Preheating and locked inflation: an analytic approach towards parametric resonance

# Preheating and locked inflation: an analytic approach towards parametric resonance

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###### Abstract

We take an analytic approach towards the framework of parametric resonance and apply it on preheating and locked inflation. A two-scalar toy model is analytically solved for the coupling for the homogenous modes. The effects of dynamic universe background and backreaction are taken into account. We show the average effect of parametric resonance to be that ’s amplitude doubles for each cycle of .

Our framework partly solves the broad resonance for preheating scenario, showing two distinct stages of preheating and making the parameters of preheating analytically calculable. It is demonstrated for slowroll inflation models, preheating is terminated, if by backreaction, typically in the 5th e-fold. Under our framework, a possible inhomogeneity amplification effect is also found during preheating, which both may pose strong constraints on some inflationary models and may amplify tiny existing inhomogeneities to the desired scale. For demonstration, we show it rules out the backreaction end of preheating of the quadratic slowroll inflation model with mass . For locked inflation, parametric resonance is found to be inhibited if has more than one real component.

a,b]Lingfei Wang Prepared for submission to JCAP

Preheating and locked inflation: an analytic approach towards parametric resonance

• Physics Department, Lancaster University, Lancaster LA1 4YB, UK

• School of Physics, Nanjing University, 22 Hankou Road, Nanjing 210093, China

E-mail: l.wang3@lancaster.ac.uk

## 1 Introduction

Inflation was first proposed in [1] 30 years ago to solve multiple problems previously encountered by the Hot Big Bang theory. It soon received public attention and subsequent works of others[2, 3, 4, 5, 6, 7, 8, 9] have greatly perfected our understanding of inflation. Inflation has become so popular that almost all high energy theories have expressed their personal views on how inflation may be achieved, e.g. [10, 11, 12, 13, 14]. Even now, after 30 years of development, articles are still being entitled “inflation” on a daily basis.

At the same time, reheating, the immediate subsequent scenario to inflation, was more or less in oblivion. It was not until 1994 that Kofman et al. came to realize[15] that the explosive particle production, which they named as preheating, should be first processed before the ordinary perturbative method is applied. Three years later, they proposed the analysis of reheating[16], covering numerous factors taken into account. (See [17] for a recent review.) Besides broad resonance, the narrow resonance also came into people’s sight in the 1990s as a different non-perturbative particle production process during reheating[18, 19]. The analytic study of preheating however remains stalled afterwards, although numerical studies and applications on inflationary models have been going on anyhow. Despite that some applications of the theory in [16] have been quite successful, it is still very mathematically involved, and therefore, some of the underlying physics are not revealed.

On the other hand, locked inflation[9] was proposed as a fastroll inflationary scenario. However, it was later found to be subject to multiple constraints, which together ruled out the whole parameter space[20]. Among them, parametric resonance is a crucial one which may terminate locked inflation much earlier than expected. Recently, locked inflation is recalled in a cyclic universe model which incorporated D-branes and curvature[21], bringing locked inflation and locked inflationary contraction to each cycle. The proposed model however faces the same parametric resonance problem as locked inflation does.

In this article, we construct an analytic framework of parametric resonance for the homogenous mode, in a way more straightforward and less involved. We then apply it on preheating and locked inflation, and discuss the possible inhomogeneity amplification effect. First, from section 2.1 to section 2.3, we solve a half cycle of parametric resonance and show the cause of the exponential boost effect. In section 2.4, we add up the effect cycle by cycle and give the exact solution to the Mathieu equation. We then put universe expansion into consideration in section 3.1 and investigate how backreaction acts as the terminator of parametric resonance in section 3.2. Such approaches then allow the discussion of preheating and locked inflation in section 4.1 and section 4.2 respectively. A brief discussion is carried out in section 5 on how parametric resonance may amplify the existing inhomogeneity and constrain inflationary models. The results are summarized in section 6.

## 2 The Simple Framework

To construct the framework, we consider the simple case first in a static background with Lagrangian density

 L=−12m2ϕ2−12M2χ2−λϕ2χ2−12∂μϕ∂μϕ−12∂μχ∂μχ. (2.1)

and are real scalar fields with bare masses and . Their interaction strength is characterized by the dimensionless parameter . Here we consider the configuration of and to neglect all backreactions. indicates the amplitude of oscillation when operating on a field. We allow to be negative, but still use to indicate the absolute value of for convenience when we compare it with other values. In many of our calculations however, we just neglect because it’s too small. Throughout this paper, we will stick to the homogenous modes of both fields, so the spatial dependence will also be neglected in the notation. For simplicity, we will also use the word “parametric resonance” to indicate only the parametric resonance within our assumed parameter space. Please note this Lagrangian doesn’t suffer from problems like negative infinite energy from the negative because it’s just the effective part we take out from the total Lagrangian, so terms like the quartic ones don’t distract us. As long as such terms are negligible, one doesn’t need to worry about the validity of eq. (2.1).

