enhancement [7] of the bispectrum. Let us stress that the presence of some degree of modulations of the potential is automatic, and is an example of the situation described above in which traces of ultraviolet physics remain in the low-energy Lagrangian. We do not introduce modulations in order to make the scalar perturbations more interesting. However, it is important to examine the typical amplitude and frequency of modulations in models that are under good microphysical control, in order to ascertain whether well-motivated models produce signatures that can be detected in practice.

To achieve this, we first investigate in detail the realization of axion monodromy inflation in string theory. We compute the axion decay constants in terms of compactification data, we assess the importance of higher-derivative terms, and we estimate the amplitude of modulations for the case of Euclidean D1-brane contributions to the Kähler potential. We also identify a potentially-important contribution to the inflaton potential, arising from backreaction in the compact space, and we present a model-building solution that suppresses this contribution.

We find that detectable modulations of the scalar power spectrum and bispectrum are possible in models that are consistent with all current data and that are under good microphysical control. In fact, we find substantial parameter ranges that are excluded not by microphysics, but by observational constraints on modulations of the scalar power spectrum.

The organization of this paper is as follows. We begin in §2 by describing the classical evolution of the homogeneous background in axion monodromy inflation with a modulated linear potential. We then solve, in §3, the Mukhanov-Sasaki equation governing the evolution of scalar perturbations, giving an analytical result for the spectrum in terms of the frequency and amplitude of the modulations of the potential. Next, we briefly discuss the bispectrum and express the amplitude of the non-Gaussianity in terms of the model parameters. We then present, in §4, an analysis of the constraints imposed on axion monodromy inflation by the WMAP5 data (for prior work constraining similar oscillatory power spectra, see e.g. [8, 9, 10, 11, 12, 13, 14]). Then, in §5 and §6, we present a comprehensive analysis of the constraints imposed by the requirements of computability and of microphysical consistency, including validity of the string loop and perturbation expansions, successful moduli stabilization, and bounds on higher-derivative terms. In §7 we combine the observational and theoretical constraints, with results presented in figure 7.

### 1.1 Review of axion monodromy inflation

In this section we will briefly review the motivation for axion monodromy inflation, as well as the most salient phenomenological features. We will postpone until §5 a more comprehensive discussion of the realization of this model in string theory.

Inflation is sensitive to Planck-scale physics: contributions to the effective action arising from integrating out degrees of freedom with masses as large as the Planck scale play a critical role in determining the background evolution, and hence the observable spectrum of perturbations (see [15] for a review of this issue). A central problem in inflationary model-building is establishing knowledge of Planck-suppressed terms in the effective action with accuracy sufficient for making predictions. The most elegant solution to this problem is to provide a symmetry that forbids such Planck-suppressed contributions. Because invoking such a symmetry amounts to forbidding couplings of the inflaton to Planck-scale degrees of freedom, it is important to understand this issue in an ultraviolet-complete theory, such as string theory.

One promising mechanism for inflation in string theory involves the shift symmetry of an axion. Axions are numerous in string compactifications and generally enjoy continuous shift symmetries that are valid to all orders in perturbation theory, but are broken by nonperturbative effects to discrete shifts . As noted in [5], the shift symmetries of axions descending from two-forms are also broken by suitable space-filling fivebranes (D5-branes or NS5-branes) wrapping two-cycles in the compact space.

In axion monodromy inflation [5], an NS5-brane wrapped on a two-cycle breaks the shift symmetry of the Ramond-Ramond two-form potential , inducing a potential that is asymptotically linear in the corresponding canonically normalized field ,

 V=μ3ϕ, (1.1)

with a constant mass scale. Inflation begins with a large expectation value for the inflaton, , and proceeds as this expectation value diminishes; note that the NS5-brane, like any D-branes that may be present in the compactification, remains fixed in place during inflation. As argued in [5], this gives rise to a natural model of inflation, with the residual shift symmetry of the axion protecting the potential from problematic corrections that are endemic in string inflation scenarios.

