Spinor field in Bianchi type-$IX$ space-time

TL;DR

This paper investigates the influence of spinor fields on Bianchi type-IX universe models, revealing that non-diagonal energy-momentum components restrict the geometry, resulting in either isotropic or locally rotationally symmetric space-times with distinct expansion behaviors.

Contribution

It demonstrates how spinor fields affect Bianchi type-IX cosmologies, especially regarding energy-momentum tensor components and resulting geometric restrictions, including oscillatory and rapid expansion modes.

Findings

Non-diagonal energy-momentum tensor components impose geometric restrictions.

Positive bb
gives oscillatory expansion mode.

Trivial bb
leads to rapid early expansion.

Abstract

Within the scope of Bianchi type-$IX$ we have studied the role of spinor field in the evolution of the Universe. It is found that unlike the diagonal Bianchi models in this case the components of energy-momentum tensor of spinor field along the principal axis are not the same, i.e. $T_1^1 \ne T_2^2 \ne T_3^3$, even in absence of spinor field nonlinearity. The presence of nontrivial non-diagonal components of energy-momentum tensor of the spinor field imposes severe restrictions both on geometry of space-time and on the spinor field itself. As a result the space-time turns out to be either locally rotationally symmetric or isotropic. In this paper we considered the Bianchi type-$IX$ space-time both for a trivial $b$, that corresponds to standard $BIX$ and the one with a non-trivial $b$. It was found that a positive $\lambda_1$ gives rise to an oscillatory mode of expansion, while a trivial $\lambda_1$ leads to rapid expansion at the early stage of evolution.

Equations

$$L=\frac{\imath}{2}\biggl{[}\bar{\psi}\gamma^{\mu}\nabla_{\mu}\psi-\nabla_{\mu}\bar{\psi}\gamma^{\mu}\psi\biggr{]}-m_{\rm sp}\bar{\psi}\psi-F,$$

$${\nabla }_{\mu }\psi$$

$${\nabla }_{\mu }\bar{\psi }$$

$${\Gamma }_{\mu }=\frac{1}{4}{\bar{\gamma }}_a{\gamma }^{\nu }{\partial }_{\mu }{e}_{\nu }^{(a)}-\frac{1}{4}{\gamma }_{\rho }{\gamma }^{\nu }{\Gamma }_{\mu \nu }^{\rho },$$

$$T_{\mu }^{\rho }$$

$$d{s}^2=d{t}^2-a_1^2(t)d{x}_1^2-[h^2(x_3)a_1^2(t)+f^2(x_3)a_2^2(t)]d{x}_2^2-a_3^{2}d{x}_3^2+2a_1^2(t)h(x_3)d{x}_{1}d{x}_2,$$

$$\delta =-\frac{f^{\prime \prime }}{f}=1\Rightarrow f^{\prime \prime }+f=0.$$

$$g_{\mu \nu }=e_{\mu }^{(a)}{e}_{\nu }^{(b)}{\eta }_{ab},$$

$$e_0^{(0)}=1,e_1^{(1)}=a_1,e_2^{(2)}=a_{2}f,e_3^{(3)}=a_3,e_2^{(1)}=-a_{1}h.$$

$${\gamma }_{\mu }=e_{\mu }^{(a)}{\bar{\gamma }}_a,{\gamma }^{\nu }=e_{(b)}^{\nu }{\bar{\gamma }}^b,$$

$$e_{\mu }^{(a)}{e}_{(b)}^{\mu }={\delta }_b^a,e_{\mu }^{(a)}{e}_{(a)}^{\nu }={\delta }_{\mu }^{\nu },$$

$${\gamma }_0$$

$${\gamma }_2$$

$${\bar{\gamma }}^0={\bar{\gamma }}_0,{\bar{\gamma }}^1=-{\bar{\gamma }}_1,{\bar{\gamma }}^2=-{\bar{\gamma }}_2,{\bar{\gamma }}^3=-{\bar{\gamma }}_3.$$

