Contents

The charged inflaton and its gauge fields:

Preheating and initial conditions for reheating

Kaloian D. Lozanov and Mustafa A. Amin

Physics & Astronomy Department, Rice University, 6100 Main Street, Houston, U.S.A.

Abstract
We calculate particle production during inflation and in the early stages of reheating after inflation in models with a charged scalar field coupled to Abelian and non-Abelian gauge fields. A detailed analysis of the power spectra of primordial electric fields, magnetic fields and charge fluctuations at the end of inflation and preheating is provided. We carefully account for the Gauss constraints during inflation and preheating, and clarify the role of the longitudinal components of the electric field. We calculate the timescale for the back-reaction of the produced gauge fields on the inflaton condensate, marking the onset of non-linear evolution of the fields. We provide a prescription for initial conditions for lattice simulations necessary to capture the subsequent nonlinear dynamics. On the observational side, we find that the primordial magnetic fields generated are too small to explain the origin of magnetic fields on galactic scales and the charge fluctuations are well within observational bounds for the models considered in this paper.

## 1 Introduction

Particle production during inflation [1, 2, 3, 4] and reheating [5, 6, 7, 8, 9, 10, 11, 12] sets up the initial conditions for the formation of observed structure and the beginning of the hot big bang. Particle production in models with gauge fields is particularly interesting because of the ubiquity of gauge fields in the Standard Model (SM) and their natural appearance in extensions beyond the SM. Gauge fields coupled to scalar fields can have important consequences for the generation of curvature [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] and charge perturbations [40, 35, 41, 42, 43, 44, 45], gravitational waves [30, 32, 46, 47, 48, 49, 50, 51, 52, 53, 54] as well as seeding primordial magnetic fields [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 41, 68, 69] during inflation. Such gauge fields can significantly affect the transition to a radiation dominated universe after inflation [70, 71] and can lead to novel non-perturbative phenomena [72, 73, 74, 75, 76, 77, 78, 79].

An analysis of particle production with gauge fields in the early universe has been undertaken in many previous studies. For example, gauge field production during inflation and its consequences was reviewed in [80]. Non-perturbative gauge field production during and after inflation was explored in [81, 82, 83, 35, 84, 85, 86, 87, 88, 89, 90], whereas nonlinear dynamics of gauge fields after inflation was considered in (for e.g.) [91, 92, 70, 71, 93, 94, 77] for Abelian fields and [72, 74, 76, 73, 75, 78] for non-Abelian ones.

In this paper we re-visit particle production in locally gauge invariant models with Abelian and non-Abelian fields coupled to charged scalar fields. In these models we assume that a component of the charged scalar field plays the role of the inflaton condensate. Care has to be taken with gauge fields because they have to satisfy certain constraint equations (along with the usual evolution equations). The natural gauge redundancy can lead to complications in quantization or to spurious gauge modes during numerical evolution. The non-zero vacuum expectation value (vev) of the inflaton condesate during inflation and reheating changes some of the common results for gauge fields coupled to scalar fields with zero vevs. Moreover, certain common gauge choices become ill-defined when the inflaton condensate starts oscillating at the end of inflation. Finally, if significant particle production occurs, back-reaction becomes important and classical lattice simulations are needed to fully explore the nonlinear dynamics of the fields. Initial conditions for such lattice simulations can be nontrivial because of the constraints on the different variables that must be satisfied. We pay special attention to all of these issues in this work. While we restrict ourselves to “minimal” models consistent with local gauge invariance, our techniques and results should carry over to more complicated scenarios such as [85, 86].

We calculate particle production during inflation using gauge invariant variables and with proper accounting for the constraints (Section 3). The use of gauge invariant variables naturally avoids issues with spurious gauge degrees of freedom and makes the quantization and subsequent evolution of perturbations particularly transparent. We provide power spectra for the electric and magnetic fields at the end of inflation, along with simple analytic estimate for their shape. In Appendix A we explain their shape via approximate analytic calculations.

While useful during inflation, gauge invariant variables become ill-defined when the inflaton starts oscillating. We argue for the use of well defined Coulumb gauge variables for analysing non-perturbative particle production during preheating at the end of inflation. In Section 4, we carry out a Floquet like analysis for the resonant production of the gauge fields. Here, we point out some minor discrepancies in the literature regarding the productions of the gauge invariant longitudinal component of the electric field. We then estimate the end of preheating by calculating the time when back-reaction of the resonantly produced gauge fields becomes important (Section 4.2). In Appendix B we provide technical details for quantizing and calculating the back-reaction in the Coulomb gauge.

