#### Abstract

We derive a stochastic wave equation for an inflaton in an environment of an infinite number of fields. We study solutions of the linearized stochastic evolution equation in an expanding universe. The Fokker-Planck equation for the inflaton probability distribution is derived. The relative entropy (free energy) of the stochastic wave is defined. The second law of thermodynamics for the diffusive system is obtained. Gaussian probability distributions are studied in detail.

#### 1. Introduction

The CDM model became the standard cosmological model since the discovery of the universe acceleration [1, 2]. It describes very well the large-scale structure of the universe. The formation of an early universe is explained by the inflationary models involving scalar fields (inflatons) [3, 4]. Such models raise the questions concerning the dynamics from their early stages till the present day. The origin of the cosmological constant as a manifestation of “dark energy” could also be explored. In models of the dark sector we hope to explain the “coincidence problem”: why the densities of the dark matter and the dark energy are of the same order today as well as “the cosmological constant problem” [5]; why the cosmological constant is so small. We assume that the dark matter and dark energy consist of some unknown particles and fields. They interact in an unknown way with baryons and the inflaton. The result of the interaction could be seen in a dissipative and diffusive behaviour of the observed luminous matter. The diffusion approximation does not depend on the details of the interaction but only on its strength and “short memory” (Markovian approximation). In this paper we follow an approach appearing in many papers (see [6–10] and references quoted there) describing the dark matter and the fields responsible for inflation (inflatons) by scalar fields. In the CDM model the universe originates from the quantum Big Bang. The quantum fluctuations expand forming the observed galactic structure. The transition to classicality requires a decoherence. The decoherence can be obtained through an interaction with an environment. The environment may consist of any unobservable degrees of freedom. In [11, 12] these unobservable variables are the high energy modes of the fields present in the initial theory. We assume that the environment consists of an infinite set of scalar fields interacting with the inflaton. The model is built in close analogy to the well-known infinite oscillator model [13–16] of Brownian motion. As the model involves an infinite set of unobservable degrees of freedom the statistical description is unavoidable. We have an environment of an infinite set of scalar fields which have an arbitrary initial distributions. We begin their evolution from an equilibrium thermal state. In such a case we obtain a random physical system driven by thermal fluctuations of the environment. We could also take into account quantum fluctuations extracting them as high momentum modes of the inflaton as is done in the Starobinsky stochastic inflation [17]. In such a framework a description of a quantum random evolution is reduced to a classical stochastic process. We can apply the thermodynamic formalism to the study of evolution of stochastic systems. In such a framework we can calculate the probability of a transition from one state to another. In particular, a vacuum decay can be treated as a stochastic process leading to a production of radiation [18].

The plan of this paper is the following. In Section 2 we derive the stochastic wave equation for an inflaton interacting with an infinite set of scalar fields in a homogeneous expanding metric. In Section 3 we briefly discuss a generalization to inhomogeneous perturbations of the metric satisfying Einstein-Klein-Gordon equations. In Section 4 we approximate the nonlinear system by a linear inhomogeneous stochastic wave equation with a space-time-dependent mass. The Fokker-Planck equation for the probability distribution of the inflaton is derived. A solution of the Fokker-Planck equation in the form of a Gibbs state with a time-dependent temperature is obtained. In Section 5 a general linear system is discussed. In Section 6 we obtain partial differential equations for the correlation functions of this system. In Section 7 Gaussian solutions of the Fokker-Planck equation and their relation to the equations for correlation functions are studied. In Section 8 we discuss thermodynamics of time-dependent (nonequilibrium) diffusive systems based on the notion of the relative entropy (free energy). In the Appendix we treat a simple system of stochastic oscillators in order to show that the formalism works well for this system.

#### 2. Scalar Fields Interacting Linearly with an Environment

The CMB observations show that the universe was once in an equilibrium state. The Hamiltonian dynamics of scalar fields usually discussed in the model of inflation do not equilibrate. We can achieve an equilibration if the scalar field interacts with an environment. We suggest a field theoretic model which is an extension of the well-known oscillator model discussed in [13–16]. We consider the LagrangianEquations of motion for the scalar fields readwhere is the metric tensor and .