$${\gamma }^0={\bar{\gamma }}^0,{\gamma }^1=\frac{1}{a_1}{\bar{\gamma }}^1+\frac{h}{a_{2}f}{\bar{\gamma }}^2,{\gamma }^2=\frac{1}{a_{2}f}{\bar{\gamma }}^2,{\gamma }^3=\frac{1}{a_3}{\bar{\gamma }}^3.$$

$${\Gamma }_1$$

$${\Gamma }_2$$

$${\Gamma }_3$$

$${\Gamma }_0$$

$${\gamma }^{\mu }{\Gamma }_{\mu }$$

$${\Gamma }_{\mu }{\gamma }^{\mu }$$

$$V=a_1{a}_2{a}_3$$

$$ı{\gamma }^{\mu }{\nabla }_{\mu }\psi -m_{\mathrm{sp}}\psi -\mathcal{D}\psi -ı\mathcal{G}{\gamma }^5\psi$$

$$ı{\nabla }_{\mu }\bar{\psi }{\gamma }^{\mu }+m_{\mathrm{sp}}\bar{\psi }+\mathcal{D}\bar{\psi }+ı\mathcal{G}\bar{\psi }{\gamma }^5$$

$$ı{\bar{\gamma }}^0\dot{\psi }+\frac{ı}{2}\frac{\dot{V}}{V}{\bar{\gamma }}^0\psi +\frac{1}{4}\frac{a_1{h}^{\prime }}{a_2{a}_{3}f}{\bar{\gamma }}^5{\bar{\gamma }}^0\psi +\frac{ı}{2}\frac{f^{\prime }}{a_{3}f}{\bar{\gamma }}^3\psi -\left[m_{\mathrm{sp}}+\mathcal{D}\right]\psi -ı\mathcal{G}{\bar{\gamma }}^5\psi$$

$$ı\dot{\bar{\psi }}{\bar{\gamma }}^0+\frac{ı}{2}\frac{\dot{V}}{V}\bar{\psi }{\bar{\gamma }}^0-\frac{1}{4}\frac{a_1{h}^{\prime }}{a_2{a}_{3}f}\bar{\psi }{\bar{\gamma }}^5{\bar{\gamma }}^0+\frac{ı}{2}\frac{f^{\prime }}{a_{3}f}\bar{\psi }{\bar{\gamma }}^3+\left[m_{\mathrm{sp}}+\mathcal{D}\right]\bar{\psi }+ı\mathcal{G}\bar{\psi }{\bar{\gamma }}^5$$

$$L$$

$$T_0^0$$

$$T_1^1$$

$$T_2^2$$

$$T_3^3$$

$$T_2^1$$

$$T_1^2$$

$$T_3^2$$

$$T_2^3$$

$$T_3^1$$

$$T_1^3$$

$$T_1^0$$

$$T_0^1$$

$$T_2^0$$

Full text

Spinor field, dark energy, anisotropic cosmological models, isotropization

I Introduction

Nonlinear spinor fields plays a significant role in explaining the evolution of the Universe at different its stages. It was shown by a number of authors that the introduction of nonlinear spinor field into the system can (i) give rise to a singularity-free Universe; (ii) accelerate the isotropization process of initially anisotropic space-time and (iii) generate late time acceleration of space-time expansion Saha1997GRG ; Saha1997JMP ; Saha2001PRD ; Saha2004aPRD ; Saha2004bPRD ; PopPLB ; PopPRD ; PopGREG ; FabIJTP ; Saha2006ECAA ; kremer1 ; Saha2006PRD ; Saha2006GnC ; Saha2007RRP ; Saha2009aECAA ; ELKO ; FabGRG . Moreover, it can simulate different types of dark energy and perfect fluid Krechet ; Saha2010CEJP ; Saha2010RRP ; Saha2011APSS ; Saha2012IJTP . Recently it was also found that the presence of non-diagonal components of energy-momentum tensor of the spinor field imposes severe restrictions to the space-time geometry as well Saha2015CJP ; Saha2015CnJP ; Saha2016IJTP ; Saha2016EPJP .