Once nonlinear effects become important, simulations become essential. Nonlinear simulations with gauge fields (especially non-Abelian fields) can be challenging because of the large number of components and the necessity of satisfying constraint equations. In Section 5 of this work we provide a simple prescription for setting up initial conditions for such lattice simulations which can be applied in any gauge. In our prescription, the lattice initial conditions naturally satisfy the necessary gauge constraints. We note that our initial conditions accurately account for metric perturbations, field interactions and gauge constraints to linear order. We provide an example of lattice initial conditions in temporal gauge which is a common choice for simulations. We arrive at these initial conditions via gauge invariant variables; this serves as a model to set up initial conditions in any gauge. We will carry out actual lattice simulations in future work.

We have tried to make the paper self-contained, so that it can be used for future reference easily. To this end, we provide the necessary equations for the perturbations of metric and matter fields (scalar and gauge fields) in position and Fourier space for gauge invariant variables. While we work with gauge invariant variables as far as possible, we also provide a dictionary to translate our results to other popular gauges.

For pedagogical purposes, we carry out the analysis for Abelian fields first (Section 2). We show in the later half of the paper (Section 6) how the non-Abelian analysis can be reduced to an analysis of multiple copies of the Abelian case in the linear regime. Hence, the analysis for the Abelian case, including the setting up of the lattice initial conditions, can be easily carried over to the non-Abelian case. We show how to apply the developed techniques to a model and its extension: a model.

We discuss the observational consequences of a charged inflaton and its gauge fields in Section 7. We summarize our results in the Conclusions section and discuss future directions.

## 2 The Abelian model

We consider an action with matter minimally coupled to gravity

 S =−m2\tiny{Pl}2∫d4x√−gR+Sm, (1)

where is the Ricci scalar, is the determinant of the metric and is the reduced Planck mass. The matter action contains a complex scalar field and the gauge field :

 Sm=∫d4x√−gLm=∫d4x√−g[(Dμφ)∗(Dμφ)−V(|φ|)−14FμνFμν], (2)

where the field tensor for the gauge fields and the gauge-covariant derivative are given by

 Fμν(A)=∇μAν−∇νAμ,Dμφ=(∇μ+igA2Aμ)φ. (3)

In the above equations, is the usual Levi-Civita connection. The action is invariant under local gauge transformations

 φ→e−igA2α(xν)φ,Aμ→Aμ+∇μα(xν), (4)

where is an arbitrary real function of space and time. The total action is also invariant under space-time differomorphisms. The gauge symmetry implies that not all of the components of the -vector and the real and imaginary parts of the scalar field are physical degrees of freedom (dof). We remedy this redundancy by working in the appropriate set of gauge invariant variables or by fixing the gauge. The redundancy due to space-time diffeomorphisms is handled in a similar fashion.

We will present our answers as power spectra of the electric and magnetic fields, which are defined in the usual way [95]:1

 Ei≡F0i=∇0Ai−∇iA0,Bi≡12ϵilmFlm=12ϵilm(∇lAm−∇mAl). (5)

### 2.1 U(1) gauge invariants

When the field , it can be written in polar co-ordinates as

 φ(xμ)=1√2ρ(xμ)eigA2Ω(xμ). (6)

Under the local gauge transformation (see eq. (4)) and . It is convenient to work in local gauge invariant variables given by the following five fields:

 ρ(xν)andGμ(xν)≡Aμ(xν)+∇μΩ(xν). (7)

In these variables the matter action becomes

 Sm=∫d4x√−g[12∇μρ∇μρ+12(gAρ2)2GμGμ−V(ρ)−14Fμν(G)Fμν(G)], (8)

where . It is worth noting that the variable appearing in eq. (6) does not make an appearance in the above action. There is no need to worry about the gauge redundancy when working with gauge invariant variables.

Note that the and fields are invariant with respect to transformations and their expressions in terms of are identical to those in terms of :

 Ei=∇0Gi−∇iG0,Bi=12ϵilm(∇lGm−∇mGl). (9)

### 2.2 Diffeomorphism invariants

We will work in a perturbed Friedmann-Roberston-Walker space-time with the metric:

 ds2 =(¯gμν+δgμν)dxμdxν (10) =(1+2ϕ)a2(τ)dτ2+2(∂iB+Si)a2(τ)dxidτ −[(1−2ψ)δij−2∂i∂jE−∂jKi−∂iKj−~hij]a2(τ)dxidxj,

where , , , are scalar perturbations, , are divergence-free -vector perturbations, and is a traceless transverse -tensor perturbation.

