Anisotropic cosmological models with spinor and scalar fields and viscous fluid in presence of a Λ term: qualitative solutions
aa r X i v : . [ g r- q c ] A ug Anisotropic cosmological models with spinor and scalar fields and viscousfluid in presence of a L term: qualitative solutions Bijan Saha and Victor Rikhvitsky
Laboratory of Information TechnologiesJoint Institute for Nuclear Research, Dubna141980 Dubna, Moscow region, Russia ∗ (Dated: January 8, 2019)The study of a self-consistent system of interacting spinor and scalar fields within thescope of a Bianchi type I (BI) gravitational field in presence of a viscous fluid and L termhas been carried out. The system of equations defining the evolution of the volume scaleof BI universe, energy density and corresponding Hubble constant has been derived. Thesystem in question has been thoroughly studied qualitatively. Corresponding solutions aregraphically illustrated. The system in question is also studied from the view point of blowup. It has been shown that the blow up takes place only in presence of viscosity. PACS numbers: 03.65.Pm and 04.20.HaKeywords: Spinor field, scalar field, Bianchi type I (BI) model, Cosmological constant,viscous fluid,qualitative analysis
I. INTRODUCTION
The problem of an initial singularity still remains at the center of modern day cosmology.Though the Big Bang theory is deeply rooted among the scientists dealing with the cosmology ofthe early Universe, it is natural to reconsider models of a universe free from initial singularities.Another problem that the modern day cosmology deals with is the accelerated mode of expansion.In order to answer to these questions a number of theories has been proposed by cosmologists. Ithas been shown that the introduction of a nonlinear spinor field or an interacting spinor and scalarfields depending on some special choice of nonlinearity can give rise to singularity free solutionsin one hand [1, 2, 3, 4], on the other hand they may exploited to explain the late time acceleration[5, 6].Why study a nonlinear spinor field? It is well known that the nonlinear generalization of clas-sical field theory remains one possible way to overcome the difficulties of a theory that considerselementary particles as mathematical points. In this approach elementary particles are modelledby regular (solitonlike) solutions of the corresponding nonlinear equations. The gravitational fieldequation is nonlinear by nature and the field itself is universal and unscreenable. These propertieslead to a definite physical interest in the gravitational field that goes with these matter fields. Weprefer a spinor field to scalar or electromagnetic fields, as the spinor field is the most sensitive tothe gravitational field.Why study an anisotropic universe? Though spatially homogeneous and isotropic, Friedmann-Robertson-Walker (FRW) models are widely considered as a good approximation of the presentand early stages of the Universe. However, the large scale matter distribution in the observable ∗ Electronic address: [email protected]; URL:
B. Saha and V. RikhvitskyUniverse, largely manifested in the form of discrete structures, does not exhibit a high degree ofhomogeneity. Recent space investigations detect anisotropy in the cosmic microwave background.The Cosmic Background Explorer’s differential radiometer has detected and measured cosmicmicrowave background anisotropies at different angular scales.These anisotropies are supposed to contain in their fold the entire history of cosmic evolutiondating back to the recombination era and are being considered as indicative of the geometry andthe content of the Universe. More information about cosmic microwave background anisotropyis expected to be uncovered by the investigations of the