In this paper we perform a careful analysis of the consequences of nonperturbative effects for the axion monodromy scenario. Such effects are generically present: specifically, Euclidean D-branes make periodic contributions to the potential in most realizations of axion monodromy inflation. However, the size of these contributions is model-dependent. It was shown in [5] that there exist classes of examples in which nonperturbative effects are practically negligible, but we expect – as explained in detail in §6.5 – that in generic configurations, periodic terms in the potential make small, but not necessarily negligible, contributions to the slow roll parameters.

Therefore, it is of interest to understand the consequences of small periodic modulations of the inflaton potential in axion monodromy inflation. In this paper we address this question in two ways: first, in §24, by studying a phenomenological potential that captures the essential effects; and second, in §5 and §6, by investigating the ranges of the phenomenological parameters that satisfy all known microphysical consistency requirements dictated by the structure of string compactifications in which axion monodromy inflation can be realized.

## 2 Background Evolution

In this section we will study the background evolution of the inflaton in the presence of small periodic modulations of the potential. We will focus on modulations in axion monodromy inflation with a linear potential, but our derivations are easily modified to account for other models with a modulated potential. We will denote the size of the modulation by , and write our potential as in [5],

 V(ϕ)=μ3ϕ+Λ4cos(ϕf)=μ3[ϕ+bfcos(ϕf)], (2.1)

where we defined the parameter . The equation of motion for the inflaton is then

 ¨ϕ+3H˙ϕ+μ3−μ3bsin(ϕf)=0. (2.2)

To solve (2.2), we begin with two approximations. Monotonicity of the potential requires1 , and as we will see in §4, for the case observational constraints in fact imply . This suggests treating the oscillatory term in the potential as a perturbation. Furthermore, the COBE normalization implies that during the era when the modes that are observable in the cosmic microwave background exit the horizon. This allows us to drop terms of higher order in .

Under these conditions, it is straightforward to solve for the evolution of the homogeneous background. Expanding the field as , the equations of motion of zeroth and first order in become

 ˙ϕ0=−√μ33ϕ0, (2.3)
 ¨ϕ1+√3μ3ϕ0˙ϕ1−μ32ϕ0ϕ1=μ3sin(ϕ0f), (2.4)

where we have neglected terms of higher order in and we have made use of the slow roll approximation for .2 Using equation (2.3), we can rewrite equation (2.4) with as an independent variable instead of , yielding

 ϕ′′1−3ϕ0ϕ′1−32ϕ1=3ϕ0sin(ϕ0f). (2.5)

where primes denote derivatives with respect to . For the period of interest, in which the modes now visible in the CMB exit the horizon, it is a good approximation to neglect the motion of everywhere except in the driving term. The inhomogeneous solution is then given by

 Missing or unrecognized delimiter for \left (2.6)

where denotes the value of the field at the time at which the pivot scale exits the horizon. Assuming 60 e-foldings of inflation, this happens around . For decay constants obeying , there is less than one oscillation in the range of modes that are observable in the cosmic microwave background, leading to an uninteresting modulation with very long wavelength. We will thus make the additional assumption that . Assuming that and , using the slow roll approximation for , and working to first order in , the solution thus becomes

 ϕ(t)=ϕ0(t)+bϕ1(t)=ϕ0(t)+bf3fϕ∗1+(3fϕ∗)2[−sin(ϕ0(t)f)+3fϕ∗cos(ϕ0(t)f)], (2.7)

with given by

 ϕ0(t)=[ϕ3/2∗−√32μ3/2(t−t∗)]2/3. (2.8)

In the absence of oscillations, i.e. for , axion monodromy provides a model of large field inflation that is easily studied using the slow roll expansion. Assuming for concreteness that the CMB scales left the horizon 60 e-foldings before the end of inflation, we are interested in the perturbations around . After imposing the COBE normalization, one finds that CMB perturbations are produced at a scale GeV with a spectral tilt and a tensor-to-scalar ratio . For reference, the Hubble constant during inflation is then GeV.

