Stokes phenomenon and gravitational particle production -- How to evaluate it in practice
RRESCEU-1/21
Stokes phenomenon and gravitational particleproduction — How to evaluate it in practice
Soichiro Hashiba, a,b
Yusuke Yamada b a Department of Physics, Graduate School of Science, The University of Tokyo,Hongo 7-3-1 Bunkyo-ku, Tokyo 113-0033, Japan b Research Center for the Early Universe (RESCEU), Graduate School of Science,The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
Abstract:
We revisit gravitational particle production from the Stokes phenomenon view-point, which is crucial to understand asymptotic behavior of mode functions with a timedependent frequency. One of our purposes of this work is to show how the analysis focusingon the Stokes phenomenon can be used to analytically estimate non-perturbative particleproduction rate. In particular, with several examples of time-dependent background, weexamine some methods that make the analysis more practical. Specifically, we considerthe particle production in simple expanding backgrounds, R inflation, and a model withsmoothly changing mass. Since some of our models show difficulties in analyzing Stokesphenomenon, we discuss simplification of the problem and the accuracy of analytic estima-tion within our approximations. We also propose an approximation to take into accountthe most important contribution among infinite number of turning points and poles, whichgreatly simplifies the problem, but still gives a good analytic estimation. a r X i v : . [ h e p - t h ] J a n ontents R model: imitation method 104 Smooth transition model: consideration of poles 175 Summary and conclusion 21A Review of WKB/phase integral method and Stokes phenomenon 23 A.1 Examples 27A.1.1 Particle production from instability 27A.1.2 Particle production induced by fast moving background 31
B Analytic estimation of particle production induced by oscillating back-ground 34
B.1 Rough estimate 34B.2 Improved estimate 35
C Improvement of particle production rate for a transition model 36
Particle production induced by nontrivial background is one of the most interesting prop-erties of quantum field theory. Sauter-Schwinger effect [1, 2], Hawking radiation [3] arewell known examples of such effects. Remarkable feature of these effects is that they arenon-perturbative in coupling constants. For instance, Sauter-Schwinger effect cannot bedescribed by sum of perturbative Feynman diagram processes, which only contain g n where g is a coupling constant and n is a positive integer.Non-perturbative particle production takes place in the cosmological context as well,and one of the examples is broad resonance regime in preheating after inflation [4–6]. Narrow resonance is not of the kind since it can be described perturbative processes with multipleinflaton particles [6]. In some literature, both narrow and broad resonance are refered to as non-perturbativeeffects. In our context, only broad resonance regime is referred to as a non-perturbative effect, where particle – 1 –t has been known that even though non-perturbative particle production seems typicallysmall effects, repeated events in preheating lead to significant impact on the evolution ofuniverse as well as inflaton dynamics [6]. Therefore, the non-perturbative processes wouldplay important roles in physics.The other interesting property of non-perturbative particle production is that rela-tively heavy particles, which would hardly be produced by perturbative processes, can beproduced. For instance, during broad resonance in preheating, large amount of particleproduction can take place even if the particle is much heavier than inflaton particle, whichcannot be the case in perturbative processes due to kinematical reasons. The resultantheavy particles may become, for instance, unwanted relic, which may eventually dominatethe universe [7]. Therefore, one has to take the non-perturbative particle production intoaccount.How should we evaluate “particle production” in time dependent background? Defi-nition of particle requires a solution of (free) equation of motion in a given background,which is a second order differential equation. There are two independent solutions, whichare typically called positive and negative frequency modes, and we expand a field in termsof these two independent solutions, put the creation and annihilation operators, and definea vacuum state. On time dependent background, a positive or a negative frequency modefunction defined at early time generally becomes a linear combination of positive and neg-ative frequency modes defined at late time. Such a change of asymptotic behavior meansthat the annihilation operator at early time is mixing of the creation and the annihilationoperator at late time, and the vacuum state at the early time is no longer a vacuum stateat late time. In this sense, asymptotic behavior of a set of mode functions has one-to-onecorrespondence with notion of particle production. Thus, we need methods to understandthe asymptotic behavior of mode functions in a given background, particularly how thepositive and the negative frequency mode mix with each other in late time.Asymptotic mixing of mode functions can be understood as the Stokes phenomenon inmathematical language. Such a behavior is well described by using WKB/phase integralmethod. The relation between particle production and the Stokes phenomenon has beenknown and applied to Sauter-Schwinger effect [8–11], Hawking radiation of blackhole [12]and in de Sitter spacetime [10, 13, 14], preheating [15], particle production in expandinguniverse [16, 17] and particle production associated with vacuum decay [18]. Understandingof the Stokes phenomenon gives us a systematic way to understand the particle productioncaused by nontrivial background.The purpose of this paper is to describe how the gravitational particle production,which is induced by time varying gravitational background [19, 20], can be evaluated an-alytically by focusing on the Stokes phenomenon, and develop some techniques to discusssome complicated systems. In particular, we discuss particle production in several non-trivial gravitational backgrounds. Curved backgrounds are one of the situations where we production rate is given by the form ∼ e − c/g , which is not a perturbative function in terms of a couplingconstant g , is typically a momentum and mass dependent quantity. The relation between broad and narrowresonance regime is similar to non-perturbative production and multi-photon processes in Sauter-Schwingereffect. – 2 –ecessarily evaluate non-perturbative particle production. In this regard, analysis of theStokes phenomenon is a mathematical method to investigate such particle production sys-tematically. In order to make the method more accessible, we will explicitly show how wecan use it with some simple examples. Also, we will reconsider whether particle productiontakes place in simple expanding backgrounds from the Stokes phenomenon viewpoint.The analysis based on the Stokes phenomenon has a rigorous mathematical background.However, it might not always be useful in practice if one would like to have an analyticestimation in the system with complicated time-dependent background. In cosmologicaland phenomenological models, we come across with such difficulties. Then, one may thinkthat the analysis of the Stokes phenomenon is not so useful for practical purposes unlessapproximation method is established. Therefore we examine some approximations explicitlyand discuss how accurately we may be able to estimate the particle production rate withinthe approximation. We will show that, depending on how accurately we analyze the Stokesphenomenon, or how accurate approximation we use for the time dependent background,we are able to improve the analytic estimation. In our examples, we find that the analysis ofthe Stokes phenomenon can actually give accurate estimation enough for practical purposesin cosmology as well as phenomenology.The rest of this paper is organized as follows. In Sec. 2, we discuss particle productionin simple expanding backgrounds. In such backgrounds, no particle production takes placeexcept one nontrivial example. We discuss the nontrivial one as the simplest example.Section 3 is devoted to develop a technique for approximation of complicated background.As a concrete example, we discuss the particle production (preheating) after R inflation [21]where Hubble parameter oscillates and curvature induced terms lead to particle production.In Sec. 4, we discuss the case that background quantity has infinite number of poles incomplex time plane besides the turning points, which makes the Stokes phenomenon analysiscomplicated. We first show the analysis without a pole contribution, and then improve itby taking into account a pole contribution. To the best of our knowledge, the Stokes lineanalysis with poles has not been shown explicitly in the context of particle production. Our analysis would tell us the importance of a pole contribution, in order to improve theestimate, or said oppositely, when we may neglect such a contribution. The detailed analysisof the pole contribution is given in Appendix C. We conclude in Sec. 5. For readers whoare not familiar with the Stokes phenomenon, we give a brief review of it in Appendix A,which contains minimal information necessary for this paper.Throughout this paper, we will take natural unit c = (cid:126) = 1 . [22] has discussed the contribution from poles in the context of quantum tunneling. In [16], the authortakes the pole contribution into account in models of nontrivial background geometries, but the formalismseems different from ours. For example, the notion of Stokes line is not discussed in [16]. – 3 – Particle production in simple expanding universe models: warm up