microwave anisotropy probe. There iswidespread consensus among cosmologists that cosmic microwave background anisotropies atsmall angular scales are the key to the formation of discrete structures. The theoretical argu-ments [7] and recent experimental data that support the existence of an anisotropic phase thatapproaches an isotropic phase leads one to consider universe models with an anisotropic back-ground.Why study a system with viscous fluid? The investigation of relativistic cosmological mod-els usually has the energy momentum tensor of matter generated by a perfect fluid. To considermore realistic models one must take into account the viscosity mechanisms, which have alreadyattracted the attention of many researchers. Misner [7, 8] suggested that strong dissipative due tothe neutrino viscosity may considerably reduce the anisotropy of the black-body radiation. Viscos-ity mechanism in cosmology can explain the anomalously high entropy per baryon in the presentuniverse [9, 10]. Bulk viscosity associated with the grand-unified-theory phase transition [11] maylead to an inflationary scenario [12, 13, 14].A uniform cosmological model filled with fluid which possesses pressure and second (bulk)viscosity was developed by Murphy [15]. The nature of cosmological solutions for homoge-neous Bianchi type I (BI) model was investigated by Belinskii and Khalatnikov [16] by taking intoaccount dissipative process due to viscosity. They showed that viscosity cannot remove the cos-mological singularity but results in a qualitatively new behavior of the solutions near singularity.They found the remarkable property that during the time of the big bang matter is created by thegravitational field.Given the importance of both viscous mechanism and nonlinear spinor field we have recentlystudied the system in question from various aspects. In [17] we have studied the evolution of aBI universe filled with viscous fluid in presence of a L term. Exact solutions to the correspondingsystem of equations were found for some special choice of viscosity parameters. This study wasfurther developed in [18], where the system was studied qualitatively. Introduction of a nonlin-ear spinor field into the system considerably changes the situation giving rise to some unexpectedresults such as Big Rip without phantom dark energy. The system in question was studied ana-lytically in [19, 20] and generalized in [21] employing both numerical and qualitative methods.Since the interacting system of spinor and scalar fields gives rise to a induced nonlinearity of thespinor field that can change the picture drastically, we plan to consider this system as well. Someexact solutions to the system of equations were obtained in [22]. Here we thoroughly study theinteracting spinor and scalar fields within the framework of a BI gravitational field in presence ofa viscous fluid and L term. In doing so we will exploit both numerical and qualitative methods. II. BASIC EQUATIONS
We consider a self-consistent system of interacting nonlinear spinor and scalar fields withinthe scope of a Bianchi type-I (BI) gravitational field filled with a viscous fluid in presence of anisotropiccosmologicalmodelswithspinorand scalarfields · · · L ss = i (cid:20) ¯ yg m (cid:209) m y − (cid:209) m ¯ yg m y (cid:21) − m ¯ yy + j , a j , a ( + l F ) , (2.1)Here m is the spinor mass, l is the coupling constant and F = F ( I , J ) with I = S = ( ¯ yy ) and J = P = ( i ¯ yg y ) . According to the Pauli-Fierz theorem among the five invariants only I and J are independent as all other can be expressed by them: I V = − I A = I + J and I Q = I − J . Therefore,the choice F = F ( I , J ) , describes the nonlinearity in the most general of its form [3]. Note thatsetting l = ds = dt − a dx − b