Quantum outofequilibrium cosmology
Abstract
In this work, our prime focus is to study the one to one correspondence between the conduction phenomena in electrical wires with impurity and the scattering events responsible for particle production during stochastic inflation and reheating implemented under a closed quantum mechanical system in early universe cosmology. In this connection, we also present a derivation of quantum corrected version of the Fokker–Planck equation without dissipation and its fourth order corrected analytical solution for the probability distribution profile responsible for studying the dynamical features of the particle creation events in the stochastic inflation and reheating stage of the universe. It is explicitly shown from our computation that quantum corrected Fokker–Planck equation describe the particle creation phenomena better for Dirac delta type of scatterer. In this connection, we additionally discuss Itô, Stratonovich prescription and the explicit role of finite temperature effective potential for solving the probability distribution profile. Furthermore, we extend our discussion of particle production phenomena to describe the quantum description of randomness involved in the dynamics. We also present computation to derive the expression for the measure of the stochastic nonlinearity (randomness or chaos) arising in the stochastic inflation and reheating epoch of the universe, often described by Lyapunov Exponent. Apart from that, we quantify the quantum chaos arising in a closed system by a more strong measure, commonly known as Spectral Form Factor using the principles of random matrix theory (RMT). Additionally, we discuss the role of out of time order correlation function (OTOC) to describe quantum chaos in the present nonequilibrium field theoretic setup and its consequences in early universe cosmology (stochastic inflation and reheating). Finally, for completeness, we also provide a bound on the measure of quantum chaos (i.e. on Lyapunov Exponent and Spectral Form Factor) arising due to the presence of stochastic nonlinear dynamical interactions into the closed quantum system of the early universe in a completely modelindependent way.
1 Introduction
In continuation with this, we discuss about the epoch of reheating which occurs after the end of inflationary stage of the universe which finally results in the stochastic random burst of particle production. The dynamics of these stochastic random bursts of particle production can be well understood by using a Fokker–Planck equation, which gives us a statistical interpretation of the number density of particles created per scattering event. Since, the number of particles created in a given nonadiabatic event is not discrete in nature but rather its random, which means that there must be a probability distribution function associated with the particle number. The various dynamical features of this type of probability distribution and its physical consequences has been studied in Ref. [42]. It has been phenomenologically proposed in Ref. [42] that such probability density function would necessarily is Gaussian one. The occupation number of the produced particles, \(n_{k}\), executes a drifting Brownian motion and a Fokker–Planck (FP) equation that evolves the probability distribution, \(P(n_{k};\tau )\), emerging out of this Brownian motion has been studied in Ref. [42]. We further compute the analytical expressions for the mean, variance and other higher order moments which are commonly known as, skewness and kurtosis and such additional statistical higher order moments are very useful to study the exact mathematical form and asymptotic limits of the probability distribution function. The evolution of mean, variance, skewness and kurtosis finally gives a coarsegrained analysis of the Fokker–Planck dynamics to more corrected orders of magnitude in quantum regime. We show in this paper explicitly that though Gaussianity is an inherent part of the probability density function, but the consideration of the higher order moments in the Fokker–Planck equation tells us that the density function may not be a Gaussian one but with some higherorder corrections entailed into it due to the quantum mechanical origin. Therefore, to a greater extent we extend the more corrected quantum version of the Fokker–Planck equation used to describe the dynamics of the probability distribution function used in Ref. [42] that tells us the dynamics of the bursts of particle production in these random scattering events. The more quantum corrected version tells us that the probability amplitude of the particle production in the scattering events is more than a Log normal distribution. The distribution profile of the probability distribution function depends largely on the profile of the scatterer, i.e., the effective potential V(x) in the Schrödingerlike equation. While calculating the Fokker–Planck dynamics we observe that the skewness gives us a clue about the rate at which the particle production occurs meaning that longer the trailing part of the profile more is the number density of particles in the scattering event for a given time in the frame of the observer, whereas, kurtosis tells us the width of the probability distribution function which is essentially the amplitude with which the particle production phenomena occurs, which more suggestively tells us about the standard deviation of the density function from Gaussianity. This may be a signature of nonGaussianity that arises in various models in early universe cosmology.
In this connection it is important to note that, such stochastic approaches to the early universe scenario have been studied in details in [48, 49], where the authors give an account of how chaos arises in the context of eternal inflation. As any rapidly oscillating classical field looses its energy by creating pairs of elementary particles, these particles interact with each other and comes to a state of showing thermal behaviour at some temperature T. This implies that we must eliminate the necessary assumption of the universe being in thermal equilibrium. This means that the inflating universe is rather thermal in the sense that the particle creation events that occurs during the quantum fluctuation in the randomly distributed scalar fields \(\phi \) which results in a chaotic model of the inflationary scenario of the universe thereby leading to a generation of stochastic idea of the particle creation events during the thermalization of the quantum states of the field randomly distributed over the spacetime. These particle creation events are more phenomenologically associated with one of the fundamental ideas in outofequilibrium statistical mechanics known as Fokker–Planck equation which gives the rate of the particle production during theses random events in stochastically emerging spacetime along with the distribution function that this rate charts out. In Ref. [42], such a phenomenon of particle creation events by the randomly spaced scatterers in due context of cosmology has been shown where the statistics of the produced particles as a function of time which is the probability distribution function \(P(n_{k},\tau )\) has been predicted to be following a LogNormal distribution. The entire process have been carried out with the deltascatterers which are localized in spacetime.
Following Ref. [42], in this paper we give a more improved quantum corrected version of the same approach to the probability density function of the particle production events and our prediction from the results show that the higher order quantum correction terms being included into the Fokker–Planck equation introduces an approximation to the theory. This tells us that the number of particles produced in a given nonadiabatic event during the reheating stage of the universe is quantized, which would mean that the rate of particle production in a given event gives rise to a discrete set of occupation number \(n_{k}\). Furthermore, the quantum corrected terms obtained by deriving the Fokker–Planck equation takes the general form, which is linear in \(n_{k}\) being the first order in \(\tau \). Using this information we calculate further the leading order, second and third order terms in the Fokker–Planck equation. Hence, we derive the analytic expression of the quantum corrected version of Fokker–Planck equation. We also calculate the various higher moments in order to get an overview of the nature of the solution of the quantum corrected Fokker–Planck equation which are  standard deviation, skewness and kurtosis which gives the hint of how the probability density function deviates from its Gaussian nature when the higher order quantum corrections are taken into account in the computation. This in turn may will be another indirect signature of the primordial nonGaussianity in cosmology other than obtaining the signatures provided by the 3point functions from scalar fluctuations.
Apart from that, we discuss about spectral form factor (SFF), which measures the random distribution of eigen values of the energy hamiltonian of a chaotic system. For this computation of SFF we use the principles of random matrix theory (RMT) in this paper. In the present context an upper bound on SFF denotes the saturation of eigen value distribution hence supports the Ref. [50] for quantum chaotic system. Within the framework of quantum physics, chaotic systems can be characterized using only some additional constraints. This theoretical approach is discussed in Refs. [51, 52, 53, 54, 55] and the authors use the theory of random matrices to characterize quantum mechanical system. In this method, any arbitrarily complicated manybody Hamiltonian can be replaced by matrix of random numbers drawn from a Gaussian statistical ensemble. This random matrix approach towards quantum mechanics help to characterize and understand the underlying features of the chaotic random system. After studying the behaviour of SFF with time one can further comment that whether it is valid for a cosmological particle production event (semiclassical) or not. For our purpose we discuss generalized version of SFF for different even order polynomial structure of random potential and then extend that result to describe the cosmological particle production events [56, 57]. For any random potential we can use this method of SFF and we can deal with scatterer of any arbitrary type. For any such scatterer we can get a bound on randomness in the chaotic system characterised by SFF. Also using specific transfer matrix for different conformal time dependent effective mass profiles which are precisely known in this paper, we can finally compute Lyapunov exponent which also measure stochastic randomness.
Also it is important to note that in Ref. [42], the scatterers were considered to be some localized potential functions in spacetime. On the contrary the choice of our specific time dependent mass profiles mimics the role of thermalized fields or effective potential functions, which are playing the role of scatterers in this context. We see that the choice of these time dependent mass profiles leads to particle production which is chaotic in nature and therefore, to determine the rise of chaos in such a system we quantify as well as analyse chaos by a well known quantities known as the, Lyapunov exponent [58] and Spectral Form Factor (SFF) [59]. Here fusing the principles of random matrix theory (RMT) we provide a generalized bound on randomness (or stochasticity) for any general random scatterer whose potential can be expressed in terms of an even polynomial. More precisely, we provide a possible method to compute the degree of randomness in a chaotic system and from that one can check the bound on chaos.
The plan of the paper is as follows – In Sect. 2 we discuss about the model which is responsible for the quantum description of chaos during the cosmological particle production and have similarities with the quantum mechanical problem of electrical conducting wire with impurities. In Sect. 3, we have presented the analytical expressions for the Bogoliubov coefficients, transmission and reflection coefficients, Lyapunov exponent, conductance, and resistance for different time dependent mass profile. We have discussed the correspondence between In Sect. 5 the specific role of Spectral Form Factor (SFF) to quantify chaos in the context of particle production rate is discussed. In Sect. 6 the particle production event with quantum corrected Fokker–Planck equation is discussed by taking contribution upto fourth order and also different higher order moments from the quantum corrected probability density function are explicitly computed. Finally, in Sect. 7 we conclude with the future prospect and physical impacts of our work.
Additionally it is important to note that, throughout this paper, we use natural system of units, \( \hbar = c = 1 \).
2 Modelling randomness in cosmology
The background model which we consider in this section to quantify quantum chaos in cosmology consists of a massless scalar field interacting with coupled with a background scalar field with conformal time dependent mass profile which in principle have heavier or comparable to the Hubble scale (\(m\ge H\)) [46, 60, 61]. It is important to note that such heavy mass profiles play significant role in finding various cosmological correlation functions and also can be treated as an additional probe to break the degeneracy between various models of inflation from the perspective of implementing cosmological perturbation theory in (quasi) de Sitter background. We know that in usual set up of primordial cosmological perturbation such heavy fields are not appearing in the low energy effective field theory action. For that case in the simplest situation we actually start with an one field set up where the kinetic term is canonical in nature and the field is minimally coupled with the background gravity which is treated to be classical usually. Also such field has an effective structure of the interaction potential which play crucial role to study the time dynamics in FLRW cosmological background. Here specifically the field is treated to be massless compared to the Hubble scale (\(m\ll H\)). However, this is not the complete story yet. To explain this let us start with a Ultra Violet (UV) complete set up of quantum field theory (QFT) such as string theory in higher dimensions. There are various examples of string theory from which one can start the computation, which are  Type II A, Type II B, Heterotic, M  theory etc. Also the low energy extension of such theories (supergravity) are also useful for the computation in the context of cosmology. Here it is important to note such all such theories contain massive (\(m\gg H\)), intermediate mass (\(m\approx H\)) and massless (\(m\ll H\)) fields in the matter multiplet. To write down an effective field theory (EFT) one need to integrate out all such heavy degrees of freedom from the UV complete version of the action.^{4} After doing dimensional reduction along with applying various compactification techniques one can derive various types of UV complete effective field theories at cosmological scale where the effective couplings of various relevant and irrelevant Wilsonian operators have time dependent profile in FLRW background and in such a case from the relevant quadratic operator one can also get the time dependent effective mass which is in general heavy (\(m\ge H\)). It is further important to mention here that, such heavy fields can give rise to non vanishing one point function for scalar (curvature) perturbation in cosmology, which carries the signature of Bell’s inequality violation in primordial universe [46, 61, 62, 63, 64, 65, 66]. Also it is important to note that such Bell violating set up can be explained using the theory of quantum entanglement in (quasi) de Sitter background and can give rise to nonvanishing quantum information theoretic measure i.e. Von Neumann entropy, R\(\acute{e}\)nyi entropy, quantum discord, logarithmic entangled negativity [67, 68, 69, 70] etc. Additionally, one can get correct expression for two point function and also the three point function from scalar (curvature) perturbation, which will show significant effect in estimating primordial nonGaussianity from single field models of inflation. Apart from this one can consider a simplest situation in four spacetime dimensions where the cosmological dynamics is explained in terms of two interacting scalar fields. The light field (\(m\ll H\)) is participating in inflation and the other heavy field (\(m\gg H\)) is participating to explain the dynamics of reheating. If we path integrate out the reheating degrees of freedom then we get an effective field theory of inflation which is exactly same as we have explained earlier. But here one can consider the other possibility as well in which one can path integrate out the light inflaton degrees of freedom and write down an effective field theory to describe reheating in terms of the heavy fields (\(m\ge H\)). In such a description this reheating field have mass and in the effective field theory description one can write down some time dependent coupling in terms of the integrated inflaton degrees of freedom and the mass of the reheating field appearing in the coefficient of the relevant quadratic operator. In this description such time dependent coupling is treated to be the time dependent effective mass parameter profile which is considered in the present discussion. So it is evident from this discussion that using both the effective field theory of inflation and reheating one can actually explain the origin of such time dependent effective mass profiles in four dimensions. However, in this paper since our objective is to study the cosmological particle production phenomena, we will mostly focus on the reheating epoch of the universe.
 1.
In EFT time dependent couplings are appearing after path integrating out the massive degrees of freedom. This prescription is usually used to construct a most generic EFT of inflation.
 2.
In EFT time dependent couplings are appearing after path integrating out the massless degrees of freedom. This prescription is usually used to construct a most generic EFT of reheating.