We consider a massive scalar φ coupled to the Ricci scalar R in expanding background ofwhich Lagrangian is given by S = − (cid:90) d x √− g (cid:2) g µν ∂ µ φ∂ ν φ + m φ + ξRφ (cid:3) , (2.1)where m and ξ are a scalar mass and a coupling constant, respectively. In flat expandingbackground ds = a ( η )( − dη + d x ) , where a is a scale factor and η is a conformal time dη = dt/a , the action of the scalar field is reduced to S = 12 (cid:90) dηd x (cid:20) χ (cid:48) − ( ∂ i χ ) − (cid:18) m a + (6 ξ − a (cid:48)(cid:48) a (cid:19) χ (cid:21) , (2.2)where a prime denotes the derivative with respect to the conformal time, and we haveintroduced a rescaled scalar χ ≡ aφ . The canonically quantized scalar χ can be expandedas follows: χ ( x , η ) = (cid:90) d k (2 π ) / (cid:16) ˆ a k v k ( η ) e i k · x + ˆ a † k ¯ v k ( η ) e − i k · x (cid:17) , (2.3)where ˆ a k (ˆ a † k ) denotes the annihilation (creation) operator that obeys the canonical com-mutation relation [ˆ a k , ˆ a † k (cid:48) ] = δ ( k − k (cid:48) ) , and v k ( η ) denotes the mode function. The equationof motion of the scalar leads to v (cid:48)(cid:48) k + (cid:18) k + m a + (6 ξ − a (cid:48)(cid:48) a (cid:19) v k = 0 . (2.4)We see that the effective frequency of the mode function is time-dependent, which impliesthat particle creation inevitably takes place due to the background a ( η ) .In order to discuss the particle production in time-dependent backgrounds, we willuse the WKB/phase integral method, which would give a universal way to analyze non-perturbative particle creation. For readers who are not familiar with this method and theStokes phenomenon, we briefly review these mathematical background in Appendix A. Inthe following, we give some qualitative review about the Stokes phenomenon. The formalsolution to the above mode equation is given by two basis functions f ± ( η ) = 1 √ ω k e ± i (cid:82) η dη (cid:48) ω k ( η (cid:48) ) , (2.5)where ω k = k + m a + (6 ξ − a (cid:48)(cid:48) a (2.6)is the effective frequency. In a sufficiently adiabatic region, ω (cid:48) k /ω k (cid:28) , the mode functionwould be well approximated by a linear combination of them as v k ( η ) = α k f − + β k f + , (2.7)where α k and β k are the Bogoliubov coefficients, which is normalized as | α k | − | β k | = 1 inscalar case. In particular, the adiabatic vacuum mode function is given by α k = 1 , β k = 0 at the initial time η = η . – 4 – latexit sha1_base64="ahMzxcvHhfg7Pdr8J4bqqCliz/Y=">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 ✓ f f + ◆
1) 2(1 − w ) H i (1 + 3 w ) ( H i η + 1) . (2.12)For concreteness, we consider the case w = 0 , / , which corresponds to matter, radiation,and kinetic energy dominated era, respectively. For a conformally coupled scalar field in matter dominated universe, w = 0 , ξ = , we havethe effective frequency ω k = k + m ( H i η + 1) . (2.13)In this case, turning points η c satisfying ω k ( η c ) = 0 appear at H i η + 1 = − (cid:114) k m (1 ± i) , (cid:114) k m (1 ± i) . (2.14)A schematic picture of the Stokes lines in this model is shown in Fig. 2. Since η = 0 is theinitial time, Stokes line crossing the real time axis before this time is unphysical. On theother hand, there is a Stokes line connecting the turning point H i η c = (cid:113) k m (1 ± i) − andits conjugate, which cross the real axis after η = 0 for k > m .– 6 – latexit sha1_base64="xtPWyD18zxNZ67brU6BKJFu/k1I=">AAACZHichVFNLwNBGH66vqqKIhKJRERDnJq3QoiTcHFUVSQ0ze4abOxXdqdNqvEHuBIHJxIR8TNc/AEHf0AijpW4OHh3u4nQ4J3MzDPPvM87z8xormn4kugpprS0trV3xDsTXcnunt5UX/+675Q9XRR0x3S8TU31hWnYoiANaYpN1xOqpZliQztYCvY3KsLzDcdek1VXFC11zzZ2DV2VTOWolEpThsIYbQbZCKQRxYqTusE2duBARxkWBGxIxiZU+Ny2kAXBZa6IGnMeIyPcFzhCgrVlzhKcoTJ7wOMer7Yi1uZ1UNMP1TqfYnL3WDmKcXqkW6rTA93RC338WqsW1gi8VHnWGlrhlnqPh/Lv/6osniX2v1R/epbYxVzo1WDvbsgEt9Ab+srheT0/vzpem6AremX/l/RE93wDu/KmX+fE6gUS/AHZn8/dDNanMtmZDOWm0wuL0VfEMYwxTPJ7z2IBy1hBgc8VOMEpzmLPSlIZUAYbqUos0gzgWygjn2eeibQ= (unphysical)
50 k / H i l n ( n k ) Figure 3 . Comparison between a numerical calculation (black dots) and analytic estimate (2.18)(blue line). The vertical line shows k = 2 m . Here, we take m = 0 . H i . We finally note that, although the “particle number” as a square of the Bogoliubovcoefficient asymptotes to a fixed value numerically and analytically, strictly speaking, the“particle production” discussed here does not make sense, since the effective mass of theparticle continuously grows during this era, and one particle state is not well defined.Nevertheless, the Bogoliubov coefficient, or more precisely, the connection matrix would beinherited by the asymptotically defined particle with stationary frequency. We discuss whether non-perturbative particle production can take place in other situa-tions. The answer is negative, and we find no particle production in the simple setup wediscuss below. The turning points show up in the following discussions correspond to knowndynamics of mode function in each case.
Minimally coupled massless scalar in matter dominated universe
Let us consider another simple case with minimally coupled massless scalar m = 0 , ξ = 0 inmatter dominated era w = 0 . In this case, the effective frequency is ω k = k − H i ( H i η + If the expansion terminates at some point, it would mean that scale factor changes its behavior, whichchanges the scale factor a ( η ) = ( H i η + 1) to some different form, which may lead to additional turningpoints and Stokes lines. Even in such a case, if the Stokes line we discussed and additional one is well sepa-rated, the connection problem we discussed is not affected, namely the Bogoliubov coefficient is preserved,but we also need to discuss the connection problem at the additional Stokes line crossing. Here we will notdiscuss such an issue since it requires to fix the scale factor evolution, namely the history of the universe. – 8 – ) − and small k modes seems to have instability ω k < . Such modes eventually findthe turning point H i η c + 1 = √ H i k . Even though this is Stokes phenomenon, we cannotinterpret it as particle production simply because we cannot define particle at the initialtime η = 0 . We notice that such an “instability” appears for the modes outside of thehorizon, which is reasonable since we cannot have a notion of particle for such modes. Wealso find that the turning point depends on k and its value becomes larger for smaller k . Wemay interpret the turning point to be the time around which the mode comes into horizon.Only after passing the turning point H i η c + 1 = √ H i k , ω k becomes real and we are able tointroduce the notion of particles. Similar behavior can be found in the kination era as wewill see below. We also note that in this case, one can solve the mode equation exactly, andcan check the behavior of the mode function. In radiation dominated universe
Let us consider the radiation dominated era w = 1 / , where the effective frequency is givenby ω k = k + m ( H i η + 1) . (2.20)In this case, the turning point is H i η c + 1 = ± i km . We find that the real part of the turningpoint is − /H i , where the scale factor is a ( η ) = 0 . If the radiation dominated era startsfrom a = 1 , this point will not be crossed and therefore we may identify this point to beunphysical turning points. Because of vanishing curvature contribution, the presence of thenon-minimal coupling does not change the structure of frequency ω k . From this observation,we can conclude that there is no gravitational particle production in radiation dominatedera. Kinetic energy dominated (kination) universe
In kination era, the equation of state parameter is w = 1 , and we have ω k = k + m ( H i η + 1) − (6 ξ − H i ( H i η + 1) − . (2.21)For conformally coupled case ξ = 1 / , a turning point appears at H i η + 1 = − k /m which is unphysical, and there is no particle production. The next simple example is thecase of ξ < / with m = 0 . In this case, again, the turning point appears at unphysicalregion, and no particle production takes place.For ξ > / with m = 0 , we find a turning point at H i η c + 1 = √ ξ − k on the real η -axis.We notice that, until that time, the square of the frequency becomes negative, which meansinstability for the mode. This is known as spinodal instability in kination era [23]. This“instability” is similar to that in matter dominated era discussed above. As is the case ofthe “tachyonic mode” in matter dominated era, until the time reaches the turning point, Strictly speaking, location of the turning points are not crucial, and the place of particle productionis determined by the point where Stokes line crosses the real axis. However, in this case, the Stokes lineconnecting two turning points are straight line, and therefore, the particle production associated with theturning points are unphysical. – 9 –e cannot have a well defined particle since the frequency is not real for H i η + 1 < √ ξ − k .Therefore, it is more appropriate to understand this turning point to be the time whenthe corresponding mode reenter the horizon. Only after the reentering, we can define aparticle, which is physically reasonable. If the scalar had quantum fluctuation prior to thekination era, its amplitude can be conserved until horizon reenter. After this point, themode start to oscillate around the minimum, which may eventually dominate the universesince the kination energy decays as a − . Such a mechanism has been applied to overcomethe graviton overproduction problem in [23]. Comments on the simple expansion models