dy − c dz , (2.2)with a , b , c being the functions of time t only. Here the speed of light is taken to be unity.For the BI space-time (2.2) on account of the L term this system has the form¨ bb + ¨ cc + ˙ bb ˙ cc = k T + L , (2.3a)¨ cc + ¨ aa + ˙ cc ˙ aa = k T + L , (2.3b)¨ aa + ¨ bb + ˙ aa ˙ bb = k T + L , (2.3c)˙ aa ˙ bb + ˙ bb ˙ cc + ˙ cc ˙ aa = k T + L , (2.3d)where over dot means differentiation with respect to t and T mn is the energy-momentum tensor ofthe material field given by T rm = i g rn (cid:18) ¯ yg m (cid:209) n y + ¯ yg n (cid:209) m y − (cid:209) m ¯ yg n y − (cid:209) n ¯ yg m y (cid:19) (2.4) +( − l F ) j , m j , r − d rm L + T nm m . Here T nm m is the energy-momentum tensor of a viscous fluid having the form T nm m = ( e + p ′ ) u m u n − p ′ d nm + h g nb [ u m ; b + u b : m − u m u a u b ; a − u b u a u m ; a ] , (2.5)where p ′ = p − ( x − h ) u m ; m . (2.6)Here e is the energy density, p - pressure, h and x are the coefficients of shear and bulk viscosity,respectively. In a comoving system of reference such that u m = ( , , , ) we have T = e , (2.7a) T = − p ′ + h ˙ aa , (2.7b) T = − p ′ + h ˙ bb , (2.7c) T = − p ′ + h ˙ cc . (2.7d) B. Saha and V. RikhvitskyWE consider the case when both the spinor and the scalar fields depend on t only. We also definea new function t = abc , (2.8)which is indeed the volume scale of the BI space-time. It was shown in [19, 20, 22] that thesolutions of the spinor and scalar field equations can be expressed in terms of t . Then for thecomponents of the energy-momentum tensor we find T = mS + C t ( + l F ) + e ≡ ˜ T , (2.9a) T = D S + G P − C t ( + l F ) − p ′ + h ˙ aa ≡ ˜ T + h ˙ aa , (2.9b) T = D S + G P − C t ( + l F ) − p ′ + h ˙ bb ≡ ˜ T + h ˙ bb , , (2.9c) T = D S + G P − C t ( + l F ) − p ′ + h ˙ cc ≡ ˜ T + h ˙ cc , . (2.9d)In account of (2.9) from (2.3) we find the metric functions [3] a ( t ) = Y t / exp (cid:20) X Z e − k R h dt t ( t ) dt (cid:21) , (2.10a) b ( t ) = Y t / exp (cid:20) X Z e − k R h dt t ( t ) dt (cid:21) , (2.10b) c ( t ) = Y t / exp (cid:20) X Z e − k R h dt t ( t ) dt (cid:21) , (2.10c)with the constants Y i and X i obeying Y Y Y = , X + X + X = . As one sees from (2.10a), (2.10b) and (2.10c), for t = t n with n > t , and the anisotropic model becomes isotropic one.So one needs to find the function t , explicitly. Corresponding equation can be derived fromEinstein equations and Bianchi identity [a detailed description of this procedure can be found in[19, 20, 22]]. For convenience, we also define the generalized Hubble constant. The system thenreads [22]: ˙ t = H t , (2.11a)˙ H = k (cid:0) x H − w (cid:1) − (cid:0) H − ke − L (cid:1) + k (cid:0) m t + n t n − ( l + t n ) (cid:1) , (2.11b)˙ e = H (cid:0) x H − w (cid:1) + h (cid:0) H − ke − L (cid:1) − h (cid:2) k (cid:0) m t + t n − ( l + t n ) (cid:1)(cid:3) . (2.11c)Here k is the Einstein’s gravitational constant, L is the cosmological constant, l is the self-coupling constant, m is the spinor mass and n is the power of nonlinearity of the spinor field (herenisotropiccosmologicalmodelswithspinorand scalarfields · · · h and x are the bulk and shear viscosity,respectively and they are both positively definite, i.e., h > , x > . (2.12)They may be either constant or function of time or energy. We consider the case when h = A e a , x = B e b , (2.13)with A and B being some positive quantities. For p we set as in perfect fluid, p = ze , z ∈ ( , ] . (2.14)Vismpl05 Note that in this case z =
0, since for dust pressure, hence temperature is zero, thatresults in vanishing viscosity. Note that a system in absence of spinor field has been studied in[17, 18]. In that case the corresponding system is analogical to the one given in (2.11) without thethird terms in (2.11b) and (2.11c).