From the Lagrangian, we can write down the potential

 V(ϕ,χ)=12m2ϕ2+12M2χ2+λϕ2χ2, (2.2)

in which dominates according to the above configuration. is visualized in figure 1(a), where the vertical lines also suggests the domination of . More details of the potential are displayed in figure 1(b) and figure 1(c).

It is not difficult to imagine the motion of the two fields under such a configuration. Under , ’s effective mass squared would be and will always be oscillating with frequency and constant amplitude, unaffected by the motion of . We can easily derive its equation of motion

 ¨ϕ+(m2+2λχ2)ϕ=0, (2.3)

where dots mean derivatives w.r.t time. After neglecting , we get the Simple Harmonic Oscillator(SHO) solution .

’s effective mass squared is much larger than ’s most of the time, so it will oscillate much faster than , as long as is not very close to zero. When crosses zero however, the motion of would become very different from the stage of large . We will model the two stages in turn, trying to find out why and how is boosted exponentially. Still, we write the equation of motion of here for future reference.

 ¨χ+(M2+2λϕ2)χ=0. (2.4)

### 2.1 Rolling Stage

The rolling stage is when is not very close to zero, therefore is oscillating much faster than . It also requires that ’s effective mass varies slowly and thus can be regarded as a constant in each cycle of ’s oscillation. For most cases in which the rolling stage takes up more than half of ’s oscillation, one only needs to take into account the second condition, and then the first one is automatically satisfied. So the condition of this stage is

 ∣∣ ∣∣dm2χm3χdt∣∣ ∣∣≪1. (2.5)

Although this stage can be easily solved with simple approximations, we still would like to work on it again here, mainly to arrive at the variables we are interested in and confirm the rolling stage has a generally vanishing total effect on them. Suppose at time , is at maximum with and , and scalar field is rolling down slowly with time derivative . We take the effective mass constant and get the zeroth order SHO solution of eq. (2.4)

 χ(0)≡|χ|cosmχt. (2.6)

We then derive the equation of motion up to first order by introducing ,

 ¨χ(1)+m2χχ(1)+4λϕ˙ϕ|χ|tcosmχt=0 (2.7)

and get the solution with the initial condition and ,

 (2.8)

At time , when reaches maximum again, we have

 (2.9)

If we take when is rolling down as an example, we will find out gets an additional amplitude correction from the decrease of . To be precise, we consider the time interval , during which rolls from one maximum to the other. ’s amplitude is affected at a rate

 Δ|χ|Δt=−χ(1)(πmχ)Δt=−λϕ˙ϕm2χ|χ|. (2.10)

Replacing with differential operator gives

 |χ|∝1√mχ, (2.11)

which is the effect on due to the variation of .

Similarly, we can calculate ’s phase change resulting from the variation of , or directly. The motion of generates an additional which changes the time needed to push back to its maximum again. The phase change is therefore

 φχ≡˙χ(1)(πmχ)mχ|χ|=π2λϕ˙ϕm3χ=π2logm′χmχ, (2.12)

in which and are the initial and final effective masses of .

The above calculation has shown how is affected by ’s motion when is away from zero. Although the effects on ’s amplitude (eq. (2.11)) and phase (eq. (2.12)) are strong, can’t get exponentially boosted in this way. This is because the effects during the upward rolling period of are exactly inverse from those during the downward period, so they generally cancel. is increased when is rolling towards zero, but it get decreased when moves away from zero. The same thing happens to ’s phase change, so the net effect vanishes.

### 2.2 Zero-Crossing Stage

When is small and is relatively large, the condition of rolling stage eq. (2.5) breaks, and the system enters the zero-crossing stage. The zero-crossing stage is the period crosses zero. During this stage, the motion of is very much different from that during the rolling stage. It’s because ’s effective mass varies very fast in this stage and can become smaller than ’s effective mass when is crossing zero.