In this perturbed space-time, we define the perturbations of the following invariant variables:

 ρ(xμ) =¯ρ(τ)+δρ(xμ), (11) Gμ(xμ) =[G0(xμ),∂iG∥(xμ)+G⊥i(xμ)],

where and are scalars and is a divergence-free -vector. Note that the gauge fields vanish at the background level: the spatial components are zero from isotropy of the Friedmann-Robertson-Walker (FRW) background and the equations of motion will set the background temporal component to zero. Hence we will work at the linear level in .

For our scenario, there are two physical scalar metric perturbations, with scalar field perturbation and scalar parts of the gauge field adding three more. We choose to work with the following five diffeomorphism invariant combinations:

 Φ =ϕ−1a∂τ[a(B−∂τE)], (12) Ψ =ψ+H(B−∂τE), δ~ρ =δρ−(∂τ¯ρ)(B−∂τE), G0 , G∥ ,

where . The first two are the standard Bardeen variables. and are diffeomorphism invariant since the gauge field vanishes at the background level.

Similarly, for the vectors we chose to work with the following diffeomorphism invariant combinations

 ~Vi≡Si−∂τKi, (13) G⊥i,

where and are divergence free.

The traceless, transverse -tensor perturbation is already diffeomorphism invariant. Similarly, the electric and magnetic fields defined in eq. (9) are already diffeomorphism invariant. It is convenient to split the electric and magnetic fields into divergence-free and curl-free parts

 Ei(xμ)=∂iE∥(xμ)+E⊥i(xμ),Bi(xμ)=B⊥i(xμ). (14)

Note that from the definition of in eq. (9).

### 2.3 Equations of motion

The general equations of motion for the matter and metric fields in curved space-time take the following form:

 DμDμφ+∂V∂φ∗=0, (15) ∇μFμσ+igA2(φ(Dσφ)∗−φ∗Dσφ)=0, Gμν=1m2\tiny{Pl}Tμν,

where is the Einstein tensor and the energy momentum tensor is given by

 Tμν=2(D(μφ)∗Dν)φ−FμαFνα−gμν[(Dαφ)∗Dαφ−V−14FαβFαβ]. (16)

In terms of the gauge invariant variables defined in Section 2.1, the above equations become

 ∇μ∇μρ+∂V∂ρ−ρ(gA2)2GμGμ=0, (17)
 ∇μFμσ(G)+(ρgA2)2Gσ=0, (18)
 Tμν =∇μρ∇νρ−(gAρ2)2GμGν−Fμα(G)Fνα(G) (19) −gμν[12∇αρ∇αρ−12(gAρ2)2GαGα−V(ρ)−14Fαβ(G)Fαβ(G)].

Equation (18) implies the following definition of the conserved 4-current:

 jμ=(ρgA2)2Gμ,∇μjμ=0. (20)

The equivalent of Maxwell’s equations for the electric and magnetic fields are

 ∇iEi=(ρgA2)2G0=j0,ϵilm∇lBm−∇0Ei=(ρgA2)2Gi=ji. (21)

Next, we write down the equations of motion for the background (space independent) fields and linearized perturbations around these background fields in terms of gauge invariant variables.

#### Background

Assuming the scalar field plays the role of the inflaton and treating the gauge fields as perturbations, the evolution of can be determined from eq. (17):

 ∂2τ¯ρ+2H∂τ¯ρ+a2∂V∂¯ρ=0, (22)

where is given by the background Einstein equation

 (23)

The electric and magnetic fields vanish at the background level.

#### Linearized perturbations in position space

From eq. (17) and eq. (18) we get the equations of motion for diffeomorphism and gauge invariant scalar perturbations:

 ∂2τδ~ρ +2H∂τδ~ρ−Δδ~ρ+a2∂2V∂¯ρ2δ~ρ−∂τ¯ρ(3∂τΨ+∂τΦ)+2a2∂V∂¯ρΦ=0, (24)
 ∂2τG∥+a2(¯ρgA2)2G∥−∂τG0=0, (25)
 ∂τΔG∥−ΔG0+a2(¯ρgA2)2G0=0, (26)

where eq. (26) is the linearized version of the Gauss constraint. Note that the scalar components of the gauge fields and do not depend on the metric perturbations.