III. QUALITATIVE ANALYSIS
The study of the behavior of dynamic system given by a system of ordinary differential equa-tions implies the survey of all possible scenarios of development for different values of the problemparameters. It is necessary to understand at least how the process of evolution comes to an end if itdoes so at infinitively large time for a given set of initial conditions which can be given anywhere.So, under the specific behavior of the system we understand the phase portrait of the system,i.e., the family of integral curves, covering the total phase space. It is easy to imagine as far asany point of the space can be declared as the initial one and at least one integral curve will passthrough it (or it will be fixed point).Certainly, it is difficult to imagine such a set of curves. In many cases, close (and not only)curves transform into each other at some diffeomorphism of space. These curves are known astopologically equivalent. The differences between them are not very important for our study.They all behave in the same manner. This relation - ”the relation of equivalence” - divides thefamily of curves into the classes of equivalence. For graphical demonstration it will be convenientto present at least one representative of each class.The change of the value of problem parameters not always results in significant change of thephase portrait. Repeating this method, we say that one family of integral curves (covering the totalspace) for the given set of parameters is equivalent to the other for another set of parameters, ifthere exists a diffeomorphism of space transforming the first family into the second. It is clearthat there occurs the division into the classes of equivalence, and we are not very interested indifferences between equivalent families. We argue that the corresponding changes in parametersdo not alter anything on principle. So it is sufficient to demonstrate only one phase portrait for agiven set of parameters underlining the features of the given class.However, for some critical relations between the parameters there occurs significant changes.These are the boundary relations of parameters, dividing, as usual, parameter space into regionsof similar behavior. Thus accomplishes the qualitative classification of the mode of evolution ofdynamic system. Now, giving the concrete value of parameters, we can define which region ofparameters they correspond to, thus define the type of behavior. Moreover, given the specificinitial conditions, we can answer the question to which region of phase space the evolution of thesystem lead in time. B. Saha and V. RikhvitskyIn our cosmological model, numerical parameters A , a , B , b are related to the viscosity, while l and L are the (self)-coupling and cosmological constants.Initially, we consider the system of Einstein and Dirac equations. Solving these equations,we find the components of the spinor field and metric functions a , b , c in terms of volume scale t = abc of the BI universe. Finally, in order to find t from Einstein equations and Bianchi identity,we deduce three first order ordinary differential equations. Further for convenience we introducea new function n inverse to t , i.e., n = / t .The fact that the system has the dimension greater than 2, strongly complicates qualitativeanalysis. Note that well known Lorentz system of three ordinary differential equations with poly-nomial right hand side with degree less or equal to 2, possesses in some region of parameter spacechaotic behavior known as a strange attractor and in that region there do not exist first integrals(i.e., globally defined invariants). Though the set of singularities is very simple, there exist onlythree singular (fixed) points: two focus and one saddle. The presence of such example does notallow us to make an optimistic conclusion on the basis of simple construction of our system (withpolynomials in the right hand side and absence of singular points the in region of space we areinterest in, which is even dynamically closed.Nevertheless, on the boundary of the the space e =
0, as well as n = t = + ¥ ), whichare dynamically closed themselves, the complete classification has been done. The dynamicalcloseness of these planes simultaneously as an obstacle for penetration from positive octant e > ∧ n > n = / t . In this case the obvious singularity that occurs at t = n = t = ¥ while n = ¥ to t =
0. The system (2.11) on account of (2.13)takes the form:˙ n = − H n , (3.1a)˙ H = k (cid:0) x H − w (cid:1) − (cid:0) H − ke − L (cid:1) + k (cid:0) m n + n n − n ( l + n − n ) (cid:1) , (3.1b)˙ e = H (cid:0) x H − w (cid:1) + h (cid:0) H − ke − L (cid:1) − h (cid:2) k (cid:0) m n + n − n ( l + n − n ) (cid:1)(cid:3) . (3.1c)Let us now study the foregoing system of equations in details. A. Behavior of the solutions on n = plane As one can see, in this case the system (3.1) takes the form:˙ H = k (cid:0) x H − w (cid:1) − (cid:0) H − ke − L (cid:1) , (3.2a)˙ e = H (cid:0) x H − w (cid:1) + h (cid:0) H − ke − L (cid:1) . (3.2b)This system of equations completely coincides with the one when the BI universe is filled withviscous fluid only. The system in question was thoroughly studied in [18], hence we skip thisstudy in the present report.nisotropiccosmologicalmodelswithspinorand scalarfields · · · B. Behavior of the solutions on e = plane The plane e = e (cid:12)(cid:12) e = =
