Nonlinear spinor fields in an anisotropic universe filled with viscous fluid: Exact solutions and qualitative analysis

A self-consistent system containing a nonlinear spinor field and a Bianchi type-I (BI) gravitational field is considered in the presence of a viscous fluid and the cosmological constant. Nonlinear terms in the Lagrangian spinor-field appear either due to a self-action, or as a result of interaction with a scalar field. They are given by power functions of the invariants I and J, constructed from the bilinear spinor forms S and P. As far as the viscosity is concerned, it is a function of the energy density ? exhibiting a power-law behavior. Self-consistent solutions of the spinor, scalar, and gravitational field equations are derived. The obtained solutions are expressed in terms of the function τ(t), where τ is the volume scale in the BI-type Universe. A system of equations for τ, H, and ? is derived, where H is the Hubble constant, and ? is the viscous-flow energy. Exact solutions of the system are found for some special choices of the nonlinearity and viscosity. A complete qualitative analysis of the evolution at the boundaries is performed, and numerical solutions are obtained in the most interesting cases. In particular, it is shown that the system has Big Rip type solutions, which is typical for systems containing a phantom matter.     

Spinor fields in Bianchi type-I universe

A system of minimally coupled nonlinear spinor and scalar fields within the scope of a Bianchi type-I (BI) cosmological model in the presence of a perfect fluid and a cosmological constant (Λ term) is studied, and solutions to the corresponding field equations are obtained. The problem of initial singularity and the asymptotical isotropization process of the Universe are thoroughly studied. The effect of the Λ term on the character of evolution is analyzed. It is shown that some special choice of spinor field nonlinearity generates a regular solution, but the absence of singularity results in violating the dominant energy condition in the Hawking-Penrose theorem. It is also shown that a positive Λ, which denotes an additional gravitational force in our case, gives rise to an oscillatory or a non-periodic mode of expansion of the Universe depending on the choice of problem parameter. The regular oscillatory mode of expansion violets the dominant energy condition if the spinor field nonlinearity occurs as a result of self-action, whereas, in the case of a linear spinor field or nonlinear one that occurs due to interaction with a scalar field, the dominant condition remains unbroken. A system with time-varying gravitational (G) and cosmological (Λ) constants is also studied to some extent. The introduction of magneto-fluid in the system generates nonhomogeneity in the energy-momentum tensor and can be exactly solved only under some additional condition. Though in this case, we indeed deal with all four known fields, i.e., spinor, scalar, electromagnetic, and gravitational, the over-all picture of evolution remains unchanged.     

Bianchi Type-I Model with Cosmic String in the Presence of a Magnetic Field: Spinor Description

A Bianchi type-I cosmological model in the presence of a magnetic flux along a cosmic string is investigated. A nonlinear spinor field is used to simulate the cosmological cloud of strings. It is shown that the spinor field simulation offer the possibility to solve the system of Einstein’s equation without any additional assumptions. It is pointed out that the present model is nonsingular at the end of the evolution and does not allow the anisotropic Universe to turn into an isotropic one.     

A homogeneous multicomponent cosmological model with interacting spinor,scalar, and vector fields in the presence of dark energy

The evolution of a homogeneous multicomponent cosmological model with interacting spinor, vector, and scalar fields in the presence of dark energy described by the ideal liquid with the corresponding state equation is considered. The source of the vector and spinor fields is the kinetic energy of the inflation (scalar) field that is modeled by introduction of Lagrangians for the spinor and vector fields interacting with the scalar field through the squared gradient. A system of the dynamic Einstein–Proca–Klein–Fock and ideal liquid equations in the presence of interaction of the cosmological model components is solved. The role of individual components in the process of model evolution is elucidated.     

Gravitational and quantum effects in rotating cosmological models

Specific effects of the dynamics of (spinor and scalar) wave fields are considered in rotating uniform Gödel-type cosmological models. It is shown that the gravitational interaction of the spinor field can be reduced to the interaction between its pseudovector current and the angular velocity of space-time rotation and is similar to its interaction with the pseudotrace of the space-time twisting. The mean values of energy-momentum tensor of the quantized scalar field in vacuum are calculated in rotating cosmological models and the difference between these values and their mean counterparts in vacuum is determined for Friedman's nonrotating cosmological models.State Education Institute, Yaroslavl'. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 35–38, June, 1992.     