One can then ask what happens once the oscillations are switched on, i.e. when . It turns out that the effect on the number of e-foldings is negligible as long as . Hence the inflationary scale is well-approximated by the slow roll analysis. On the other hand, the detailed properties of the perturbations are very different from the slow roll case and cannot be calculated in that expansion. We turn to this issue in the next section.

## 3 Spectrum of Scalar Perturbations

Having understood the background evolution, we are now in a position to calculate the power spectrum in axion monodromy inflation. One might be tempted to do this by brute-force numerical calculation, but we find it more instructive to have an analytic result. We will show that under the same assumptions made in calculating the background evolution, i.e. slow roll for , , , and to first order in , the scalar power spectrum is of the form

 Δ2R(k)=Δ2R(k∗)(kk∗)ns−1[1+δnscos(ϕkf)]≈Δ2R(kk∗)ns−1+δnsln(k/k∗)cos(ϕkf), (3.1)

where the quantity parameterizes the strength of the scalar perturbations and will be introduced in detail in the next subsection. The second equality is valid as long as , and is given by

 δns=12b√(1+(3fϕ∗)2)√π8coth(π2fϕ∗)fϕ∗, (3.2)

where

 ϕk=√ϕ2∗−2lnk/k∗≃ϕ∗−lnk/k∗ϕ∗ (3.3)

is the value of the scalar field at the time when the mode with comoving momentum exits the horizon.

In §3.1 we will give a derivation of this result that makes no further approximations. In §3.2 and §3.3 we will present two additional derivations of (3.1) that are valid only as long as but that lead to a better understanding of the relevant physical effects behind the power spectrum (3.1). Let us at this point briefly summarize the scales that will be relevant for our discussion in the next subsections.

Given the potential (2.1), the time frequency of the oscillations of the inflaton is . This is also the time frequency of the oscillations of the background. Perturbations around this background can be quantized in terms of the solutions of the Mukhanov-Sasaki equation, assuming an asymptotic Bunch-Davies vacuum. Every perturbation mode with comoving momentum oscillates with a time frequency that is redshifted by the expansion of the universe until the mode exits the horizon and freezes when .

Then, if , every mode will at a certain time resonate with the background, as stressed by Chen, Easther, and Lim in [7]. Using the slow roll equation of motion and the COBE normalization,

 3H˙ϕ≃−V′(ϕ),˙ϕ2≃23ϵV,V≃5⋅10−7ϵM4p, (3.4)

the requirement can be re-expressed as

 ωH ≃ M2pϕf≃√2ϵMpf>1, (3.5) ωMp ≃ √2ϵV31fMpl<1, (3.6)

hence defining a range of values for the axion decay constant for which resonances occur. Using , we obtain . We will show in §5 and §6 that falls in this range in a class of microphysically well-controlled examples.

Going beyond our approximations, the model also predicts a small amount of running of the scalar spectral index, of order , from terms of higher order in the expansion. Furthermore, develops a very mild momentum dependence. We will neglect these effects because these will most likely not be observable in current or near-future CMB experiments.

### 3.1 Analytic solution of the Mukhanov-Sasaki equation

We begin our study of the spectrum by choosing a gauge such that the scalar field is unperturbed, , and the scalar perturbations in the spatial part of the metric take the form

 δgij(x,t)=2a(t)2R(x,t)δij. (3.7)

The quantity is a gauge-invariant quantity and in the case of single-field inflation is conserved outside the horizon. It is closely related to the scalar curvature of the spatial slices, but we will not need its precise geometric interpretation at this point.

The translational invariance of the background and thus the equations of motion governing the time evolution of the perturbations make it convenient to look for solutions of the linearized Einstein equations in Fourier space. One defines

 R(x,t)=∫d3k(2π)3/2[Rk(t)eik⋅xα(k)+Rk(t)∗e−ik⋅xα∗(k)], (3.8)

where is the comoving momentum, and is its magnitude. The rotational invariance of the background ensures that can depend only on the magnitude of the comoving momentum but not on its direction. Directional dependence can only be contained in the stochastic parameter that parameterizes the initial conditions and is normalized so that