As we have seen above, most of simple expanding universe models would not lead to non-perturbative particle production. One of the reason for the absence of the particle pro-duction is that there are only few turning points. As we will see in the following section,more nontrivial time dependent background will lead to non-perturbative particle produc-tion. One may expect it simply because more complicated time dependence may lead tomore turning points and Stokes lines, namely particle production “events”. Such a naiveexpectation is actually true in the following examples. R model: imitation method In this section, we discuss scalar particle production in R inflation model [21], which issupported by CMB observations [24]. The main purpose of this section is to show howapproximation of backgrounds and the Stokes phenomenon analysis based on the approx-imated expression can give us an analytic estimate with a good accuracy. Indeed, if onewants to analyze this model exactly, the Hubble parameter becomes very complicated andaccordingly the Stokes lines cannot be discussed analytically.In this model, one can use the Jordan frame as well as the Einstein frame by introducingscalaron, but for our purpose, we use the Jordan frame since all dynamics can be writtenin terms of gravity. After inflationary phase, the Hubble parameter and the Ricci scalarstarts to oscillate, which corresponds to scalaron oscillation in Einstein frame. We focusespecially on the case with a sizable non-minimal coupling to Ricci scalar, where the particleproduction in this phase becomes efficient. For the gravitational part of the action, weassume S g = M (cid:90) d x √− g (cid:18) R + R M (cid:19) , (3.1)where M denotes a mass parameter, which becomes scalaron mass in Einstein frame. The Since we are assuming non-minimal coupling, there are mixing terms between scalaron and the scalar φ besides the terms coming from metric. In Jordan frame, the scalaron mixing terms are included ingravitational coupling. – 10 –quations of motion for Hubble parameter and Ricci scalar are (see e.g. [25]) ¨ H − ˙ H H + 12 M H = − H ˙ H, (3.2) ¨ˆ R + (cid:18) M − H −
32 ˙ H (cid:19) ˆ R = 0 , (3.3)where ˆ R ≡ a / R . Here we have taken spacetime metric to be ds = − dt + a ( t ) d x ratherthan conformal one by the reason explained below, and the Hubble parameter is H = ˙ a/a .Inflation ends around t = t os where ˙ H = − M / , which is the initial condition of theoscillating phase. With such an initial condition and the equation of motion, we find H ( t ) = (cid:18) M + 34 ( t − t os ) + 34 M sin M ( t − t os ) (cid:19) − cos (cid:18) M t − t os ) (cid:19) . (3.4)Hereafter, we will take t os = 0 without loss of generality.We will consider particle production in this background. The action of the scalar φ isgiven in (2.1). So far, we have used conformal time to analyze the particle production. However, at this stage, usual time coordinate t is more appropriate for our analysis sincerewriting time evolution with conformal time makes various quantity more complicated.With the flat expanding coordinate ds = − dt + a ( t ) d x , the canonically normalizedscalar field action is given by S = 12 (cid:90) dtdx (cid:20) ˙ ψ − a ( ∂ i ψ ) − (cid:18) m + 32 (4 ξ −
1) ˙ H + 34 (16 ξ − H (cid:19) ψ (cid:21) , (3.5)where ψ ≡ a / φ . Accordingly, we quantize the rescaled scalar field as ψ = (cid:90) d k (2 π ) / (cid:16) ˆ a k u k ( t ) e i k · x + ˆ a † k ¯ u k ( t ) e − i k · x (cid:17) . (3.6)The mode equation for u k ( t ) becomes ¨ u k ( t ) + Ω k u k ( t ) = 0 , (3.7)where the effective frequency is Ω k = k a + m + 6 (cid:18) ξ − (cid:19) ˙ H + 12 (cid:18) ξ − (cid:19) H . (3.8)The effective frequency Ω k exhibits damped oscillation as shown in Fig. 4. When theeffective frequency exhibits damped oscillation, turning points and Stokes segments appear The reason why we have used the conformal time in previous discussion is that such coordinate simplifiesanalysis in cases where a scale factor is given by a simple power law. There, conformal time expressionavoids some branch cut appearing in effective frequency ω k , which leads to unnecessary complication. Equation (3.8) has the curvature induced term even when φ is ‘conformally’ coupled ( ξ = 1 / ). This isdue to our choice of frame. If we take the comoving frame and rescale the field as χ = aφ , the curvature-dependent term vanishes at ξ = 1 / . However, this difference is negligible when we consider a large ξ case. – 11 – ● ● ● ● ● Im t ●● ●● ●●
Re t Ω k Re t
Figure 4 . Lower panels: Damped oscillation of effective frequency. Upper panels: Correspondingturning points (black dots) and Stokes segments (red lines) in complex time plane. Dashed linesshow zeros (left panels) and minima (right panels) of the effective frequency. When the amplitudeof the effective frequency is so large that it can become tachyonic, its Stokes segments appear alongreal time axis. as depicted in Fig. 4. In case the amplitude of the effective frequency is still so large thatit can become tachyonic, Stokes segments lie along the real time axis where the effectivefrequency is tachyonic, and become shorter and shorter at later oscillation (left panels ofFig. 4). When the effective frequency can no longer become tachyonic after some times ofoscillation, Stokes segments are almost perpendicular to the real time axis, cross it near theminima of the effective frequency and become longer and longer at later oscillation (rightpanels of Fig. 4). Each Stokes line crossing corresponds to particle production “event”, andthe amount of production at each one can be evaluated as explained in Appendix A.1.1and A.1.2, respectively. The physical interpretation of this mathematical structure is thatparticles are produced mostly when they become tachyonic or lightest as naturally expected.Note that this model has a similar structure as that discussed in [26], where preheating fromtrilinear interaction is discussed and the tachyonic regime dominates particle production. Since the particle production is more efficient in the tachyonic regime, in the following,we will focus only on that regime. The WKB/phase integral solution is u k ( t ) = α k ( t ) √ k e − i (cid:82) t dt (cid:48) Ω k ( t (cid:48) ) + β k ( t ) √ k e i (cid:82) t dt (cid:48) Ω k ( t (cid:48) ) . (3.9)Here, the Bogoliubov coefficient β k is important to evaluate the particle number. In Ap-pendix A.1.1 we show how the Bogoliubov coefficient β k is calculated after passing a Stokes Our model has a more complicated effective frequency, with which we will discuss how to make ananalytic estimation. – 12 –egment in tachyonic regime. With the approximate formula (A.31), β k is evaluated by theintegral during a tachyonic region: | β k | = exp (cid:32)(cid:90) t c, n +1 t c, n | Ω k | dt (cid:33) , (3.10)where t c, n and t c, n +1 ( n = 0 , , , · · · ) denote the ( n + 1) -th pair of turning points. There-fore, particle number can be simply given by integrating the effective frequency (3.8) alongthe tacyonic Stokes segments, in other words, the area of the red shaded region in thelower panel of Fig. 5. The particle number derived from the Bogoliubov coefficient (3.10) isdepicted in Fig. 6. In this figure, we calculate the Bogoliubov coefficient (3.10) by numer-ical integration of the effective frequency (3.8) and compare the resultant particle numberwith the one obtained by solving the equation of motion (3.7) numerically. Both coincidewith each other, which means that analytic estimation can be given by performing theintegral (3.10). Re t Ω k Re tIm Ω k Figure 5 . The time evolution of the squared value (upper panel) and the imaginary part (lowerpanel) of the effective frequency Ω k . The right-hand side of (3.10) corresponds to the area of thered shaded region. Let us discuss how to make an analytic estimate for the particle production in thissituation. Since the Hubble parameter given in (3.4) is too complicated, it seems difficultto find turning points and to integrate the effective frequency along Stokes lines analytically,although numerical evaluation is possible as shown in Fig. 6. We note that such a situationwould always be the case in realistic models, where time dependent backgrounds takevery complicated form. Therefore, we should make a strategy to overcome this difficulty:First, we look for an approximated frequency, which has an analytically accessible formand imitates the shape of the exact one. This strategy is justified by the fact that (A.31)depends not on the detail (e.g. higher order derivative) of the effective frequency but onlyon its shape. Therefore, we use an approximated form of Hubble parameter as well as scalefactor instead of the exact expression (3.4). We numerically integrated (3.4) and found that– 13 –
10 20 30 40024681012 k / m l n ( n k ) Figure 6 . The particle number produced by the first oscillation of the Ricci scalar. Black dotsand the blue solid line depict the numerical result obtained by solving the equation of motion withthe effective frequency (3.8) and the approximation (A.31), respectively. In this graph, we take M = 15 . m, ξ = 10 . . This shows that our approximation formula (A.31) actually well describesthe particle production rate. a ( t ) = (1 + M t/ is a very good approximation. Also, the effective frequency (3.8) canbe approximated as Ω k ( t ) ≈ k (1 + M t/ + m − (cid:18) ξ − (cid:19) M ( M t + 4) sin(
M t ) (3.11)up to the order of /t in the curvature induced term (the last term in right-hand side).Although we see a some discrepancy between this approximated form and the exact onederived from (3.4) as shown in Fig. 7 for small M t , this approximation becomes better andbetter in later time