We consider the flat expanding metric:In the metric (4), (3) readsWe may chooseWe write

ThenwhereWe consider large so that for a large time we may neglect term. Moreover, we assume that and that is approximately constant (this is exactly so for the de Sitter space and approximate for power-law expansion when the dependent term decays as ). Then, the solution of (8) isInserting the solution of (3) in (2) we obtain an equation of the following form:whereand the noise depends linearly on the initial conditions :If the correlation function of the noise is to be stationary (depend on time difference) then we needThen, (assuming that , true in the classical Gibbs state) we getWe assume a certain probability distribution for initial values. Relation (14) is satisfied for classical as well as quantum Gibbs distribution with the Hamiltonian . In the classical field theory in the Gibbs state the covariance of the fields in (14) is . If then this covariance is approximated by [19] . We chooseUnder assumption (16) and a continuous spectrum of in (11) we shall haveHere, comes from ; the term is (an infinite) mass renormalization which appears already in the Caldeira-Leggett model [15]; it could be included in . The last term can be neglected when tends to . We shall omit these terms in further discussion.

In an expanding metric, (11) takes the following form:

whereHere, comes from the expectation value of the initial values and with the neglect of . If we do not neglect in (8) then the form of (19) would be much more complicated (we would obtain a nonlocal equation). We make the approximation (19) which preserves the Markov property and provides stochastic fields which are regular functions of ( it can be considered as a cutoff ignoring high momenta components of ). Equation (18) has been derived earlier in [19]. It is applied in the model of warm inflation [20].

#### 3. Stochastic Equations in a Perturbed Inhomogeneous Metric

We did not write yet equations for the metric which result from Lagrangian (1). The power spectrum of inflaton perturbations depends on the metric [3, 4, 21–23]. In general, the equations for the metric are difficult to solve. They are solved in perturbation theory. We consider only scalar perturbations of the homogenous metric using the scalar fields . Then, the metric is expressed in the following form [3, 23]:Inserting the metric in (2)-(3) we obtainandwhereLetwithThen, in the approximation neglecting higher powers of in whereWe again assume that . Then, in the derivation of (18) for the only change comes from the differentiation of inside the integral (11) and the factor multiplying the fields. So,and

Hence, our final equation is (for a general theory of a stochastic wave equation on a Riemannian manifold see [24] and references cited there)

#### 4. A Simplified System of a Decaying Inflaton

The metric can be expressed by from Einstein equations resulting from the Lagrangian (1). We expand inflaton equation with a potential around the classical solution of Klein-Gordon-Einstein equation. The linearized version of the equation for fluctuations takes the form of the Klein-Gordon equation with a space-time-dependent mass [3, 4, 23, 25, 26]:where we write and treat (31) as Ito stochastic differential equation [27]. The function depends on the potential in (18) and on the choice of coordinates (the choice of gauge [28]). We do not discuss in this paper. We consider in this section the simplified version of (31) without terms:

We define the energy density:Then, from (32) applying the stochastic calculus [27, 29] and (32), we obtainWe may first integrate this equation and use for the Ito integral. Differentiating the expectation value over we obtainEquation (35) describes the inflaton density with and a cosmological term varying with the speed . The term violates the energy conservation of the inflaton. It describes a decay of the inflaton into the fields. If we couple the fields to radiation then if fields are invisible the observable effect will be detected as a production of radiation from the decay of the inflaton [18, 30].

For the stochastic system (32) the Fokker-Planck equation readsThen, the Gaussian solution iswhere . It can be checked thatandThis normalization factor is infinite (needs renormalization) but the value of does not appear in the expectation values .

has the meaning of the inverse temperature. The dependence (38) of the temperature of the diffusing system on the scale factor has been derived (for and arbitrary ) in [31–33] for any system with and the correction (35) to the cosmological term (in eq. (38) ) (for time-dependent cosmological term see [5, 34, 35]).

#### 5. The Linearized Wave Equation

We can rewrite (4)-(5) in a way that they do not contain first-order time derivatives of fields. LetThen, the linearized version of the inflaton equation expanded around the classical solution (with an account of metric perturbations ) readswhereFor the stochastic system (42) the Fokker-Planck equation readswhereand depends on the classical solution in the potential [23, 25].

We may write the noise in the Fourier momentum space:Then, (42) is rewritten as an ordinary (instead of partial) differential equation.

#### 6. A Differential Equation for Correlations

Let us consider (42) expressed in the following form:whereLet us denoteUsing the stochastic calculus [27, 29] and taking the expectation value we get a system of differential equations for the correlation functions:If the system is translation invariant then we can Fourier transform these equations obtaining a system of ordinary differential equations for Fourier transforms:where is defined in (46).

#### 7. Gaussian Solutions of the Fokker-Planck Equation

We look for a solution of the Fokker-Planck equation (44) in the following form:or in the momentum spaceIn the configuration space is an operator and in the momentum space a function of . is determined by normalization or directly from the Fokker-Planck equation:The equations for readwhereWe skip the equations for and .