 Missing or unrecognized delimiter for \right (3.9)

where the average denotes the average over all possible histories. With this ansatz, the Einstein equations turn into an ordinary differential equation, the Mukhanov-Sasaki equation, governing the time evolution of . We will use it in the form3

 d2Rkdx2−2(1+2ϵ+δ)xdRkdx+Rk=0, (3.10)

where , with the conformal time given as usual by . Outside the horizon, i.e. for or equivalently , the quantity approaches a constant which we denote by . In terms of we define the primordial power spectrum for the scalar modes as

 ∣∣R(o)k∣∣2=2π2Δ2R(k)k3. (3.11)

To evaluate this quantity, it will again turn out to be sufficient to solve to first order in . We therefore expand the slow roll parameters,

 ϵ=ϵ0+ϵ1+O(b2), (3.12)
 δ=δ0+δ1+O(b2). (3.13)

For the background solution (2.7), the first-order terms are given by

 Missing or unrecognized delimiter for \right (3.14)
 δ1=−3b[1+(3fϕ∗)2][sin(ϕ0f)−(3fϕ∗)cos(ϕ0f)]. (3.15)

We now consider an ansatz of the form

 Rk=R(o)k,0[i√π2xν0H(1)ν0(x)+g(x)]. (3.16)

Here the index on the Hankel function, , is given by , is a perturbation of order , and is the value of outside the horizon in the absence of modulations, i.e. for . To be explicit, it is given by4

 R(o)k,0=∓i√μ3ϕ3k61k3/2, (3.17)

where once again is the value of the scalar field at the time the mode with comoving momentum exits the horizon. The quantity of interest to first order in is then

 ∣∣R(o)k∣∣2=∣∣R(o)k,0∣∣2[1+2Reg(0)]≈∣∣R(o)k,0∣∣2e2Reg(0)=∣∣R(o)k,0∣∣2(kk∗)2Reg(0)ln(k/k∗). (3.18)

Our ansatz automatically solves the equation of order . To first order in and in the slow roll parameters, the Mukhanov-Sasaki equation leads to an equation for of the form

 d2gdx2−2xdgdx+g=2eix(2ϵ1+δ1). (3.19)

In writing this equation, we have dropped terms of order , which amounts to setting . Next, we notice that is suppressed relative to by a factor . Since we are interested in the regime , we can thus drop the term proportional to on the right hand side of equation (3.19). Furthermore, it turns out to be convenient to rewrite using trigonometric identities. Ignoring an unimportant phase, one finds

 δ1=−3b√1+(3fϕ∗)2cos(ϕ0f). (3.20)

It will be convenient to write as . Introducing , equation (3.19) becomes

 d2rdx2−2xdrdx+r=−6b√1+(3fϕ∗)2cos(x)cos(ϕkf+lnxfϕ∗). (3.21)

The solution to this equation can be found e.g. using Green’s functions. We are particularly interested in the inhomogeneous solution at late times, i.e. in the limit of vanishing . Using more trigonometric identities, we find that the solution in this limit can be brought into the form

 r(0)=6b|I(fϕ∗)|√1+(3fϕ∗)2cos(ϕkf+β(fϕ∗)), (3.22)

where is an unimportant phase that we will ignore, and is the integral

 I(fϕ∗)=π2∫∞0dxJ32(x)J−12(x)xifϕ∗. (3.23)

Written in this form, the integral can be recognized as a Weber-Schafheitlin integral and can be done analytically (see e.g. [20]). One finds

 |I|=√π8coth(π2fϕ∗)fϕ∗. (3.24)

Combining equations (3.18), (3.22) and (3.24), we finally obtain an expression for ,

 δns=2r(0)cos(ϕkf)=12b√1+(3fϕ∗)2√π8coth(π2fϕ∗)fϕ∗. (3.25)

Once again, this derivation is valid to first order in and assumes slow roll for , , and . In particular, it makes no use of an expansion, although this approximation will be needed in the derivations in §3.2 and §3.3. A comparison between our analytical result for as a function of for a fixed value of and the result of a numerical calculation using a slight modification of the code described in [21] is shown in Figure 1.