M t (cid:29) . In the following, we discuss the particle production withthis approximated formula and will show that an analytic estimation based on (3.11) is inagreement with particle number derived from numerical integration with a more complicatedform (3.4).At late time oscillation M t (cid:29) , where the effective frequency (3.11) is a good approx-imation (Fig. 7), we can further approximate (3.11): Ω k ( t ) ≈ k ( M t/ + m − (cid:18) ξ − (cid:19) Mt sin( M t ) (3.12)for simplicity. Since we are now considering the tachyonic regime, in other words, the largeamplitude case, we can take ∆ ≡ (cid:18) ξ − (cid:19) − k p + m M M t, (3.13)where k p ≡ k ( Mt/ is a physical wave number, as a perturbation parameter much smallerthan unity. Under such assumptions, we can analytically evaluate the integration (3.10)– 14 – - - - Mt Ω k = / M Figure 7 . Comparison between the curvature induced terms of the exact effective frequencyderived from the Hubble parameter (3.4) (blue dashed line) and one of the approximated form (3.11)(orange solid line) In this graph, we take k = 0 , m = 0 , ξ = 10 . . The approximation becomes betterand better as time passes. and obtain approximated particle number produced at n -th oscillation as n k,n ≈ exp (cid:34) (cid:115) ξ − n + ) π (cid:16) (cid:16) π (cid:12)(cid:12)(cid:12) (cid:17) − F (cid:16) π (cid:12)(cid:12)(cid:12) (cid:17) ∆ n + (cid:17)(cid:35) , (3.14)where ∆ n + is the value of the small parameter (3.13) at M t = (2 n + ) π , E( ϕ | k ) and F( ϕ | k ) the incomplete elliptic integral of the second kind and the first kind, respectively.The derivation of this formula is given in appendix B.1. This analytic estimate showsa good agreement with the numerically evaluated exact one at late time oscillation asdepicted in Fig. 8 as long as M t (cid:29) , | ξ | (cid:29) and the amplitude of the Ricci scalar is large.The discrepancy between numerical result and analytic estimate (3.14) for low momentummode is at most O (1) , which originates mainly from the discrepancy between our simplifiedeffective frequency (3.12) and the exact one given by (3.4) as explicitly shown in Fig. 7.At early time oscillation, especially at the first oscillation, the particle number (3.14) isnot a good approximation as shown by the blue dashed line in Fig. 10. This is simply becausethe approximated effective frequency (3.11) no longer approximates the exact one (3.8) well(Fig. 7). In such a situation, we can improve our estimation by adopting a better effectivefrequency approximated up to the order of /t : Ω k = k (1 + M t/ / + m − (4 ξ − M M t + 4 sin(
M t ) + 4 ξM ( M t + 4) (3.15)instead of the previous one (3.11). This improved approximation shows even better agree-ment with the exact one as depicted in Fig. 9. We can make an analytic estimate in thesame way as done in previous case, and the analytic estimate agrees with the numericalresult even at the first oscillation as shown in Fig. 10. In appendix B.2, we put the detailedderivation of analytic estimation used in Fig. 10. Thus, we have explicitly shown that we are– 15 –
20 40 60 80 100 - - - k / m l n ( n k ) Figure 8 . Comparison between the numerical result from (3.4) and (3.8) (black dots) and (3.14)(blue solid line) at the sixth oscillation n = 5 . In this graph, we take M = 15 . m, ξ = 10 . . able to find an analytic estimate even if the background is very complicated, and accuracyof the estimate is directly related to that of the effective frequency as we naively expected.We note that the reason why we did not take (3.15) from the beginning is that we wouldlike to explicitly show how the accuracy of the approximation of the effective frequencyaffects the estimate. - - - Mt Ω k = / M Figure 9 . Comparison between the curvature induced terms of the exact effective frequencyderived from the Hubble parameter (3.4) (blue dashed line), one of the approximated form (3.11)(orange dotted line) and the improved approximation (3.15) (red solid line). In this graph, we take k = 0 , m = 0 , ξ = 10 . . The improved one (3.15) well approximates the exact one even at the firstoscillation. This particular example gives an important lesson: In estimating produced particlenumber, we have first used an approximated form of the effective frequency (3.11), whichgives a similar shape to one given by a (more complicated) exact one (3.4). Although– 16 –
10 20 30 400510152025 k / m l n ( n k ) Figure 10 . Comparison between the first approximation (3.14) (blue dashed line) and the im-proved one (B.15) (red solid line) at the first oscillation n = 0 . Black dots denote the numericalresult from (3.4) and (3.8). In this graph, we take M = 15 . m, ξ = 10 . . The formula (3.14) over-estimates the particle number, whereas (B.15) is a good approximation even at the first oscillation. the detailed functional forms of the exact one and the approximation are different, ourestimation is in good agreement with numerical results. This means that as long as theeffective frequency behaves similarly, one may use approximated expressions. Actually,the particle number (3.10) depends only on the area of tachyonic region of the effectivefrequency. This seems very reasonable since a particle φ feels the similar background, butit is not so trivial since the detailed Stokes line structure could be different. In general,complicated background such as (3.4) may give multiple turning points, which do notshow up in an approximated one. What we have shown in this example implies thatas long as we could capture important Stokes lines and turning points, the estimation ispossible by approximating the background. This lesson is very important for practicalpurposes since in various models, the time dependence is very complicated and it is almostimpossible to analyze the Stokes line structure. We should, however, note that the accuracyof the approximation of the background is very important since the particle density isbasically given by exponential of phase integral, namely the number density has exponentialsensitivity to the effective frequency. This point is clear from Fig. 10. In this section, we discuss particle production associated with the transition of the domi-nant energy components in the universe. When the dominant energy component changese.g. from matter to radiation, or vise versa, the expanding metric changes and such a tran-sition is known to induce gravitational particle production [19, 20]. In particular, in suchparticle production, the transition time scale plays a crucial role in determining the num-ber of produced particle; it yields the exponential suppression for heavy or high momentumparticles [27–29]. – 17 –hen we consider a transition e.g. from inflation to kination dominated era, a modelwith some finite transition time scale ∆ η is considered, where the scale factor is given by a ( η ) = 12 (cid:18) − tanh η ∆ η (cid:19) f ( η ) + 12 (cid:18) η ∆ η (cid:19) g ( η ) . (4.1)Here f ( η ) and g ( η ) describe the expansion law in old and new phase, respectively. Thetransition law changes around η = 0 within a (conformal) time scale ∆ η . For instance,production of a conformally coupled massive scalar at the transition from inflation to kina-tion was discussed in [28, 29]. For a conformally coupled massive scalar case, the effectivefrequency is given by ω k = k + m a = (cid:18) k + m f + g (cid:19) − m f − g η ∆ η . (4.2)If ∆ η is sufficiently small compared to the Hubble scale at the transition, the particleproduction is dominated by the last term.We would like to reconsider such a particle production, particularly focusing on Stokesphenomenon. However, in general, such a transition model is not so easy to study analyti-cally. Besides that, it would be rather instructive to explain in detail how the WKB/phaseintegral method can be applied to such a model. Therefore, instead of more involved tran-sition models, we consider a well known toy model with the following effective frequency,which was first investigated in [30]: ω k = k + m e − t ∆ t . (4.3)The effective frequency changes from ω i = k to ω f = k + m around t = 0 within thetime interval ∆ t . This toy model would give us an insight on the transition of the scalefactor between different era. The time interval ∆ t would correspond to the transition timescale, which may be very short, and then the effective mass coming from e.g. a curvatureterm can change suddenly. In this toy model, even if the transition is very sudden, we areable to discuss the particle production rate since the exact solution of this model is known,which yields β k = (cid:114) ω f ω i Γ[1 − (2i ω i ∆ t )]Γ(2i ω f ∆ t )Γ(i ω − ∆ t )Γ[1 + (i ω − ∆ t )] , (4.4) n k = | β k | = sinh ( πω − ∆ t )sinh(2 πω i ∆ t ) sinh(2 πω f ∆ t ) , (4.5)where ω − = ω f − ω i . We note that the effective frequency (4.3) is realized as that of aconformally coupled massive scalar field in a expanding universe where the metric is givenby ds = a ( η )( − dη + d x ) and a ( η ) = e − η ∆ η . This interpretation would not be realistic, since in such a spacetime, η → −∞ is an initial singularity,and we are not sure if a scalar particle is well defined. However, our analysis can be applied to the casewhere ω k = k + M + ∆ m ( A + B tanh η ∆ η ) by a simple replacement. This corresponds to the scale factor a ( η ) = ( A + B tanh η ∆ η ) . There, the initial singularity is absent unless A ± B > . – 18 –ith possible generalization to expanding universe in mind, let us calculate the Bogoli-ubov coefficient β k with the WKB method, and check whether we can capture the particleproduction from the transition. In particular, we first discuss a simple approximated for-mula, which is often used and gives a certain limit of the exact one (4.5). However, we willfind an obvious contradiction. We also show how to improve the formula by careful analysisof the Stokes lines.In this case, turning points appear at t c = (cid:20) i(2 n + 1) π − log (cid:18) k + m k (cid:19)(cid:21) ∆ t, (4.6)where n is an arbitrary integer. Besides the turning points, the effective frequency (4.3) hasan infinite tower of poles at t = i(2 n + 1) π ∆ t . We should emphasize that the appearance ofinfinite number of poles and turning points lead to infinite number of Stokes lines as shownbelow, which make analysis quite complicated in general.How can we evaluate the particle production in the presence of infinite number ofturning points? Let us take the following first approximation: Although an infinite numberof turning points and poles appear, the most relevant one for particle production is a pair ofturning points closest to the real time axis [31]. These turning points, poles and Stokes linesare depicted in Fig. 11. Neglecting the Stokes lines associated with poles and turning points ●●● ××× - - - - - - Re ( t / t ) I m ( t / Δ t ) Figure 11 . Structure of turning points (black dots), poles (black cross marks) and Stokes lines(blue lines) in the model with (4.3). In this plot, we take ω f = 2 ω i . The only Stokes line crossingreal time axis is the one connecting the pair of the turning points closest to real t -axis. with n (cid:54) = 0 , − , the situation is the same as the model in Appendix A.1.2. As explained One would need to modify the frequency so that Stokes segment disappears as explained in Ap- – 19 –n Appendix A.1.2, integration along the Stokes line gives the Bogoliubov coefficient