It is useful to introduce instead of the variables (defined by Fourier transforms of ):We can invert these relations:

can be expressed asFrom (72) it can be seen that the probability distribution is diagonal in the variables and . So, we obtain the expectation value .

Assume that we calculate the expectation values at time : can be expressed by as

If we know then we can expressNote that means .

The relations (67)-(81) allow relating the solutions of the stochastic equation (42) with the solutions of the differential equations (62)-(64) and the solution of the Fokker-Planck equation (44). In fact, the solution can be expressed by the Fokker-Planck transition function (which is defined by the solution of the stochastic equation (42) [27]) as follows:The expectation values of the solution of the stochastic equation with the initial condition isHence,Note that the initial value in (84) according to (59) is determined by the initial values of . A possible choice for the initial value is the thermal Gibbs distribution:which corresponds to the initial condition , and . We could also consider the initial probability distribution:describing the field concentrated at . Then, in (59) . In such a case, the probability distribution (59) describes the probability of the transition from to (see [36] for such calculations in quantum mechanics).

#### 8. The Relative Entropy

Assume we have a functional equation of the form (like (44)):Let us assume that we have two solutions and of this equation. Define the relative entropywhereFrom the definition of it follows that [37]Calculation of the time derivative of givesWe choose (then because is the normalization factor). Then, we define the entropy (the entropy of inflaton and gravitational perturbations has been discussed earlier in [38, 39]):Using (87) we calculate the time derivative:The second term is zero, whereas the first term is equal toIn (95) the first term is positive whereas the second term depends on the dynamics (it is vanishing for Hamiltonian dynamics). Using formula (59) for we obtainFormula (96) has a thermodynamic meaning relating the sum of free energy and entropy to the internal energy expressed by the rhs of (96). At the initial time (with the initial conditions discussed at the end of Section 7) the rhs of (96) is the mean value of the energyIn the static universe we would have an equilibrium distribution as . In such a case the thermodynamic relation (95) would describe the standard version of the second law of thermodynamics of diffusing systems. with in (92) would show the approach to equilibrium. In the expanding universe the relation (96) can serve for a comparison of various probability measures starting from different initial conditions.

#### 9. Summary

The main source of observational data [1, 2] comes from measurements of the cosmic microwave background (CMB) and observations of galaxies evolution (including galaxies distribution). The CMB spectrum and its fluctuations are the test ground for models involving quantum and thermal fluctuations. A simplified description of an interaction of a relativistic system with an environment leads to a stochastic wave equation for an inflaton generating the expansion (inflation) of the universe. We considered a linearization of the wave equation. We discussed the Fokker-Planck equation for the probability distribution of the inflaton. Gaussian solutions of the Fokker-Planck equation for linearized systems can be treated as Gibbs states with a time-dependent temperature. The model leads to a formula for density and temperature evolution. We have derived the density evolution law in (35) and the temperature evolution in (38) in a simplified model. In order to obtain the results in the complete model we would have to solve (numerically) equations of Section 7. The comparison of density evolution (38) with observations is discussed in [32] and in similar models with the decaying vacuum ( see [34, 35] and references cited there). The model allows calculating (and compare with observations) the power spectrum resulting from thermal fluctuations which may go beyond the approximations applied in the warm inflation of [20]. We have introduced a thermodynamic description of the expanding diffusive systems in terms of the relative entropy (free energy) and entropy. The state of a stochastic system can be identified with its probability distribution. The relative entropy allows comparing the evolution of the probability distributions with different initial conditions. In this sense relative entropy can be treated as a quantitative measure of a decay of one state into another state (as an alternative to a quantum description of vacuum decay in cosmology [40, 41]).

#### Appendix

#### A. Statistical Physics of a Static Finite Dimensional Model

A finite dimensional analog of the wave equation is ():The Fokker-Planck equation readsThe stationary solution iswhere is the energy of the oscillator. It describes a Gibbs state with the temperatureWe look for a solution of (A.3) in the following form:ThenLet us writeThen, the stochastic equation for readswhereThe solution of (A.12) isWe can easily calculate the correlation functions of and in two ways: either from the stochastic equations or using the probability distribution resulting from the solution of the differential equations (A.8)-(A.10).

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The author declares that there are no conflicts of interest.

#### Acknowledgments

Interesting discussions with Zdzislaw Brzezniak on stochastic wave equations during my stay at York University are gratefully acknowledged