A brief overview of the connection between the scattering problem in quantum mechanics to that of cosmological particle creation events
Scattering in conduction wire  Cosmological particle creation  

Symbol  Physical interpretation  Symbol  Physical interpretation 
x  Distance  \(\tau \)  Conformal time 
V(x)  Potential  \(m^{2}(\tau )\)  Time dependent mass parameter 
\(\Psi (x)\)  Wave function  \(\phi _{k}(\tau )\)  Mode function in Fourier space 
\(N_{s}\)  No. of scatterers  \(N_{s}\)  No. of nonadiabatic events 
\(\Delta \)x  Distance between scatterers  \(\Delta \tau \)  Time between nonadiabatic events 
\(\xi \)  Localization length  \(\mu _{k}\)  Mean particle production rate 
\(\rho (x)\)  Resistance  \(n_{k}(\tau )\)  Particle occupation number 
E  Energy eigen value  \(k^2\)  Wave number of Fourier modes 
\(N_c\)  Number of channels  \(N_f\)  Number of fields 
3 Randomness from conduction wire to cosmology: dynamical study with time dependent protocols
 1.Lyapunov exponent: It actually quantify the amount of chaos appearing in the quantum mechanical systems that we are studying in the context of early universe cosmology. In our discussion it tells us the degree of randomness in the stochastic particle production. In our case, the chaos emerges due to the random scattering events which are non adiabatic and we call these as cosmological scattering events leading to particle production. In this section, we discuss about Lyapunov exponent and try to discuss their behaviour for the different time dependent mass profiles. In thus context, Lyapunov exponent is defined as [42, 79]:where, T is the transmission coefficient given by the following expression:$$\begin{aligned} \lambda =  \mathrm{log}~ T, \end{aligned}$$(3.1)with t and \(t^{*}\) being the transmission amplitude of the incoming and the outgoing wave. In the present discussion, the transmission coefficient can be expressed as:$$\begin{aligned} T = t^{*}t=t^2, \end{aligned}$$(3.2)where \(\beta \) and \(\alpha \) are the Bogoliubov coefficients. Also, it is important to note that, in the present context one can define the reflection coefficient as:$$\begin{aligned} T = t^2=\frac{1}{\alpha ^{2}}, \end{aligned}$$(3.3)where \({\tilde{r}}\) and \({\tilde{r}}^{*}\) being the reflection amplitude of the incoming and the outgoing wave. Finally from Eqs. (3.3) and (3.4), we get the following conservation equation:$$\begin{aligned} R={\tilde{r}}^{*}{\tilde{r}}={\tilde{r}}^2=\frac{\beta ^2}{\alpha ^{2}}, \end{aligned}$$(3.4)where we have used the following normalization condition for the Bogoliubov coefficients, as given by:$$\begin{aligned} R + T={\tilde{r}}^2 + t^2=\frac{1+\beta ^2}{\alpha ^{2}}=1, \end{aligned}$$(3.5)$$\begin{aligned} \alpha ^2\beta ^2=1. \end{aligned}$$(3.6)
 2.Conductance: It quantify the degree of support of the flow of electron inside an electrical conduction wire. this is exactly reciprocal of resistance. In the present context, conductance refers to the ability of the massless scalar fields to transmit through the massive fields which are the specific heavy mass profiles that we have discussed above. This may be more suggestive in telling us about the interaction of the massless scalar field with the massive fields. More value of conductance refers to the larger transmitivity of the background fields through the scatterers and viceversa. Thus, conductance also carries a valuable information about the transmission coefficient of the scalar field interacting with the scatterer. In this context, the conductance can be expressed as:where \(\lambda \) is the Lyapunov exponent, T is the transmission coefficient, t is the transmission amplitude of the incoming/outgoing wave and \(\beta ,~\alpha \) are the Bogoliubov coefficients as mentioned above.$$\begin{aligned} G = \exp \left( 2\lambda \right) =T^2=t^4=\frac{1}{\alpha ^{4}}, \end{aligned}$$(3.7)
 3.Resistance: It quantify the degree of oppose of the flow of electrons inside an electrical conduction wire. It is the property by the virtue of which the scatterers (which are the time dependent mass profiles in our case) resist the massless scalar field to tunnel through them. In other words, it is the same Schrödinger formulation in quantum mechanics where the incoming wave interacts with a potential barrier and the strength of the barrier is the measurement of resistance to the tunneling of the incoming particle through it. This means that more the resistance to the incoming wave, more is the lower is the transmission probability across the barrier. Resistance is defined as the reciprocal of conductance G(k), which gives:$$\begin{aligned} r(k) = \frac{1}{G(k)}=\exp \left( 2\lambda \right) =\frac{1}{T^2}=\frac{1}{t^4}=\alpha ^{4}. \end{aligned}$$(3.8)
To study the cosmological particle creation problem during early epoch of universe (specifically during reheating) we use the analogy with the quantum mechanical scattering problem inside an electrical conduction wire in presence of time dependent effective mass profile we will perform the computation in (quasi) de Sitter space using FLRW spatially flat metric.
3.1 Protocol I: \(m^{2} (\tau ) = m^2_{0}(1 \tanh (\rho \tau ))/2\)
3.1.1 Bogoliubov coefficients
In Fig. 6a, b, we have shown the variation of the Bogoliubov Coefficients with wave number k.
3.1.2 Optical properties: reflection and transmission coefficients
In Fig. 7a, b, we have shown the variation of the transmission and reflection coefficients with wave number k.
3.1.3 Chaotic property: Lyapunov exponent
In Fig. 8, we observe that with increase in wave number k the Lyapunov exponent decreases. This shows that the Lyapunov exponent is dependent on the momenta values of the fields interacting with the massive field acting as a scatterer. Furthermore, we discover that for the mass profile I, the chaos in the event reduces with increase in the wave number. This suggests that lesser the number of fields interacting with the massive field more is the chaos in the quantum system considered in this paper. Since, a negative value of Lyapunov Exponent pulls a system out of chaos, this further tells us that the Lyapunov exponent is inversely related to the number of background fields interacting with the scatterer or the massive field. This may be interpreted in the following way in the context of Schrödinger problem in quantum mechanics that a higher value of wave number k of the incoming wave would be able to cross a potential barrier of a given strength and would be able to get transmitted through the barrier and the pulse won’t damp easily than that of a wave with lower k value. This means that the scatterer acts as a definitive medium which allows only certain wave numbers to pass through thus reducing the chaos in the system.
3.1.4 Conduction properties: conductance and resistance
In Fig. 9a we have shown the variation of conductance with wave number k. This figure shows that with increase in the momenta value of the massless scalar field, the conductance also increases. Now, accounting for \(m_{0}\) values, we see that for \(m_{0} = 1\) the conductance shoots up at a much lower k value than that of \(m_{0} = 2\) and \(m_{0} = 3\). This suggests that for \(m_{0} = 1\) the field has a much higher transmission probability than that of \(m_{0} = 2\) and \(m_{0} = 3\). An increase in transmission probability gives a direct evidence of the conductance value. Therefore, we conclude that for \(m_{0} = 1\) the field has more conductance value in comparison to \(m_{0} = 2\) and \(m_{0} = 3\). We also conclude that larger the momenta value, more is the transmission coefficient and thereby shoots up the conductance of the system. This means that an incoming wave with large momenta value would eventually cross a barrier potential field thereby increasing the conductance of the system as the transmission probability would be much higher than an incoming wave with lower momenta value.
In Fig. 9b we have shown the variation of resistance with wave number. We observe that with an increase in the value of k the resistance starts decreasing which suggests that with an increase in momenta value the transmission probability across the scatterer. This may be viewed in accordance with the potential barrier in the Schrödinger equation in quantum mechanics also starts increasing thereby allowing the incoming wave to tunnel through the barrier thereby increasing the transmission probability and hence,reducing the resistance. We also observe that with an increase in k value the resistance reduces less rapidly for \(m_{0} = 1\) than that of \(m_{0} = 3\) and \(m_{0} = 2\). Whereas, it reduces more rapidly for \(m_{0} = 3\) suggesting that higher the value of the constant \(m_{0}\) lower is the value of resistance offered.
3.2 Protocol II: \(m^{2} (\tau ) = m_{0}^{2}~ \mathrm{sech}^2(\rho \tau )\)
3.2.1 Bogoliubov coefficients
In Fig. 11a, b, we have shown the variation of the Bogoliubov coefficients with wave number k.
3.2.2 Optical properties: reflection and transmission coefficients
In Fig. 12a, b, we have shown the variation of the transmission and reflection coefficients with wave number k.
3.2.3 Chaotic property: Lyapunov exponent
In Fig. 13, we have shown the wave number dependence of Lyapunov exponent. Here we observe that with increase in k value the Lyapunov exponent decreases. This implies that the Lyapunov exponent is dependent on the momenta values of the fields interacting with the massive field acting as a scatterer. Furthermore, we also discover that for this mass profile II, the chaos in the event reduces with increase in the k value.
3.2.4 Conduction properties: conductance and resistance
In Fig. 14a we have shown the wave number dependence of conductance. We observe that for \(m_{0} = 1\) conductance starts increasing at a larger value of k than that of \(m_{0} =2\) and \(m_{0} = 3\). But, in contrast to the variation of conductance with momenta k in the above figure, here the conductance starts increasing rapidly for \(m_{0} = 3\) than that for \(m_{0} = 1\) which suggests that the transmission probability for \(m_{0} = 3\) is much higher than \(m_{0} = 1\) and \(m_{0} =2\), thereby making it more conductive than the other two.
In Fig. 14b, we have shown the wave number dependence of resistance. Here like the first mass profile the resistance for \(m_{0} = 3\) falls more rapidly than that of \(m_{0} = 2\) and \(m_{0} = 1\). This suggests that for the given mass profile II, as the value of \(m_{0}\) increases, the value of resistance also decreases. But unlike the first mass profile, the resistance for \(m_{0} = 3\) falls more rapidly suggesting that for \(m_{0} = 3\) this specific mass profile offers more resistance than the first one. Therefore, we conclude that for the same values of \(m_{0}\) this mass profile offers less resistance in comparison to the first mass profile.
3.3 Protocol III: \(m^{2} (\tau ) = m_{0}^{2}~\Theta (\tau )\)
3.3.1 Bogoliubov coefficients
In Fig. 16a, b, we have shown the variation of the transmission and reflection coefficients with wave number k.
3.3.2 Optical properties: reflection and transmission coefficients
In Fig. 17a, b, we have shown the variation of the transmission and reflection coefficients with wave number k.
3.3.3 Chaotic property: Lyapunov exponent
From Fig. 18 we observe that with increase in wave number the Lyapunov exponent decreases more like a rectangular hyperbolic fashion. In comparison to the other two mass profiles where the reduction in the value of the Lyapunov exponent is much less rapid in comparison to this mass profile discussed here. This suggests that since, the mass profile is a heavy side theta function, which is a quenched mass protocol, the Lyapunov exponent also gives a similar like profile. This shows that the Lyapunov exponent is dependent on the wave number of the fields interacting with the massive field acting as a scatterer. Furthermore, we discover that for this given mass profile, the chaos in the event reduces with increase in the k value. So, in this case the Lyapunov exponent decays much rapidly than the first two mass profiles. Next, we will try to find an upper bound of Lyapunov exponent using the definition of [81].
3.3.4 Conduction properties: conductance and resistance
In Fig. 19a, we have shown the wave number dependence of conductance. This figure shows that with increase in the wave number of the massless scalar field, the conductance also increases. Now, accounting for \(m_{0}\) values, we see that for \(m_{0} = 1\) the conductance shoots up at a much lower k value than that of \(m_{0} = 2\) and \(m_{0} = 3\). This suggests that for \(m_{0} = 1\) the field has a much higher transmission probability than that of \(m_{0} = 2\) and \(m_{0} = 3\). An increase in transmission probability gives a direct evidence of the conductance value. Therefore, we conclude that for \(m_{0} = 1\) the field has more conductance value in comparison to \(m_{0} = 2\) and \(m_{0} = 3\).
In Fig. 19b, unlike the mass profile I the resistance for \(m_{0} = 1\) falls more rapidly than that of \(m_{0} = 2\) and \(m_{0} = 3\). This suggest that for the given mass profile, as the value of \(m_{0}\) increases, the value of resistance also increases suggesting that heavier the field gets lesser is the transmission probability of the incoming wave to tunnel through it thereby reducing the value of conductance for this specific mass profile.
4 Quantum chaos from out of time ordered correlators (OTOC)
4.1 Chaos bound in outofequilibrium quantum field theory (OEQFT) and its application to cosmology
 1.Let us start with a completely mathematical problem described by a time dependent function \(g(\tau )\), which satisfy the following set of properties:
 (a)
In the complex plane \(g(\tau +iT)\) is analytic in the half strip described within \(\tau >0\) and \(\frac{\beta }{4}\le T \le \frac{\beta }{4}\). In this context, \(\tau \) and T represent the real and imaginary part of the complex number \(\tau +iT\) after analytical continuation in complex plane.
 (b)
The function \(g(\tau )\) is completely real at \(T=0\).
 (c)After analytical continuation the function in the complex plane satisfy the following constraint:which is perfectly valid in the complete half strip.$$\begin{aligned} g(\tau +iT)\le 1, \end{aligned}$$(4.4)
 (a)
 (c)Next, we actually conformally map the entire half strip to a unit thermal circle in the complex plane, which can be done using the following Möbius transformation:(4.5)In Fig. 20a, we have shown the behaviour of the amplitude of the complex number z with respect to the parameters (\(\tau ,T\)) in 3D plot. Finally, to check the consistency with Schwarz–Pick inequality we have plotted the complex number z at \(T=0\) in Fig. 20b.$$\begin{aligned} \Delta _{\beta }(\tau +iT):=\sinh \left( \frac{2\pi }{\beta }(\tau +iT)\right) . \end{aligned}$$(4.6)
 3.
Further using Eq. (4.4) one can further say that the complex function g(z) is an analytic function from one to one conformal map from unit disk to unit disk.
 4.
A variant of the Schwarz lemma can be represented as a invariant contribution under analytic automorphisms on the unit disk, which implies the bijective holomorphic mappings of the unit disc to itself. This specific variant is known as the Schwarz–Pick theorem.