arisingfrom the Stokes line crossing, and accordingly a produced particle number density, whichin this case is given by n k = exp (cid:32) (cid:90) ¯ t c t c ω k dt (cid:33) = e − πω i ∆ t . (4.7)We find this corresponds to (4.5) in the adiabatic limit ∆ t → ∞ while ω i ∆ t fixed. If thetransition is abrupt ω i ∆ t (cid:46) , this WKB solution overestimates ( ω f (cid:38) ω i ) or underesti-mates ( ω f (cid:46) ω i ) the particle number (see Fig. 12). Nevertheless, this result is consistentwith high momentum behavior of the exact solution. The reason can be understood interms of Stokes lines as follows: In the limit ∆ t → ∞ , the Stokes line emanating from theturning points nearest to the real axis seems like a single line and the contributions fromother turning points and poles are negligible because the interval is proportional to ∆ t .Physical interpretation is that high momentum modes do not feel any event caused by thetransition of the mass, and its production rate is independent of mass parameter m . ω i Δ t W(cid:0)(cid:1) / e(cid:2)(cid:3)(cid:4)(cid:5) ω f = ω i ω f = ω i ω f = ω i ω f = ω i Figure 12 . Ratio of the WKB solution (4.7) to the exact one (4.5) with various ω f . At theadiabatic limit ∆ t → ∞ , the WKB solution corresponds with the exact one. Although (4.7) gives an approximate formula consistent with exact solution in adiabaticlimit, the approximated formula given by the integration along the Stokes line connectingturning points gives non-vanishing value even in the limit m → , which is obvious contra-diction. In this sense, the formula (4.7) needs to be improved.How can we improve the formula at least to avoid the obvious contradiction? Themissing contribution here is that of poles neighboring turning points. Actually, takingmore careful treatment of pole contribution to Stokes lines and Stokes constants, we areable to obtain an improved formula n k = (cid:0) − e − πω − ∆ t (cid:1) e − πω i ∆ t . (4.8)Since the derivation of the improved formula is more involved, we put the details of itsderivation in Appendix C. The improved formula (4.8) shows at least a consistent behavior pendix A.1.2. In the following, we simply use the result in Appendix A.1.2. – 20 –n the limit m → , at which the produced particle number vanishes as expected sincenothing happens in the limit. In the adiabatic limit k ∆ t > , we find an expected behavior n k ∼ e − πω i ∆ t . (4.9)Therefore, the formula (4.8) in some limits shows the expected behavior. We show thebehavior of the improved particle density (4.8) and exact one (4.5) with m = 1 , ∆ t = 0 . and m = 1 , ∆ t = 1 in Fig. 13. As expected, the high momentum behavior is preciselyreproduced. However, for small ∆ t or low momentum modes, the WKB one underestimatesthe particle density. This is reasonable because in both cases, the adiabaticity is violatedsignificantly. Nevertheless, for relatively large momentum limit, the improved WKB formulagives a good approximation for the estimation of the produced particle density.We note that similar models were discussed in [16] where the author used a differentmethod to evaluate the amount of particle production due to the Stokes phenomenon, withfocusing on the contribution coming from the poles in complex time plane. In our analysis,we followed a standard method in evaluating the Stokes constant and continuation of modefunctions.As we have shown, even in this simple model, one needs to evaluate the Stokes constantand to consider the connection problem in order to find out the correct particle productionrate. Nevertheless, we have also shown that such an analysis can in principle give a verygood approximation for the produced particle number. Even in more involved situation,similar discussion would enable us to have an analytic estimation.However, we should mention that in the limit ∆ t → , even the improved formula (4.8)fails to reproduce the exact one (4.5) as can be seen e.g. from m ∆ t = 0 . case in Fig. 13. Inappendix C, we have taken into account only the nearest two turning points and a pole. Thisis not enough, however. In the sudden transition limit ∆ t → , we can no longer neglectother turning points and poles since the distance between them is scaled by ∆ t . If onewould like to consider such a limit, the evaluation with Stokes lines seem not appropriateand different methods would be necessary. Nevertheless, we have shown that WKB/phaseintegral method provides an analytic estimate with a good accuracy unless the transitiontime scale is too small. In this paper, we have demonstrated how we can analytically evaluate particle productionin time-dependent backgrounds by focusing on the Stokes phenomenon. The Stokes phe-nomenon analysis clarifies how we should connect solutions of a second order differentialequation in regions that have different asymptotic forms of solutions. Since the origin ofparticle production is nothing but the difference between the asymptotic behaviors of modefunctions at early and late time, particle production can be systematically analyzed fromthe Stokes phenomenon viewpoint. More concretely, in most cases, the amount of producedparticle can be evaluated by the phase integral along a Stokes line connecting a pair ofturning points, at which the effective frequency vanishes, as shown e.g. in Eqs. (3.10) (intachyonic case) and (4.7) (in non-tachyonic case).– 21 – - - k / m Log n k m Δ t = - - - k / m Log n k m Δ t = Figure 13 . Comparison of particle number density derived by WKB method (4.8) and the exactone (4.5) for and m ∆ t = 0 . (left) or m ∆ t = 1 (right).The blue solid line is our improved WKBresult (4.8) and the red dashed line is exact one (4.5). Exact forms of effective frequency in most realistic models such as that in Sec. 3 aretoo complicated to be integrated analytically. Furthermore, even if we find turning pointsand Stokes lines, the number of them can be infinite as the model in Sec. 4. In such cases,complete analysis of Stokes phenomenon seems impossible or impractical. Therefore, forpractical purposes, we need to know whether we can use approximations in order to obtainanalytic estimations.In this paper, we have addressed this issue and discussed possible approximationsfor practical use of Stokes phenomenon analysis. In Sec. 3, we have discussed whetherthe approximation of the effective frequency can be used to make an estimate of particleproduction. We have found that even simplified effective frequency can give us an estimationwith satisfactory accuracy. We have also shown that our estimation can be further improvedby making the shape of an approximated effective frequency more resemble that of the exactone. In Sec. 4, we have discussed how to treat an infinite number of the turning pointsas well as poles. Despite the infinite number of them, we have found that the Stokes lineconnecting the turning points nearest to the real time axis gives a main contribution tothe particle production. As shown in Sec. 4, taking into account only this Stokes line givesus an estimate consistent with adiabatic limit. Furthermore, we have also found that theestimate can be improved by taking into account the contribution from poles . Although wehave only taken into account one pole contribution among infinite number of them, suchan analysis still gives satisfactory estimation.Our observations mean that, for analytic estimates, we do not need exact functionalform of background nor analysis of all the Stokes lines and poles. In order to make ananalytic estimation, an approximated effective frequency and the contribution from theStokes line and pole that are closest to the real time axis are enough to evaluate theamount of particle production. Our result enables us to analyze particle production evenin more complicated situation including not only gravitational particle production but alsogeneral non-perturbative production such as Schwinger effect and Hawking radiation.In conclusion, produced particle number can be well estimated in many cases by thefollowing recipe: Approximate the effective frequency so that it can be integrated analyti-– 22 –ally. An approximation works as long as a shape is similar to the exact one. Find turningpoints and Stokes lines of the approximated effective frequency in the complex time plane.Evaluate the phase integral of the approximated effective frequency along the Stokes seg-ments and we will obtain an estimate of produced particle number. If there are infinitenumber of Stokes lines, take into account the one nearest to the real time axis, as firstapproximation. In the presence of poles, as first approximation, take the one closest to areal axis into account as we have done in appendix C.We admit that this method fails at non-adiabatic limit such as abrupt transition(e.g. ∆ t → limit in Figs. 12 and 13) since this method starts from the WKB solu-tion, namely, adiabatic approximation. Although even in such a case we can improve ourestimation by taking into account the contribution from poles and other Stokes lines, theremight be more appropriate mathematical method for this situation. We leave it for ourfuture study.Finally, we would like to mention possible future directions. The time evolution of theBogoliubov coefficients even in intermediate regime e.g. during transition, can be describedby an optimal approximation shown by Dingle and derived by Berry [32] (for review, seee.g. [10, 17]). With our approximation methods, such a time dependent formula allows usto discuss, for instance, backreaction from particle production to the background. It wouldalso be interesting to see the relation to world line instanton method [33]. We expect thatsuch a different formulation gives us a complementary insight on the particle production incosmological models. We will address these possibilities in future. Acknowledgement
We would like to thank Jun’ichi Yokoyama for discussion. SH is supported by JSPS KAK-ENHI, Grant-in-Aid for JSPS Fellows 20J10176 and the Advanced Leading Graduate Coursefor Photon Science (ALPS). YY is supported by JSPS KAKENHI, Grant-in-Aid for JSPSFellows JP19J00494.
