Stochastic Cosmology and the Vacuum Energy Parameter

The Vacuum Energy Parameter (VEP) of standard cosmology ( L W ) denotes the fraction of the critical density attributed to the accelerated expansion of the Universe. Astrophysical evidence sets the numerical range of VEP at exp 0.692 0.012 L W ± = , yet the root cause of these values is currently unknown. Drawing from the stochastic interpretation of early-Universe cosmology, we develop here a derivation of the VEP based on classical diffusion theory and the Langevin equation. Predictions are shown to be in reasonable agreement with observations.


Introduction
It is known that the Friedmann-Robertson-Walker (FRW) model is based on the cosmological principle, according to which matter distribution in the large-scale structure of the Universe is homogeneous and isotropic [1]. Besides homogeneity and isotropy, the FRW model implicitly assumes that cosmological processes are governed by deterministic laws excluding -by default -concepts such as randomness and probability distributions.
Over the years, several studies have advocated extending the basis of FRW cosmology, arguing that a stochastic description may be better equipped to model the early evolution of the Universe [3-6]. We fully subscribe to this viewpoint and go even further in suggesting that stochastic cosmology is, in fact, an integral part of complex dynamics. The goal of complex dynamics is to explore the evolution of large interacting ensembles that are out-of-equilibrium and capable of sustaining a wide spectrum of collective behaviors. As the dynamics of complex systems includes non-deterministic processes, it provides a natural foundation for the ideas of stochastic cosmology.
Our approach starts from the premise that the cosmic fluid density of the FRW model may be mapped to a classical scalar field. Following the philosophy of stochastic cosmology, we show that behavior of this field matches the dynamics of a damped harmonic oscillator in contact with a reservoir of random fluctuations. Universe expansion supplies the damping mechanism and is embodied in the Hubble parameter.
Setting the gravitational self-interaction scale as the upper bound energy of the scalar field leads to predictions in reasonable agreement with observations. The paper is partitioned in the following way: next section delves into the stochastic interpretation of early Universe cosmology while section 3 defines the set of assumptions and conventions underlying our work. The construction of the Langevin equation and the derivation of the VEP form the object of sections 4 and 5. Concluding remarks and follow up developments are discussed in the last section.

Stochastic interpretation of early Universe cosmology
The mass dimension of fluid density in natural units is given by (2) whose mass dimension in 3+1 spacetime is (see [2] for example) It is important to draw an upfront distinction between (2) and the hypothetical fields of cosmic inflation, such as "inflaton", "dilaton" or "quintessence". These fields supplement the standard framework of cosmology but do not share common roots with the fluid density concept of the FRW model. Despite their difference in physical nature, both (1) and the postulated fields of inflationary cosmology satisfy the scalar field equation [1] ( ) 0 where ( ) V j stands for a generic potential function. To understand why this is the case, consider the Lagrangian density of any scalar field in curved spacetime [1] 1 The corresponding field equation is Space derivatives can be safely ignored in comparison with the time derivative, under the plausible assumption that the field is spatially homogeneous. If, in addition, the geometry is close to being spatially flat, the line element may pe cast in a form compatible with the FRW cosmology, i.e.
It is not unreasonable to think that (4) may be generalized to a scenario including random fluctuations present in the evolution of the early Universe. The most straightforward path to this scenario consists of adding a "thermal-like noise" term in the right-hand side of (4). This leads to For simplicity, we take the statistics of the noise term to be delta correlated as in where D denotes the strength of noise correlations. In line with classical diffusion theory in a harmonic potential, we posit that ( ) V j is a quadratic function Next section defines the main set of assumptions and conventions used below in the analysis of (8).

Assumptions and conventions
A1) In section 5, both scalar field and time parameters are cast in dimensionless form and Carrying out this operation means normalizing x with the appropriate powers of the mass scale m . The rationale for the most natural choice of m in the context of our paper is also given in section 5.
A2) To fix the initial conditions, we posit that the amplitude of the scalar field starts from an arbitrarily chosen lower value and develops larger values at later times. Hence, we A3

Vacuum state and the stationary regime of the Langevin equation
The evolution of field expectation value in the long-time limit of classical diffusion theory and leads to the exponential decay equation Here, so that Direct substitution of (16) in (8) -(11) and on account of A2) gives It is instructive to evaluate next the analogy between (17) The limit In closing this section, we mention that there are two key relationships linking the coefficient D of (9), the damping coefficient where B k is the Boltzmann constant and T the temperature.
b) The second relationship stems from the equipartition theorem of Thermodynamics and takes the form

Derivation of the Vacuum Energy Parameter (VEP)
To further proceed with our derivation, we recall that the Einstein-Hilbert action in the absence of matter or radiation takes the form [1,10] where R represents the scalar curvature, N G is Newton's constant and It is also reasonable to expect that, in the asymptotic vacuum state defined by (33), the energy carried by the scalar field j is upper limited by the gravitational scale G M . On account of these considerations and by (28) ). Recall that, by assumption A3), the instantaneous and expectation values of the vacuum density r L are set to be identical.

Conclusions and outlook
Our work has developed from the premise that the cosmic fluid density of the FRW model may be mapped to a classical scalar field. Following in the footsteps of stochastic cosmology, we have found that the behavior of this field corresponds to the dynamics of a damped harmonic oscillator in contact with a reservoir of random fluctuations. In this picture, Universe expansion supplies the damping mechanism as embodied in the Hubble parameter. Setting the gravitational self-interaction scale as the upper bound energy of the scalar field yields a VEP prediction in reasonable agreement with observations.
In line with [11][12], a sequel to this work will attempt bridging the gap between stochastic cosmology and self-organized criticality (SOC). The plan is to show that SOC sheds new light on the inner workings of the early-Universe cosmology [13].

APPENDIX
The goal of the Appendix section is to recover (13) from the classical diffusion theory applied to Brownian motion [8]. The evolution of the field expectation value is described