The Peculiarities of the Cosmological Models Based on Nonlinear Classical and Phantom Fields with Minimal Interaction. II. The Cosmological Model Based on the Asymmetrical Scalar Doublet††thanks: The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University
A detailed comparative qualitative analysis and numerical simulation of evolution of the cosmological models based on the doublet of classical and phantom scalar fields with self-action. The 2-dimensional and 3-dimensional projections of the phase portraits of the corresponding dynamic system are built. Just as in the case of single scalar fields, the phase space of such systems, becomes multiply connected, the ranges of negative total effective energy unavailable for motion, getting appear there. The distinctive feature of the asymmetrical scalar doublet is the time dependency of the prohibited ranges’ projections on the phase subspaces of each field as a result of which the existence of the limit cycles with null effective energy depends on the parameters of the field model and initial conditions. The numerical models where the dynamic system has limit cycles on hypersurfaces of null energy, are built. It is shown, that even quite weak phantom field in such model undertakes functions of management of the dynamic system and can significantly change the course of the cosmological evolution.
osmological model, phantom and classical scalar fields, quality analysis, asymptotic behavior, numerical simulation
4.20.Cv, 98.80.Cq, 96.50.S 52.27.Ny
- 1 The Basic Relations of the Cosmological Model Based on the Asymmetrical Scalar Doublet
2 Qualitative Analysis of the Cosmological Model
- 2.1 Reducing the System of Equations to the Canonical Form
- 2.2 Areas of the Reality of the Solution and the Motion in the Neighbourhood of Energy Hypersurfaces
- 2.3 The Singular Points of the Dynamic System
- 2.4 The Character of Singular Points of the Dynamic System of Assymetrical Scalar Doublet
- 3 The Numerical Simulation of the Dynamic System:
- 4 The Analysis of the Results
A Appendices. The Results of Numerical Simulation
- A.1 Appendix A. The Case of Inaccessibility of the Singular Points and
- A.2 Appendix B. The Case of Inaccessibility of the Singular Points and
- A.3 Appendix C. The Case of Inaccessibility of All Singular Points Except
- A.4 Appendix D. The Impact of the Cosmological Constant on the Behavior of the Dynamic System. The Positive Values of the Cosmological Constant
- A.5 Appendix E. The Impact of the Cosmological Constant on the Behavior of the Dynamic System. Negative Values of the Cosmological Constant
1 The Basic Relations of the Cosmological Model Based on the Asymmetrical Scalar Doublet
1.1 The Lagrangian Function and the Potential of Self-Action
In the previous paper of the Author  the cosmological models based on single classical and phantom scalar fields with self-action were investigated. Let us now consider a system comprising two minimally interacting scalar fields, classical and phantom ones, which we will call an asymmetrical scalar doublet. The Lagrangian function of the scalar doublet comprising classical and phantom scalar fields with self-action in the Higgs form with minimal coupling has the following form [3, 4]:
– the potential energy of the corresponding scalar fields, and are the constants of their self-action, and are their quanta masses. Introducing summary potential
we can make the following conclusions:
1. The potential possess the following symmetries:
2. The function at has the absolute maximum in the origin of coordinates of the phase space , and at it has the absolute minimum: everywhere, it has a conditional extremum in these points (saddle points) at .
3. The function has its absolute maximum at and in and at (i.e. ) and the absolute minimum in these points at (i.e. ): everywhere; it has a conditional extremum in these points (saddle points) at and : everywhere.
4. The function has its absolute maximum at and in the points and at (i.e. ) and the absolute minimum in these points at (i.e. ): everywhere; it has a conditional extremum in these points (saddle points) at and , i.e., at .
5. The function has its absolute maximum at , and in the points , , and at (i.e. ) and the absolute minimum in these points at : everywhere; it has a conditional extremum in these points (saddle points) at , i.e. at . \ruleup\figcaption
On the left-hand side: the graph of the potential , on the right-hand side: the graph of the potential\ruledown
The typical graphs of the potential function , corresponding to two opposite cases described in (1.2) are shown on Fig. 1.1. On the left graph one can see 4 maximums, 1 central minimum and 4 saddle points while on the right one – 4 minimums and one central maximum as well as 4 saddle points. The right figure is obviously obtained by mirroring of the left one from the plane .
Thus, taking into account the fact that stationary points of the dynamic system with the Lagrangian function of the form (1) coincide with the stationary points of the potential , we can state the following: depending on the signs of the parameters potential , the corresponding dynamic system should have 1, 3 or 9 stationary points among which there should be the attractive (absolute minimum), repulsing (absolute maximum) and saddle (conditional extremum) points. This result fully coincides with the conclusions of the paper , where it was obtained with the help of qualitative theory of the dynamic systems.