 5.Now the hyperbolic metric in complex plane is defined as:Further using this metric and applying Schwarz–Pick theorem one can write:$$\begin{aligned} ds^2=4\frac{dz d{\bar{z}}}{\left( 1z^2\right) ^2}=\left( \frac{2dz}{\left( 1z^2\right) }\right) ^2. \end{aligned}$$(4.7)$$\begin{aligned}&{{{\mathbf {\underline{SchwarzPick~inequality:}}}}}\nonumber \\&\quad \frac{dg}{\left( 1g(z)^2\right) }\le ds=\frac{2dz}{\left( 1z^2\right) }. \end{aligned}$$(4.8)
 6.Further applying the fact that the function \(g(\tau )\) is real at \(T=0\) and using Eq. (4.8) we get the following simplified result:$$\begin{aligned}&\frac{1}{1g^{2}(\tau )}\left \frac{dg(\tau )}{d\tau }\right \le \left[ \frac{1}{1z^2}\left \frac{dz}{d\tau }\right \right] _{T=0}\nonumber \\&\quad =\frac{\pi }{\beta }\coth \left( \frac{2\pi \tau }{\beta }\right) . \end{aligned}$$(4.9)
 7.Further, rearranging Eq. (4.9) we get the following final result:which is the outcome of SchwarzPick inequality and very very useful to prove the universal chaos bound in OEQFT.$$\begin{aligned}&\frac{1}{(1g(\tau ))}\left \frac{dg(\tau )}{d\tau }\right \nonumber \\&\quad \le \frac{1}{2}(1+g(\tau ))\frac{2\pi }{\beta }\coth \left( \frac{2\pi \tau }{\beta }\right) , \end{aligned}$$(4.10)Now it is important to note that in this context,This further implies that:$$\begin{aligned}&\frac{1}{2}(1+g(\tau ))\coth \left( \frac{2\pi \tau }{\beta }\right) \nonumber \\&\quad \le 1+\frac{\beta }{2\pi }{{\mathcal {O}}}\left( \exp \left( \frac{4\pi \tau }{\beta }\right) \right) . \end{aligned}$$(4.11)Now at very large time scale (\(\tau \rightarrow \infty \)) or at very high temperature (\(\beta =1/T\rightarrow 0\)) one can neglect the contribution from the second subleading term. As a result we get the following inequality:$$\begin{aligned} \frac{1}{(1g(\tau ))}\left \frac{dg(\tau )}{d\tau }\right \le \frac{2\pi }{\beta }+{{\mathcal {O}}}\left( \exp \left( \frac{4\pi \tau }{\beta }\right) \right) .\nonumber \\ \end{aligned}$$(4.12)$$\begin{aligned} \frac{1}{(1g(\tau ))}\left \frac{dg(\tau )}{d\tau }\right \le \frac{2\pi }{\beta }, \end{aligned}$$(4.13)
 8.Further, we take the following phenomenological function:where k is constant and \(\lambda \) is the Lyapunov exponent. This function satisfy all the requirements that we have mentioned earlier explicitly. Further substituting this function in the result obtained in Eq. (4.10) we get the following simplified result^{16}:$$\begin{aligned} g(\tau )=1k\exp [\hbar \lambda \tau ], \end{aligned}$$(4.14)which proves the Universal chaos bound in OEQFT.$$\begin{aligned} \lambda \le \frac{2\pi }{\hbar \beta }, \end{aligned}$$(4.15)
 9.This bound on the Lyapunov exponent is an unique feature of all classes of OEQFT set up. It has a very strong impact in the context of early universe cosmology, specifically during reheating epoch. By knowing specific time dependent couplings in the context of effective field theory (EFT) it is possible to give an estimate of Lyapunov exponent in such OEQFT set up. We will show this feature in the next section for three known model of interactions appearing in EFT. In such a situation one can give an estimate of the upper bound on reheating temperature using this bound, which is again obviously an universal bound itself. The earlier study in the context of reheating actually predicts a very crude estimate of reheating temperature which is based on the assumption that reheating is extremely model dependent. It actually means that to write an EFT of reheating we need to know the all interacting relativistic degrees of freedom in a specific model. In this framework the total energy density during reheating can be expressed in terms of total number of relativistic degrees of freedom by the following expression:Using this expression of energy density during reheating epoch one can able to express the reheating temperature as:$$\begin{aligned} \rho _{\mathrm{reh}}=\frac{\pi ^2}{30}g_{*}(T_{\mathrm{reh}})T^4_{\mathrm{reh}}. \end{aligned}$$(4.16)where \(g_{*}(T_{\mathrm{reh}})\) is the effective number of total relativistic degrees of freedom present in the thermal bath at temperature \(T=T_{\mathrm{reh}}\) and \(V_{reh}\) is the scale of reheating which can be obtained by fixing the field value at \(\phi =\phi _{\mathrm{reh}}\) for a specific model. Counting all the degrees of freedom in the particle physics model one can fix \(g_{*}(T_{\mathrm{reh}})\) in the present context. To find the reheating constraint from the prescribed set up let us further introduce the number of efoldings at the epoch of reheating, which is defined as:$$\begin{aligned} T_{\mathrm{reh}}= & {} \left( \frac{30}{\pi ^2 g_{*}(T_{\mathrm{reh}})}\right) ^{1/4}\rho ^{1/4}_{\mathrm{reh}}\approx \left( \frac{30}{\pi ^2 g_{*}(T_{\mathrm{reh}})}\right) ^{1/4}V^{1/4}_{\mathrm{reh}},\nonumber \\ \end{aligned}$$(4.17)where \({{\mathcal {N}}}_{\mathrm{total}}\) is the total number of efoldings which is defined as:$$\begin{aligned}&{{\mathcal {N}}}_{\mathrm{reh}}=\int ^{t_{e}}_{t_{\mathrm{reh}}}H~dt\nonumber \\&\quad ={{\mathcal {N}}}_{\mathrm{total}}\widetilde{\Delta {{\mathcal {N}}}}\approx \frac{1}{M^2_p}\int ^{\phi _e}_{\phi _{\mathrm{reh}}}\frac{V(\phi )}{V^{\prime }(\phi )}~d\phi , \end{aligned}$$(4.18)Here \(t_{e}\), \(t_{i}\) and \(t_{\mathrm{reh}}\) are the representative time to specify end of inflation, starting of inflation and time scale at the end of reheating respectively. Similarly \(\phi _{e}\) and \(\phi _{\mathrm{reh}}\) are the field values at the end of inflation and reheating respectively, which can be computed for a given known model of inflation. Also it is important to note that in this context, \(\widetilde{\Delta {{\mathcal {N}}}}\) is defined as:$$\begin{aligned} {{\mathcal {N}}}_{\mathrm{total}}=\int ^{t_{e}}_{t_{i}}H~dt \frac{1}{M^2_p}\int ^{\phi _e}_{\phi _{i}}\frac{V(\phi )}{V^{\prime }(\phi )}~d\phi \sim \underbrace{{{\mathcal {O}}}(6070)}_{\mathbf{From~Planck~observation}}~. \end{aligned}$$(4.19)Here \(\Delta {{\mathcal {N}}}\) is defined as:$$\begin{aligned} \widetilde{\Delta {{\mathcal {N}}}}= & {} {{\mathcal {N}}}_{\mathrm{total}}{{\mathcal {N}}}_{\mathrm{reh}}=\Delta {{\mathcal {N}}}\left( {{\mathcal {N}}}_{\mathrm{reh}}{{\mathcal {N}}}_{\mathrm{cmb}}\right) \nonumber \\&\Longrightarrow \Delta {{\mathcal {N}}}\widetilde{\Delta {{\mathcal {N}}}}=\left( {{\mathcal {N}}}_{\mathrm{reh}}{{\mathcal {N}}}_{\mathrm{cmb}}\right) . \end{aligned}$$(4.20)From different models of inflation one can explicitly compute efoldings at horizon exit, which is given by the following expression:$$\begin{aligned} \Delta {{\mathcal {N}}}={{\mathcal {N}}}_{\mathrm{total}}{{\mathcal {N}}}_{\mathrm{cmb}}. \end{aligned}$$(4.21)Consequently, the value of \(\Delta {{\mathcal {N}}}\) from observation can be estimated as:$$\begin{aligned} {{\mathcal {N}}}_{\mathrm{cmb}}= & {} \int ^{t_e}_{t_{\mathrm{cmb}}}H~dt\approx \frac{1}{M^2_p}\int ^{\phi _e}_{\phi _{\mathrm{cmb}}}\frac{V(\phi )}{V^{\prime }(\phi )}~d\phi \nonumber \\&\sim \underbrace{{{\mathcal {O}}}(810)}_{\mathbf{From~Planck~observation }}~. \end{aligned}$$(4.22)Now, to give a numerical estimate of the reheating temperature let us consider the following simplest monomial model:$$\begin{aligned} \Delta {{\mathcal {N}}}\sim {{\mathcal {O}}}(52{}60). \end{aligned}$$(4.23)where \(V_0\) fix the overall scale of the potential and p is the degree of the monomial which depends on the characteristic of the model. For this model the field value during reheating can be expressed as:$$\begin{aligned} e V(\phi )=V_0 \left( \frac{\phi }{M_p}\right) ^p, \end{aligned}$$(4.24)The reheating scale is quantified in terms of the number of efoldings as:$$\begin{aligned} \phi _{\mathrm{reh}}=\sqrt{2p{{\mathcal {N}}}_{\mathrm{reh}}+\left( \frac{\phi _e}{M_p}\right) ^2}~M_p. \end{aligned}$$(4.25)Consequently, for the monomial model the reheating temperature can be quantified as:$$\begin{aligned} V(\phi _{\mathrm{reh}})=V_0 \left( \frac{\phi _{\mathrm{reh}}}{M_p}\right) ^p=V_0~\left[ 2p{{\mathcal {N}}}_{\mathrm{reh}}+\left( \frac{\phi _e}{M_p}\right) ^2\right] ^{p/2}.\nonumber \\ \end{aligned}$$(4.26)Here \(V_{\mathrm{inf}}\) is the scale of inflation which is quantified by the following expression:$$\begin{aligned}&{{{\mathbf {\underline{Reheating~bound~from~model:}}}}}\nonumber \\&T_{\mathrm{reh}}=\left( \frac{30}{\pi ^2 g_{*}(T_{\mathrm{reh}})}\right) ^{1/4}V^{1/4}_0\left[ 2p{{\mathcal {N}}}_{\mathrm{reh}}+\left( \frac{\phi _e}{M_p}\right) ^2\right] ^{p/8}\nonumber \\&\quad <V^{1/4}_{\mathrm{inf}}. \end{aligned}$$(4.27)As a result, we get the following bound on the reheating temperature:$$\begin{aligned}&{{{\mathbf {\underline{Upper~bound~on~inflationary~scale:}}}}}\nonumber \\&\quad V^{1/4}_{\mathrm{\inf }}\le 1.67\times 10^{16}\,\mathrm{GeV}\left( \frac{r(k_*)}{0.064}\right) ^{1/4}. \end{aligned}$$(4.28)which is true for any models of inflation. From the Planck 2018+BICEP2/Keck Array BK14 data the upper bound on the tensortoscalar ratio (primordial gravitational waves) is restricted to:$$\begin{aligned}&{{{\mathbf {\underline{Upperbound~on~reheating~ temperature~from~inflation:}}}}}\nonumber \\&\quad T_{\mathrm{reh}}\le 1.67\times 10^{16}\,\mathrm{GeV}\left( \frac{r(k_*)}{0.064}\right) ^{1/4}, \end{aligned}$$(4.29)where \(k_*\sim 0.05\,\mathrm{Mpc}^{1}\) is the pivot scale of momentum. This implies that the upper bound of reheating temperature from the Planck 2018+BICEP2/Keck Array BK14 data is given by:$$\begin{aligned} r(k_*)<0.064, \end{aligned}$$(4.30)Here to writing down this expression for reheating temperature it is important to consider the following assumption:$$\begin{aligned} T_{\mathrm{reh}}\le 1.67\times 10^{16}\,\mathrm{GeV}. \end{aligned}$$(4.31)This further implies that depending on the background particle physics model reheating temperature actually varies in a wide range and one cannot able to determine exactly its value as there is no such universal bound available earlier in this context. This is the main shortcoming of the phenomenological prediction of reheating temperature in the context of early universe cosmology.
 (a)
Contribution from the kinetic term of the field which is mainly responsible for reheating is neglected.
 (b)
We also assume that reheating is described by scalar field.
On the other hand, just only considering the dynamical details of quantum chaos one can express the reheating temperature in terms of the Lyapunov exponent:which is an universal lower bound on reheating temperature in the present context of discussion as it is not involve any model dependence from the background theory. This implies that the universal bound on quantum chaos in OEQFT restrict us to fix an universal model independent lower bound on reheating temperature. Combing the obtained bound in this paper and the upper bound obtained from inflation one can restrict the reheating temperature within a specified range. Additionally, the present analysis helps us put an unique upper bound on the Lyapunov exponent in terms of the scale of inflation (or tensortoscalar ratio) as:$$\begin{aligned}&{{{\mathbf {\underline{Universal~lowerbound`on~reheating~temperature:}}}}}\nonumber \\&\quad T_{\mathrm{reh}}\ge \frac{\lambda }{2\pi }, \end{aligned}$$(4.32)$$\begin{aligned} \lambda \le V^{1/4}_{\mathrm{inf}}=1.67\times 10^{16}\,\mathrm{GeV}\left( \frac{r(k_*)}{0.064}\right) ^{1/4}. \end{aligned}$$(4.33)  (a)
4.2 Out of time ordered correlators (OTOC) in OEQFT
4.2.1 What is OTOC?
4.2.2 Estimation of scrambling and dissipation time scales from OTOC
 1.
Scrambling time:
Here the associated time scale where the operator \({{\mathcal {C}}}(\tau )\) is relevant is known as the scrambling time scale \(\tau _{*}\). Sometimes in literature this is known as the Ehrenfest time scale. A possible distinction between the classical and quantum description of chaos can be described by the Ehrenfest time scale in which the previously mentioned OTOC don’t grow with respect to the associated time scale and saturates at the same scale. In the next section we have provided a alternative chaos bound on OTOC (i.e. SFF in our case) from which we have further give an estimate of the bound on the Ehrenfest time scale.
 2.
Dissipation time:
Another time scale for chaos is the exponential decay time scale \(\tau _d\) in which the two point thermal correlation function behaves like \(\langle V(0)V(\tau )\rangle \). Sometimes in this literature it is known as the dissipation time scale or the collision time scale. In the context of strongly coupled quantum field theories at finite temperature it is expected that the dissipation time scale \(\tau _d\sim \beta \). It is also expected that for large time limit the more general form of the OTOC during this time scaled as:where \(\cdots \) represent higher order terms which are more suppressed by the dissipation time scale \(\tau _d\).$$\begin{aligned}&\langle V(0)V(0)W(\tau )W(\tau )\rangle \sim \langle V(0)V(0) \rangle \langle W(\tau )W(\tau ) \rangle \nonumber \\&\quad +\,{{\mathcal {O}}}(\exp [\tau /\tau _d])+\cdots , \end{aligned}$$(4.41)
In Fig. 21a, b, we have shown the variation of the time dependent behaviour of semiclassical and classical OTOC for billiards, which show they are different in both the cases.
5 Quantum chaos from RMT: an alternative treatment in cosmology
Properties of Gaussian matrix ensemble in random matrix theory (RMT)
Element of matrix  Type of ensemble  Relation 

Elements are real  Gaussian orthogonal ensemble  Time reversal symmetric Hamiltonian 
Elements are complex  Gaussian unitary ensemble  Broken time reversal symmetric Hamiltonian 
Elements are quaternion  Gaussian symplectic ensemble  – 
5.1 Quantifying chaos using RMT
Gaussian matrix ensemble is a collection of large number of matrices which are filled with random numbers picked arbitrarily from a Gaussian probability distribution. See refs. [93, 94] for more details.
In Table 2, we have explicitly mentioned the properties of the each elements of the Gaussian matrix ensemble in random matrix theory (RMT).
5.2 OTOC in random matrix theory (RMT)
In earlier section we have introduced OTOC and its application to cosmology. In this subsection, we will discuss about OTOC appearing in the context of RMT.
5.2.1 Two point OTOC
5.2.2 Four point OTOC
5.3 Spectral form factor (SFF) from OTOC
From the traditional perspective the idea of quantum chaos is used in the context of study of spectral aspects of statistical field theory. Recent developments are made in the context of black hole theory and quantum information theory where using OTOC one can quantify quantum chaos. However in this paper our one of the prime objective to apply the concept of quantum chaos to study early universe cosmology, which is obviously another new direction of future research area. In this subsection, our air is to give a formal proof which establish the connection between Spectral Form Factor (SFF) and OTOC in OEQFT. First of all we consider a limit where \(\beta =1/T=0\) in which distribution of quantum operator insertions around a thermal circular path is very straightforward.