### 3.2 Saddle-point approximation

As we have seen in the last subsection, it is possible to calculate the power spectrum analytically to first order in , assuming slow roll for , , and , but the derivation sheds little light on the physics behind the results. To get a better understanding, it is instructive to look at the integral (3.23) more explicitly. For this purpose, it is convenient to separate into its real and imaginary parts, , with

 Ic = ∫∞0dx(sinx−xcosx)cosxx2cos(lnxfϕ∗), (3.26) Is = ∫∞0dx(sinx−xcosx)cosxx2sin(lnxfϕ∗). (3.27)

For ranges of the axion decay constant such that , these integrals can be done in a stationary phase approximation. Using trigonometric identities to rewrite the products of trigonometric functions appearing in the integrands into sums of trigonometric functions with combined arguments, one finds that the stationary phase occurs at . Expanding around the stationary point and performing the integral as usual, one finds to leading order in

 Ic = √π8fϕ∗sin[1+ln(2fϕ∗)fϕ∗−π4], (3.28) Is = √π8fϕ∗cos[1+ln(2fϕ∗)fϕ∗−π4], (3.29)

which leads to

 |I|=√Ic2+Is2=√π8fϕ∗. (3.30)

This agrees with our previous result, equation (3.24), as long as . We have not only reproduced our earlier results, however: we also learn that at least for small , the integral is dominated by a period of time around . Up to the factor of two in the denominator, this corresponds to the period when the frequency of the oscillations of the scalar field background equals the frequency of the oscillations of a mode with comoving momentum .5 The stationary phase approximation thus captures a resonance between the oscillations of the background and the oscillations of the fluctuations, and is good as long as , i.e. as long as the resonance occurs while the mode is still well inside the horizon. One might suspect that this has an interpretation in terms of particle production, and we shall make this more precise in what follows.

Recall that our ansatz for was given in (3.16), where is the solution of the equation

 d2gdx2−2xdgdx+g=2eixδ1, (3.31)

with again given by

 δ1=−3b√1+(3fϕ∗)2cos(ϕkf+lnxfϕ∗), (3.32)

and initial conditions given by and . As we have just learned, the effect of the driving term can be ignored long after the resonance has occurred, i.e. for .6 This implies that at late times, must be a solution of the homogeneous equation which can be written as

 g(x)=c(+)k(i√π2x32H(1)3/2(x))+c(−)k(−i√π2x32H(2)3/2(x)), (3.33)

where are momentum dependent coefficients. The solution for equation (3.31) can also be written explicitly as

 g(x) = (xcosx−sinx)∞∫x2eiy(cosy+ysiny)y2δ1 (3.34) +(cosx+xsinx)∞∫x2eiy(siny−ycosy)y2δ1.

For we can take the lower limit in the integrals to zero and this can be brought into the form

 g(x)=12(I2+iI1)(i√π2x32H(1)3/2(x))+12(I2−iI1)(−i√π2x32H(2)3/2(x)), (3.35)

where the integrals and are given by

 I1 = −6b√1+(3fϕ∗)2∞∫0eiy(cosy+ysiny)y2cos(ϕkf+lnxfϕ∗), (3.36) I2 = −6b√1+(3fϕ∗)2∞∫0eiy(siny−ycosy)y2cos(ϕkf+lnxfϕ∗). (3.37)

In the saddle point approximation these evaluate to

 I1=iI2=−6b√(1+(3fϕ∗)2)√π8fϕ∗e−i(ϕkf−1+ln2fϕ∗fϕ∗+π4). (3.38)

Combining equations (3.16), (3.35), and (3.38), we finally find that the curvature perturbation for takes the form

 Rk=R(o)k,0(i√π2xν0H(1)ν0(x)−c(−)ki√π2xν0H(2)ν0(x)), (3.39)

with given, up to an unimportant momentum-independent overall phase, by

 c(−)k=6b√(1+(3fϕ∗)2)√π8fϕ∗e−i(ϕkf). (3.40)

One might now interpret the coefficient of the negative frequency mode as a Bogoliubov coefficient that measures the amount of particles with comoving momentum being produced while this mode is in resonance with the background. It seems hard to make this precise as one really is comparing mode solutions of different backgrounds rather than mode solutions of different asymptotically Minkowski regions in the same background.