0. Depending on the sign of H this plane iseither attractive or repulsive, namely, for H > H < ¶ ˙ e¶e = − H ( + z ) < . In presence of of spinor and scalar fields the system (3.1) at e = n = − H n , (3.3a)˙ H = − H + L + (cid:16) m n + n n − n ( l + n − n ) (cid:17) . (3.3b)The system (3.3) has the following integral curves6 H = L + m n + C n − n + n ( ln n + ) (3.4a)where C is some arbitrary constant.The characteristic equation of nontrivial singular points on e = m l n n + + L l n n + n n n + + m ln n + + L ln n + m n + L = . (3.5)Depending on changes of signs in the sequence of l , m , L it has one, two or no solutions.In Tables A1, B1, C1, D1 we illustrated the phase-portrait on e = L , respectively for n = , , , l <
0. In Tables A2, B2, C2, D2 we illustratedthe phase-portrait on e = L , respectively for n = , , , l =
0. In Tables A3, B3, C3, D3 we illustrated the phase-portrait on e = L , respectively for n = , , , l > m < H =
0, and therefore, the trajec-tory of oscillation partially passes in the region which is attractive to the plane e = e becomes dominant. It results in the fact that e becomes infinity within a finite range of time. B. Saha and V. Rikhvitsky L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table A1. Case with e = n = l < L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table E1. Case with e = n = l = L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table A2. Case with e = n = l > · · · L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table B1. Case with e = n = l < L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table E2. Case with e = n = l = L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table B2. Case with e = n = l > L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table C1. Case with e = n = l < L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table E3. Case with e = n = l = L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table C2. Case with e = n = l > · · · L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table D1. Case with e = n = l < L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table E4. Case with e = n = l = L < L = L > m < –1–0.500.51 H 1 2 3 4 a –1–0.500.51 H 1 2 3 4 b –1–0.500.51 H 1 2 3 4 c m = –1–0.500.51 H 1 2 3 4 d –1–0.500.51 H 1 2 3 4 e –1–0.500.51 H 1 2 3 4 f m > –1–0.500.51 H 1 2 3 4 g –1–0.500.51 H 1 2 3 4 h –1–0.500.51 H 1 2 3 4 i //Table D2. Case with e = n = l > C. Qualitative analysis of the complete system
The system (3.1) in absence of viscosity, i.e., under h = x = F = en + z , (3.6a) F = ( H − e − L − m n ) n − l ( ln n + ) . (3.6b)The second of them (3.6b) remains to be the first integral even after the introduction of bulkviscosity x . The first one, i.e., Eq. (3.6a) under x = FIG. 1: Evolution of func-tion inverse to volumescale FIG. 2: Evolution ofvolume scale FIG. 3: 3D view in n , H , e spaceFIG. 4: Evolution of func-tion inverse to volumescale FIG. 5: Evolution ofvolume scale FIG. 6: 3D view in n , H , e space Harnessing the Tables 1, 2 and 3, helps one to understand the 3D phase portrait leaning on thecontinuous dependence of the velocity fields of the coordinates n , H , e of phase space.nisotropiccosmologicalmodelswithspinorand scalarfields · · · FIG. 7: Evolution of func-tion inverse to volumescale FIG. 8: Evolution ofvolume scale FIG. 9: 3D view in n , H , e spaceFIG. 10: Evolution offunction inverse to volumescale FIG. 11: Evolutionof volume scale FIG. 12: 3D view in n , H , e space In order to cover the infinite phase space completely, it is mapped on coordinate parallelepipedwith its axes being the the arc-tangent of the corresponding coordinates. The lower horizontalplane always represents the e = H < H > H repeats the evolution, approaching to the n = e = n = H > e = e =
0, though not necessarily reach it.In the Figs. 10 - 3 we have illustrated functions inverse to the volume scale n ( t ) [Figs. 10,7,4,1],volume scale t ( t ) [Figs. 11,8,5,2] and phase portrait in n , H , e space [Figs.12,9,6,3], for a = b = z = / A = B = n = m = L < l there occur the following situations: (i) for l < l > L , which is infact the additional gravitational field, generates oscillatory regime of expansion.In case of L > l < l > IV. EVOLUTION WITH BLOW UP
Studying the system of ODE let us imagine the integral curves in the space. It is very importantto know the directional field given by this system. It is more important than the correspondingvector field. First of all, the integral curves, by definition, are tangent to the vector field, hence tothe directional field, at those (peculiar) points, where vector field becomes trivial with the directionbeing indefinite. Secondly, like the vector field the directional field is also continuous (excludingthe peculiar points), but it may be continuously continued at the boundary where the vector fieldmight be infinity.We are interested in two aspects: how rapidly the solution can tend to infinity at the distantboundary (simply infinity) and how does it behave at the infinity. The way the problem is posedbecomes reasonable when the space is closed by means of infinitely remote points in any giveninterpretation.We will follow the Pensele’s ( ) principle of continuity - the properties of a system at continuouschange from one common position to another without losing generality. We are interested inqualitative properties in solving the system of ODE. Deforming the vector field continuously, atthe same time leaving the peculiar points unaltered, we don’t change the qualitative behavior ofthe integral curves with an accuracy of topological equivalent. In this way we can simplify theanalysis, substituting the initial system by a simpler one, constructed from convenient elements.