Because , we still adopt the approximation that is not affected by the motion of , and is still negligible. Here we set at . In most applications of parametric resonance, holds throughout the zero-crossing stage. We then get the motion of the two fields

 ϕ=|ϕ|mt, (2.13)

and thus

 ¨χ+2λ|ϕ|2m2t2χ=0. (2.14)

Here we use as the scaled time, and primes as derivatives w.r.t . The equation of motion of then becomes

 χ′′+τ2χ=0. (2.15)

The breaking of condition eq. (2.5) corresponds to . If one is only interested in the behavior of the system in , one can simply expand in Taylor series of at and only take the first few terms. Numerical simulation tells us, however, the energy boost on spans not only , but at an even broader range. To precisely calculate the boost effect, we need to take the analytic solution of eq. (2.15)

 χ(τ)=0F1(34,−116τ4)χ0+0F1(54,−116τ4)χ′0τ, (2.16)

where is the confluent hypergeometric (limit) function, and the subscript 0 denotes the value of variables at .

Given the initial condition and , one can already calculate the effect of parametric resonance. In order to enable our further calculation, we take out the leading term of for (but still holds). To do this, we expand in series of . The leading term is

 0F1(a,−z)≈Γ(a)√πsin(π4(2a+1)−2√z)z14−a2,z≫1. (2.17)

Here we want to calculate the energy boost effect from parametric resonance, so we characterize the boost rate with

 η≡12limτ→+∞logEs(τ)Es(−τ). (2.18)

The scaled energy density of is defined as

 Es(τ)≡τ2χ2(τ)+χ′2(τ), (2.19)

with its relation with the (unscaled) total energy density of

 E≡λ|ϕ|2m2t2χ2+12˙χ2=√λ|ϕ|2m22Es. (2.20)

Substituting eq. (2.16) and eq. (2.17) into eq. (2.18), we arrive at

 η=12log(1+4√2Γ(34)Γ(54)χ0χ′0Γ2(34)χ20−2√2Γ(34)Γ(54)χ0χ′0+4Γ2(54)χ′20). (2.21)

In the above equation, the limit has already been taken. As a result, all non-leading terms of are removed. The in the sin function of approximation is removed too.

From eq. (2.21) we find the boost rate is hardly dependent on any parameter of the system, (, , etc,) or the energy density of any component, as long as our previous assumptions stay valid. It is also easily verified by numerical simulations. This constant property is very helpful when we sum the boost rates across cycles of in subsequent calculations. The fraction in the log function of eq. (2.21) should be of order unity, so the boost effect is significant and exponential. The boost rate acquires the same sign with , which means the energy of can either be boosted or dampened, depending on the sign of .

### 2.3 Phase Delay

In the above subsection, we have represented with and , the motion state of at . The previous calculation however only gives half of the analytic solution — we can calculate with any given and , but we haven’t developed any method in deriving them two, except numerical ways. Therefore we will derive and analytically in this subsection and provide a completely analytic solution. But before that, it’s quite important to first clarify what “phase” is referred to here.

The system has two degrees of freedom, and , which means we need four independent parameters to fully describe the motion state of the system at any specific time. We are interested in the two parameters characterizing , which can either be chosen as and , or its amplitude and phase. From eq. (2.21), we have already learned the boost rate is a function of and . Although is not strictly a SHO at the zero-crossing stage, if we manage to represent the motion of at with an amplitude-like and a phase-like variable however, the amplitude-like variable would cancel and would then only depend on the phase-like variable. It’s obvious may not stay constant in every half cycle of , so reducing to a single-parameter function allows us to track better when studying the boost effect across many cycles of .

In order to find the phase representation of at , we attempt to solve and reversely. It is reasonable to believe is SHO-like at and set the initial condition as

 χ(τ)=√Es(τ)|τ|sinβ,χ′(τ)=√Es(τ)cosβ (2.22)

where is the variable we use to represent the phase of at . We are only interested in here, so we come up with the combination

 |τ|χcosβ−χ′sinβ=0. (2.23)

By putting in the solution eq. (2.16) and eq. (2.17), it simplies to

 Γ(34)sin(β+12τ2−58π)χ0=2Γ(54)sin(β+12τ2−78π)χ′0. (2.24)

For a specific process with initial conditions given, and should be definite and thus independent of the choice of in eq. (2.24). For this reason, we contract the into and redefine as

 β=β(τ)≡β0−12τ2+78π, (2.25)

where is the variable for phase but it’s now independent of time . For short, we will however still use to indicate . In this way, eq. (2.24) also becomes independent of , and simplifies to

 (2.26)

This immediately gives the solution

 χ0 = 2Γ(54)sinβX, (2.27) χ′0 = Γ(34)cos(β−π4)X, (2.28)

where is a function of and and is not important in our calculation.