The evolution of the gauge and diffeomorphism invariant vector perturbations involving matter fields can be obtained from

 ∂2τG⊥i−ΔG⊥i+a2(¯ρgA2)2G⊥i=0. (27)

The perturbations do not couple to metric perturbations. As mentioned earlier, the tensor perturbations are also decoupled from the matter fields at the linear level. We shall not consider tensor perturbations any further in this paper.

We now turn to the Einstein equations. The energy-momentum tensor in eq. (19) is quadratic in the (with at the background level). Hence, to linear order in the perturbations, the energy-momentum tensor depends only on the perturbations in and the metric. Also for ; there is no anisotropic stress. The linearised Einstein equations (for scalar perturbations) yield

 Φ=Ψ, (28) (∂τH−H2−Δ)Ψ=12m2pl[−∂τ¯ρ(∂τδ~ρ+Hδ~ρ)+δ~ρ∂2τ¯ρ], ∂τΨ+HΨ=12m2plδ~ρ∂τ¯ρ.

Note that the gauge fields are completely decoupled from the scalar metric perturbations.

The picture for vector perturbations is even simpler - the linearized Einstein equations for the vector perturbations involve only metric perturbations:

 (29)

which do not affect the matter vector perturbations.

At the linearized level, the equations of motion for the electric and magnetic fields defined in eq. (9) are simple2

 −ΔE∥=a2(¯ρgA2)2G0=a2j0, (30) −∂τE∥=a2(¯ρgA2)2G∥≡a2j∥, ϵilm∂lB⊥m−∂τE⊥i=a2(¯ρgA2)2G⊥i≡a2j⊥i.

On the right-hand sides of the last two equations, we have defined the scalar perturbations and divergence-free vector perturbations in the 3-current density respectively.

#### Linearized perturbations in Fourier space

For calculational purposes, we move to Fourier space. Fourier space is also particularly convenient for solving the various constraint equations, which essentially become algebraic in the relevant variables. Our Fourier convention is .

We begin with the equations of motion for the scalar perturbations. From the (Fourier space version of the) constraints, eqns. (28), we can substitute the gravitational potential and its derivative into the evolution equation for (cf. eq. (24)) to obtain an equation of motion which only involves :

 ∂2τδ~ρ{k}+2H∂τδ~ρ{k}+k2δ~ρ{k}+a2∂2V∂¯ρ2δ~ρ{k} (31) +2m2\tiny{Pl}⎡⎢ ⎢⎣(H∂τ¯ρ+a22∂V∂¯ρ)δ~ρ{k}∂2τ¯ρ−∂τ¯ρ(∂τδ~ρ{k}+Hδ~ρ{k})∂τH−H2+k2−(∂τ¯ρ)2δ~ρ{k}⎤⎥ ⎥⎦=0.

The remaining scalar perturbations, and , are governed again by an evolution equation, eq. (25), and a constraint, eq. (26). Before moving to the Fourier transformed versions of these equations we define the longitudinal (i.e. curl-free) component of the space-like part of :

 (GL)i=∂iG∥. (32)

The Fourier transform of can be expressed in terms of a longitudinal polarisation vector, , as follows:

 GL{k}=ϵL{k}GL{k}, (33)

where we shall call the scalar , the longitudinal mode. The polarisation vector has the following properties:

 ϵL{k}=ϵL∗−{k},ϵL∗{k}⋅ϵL{k}=1,i{k}⋅ϵL{k}=k,i{k}×ϵL{k}=0. (34)

The Fourier transformed equations, eqns. (25) and (26), then take the form

 (gA¯ρa2)2G0{k}=−k2G0{k}−k∂τGL{k}, (35)
 ∂2τGL{k}+k∂τG0{k}+(gA¯ρa2)2GL{k}=0. (36)

There is a similarity between the pairs and . and are both dynamical fields, evolved according to second order in time equations of motion, eq. (36) and eq. (24), respectively. Each of these perturbations has its own auxiliary field, and respectively, determined by a constraint equation. Substituting the auxiliary field (i.e. ) into the equation of motion we obtain the following expression in terms of only:

 ∂2τGL{k}+2(H+∂τ¯ρ¯ρ)∂τGL{k}1+(gA¯ρa2k)2+[k2+(gA¯ρa2)2]GL{k}=0. (37)

We now turn to vector perturbations. The matter vector perturbations are decoupled from the metric vector perturbations. Similarly to the longitudinal case, we introduce a pair of transverse polarisation vectors which satisfy