Equation (3.39) also shows that instead of starting in the Bunch-Davies state and then following the mode through the resonance, one may start the evolution after the resonance has occurred but use a state that is different from the Bunch-Davies state, which is similar to what is considered in [22, 23, 8, 9, 10]. The departure from the Bunch-Davies state is of course quantified by .

### 3.3 Particle production and deviations from the Bunch-Davies state

Here we will deal with a conceptual question that generically arises in inflationary models with oscillations in the scalar potential. Driven by the background motion of the inflaton, the oscillating contributions constitute a time-oscillating perturbation to the Hamiltonian of the system. Now, perturbations oscillating in time will generically induce transitions, in our case from the original vacuum state to some excited states. This implies that the vacuum state of the full system will deviate from the Bunch-Davies vacuum of the homogeneous background inflationary evolution. We will now estimate the resulting quantity of particle production and relate the result to the derivation of the scalar power spectrum given in the preceding sections.

To lowest order the oscillating perturbation is given by

 ΔH(2)=12V′′(ϕ0(t))⋅δϕ2, (3.41)

implying that the lowest-order transitions will be from the vacuum to two-particle states . As the physical momentum corresponding to a given comoving momentum is exponentially decaying in the inflationary regime, any two-particle state with given comoving momentum will be in resonance with the oscillating perturbation only for a short period of time which we will have to estimate in due course.

In transforming the Hamiltonian of the fluctuations into Fourier space

 H[δϕ→p]=12δ˙ϕ2→p+12→p2δϕ2→p+12V′′(ϕ0(t))⋅δϕ2→p, (3.42)

we find that the system takes the form of a perturbed harmonic oscillator with eigenfrequency for each momentum mode separately,

 H[δϕp]=12δ˙ϕ2p+12ω2pδϕ2p+12V′′(ϕ0(t))⋅δϕ2pΔH(2)[δϕp]. (3.43)

Now we compare this to the perturbed harmonic oscillator in one-dimensional quantum mechanics,

 H=12˙x2+12ω20x2+12δω(t)2x2. (3.44)

In going to dimensionless variables we can write this as

 H=12ω0(p2+q2)+δω(t)22ω0q2ΔH(2), (3.45)

where for our case of a periodic perturbation periodic with frequency we have

 δω(t)2=δω2cos(ωt). (3.46)

We want to determine the time-dependent transition matrix element in time-dependent perturbation theory for a periodic perturbation. To do so, we first write the perturbation in standard form for time-dependent perturbation theory as

 ΔH(2)(t)=δω(t)22ω0q2=δω24ω0q2(eiωt+e−iωt)≡F(eiωt+e−iωt), (3.47)

in the notation of equations (40.1) through (40.9) of [24]. The Hamiltonian and the transition matrix elements can be written in terms of creation and annihilation operators and using and . Then, canonical quantization of the unperturbed part yields a discrete spectrum of eigenstates with energy spectrum .

If we compare this with our actual case above, we see that for each momentum mode , and are replaced by appropriate dimensionless fields and . In complete analogy to the simple quantum mechanical oscillator, there will be a tower of discrete states with energies . In particular, labels the two-particle state which has energy difference with respect to the ground state. We thus have for the perturbation in our actual case

 ΔH(2)(t)=δω(t)22ωpδφ2→p=δω24ωpδφ2→p(eiωt+e−iωt)≡F(eiωt+e−iωt). (3.48)

For the transition matrix element one then finds

 ⟨p,−p|ΔH(2)|0⟩=F20(eiωt+e−iωt)withF20=δω24ωp⟨0|a2√2(a+a†√2)2|0⟩=δω24√2ωp. (3.49)

Here we have used that

 |p,−p⟩=(a†)2√2|0⟩, (3.50)

and

 ⟨0|a2(a+a†)2|0⟩=⟨0|a2(a†)2|0⟩=2. (3.51)

