Noise-Induced Origin of the Fundamental Scalar Field

The physical nature of the fundamental scalar field generation and hence the origination of the Universe is a matter of the discussions for many years. We propose to use the statistical approach to the description of the steady states of the quasi stationary systems with the elements of the quantum field theory methods as a basis to explain the appearance of the cosmological scalar field. Particularly, we apply two fundamental principles, i.e., the H-theorem and least-energy principle to show principal possibility of the scalar field origination. Along with the basic statement that in the presence of the fundamental scalar field, the energy of the vacuum ground state is lower than the ground state energy of the vacuum with no scalar field (primary vacuum), and with regard to the nonlinear interaction of fluctuating physical fields with the scalar field, these principles are employed to reveal probable phase transitions that may be associated with origin and further evolution of the Universe. Thus, we propose the possible physical justification of the spontaneous cosmological scalar field generation.


Introduction
Modern notions (rather hypotheses) on the reason for the formation of the Universe imply an instability of some (hypothetical) scalar fields associated with the quantum nature of the matter [1]. The reasons and physical mechanism of the appearance of this field, and hence of the origination of the Universe, remain for many years a question open for a discussion. We propose one more approach similar to that proposed in [2] [3] [4]-to describe the origination and evolution of the Universe in terms of the first principles of statistical mechanics and quantum How to cite this paper: Lev, B.I. and Zagorodny, A.G. (2020) Noise-Induced Ori-theory. Our assumptions are given in what follows.
In the case of spontaneous generation of the scalar field in vacuum, the ground-state energy of the "new" vacuum (i.e. the initial vacuum plus the scalar field) for the fields of other nature should be lower than the ground state energy of the "initial" vacuum. Moreover, the self-consistent interaction of the scalar field with fluctuations of any other field provides energy conservation for the new state of the system. Calculations of the partition function for this system reveal a probability of a phase transition from the state with zero scalar field to the state with finite spontaneously generated scalar field.

Statistical Distribution in Energy Space
According to Gibbs [5], we can always pass from the description in terms of phase variables to the description in terms of energy. Hence, we may treat the entropy as a function of energy and employ the quasi-equilibrium Gibbs distribution to calculate the partition function. We can begin with the statistical description of the Universe based on the Gibbs distribution in the energy representation [5]. The canonical Gibbs distribution in the phase space is given by is an element of the phase space, kT Θ = , T is the temperature, and F is the free energy that can be found from the normalization condition ( ) The phase space is known [5] to be determined by the energy of the system and by external parameters. We introduce the quantity d ln dE Then we can pass to the distribution in the energy space The normalization condition yields In order to select the states with dominant contributions in the partition function, we employ the condition for the temperature given by We assume that the relation between the changes of the value of the phase space from the energy E is known. In terms of this definition and within the context of fundamental principles of statistical mechanics [6] that reproduces the entropy of the system bearing in mind that the temperature describes dependence of entropy only on energy but not on the other thermodynamic functions. It also follows that integration over energy in the continual sense yields an expression for the partition function. It is obvious that the extreme contribution in the partition function is associated with the states for and that for any deviations from the latter condition the contribution in the partition function is negligibly small similarly to the contribution of quantum corrections to the classical trajectories [7] [8].
The Universe is non-equilibrium from origination, so in order to describe its evolution we introduce an additional intrinsic parameter "time". We assume that both the statistical distribution and the evolution of the Universe can be described in terms of the distribution function that depends only on energy. An example of how this idea is applied to describe the properties of the statistical distribution is given in [7] [8]. It seems quite natural to suggest the evolution of the system in the energy space to be analogous to the Brownian motion in such a space. This raises the question which system can serve for the Universe as a thermostat. It is reasonable to suggest that such thermostat is the vacuum with fluctuations of all physical fields that interact with the fundamental scalar field and thus influence even the ground state of the vacuum. On the other hand, this suggestion opens the possibility to describe the Universe evolution in the energy space by the appropriate distribution function governed by the Fokker-Planck equations with nonlinear energy dependence of the diffusion and dissipation coefficients associated with relevant nonlinear Langevine equations [7] [8]. Just this assumption shows the way to describe the evolution of the Universe both before and after origin. Now let us apply the above speculations to the description of the Universe. First, we suggest that the vacuum ground state possesses energy. We also assume that fluctuations of all fields existing in vacuum can occur and thus we can write the equation of state for the vacuum. The thermodynamic relations yield the pressure given by d d