1.2 The Equations of the Cosmological Model
The energy-momentum tensor of the scalar field relative to the Lagrangian function (1) takes the standard form:
The Lagrangian function’s variation (1) leads to the next field equations:
Carrying out renormalization of the Lagrangian function (1), and adding a constant to it, (see ), let us reduce it to the following form:
The corresponding renormalzation of the energy-momentum tensor gives us
The equations of free classical and phantom fields can be found by means of the standard variational procedure over the Lagrangian function in the form (1.2):
where and are effective masses of scalar bosons:
Let us further consider a self-consistent system of equations of the cosmological model (10), (11) and Einstein equations (13) 111Here we use the Planck system of units: ; the Ricci tensor is obtained by means of convolution of the first and fourth indices ; the metrics has the signature .
where is the cosmological constant based on free assymetrical scalar doublet and space-flat Friedmann metrics (14)
assuming The energy-momentum tensor at that (9) takes a structure of the energy-momentum tensor of the isotropic liquid with energy density and pressure p:
Herewith the followind identity law is fulfilled:
The considered system comprises one Einstein equation
and two equations of the scalar field:
2 Qualitative Analysis of the Cosmological Model
2.1 Reducing the System of Equations to the Canonical Form
Proceeding to the dimensionless Compton time: () and carrying out a standard change of variables:
where the total energies for classical and phantom fields are introduced:
Let us also introduce the effective reduced pressure of the scalar doublet:
so that it is
This, let us reduce the Einstein equation (1.2) to the dimensionless form:
Beforehand, let us notice that the conclusions of the qualitative theory relative to stationary points of the dynamic system and their character can’t differ from the conclusions obtained on the basis of the potential function’s analysis. However, the qualitative theory provides more details of the dynamic system’s behavior in the neighbourhood of stationary points.
In order the system of differential equations (31) to have a real solution, the non-negativeness of the expression beneath the radical in the equations i.e. non-negativeness of the system’s effective energy with an account of the cosmological term, is required:
The inequality (2.1) can lead to breaking of the simple connectedness of the phase space and generation of closed lacunae in it, which are limited by surfaces with null effective energy. To reduce the system (31) to the standard notation of the qualitative theory (see, e.g. )
let us accept the following denotations:
The corresponding normal system of equations in the standard notation (33) has the following form:
The necessary condition of the solution’s reality (2.1) can be re-written in the following form:
2.2 Areas of the Reality of the Solution and the Motion in the Neighbourhood of Energy Hypersurfaces
As was mentioned above, the unique property of the considered system is change of the topology of the phase space as a consequence of appearance of the ranges in it, where the motion is not possible. These ranges stand out by the condition of non-negativeness of the effective total energy (2.1) while ranges that are available for the phase trajectories are defined by the condition (2.1). The null effective energy hypersurfaces , splitting the phase space to the admitted and forbidden regions of the dynamic variables, are described by the equations:
Let us notice that as a consequence of definition (27) renormalized value of the cosmological constant can take generally speaking negative values as well. Let us notice then that the field equations (19) – (20) have the form of equations of free oscillations in a field of potential of the 4th order
with a nonlinear ‘‘coefficient of friction’’:
According to the theory of oscillations the corresponding dynamic system, losing its total energy as a consequence of dissipative process corresponding to the friction force in (38), should go down with time to the minimum of the potential energy if it exists; it should be rolling ‘‘down’’ infinitely in the case if does not. The specific character of the problem is concluded in, first of all, the significant dependence of the friction force on the total energy of the system, and second, in the nonlinear coupling of the subsystems by the ‘‘friction force’’ and third, in the factor of the negativeness of the kinetic energy of the phantom component. Let us consider the motion of the null effective energy on the hypersurface (2.2). The equations (31) on this hypersurface take the following form:
Thus, multiplying even equations (39) by Z and z we correspondingly obtain the integrals of the total energy at motion along the hypersurface :
This fact exactly shows that we deal with free oscillations. As a consequence of (2.2) the integrals of the total energy on this trajectory should be coupled by the following relation:
Thus we can state that the phase trajectories on the null energy surface which are described by the equations (39) with the integrals of the total energy (2.2) – (41) are the exact solution of the complete equations of motion (31). To obtain a certain phase trajectory one of the constants ( or ) should be set while the second one should be obtained from the relation (41). Thus, the desired phase trajectories are cross-sections of the surface of null effective energy (2.2). We can also obtain explicit solutions of the field equations on the surface (2.2), integrating (2.2):
The result of integration is expressed by means of elliptical functions. In the case of residence of the attractive centers inside the forbidden regions it is expected that the phase trajectories will asymptotically adhere to the surfaces of null effective energy while when saddle points can be found inside these regions it is expected that the phase trajectories will repulse from the surface of null effective energy.