In this paper we plan to extend our previous study to Bianchi type-$IX$ cosmological model. One of the reasons to consider this model is familiar solutions like the FRW Universe with positive curvature, the de-sitter Universe , the Taub-Nut solutions etc. are of Bianchi type-IX space-times. It should be noted that due to its importance many authors have studied the evolution of the Universe within the scope of a Bianchi type-$IX$ model. Bali et. al. have studied the Bianchi type-$IX$ string cosmological models filled with bulk viscous fluid bali2001 ; bali2003 , whereas such a model for perfect fluid was investigated by Tyagi et. al. in tyagi . Analogous system with a time varying $\Lambda$-term was studied in pradhan . A scalar tensor theory of gravitation within the framework of Bianchi type-$IX$ was studied by Reddy and Naidu reddi .

II Basic equation

Let us consider the spinor field Lagrangian in the form

[TABLE]

where the nonlinear term $F$ describes the self-interaction of a spinor field and can be presented as some arbitrary functions of invariants generated from the real bilinear forms of a spinor field. We consider the case when $F=F(K)$ with $K$ taking one of the followings values $\{I,\,J,\,I+J,\,I-J\}$. By virtue of Fierz theorem this comes out to be the most general form of spinor field nonlinearity.

Here $\nabla_{\mu}$ covariant derivative of the spinor field having the form

[TABLE]

where $\Gamma_{\mu}$ is the spinor affine connection defined as

[TABLE]

where $\bar{\gamma}_{a}$ are the Dirac matrices in flat space-time, $\gamma_{\nu}$ are the Dirac matrices in curved space-time, $e^{(a)}_{\nu}$ are the tetrad and $\Gamma^{\rho}_{\mu\nu}$ are the Christoffel symbols.

The energy momentum tensor of the spinor field is given by

[TABLE]

Bianchi type $IX$ space-time ($BIX$) is given by

[TABLE]

with $a_{1},\,a_{2},\,a_{3}$ being the functions of $t$. Here $h$ and $f$ are the functions of $x_{3}$. Here we consider the Bianchi type $IX$ space-time, which imposes the following restriction of $f$, namely

[TABLE]

It should be noted that it is customary to assume $f(x_{3})=\sin(x_{3})$ and $h(x_{3})=\cos(x_{3})$. We don’t write these concrete expressions for the functions $f(x_{3})$ and $h(x_{3})$ here, but do it later in due course.

To find the spinor affine connection (3) we have to know the tetrad corresponding to the metric (5). Exploiting the well known relation

[TABLE]

we choose the tetrad corresponding to (5) as follows:

[TABLE]

From

[TABLE]

such that

[TABLE]

one now finds

[TABLE]

where we take into account that

[TABLE]

Using the laws of raising and lowering the indices one also finds

[TABLE]

Now we are ready to compute spinor affine connections using (3):

[TABLE]

Then one finds

[TABLE]

where we introduce the volume scale

[TABLE]

and $\bar{\gamma}^{5}=-\imath\bar{\gamma}^{0}\bar{\gamma}^{1}\bar{\gamma}^{2}\bar{\gamma}^{3}.$

The spinor field equations corresponding to the Lagrangian (1) has the form

[TABLE]

where ${\cal D}=2SF_{K}K_{I}$ and ${\cal G}=2PF_{K}K_{J}$.

In view of (14a) and (14b) the system (16) can be rewritten as

[TABLE]

In view of (16) the spinor field Lagrangian can be written as

[TABLE]

from (4) one finds the following components of energy-momentum tensor

[TABLE]

From (19) we see that the spinor field distribution along the main axis is anisotropic, i.e. $T_{1}^{1}\neq T_{2}^{2}\neq T_{3}^{3}$ and these components do not vanish even in absence of spinor field nonlinearity.

The components of Einstein tensor corresponding to (5) are

[TABLE]

From (20) one finds the following relations between its components:

[TABLE]

From (19) it can be shown that

[TABLE]

Moreover from (19) it follows that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Then the system of Einstein equations

[TABLE]

on account of linearly dependent components takes the form

[TABLE]

Then in view of $\frac{f^{\prime}}{f}\neq 0$ from (28f) we find $\left(\frac{\dot{a}_{2}}{a_{2}}-\frac{\dot{a}_{3}}{a_{3}}\right)=0$. On the other hand for same reason (28j) yields $A^{1}=0$, whereas inserting $A^{1}=0$ into (28i) we obtain $A^{2}=0$. Thus in this case from (28f) - (28j) we have

[TABLE]

In view of $A^{2}=0$ the equation (28h) yields two possibilities:

[TABLE]

which means the model is rotationally symmetric, or

[TABLE]

which means the model is isotropic. Note that in case of diagonal energy-momentum tensor we have strictly locally rotationally symmetric space-time SahaGnC2013 .