Spinors, Inflation, and Non-Singular Cyclic Cosmologies

We consider toy cosmological models in which a classical, homogeneous, spinor field provides a dominant or sub-dominant contribution to the energy-momentum tensor of a flat Friedmann-Robertson-Walker universe. We find that, if such a field were to exist, appropriate choices of the spinor self-interaction would generate a rich variety of behaviors, quite different from their widely studied scalar field counterparts. We first discuss solutions that incorporate a stage of cosmic inflation and estimate the primordial spectrum of density perturbations seeded during such a stage. Inflation driven by a spinor field turns out to be unappealing as it leads to a blue spectrum of perturbations and requires considerable fine-tuning of parameters. We next find that, for simple, quartic spinor self-interactions, non-singular cyclic cosmologies exist with reasonable parameter choices. These solutions might eventually be incorporated into a successful past- and future-eternal cosmological model free of singularities. In an Appendix, we discuss the classical treatment of spinors and argue that certain quantum systems might be approximated in terms of such fields.     

Rotating quantum cosmological models with wave fields

The quantum evolution of homogeneous rotating cosmological models of the Gödel type with spinor and scalar fields is considered within the framework of the formalism of superspace quantization. De Witt's equation for a rotating cosmological model is shown to take the form of a Schrödinger equation in which the role of time is played by the phase of the spinor field, and it becomes possible to determine correctly the probability that rotating models lacking an initial singularity exist.K. D. Ushinskii Yaroslav Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 41–44, January, 1993.     

Extended Einstein–Cartan theory à la Diakonov: The field equations

Diakonov formulated a model of a primordial Dirac spinor field interacting gravitationally within the geometric framework of the Poincaré gauge theory (PGT). Thus, the gravitational field variables are the orthonormal coframe (tetrad) and the Lorentz connection. A simple gravitational gauge Lagrangian is the Einstein–Cartan choice proportional to the curvature scalar plus a cosmological term. In Diakonov?s model the coframe is eliminated by expressing it in terms of the primordial spinor. We derive the corresponding field equations for the first time. We extend the Diakonov model by additionally eliminating the Lorentz connection, but keeping local Lorentz covariance intact. Then, if we drop the Einstein–Cartan term in the Lagrangian, a nonlinear Heisenberg type spinor equation is recovered in the lowest approximation.     

The First Quantization of Spin \frac32\frac{3}{2} Field in de Sitter Space

The de Sitter solution to the positive cosmological Einstein field equation has been viewed as a one-sheeted hyperboloid embedded in a five dimensional Minkowski space. To find Lagrangian equation of supersymmetry-group in the de Sitter space, the different spinor field’s quantization have been demonstrated. In this work, the first quantization of spin field in the time-space de Sitter universe with ambient space notation has been done.     

Early inflation,isotropization, and late time acceleration in a Bianchi type-I universe

A system of Einstein equations is solved for the Bianchi type-I metrics that describes a homogeneous and isotropic Universe. The system contains nonlinear differential equations of the second-order, which depend only on time. The method of solution is described, and the general form of the solution is found. Explicit analytical expressions are obtained in some particular cases. Numerical integration is used to describe possible solution types in the general case. The evolution of the Universe has been investigated in the presence of different types of sources, namely, a perfect fluid, a van der Waals fluid, the cosmological constant, quintessence, a Chaplygin gas, a modified quintessence, and a nonlinear spinor field. It is shown that the presence of a van der Waals fluid leads to inflation in the early stage of evolution, while the modified quintessence leads to a cyclic or oscillating Universe. It has been shown, that for some special choice of parameters the late time acceleration can be attributed to the influence of a nonlinear spinor field.     

CMB anisotropies and inflation from non-standard spinors

The apparent alignment of the cosmic microwave background multipoles on large scales challenges the standard cosmological model. Scalar field inflation is isotropic and cannot account for the observed alignment. We explore the imprints, a non-standard spinor driven inflation would leave on the cosmic microwave background anisotropies. We show it is natural to expect an anisotropic inflationary expansion of the Universe which has the effect of suppressing the low multipole amplitude of the primordial power spectrum, while at the same time to provide the usual inflationary features.     

Separation of Spin 0, 1/2, 1 Field Equations in Lemaître-Tolman-Bondi Cosmological Model with Cosmological Constant

The problem of variable separation of the scalar field equation is approached within the Lemaître-Tolman-Bondi (LTB) cosmological model with cosmological constant Λ. Parametric solutions of the cosmological Newton-like equation of the model are preliminary determined that result factorized in the parameter and in the radial dependence. The result holds on a sufficient condition that relates the two arbitrary integration functions of the model. The condition is of the same type of the one that ensures, in absence of cosmological term, the separability of the spin field equations for spin 0, 1/2, 1. It is then shown that the scalar field equation results automatically separable in the class of LTB models determined. The separated radial equation results independent of Λ, while the separated time equation strictly depends on Λ. The separability of the field equations is then checked to hold, in the same context, for spinor field equation of spin 1/2 and spin 1.     

Dual models in four dimensions with internal symmetries

Internal symmetries are introduced in dual models by compactifying N of the spacetime dimensions. Within the context of field theory, we have proved that this proposal is compatible with everyday experience. Dual models obtained in this way are shown to be mathematically self-consistent. In the dual spinor model a four-dimensional formalism is obtained by compactifying six of the directions, implying the existence of an SU(4) symmetry.     

Spinor brane

The thick brane model supported by a nonlinear spinor field is constructed. The different cases with the various values of the cosmological constant ${\Lambda \left( {l} < \\ =\\ > \right) 0}${\Lambda \left( \begin{array}{l} < \\ =\\ > \end{array} \right) 0} are investigated. It is shown that regular analytical spinor thick brane solutions with asymptotically Minkowski (at Λ = 0) or anti-de Sitter spacetimes (at Λ <  0) do exist.     

Smoothing regularization and particle creation in curved spacetime

A regularization procedure is given for the stress tensor of a quantized field in a background metric. This regularization is shown to be equivalent to a covariant renormalization of constants in the generalized Einstein equations. An example of the massive spinor field in Robertson-Walker universe is considered. Regular values of the stress tensor near the cosmological singularity are found.     

Spinor Field Nonlinearity and Space-Time Geometry

Within the scope of Bianchi type VI,VI0,V, III, I, LRSBI and FRW cosmological models we have studied the role of nonlinear spinor field on the evolution of the Universe and the spinor field itself. It was found that due to the presence of non-trivial non-diagonal components of the energy-momentum tensor of the spinor field in the anisotropic space-time, there occur some severe restrictions both on the metric functions and on the components of the spinor field. In this report we have considered a polynomial nonlinearity which is a function of invariants constructed from the bilinear spinor forms. It is found that in case of a Bianchi type-VI space-time, depending of the sign of self-coupling constants, the model allows either late time acceleration or oscillatory mode of evolution. In case of a Bianchi VI₀ type space-time due to the specific behavior of the spinor field we have two different scenarios. In one case the invariants constructed from bilinear spinor forms become trivial, thus giving rise to a massless and linear spinor field Lagrangian. This case is equivalent to the vacuum solution of the Bianchi VI₀ type space-time. The second case allows non-vanishing massive and nonlinear terms and depending on the sign of coupling constants gives rise to accelerating mode of expansion or the one that after obtaining some maximum value contracts and ends in big crunch, consequently generating space-time singularity. In case of a Bianchi type-V model there occur two possibilities. In one case we found that the metric functions are similar to each other. In this case the Universe expands with acceleration if the self-coupling constant is taken to be a positive one, whereas a negative coupling constant gives rise to a cyclic or periodic solution. In the second case the spinor mass and the spinor field nonlinearity vanish and the Universe expands linearly in time. In case of a Bianchi type-III model the space-time remains locally rotationally symmetric all the time, though the isotropy of space-time can be attained for a large proportionality constant. As far as evolution is concerned, depending on the sign of coupling constant the model allows both accelerated and oscillatory mode of expansion. A negative coupling constant leads to an oscillatory mode of expansion, whereas a positive coupling constant generates expanding Universe with late time acceleration. Both deceleration parameter and EoS parameter in this case vary with time and are in agreement with modern concept of space-time evolution. In case of a Bianchi type-I space-time the non-diagonal components lead to three different possibilities. In case of a full BI space-time we find that the spinor field nonlinearity and the massive term vanish, hence the spinor field Lagrangian becomes massless and linear. In two other cases the space-time evolves into either LRSBI or FRW Universe. If we consider a locally rotationally symmetric BI(LRSBI) model, neither the mass term nor the spinor field nonlinearity vanishes. In this case depending on the sign of coupling constant we have either late time accelerated mode of expansion or oscillatory mode of evolution. In this case for an expanding Universe we have asymptotical isotropization. Finally, in case of a FRW model neither the mass term nor the spinor field nonlinearity vanishes. Like in LRSBI case we have either late time acceleration or cyclic mode of evolution. These findings allow us to conclude that the spinor field is very sensitive to the gravitational one.