By comparing it with the SHO solution, or the SHO-like solution eq. (2.22), one can find out the motion of at is indeed non-SHO. The phase of its kinetic term is delayed by . This is also confirmed numerically for all in figure 2. Such a phase delay can be understood from the decreasing transfer efficiency from ’s potential energy to kinetic energy as is approaching zero. Given the that makes , i.e. at the bottom of its potential at , such doesn’t induce the largest any more. To understand it better, we compare such with , where . The difference caused by is two-fold. On one hand, the phase makes reach every value slightly earlier than phase , and such difference in timing gives a larger effective mass of when compare at the same value of , and in consequence, a larger energy transfer efficiency of order . On the other hand, for , gets an extra up-climbing period just before . This effect decreases the energy transfer efficiency of , but only has order . As a result, the net effect is has a larger energy transfer efficiency and therefore a larger kinetic energy. The maximum of kinetic energy is then shifted towards a larger , and its phase gets delayed relatively.

We continue our calculation by substituting the set of solution eq. (2.27) and eq. (2.28) into the expression of , eq. (2.21), and it becomes a very simple function of

 η(β)=12log(1+4√2sinβcos(β−π4)). (2.29)

We are now able to compare our results with numerical ones. In figure 3, three curves of are drawn — the analytical result eq. (2.29), the numerical solution, and the semi-analytical result which is acquired by substituting the numerically derived and into eq. (2.21). All three curves match perfectly.

So now we have successfully reduced the parametric resonance effect to a completely constant one — it depends only on , the phase of . Ths boost rate is however independent of any of the parameters of the system, , , etc, or even the energy density of . This is indeed the case from numerical simulations. Because the boost rate is independent of the energy of , we can simply sum the boost rates when calculating the total boost rate for many cycles of .

If we naively assume is evenly distributed in , and observe many enough independent half cycles of parametric resonance, the average boost rate can then be calculated, which is

 ⟨η⟩≡12π∫2π0η(β)dβ=0.346. (2.30)

This actually means for every half cycle of , ’s energy density becomes times of its original. For future reference, here we also define the maximum boost rate

 ηm≡maxβ∈[0,2π)η(β)=η(38π)=log(1+√2)=0.881. (2.31)

### 2.4 Long Time Evolution

We now continue to study the long time evolution of across cycles of . Suppose the initial phase of is , by which we mean initially, when reaches its maximum, there is

 χ=√Eλ|ϕ|2sinβ,˙χ=√2Ecosβ. (2.32)

We can then use the boost rate acquired in the last subsection to evolve the system from one maximum of to the next. To characterize the state of , at every maximum of we use a polar coordinate system as the phase diagram of . The angular coordinate is chosen to be , and the radial coordinate is , where is the total boost rate. Therefore the and coordinates are proportional to and respectively. If we choose the initial condition as a constant energy density with evenly distributed , it then corresponds to a smooth circle with unit radius on the phase diagram. For a general overview, numerical results are given in figure 4, figure 5 and figure 6.

These figures suggest a set of linear operations be applied on each “–” mark on the phase diagram, in order to get to its corresponding “+” mark with analytical methods. The linearity is also determined by the linear equation of motion of . To clearly demonstrate the operations are linear, we represent plots on the phase diagram with matrices of their Cartesian coordinates, and model the total evolution of field between neighboring maxima of as a linear transformation matrix, which we call here as the transformation matrix. The transformation matrix should then be decomposed into two parts:

1. During the zero-crossing stage, is boosted with the boost rate a function of phase . So for an initial circle on the phase diagram, it is distorted to an ellipse because of the phase dependence of . For convenience, we rotate the ellipse by shifting to align its major axis with -axis. If the initial circle has a radius , we know at once the semi-major axis has length , and the length of semi-minor axis is . The largest negative boost rate comes from the time-reversibility of ’s equation of motion. When time-reversed, the largest (positive) boost rate always corresponds to the largest negative boost rate and proves its existence. During this step, the plot in the phase diagram is transformed to which satisfies

 (x′y′)=(eηm00e−ηm)(xy). (2.33)
2. The ellipse is rotated with an angle which is generated from the difference of the periods of and . This is because the period of is not an integer times of the period of . Therefore doesn’t return to the exactly same position when reaches maximum (or zero) again as where it was at ’s last maximum (or zero). One can easily confirm the rotation angle is independent of phase , so the elliptical shape is preserved. Here we suppose the rotation angle that should be added to is , so the plot is transformed to

 (x′′y′′)=(cosδ−sinδsinδcosδ)(x′y′). (2.34)

By definition, is the phase goes through from one maximum of to the next. So we can simply calculate the rotation angle as

 δ≡2∫π2m0√2λ|ϕ|sinmtdt=2√2λ|ϕ|m, (2.35)

in which the bare mass has been neglected. Therefore the total transformation matrix is

 (cosδ−sinδsinδcosδ)(eηm00e−ηm) = (eηmcosδ−e−ηmsinδeηmsinδe−ηmcosδ) (2.36) = B−1(b−√b2−100b+√b2−1)B,

where and is a matrix of eigenvectors which is not important.111To be more precise, one may want to calculate the transformation matrix via a three-step method — two rotations with angle separated by a boost. The difference however only shows in the eigenvectors , but the eigenvalues remain the same.

We are mostly interested in the total boost rate during many cycles of , which is represented by . Since is multiplied on the left by the transformation matrix for every half cycle of , the total boost rate is actually determined by many powers of the transformation matrix. For this purpose, we have derived eq. (2.36) which allows us to only consider the power of the eigenvalues of the transformation matrix.

For i.e. , we have . Because one eigenvalue is larger than the other, the overall effect is the system converges to the eigenstate with the larger eigenvalue. This means all initial phases converge to the same one after enough long time. Because the larger eigenvalue is greater than 1, ’s total energy also increases gradually.

For i.e. , the two eigenvalues have unit magnitude and opposite arguments. Consequently there is no such phase convergence, and no overall change in ’s total energy should be expected.

Such difference is also quite clear from the figures. The boost rate has dependence on the initial phase and transforms a circle in the phase diagram to an ellipse. Because of such a transformation, the phases converge towards -axis. On the other hand, rotates the ellipse and this may ruin the convergence if it’s large enough. These two effects compete and, in consequence, the above two distinct behaviors of the system may occur.

This also explains the exact solution of eq. (2.4), the so-called Mathieu equation. The solution of Mathieu equation has bands of stable solutions in which the amplitude of generally remains at its initial scale as time passes, and bands of unstable solutions which have an exponentially growing w.r.t time. Those bands correspond exactly to the non-convergent and convergent cases respectively.

For better understanding, we can also calculate the phase that is converged to. The fixed points under transformation matrix eq. (2.36) satisfy , i.e.

 yx=y′′x′′=xe2ηmsinδ+ycosδxe2ηmcosδ−ysinδ. (2.37)

The solutions are

 yx=sinhηm±√sinh2ηm−tan2δeηmtanδ. (2.38)

We can see how the solution is represented in the phase diagram in figure 7. From the figure we know at once we should take the minus sign in eq. (2.38) as the phase converged to. Based on these results, we can further derive ’s long-time boost rate (which takes time far more than sufficient for convergence to take place, if there is). For the convergent case, the long time boost rate is simply the of the larger eigenvalue in eq. (2.36), and for the non-convergent case, no boost occurs so the long time boost rate is zero. We thus define the long time boost rate as

 (2.39)

It’s also visualized in figure 8.

In figure 8, we can see the convergent-non-convergent boundary is at , which is exact and can be confirmed from the expression of . This indicates only half of the values give a long-time boost effect. If we consider the case with a slowly scanning over at a constant speed, we would get the average long-time boost rate

 ⟨ηl⟩≡12π∫2π0ηl(δ)dδ≈0.346. (2.40)

Please note there is relation .

It’s quite interesting here to notice “happens” to be very close to . Although the expressions of and are very different, this however is not a coincidence. and are actually identical and numerical integrals have confirmed that. It is not yet very clear why they are equal, but the following might be a reasonable explanation. Every value of is actually hit the same number of times during the calculation of and . Consequently the transformation matrices of all ’s operate on the initial state the same number of times, although the order of the operations is different for and . The transformation matrices do not commute, but their commutation is only a rotation. The commutation could result in an either slightly larger or smaller boost rate, depending on whether the rotation is towards the major axis of the total transformation matrix or away from it. A huge number of such commutations is needed to make the two sequences the same. Some of the commutations contributes positively to the total boost rate while others negatively. When summed up, their effects cancel, and maybe one might be willing to think this as the reason of the equivalence between and .

## 3 Steps Towards Cosmological Applications

So far, we have established the framework to consider the boost effect of parametric resonance in a static background. To apply it practically on cosmology, there are still a few steps should be taken first. One is the need to put our framework into the expanding background; the other is the consideration of backreaction. We will discuss them in this section.