It should be noted that for the diagonal Bianchi models the volume scale $V$ plays crucial role in the evolution of the Universe. $V$ plays important role for non-diagonal Bianchi models too. So let us now write the euqtion for $V$. Summation of (28), (28), (28c) and 3 times (28d) gives

[TABLE]

As one sees, to find the solution of (32) one need to know $f$, $h$, $A_{0}$, the spinor field nonlinearity as well as the metric functions, or at least their expressions in terms of $V$.

To find the metric functions in terms of $V$ we use the proportionality condition. In doing so let us compute the expansion and shear corresponding to metric (5). Let the four-velocity is given by $u^{\mu}=(1,\,0,\,0,\,0)$. Then for the expansion we have

[TABLE]

The shear is given by

[TABLE]

Taking into account that $u_{\mu}=u^{\nu}g_{\nu\mu}=u^{0}g_{0\mu}=(1,\,0,\,0,\,0)$,  $u_{\alpha;\nu}=u_{\alpha,\nu}-\Gamma^{\mu}_{\alpha\nu}u_{\mu}=-\Gamma^{0}_{\alpha\nu}$,  $P_{\alpha\beta}=g_{\alpha\beta}-u_{\alpha}u_{\beta}$ and $P^{\alpha}_{\beta}=\delta^{\alpha}_{\beta}-u^{\alpha}u_{\beta}$ from (34) we find

[TABLE]

Using (35) we find

[TABLE]

Thus we see that the diagonal components of shear tensor $\sigma_{\alpha}^{\beta}$ does not depend on $f$ and $h$.

Let us assume the proportionality condition

[TABLE]

which after inserting (36a) and (33) gives

[TABLE]

In view of $\vartheta=\dot{V}/V$ from (38) in one hand we find

[TABLE]

with

[TABLE]

On the other hand from (28f) one finds

[TABLE]

Then from (15) one finds

[TABLE]

So finally we can write the expressions for metric functions in terms of $V$ as

[TABLE]

In this concrete case we have $X_{1}=q_{2}$,   $X_{2}=\sqrt{{q_{3}}/{q_{2}}}$,    $X_{3}={1}/{\sqrt{q_{2}q_{3}}}$,   $Y_{1}=q_{1}+1/3,$   and   $Y_{2}=Y_{3}=1/3-q_{1}/2.$

From (43) one finds that ${a_{1}}/{(a_{2}a_{3})}=q_{2}^{2}V^{2q_{1}-1/3}$. Then the equation for $V$ (32) can be rewritten as

[TABLE]

From the spinor field equations (16) we find that the bilinear forms of the spinor field in this case obey the following system of equations:

[TABLE]

From (45) it can be easily shown that

[TABLE]

On the other hand from Fierz theorem we have

[TABLE]

Now taking into account that $A^{1}=0$ and $A^{2}=0$ from (47) one finds

[TABLE]

Then inserting (48) into (46) one finds

[TABLE]

It should be emphasized that in case of diagonal Bianchi space-time we obtain the expression (49) only when the spinor field nonlinearity depends on $K=I=S^{2}$. Thus we see that in case of $BIX$ space-time $S=V_{0}/V$ independent to our choice of $K$.

Let us now see, what happens if $K$ takes any of the following expressions $\{J,\,\,I+J,\,\,I-J\}$. In case of diagonal Bianchi models exact expressions were found for massless spinor field only. So here we consider massless spinor field. Then in case of $K=J$ from (45b) we find

[TABLE]

From (50) one can formally express $K=J=P^{2}$ in terms of $V$.