 ϵT±{k}=ϵT±∗−{k},ϵTλ′∗{k}⋅ϵTλ{k}=δλ′λ,i{k}⋅ϵT±{k}=0,i{k}×ϵT±{k}=±kϵT±{k}. (38)

The divergence-free perturbations in terms of these polarization vectors are

 G⊥{k}(τ)=∑λ=±ϵTλ{k}GTλ{k}(τ), (39)

with the equations of motion

 ∂2τGT±{k}+[k2+(gA¯ρa2)2]GT±{k% }=0. (40)

The fact that Hubble friction does not appear in the evolution equations for the transverse modes is because of conformal invariance of massless gauge fields. However, the longitudinal components (which exist when the gauge field is effectively massive) do feel Hubble friction.

One can also rewrite the electric and magnetic fields, and the 3-current current, in terms of longitudinal and transverse modes, e.g.

 Ei=∂iE∥+E⊥i=(EL)i+E⊥i, (41)

which in terms of the polarization vectors in Fourier space becomes

 E{k}=ϵL% {k}EL{k}+∑λ=±ϵTλ{k}ETλ{k}. (42)

Similar expansions hold for and with the exception . Below we give the expressions used in the subsequent sections to calculate the primordial power spectra of the longitudinal and transverse modes of and

 EL{k}=kG0{k}+∂τGL{k}=(¯ρgA2)2k2a2+(¯ρgA2)2∂τGL{k}, (43) ET±{k}=∂τGT±{k},BT±{k}=±kGT±{k}.

The charge and current densities can also be expressed in terms of the field’s longitudinal and transverse modes

 (44)

The first identity is simply , i.e. the Gauss constraint. In a consistent quantum analysis of the perturbations during and after inflation, one cannot set and to zero by hand (this differs from the treatment in [62][96]). This is because the vacuum fluctuations in can be enhanced due to horizon crossing during inflation or non-adiabatic particle production during preheating. We shall see this aspect in detail in the subsequent sections.

### 2.4 Gauge transformations

In the upcoming sections we will use either the gauge invariant variables discussed above or work in some particular gauge depending on which approach is best for the problem at hand. In this short section we provide the relationships between variables in some of the popular gauges used in the literature and the gauge invariant ones. These relationships can also be used to move from one gauge to another. When the variables are well defined, we can use the equations of motion and the solutions in the gauge invariant case to recover the corresponding expressions in our gauge of choice. Occasionally, some of the variables become ill-defined or the relationships between variables require a patching up of co-ordinate maps (for example during reheating). Such cases can be dealt with on an individual basis, or one simply derives the equations of motion and solutions directly from the equations of motion themselves.

In different gauges (but not the gauge invariant case) the complex scalar field is represented as , and the gauge fields by . For perturbations, we will always work around an FRW background. Using the global invariance, we set the homogeneous, imaginary part of , . That is . For the homogeneous part with the sign accounting for the change in sign during an oscillation through zero. In Fourier space, the transverse modes and the scalar perturbations along the direction of motion of the homogeneous field (in all the gauges discussed below) are related to the gauge invariant variables as follows:

 δφ0{k}=δρ{k},AT±{k}=GT±{k}, (45)

with the equations of motion for and being identical to the gauge invariant case with .

Coulomb gauge:
In this gauge . In Fourier space we get

 δφ1{k}=−¯ρgA2kGL{k},A0{k}=G0{k}+1k∂τGL{k},AL{k}=0. (46)

Unitary gauge:
In this gauge , which yields

 δφ1{k}=0,A0{k}=G0{k},AL{k}=GL{k}. (47)

The equations in the Unitary gauge are identical to those in the gauge invariant one.

Temporal gauge:
In this gauge . However, the theory is still invariant under the time-independent transformation and , which in Fourier space translates to and . Hence, we completely fix the gauge by choosing an such that at some moment of time, , . With this condition, we have

 δφ1{k}=¯ρgA2∫ττinG0{k}(η)dη,A0{k}=0,AL{k}=GL{k}+k∫ττinG0{k}(η)dη. (48)

Lorenz gauge:
In this gauge . In this case

 δφ1{k} =¯ρgA(AL{k}−GL{k})/(2k),A0{k}=G0{k}+∂τ(GL{k}−AL{k})/k, (49) AL{k} =∫G(τ,η)(∂η+2H(η))(kG0{k}(η)+∂ηGL{k}% (η))dη,

where is the Greens function of the linear operator . Arriving at the above form of the relationship between variables requires a bit of explanation. In Fourier space the Lorenz gauge condition translates to , and the equation governing yields . This equation yields the particular solution above only if we can set the complementary solution to zero. This can always be done since there is a residual degree of freedom such that under the transformations and , the theory remains invariant; provided obeys . In Fourier space we get . Since the operator evolving is identical to the one evolving , we can always choose such that the complementary part of vanishes, thus arriving at the particular solution provided above.