A Review of WKB/phase integral method and Stokes phenomenon
In this appendix, we give a brief review of the phase integral method with particular focuson the Stokes phenomenon, which is crucial for understanding non-perturbative particleproduction. For a more comprehensive review, see e.g. [34]. We also note that recently theexact WKB analysis is also applied to particle production context [11, 15]. The exact WKBanalysis is mathematically more rigorous, but for our practical purposes, the usual WKBmethod seems sufficiently useful, and we will use it here. The phase integral method can bethought of generalization of the WKB method. To discuss the particle production in time-dependent background including curved spacetime as well as an oscillating scalar field, weneed to solve the mode equation of target particle, but in general finding exact solution isquite difficult or impossible. Let us consider the following second order differential equation, ¨ ψ ( t ) + ω ( t ) ψ ( t ) = 0 , (A.1)– 23 –here ω ( t ) corresponds to a time dependent frequency. Let us consider the following formalsolution ψ ( t ) = 1 (cid:112) q ( t ) exp (cid:18) ± i (cid:90) t dt (cid:48) q ( t (cid:48) ) (cid:19) . (A.2)Here, q ( t ) is an unspecified function, which should satisfy q − / d q − / dt + ω q − . (A.3)If such q ( t ) is found, this means that we find an exact solution, but as mentioned above, itis not always the case. Instead, let us consider an approximate solution. We consider anapproximated solution in the form of ψ ( t ) ≈ (cid:112) Q ( t ) exp (cid:18) ± i (cid:90) t dt (cid:48) Q ( t (cid:48) ) (cid:19) , (A.4)where Q ( t ) is a chosen function. In order to quantify how small the error from the exactone is, we introduce the following quantity ε ( t ) ≡ Q − / d Q − / dt + ω Q −
1= 116 Q (cid:34) (cid:18) dQ dt (cid:19) − Q d Q dt (cid:35) + ω − Q Q . (A.5)If | ε | (cid:28) , the approximated solution (A.4) is a good approximation. In principle, onecan choose Q ( t ) to be whatever function as long as the approximation is valid, | ε | (cid:28) .The WKB method corresponds to the special choice Q ( t ) = ω ( t ) . Sometimes, this freedomof the choice of Q ( t ) is useful to have a better approximation than WKB one. We mayinclude the terms higher order in ε and its time derivatives to improve the approximation.The role of higher-order terms are discussed e.g. in [35]. However, for simplicity, we donot discuss the improvement with higher-order contributions here. We note that one wouldbe able to construct a mode function including infinite series of higher order terms, whichseems an exact solution. This is not true since it is an asymptotic series. In this regard,one needs to take into account Stokes phenomenon, which is explained below. See [34] formore detailed discussion about higher order terms.Once we find a good approximated solution, we expect that the solution can be contin-ued along whole (real) time axis. However, this is not really true in general. If we extend thetime t to be complex, generally there are points Q ( t ) = 0 , which are called turning points.At turning points, the error (A.5) becomes large and the approximated solution (A.4) isinvalid there. Such behavior may also happen if Q ( t ) has poles in complex t plane. Moreimportantly, from turning points, the so-called Stokes and anti-Stokes lines emanate, whichgo to infinity or end up at the poles or other turning points. The Stokes and anti-Stokeslines are defined as follows: Stokes line: Q ( t ) dt = pure imaginary , (A.6)anti-Stokes line: Q ( t ) dt = real . (A.7)– 24 –ote that all the Stokes and anti-Stokes lines from the same turning point never inter-sect with each other. Stokes line is particularly important for discussion of the particleproduction. On the Stokes line, exp( ± i (cid:82) Q ( t ) dt ) increases or decreases significantly. It iswell known that WKB/phase integral solution is given as asymptotic series and definedonly locally. The Stokes line can be thought of the boundary of the regions, each of whichhas locally defined solutions. It is what we call
Stokes phenomenon how locally definedsolutions are connected around the Stokes line. It is important to note that even if turningpoints are not on the real time axis, Stokes lines may cross the real time axis and lead tothe change of the mode function behavior in real time, which can be understood as “particleproduction” as we will discuss in detail later.We briefly summarize the Stokes phenomenon as follows. The two basis solutions aregiven by f ± ( t ) = 1 (cid:112) Q ( t ) exp (cid:18) ± i (cid:90) tt Q ( t (cid:48) ) dt (cid:48) (cid:19) , (A.8)and t is some point in complex t -plane. Suppose there is a simple turning point aroundwhich Q ( t ) can be expanded as Q ( t ) ∼ A ( t − t c ) + O (( t − t c ) ) , (A.9)where A is a non-vanishing constant and t c is a turning point. Let us take the lower limitof phase integral to be t c , and then we find i (cid:90) tt c Q ( t (cid:48) ) dt (cid:48) ∼ i √ A ( t − t c ) . (A.10)Therefore, there are three Stokes and three anti-Stokes lines respectively. More specifically,parametrizing t as t − t c = re i θ where r and θ are real ( < r, ≤ θ < π ), the Stokesline correspond to θ = π , π, π , and anti-Stokes lines θ = 0 , π , π . Note that this isapproximation valid only around t = t c and the lines are in general not a straight line.Schematically, the structure of Stokes and anti-Stokes lines around a turning point aredepicted in Fig. 14. The Stokes lines separates the surrounding region into three parts ingeneral. As we show in Fig. 14, each Stokes line is characterized by whether i (cid:82) tt c Q ( t (cid:48) ) dt (cid:48) increases or decreases on the line, which is specified by + or − signs in Fig. 14. Thesesigns are determined by our choice of the phase of Q ( t ) around the turning points. Forinstance, along the minus Stokes line, f + decreases exponentially whereas f − increases. Inthis case, we call f − to be dominant, and f + subdominant. For the plus Stokes line, thedominant and subdominant components are opposite. When crossing a anti-Stokes line, thedominant and subdominant component changes. For example, in Fig. 14, f − is dominant inthe region 2 and becomes subdominant in region (cid:48) . One may also think that f + becomes This is more precisely discussed in the context of the exact WKB analysis. In the exact WKB analysis,one considers the Borel resummation of the asymptotic series, which gives the exact solution. However, onthe Stokes lines, the solutions are not Borel summable, and in this sense, the Stokes line is actually theboundary of the locally defined solutions. – 25 – egion 1region 2region 3region 2’ region 3’ region 1’ ー+ ー
Figure 14 . Stokes and anti-Stokes lines around a turning point. The center dot is a turningpoint, red dotted (blue dashed) lines represent (anti-)Stokes lines. The shaded line correspond to abranch cut. The plus and minus signs mean that i (cid:82) tt c Q ( t (cid:48) ) dt (cid:48) increases ( + ) or decreases ( − ) alongthe Stokes line. dominant because it is near a plus Stokes line. When we consider the continuation ofthe solution in regions separated by a Stokes line, the Stokes phenomenon takes place. Webriefly summarize the rules as follows:1. When we continue the solution separated by a minus (plus) Stokes lines in a counter-clockwise sense, the basis solutions change as f ∓ → f ∓ + Sf ± , f ± → f ± , (A.11)where the upper (lower) sign is for a minus (plus) Stokes line. Here S is called theStokes constant, which is determined by consistency conditions. We also note thatthe Stokes constants are assigned to each Stokes line. For instance, the connectionfrom region 1 to region 2 leads to f − → f − + Sf + whereas f + → f + . When solutionsare continued in clockwise direction, the sign of S becomes opposite S → − S .2. When crossing a Stokes line emanating from a different turning point, one has tochange the lower limit of the phase integration, and then to apply the above rule.More specifically, suppose we have two turning points t a and t b . When we considerthe Stokes line crossing, which emanates from t a , we take the basis solutions to be f ± = √ Q exp( ± i (cid:82) tt a Qdt (cid:48) ) . If we next consider the continuation of a solution to theregion separated by a different Stokes line emanating from t b , we should change the In this sense, only Stokes lines are important to know which component is dominant as well as to knowwhere Stokes phenomenon takes place. Therefore, one can focus only on Stokes lines. Here, we have shownanti-Stokes lines to clarify its role. – 26 –asis solutions as f ± = e ± K exp (cid:18) i (cid:90) tt b dtQ ( t ) (cid:19) ≡ e ± K ˜ f ± , (A.12)where K = i (cid:82) t b t a Q ( t ) dt connecting the two turning points. When crossing the Stokesline emanating from t b , the connection rule explained above is applied to ˜ f .3. When we consider the crossing of a branch cut in a counterclockwise sense, the basissolutions change as f ± → − i f ∓ . (A.13)One of the important consequences of the first and the third rule is the following: Supposea turning point is well separated from any other turning points and poles, and the corre-sponding Stokes lines are of the form shown in Fig. 14. Let us consider the connection fromregion 1 to region (cid:48) in the counterclockwise direction. One can easily confirm that theconnection is given as (cid:32) f + f − (cid:33) → (cid:32) S (cid:33) (cid:32) S (cid:48) (cid:33) (cid:32) S (cid:48)(cid:48) (cid:33) (cid:32) f + f − (cid:33) , (A.14)where S , S (cid:48) and S (cid:48)(cid:48) are Stokes constants of each Stokes line. On the other hand, we alsoknow that when we cross the cut, the connection is given by (cid:32) f + f − (cid:33) → (cid:32) (cid:33) (cid:32) f + f − (cid:33) . (A.15)Since these two connection rules should be the same, we find a consistent solution to be S = S (cid:48) = S (cid:48)(cid:48) = i . (A.16)This is one of the examples how we fix the Stokes constants from consistency conditions.We should emphasize that this result is very much used when we take the well-separatedturning points approximation. If the turning points are well separated, we may simplytake the Stokes constant to be i for a counter-clockwise connection (or − i for a clockwiseconnection). A.1 ExamplesA.1.1 Particle production from instability