In case of $K=I+J$ we have

[TABLE]

Summation of (51a) multiplied by $S_{0}$ and (51b) multiplied by $P_{0}$ gives

[TABLE]

Further in view of $S_{0}=V_{0}$ from (52) one finds equation for $P_{0}$ analogous to (50). Knowing $J=P^{2}$ we obtain the expression for $K=I+J$ in terms of $V$.

Finally, for $K=I-J$ we have

[TABLE]

Subtraction of (53b) multiplied by $P_{0}$ from (53a) multiplied by $S_{0}$ gives

[TABLE]

with the solution

[TABLE]

Here it is interesting note that in absence of spinor field nonlinearity the system (45) takes the form

[TABLE]

One can easily find that this system too allows the first integral (46). Moreover, in case of massless spinor field we find

[TABLE]

Now in view of (29) and (49) Eq. (57) can be rewritten as

[TABLE]

which gives

[TABLE]

Hence the behavior of invariants, constructed from bilinear spinor forms categorically differs from that we obtain for diagonal Bianchi models.

Let us now recall that the functions $f(x_{3})$ can be concretize using the restriction imposed on it, namely, from the (6) one finds the following solutions for $f:f=\sin(x_{3})$ and $f=\cos(x_{3})$. Following a number of authors we choose $f=\sin(x_{3})$. As far as $h$ is concerned, it should be determined from (28e). In doing so we go back to (28e) which can be rewritten as

[TABLE]

In view of (45d) this equations can be written as

[TABLE]

Now the left hand side of (61) depends of $x_{3}$ only, while the right hand side depends on only $t$. So we can finally write the following system

[TABLE]

From the foregoing equations in view of $f=\sin(x_{3})$ and (43) we finally obtain

[TABLE]

So finally in view of (45d) equation (44) $V$ can be rewritten as

[TABLE]

To find the solution to the equation (64) we have to give the concrete form of spinor field nonlinearity. Following some previous papers, we choose the nonlinearity to be the function of $S$ only, having the form

[TABLE]

Recently, this type of nonlinearity was considered in a number of papers Saha2015CJP ; Saha2015CnJP ; Saha2016IJTP ; Saha2016EPJP .

In what follows, we thoroughly study the cases with trivial and non-tribal $b$.

Case 1. $b=0$

Here we consider the simplest possible case setting $b=0.$ This case coincides with the one that corresponds to the diagonal energy-momentum tensor.

From (6) we have $f=\sin(x_{3})$. Solving (62a) in this case one finds $h=\cos(x_{3})$. It should be emphasized that this form of Bianchi type-$IX$ metric with $f=\sin(x_{3})$ and $h=\cos(x_{3})$ is generally considered in literature. In this case for $A_{0}^{3}$ from (63b) we also find

[TABLE]

with the solution

[TABLE]

Now on account of $S=V_{0}/V$ equation (64) together with (63b) can be written as

[TABLE]

[TABLE]

In what follows, we solve the foregoing equation numerically. For this reason we first rewrite it as a system of equations in the following way:

For simplicity we consider only three terms of the sum. We set $n_{k}=n_{0}:1-2n_{0}=0$ which gives $n_{0}=1/2$. In this case the corresponding term can be added with the mass term. We assume that $q_{1}$ is a positive quantity, so that $4q_{1}+1/3$ is positive too. For the nonlinear term to be dominant at large time, we set $n_{k}=n_{1}:1-2n_{1}>4q_{1}+1/3$, i.e., $n_{1}<1/3-2q_{1}$. And finally, for the nonlinear tern to be dominant at the early stage we set $n_{k}=n_{2}:1-2n_{2}<0$, i.e., $n_{2}>1/2$. Since we are interested in qualitative picture of evolution, let us set $q_{2}=1,\,q_{3}=1$ and $\kappa=1$. We also assume $V_{0}=1$. Then we have

[TABLE]

We set $m_{\rm sp}=1$ and $l_{0}=2$. As far as $q_{1}$, $n_{1}$ and $n_{2}$ are concerned, in line of our previous discussions we choose them in such a way that the power of nonlinear term in the equations become integer. We have also studied the case for some different values, but they didn’t give any principally different picture. We choose $q_{1}=2/3$,  $n_{1}=-3/2<1/3-2q-1=-1$ and $n_{2}=3/2>1/2$. It should be noted that we have taken some others value for $q_{1}$ such as $q_{1}=-1$, but it does not give qualitatively different picture. We have also set $x_{3}=[0,\,\pi]=k\pi/5$ with step $k=0..5$. Finally we have considered time span $[0,\,2]$ with step size $0.001$. Here we consider different values of $\lambda_{i}$ both positive and negative. We choose the initial values for $V(0)=1$, $Y(0)=\dot{V}(0)=0.1$, and $A_{0}^{3}(0)=1$, respectively.

In Figs. 1 and 2 we have plotted the phase diagram of $[V,\,\dot{V},\,A_{0}^{3}]$ for both positive and negative $\lambda_{2}$, respectively. In both cases $\lambda_{1}=1$. Analogical picture was found for $\lambda_{2}=0$ and $\lambda_{1}=1$.

In Figs. 3 and 4 evolution of $V$ corresponding to Figs. 1 and 2 are demonstrated. As one sees, in both cases we have oscillatory mode of expansion.

In Figs. 5 and 6 we have illustrated the evolution of $V$ for $\lambda_{1}=0$ and $\lambda_{2}=0$ (linear spinor field) and for $\lambda_{1}=0$ and $\lambda_{2}=1$, respectively. This shows that in case of $\lambda_{1}=0$ we have a rapid expansion of $V$ at a very early stage.

In the Figures each color corresponds to a concrete value of $x_{3}=k\pi/5$, namely, red, green, yellow, blue, magenta and black color corresponds to $k=0,1,2,3,4,5$

Case 2. $b\neq 0$

Let us consider the case with non-trivial $b$. Inserting $f=\sin(x_{3})$ into (62a) one finds

[TABLE]

The equations (64) together with (63b) in this case can be written as

[TABLE]

where we denote

[TABLE]

In what follows, we solve the foregoing system numerically for $b=1$ with all other parameters taking same value as corresponding cases for a trivial $b$.

In Figs. 7 and 8 we have plotted the phase diagram of $[V,\,\dot{V},\,A_{0}^{3}]$ for both positive and negative $\lambda_{2}$, respectively. In both cases $\lambda_{1}=1$. Analogical picture was found for $\lambda_{2}=0$ and $\lambda_{1}=1$. As one sees, while in case of a trivial $b$ we have a focus like phase diagram, in this case with non-zero $b$ the phase diagram is a spiral.

In Figs. 9 and 10 evolution of $V$ corresponding Figs. 7 and 8 are demonstrated. As one sees, in both cases we have oscillatory mode of expansion.

In Figs. 11 and 12 we have illustrated the evolution of $V$ for $\lambda_{1}=0$ and $\lambda_{2}=0$ (linear spinor field) and for $\lambda_{1}=0$ and $\lambda_{2}=1$, respectively. This shows that in case of $\lambda_{1}=0$ we have a rapid expansion of $V$ at a very early stage.

III Conclusion

Within the scope of Bianchi type-$IX$ cosmological model we have studied the role of spinor field in the evolution of the Universe. It is found that unlike the diagonal Bianchi models in this case the components of energy-momentum tensor of the spinor field along the principal axis are not the same, i.e. $T_{1}^{1}\neq T_{2}^{2}\neq T_{3}^{3}$, even in absence of spinor field nonlinearity. The presence of nontrivial non-diagonal components of energy-momentum tensor of the spinor field imposes severe restrictions both on geometry of space-time and on the spinor field itself. As a result the space-time turns out to be either locally rotationally symmetric or isotropic. In this paper we considered the Bianchi type-$IX$ space-time both for a trivial $b$, that corresponds to standard $BIX$ and the one with a non-trivial $b$. It was found that a positive $\lambda_{1}$ gives rise to an oscillatory mode of expansion, while a trivial $\lambda_{1}$ leads to rapid expansion at the early stage of evolution.

**Acknowledgments

**This work is supported in part by a joint Romanian-LIT, JINR, Dubna Research Project, theme no. 05-6-1119-2014/2016.

I would also like to thank Victor Rikhvitsky for constant help in numerical solutions.