Let us notice, however, another circumstance which qualitatively differs the cosmological model with asymmetric scalar doublet from the corresponding model with a single scalar field, considered above. The hypersurface of the phase space (2.2) is defined through the parameters of the field model of the scalar doublet and does not depend on the time variable . However, the intersections of the 2-dimensional phase planes of the single fields and with hypersurface can be 2-dimensional curves and (closed or open ones), essentially depending on the values of the dynamic variables of another scalar field and therefore depending on the time variable:
As a consequence of (2.2) the topology of 2-dimensional phase subsrufaces and can significantly change with time looking through all the cases considered in the previous article . This factor is essentially new and significant for the cosmological model.
2.3 The Singular Points of the Dynamic System
Note 1. Let us notice the following important circumstance. In the case when in certain singular point the reality condition is fulfilled (36), then, basically the phase trajectories of the dynamic system (33) – (34) can come into such singular point or come out from it. In case singular point is situated in the forbidden range of the phase space, the phase trajectories of the dynamic system can’t pass through that singular point and only can be attracted to the boundary of the forbidden range or repulse from it depending on the character of the singular point.
Thus, as was noted above, the dynamic system (33) – (34) has the following 9 singular points.
1. : The system of algebraic equations (43) always has the following trivial solution at any values of and :
2. , : We have two more solutions symmetrical in at any values of and :
The necessary condition of reality of the solutions in singular points , is:
3. , : We have two more solutions which are symmetrical in at any and :
The necessary condition of reality of the solutions in singular points , is:
4. , , , : We have 4 more solutions symmetrical in and at and :
The necessary condition of reality of the solutions in singular points , , , is:
2.4 The Character of Singular Points of the Dynamic System of Assymetrical Scalar Doublet
The determinant of this matrix is equal to:
Notice 2. Let us notice that since all the dynamic variables are real values, all functions with their partial derivatives in the admitted regions of the phase space are also real values. However, in the forbidden ranges of the phase space, i.e. in ranges with negative effective energy , the derivatives over the dynamic variables can become imaginary values. This means that this singular point is situated in the inaccessible region of the phase space. Let us notice that condition of reality in the matrix of the dynamic system can be even simultaneously violated in the derivatives of type .
Let us consider equations for eigenvectors and eigenvalues of the dynamic system’s matrix:
where is an identity matrix. Due to block-diagonal structure of the dynamic system’s matrix its eigenvalues are defined by the characteristic equations in corresponding planes while eigenvectors , corresponding to these eigenvalues, lie pairwise in different phase planes: , . This fact allows to significantly simplify the qualitative analysis of the phase trajectories in the neighbourhood of the singular point and reduce it to enumeration of combinations of the dynamic system’s oscillations in 2-dimensional planes . If a singular point is situated in the admitted region, then as a consequence of reality of the elements of the dynamic system’s matrix, each of its complex eigenvalues should be corresponded by a complex conjugated value , so that . If a singular point is situated in the inaccessible region of the phase space, the last condition may not hold. In this case the conclusions of the qualitative theory are only conditionally applicable to the extent that the prohibited region’s radius is small. A certain behavior of the phase trajectory in these cases should be specified with the help of numerical integration of the dynamic equations. According to the qualitative theory of differential equations (see [16, 15]) the radius-vector of the phase trajectory in the neighbourhood of the singular point is decribed by the following equation:
where are arbitrary constants which are defined by the initial conditions, are eigenvalues of the dynamic system’s matrix , are eigenvectors of this matrix corresponding to eigenvalues . In the cases when a singular point is inaccessible, the evaluation formula (54) is nevertheless good enough approximation of the phase trajectory. We will rely on this evaluation in the cases when the results of the standard qualitative theory which could be suitable for real matrices of the dynamic system, are absent.
Let us briefly state the results of the qualitative analysis of the dynamic system (31). First, these calculations show that all the singular points of the dynamic system are split into 4 groups where the character of points inside each group is the same.
2.4.1 The Characteristic Equation and the Qualitative Analysis In The Neighbourhood of Null Singular Point
the following single singular point is included in the first group:
and its determinant is equal to:
As a consequence of the notice made above on page 2.4, point is accessible at and is inaccessible at . The characteristic equation for matrix has the following form:
thus the eigenvalues of the matrix are equal to:
herewith it is
The character of the singular point in the plane . type Center Saddle Saddle Attractive Node Attractive Focus Inaccessible Center Inaccessible Saddle Inaccessible Center
Note. Let us consider examples of such trajectories, selecting corresponding parameters in (54).\figcaption
Examples of phase trajectories (54) in the neighbourhood of singular points. On the left-hand side – inaccessible center in the plane : ; on the right-hand side – inaccessible saddle in the plane : (54) .\tabcaption
The character of the singular point in the plane . type Saddle Center Saddle Attractive node Attractive focus Inaccessible saddle Inaccessible center Inaccessible center
2.4.2 The Characteristic Equation and the Qualitative Analysis in the Neighbourhood of Singular Points
Two symmetrical singular points (2.3) are included in the second group providing and :
the dynamic system’s matrices and in these points coincide and are equal to:
In the opposite case the singular points are situated in inaccessible regions. The eigenvalues of the matrix are equal to: