Non-minimally coupled nonlinear spinor field in FRW cosmology

TL;DR

This paper investigates how a non-minimally coupled nonlinear spinor field influences the evolution of the universe in FRW cosmology, highlighting its role in rapid expansion or dark energy domination depending on the nonlinearity type.

Contribution

It introduces analysis of non-minimal coupling of nonlinear spinor fields in FRW cosmology, exploring their effects on universe evolution with different nonlinearities.

Findings

Non-minimal coupling leads to rapid expansion with radiation-like nonlinearity.

Dark energy-like nonlinearity results in universe evolution dominated by the spinor field.

Differences between minimal and non-minimal cases are negligible for dark energy scenarios.

Abstract

Within the scope of a FRW cosmological model we have studied the role of spinor field in the evolution of the Universe when it is non-minimally coupled to the gravitational one. We have considered a few types of nonlinearity. It was found that if the spinor field nonlinearity describes an ordinary matter such as radiation, the presence of non-minimality becomes essential and leads to the rapid expansion of the Universe, whereas if the spinor field nonlinearity describes a dark energy, the evolution of the Universe is dominated by it and the difference between the minimal and non-minimally coupled cases become almost indistinguishable.

Figures

Equations

$$S=\int \sqrt{-g}\left[\left({\kappa }_1+{\lambda }_{1}S\right)R+L_{\mathrm{sp}}\right]d\Omega ,{\kappa }_1=\frac{1}{2\kappa }.$$

$$L_{\mathrm{sp}}=\frac{ı}{2}\left[\bar{\psi }{\gamma }^{\mu }{\nabla }_{\mu }\psi -{\nabla }_{\mu }\bar{\psi }{\gamma }^{\mu }\psi \right]-m\bar{\psi }\psi -\lambda F(S).$$

$${\nabla }_{\mu }\psi ={\partial }_{\mu }\psi -{\Gamma }_{\mu }\psi ,{\nabla }_{\mu }\bar{\psi }={\partial }_{\mu }\bar{\psi }+\bar{\psi }{\Gamma }_{\mu }.$$

$$R_{\mu }^{\nu }-\frac{1}{2}{\delta }_{\mu }^{\nu }R=\frac{1}{\left({\kappa }_1+{\lambda }_{1}S\right)}\left[T_{\mu }^{\nu }+{\lambda }_1\left(g^{\nu \tau }{\nabla }_{\mu }{\nabla }_{\tau }-{\delta }_{\mu }^{\nu }\square \right)S\right].$$

$$ı{\gamma }^{\mu }{\nabla }_{\mu }\psi -m\psi -\lambda F_S\psi +{\lambda }_{1}R\psi$$

$$ı{\nabla }_{\mu }\bar{\psi }{\gamma }^{\mu }+m\bar{\psi }+\lambda F_S\bar{\psi }-{\lambda }_{1}R\bar{\psi }$$

$$d{s}^2=d{t}^2-a^2\left(d{x}_1^2+d{x}_2^2+d{x}_3^2\right),$$

$$e_0^{(0)}=1,e_i^{(i)}=a,i=1,2,3.$$

$${\Gamma }_{\mu }$$

$${\Gamma }_0=0,{\Gamma }_1=\frac{\dot{a}}{2}{\bar{\gamma }}^1{\bar{\gamma }}^0,{\Gamma }_2=\frac{\dot{a}}{2}{\bar{\gamma }}^2{\bar{\gamma }}^0,{\Gamma }_3=\frac{\dot{a}}{2}{\bar{\gamma }}^3{\bar{\gamma }}^0.$$

$$ı{\bar{\gamma }}^0\left(\dot{\psi }+\frac{3}{2}\frac{\dot{a}}{a}\psi \right)-m\psi -\lambda F_S\psi +{\lambda }_{1}R\psi$$

$$ı\left(\dot{\bar{\psi }}+\frac{3}{2}\frac{\dot{a}}{a}\bar{\psi }\right){\bar{\gamma }}^0+m\bar{\psi }+\lambda F_S\bar{\psi }-{\lambda }_{1}R\bar{\psi }$$

$$S=\frac{C_0}{a^3},C_0=Const.$$

$$T_{\mu}^{\,\,\,\rho}=\frac{\imath}{4}g^{\rho\nu}\biggl{(}\bar{\psi}\gamma_{\mu}\nabla_{\nu}\psi+\bar{\psi}\gamma_{\nu}\nabla_{\mu}\psi-\nabla_{\mu}\bar{\psi}\gamma_{\nu}\psi-\nabla_{\nu}\bar{\psi}\gamma_{\mu}\psi\biggr{)}\,-\delta_{\mu}^{\rho}L_{\rm sp},$$

$$T_0^0$$

$$T_1^1$$

$${\nabla }_{\mu }{\nabla }_{\nu }S$$

$$\square S$$

$$2\frac{\ddot{a}}{a}+\frac{{\dot{a}}^2}{a^2}$$

$$3\frac{{\dot{a}}^2}{a^2}$$

$$\dot{a}=\sqrt{\frac{\left(mS+\lambda F\right)}{3\left({\kappa }_1-2{\lambda }_{1}S\right)}}a=\sqrt{\frac{\left(m{C}_0+\lambda a^{3}F\right)}{3\left({\kappa }_1{a}^3-2{\lambda }_1{C}_0\right)}}a.$$