Let us show some examples of connection problems in order to understand how the aboverules. As a first example, we consider the following effective frequency ω ( t ) = t − a , (A.17) This rule originates from the single-valued-ness of the original differential equation. We introducedthe branch cut is for Q ( t ) , which was introduced to construct our approximate solution. Such a branchcut, however, does not appear for an exact solution. Therefore, the basis solutions e.g. in the first and thesecond Riemann sheet should be related to each other. – 27 –here a is a real constant. This toy model describes a particle which experiences instabilityduring time interval ( − a, a ) . Although the notion of the particle is not well-defined duringinstability phase, we think of the amplification of the mode function as “particle production”.When we discuss Stokes lines associated with the effective frequency ω ( t ) which is realon the real t -axis, there can be the Stokes lines connecting two turning points, which we calla Stokes segment . Actually, in this toy model, there is a Stokes segment connecting t = − a and t = a on the real t -axis. As discussed e.g. in [11], such a segment makes it difficultto analyse the connection problem. Therefore, we consider a prescription (see e.g. [11]),where we introduce an infinitesimal phase to the parameter a → ae i η and η (cid:54) = 0 is a real(small) parameter. Such a modification leads to deformation of the Stokes line structureand as a result, the Stokes segment is decomposed into several Stokes lines. Dependingon the sign of η , we have two different types of diagrams as shown in Figs. 15 and 16.As we will see below, the difference of the two choices disappear at the leading order ofthe semiclassical analysis, as shown in [11]. In other words, such a difference cannot beremoved if the semiclassical/adiabatic approximation is invalid, and we may think of it asthe limitation on the WKB/phase integral method.Let us discuss the connection problems for small positive and negative η separately.The sketch of Stokes lines for each case are shown in Figs. 15 and 16, respectively. Inboth cases, we take the initial adiabatic vacuum mode function to be ψ ( t ) = 1 (cid:112) ω ( t ) exp (cid:18) − i (cid:90) tt ω ( t (cid:48) ) dt (cid:48) (cid:19) = ( t, t ) , (A.18)where t ( (cid:28) − Re( ae i η )) denotes the initial time, and we have used a shorthand notation ( x, y ) ≡ √ ω exp (cid:0) i (cid:82) yx ωdt (cid:1) introduced in [36].Let us first consider the case with η > ( | η | (cid:28) ) shown in Fig. 15. In this case, theStokes line we first cross emanates from a turning point t = − ae i η ≡ − t c . Then, one needsto change the lower limit of the phase integral according to the second rule, and the modefunction is written as ψ = [ − t c , t ]( t, − t c ) , (A.19)where we have introduced a shorthand notation [ a, b ] = exp (cid:16) i (cid:82) ba ωdt (cid:17) . Since the firstStokes line has a positive sign, ( t, − t c ) is a subdominant, and therefore, the continuation isdone without any change, ψ → ψ (cid:48) = [ − t c , t ]( t, − t c ) . (A.20)The second Stokes line we come across with emanates from the turning point t = t c , so weneed to change the lower limit of the phase integration. Thus, we change ψ (cid:48) = [ t c , t ]( t, t c ) . (A.21) In [11], this statement is explained in the context of the exact WKB analysis. We also find the similar(or essentially the same) difficulty. The Stokes lines in these figures are described as straight lines, but actual ones are not straight butcurved. They are shown only for an illustrative purpose. – 28 – latexit sha1_base64="7+qSLhx25l0hZUMNztrMTiMMYfY=">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
Next example we show is more realistic and well known one. Let us consider the followingeffective frequency ω ( t ) = k + v t , (A.32)where k and v are real constants. This effective frequency describes the model where a scalarfield χ couples to a coherently oscillating scalar field φ ( t ) after the end of inflation throughan interaction λ φ χ , which is known as the broad resonance regime in preheating. Near φ ∼ , such an effective mass looks m ∼ λ ˙ φ ∗ t where ˙ φ ∗ is approximated as a constant,and we identify v ∼ λ ˙ φ ∗ [6, 37].As the previous example, turning points appear at t = ± i kv , from which Stokes linesemanate. Therefore, we introduce a phase shift for k such that the Stokes segment isdecomposed. We illustrate the Stokes line structure in Figs. 17 and 18. We denote theturning points as t = ± i kv ≡ ± t c . We take the lower limit of the phase integration to besome large negative point t = t < , and consider the initial mode function to be ψ = ( t, t ) . (A.33)In the following, we discuss the connection problem of two cases separately.Let us first discuss the case shown in Fig. 17. We change the integration limit as ψ = [ t c , t ]( t, t c ) and the first Stokes line crossing does not change the form since ( t, t c ) issubdominant: ψ → ψ (cid:48) = [ t c , t ]( t, t c ) . For the second crossing, we change the lower limitas ψ (cid:48) = ab ( t, − t c ) where a = [ t c , t ] , b = [ − t c , t c ] where we have separated the two factorsto specify how we perform the integration. The second crossing leads to the followingcontinuation ψ (cid:48) → ˜ ψ = ab { ( t, − t c ) + S ( − t c , t ) } = ( t, t ) + Sa b ( t , t ) . (A.34)We note that in the limit, we can separate a as a = e ( κ (cid:48) +i ζ (cid:48) ) where κ (cid:48) and ζ (cid:48) are real, andgiven by e κ (cid:48) = [ t c , t s ] and e i ζ (cid:48) = [ t s , t ] . Here, t s is the point where the Stokes segmentintersects with the real t -axis. We also note that κ (cid:48) > since the Stokes line connecting t s and t c is a positive one. We also find that b = e − κ (cid:48) . Thus, with the approximation S ∼ i ,we find the resultant mode function in the large positive t on the real axis as ˜ ψ = ( t, t ) + i e − κ (cid:48) +i ζ (cid:48) ( t , t ) . (A.35)– 31 – latexit sha1_base64="7+qSLhx25l0hZUMNztrMTiMMYfY=">AAACZHichVHLSsNAFD2Nr1pf1SIIghRLxY3lVhTFVdGNS2vtA6qUJI41mCYhSQu1+AO6VVy4UhARP8ONP+CiPyCIywpuXHibBkRFvcPMnDlzz50zM4qla45L1AxIXd09vX3B/tDA4NDwSHh0LOeYVVsVWdXUTbugyI7QNUNkXc3VRcGyhVxRdJFXDtba+/masB3NNLbcuiV2KnLZ0PY0VXaZSs+VwjFKkBfRnyDpgxj82DDDN9jGLkyoqKICAQMuYx0yHG5FJEGwmNtBgzmbkebtCxwhxNoqZwnOkJk94LHMq6LPGrxu13Q8tcqn6NxtVkYRp0e6pRY90B090/uvtRpejbaXOs9KRyus0sjxRObtX1WFZxf7n6o/PbvYw7LnVWPvlse0b6F29LXD81ZmZTPemKEremH/l9Ske76BUXtVr9Ni8wIh/oDk9+f+CXLzieRigtILsdSq/xVBTGIas/zeS0hhHRvI8rkCJzjFWeBJGpQi0ngnVQr4mgi+hDT1AWGeibE=
B.1 Rough estimate