5.4 Two point SFF and thermal Green’s function in RMT
In this subsection our prime objective is to explicitly compute the expression for SFF for different even polynomial potential of random matrices. This is very useful to quantify chaos when we have no information about the interaction or time dependent effective mass profile which will finally give rise to scattering in conduction wire in presence of impurity or cosmological particle creation during reheating.
In usual prescriptions, SFF is averaged over an statistical ensemble of random matrix. This is a very particular feature of SFF and can be directly linked to the quantification of quantum chaos. Before going to discuss further here it is important to note that, all distribution for eigenvalues are different from each other but quite similar at small scales. This is a very crucial information for the computation of SFF to quantify chaos.
 1.
\(1/N^{2}\) part with sine squared function gives the ramp and have subdominant contribution.
 2.
1 / N part with Delta function gives the plateau and dominant.
 1.
For large \(x(\gg 1)\), \(\frac{\sin x}{x}\rightarrow 0\) and only the Dirac Delta function remains intact. So in this specific limit the vanishing of \(\sin \) term implies that the oscillatory fluctuations don’t contribute in the final expression for SFF. This limiting situation is called spectral rigidity.
 2.
For small \(x(\ll 1)\), \(\frac{\sin x}{x}\rightarrow 1\). In this limiting situation the integral gets maximum contribution from the \(\omega =0\) region. And this part contributes in ramp region.
5.5 SFF for even polynomial random potentials
5.5.1 For Gaussian random potential
From Fig. 23a–d we see that SFF at finite temperature decays with increasing \(\tau \) and reach zero. But with changing \(\beta \), SFF values remains almost same initially (for higher \(\beta \) or lower temperature).
5.5.2 For quartic random potential
From Fig. 26a, b, we see that SFF at finite temperature decays with increasing \(\tau \) and reach zero. But with changing \(\beta \) SFF values remains almost same initially (for higher \(\beta \) or lower value of temperature).
5.5.3 For sextic random potential
In Fig. 28a, b for sextic potential behaviour of density function \(\rho (\lambda )\) is shown. The curve follows from Eq. (5.185). Again choosing \(g=h=0\) will produce the Wigner law. Deviating g and h by small amount shows deviation from Wigner semicircle law. For \(g>0,h>0\) the curve shows plateau region though merge with semicircle at end points. But choosing \(g<0,h<0\) and \(g=0\) and \(h<0\) show deviation from semicircle and don’t converge even at end points.
In Fig. 30a–c, it is observed that SFF with variation in N get saturated at different value of \(\tau \). But with increasing N the value of the saturation point, will decrease. Subtracting the change of axis[\(SFF_{\tau =0}\)] we get the predicted bound of SFF.
5.5.4 For octa random potential
In Fig. 31a, b for octic potential behaviour of density function \(\rho (\lambda )\) is shown. The curve follows from Eq. (5.212). Again choosing \(g=h=k=0\) will produce the Wigner law. Deviating g, h and k by small amount shows deviation from Wigner semicircle law. For \(g=0,h>0,k>0\) the curve shows plateau region though merge with semicircle at end points. But choosing \(g>0,h<0,k<0\) and \(g>0,h>0,k<0\) show deviation from semicircle and don’t converge even at end points. On the other hand, if we choose \(g>0,h>0,k>0\) then we get a valley region lying between two peaks of the maxima of the density distribution of eigen values of the random matrices under consideration. The same behaviour can be obtained by fixing \(g>0,h<0,k>0\), \(g=h=k=1\) and \(g=0,h\gg 0,k\gg 0\). Only slight difference can be visualised in the peak heights of the maxima and also in the spread in the valley region. But in all such cases in between it will not at all match with the Wigner semicircle law, but converge to the end points of the Wigner semicircle, which is obtained by setting \(g=h=k=0\).
In Fig. 33a–c, it is observed that SFF with variation in N get saturated at different value of \(\tau \). But with increasing N the value of the saturation point, will decrease.Subtracting the change of axis[\(SFF_{\tau =0}\)] we get the predicted bound of SFF. From these plots we can say that time variation of SFF follow oscillatory pattern initially but after certain time it has linear decaying amplitude for dominance of linear part. Then after \(\tau >2\pi N\) region SFF abruptly saturated due to second part of the connected part of the total Green’s function \(G_{c}\). On the other hand, for \(\tau <2\pi N\) region SFF is decaying in amplitude and increasing with time. After \(\tau >2\pi N\) region the function will be constant thereafter.
Here it is important to note that, depending on the specific structure of the even polynomial random potential the upper bound on chaos very slightly changes (i.e. the amplitude for saturation of SFF is almost at the same order of magnitude for different even polynomial random potentials). But the late time behaviour for different random potentials are almost same as it shows complete saturation with respect to time. The saturation depends only on value of N. Also it is import to note from the plots that, for each even polynomial potential sudden transition from the random oscillatory behaviour to the perfect saturation of SFF take place at the unique time, \(\tau =2\pi N\).
5.5.5 Estimation of diptime scale from SFF
Now we introduce the concept of diptime which denotes the change in falloff behaviour of SFF near the critical points. It is estimated by comparing the initial falloff behaviour with late time behaviour of the curve from which it starts the linear increase (ramp part).
Falloff behaviour near critical points for different even polynomial random potential
Potential  1st critical point  2nd critical point  3rd critical point  4th critical point 

Gaussian  \(\tau ^{3}=\frac{\tau }{N^{2}}\)  –  –  – 
Quartic  \(\tau ^{3}=\frac{\tau }{N^{2}}\)  \(\tau ^{5}=\frac{\tau }{N^{2}}\)  –  – 
Sextic  \(\tau ^{3}=\frac{\tau }{N^{2}}\)  \(\tau ^{5}=\frac{\tau }{N^{2}}\)  \(\tau ^{7}=\frac{\tau }{N^{2}}\)  – 
Octa  \(\tau ^{3}=\frac{\tau }{N^{2}}\)  \(\tau ^{5}=\frac{\tau }{N^{2}}\)  \(\tau ^{7}=\frac{\tau }{N^{2}}\)  \(\tau ^{9}=\frac{\tau }{N^{2}}\) 
Order of magnitude estimation of conformal time (\(\tau \)) and physical time (t) in terms of the order of polynomial N
Equation of \(\tau \)  \(\tau \) in order of N  t in order of N 

\(\tau ^{3}=\frac{\tau }{N^{2}}\)  \(\tau =O(\sqrt{N})\)  \(t=O(1)\) 
\(\tau ^{5}=\frac{\tau }{N^{2}}\)  \(\tau =O(N^{\frac{1}{3}})\)  \(t=O(N^{\frac{1}{6}})\) 
\(\tau ^{7}=\frac{\tau }{N^{2}}\)  \(\tau =O(N^{\frac{1}{4}})\)  \(t=O(N^{\frac{1}{4}})\) 
\(\tau ^{9}=\frac{\tau }{N^{2}}\)  \(\tau =O(N^{\frac{1}{5}})\)  \(t=O(N^{\frac{3}{10}})\) 
5.6 Universal bound on quantum chaos from SFF and its application to cosmology
In Fig. 34a–c we have shown the nature of \(G_{dc}\) with different parameter. For all the cases \(G_{dc}\) decays to zero at \(\tau \rightarrow \infty \) which matches our analytical conclusion. Here we differentiate the \(\tau <2\pi N\) and \(\tau >2\pi N\) region with different color. In Fig. 35 we show the behaviour of SFF with temperature and time for large and small N. At higher \(\tau \) (Fig. 32a–c) SFF decays with \(\tau \) and at last goes to \(\frac{1}{\pi N}\). Else (Fig. 35a–d) SFF increases with \(\tau \) for \(\tau < 2\pi N\) and at last saturate to \(\frac{1}{\pi N}\). Now from the analytical solution we know that for large \(\tau \) or large \(\beta \) \(G_{dc}\) doesn’t contribute to the SFF as the \(G_{c}\) has \(\frac{\tau }{(2\pi N)^{2}}\) term.But for higher N and low \(\tau \) there should be a minima(\(\frac{1}{\pi N}\frac{1}{N}\)) for this function within the range \(\tau < 2\pi N\). As soon as the point \(\tau =2\pi N\) is crossed the function change its form and get saturated. This way of calculation of bound conclude that nature of SFF at late \(\tau \) shows same nature independent of type of potential. For infinite temperature SFF saturate at same level and at same \(\tau \) value for different potential with same N. For finite temperature it decays to zero irrespective of nature of potential. SFF is a measure of quantum chaos in a dynamical system. Bound on SFF prove that whatever be the interaction, every system at infinite temperature with same N saturated at same value. But at finite temperature randomness in the system decays to zero at late time and the system equilibrate within itself.
Here we consider Gaussian (Eq. (5.154)), quartic (Eq. (5.174), Eq. (5.176), Eq. (5.175)), sextic (Eq. (5.197), Eq. (5.198), Eq. (5.199)), octa (Eq. (5.219), Eq. (5.220), Eq. (5.221)) potential and applying same limit to get the SFF.
6 Randomness from higher order Fokker–Planck equation: a probabilistic treatment in cosmology
6.1 Cosmological scattering problem
Here we discuss about the cosmological scattering problem due to the particle creation in the context of early universe physics (mostly during reheating epoch). For detailed derivation of the results see Refs. [42, 99], which we have followed in this discussion mostly. As we have already discussed in the first half of the paper that, the Klein–Gordon equation, which is the dynamical master equation of the particles created during reheating can be solved in the same way as Schrödinger problem by formulating it as scattering problem in presence of an impurity potential inside a conduction wire [100] and can be related to the phenomena of chaos [43, 44, 101, 102]. In this section our prime objective is to establish this connection including the possible quantum effects (corrections) and we will try to develop a formalism to explain the quantum analogue of the chaos during cosmological particle creation.
6.2 Fokker–Planck Equation
 1.
First of all, we identify, \(M_{1}=M_{\tau }\) and \(M_{2}=M_{\delta \tau }\).
 2.Then we apply the composition law$$\begin{aligned} M=M_{2}M_{1}. \end{aligned}$$(6.37)
 3.Finally, we write \(M_{1}=M_{\tau }\) as:This implies:$$\begin{aligned} M_{1}=M_{2}^{1}M=M+\delta M(M,M_{2}). \end{aligned}$$(6.38)$$\begin{aligned} \delta M=(M^{1}_21)M. \end{aligned}$$(6.39)
 1.
Constraint I:
First of all, we talk about the Constraint I, which will fix the normalization condition of the probability density distribution as given by:This is obtained by setting the coefficient of the Lagrange multiplier \(g_1\) to zero.$$\begin{aligned} \langle 1 \rangle _{\delta \tau }=1. \end{aligned}$$(6.51)  2.
Constraint II:
Using the Constraint II, it is possible to fix the local mean particle production rate, which is quantitatively defined as:This is obtained by setting the coefficient of the Lagrange multiplier \(g_2\) to zero.$$\begin{aligned} \frac{\langle n_{2}\rangle _{\delta \tau }}{\delta \tau } = \mu . \end{aligned}$$(6.52)  3.
Constraint III:
Finally the Constraint III demands that:which basically implies that the addition of infinitesimal interval can’t correspond to a finite significant change in transfer matrix.$$\begin{aligned} \lim _{\delta \tau \rightarrow 0}M_{\tau +\delta \tau } \rightarrow M_{\tau }, \end{aligned}$$(6.53)To establish this statement we start with the following transfer matrix written in the polar form for \(j=2\):Now in the limit \(\delta \tau \rightarrow 0\) we have:$$\begin{aligned} M_2= \left( {\begin{array}{cc} e^{i \theta _2 } \sqrt{1+n_2} &{} e^{i(2 \phi _2  \theta _2) } \sqrt{1+n_2} \\ e^{i(2 \phi _2  \theta _2) } \sqrt{1+n_2} &{} e^{i \theta _2 } \sqrt{1+n_2} \end{array}} \right) . \end{aligned}$$(6.54)Consequently the transfer matrix can be simplified as:$$\begin{aligned}&\lim _{\delta \tau \rightarrow 0}n_2=0,\quad \lim _{\delta \tau \rightarrow 0}e^{\pm i\theta _2}=1,\nonumber \\&\quad \lim _{\delta \tau \rightarrow 0}e^{\pm i(2\phi _2\theta _2)}=0. \end{aligned}$$(6.55)To impose this specific nontrivial constraint we assume that the following condition is satisfied:$$\begin{aligned} \lim _{\delta \tau \rightarrow 0} M_2= \left( {\begin{array}{cc} 1~~~ &{}~~~ 0 \\ 0~~~ &{}~~~ 1 \end{array} } \right) =\mathbf{I}. \end{aligned}$$(6.56)where \( U(\theta _2)\) is a real valued and positive definite arbitrary function. This is possible if the function \(U(\theta _2)\) has an extremum at \(\theta _2=0\) where \(e^{i\theta _2}=1\). One can choose various types of function which can satisfy these constraints. For an example, as a phenomenological choice one can consider the following functional form:$$\begin{aligned} \langle U(\theta _2)\rangle _{\delta \tau }=\alpha \delta \tau \Longrightarrow \lim _{\delta \tau \rightarrow 0} \langle U(\theta _2)\rangle =\mathrm{fixed}, \end{aligned}$$(6.57)As a result, the probability density function reaches its maximum at \(\theta _2=0\) when the time interval \(\delta \tau \rightarrow 0\). Further, extremizing the expression for the Shannon entropy we get the following expression for the probability density distribution function \(P({n_{2},\theta _{2},\phi _{2}};\delta \tau )\) as given by:$$\begin{aligned} U(\theta _2)= & {} \left[ (e^{i\theta _2}1)(e^{i\theta _2}1)\right] ^{p}=e^{i\theta _2}1^{2p}\nonumber \\= & {} 4\sin ^{2p}\frac{\theta _2}{2}\quad \forall ~p=1,2,3,\ldots . \end{aligned}$$(6.58)where we introduce two new functions \({{\mathcal {K}}}(g_2)\) and \({{\mathcal {K}}}(g_3)\) which are defined as:$$\begin{aligned} P_2= & {} P({n_{2},\theta _{2},\phi _{2}};\delta \tau )=\left( \frac{1}{{{\mathcal {K}}}(g_2)}~e^{g_2n_2}\right) \nonumber \\&\times \left( \frac{1}{{{\mathcal {K}}}(g_3)}~e^{g_3 U(\theta _2)}\right) , \end{aligned}$$(6.59)$$\begin{aligned} {{\mathcal {K}}}(g_2)\equiv & {} \int ^{\infty }_{0} dn_2~e^{g_2n_2}=\frac{1}{g_2},\end{aligned}$$(6.60)Further using Eqs. (6.52) and (6.57), we get the following simplified expression for the probability density function: \(P({n_{2},\theta _{2},\phi _{2}};\delta \tau )\) as given by:$$\begin{aligned} {{\mathcal {K}}}(g_3)\equiv & {} \int \frac{d\theta _2}{2\pi }~e^{g_3 U(\theta _2)}. \end{aligned}$$(6.61)which implies that the probability density function is independent of \(\phi _{2}\) after applying the maximum entropy ansatz. For weak scattering, this corresponds to scattering events being uniformly distributed. Now if we consider large number of scatterings, then applying Central Limit Theorem one can show that the final result is not sensitive to the probability density function \(P_2\). In this discussion we have explicitly provided the mathematical form of the probability density function, which is not very important to derive the Fokker–Planck equation.$$\begin{aligned}&{{{\mathbf {\underline{Maximum~Entropy~Ansatz:}}}}}\quad P_2=P({n_{2},\theta _{2},\phi _{2}};\delta \tau )\nonumber \\&\quad =\left( \frac{1}{\mu \delta \tau }~e^{\frac{n_2}{\mu \delta \tau }}\right) \left( \frac{1}{{{\mathcal {K}}}(\alpha \delta \tau )}~e^{g_3 \alpha \delta \tau U(\theta _2)}\right) \nonumber \\&\quad =P(n_2;\delta \tau )P(\theta _2;\delta \tau )\nonumber \\&\quad =P(n_2,\theta _2;\delta \tau ), \end{aligned}$$(6.62)
 1.