Origin of Classical Fundamental Scalar Field
The above consideration and relations are well known. Now we employ them to propose one more possible explanation of the origin of fundamental scalar field.
We begin with the assumption that the phase transition from the "initial" vacuum to the new vacuum state is accompanied by the appearance of new scalar field. This means that the presence of the scalar field makes the "new" vacuum different from the "primary" vacuum for any field that may exist. The scalar field decreases the energy of the "new" vacuum with respect to the energy of the "primary" vacuum. Hence, the ground-state energy of the "new" vacuum is given by Here the second term is the scalar field energy; the coefficient 2 0 µ describes the coupling of the new field and the "primary" vacuum, i.e., the self-consistent interaction of the new field with the fluctuations of all other fields that can exist in the "primary" vacuum. Notice, that the coupling coefficient is now positive so there is no need to use the explanations accepted in the standard approach. The contribution of the above interaction to the partition function (2) is given by where the coupling coefficient is presented in terms of its average value plus the fluctuation caused by the nonlinear coupling of the scalar field with a fluctuation field of other nature. We also assume that the mean-square value of the fluctuation is equal to This means that we have a system with the effective energy (averaged over the fluctuations of the other field coupled with the scalar field) given by where the last two-term is the well-known expression for the energy of the fundamental scalar field ( ) The total effective energy of the "new" vacuum with the fundamental scalar field is given by In the case of no scalar field 0 ϕ = , This relation can be applied to estimate the maximum dispersion of the field fluctuations. If 2 σ tends to infinity, then the energy of the new state tends to the initial energy of the ground state. Thus, we come to the standard form of the energy of the fundamental scalar field, but with different behavior of the energy of vacuum at the presence of the scalar field. The coefficient of non-linearity in the potential energy is determined by the coupling of the fundamental scalar field with the fluctuations of the field of different nature. This means that there could be a new scenario of the Universe formation. In this scenario, the energy of pure vacuum does not contribute to the energy-momentum tensor and thus we cannot introduce dynamic presentation (and geometry) for such state. Only, if the fundamental scalar field appears and the matter is originated we can tell about the geometry. In this sense, the potential of scalar field determines the vacuum state of the Universe.

Conclusions
Standard cosmological models involve a scenario of the Universe nucleation and expansion based on a scalar field which is of fundamental importance for the unified theories of weak, strong, and electromagnetic interactions with spontaneous symmetry breaking [1]. A theory of new-phase bubble nucleation and expansion was proposed in Ref. [1]. Various cosmological models describe tunneling through the potential barrier in terms of the potentials ( ) V ϕ of arbitrary forms. Here we propose a modification of the standard cosmological model. As was mentioned above, we assume that the fundamental scalar field interacts with possible fluctuations of fields of the other nature. To proceed further and to calculate the size of the bubble, we have to violate the equivalence of the local minima. Attempts have been made [1] to obtain non-linearity of such type associated with the fluctuations of the medium and produced by the interaction with the fields of different nature [9] [10]. Similar ideas are used in the description of phase transitions in condensed media, e.g., liquid crystals, superconductors etc. [11]. In order to explain the fact that the transition in such systems is the firstorder one, the physical mechanism has been reduced to the interaction of the scalar order parameter with the vector field that includes information on possible fluctuations in the system [11] [12]. This means that the contribution of all existing configurations of such fields results in the additional part in the potential energy which proportional 3 ϕ in turn violates the equivalence of the local minima and opens the possibility to determine the bubble size of new phase with nonzero fundamental scalar field [12]. Thus the "condensing" value of fluctuations of the field that is "external" with respect to the scalar field completely determines both the mean critical size of the new phase bubble and the probability of its formation. In this case characteristics of such formation have no free parameters other than fluctuation dispersion.
As mentioned above, only the fundamental scalar field and its symmetry breaking generate the matter, and equilibrium distribution of this matter deter-