If the energy of the two-particle state were not too close to the perturbation frequency , we could use time-dependent perturbation theory with the above matrix element and obtain the first order transition probability ,

 P0→2k = ∣∣∣−i∫tdt′⟨k,−k|ΔH(2)(t′)|0⟩eiω20t′∣∣∣2 (3.52) = 2|F20|2(2ka)2+ω2+[(2ka)2−ω2]cos(2ωt)[(2ka)2−ω2]2 = δω416(k/a)2(2ka)2+ω2+[(2ka)2−ω2]cos(2ωt)[(2ka)2−ω2]2,

where . This gives the resonance line feature characteristic of transition processes.

However, as for any given the physical momentum and frequency will decrease extremely rapidly with , we can approximate the amount of transition happening in the short time interval during which the two-particle state of given is in near-resonance . Close to resonance, time-dependent perturbation theory breaks down (visible in the singularity of the above result for ); however, for periodic perturbations one can solve the Schrödinger equation of the coupled two-state system exactly [24]. One finds that on resonance the transition probability is

 P0→2k=12[1−cos(2Ωt)]=12[1−cos(δω22√2k/at)],whereΩ≡F20. (3.53)

That is, near resonance the system effectively oscillates with frequency between the vacuum and the two-particle state.

We now have to estimate the time during which a two-particle state of comoving momentum stays in near-resonance. We will follow the analysis in [7] and look at the interference terms induced between the perturbation and the periodicity of the interaction matrix element in (3.52). We note that, on the one hand, the two-particle state with frequency stays in resonance with the perturbation with frequency only for a time roughly estimated to be (for the relative phase shifting from to )

 Δt1∼2πΔω. (3.54)

On the other hand, in the inflating universe it takes very roughly a time

 Δt2∼2ΔωωH (3.55)

to change the frequency of the two-particle state from, say, to . Equating the two provides us with the effective duration of near-resonance,

 Δtres≡Δt1=Δt2∼2√πHωH−1. (3.56)

Plugging this into the above transition result and remembering that near resonance , we get

 P0→2k≃12[1−cos(√2δω2ω√πHωH−1)]. (3.57)

Now, in our case above we see that in terms of comoving momenta , and further

 δω(t)2=V′′(ϕ0(t))=Λ4f2cos(ϕ0(t)f)=δω2cos(ωt),ω=Hfϕ0. (3.58)

Noting that in our scenario of interest we have and that , we can expand the argument of the cosine around zero. If we then plug in the microscopic definitions of the quantities and , we get

 P0→2k≃π2Λ8f4ω2H2⋅Hω=π2Λ8fϕ3∗f2H4=π29Λ8fϕ3∗f2μ6ϕ2∗=36π8b2fϕ∗, (3.59)

where denotes the vev of the inflaton field around 60 e-foldings before the end of inflation.

Next, because characterizes the transition probability to the two-particle states, it may be related to the negative frequency Bogoliubov coefficient that relates the out-vacuum to the in-vacuum. Specifically, the out-vacuum is specified by the modes

 uk(out)=c(−)u−k+c(+)uk, (3.60)

whereas the original Bunch-Davies in-vacuum had modes

 uk(in)=uk,c(+)(in)=1. (3.61)

We therefore find that

 |c(−)|≃√P0→2k=6b√π8fϕ∗. (3.62)

In comparing these results with the general treatment of the Mukhanov-Sasaki equation above, we see by looking at (3.39) and (3.18) that we can identify

 uk = i√π2xν0H(1)ν0(x) (3.63) u−k = −i√π2xν0H(2)ν0(x) (3.64)

and thus from (3.18) we conclude that

 δns=2Reg(x)cos(ϕkf)=x→02Re(c(+)c(−))cos(ϕkf)≃2|c(−)|≃12b√π8fϕ∗ (3.65)

which agrees with the general result (3.25) in the appropriate limit and , corresponding to , where .

Note that in calculating the transition probability we lose information about the phase of the transition matrix element as given in (3.49). Therefore, if we estimate the population coefficient from