Itô prescription:
According to this prescription one can write:which is true for \(Q=1\). Now using the ChapmanKolmogorov equation in the present context we get:$$\begin{aligned} n(\tau +\epsilon )= & {} y+\epsilon a(y)+\sqrt{D(y)}\int ^{\tau +\epsilon }_{\tau }d\tau ^{\prime }~b(\tau ^{\prime })\nonumber \\= & {} y+\epsilon a(y)+\sqrt{D(y)}~{{\mathcal {B}}}_{\epsilon }, \end{aligned}$$(6.85)Here upto the order \(\epsilon \) we use the following fact:$$\begin{aligned}&P(n,\tau +\epsilon y,\tau )=\left\langle \delta \left( ny\epsilon a(y)\sqrt{D(y)}~{{\mathcal {B}}}_{\epsilon }\right) \right\rangle \nonumber \\&\quad \simeq \left( 1\epsilon \frac{\partial a(n)}{\partial n}{{\mathcal {B}}}_{\epsilon }~\frac{\partial (\sqrt{D(n)})}{\partial n}+\frac{{{\mathcal {B}}}^2_{\epsilon }}{2}\frac{\partial ^2 D(n)}{\partial n^2}\right) \nonumber \\&\qquad \times \left\langle \delta \left( ny\epsilon a(y)\sqrt{D(y)}~{{\mathcal {B}}}_{\epsilon }\right) \right\rangle . \end{aligned}$$(6.86)Also we have used the following well known identity of Dirac Delta function, as given by:$$\begin{aligned} a(n)=a(y)+{{\mathcal {O}}}(\epsilon ). \end{aligned}$$(6.87)Now, we expand the Dirac Delta function in the powers of \(\epsilon \), as given by:$$\begin{aligned} \delta (f(y))=\frac{1}{f^{\prime }(y)}\delta (yy_0), \quad \mathrm{where}~~f(y_0)=0.\nonumber \\ \end{aligned}$$(6.88)where it is important to note that we have Taylor expanded the Dirac Delta function of the order of \({{\mathcal {B}}}^2_{\epsilon }\), this is because of the reason that \({{\mathcal {B}}}_{\epsilon }\sim {{\mathcal {O}}}(\sqrt{\epsilon })\). Hence we use this result in ChapmanKolmogorov equation and we get the following simplified integral:$$\begin{aligned}&\delta \left( ny\epsilon a(y)\sqrt{D(y)}~{{\mathcal {B}}}_{\epsilon }\right) \nonumber \\&\quad =\delta (ny)+\left[ \epsilon a(n)+\sqrt{D(y)}~{{\mathcal {B}}}_{\epsilon }\right] \delta ^{\prime }(xy)\nonumber \\&\qquad +\frac{1}{2}\left[ \epsilon a(n)+\sqrt{D(y)}~{{\mathcal {B}}}_{\epsilon }\right] ^2\delta ^{\prime \prime }(xy)+\cdots \nonumber \\ \end{aligned}$$(6.89)Further, performing the integration and using Eq. (6.83) and Eq. (6.84) we finally get the following simplified result of this integral:$$\begin{aligned}&P(n;\tau +\epsilon n_0)=\int dy~P(y,\tau n_0)~\left\langle \left[ \left( 1\epsilon \frac{\partial a(n)}{\partial n}\right. \right. \right. \nonumber \\&\quad \left. \left. \left. {{\mathcal {B}}}_{\epsilon }~\frac{\partial (\sqrt{D(n)})}{\partial n}+\frac{{{\mathcal {B}}}^2_{\epsilon }}{2}\frac{\partial ^2 D(n)}{\partial n^2}\right) \delta (yn)\right. \right. \nonumber \\&\quad +\left( \epsilon a(n)+\sqrt{D(n)}{{\mathcal {B}}}_{\epsilon }\right) \delta ^{\prime }(yn)\nonumber \\&\quad \left. \left. +\frac{D(n) {{\mathcal {B}}}^2_{\epsilon }}{2}\delta ^{\prime \prime }(yn)\right] \right\rangle \end{aligned}$$(6.90)So for \(Q=1\) the Fokker–Planck equation can be written starting from Langevin equation II in the following form:$$\begin{aligned}&P(n;\tau +\epsilon n_0)=P(n;\tau n_0)+\epsilon \frac{\partial P(n;\tau n_0)}{\partial \tau }\nonumber \\&\quad =P(n;\tau n_0)+\epsilon \left\{ \frac{\partial }{\partial n}\left( a(n)P(n,\tau n_0)\right) \right. \nonumber \\&\qquad \left. +\frac{\partial ^2}{\partial n^2}\left( D(n)P(n,\tau n_0)\right) \right\} +{{\mathcal {O}}}(\epsilon ^2). \end{aligned}$$(6.91)(6.92)  2.
Generalized Itô prescription:
In this situation for general Q one can write:Further, we have to make the following substitution:$$\begin{aligned} \langle n(\tau +\epsilon )y\rangle =\epsilon \left[ a(y)+(1Q)D^{\prime }(y)\right] . \end{aligned}$$(6.93)This will finally lead to the following Fokker–Planck equation:$$\begin{aligned} a(y)\longrightarrow a(y)+(1Q)D^{\prime }(y). \end{aligned}$$(6.94)(6.95)
6.3 Corrected probability distribution profiles: quantum effects from nonGaussianity
In this subsection we get different order correction to the Fokker–Planck equation that we have derived by Taylor expansion. As we already know that the Taylor expansion of the probability density distribution function is taken with respect to time \(\tau \). On the other hand, using Maximum entropy ansatz we have considered the Taylor expansion of the ensemble average of the distribution function with respect to the occupation number n. After that we equate both the results and comparing the coefficient of \(\delta \tau \) from the both sides of the expansion (see previous Eq. (6.68) for more details). Now without truncating both the sides of this expression one can get additional contributions in \(\delta \tau \) and in its higher order. If we do the comparison including such additional contributions then it will give rise to corrected version of the Fokker–Planck equation valid upto higher orders. Further solving these sets of differential equation order by order one can explicitly justify the validity of all such corrections in the Fokker–Planck equation. In this paper we have investigated this possibility by considering the contributions upto fourth order. All such higher order correction terms are very useful to describe the nonGaussian effects appearing during the process of cosmological particle production during reheating phase of early universe. On top of that, one can explain the origin of such higher order contributions in the quantum mechanical ground as it produces non vanishing significant effects in the expression for the higher order statistical moments directly originating from the various quantum mechanical correlations (onepoint, twopoint, threepoint etc.) computed during cosmological particle production at the epoch of reheating of early universe. More precisely, the deviation from Gaussianity (in other words the deviation from lognormal distribution) in the present context can be directly linked with the quantum mechanical effects appearing during reheating epoch of early universe and for this reason one can interpret the higher order corrected version of the Fokker–Planck equation as a quantum corrected Fokker–Planck equation. Since in this paper we have provided the analytical correction upto the fourth order, one can say that in this derivation we have actually provided the fourth order quantum corrected Fokker–Planck equation. The details of this derivations are explicitly discussed in the following sub sections, where doing the analysis we justify order by order that how such specific corrections will modify the lognormal distribution and its impact in the quantum mechanical ground.
6.3.1 First order contribution
In Fig. 38, we have shown the evolution of the probability density function with respect to the occupation number per mode, for a fixed time (\(\mu _k\tau \)=fixed). For very small values of the parameter n we have observed from the plot that the deviation from Gaussian feature is observed.
6.3.2 Second order contribution
From Fig. 39 we can say that for \(m_{2}=\pm 2,\pm 3\) second order corrected probability distribution function almost overlap at lower values of the occupation number n but deviate significantly as n increases to large number. As a consequence, for low value of n particle production rate is independent on \(m_{2}\) but as n increases they significantly deviate. It also implies that for higher values of n the integer \(m_{2}\) constrains particle production rate. For \(m_2=\pm 1\) we found that the second order corrected probability distribution function significantly deviates from log normal (Gaussian) distribution and both of them explicitly show the signature of nonGaussianity is the second order corrected distribution function. Finally, we have shown that for \(m_2=0\) the amount of deviation from log normal (Gaussian) distribution is small compared to results obtained from the other values of \(m_2\).
In Fig. 40, we have shown the probability density function for second order correction with respect to the occupation number per mode, for a fixed time (\(\tau =\frac{0.15}{\mu _{k}}\)). From this plot we have observed irregular oscillations with deviation from Gaussian feature.We have selected a high value for momentum cutoff[\(\lambda _{c}\)] for this particular plot.
6.3.3 Third order correction
From Fig. 41 one can say that the third order corrected probability distribution function for different value of \(m_{3}=0\) overlap at lower n limit (n \(\rightarrow 0\)) though deviate significantly at large n limit. At lower values of n, particle production probability is independent of \(m_{3}\) and almost flat. But as soon as n reaches values greater than unity the distribution increase exponentially and for different \(m_{3}\) they differ from each other.
In Fig. 42, we have shown the third order correction of probability density function with respect to the occupation number per mode, for a fixed time (\(\mu _k\tau \)=fixed). From this plot we have observed primary gaussian feature followed by an exponential type increase.We consider the perturbative expansion to be valid. So the higher order correction contribute less than the previous order. To normalize the analytical solution we have used this. In latter part we can see the exponential type increase contribute in longer tail effect [higher kurtosis].
6.3.4 Fourth order correction
During the analysis we assume that the particle production probability and all its derivative has a constant same value for \(n=0.0001\), which is very very helpful for us to deal with the initial conditions during performing numerical techniques to solve the Eq. (6.225).
From Fig. 43 we observe that the fourth order corrected probability distribution function for different \(m_{4}\) is almost flat upto \(n=1\) and after that the distribution function suddenly increases. Additionally, we observe that the fourth order correction has deviation from Gaussianity at small values of the occupation number. On the other hand, for large values of the occupation number we get a Gaussian like feature and that is shown explicitly in the mentioned plot.
In Fig. 44, we have shown the fourth order correction to the probability density function with respect to the occupation number per mode, for a fixed time (\(\mu _k\tau \)=fixed) flow the initial gaussian then exponential decay type distribution.This decreasing feature suggest very low effect from fourth order correction which is also supported by perturbative expansion assumption.
6.3.5 Total solution considering different order correction
Now we plot total solutions with different order of correction with main solution.Then we check validation of different order of correction altogether and how they merge with each other at what limit.
A. Upto second order correction:
From Fig. 45a, b we observe that at low values of n, \(P_{1}+P_{2}\) and \(P_{2}\) are significantly different, but as we increase n they overlapped and \(P_{2}\) effect is more over \(P_{1}+P_{2}\) so the second order solution dominate over the first order solution.On the other hand Fig. 45c shows how the second order contribution effect the primary gaussian curve. Distinct oscillation effect on gaussian curve shows the effect of second order correction which is not available from the numerical solution.
B. Upto third order correction:
From Fig. 46a, b we observe that \(P_{1}+P_{2}+P_{3}\), \(P_{2}+P_{3}\) and \(P_{3}\) overlap at higher n limit though separated. At low n limit and \(P_{1}+P_{2}\) and \(P_{2}\) overlap with each other but remain separated from \(P_{1}+P_{2}+P_{3}\) for the complete range. This implies that third order contribution is dominant over the other two due to the nonlinearities in the differential equation and behaves like a nonperturbative quantum effect at the level of solution. From Fig. 46c we show the third order correction affects the tail of the gaussian and shift it bit higher.It is clear from Fig. 46d.
C. Upto fourth order correction:
From Fig. 47a we observe the final curve represented by \(P_{1}+P_{2}+P_{3}+P_{4}\) shifted the mean of the gaussian from its value at \(P_{1}\). Initial gaussian feature for \(P_{1}+P_{2}+P_{3}\) and \(P_{1}+P_{2}\) are deviated at high n limit . Here \(P_{1}+P_{2}+P_{3}+P_{4}\) and \(P_{1}\) follow exact gaussian nature but they are mirror image of one another. \(P_{1}+P_{2}+P_{3}\) and \(P_{1}+P_{2}\) don’t have this gaussian nature due to their divergence property as n increases. In Fig. 47b and Fig. 47c the effect of all order correction can be observed but effect of different order correction become evident from Fig. 47d. Second order correction introduce the oscillating feature whereas third order correction increase the tail and responsible for higher kurtosis. Fourth order correction add a small positive effect to the previous correction without any shape change.
Previously it is shown that in Ref. [42] the distribution will be lognormal at large n, considering the lowest order contribution coming from the solution of Fokker–Planck equation. We extend this result upto fourth order and shown the effect of different order correction.The oscillating nature and long tail can be specific feature of stochastic particle production in inflation epoch.This may be the effect of background field or noise at that time.The numerical solutions give different quantum numbers which support the quantum nature. In the next subsection we will calculate various statistical moments and from that we can discuss about the role of quantum effects and nonGaussianity from the Probability distribution profile for particle production in the context of early universe cosmology (mostly during reheating).
