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Non-minimally coupled complex scalar field in classical and quantum cosmology

We investigate the cosmological model with complex scalar self-interacting inflaton field non-minimally coupled to gravity. The different geometries of the Euclidean classically forbidden regions are represented. The instanton solutions of the corresponding Euclidean equations of motion are found by numerical calculations. These solutions give a rather non-trivial examples of real tunnelling geometries. Possible interpretation of obtained results and their connection with inflationary cosmology is discussed.

It is widely recognized that inflationary cosmological models give a good basis for the description of the observed structure of the Universe. Most of these models include the so called inflaton scalar field possessing non-zero classical average value which provides the existence of an effective cosmological constant on early stage of the cosmological evolution. In the series of recent papers [1][2][3] it was suggested to consider the cosmological models with the complex scalar field φ = x exp(iθ), that is equivalent to the inclusion into the theory the new quasi-fundamental constant -charge of the Universe Q.
Appearance of this new constant essentially modifies the structure of the Wheeler-DeWitt equation. Namely, the form of the superpotential U(x, a) displays now a new and interesting feature: the Euclidean region, i.e. the classically forbidden region where U > 0, is bounded by a closed curve in the minisuperspace (x, a) for a large range of parameters. Thus, in contrast with the picture of "tunneling from nothing" [4] and with the "no-boundary proposal" for the wave function of the Universe [5,6] we have Lorentzian region at the very small values of cosmological radius a and hence a wave can go into Euclidean region from one side and outgo from the other. These new features of the model require reconsideration of traditional scheme [4][5][6] and give some additional possibilities.
Here, it is necessary to stress that speaking about the "Euclidean" or "classically forbidden" regions one should understand that in the case of the quantum gravity and cosmology these terms can be used only in "loose" sense, because due to indefiniteness of the supermetric the "Euclidean" region is not impenetrable for Lorentzian trajectories. It is well-known that in the cosmological models with inflaton scalar field the Lorentzian trajectories can penetrate into Euclidean region just like as the trajectories corresponding to the Euclidean equations of motion can leave Euclidean region for the Lorentzian one (see, for example, Refs. [7][8][9][10][11][12][13][14]. However, one can use such terms as Euclidean and Lorentzian regions as people usually do investigating the tunneling type processes in cosmology [7][8][9][10][11][12][13][14] and in instanton physics [15] even in the case of indefinite supermetric. Moreover, one can give to the term "Euclidean region" quite definite value as the region where the points of the minimal contraction and maximal expansion of the Universe can exist [16].
In recent years quite a few papers were devoted to the investigation of the cosmological models with the non-minimal coupling between inflaton scalar field and gravity [17][18][19]. On one hand such models give a lot of opportunities for matching with the observational data and ideas of particle physics, on the other hand they can be treated as more consistent from the point of view of quantum gravity [20]. This letter is devoted to the investigation of the cosmological models with the complex scalar field non-minimally coupled to gravity.
We shall consider the model with the following action: Here ξ is the parameter of non-minimal coupling (we choose for convenience the sign which is opposite to generally accepted), λ is the parameter of the self-interaction of the scalar field, Λ is cosmological constant, m is the mass of the scalar field. The complex scalar field φ can be represented in the form We shall consider the minisuperspace model with the spatially homogeneous variables a (cosmological radius), x and θ. Taking into account that the variable θ is cyclical one can assume that the corresponding conjugate momentum is frozen and does not subject to quantization [1][2][3]. In this case the only influence of the variable θ on the dynamics of the system consists in the appearance of the new constant of the model: classical charge Q which is defined as Let us notice that the another approach to the complex scalar field in cosmology was discussed in [21] mainly in the context of wormhole solutions.
The super-Hamiltonian constraint equation in minisuperspace variables (a, x) has the following form: where p a and p x are the momenta canonically conjugate to a and x respectively and superpotential U(a, x) 1 looks as follows: 1 In papers [1,2] instead of superpotential the effective mass m 2 (ln a, Let us now write down the equations of motion for our minisuperspace system including gravity and scalar field. Varying the action (1) in respect to a and x and choosing for the lapse function the value N = 1 we obtain Besides we can write down the first integral of motion of our dynamical system which can be obtained from the super-Hamiltonian constraint (2): It is obvious that Euclidean counterparts of the Eqs. instantons do not exist (see Fig. 2b).
Now we are in a position to say some words about the interpretation of the obtained results. All the models describing "the birth of the Universe from nothing" in the form of tunnelling wave function [4] or due to no-boundary mechanism [5,6] are grounded on the assumption that it is possible to provide the analytical continuation from Lorentzian geometry to Euclidean one [11][12][13][14][15][16]. Here, it is necessary to stress that there is essential difference between field without interaction [6], where it is impossible to have solutions possessing the simple matching between Euclidean and Lorentzian regions [6].
The models in which the one can match the Euclidean and Lorentzian trajectories at some points where all the velocities turn to zero (so called real tunnelling geometries) are usually the different theories without matter but with the cosmological constant and they investigated in some detail [11]. In the more realistic cases we encounter the necessity to consider complexified classical trajectories in the minisuperspace [9][10][11][12][13][14]