## 3 Inflationary dynamics

The background dynamics are relatively straightforward during inflation. At the phenomenological level, with an appropriate choice of the potential and initial conditions we can easily arrange for for sufficient number of -folds. For simple models, this corresponds to const and const during inflation. Inflation ends when , when accelerated expansion stops and the field starts rolling quickly. Assuming such a background solution has been found, we focus on the quantum fluctuations around this classical background. Quantization of constrained systems, like the problem at hand, can be tricky. We find that by working in Fourier space with gauge invariant variables and substituting the constraints before quantizing, the process becomes straightforward. Once the appropriate quantized solutions for the scalar and gauge fields are available, we construct the power spectra of the electric and magnetic fields at the end of inflation.

### 3.1 Quantized scalar and vector perturbations

For the purposes of quantization, it is convenient to write down the action for the Fourier components of the dynamical perturbation variables left after imposing the constraints. The equations of motion for these variables and were provided in the previous section (see eqns. (31), (37) and (40)). The total quadratic action of these variables naturally splits into four parts:

 S(2)m=Sρ+SL+ST++ST−=∑ISI, (50)

where are the quadratic actions for the perturbations in the -th variable with

 SI=∫dτLI(τ)=∫dτ∫d3kbI(k,τ)[12|∂τfI{k}|2−12ω2I(k,τ)|fI{k}|2]. (51)

The explicit forms of and will be provided below for the different components. We first outline the general quantization procedure common to all of the components. The conjugate momentum density of the -th variable

 πI{k}(τ)=δ(LI(τ))δ(∂τfI−{k})=bI∂τfI{k}, (52)

where we have made use of . The field operators along with their conjugate momenta operators must obey the standard equal time commutators:

 [^fI{k}(τ),^fJ{q}(τ)]=0,[^πI{k}(τ),^πJ{q}(τ)]=0,[^fI{k}(τ),^πJ{q}(τ)]=i(2π)−3δIJδ({k}+{q}). (53)

Note that our fields are not canonically normalized. The non-vanishing commutator and eq. (52) imply

 [^fI{k}(τ),∂τ^fJ{q}(τ)]=i(2π)−3(bI(k,τ))−1δIJδ({k}+{q}). (54)

The field operators can be expanded in terms of operators and mode functions as

 ^fI{k}(τ)=^aI{k}uIk(τ)+^aI†−{k}uI∗k(τ), (55)

where each of the mode functions satisfies the corresponding field equations of motion obtained by varying the action (same equations as those satisfied by )

 (56)

The final ingredient needed for evolving the mode function (and hence the field operators), are the initial conditions for the mode functions. Their normalization will in turn also determine the commutation relation for the operators . We will determine the initial conditions by constructing WKB solutions for the mode functions satisfying eq. (56) at very early times.

To proceed to the WKB solutions for the initial conditions we need the explicit forms of and , which are provided below. For , i.e. when we have

 bρ(k,τ) =a2exp⎡⎢⎣12m2\tiny{Pl}∫ττidτ(∂τ(∂τ¯ρ)2∂τH−H2+k2)⎤⎥⎦, (57) ω2ρ(k,τ) =k2+a2∂2¯ρV−1m2\tiny{Pl% }[∂2τ¯ρ(∂2τ¯ρ−H∂τ¯ρ∂τH−H2+k2)+2(∂τ¯ρ)2].

Similarly, for , i.e. when we have

 bL(k,τ)=[1+(2k¯ρgAa)2]−1,ω2L(k,τ)=k2+(¯ρgAa2)2, (58)

One can check that extremising the action in eq. (50) with respect to each of the field perturbations gives the corresponding equations of motion from the previous section, namely eqns. (31), (37) and (40).

At early enough times during inflation as , a given mode of interest will be deep inside the horizon and will dominate all other physical scales (for example, ) as . At such early enough times during inflation . This hierarchy can be verified by noting that for each component and . With this information at hand, the WKB solution of eq. (56) at early enough times during inflation is3

 uIk(τ)→1(2π