6.4 Calculation of statistical moments (or quantum correlation functions) from corrected probability distribution function
Here our prime objective is to compute the different statistical moments from the quantum corrected probability distribution function as obtain by Taylor expanding in order by order from Eq. (6.68). From is corrected probability distribution function we compute the expression for \(\langle n\rangle \),\(\langle n^{2}\rangle \),\(\langle n^{3}\rangle \) and \(\langle n^{4}\rangle \) and then calculate standard deviation, skewness and kurtosis for a given time. We have explicitly shown that the non vanishing contributions of skewness and kurtosis carries the signature of significant effect of nonGaussianity. In this analysis the values of these moments are compared with predicted results obtained from lognormal (Gaussian) distribution and non zero values of kurtosis and skewness define the deviation from that.
 1.
Step I:
First of all, we use the first order master evolution equation. Then we replace the function F by the occupation number n. Consequently, we get the following time evolution equation of the first moment or one point function \(\langle n\rangle \), given by:$$\begin{aligned} \frac{1}{\mu _{k}} \frac{\partial \langle n\rangle }{\partial \tau } =\langle (1+2n)\rangle = 1 + 2 \langle n\rangle . \end{aligned}$$(6.250)  2.
Step II:
Secondly, we want to compute the expression for \(\langle n^2\rangle \). To compute this we consider here the first and second order master equations, as mentioned earlier. Considering only the first order master equation we get the following analytical expression:On the other hand, using the second order master equation we get the following analytical expression for the time evolution of the second moment or two point correlation:$$\begin{aligned}&\frac{1}{\mu _{k}} \frac{\partial \langle n^2\rangle }{\partial \tau } =\langle 2n(1+2n)+2n(1+n)\rangle \nonumber \\&\quad = \langle 4n + 6n^2\rangle =4\langle n\rangle +6\langle n^2 \rangle . \end{aligned}$$(6.251)$$\begin{aligned} \frac{1}{\mu _{k}^{2}}\frac{\partial ^{2} \langle n^2\rangle }{\partial \tau ^{2}}=\langle 2(1+6n+6n^2)\rangle =12 \langle n\rangle +12 \langle n^2\rangle +2. \end{aligned}$$(6.252)  3.
Step III:
Next, we want to compute the expression for \(\langle n^3\rangle \). To compute this we consider here the first, second and third order master equations, as mentioned earlier. Considering only the first order master equation we get the following analytical expression:On the other hand, using the second order master equation we get the following analytical expression for the time evolution of the third moment or three point correlation:$$\begin{aligned}&\frac{1}{\mu _{k}} \frac{\partial \langle n^3\rangle }{\partial \tau } =\langle 3n^2(1+2n)+6n(1+n)\rangle \nonumber \\&\quad = \langle 6n + 9n^2 + 6n^3\rangle =6\langle n\rangle +9\langle n^2 \rangle +6\langle n^3 \rangle .\nonumber \\ \end{aligned}$$(6.253)Finally, using the third order master equation we get the following analytical expression for the time evolution of the third moment or three point correlation:$$\begin{aligned}&\frac{1}{\mu _{k}^{2}}\frac{\partial ^{2} \langle n^3\rangle }{\partial \tau ^{2}}=\langle 12n(1+3n+3n^2)+6n(1+6n+6n^2)\rangle \nonumber \\&\quad =18 \langle n\rangle +72 \langle n^2\rangle +60 \langle n^3\rangle . \end{aligned}$$(6.254)$$\begin{aligned}&\frac{1}{\mu _{k}^{3}}\frac{\partial ^{3} \langle n^3\rangle }{\partial \tau ^{3}}=\langle 6(1+2n)(1+10n+10n^2)\rangle \nonumber \\&\quad =72 \langle n\rangle +180 \langle n^2\rangle +120\langle n^3\rangle +6. \end{aligned}$$(6.255)  4.
Step IV:
Next, we want to compute the expression for \(\langle n^4\rangle \). To compute this we consider here the first, second, third and fourth order master equations, as mentioned earlier. Considering only the first order master equation we get the following analytical expression:On the other hand, using the second order master equation we get the following analytical expression for the time evolution of the fourth moment or four point correlation:$$\begin{aligned}&\frac{1}{\mu _{k}} \frac{\partial \langle n^4\rangle }{\partial \tau } =\langle 4n^3(1+2n)+12n^2(1+n)\rangle \nonumber \\&\quad = \langle 16n^3 + 20n^4 \rangle =16\langle n^3\rangle +20\langle n^4 \rangle . \end{aligned}$$(6.256)Then, using the third order master equation we get the following analytical expression for the time evolution of the fourth moment or fourth point correlation:$$\begin{aligned}&\frac{1}{\mu _{k}^{2}}\frac{\partial ^{2} \langle n^4\rangle }{\partial \tau ^{2}}=\langle 12n^2(1+n)^2+48n^2(1+3n+2n^2)\nonumber \\&\qquad +12n^2(1+6n+6n^2)\rangle \nonumber \\&\quad =72 \langle n^2\rangle +240 \langle n^3\rangle +180 \langle n^4\rangle . \end{aligned}$$(6.257)Finally, using the third order master equation we get the following analytical expression for the time evolution of the fourth moment or fourth point correlation:$$\begin{aligned}&\frac{1}{\mu _{k}^{3}}\frac{\partial ^{3} \langle n^4\rangle }{\partial \tau ^{3}}=\langle 72n(1+n)(1+5n+5n^2)\nonumber \\&\qquad +24n(1+2n)(1+10n+10n^2)\rangle \nonumber \\&\quad =96 \langle n\rangle +720 \langle n^2\rangle +1440\langle n^3\rangle +840\langle n^4\rangle . \end{aligned}$$(6.258)$$\begin{aligned}&\frac{1}{\mu _{k}^{4}}\frac{\partial ^{4} \langle n^4\rangle }{\partial \tau ^{4}}=\langle 24(1+20n+90n^2+140n^3+70n^4)\rangle \nonumber \\&\quad =480 \langle n\rangle +2160 \langle n^2\rangle +3360\langle n^3\rangle \nonumber \\&\qquad +1680\langle n^4\rangle +24. \end{aligned}$$(6.259)  5.
Step V:
Further we apply the boundary conditions, i.e. \(\langle n\rangle \), \(\langle n^{2}\rangle \), \(\langle n^{3}\rangle \), \(\langle n^{4}\rangle \), \(\frac{d\langle n^{2}\rangle }{d \tau ^2}\), \(\frac{d\langle n^{3}\rangle }{d \tau ^3}\) and, \(\frac{d\langle n^{4}\rangle }{d \tau ^4}\) are vanishingly small at \(\tau =0\). Using these conditions we get expressions for \(\langle n\rangle \), \(\langle n^{2}\rangle \), \(\langle n^{3}\rangle \) and \(\langle n^{4}\rangle \).
 6.
Step VI:
Using the result obtained in Step I and using the previously mentioned boundary condition we get the following expression for the one point function^{30} (or first moment) of occupation number:which is further used to compute all the higher order moments from master evolution equation considering higher order Taylor expansion. Additionally, it is important to note that if we use higher order equations for the first moment then after imposing the boundary conditions we get the following results:$$\begin{aligned}&{{{\mathbf {\underline{First~Moment~(First~Order):}}}}}\nonumber \\&\quad \langle n\rangle _{\mathbf{I}}=\frac{1}{2} (e^{2 \tau \mu _{k}}1), \end{aligned}$$(6.260)$$\begin{aligned}&{{{\mathbf {\underline{First~Moment~(Second~Order):}}}}}\nonumber \\&\quad \langle n\rangle _{\mathbf{II}}=0, \end{aligned}$$(6.261)$$\begin{aligned}&{{{\mathbf {\underline{First~Moment~(Third~Order):}}}}}\nonumber \\&\quad \langle n\rangle _{\mathbf{III}}=0, \end{aligned}$$(6.262)Consequently, the total first moment can be written as:$$\begin{aligned}&{{{\mathbf {\underline{First~Moment~(Fourth~Order):}}}}}\nonumber \\&\quad \langle n\rangle _{\mathbf{IV}}=0. \end{aligned}$$(6.263)In Fig. 48, we have explicitly shown the time dependent behaviour of first moment or one point function of the occupation number \(\langle n\rangle \). As there is no contributions are coming from the second, third and fourth order moment generating master evolution equation for \(\langle n\rangle \), the only contribution is coming from the first order master evolution equation. From this plot we see that for a fixed value of the parameter \(\mu _k=1\), at the lower values of the time the first moment or the one point function of the occupation number initially increase with time very very slowly. Then after a certain time when \(\tau \gg 1\) it shows suddenly huge increment in the behaviour. Most importantly, this plot shows the first moment or one point function of the occupation number is not zero. This shows the first signature of the nonGaussianity as we know for Gaussian probability distribution profile this is exactly zero.$$\begin{aligned}&{{{\mathbf {\underline{Total~First~Moment:}}}}}\nonumber \\&\quad \langle n\rangle =\langle n\rangle _{\mathbf{I}}+\langle n\rangle _{\mathbf{II}}+\langle n\rangle _{\mathbf{III}}+\langle n\rangle _{\mathbf{IV}}\nonumber \\&\qquad =\langle n\rangle _{\mathbf{I}}=\frac{1}{2} (e^{2 \tau \mu _{k}}1). \end{aligned}$$(6.264)  7.
Step VII:
Using the results obtained in Step II and using the previously mentioned boundary condition we get the following expression for the two point function^{31} (or second moment) of occupation number:$$\begin{aligned}&{{{\mathbf {\underline{Second~Moment~(First~Order):}}}}}\nonumber \\&\quad \langle n^{2}\rangle _{\mathbf{I}}=\frac{1}{6} e^{6 \tau \mu _{k}} + \frac{1}{6} (2  3 e^{2 \tau \mu _{k}}) ~.\end{aligned}$$(6.267)which is further used to compute all the higher order moments from master evolution equation considering higher order Taylor expansion. Additionally, it is important to note that if we use higher order equations for the second moment then after imposing the boundary conditions we get the following results:$$\begin{aligned}&{{{\mathbf {\underline{Second~Moment~(Second~Order):}}}}}\nonumber \\&\langle n^{2}\rangle _{\mathbf{II}}=\frac{1}{24} e^{2 \sqrt{3} \mu _k \tau } \left[ 8 e^{2 \sqrt{3} \mu _k \tau }18 e^{2 \left( \sqrt{3}+1\right) \mu _k \tau }\right. \nonumber \\&\quad \left. +\left( 3 \sqrt{3}+5\right) e^{4 \sqrt{3} \mu _k \tau }3 \sqrt{3}+5\right] , \end{aligned}$$(6.268)$$\begin{aligned}&{{{\mathbf {\underline{Second~Moment~(Third~Order):}}}}}\nonumber \\&\quad \langle n^2\rangle _{\mathbf{III}}=0, \end{aligned}$$(6.269)Consequently, the total second moment can be written as:$$\begin{aligned}&{{{\mathbf {\underline{Second~Moment~(Fourth~Order):}}}}}\nonumber \\&\quad \langle n^2\rangle _{\mathbf{IV}}=0. \end{aligned}$$(6.270)In Fig. 49, we have explicitly shown the time dependent behaviour of second moment or amplitude of the two point function of the occupation number \(\langle n^2\rangle \). As there is no contributions are coming from the third and fourth order moment generating master evolution equation for \(\langle n^2\rangle \), the only contribution is coming from the first and second order master evolution equation. From this plot we see that for a fixed value of the parameter \(\mu _k=1, 10, 100\), at the lower values of the time the second moment or the amplitude of the two point function of the occupation number initially increase with time very very slowly. Then after a certain time when \(\tau \gg 1\) it shows suddenly huge increment in the behaviour.$$\begin{aligned}&{{{\mathbf {\underline{Total~Second~Moment:}}}}} \nonumber \\&\quad \langle n^{2}\rangle =\langle n^{2}\rangle _{\mathbf{I}}+\langle n^{2}\rangle _{\mathbf{II}}+\langle n^{2}\rangle _{\mathbf{III}}+\langle n^{2}\rangle _{\mathbf{IV}}\nonumber \\&\qquad =\langle n^{2}\rangle _{\mathbf{I}}+\langle n^{2}\rangle _{\mathbf{II}}. \end{aligned}$$(6.271)  8.
Step VIII:
Using the results obtained in Step III and using the previously mentioned boundary condition we get the following expression for the three point function^{32} (or third moment) of occupation number:$$\begin{aligned}&{{{\mathbf {\underline{Third~Moment~(First~Order):}}}}}\nonumber \\&\quad \langle n^{3}\rangle _{\mathbf{I}}=\frac{1}{8 \mu _k }\left[ e^{6 \mu _k \tau } \left( 12 \mu ^2_k \tau +2 \mu _k 5\right) \right. \nonumber \\&\left. \quad +(96 \mu _k ) e^{2 \mu _k \tau }+4 (\mu _k 1)\right] ~.\end{aligned}$$(6.274)$$\begin{aligned}&{{{\mathbf {\underline{Third~Moment~(Second~Order):}}}}}\nonumber \\&\quad \langle n^{3}\rangle _{\mathbf{II}}=\frac{1}{560} \left[ 35 \left( 3 \sqrt{3}5\right) e^{2 \sqrt{3} \mu _k \tau }+450 e^{2 \mu _k \tau }\right. \nonumber \\&\left. \qquad 35 \left( 3 \sqrt{3}+5\right) e^{2 \sqrt{3} \mu _k \tau }+\left( 6 \sqrt{15}+20\right) e^{2 \sqrt{15} \mu _k \tau }\right. \nonumber \\&\left. \qquad +\left( 206 \sqrt{15}\right) e^{2 \sqrt{15} \mu _k \tau }140\right] ~.\end{aligned}$$(6.275)which is further used to compute all the higher order moments from master evolution equation considering higher order Taylor expansion. Additionally, it is important to note that if we use higher order equations for the third moment then after imposing the boundary conditions we get the following results:$$\begin{aligned}&{{{\mathbf {\underline{Third~Moment~(Third~Order):}}}}}\nonumber \\&\quad \langle n^{3}\rangle _{\mathbf{III}}=\frac{1}{36960}\left[ \frac{2 }{\sqrt{3}3 i}\right. \nonumber \\&\quad \times \left( 6135 i+2045 \sqrt{3}1654\ 3^{5/6} \root 3 \of {5}+1011 \root 6 \of {3} 5^{2/3}\right. \nonumber \\&\quad \left. +1011 i 15^{2/3}\right) e^{\root 3 \of {15} \left( 1+i \sqrt{3}\right) \mu _{k} \tau }+32670 e^{2 \mu _{k} \tau }\nonumber \\&\quad 1050 \left( 10 \sqrt{3}+17\right) e^{2 \sqrt{3} \mu _{k} \tau }\nonumber \\&\quad +\left( 4090+827 i 3^{5/6} \root 3 \of {5}1011 i \root 6 \of {3} 5^{2/3}\right. \nonumber \\&\quad \left. 827 \root 3 \of {15}337\ 15^{2/3}\right) e^{i \root 3 \of {15} \left( \sqrt{3}+i\right) \mu _{k} \tau }\nonumber \\&\quad +1050 \left( 10 \sqrt{3}17\right) e^{2 \sqrt{3} \mu _{k} \tau }9240\nonumber \\&\quad +\frac{2 }{\sqrt{3}3 i}\left( 6135 i+2045 \sqrt{3}\right. \nonumber \\&\quad +827\ 3^{5/6} \root 3 \of {5}+1011 \root 6 \of {3} 5^{2/3}\nonumber \\&\quad \left. \left. 2481 i \root 3 \of {15}1011 i 15^{2/3}\right) e^{2 \root 3 \of {15} \mu _{k} \tau }\right] , \end{aligned}$$(6.276)Consequently, the total third moment can be written as:$$\begin{aligned}&{{{\mathbf {\underline{Third~Moment~(Fourth~Order):}}}}}\nonumber \\&\quad \langle n^2\rangle _{\mathbf{IV}}=0. \end{aligned}$$(6.277)$$\begin{aligned}&{{{\mathbf {\underline{Total~Third~Moment:}}}}}\nonumber \\&\quad \langle n^{3}\rangle =\langle n^{3}\rangle _{\mathbf{I}}+\langle n^{3}\rangle _{\mathbf{II}}+\langle n^{3}\rangle _{\mathbf{III}}+\langle n^{3}\rangle _{\mathbf{IV}}\nonumber \\&\qquad =\langle n^{3}\rangle _{\mathbf{I}}+\langle n^{3}\rangle _{\mathbf{II}}+\langle n^{3}\rangle _{\mathbf{III}}~. \end{aligned}$$(6.278)In Fig. 50, we have explicitly shown the time dependent behaviour of third moment or three point function of the occupation number \(\langle n^3\rangle \). As there is no contributions are coming from the fourth order moment generating master evolution equation for \(\langle n^3\rangle \), the only contribution is coming from the first, second and third order master evolution equation. From this plot we see that for a fixed value of the parameter \(\mu _k=1,10,100\), at the lower values of the time the third moment or the equal time amplitude of the three point function of the occupation number initially increase with time very slowly. Then after a certain time it shows suddenly huge increment in the behaviour. Most importantly, this plot shows the third moment or equal time amplitude of the three point function of the occupation number is not zero. This shows the second signature of the nonGaussianity as we know for Gaussian probability distribution profile this is exactly zero.