A. Blow up
The history of studying the regime with blow up is associated with S.P. Kurdiumov [23]. Thestudy of the process of heat distribution in active and nonlinear medium led to an extremely dis-tinguishing feature, namely wave and localization. Mathematical models of demography detectscritical moments: solution to the (time dependent) ODE may reach its limit within a finite time.The processes in the chromosphere of the sun possess a flashing (eruptive) character, but the mech-anism of energy transference does not detect the presence of predefined scale of time.To illustrate the detection of a characteristic time in the system with no explicit time-dependence, let us consider the following example.˙ x = − x a , x ∈ R + (4.1)It has two solutions: a) x ( t ) = x ( t ) = [ x ( ) − a − t ( − a )] − a .In case of b) the limiting value x ( t ∗ ) = t ∗ = x ( ) − a − a , if a <
1. Thenboth solutions mix up. At moment t ∗ the uniqueness condition (precisely, Lipshits condition)breaks down.The power law dependencies are typical for different types of catastrophes: from earth quakesand flood to stock exchange collapse and accidents in atomic power energy.nisotropiccosmologicalmodelswithspinorand scalarfields · · · B. Infinity
The joining of infinitely remote point to the space of ODE˙ x = F ( x ) , x ∈ R + (4.2)we execute in the following way: let us make the change of variables x = sc , s + c =
1. We callthe point ± ¥ = infinitely remote one.As a result we obtain a system of equations˙ sc − ˙ cs = c F ( sc ) , ˙ ss + ˙ cc = , (4.3)which on account of s + c = s = c F ( sc ) , ˙ c = − sc F ( sc ) . (4.4)Reducing the right hand side of the system to a common denominator in the vicinity of the point s = , c = n = − H n , (4.5a)˙ H = (cid:0) x H − ( e + p ) (cid:1) − (cid:0) H − e − L (cid:1) + f ( n ) , (4.5b)˙ e = H (cid:0) x H − ( e + p ) (cid:1) + h (cid:0) H − e − L (cid:1) − hf ( n ) , (4.5c)where f and f are the functions of t .In case of a spinor field only we have f ( n ) = m n + l ( n − ) , f ( n ) = m n − ln n .Introduction of a scalar field gives f ( n ) = m n + n n n + ( + ln n ) , f ( n ) = m n + n ( + ln n ) .Near the point e = ¥ we make the following substitution e = / m . Then the system takes theform ˙ n = − H n , (4.6a)˙ H = BH m − b + ( − z ) m − − H + L + f ( n ) , (4.6b)˙ m = A (cid:0) − H + L + f ( n ) (cid:1) m − a − BH m − b + A m − a + H ( + z ) m . (4.6c)As it is seen from (4.6c) in the absence of viscosity ( A = B =
0) the blow up along the energydensity is impossible.The answer, whether the blow up takes place in the past or in the future, depends on the sign ofthe coefficient at m with the lowest power.Let A =
0. In order to the blow up takes place at finite H , it is necessary that b >
1. In this casethe singularity will be in the future, i.e., we have Big Rip.Now consider the case with B =
0. In this case the blow up takes place in the past (Big Bang)if a > e isachieved in a finite time. The blue line indicates past while the red one the future.In the figures 14 and 13 we show the evolution of t , H and e relative to each other. In bothcases there exists possibility for infinite growth of energy density at infinitely large volume, i.e.,there occurs so-called Big Rip.6 B. Saha and V. Rikhvitsky
1 Η0−1 ν 1.00.50.00.00.5 ε 1.01.5