First, we calculate the Bogoliubov coefficient (3.10): | β k | = exp (cid:32)(cid:90) t c, n +1 t c, n | Ω k | dt (cid:33) , (B.1)where t c, n and t c, n +1 ( n = 0 , , , · · · ) denote the ( n + 1) -th pair of turning points and theeffective frequency Ω k is the approximated one (3.12): Ω k ( t ) ≈ k ( M t/ + m − (cid:18) ξ − (cid:19) Mt sin( M t ) (B.2)at late time oscillation M t (cid:29) . Here, t c, n ( t c, n +1 ) also corresponds to the start (theend) of ( n + 1) -th tachyonic region. Even with this simplified form, it is not possible tofind turning points, and therefore, we consider the following approximation: Since we arenow considering the large amplitude case and hence the curvature induced term, morespecifically, sin( M t ) dominates the time dependence of the effective frequency during theintegration range, the turning points corresponds to zeros of sin( M t ) , namely t = nπM − ,at the leading order. Therefore, we can assume that turning points are t c,n = ( nπ + ε n ) M − ,where | ε n | (cid:28) and then obtain turning points as t c,n ≈ (cid:0) nπ ± sin − ∆ n (cid:1) M − , (B.3)where ∆ n is the value of the small parameter defined by (3.13) at t = nπM − and theplus-minus sign takes plus (minus) when n is even (odd). Stokes segments connect the n -th and the (2 n + 1) -th turning points, between which φ is tachyonic. Now the Bogoliubovcoefficient (B.1) is approximated as ln | β k,n | ≈ (cid:90) π − sin − ∆ n +1 sin − ∆ n (cid:115)(cid:18) ξ − (cid:19) M nπ + x sin x − ( k p + m ) dxM , (B.4)where x ≡ M t − nπ . This integral is, however, still too complicated and we cannotperform it analytically simply because there are too many time dependent terms inside thesquare root. Here we adopt the following useful method: We ignore all time dependenceexcept sin x , which dominates the time dependence of the effective frequency as mentionedabove. Between the two turning points t c, n and t c, n +1 , except for the oscillating part sin x , we fix the time dependence of the effective frequency to be an intermediate value at M t = (2 n + ) π , and now we can calculate the Bogoliubov coefficient (B.1) analytically: ln | β k,n | ≈ (cid:115) ξ − n + ) π (cid:90) π − sin − ∆ n + 12 sin − ∆ n + 12 (cid:113) sin x − ∆ n + dx. (B.5)– 34 –his integral can be performed analytically and we find ln | β k,n | = (cid:115) ξ − n + ) π × (cid:113) − ∆ n + E (cid:32) cos − ∆ n + (cid:12)(cid:12)(cid:12)(cid:12) − ∆ n + (cid:33) , (B.6)where E( ϕ | k ) the incomplete elliptic integral of the second kind. Since ∆ n + (cid:28) , wemay expand this expression in terms of ∆ n + and find ln | β k,n | = 2 (cid:115) ξ − n + ) π (cid:16) (cid:16) π (cid:12)(cid:12)(cid:12) (cid:17) − F (cid:16) π (cid:12)(cid:12)(cid:12) (cid:17) ∆ n + (cid:17) + O (cid:16) ∆ n + (cid:17) , (B.7)where F( ϕ | k ) the incomplete elliptic integral of the first kind. Finally, we obtain thefollowing estimate by taking terms up to the leading order of ∆ n + in (B.7): n k,n = | β k,n | ≈ exp (cid:34) (cid:115) ξ − n + ) π (cid:16) (cid:16) π (cid:12)(cid:12)(cid:12) (cid:17) − F (cid:16) π (cid:12)(cid:12)(cid:12) (cid:17) ∆ n + (cid:17)(cid:35) . (B.8)As a summary of derivation, we mention the essential point of our approximation. Inderiving the analytic formula (B.8), we have focused only on the time dependence of theoscillating part sin( M t ) and ignored the rest. This enables us to perform the integration,while keeping the result as accurate as possible. This idea of approximation would beuseful for other models as well if the time dependence is dominated by a few part of thefull effective frequency. B.2 Improved estimate
Next, we calculate the Bogoliubov coefficient (B.1) with the improved approximated effec-tive frequency (3.15): Ω k ( t ) = k (1 + M t/ + m − (cid:18) ξ − (cid:19) M ( M t + 4) sin(
M t ) + 4 ξM ( M t + 4) . (B.9)Since the time dependence of this frequency is even more complicated than the previous one,we again apply the idea used in the previous case, namely we ignore all time dependenceexcept sin( M t ) , which dominates the time dependence of the effective frequency. Thisapproach reduces the problem to the one in the previous case. We take the same steps andfind turning points at M t c, n ≈ nπ + (cid:15) n + and M t c, n +1 ≈ (2 n + 1) π − (cid:15) n + , where thesmall quantity (cid:15) n satisfies sin (cid:15) n + = (cid:0) n + (cid:1) π + 44 ξ − ω k, n + M + 4 ξ (cid:0)(cid:0) n + (cid:1) π + 4 (cid:1) ≡ δ n + , (B.10)where ω k, n + is the ‘bare’ energy k p + m at M t = (cid:0) n + (cid:1) π , namely: ω k, n + = k (cid:0) (cid:0) n + (cid:1) π/ (cid:1) + m . (B.11)– 35 –ote that, we have substituted M t = (cid:0) n + (cid:1) π except for the oscillating part sin M t as ex-plained above. Solving this equation yields the perturbative quantity (cid:15) n . The integral (B.1)under our approximation is now can be written as ln β k,n ≈ (cid:115) ξ −
14 + (2 n + ) π (cid:90) π − (cid:15) n + 12 (cid:15) n + 12 (cid:113) sin τ − sin (cid:15) n + dτ. (B.12)This integral can be performed analytically and we find ln β k,n = (cid:115) ξ −
14 + (2 n + ) π × (cid:113) − δ n + E (cid:32) cos − δ n + (cid:12)(cid:12)(cid:12)(cid:12) − δ n + (cid:33) . (B.13)Since δ n + (cid:28) , we may expand this expression in terms of δ n + and find ln β k,n = 2 (cid:115) ξ −
14 + (2 n + ) π (cid:16) E (cid:16) π (cid:12)(cid:12)(cid:12) (cid:17) − F (cid:16) π (cid:12)(cid:12)(cid:12) (cid:17) δ n + (cid:17) + O (cid:16) δ n + (cid:17) , (B.14)Finally, we obtain the following estimate by taking terms up to the leading order of ∆ n + in (B.14): n k,n = | β k,n | ≈ exp (cid:34) (cid:115) ξ −
14 + (2 n + ) π (cid:16) E (cid:16) π (cid:12)(cid:12)(cid:12) (cid:17) − F (cid:16) π (cid:12)(cid:12)(cid:12) (cid:17) δ n + (cid:17)(cid:35) . (B.15)We show the comparison between the analytic result and the numerical solution with exactexpression (3.4) in Fig. 19 for the first oscillation n = 0 , which shows excellent agreementbetween them. C Improvement of particle production rate for a transition model
In this section, we show how the improved particle number formula (4.8) is derived. First,we derive a constraint on the Stokes constants by considering the connection around thepole. The schematic figure is shown in Fig. 20. We consider the mode function in large realpositive t region and take the lower limit of a phase integral to be the pole t p , namely ourmode function is f = A (cid:112) ω k ( t ) exp (cid:32) − i (cid:90) tt p ω k ( t (cid:48) ) dt (cid:48) (cid:33) + B √ ω k exp (cid:32) i (cid:90) tt p ω k ( t (cid:48) ) dt (cid:48) (cid:33) (C.1)where A and B are constants. The first Stokes line crossing leads to f = Ac ( t, t c ) + Bc − ( t c , t ) → f (cid:48) = Ac ( t, t c ) + ( Bc − + S − Ac )( t c , t ) , (C.2)where c = [ t c , t p ] and S − denotes the Stokes constant, which will be determined later. Notethat c is evaluated under the cut and c > . At the second crossing, the mode function isanalytically continued to f (cid:48) = Ac ( t, t c ) + ( Bc − + S − Ac )( t c , t ) → ˜ f = ( Bc − + S − Ac )( t c , t ) + ( S + Bc − + Ac (1 + S + S − ))( t, t c ) . (C.3)– 36 –
10 20 30 40 50 60 - k / m l n ( n k ) Figure 19 . Comparison between the analytic estimation (B.15) (blue line) and numericallycalculated values (black dots) at the first oscillation n = 0 . For analytic estimation, we use (B.14),and for numerical evaluation, we solve the equation of motion with (3.4), but we used approximatedscale factor to be a = (1 + M t/ / , which is a good approximation as the solution of (3.4). Wetake M = 15 m, ξ = 10 in this graph. Note that the first oscillation shows the largest deviation fromthe numerical result. For larger n , the agreement becomes even better. Nevertheless, the agreementfor n = 0 is already excellent. - - ++ Figure 20 . A schematic figure of the Stokes line (black solid lines) around one turning point.We consider the connection along the arrow (blue dashed line). The blue dot, the blue cross markand red thick line denote a turning point, a pole and a branch cut, respectively. Solid lines areStokes lines and the plus and minus signs mean that, moving away from the turning point alongplus (minus) lines, i ω k dt increases (decreases). Finally, taking the lower limit of the integration to be t p , we find ˜ f = ( Bc − + S − A )( t p , t ) + ( S + B + Ac (1 + S + S − ))( t, t p ) . (C.4)– 37 –et us denote ˜ f = A (cid:48) ( t, t p ) + B (cid:48) ( t p , t ) , and then the relation between A (cid:48) , B (cid:48) and A, B iswritten as (cid:32) A (cid:48) B (cid:48) (cid:33) = (cid:32) c (1 + S + S − ) S + S − c − (cid:33) (cid:32) AB (cid:33) ≡ R (cid:32) AB (cid:33) . (C.5)This matrix R satisfies det R = 1 and for the case with a first order simple pole, it is knownthat Tr R = 2 [22]. Therefore, we find a constraint S + S − = − (1 − c − ) . (C.6)Since the effective frequency is real on the real t -axis, the following condition holds [22], S + = − ¯ S − . (C.7)This relation leads to | S + | = | S − | = (1 − c − ) . Thus, we have fixed the absolute valueof the Stokes constant. We note that both S + and S − vanish when c → and indeed thisis the limit of m → as we show below.Next, let us consider the connection problem at the Stokes line crossing. As we show inFig. 11, there is one Stokes segment that crosses real t -axis. If we take into account only thisStokes segment, the connection problem is the same as that discussed in appendix A.1.2.However, there is one important difference: As we have discussed above, the Stokes constantis no longer i , but its absolute value is given by | S ± | = | − c − | . Except this point, we cantake the same step to obtain the particle number as done in A.1.2. The result is given by n k = | β k | ∼ d | S − | = d (1 − c − ) , (C.8)where d = [ t c , ¯ t c ] = e − πk ∆ t (C.9)and c = [ t c , t p ] = exp (cid:16) π ∆ t ( (cid:112) k + m − k ) (cid:17) . (C.10)As we have mentioned above, m → limit reads c → and the particle number vanishesin this limit as physically expected. Thus, we have derived the improved formula (4.8). References [1] F. Sauter,
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