 9.
Step IX:
Using the results obtained in Step IV and using the previously mentioned boundary condition we get the following expression for the four point function^{33} (or fourth moment) of occupation number:$$\begin{aligned}&{{{\mathbf {\underline{Fourth~Moment~(First~Order):}}}}}\nonumber \\&\langle n^{4}\rangle _{\mathbf{I}}\nonumber \\&=\frac{6 (\mu 1) e^{2 \mu \tau }+3 (5 \mu 2) e^{8 \mu \tau }{}2 e^{6 \mu \tau } (3 \mu (4 \mu \tau + 3){}5)3 \mu + 2}{2 \mu }.\end{aligned}$$(6.281)The third and the fourth order corrected version of the fourth order moment equations are not exactly solvable analytically. For this reason we have applied numerical techniques to solve these differential equations.$$\begin{aligned}&{{{\mathbf {\underline{Fourth~Moment~(Second~Order):}}}}}\nonumber \\&\langle n^{4}\rangle _{\mathbf{II}}=\frac{\left( 6745 \sqrt{5}\right) e^{6 \sqrt{5} \mu \tau }}{10080}\nonumber \\&\quad +\frac{\left( 45 \sqrt{5}+67\right) e^{6 \sqrt{5} \mu \tau }}{10080}+\frac{1}{5040}(5265 \sinh (2 \mu \tau )\nonumber \\&\quad 216 \sqrt{15} \sinh \left( 2 \sqrt{15} \mu \tau \right) +2610 \sqrt{3} \sinh \left( 2 \sqrt{3} \mu \tau \right) \nonumber \\&\quad 5265 \cosh (2 \mu \tau )\nonumber \\&\quad +4350 \cosh \left( 2 \sqrt{3} \mu \tau \right) 720 \cosh \left( 2 \sqrt{15} \mu \tau \right) +1568). \end{aligned}$$(6.282)Consequently, the total fourth moment can be written as:In Fig. 51, we have explicitly shown the time dependent behaviour of fourth moment or amplitude of the four point function of the occupation number \(\langle n^4\rangle \). From this plot we see that for a fixed value of the parameter \(\mu _k=1,10,100\), at the lower values of the time the third moment or the equal time amplitude of the four point function of the occupation number initially increase with time very slowly. Then after a certain time it shows suddenly huge increment in the behaviour.$$\begin{aligned}&{{{\mathbf {\underline{Total~Fourth~Moment:}}}}}\nonumber \\&\quad \langle n^{4}\rangle =\underbrace{\langle n^{4}\rangle _{\mathbf{I}}+\langle n^{4}\rangle _{\mathbf{II}}}_{{{\mathbf {\underline{Analytical}}}}} +\underbrace{\langle n^{4}\rangle _{\mathbf{III}}+\langle n^{4}\rangle _{\mathbf{IV}}}_{Numerical}~. \end{aligned}$$(6.283)
6.4.1 Standard deviation
From the Fig. 52, we can see that the uncorrected Standard Deviation (first order) and corrected Standard Deviation (second order) has significant difference in low \(\mu _{k} \tau \) limit and second order overlapped as they approach higher \(\mu _{k} \tau \). So for lower limit this second order correction is significant and for this reason during the computation of Kurtosis and Skewness we use total solution of standard deviation over the uncorrected one alone. In Fig. 52a the variance is with in the value 1 but in Fig. 52b variance has a enormous value.It is mere effect of tuning.The plots or values of variance can always be tuned to be within 1 using proper prefactor.
6.4.2 Skewness
In this subsection, our prime objective is to computed the expression for the Skewness from the corrected probability distribution function. Skewness actually measure the asymmetry of the probability distribution function of a realvalued random variable about its mean value. This measure can be positive or negative, or undefined. From positive skewness (for unimodal distribution) we can say normal curve has longer right tail.
Now from Fig. 53, we can say that the corrected Skewness deviate significantly from the uncorrected one at low \(\mu _{k} \tau \) limit. But we can see that at higher limit they overlap. Also Skewness is positive for the whole range which implies that the normal distribution curve has longer right tail. Moreover, there is a discontinuity of third order corrected Skewness in between the range \(0.1<\tau <1\) and for the rest of the whole range of time Skewness decreased upto unity and then it is increased.
6.4.3 Kurtosis
Kurtosis is a measure of the tailedness of the probability distribution of a realvalued random variable. This is actually a descriptor of the shape of a probability distribution function and there are specific ways of quantifying it for a theoretical probability distribution and corresponding ways of estimating it from a sample from a population. It is important to note that, the Kurtosis of any univariate normal distribution is 3. For practical purposes it is common practice to compare the expression for Kurtosis of a probability distribution function to 3. Probability distributions with Kurtosis less than the value 3 are identified as platykurtic, although this information does not imply the distribution is flattopped in nature. Rather, it implies that the probability distribution produces fewer and less extreme outliers than does the normal probability distribution. Probability distributions with Kurtosis greater than the value 3 are said to be leptokurtic. It is also common practice to use, the excess Kurtosis, which is the Kurtosis minus 3, to provide the comparison to the normal probability distribution profile. Like Skewness here also we calculate kurtosis from different distribution and get it at different order of correction.
 1.
The Standard Deviation is significantly large for higher \(\mu _{k} \tau \), though very small for lower regime.
 2.
Skewness is positive throughout the time regime, though becomes vanishingly small at a specific time interval (\(0.1<\tau <0.7\)).
 3.
Kurtosis is greater than 3 for the whole time regime.
 4.
The predicted LogNormal Gaussian Distribution shows deviations at significant levels. Effects of the nonGaussianity in the distribution function is clearly visualized.
 5.
The probability distribution has longer trailing ends and the trails go broad higher.
 6.
The probability distribution has a very low spread at lower \(\mu _{k} \tau \) limit though highly spread out in larger limit.
7 Conclusion

In this paper, we have provided the analogy between particle creation in primordial cosmology and scattering problem inside a conduction wire in presence of impurities. Such impurities are characterized by effective potential in the context of quantum mechanical description. On the other hand, in the context of primordial cosmology time dependent mass profile of created particles (couplings) mimics the same role.

Specific time dependence of mass profile actually restricts the structure of the scattering effective potential. To establish the analogy between two theoretical frameworks we have further computed various characteristic features of conduction wire i.e. resistance, conductance (electrical properties), Lyapunov exponent (dynamical property), reflection and transmission coefficients (optical properties), occupation number and energy density (energetics) using the expression for Bogoliubov coefficients for different mass profiles which connects the ingoing and outgoing solution of the mode functions obtained in the context of particle creation process in cosmology.
 We have solved this particle creation problem using the following crucial steps:
 1.
First of all assuming that the interactions are well known we have studied the one to one correspondence between the particle creation problem in early universe cosmology with the scattering problem inside a conduction wire. Here we have additionally neglected the effect of the expansion of our universe and this is perfectly justifiable during the epoch of reheating. For this reason we call it as Reheating Approximation.
 2.
Secondly we have studied the same problem where the particle interactions are not known at all at the level of action. In such a situation, assuming the gravitational background is classical in nature and also assuming the previously mentioned Reheating Approximation we have demonstrated the problem with the help of Random matrix theory.
 3.
Further we have solved the dynamics of the particle creation problem by studying the higher order corrections in the Fokker–Planck equation for previously mentioned random system where the interactions are not easily quantifiable at the level of action. We have constructed the fourth order corrected Fokker–Planck equation from which we have provided the solution of the random probability distribution function. Such distributions are very very useful to study the dynamical systems when particle interactions are not well known. In our analysis we have identified all of these modifications as the quantum correction to the Fokker–Planck equation, the physical implications of which we have studied in detail in this paper.
 1.

In this work, we have shown that the Lyapunov exponent varies inversely with the number of scatterers. Therefore, with an increase in the number of scatterers the Lyapunov exponent also reduces thereby reducing the amount of randomness in the system. This may be a hint to the fact that the Lyapunov exponent has a dependence on the momenta values of the incoming wavefunction of the scalar field. Additionally, it is important to note that the upper bound of Lyapunov exponent is restricted by the constraint \(\lambda \le 2\pi /\beta \) (where \(\beta =1/T\)), which is a generic bound on chaos obtained in the context of quantum field theory. As a consequence, one can find restriction on the upper bound on the reheating temperature for the different quenched mass profiles for which the chaos bound saturates. This is obviously a remarkable result in the present context as it can able to provide the explicit expression for the reheating temperature for a specified momentum scale, which was not predicted earlier in the detailed study of reheating. Most importantly, the bound on quantum chaos in terms of Lyapunov exponent directly restrict the value of reheating temperature without explicitly knowing the details of the particle interactions as appearing in the action. Just the knowledge of time dependence of the quenched mass profiles (in other words the knowledge of effective impurity potential as appearing inside the conduction wire) is sufficient enough to restrict the upper bound of reheating temperature due to quantum chaos.

In this context we have also provided the expression for the two point quantum correlation function, which is known as Spectral Form Factor (SFF) for both in finite and zero temperature. Spectral Form Factor is actually a more strong measure to find chaotic behaviour of a dynamical system compared to Lyapunov exponent. We get saturating behaviour of SFF at late time scale, which indicates that it has an upperbound. We can relate SFF for any potential (Even Polynomial Potential in this case). In the calculation of the Lyapunov Exponent for the specific time dependent mass profiles, we choose three different quenched protocols for mass profiles. Potential functions which can be represented by polynomial potential (Even only in our case) can be used to get the SFFsaturation. In this connection, we have provided a model independent upper and lower bound of SFF, which is treated as the significant bound of quantum chaos (\(1/N\left( 11/\pi \right) \le {\mathbf{SFF}} \le 1/\pi N\)) in the context of particle production event in cosmology. This is obviously a remarkable result which we have explicitly computed in this paper.In [105] this same has been calculated for general polynomial with GUE.

We have also presented the computation of quantum corrected Fokker Planck equation which corresponds to the deltascatterers. From this computation we have derived the corrected statistical distribution of the particle production events in cosmology. The distribution which has been predicted in [42] to be Gaussian doesn’t retain its form when more correction terms are taken into account. This may be treated as a signature of nonGaussian in particle production events during reheating (in cosmology).

In this paper for our study of quantum chaos in the context of cosmology we have used a closed quantum system. As we have mentioned that the present computation has been performed for a massless scalar field which interacts with the heavy fields (which acts like scatterers inside the conduction wire). The entire calculation is being done for the set up when there is only a single massless scalar field that interacts with the scatterer. One may repeat the calculation for a large number of these scalar fields interacting with the scatterers which needs the introduction of the random matrix approach in a more generalized fashion.

The system we have studied in this paper have no interactions with the background as the definition of the background in this setup is itself an illdefined one during reheating. To treat the entire system having being interacted with a background one needs to have a detailed description of the nature of background in the cosmological scenario. Then it will be possible to introduce the other nonlinear and dissipative effects into the system introduced by the background itself. Such a treatment will be studied within the framework of an open quantum system interacting with the defined background setup. One then needs to consider the entire system having being interacted with the background under a weak coupling limit. One such model as has been studied in [106].

We have calculated the Lyapunov exponent and Spectral Form Factor in this paper which is a measure of chaos or nonlinearity into the system. With the system prescribed in this work being treated as an open quantum system one may study the effects of dissipation being introduced into such a system which renders the system to be a stochastic one. With this, one may be able to study the effects of the nonlinearity being introduced into the system which may well be a good study to look for the behaviour of Lyapunov exponent and Spectral Form Factor.
 During the study of quantum correction in the Fokker–Planck equation and the deviation from log normal distribution we have followed a specific approach in which we have considered the following possibilities:
 1.
We have neglected the contribution from the damping term in the Fokker–Planck equation. One can include such effect and study its role in the context of cosmology (specifically during reheating).
 2.
During the computation we have followed a specific approach in which we have also neglected the effect of impurity potential at very high temperature during reheating. This will give rise to a simplest form of the Fokker–Planck equation where only diffusion and drift contributions are appearing explicitly. But if we include the effect of impurity potential in presence of finite temperature then it will surely effect the final solution of the probability distribution function. One can include such additional effects and study its impact during reheating epoch of the early universe.
 3.
Furthermore, during the construction of the Fokker–Planck equation from the basic principles we have followed a special approach in which the effect of diffusion and drift is appearing in a very simplified manner. However, in the study of statistical field theory Itô and Stratonovitch or more generalized prescriptions are used commonly to construct the Fokker–Planck equation. Here it is important to note that in each case it will give rise to different mathematical structure of Fokker–Planck equations. In the present context of discussion, one can follow such well known prescriptions to see its physical outcomes to solve the probability distribution function for the particle production and compare the results to check the appropriateness of these approaches during reheating.