### 3.1 Dynamic Background

When we apply this model in cosmology, a dynamic background is required.222Besides expansion, this model is also applicable in contracting phases without any prejudice. Because we are more concerned with the expanding phase, we will only discuss the effect of universe expansion here. The effect of contraction however, can be derived in the same way, and stages are simply reversely sequenced in most cases. We will still neglect the backreaction in this section, so the effect of expansion on is simply a damping factor , where is the scale factor. The equation of motion of however becomes

 ¨χ+3H˙χ+2λϕ2χ=0, (3.1)

where is the Hubble rate of the universe, and can be regarded as a constant for each half cycle of . Writing simplifies eq. (3.1) to

 ¨Ψ+2λϕ2Ψ=0, (3.2)

in which we have neglected the effective mass contribution from universe expansion for . Therefore we can see the universe expansion also gives an term.

However the universe expansion has other impacts on our framework. As is damped by expansion, the rotation angle no longer remains constant across cycles of . Therefore changes should be made to the calculations in section 2.4. For convenience, we use the notation to indicate the variable of the th half cycle of , or at the time when reaches its maximum the th time. When all the subscripts are equal in one equation, we may neglect some or all of them, as long as no confusion might be caused. Primes here are defined as the difference

 x′(n)≡x(n+1)−x(n). (3.3)

Different from the static version eq. (2.35), in an expanding background, should be redefined as

 δ(n)≡∫t(n+1)=t(n)+π2mt(n)2√2λ|ϕ|(t)cosmtdt. (3.4)

To finish the integral, we manually fix at , so it gives the same result with eq. (2.35). This approximation however gives an error

 Δδ(n)<∫t(n+1)t(n)2√2λ||ϕ|(n+1)−|ϕ|(n)|cosmtdt=2√2λ||ϕ|′(n)|m (3.5)

where the outer is to take absolute values. For short, we have

 Δδ<2√2λ||ϕ|′|m. (3.6)

Similarly we get

 δ′ ≡ δ(n+1)−δ(n)=2√2λ|ϕ|′m, (3.7) δ′′ ≡ δ′(n+1)−δ′(n)=2√2λ|ϕ|′′m, (3.8) Δδ′ < 2√2λ||ϕ|′′|m=|δ′′|. (3.9)

The universe expansion redshifts , giving

 |ϕ|′|ϕ|=e−3πH2m−1=−3πH2m, (3.10)

which consequently redshifts . The average effect of parametric resonance is then determined by the scales of , and .

When , changes very slowly compared with the oscillation of . The system therefore has enough time to converge to the eigenstate whenever there’s any. Then the system can be regarded as trapped in its eigenstate which is slowly varying. The boost rate changes correspondingly as slowly decreases, and the average boost rate in this case should then be taken as the long-time boost rate, , defined in eq. (2.40). As is slowly redshifted by universe expansion, the convergent and non-convergent cases are met in turn. This makes ’s evolution stair-like at this stage. Half of the time, the amplitude of enjoys flat platforms lasting for cycles of during which it doesn’t grow at all. The platforms are separated by steep growing regions which fills the other half of the time. The two distinct stages come in turn to get the stair-like curve of ’s amplitude.

When but , varies fast but varies very slowly. In this case, changes significantly between neighboring half cycles of , but the amount changed stays almost fixed. For every half cycle of , an almost constant adds to the initial rotation angle . Therefore for time long enough, the average effect is has an even probability at each value in . This corresponds to an even distribution of in and, results in the average boost rate which should be chosen as in eq. (2.30).

When , the relevance between neighboring ’s or ’s is small, so one usually considers as random in , being irrelevant with the history of evolution. When long enough time is experienced, the random distribution of gives the same results as the case of and so should also be chosen as the average boost rate.

In realistic cases however, it is rare to tell the last two stages apart. They share the same average boost rate and most of their characteristics are alike, so there is no need to distinguish them. Moreover, the second category has a but non-vanishing. To demonstrate the difference between the last two categories, the even probability of needs to be shown. This however requires a prolonged constant value of , which in most cases is very difficult to achieve with a non-vanishing , even though it’s much smaller than 1. For these reasons, we will treat them as a whole () in our following discussions and call it the large stage, in which is randomly distributed and independent of the history. In the same way, we name as the small stage.

We now fall back to check the approximation we made previously in eq. (3.6) and eq. (3.9), where and are regarded as constants in every half cycle. They give errors and , which fit perfectly with the three stages above. Therefore this approximation is applicable here.

Before we end this discussion on dynamic background, we would like to briefly calculate the statistical results from the random for the large stage. In cycles of , the boost effect of parametric resonance occurs times. The total boost rate is

 ´η(n)(δ)≡n∑i=1η(δ(i)). (3.11)

Assuming absolute randomness of all , we get the expectation value of as

 (3.12)

This is obvious because it’s just how is defined. To further compute the statistical error, we calculate

 ⟨´η2(n)⟩ ≡ 1(2π)n∫dnδn∑i=1η2(δ(i)) (3.13) = n2π∫2π0η2(δ)dδ+n(n−1)4π2(∫2π0η(δ)dδ)2 = n⟨η2⟩+n(n−1)⟨η⟩2.

and thus the statistical error can be estimated as

 Δ⟨´η(n)⟩≡√⟨´η2(n)⟩−⟨´η(n)⟩2=√n√⟨η2⟩−⟨η⟩2≈0.539√n. (3.14)

To justify both the statistical calculation and the definition of the large and small stages, here we also give figure 9 through multiple numerical simulations.

### 3.2 Backreaction

In this section, we discuss the backreaction of by taking into account the term in , the effective mass squared of . Since we assumed homogeneity, we will not discuss other backreactions here. In most cases of parametric resonance, the assumption starts valid. However as ’s amplitude gets boosted by resonance, this assumption would finally break at , provided the boost effect hasn’t yet been halted by other causes. Because , for every half cycle of , reaches its maximum when . From this point of view, the violation of this approximation first starts small — only at — and then expands to larger gradually while is being further boosted.

From the relation , we can find the effective potential for , which is actually

 (3.15)

where is ’s amplitude at maximum , and means taking the absolute value (and still indicates taking the amplitude). Here we can see the additional part of is proportional to the absolute of . Compared with which is proportional to , the effective potential from interaction agrees with our previous instinct that the backreaction first comes to important at small .

So we consider the motion of for half a cycle, starting from , , , and . We will use for simplicity. When , ’s equation of motion is

 ¨ϕ+m2ϕ−λ|χ|2m|ϕ|=0. (3.16)

We can analytically solve this equation and arrive at . The solution is

 t=t1≡1marccosγ2γ2+1, (3.17)
 ˙ϕ(t=t1)=(γ2+1)m|ϕ|sinmt1. (3.18)

At , is boosted by parametric resonance. This gives the interaction part of a factor of . Here is still the boost rate. For this reason, the equation of motion is slightly modified for ,

 ¨ϕ+m2ϕ+λ|χ|2m|ϕ|e2η=0. (3.19)

We then solve it again to arrive at and finish the half cycle. At the side when reaches maximum again at time , we have

 t2=t1+π2m−1marctanγ2e2η√γ2+1, (3.20)
 ϕ(t=t2)=|ϕ|(√γ4e4η+2γ2+1−γ2e2η), (3.21)

and

 |χ|=eη|χ|m|ϕ|ϕ. (3.22)

We can clearly see when , there is no boost effect and the motion of the system becomes symmetric w.r.t .

When the backreaction is weak, i.e. , we can expand the results to first order and get

 t2 = πm−e2η+1mγ2, (3.23) ϕ(t2) = |ϕ|(1−(e2η−1)γ2), (3.24) |χ| = eη|χ|m(1+(e2η−1)γ2), (3.25) δ = 2√2λ|ϕ|m(1−(π2(e2η+1)−2)γ2). (3.26)

Therefore, in case the backreaction presents, the oscillation of becomes faster and that of is compared slower. The non-vanishing gives a boost effect. Amplitudes and are thus affected and become slightly different with their original values at , suggesting an energy transfer.

Whenever one takes this backreaction into account, one should adopt these corrections. However for weak backreactions in which , the corrections actually can be neglected. Such backreaction only comes to important when . In some cases, however, the exponential boost effect is terminated by other causes before reaches , so it never comes to important.

If the system does evolve to , one then needs to take it seriously. The most significant effect is oscillates faster and faster, due to the effective mass contribution from the backreaction. This has several consequences. First and most obviously, because the boost rate is counted per half cycle of , a faster rate of ’s oscillation also leads to a faster boost to . The faster boost acts back on ’s effective mass and causes to oscillate even faster. The positive feedback gives an explosive boost effect to once the backreaction comes to important. The explosive boost effect is soon shut down when reaches the same scale with . This is because and would then have similar frequencies and the approximate replacement of with would be no longer applicable. This immediately ends the exponential boost effect of parametric resonance.