FIG. 13: The trajectory of evolution in case of an interactingspinor and scalar fields with a = , b = , z = / , A = , B = , m = , L = − , l = − V. CONCLUSION
Recently a self consistent system of nonlinear spinor and gravitational fields in the frameworkof Bianchi type-I cosmological model filled with viscous fluid was considered by one of the au-thors [19, 20]. The spinor filed nonlinearity is taken to be some power law of the invariants ofbilinear spinor forms, namely I = S = ( ¯ yy ) and J = P = ( i ¯ yg y ) . Solutions to the corre-sponding equations are given in terms of the volume scale of the BI space-time, i.e., in terms of t = abc , with a , b , c being the metric functions. This study generates a multi-parametric system ofordinary differential equations [19, 20]. Given the richness of the system of equations in this papera qualitative analysis of the system in question has been thoroughly carried out. A complete qual-itative classification of the mode of evolution of the universe given by the corresponding dynamicsystem has been illustrated. In doing so we have considered all possible values of problem param-eters independent to their physical validity and graphically presented the most distinguishable inour view results.The system is studied from the view point of blow up. It has been shown that in absence ofviscosity the blow up does not occur. It should be emphasized that phenomena similar to one inquestion can be observed in other discipline of physics and present enormous interest from thenisotropiccosmologicalmodelswithspinorand scalarfields · · · FIG. 14: The trajectory of evolution in case of a spinor field withself-action at a = , b = , z = / , A = , B = , m = , L = − , l = − point of catastrophe, demography etc. [1] B. Saha and G.N. Shikin, Journal of Mathematical Physics , 5305 (1997).[2] B. Saha, and G.N. Shikin, General Relativity and Gravitation , 1099 (1997).[3] Bijan Saha, Physical Review D , 123501 (2001).[4] Bijan Saha, Physics of Particles and Nuclei Suppl. 1, S13-S44, (2006).[5] Bijan Saha, Physical Review D , 124030, (2006).[6] Ribas, M.O., Devecchi, F.P., and Kremer, G.M., Phys. Rev. D (2005) 123502.[7] C.W. Misner, Astrophys. J. , 431 (1968).[8] W. Misner, Nature , 40 (1967).[9] S. Weinberg, Astrophysical Journal , 175 (1972).[10] S. Weinberg, Gravitation and Cosmology (New York, Wiley, 1972)[11] P. Langacker, Physics report , 185 (1981).[12] L. Waga, R.C. Falcan, and R. Chanda, Physical Review D , 1839 (1986). [13] T. Pacher, J.A. Stein-Schabas, and M.S. Turner, Physical Review D , 1603 (1987).[14] Alan Guth, Physical Review D , 347 (1981).[15] G.L. Murphy, Physical Review D , 4231 (1973).[16] V.A. Belinski and I.M. Khalatnikov, Journal of Experimantal and Theoretical Physics , 401, (1975).[17] Bijan Saha, Modern Physics Letters A (28) 2127-2143, (2005); [arXiv: gr-qc/0409104].[18] Bijan Saha and V. Rikhvitsky, Physica D , 168-176, (2006); [arXiv: gr-qc/0410056].[19] Bijan Saha, Romanian Report of Physics (1),7-24, (2005).[20] Bijan Saha, Astrophysics and Space Science , 3-11, (2007) [arXiv: gr-qc/0703085].[21] Bijan Saha and Victor Rikhvitsky, Journal Physics A: Mathematical and Theoretical Interacting spinor and scalar fields in Bianchi type-I Universe filled with viscous fluid:exact and numerical solutions [arXiv: gr-qc/0703124].[23] A.A. Samarsky, V.A. Galaktionov, S.P. Kurdimov and A.P. Mikhailov,