 1.
Footnotes
 1.
In the context of cosmology conformal time dependent effective mass profile exactly mimics the role of impurity potential in electrical conduction wire. Due to such one to one correspondence the time evolution equation (i.e. Klien Gordon equation) of the Fourier modes corresponding to the quantum fluctuation in the context of primordial cosmology can be described in terms of the Schrodinger equation in electrical conduction wire with specific impurity potential. We have investigated this possibility in detail in this paper. Additionally, it is important to note that such time dependent effective mass profiles are also important to study the role of quantum critical quench and eigen state thermalization [] during the reheating epoch of universe.
 2.
For an example, one can generalize the same prescription in three space dimensions.
 3.
 4.
Important notes:
Here we note the following points which are very useful to study the consequences from EFT set up:. 1.
In this context, one can construct an EFT by utilizing the underlying symmetries appearing in the field theoretic set up. In such a generalized description where EFT is constructed by following the top down approach, we really don’t care about the exact UV completion of the parent theory i.e. detailed quantum field theory origin at high energy scale of such effective constructions are not important in this case. See Refs. [4, 24] for more technical details.
 2.
In a more generalized prescription of EFT one can construct the set up which requires to correctly account for all relevant self interactions of adiabatic modes around and after the cosmological horizon crossing. Specifically the adiabatic mode contains all types of EFT relevant operators, including transient reductions in the effective sound speed \(c_S\) each time the background field undertakes nongeodesic motion in background target space. In an EFT framework with single field setting, where heavy directions are such that the mass of the field under consideration is heavy compared to the Hubble scale i.e. \(m\gg H\), one gets transient drops in the effective sound speed \(c_S\) during slow roll if the potential is such that the field traverses a bend even if the parent theory consists of canonically normalized scalar fields. So for general consideration one can allow many more possibilities without following any restriction to time dependent mass profile. However, these three specific types of time dependent mass profiles are very popular in the context of the study of quantum critical quench in a analytical fashion. We have considered these profiles particularly as our future objective is to apply the idea of quantum quench in the context of De Sitter space to quantify randomness during reheating phase. It is important to note that, using a simple field redefinition at the level of quantum fluctuation to the Mukhanov–Sasaki variable which results in a time dependent mass for the rescaled variable appearing with additional contributions of the mathematical form, \(\dot{c_S}/c_S\sim SH\). Here \(S=\dot{c_S}/Hc_S\) is the associated slow roll parameter with the effective time dependent sound speed \(c_S\) and H is the Hubble parameter and it is associated with the changes in the radius of curvature of the inflaton trajectory. In this case the effective sound speed is given by, \(c^{2}_S\equiv 1+4{\dot{\phi }}^2/\kappa ^2 M^2\), where \(\kappa \) is the radius of curvature of the background inflaton (\(\phi \)) trajectory and M is the effective cutoff scale of the EFT at high energy (UV) regime. Equivalently, it refers to the degree to which effective sound speed \(c_S\) is reduced, which actually quantify the distance from the adiabatic minimum of the potential in the background inflaton trajectory is forced by its evolution. Each of these possibilities has different applications in the low energy limiting region of EFT. When the effective sound speed \(c_S\ll 1\) and \({\dot{c}}_S\sim 0\) is fixed over few efolds of expansion then it is extremely difficult to maintain a meaningful derivative expansion without considering other types of special symmetries appearing in the set up. However, as mentioned earlier, within certain limits one can consider an adiabatic region where the effective sound speed \(c_S\ll 1\) and \({\dot{c}}_S\sim c_S H\) and \(S=\dot{c_S}/Hc_S\sim 1\) is fixed over a very small efold of expansion and this in turn generate all possible consistent transient strong coupling parameters without violating perturbative uniterity and these terms are explicitly appearing in the derivative expansion in the EFT. Consequently, the nature of these two types of features in the effective sound speed \(c_S\) give rise to distinctive contributions to the physical observables studied in the EFT set up. The positive detection of these physical observables in different experiments allow to extract the underlying nontrivial physics from the EFT set up. In the technical ground the adiabatic mode is identified with the Goldstone boson, which is appearing due to spontaneously broken time translational symmetry prior to the path integration of the heavy fields. In this context, the invariance of the parent theory completely fixes the entire nonperturbative structure of all possible Wilsonian EFT operators and the associated coupling parameters can be expressed entirely by the effective sound speed \(c_S\) of adiabatic perturbations, where the adiabaticity conditions \(c_S\ll 1\) and \({\dot{c}}_S\sim c_S H\) are respected. In principle, \(c_S\) can be computed in terms of the parameters of the parent theory. Thus the additional contributions appearing in the adiabatic limit \(c_S\ll 1\) and \({\dot{c}}_S\sim c_S H\) directly justifies the validity of our treatment in this paper. For further technical details of this EFT set up see Refs. [71, 72, 73].
 1.
 5.
Here it is important to note that, for inflation this scalar field is actually massless and in the effective field theory description one can construct the time dependent effective mass profile. On the other hand, in the context of reheating the scalar field is massive and in the effective field theory description one can construct time dependent effective mass in terms of the original mass of the reheating field and other degrees of freedom which are integrated out from the original theory.
 6.
Here we have assumed that the effective sound speed parameter, \(c_S=1\), which indirectly implies the fact that for background time evolution we are considering a single scalar field with canonical kinetic term minimally coupled to the gravity. Effective mass of the scalar field is \(m(\tau )\), which has time dependent profile. However, one can generalize this prescription for any general noncanonical single field (i.e.\(P(X,\phi )\) theory) theoretic framework where the effective sound speed parameter \(c_S\ne 1\).
 7.Following this discussion, one can generalize this statement for \(N_s\) number of scatterers as:
 8.
Equivalently, here one can say that it exponentially decays with the number of scatterers.
 9.
 10.In de Sitter and quasi de Sitter space one can compute the relation between the conformal time (\(\tau \)) and the physical time (t) as given by the following expressions:
 11.
Important note: In the present context, the analysis is perfectly valid for the highly localized particle production events after neglecting the cosmological expansion during reheating approximation. But this approximation fail for the events that are sufficiently spaced out. If we don’t neglect the cosmological expansion in this computation then the conformal time dependent mass term of the form \(2\tau ^2\) is restored from the background cosmological background. This actually implies that the scattering problem is being performed on a conformal time dependent potential of the form \(1/r^2\) (inverse square), which makes the analytic computations of the Bogoliubov coefficients and all the other derived physical quantities to quantify quantum randomness from the present set up extremely difficult. Here it is important to note that, for long wavelength cosmological observables particle production appears more than an efold apart and consequently the corrections appearing due to the cosmological expansion seem certainly relevant in the computation as the incoming and outgoing wave functions depart from plane waves. Although for localized particle production events, the reheating approximation considered in this paper perfectly holds good. In the present context this approximation breaks when we consider the particle production events for a sustained period of time or may be separated by times approaching an efold expansion.
 12.Here it is important to note that, since the scale factor \(a(\tau )\) is approximately a constant during reheating (reheating approximation), then conformal time (\(\tau \)) and the physical time (t) is related through the following coordinate rescaling transformation:
 13.
For this specific discussion only we keep the Planck’s constant \(\hbar \) and the Boltzmann constant \(k_B\) in our computation. But for the rest of the paper we fix \(\hbar =1\) and \(k_B=1\) for which the parameter \(\beta \) can be written as, \(\beta =1/T\). In such a situation chaos bound is given by, \(\lambda <2\pi /\beta \).
 14.In the context of weakly coupled gauge theory one can introduce ’t Hooft coupling \(\lambda _T\) which is independent of N and in such a theory the Lyapunov exponent is given by the following expression:
 15.Considering the bulk contribution weakly coupled with string theory with large radius of curvature one can show that the perturbative stringy correction to the Einstein gravity computation of the scrambling can give rise to the following first order corrected expression for the Lyapunov exponent []: where \(L_s\) is the stringy length scale and \(\mu ^2\) is a specific constant which is appearing in the shock wave equation propagating along the horizon.
 16.
Henceforth we set \(\hbar =1\) for which the bound is translated to \(\lambda \le \frac{2\pi }{\beta }\), which we will use for the further application purposes.
 17.
Thermal averaging is a very important concept in the context of AdS/CFT correspondence as the dual description of the quantum field theory of black holes can be treated as a thermal bath which have Hawking temperature.
 18.Classical result: Here one can perform the exact classical computation of OTOC to check whether the quantum and classical descriptions give the same result or not. In the case of billiards, the Poisson bracket is given by, \(\left\{ q(\tau ),p(0)\right\} _\mathbf{PB}\sim \exp [\lambda \tau ].\) One can explicitly show that in this context the Lyapunov exponent can be expressed as, \( \lambda \sim \frac{v}{\sqrt{A}}=\frac{p(0)}{\sqrt{A}},\) where \(A=\pi R^2+4aR\) is the area of the stadium and v is the velocity of the particle. Then the classical OTOC can be expressed as: where \(Z_{\mathrm{cl}}\) is the classical partition function defined as: Further taking \(A=1\) for simplicity we get: Further taking the limit \(t\gg \sqrt{\beta }\) we get the following simplified answer for classical OTOC for billiards: This result implies that in classical OTOC and in semiclassical (or quantum) OTOC the time dependence is completely different. In the case of classical OTOC it shows faster growth with respect to the result obtained for quantum OTOC.
 19.
In the present context large OTOC (\({{\mathcal {C}}}(\tau )\) implies that the quantum operator for chaos \(W(\tau )\) completely destroy the effect of the initial factor \(\exp [iH\tau ]\) and the final factor \(\exp [iH\tau ]\) to cancel their contribution in the definition of the operator \(W(\tau )\).
 20.
In the context of integrable and nonintegrable quantum mechanical system nearest neighbour distribution (NDD) is described by Poisson and Wigner functional.
 21.
This formalism is very useful when we can’t able specify the particle interaction in the effective action. More precisely, in this situation when we really don’t have any information about the particle interaction one can’t able to define the action in terms of the usual language. Additionally it is important to note that, in our computation we consider that gravitational background is classical and non dynamical. However it will not explicitly appearing in the action for the distribution of eigen values of random matrices. Also during reheating since one can neglect the contribution from the expansion of our universe, then considering only the representative action for random distribution is sufficient enough for our discussion when we don’t have any knowledge about the particle interactions at the level of action. In such a situation gravitational background is treated to be not evolving with time during reheating.
 22.Additionally, it is important to note that the density function satisfy the following normalization condition:
 23.Here to define the new time scale \({{\mathcal {T}}}\) we assume that the reheating approximation is perfectly valid. This implies that we can really neglect the expansion of the universe. This further pointing towards the fact the conformal are related by the following expression: where during reheating we have assumed that the conformal and physical time are almost same and the proportionality constant is the inverse of the scale factor \(a^{1}\), which is independent of both the time scales discussed in this paper.
 24.In case of It\({\hat{o}}\) prescription one can recast the integral \({{\mathcal {I}}}(n;\tau \epsilon )\) in to the following form:
 25.
Here it is important to note that, in one dimension J = constant directly implies \(J=0.\) But in the case of higher dimension one can get stationary outofequilibrium currents.
 26.
In the present context, N actually mimics the role of principle quantum number. Also \(m_1\) is another quantum number which is derived from N using Eq. (6.162).
 27.
Including the contributions from second order we will see that the Fokker–Planck equation can not solvable analytically.
 28.
Including the contributions from third order we will see that the Fokker–Planck equation can not solvable analytically.
 29.
Including the contributions from fourth order we will see that the Fokker–Planck equation can not solvable analytically.
 30.
Here it is important to note that as far as quantum mechanical computation is concerned, it produces same result for the one point function and first moment of the occupation number and both of them is equal to the expectation or average value of the occupation number in this context.
 31.It is important to note that, as far as quantum mechanical computation is concerned, it produces not exactly same result for the one point function and first moment of the occupation number. For two point function we actually get the following result: where \(A(\tau )\) is the amplitude of the two point function as given by the following expression: This implies that the amplitude part is exactly matches with the second moment of the occupation number in this context.
 32.It is important to note that, as far as quantum mechanical computation is concerned, it produces not exactly same result for the three point function and third moment of the occupation number. For three point function we actually get the following result: where \( B(\tau ,\tau ^{\prime },\tau ^{\prime \prime })\) is the amplitude of the three point function. If we fix \(\tau =\tau ^{\prime }=\tau ^{\prime \prime }\) (equal time) then we get the following expression: This implies that the equal time amplitude part is exactly matches with the third moment of the occupation number in this context.
 33.It is important to note that, as far as quantum mechanical computation is concerned, it produces not exactly same result for the three point function and third moment of the occupation number. For three point function we actually get the following result: where \(C(\tau ,\tau ^{\prime },\tau ^{\prime \prime },\tau ^{\prime \prime \prime })\) is the amplitude of the four point function. If we fix \(\tau =\tau ^{\prime }=\tau ^{\prime \prime }=\tau ^{\prime \prime \prime }\) (equal time) then we get the following expression: This implies that the equal time amplitude part is exactly matches with the third moment of the occupation number in this context.
Notes
Acknowledgements
SC would like to thank Quantum Gravity and Unified Theory and Theoretical Cosmology Group, Max Planck Institute for Gravitational Physics, Albert Einstein Institute for providing the PostDoctoral Research Fellowship. SC would like to thank IUCAA, Pune, India where the problem was formulated and part of the work has been done during post doctoral tenure. SC take this opportunity to thank sincerely to JeanLuc Lehners, Shiraz Minwalla, Sudhakar Panda and Varun Sahni for their constant support and inspiration. SC thank the organisers of Summer School on Cosmology 2018, ICTP, Trieste, 15 th Marcel Grossman Meeting, Rome, The European Einstein Toolkit meeting 2018, Centra, Instituto Superior Tecnico, Lisbon, The Universe as a Quantum Lab, APC, Paris, LMU Quantum Gravity School, Munich, Nordic String Meeting 2019, AEI, Summer meeting on String Cosmology, DESY, Zeuthen, Kavli Asian WInter School on Strings, Particles and Cosmology 2018 for providing the local hospitality during the work. SC also thank DTP, TIFR, Mumbai, ICTS, TIFR, Bengaluru, IOP, Bhubaneswar, CMI, Chennai, SINP, Kolkata and IACS, Kolkata for providing the academic visit during the work. AM, PC and SB are thankful to IUCAA, Pune for the visit during the work for winter project. Last but not the least, We would all like to acknowledge our debt to the people of India for their generous and steady support for research in natural sciences, especially for theoretical high energy physics, string theory and cosmology.
Supplementary material
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