Moreover, this backreaction may also offer a major contribution to . Here we give a simple estimate based on eq. (3.26). By definition, we have . So the correction to up to first order of is

 Δδ′=−2√2λ|ϕ|γ2m(π2(e4η+2η′−1)−e2η+1). (3.27)

When it has the same magnitude with the originally defined in eq. (3.7) in an expanding universe, we can solve the magnitude of ,

 γ2∼Hm<0.1. (3.28)

This means the impact of backreaction on rotation angle is even more noticeable than that on ’s period.

In the same way, it’s easy to derive has unit magnitude when . Normally in an expanding universe where the backreaction is negligible, is always decreasing so the resonance would gradually undergo a shift from the large stage to the small stage. The relation of however tells that the parametric resonance may be subject to a shift back to the large stage from the small stage, or even, the small stage may not appear at all. The back shift, whenever exists, always lies ahead of the explosive boost. So for deep discussions of the ending conditions of the exponential boost, one should always take it into account.

## 4 Cosmological Applications

### 4.1 Preheating

Preheating is a possible stage after inflation, during which the inflaton decays into particles through parametric resonance. Preheating was first realized in [15, 16], and the parameter space is the same with what we have discussed. Therefore, during preheating, the number of preheated particles grows exponentially, much faster than the redshift rate from universe expansion.

Although preheating should be followed by particle thermalization, and possibly a second stage of reheating between them, they are not within our consideration in this paper. Instead of calculating the reheating temperature and how long reheating and thermalization last, this section is devoted to showing how this framework may be applied on specific preheating calculations. Details about the second stage of reheating and thermalization can be found in [16] and [22]. For the same reason, we don’t care the reheating ratio between the dark and visible sector. To solve the dark radiation problem, one might want to refer to inflation models with more particle physics foundation, e.g. SUSY inspired inflation and the subsequent reheating and thermalization studies[23, 24, 25].

During preheating, the universe is dominated by in our model, which is usually the inflaton. As a demonstration, we here just consider the simple case of the preheating of a generic slowroll inflation with inflaton . To solve problems like horizon and flatness, inflation requires an e-folding at least . At the beginning of such slowroll inflations, there should be . Otherwise there would be a significant decrease in ’s effective mass and thus no inflation although would still be slowrolling.

In general, the end of slowroll inflation is signaled by the destruction of the slowroll condition of , after which begins to oscillate and preheating takes place. We would like to neglect the first cycle or so of in the beginning of preheating here, so holds for the rest of the preheating period. Because holds in the beginning of inflation, and is redshifted greatly during inflation, this relation is expected to still hold for a very long period in preheating. Because the backreaction terminates parametric resonance once it comes to important, it acts as an ending condition of the preheating scenario. Therefore we can first neglect the presence of backreaction, and then calculate the take-over of backreaction separately.

We label the beginning of inflation as , the end of inflation (i.e. the start of preheating) as , and the point we start to consider in preheating as . The evolution of the universe is then divided by such points, whose labels we use as subscripts, into intervals such as for the convenience of future references. To be exact, we define the point as

 |ϕ2|≡e−1|ϕ1|. (4.1)

is redshifted during inflation (), but the boost and redshift effect approximately cancel during . We therefore get

 |χ2|∼|χ1|=e−32N|χ0|. (4.2)

The universe is dominated by during preheating so the Hubble rate writes

 H2=4πm2|ϕ|23M2p (4.3)

in which for point 1 and after,

 |ϕ|=|ϕ1|(a1a)32. (4.4)

After point 2, the system starts at the large stage. There may be a transition from large to small stage during the evolution, but the termination of parametric resonance by backreaction is certain to take place at the large stage, which is ensured by its influence on shown in eq. (3.27), slightly before the positive feedback ends the exponential boost. We thus also label the point after which the large is ensured by backreaction, and the point backreaction becomes important and consequently terminates parametric resonance as points 3 and 4, also in chronological order.

After defining the timeline, we now work on the evolution of . The large and small stages share the same average boost rate , so after point 2, field is affected by parametric resonance and becomes proportional to . Due to universe expansion, is redshifted and thus acquires the factor . As we have demonstrated in section 2.1, , this gives , from the redshift of field, another factor . We multiply them all together to get the statistically average evolution of field

 |χ|=|χ2|(a2a)34e⟨η⟩πm(t−t2). (4.5)

In a matter-like universe, we have for any time and ,

 (a(tb)a(ta))32=1+√6πρ(ta)M2p(t