$$\ddot{a}=\left[\frac{\lambda \left(F-S{F}_S\right)}{\left(2{\kappa }_1-{\lambda }_{1}S\right)}-\frac{\left({\kappa }_1+7{\lambda }_{1}S\right)\left(mS+\lambda F\right)}{3\left(2{\kappa }_1-{\lambda }_{1}S\right)\left({\kappa }_1-2{\lambda }_{1}S\right)}\right]a,$$

$$\ddot{a}=\left[\frac{\lambda \left(a^{3}F-C_0{F}_S\right)}{\left(2{\kappa }_1{a}^3-{\lambda }_1{C}_0\right)}-\frac{\left({\kappa }_1{a}^3+7{\lambda }_1{C}_0\right)\left(m{C}_0+\lambda a^{3}F\right)}{3\left(2{\kappa }_1{a}^3-{\lambda }_1{C}_0\right)\left({\kappa }_1{a}^3-2{\lambda }_1{C}_0\right)}\right]a.$$

$$F=S^{1+W},W=1/3.$$

$$F=S^{1+W},W<-1/3.$$

$$F={\left(A+S^{\left(1+\alpha \right)}\right)}^{1/\left(1+\alpha \right)},$$

$$F=S^{1+W}+\frac{W}{1+W}{\epsilon }_{\mathrm{cr}},$$

$$F={\left(\frac{A}{1+W}+S^{\left(1+W\right)\left(1+\alpha \right)}\right)}^{1/\left(1+\alpha \right)},$$

Full text

Spinor field, dark energy, anisotropic cosmological models, isotropization

I Introduction

The discovery and further confirmation of the accelerated expansion of the Universe led to reconsider the existing theories of cosmology. One of the straight forward ways was to introduce some additional component into the right hand side of the Einstein equations with negative pressure which would work as repulsive force thus giving rise to the accelerated mode of expansion. A number of models were proposed by different authors. Model exploiting the spinor field was one of them. For more than two decades spinor field is being widely used in cosmology mainly thanks to its specific behavior in presence of gravitational field. In a number of papers the authors have shown that the nonlinear spinor field can give rise to regular solutions as well as explain the late-time accelerated mode of expansion of the Universe Saha2001PRD ; Saha2006PRD ; Saha2009aECAA ; ELKO ; kremer ; Saha2018ECAA . But most of those papers considered the minimal coupling of spinor and gravitational field. It should be noted that along with the dark energy models many authors suggested the modification of the Einstein equations itself. Scalar tensor theory citeBrans-Dicke, theory with non-minimal coupling, $F(R)$ theory Starobin , $F(R,\,T)$ theory with $T$ being the trace of energy-momentum tensor (EMT) Harko , $F(T)$ theory with $T$ being the torsion Li , $f(G)$ theory are the few to name. The motivation behind this research was to study the influence of spinor field in the evolution of the universe when it is non-minimally coupled to the gravitational field. Since spinor field is more sensitive to the gravitational field than the scalar one, in our view it may give rise to some unexpected results. Recently, Carloni et al Astro-Phys/1811.10300 has considered non-minimally coupled spinor field with the gravitational one. Non-minimally coupled spinor and gravitational fields within the scope of Bianchi type-I metric was studied in SpinBInm . In this report we plan to continue that study for an isotropic and homogeneous space-time given by a FRW metric.

II Basic equations

We consider the action in the form

[TABLE]

where $S=\bar{\psi}\psi$ is a scalar constructed from spinor fields, $\lambda_{1}$ is the coupling constant. Here $\kappa$ is the Einstein’s constant defined as $\kappa=8\pi G$, with $G$ being the Newton’s gravitational constant. The spinor field Lagrangian takes the form

[TABLE]

Note that in general the nonlinear term $F$ may be the arbitrary function of invariant $K$ which takes one of the following expressions: $\{I,\,J,\,I+J,\,I-J\}$. Here $I=\bar{\psi}\psi$ and $J=\imath\bar{\psi}\bar{\gamma}^{5}\psi$. Here $m$ is the spinor mass. $\lambda$ is the self coupling constant that can be positive or negative. Here $\nabla_{\mu}$ is the covariant derivative of the spinor field

[TABLE]

Here $\Gamma_{\mu}$ is the spinor affine connection.

Variation with respect to metric functions give SpinBInm

[TABLE]

where $T_{\mu}^{\nu}$ is the energy-momentum tensor of the spinor field. The corresponding equations for spinor field we find varying the action with respect to $\psi$ and $\bar{\psi}$ SpinBInm

[TABLE]

From (5) one finds that $L_{\rm sp}=SF_{S}-F.$

We consider the isotropic FRW space-time is given by

[TABLE]

with the scale factor $a$ is the functions of time only.

For the metric (6) we choose the tetrad such that they have the following nontrivial components:

[TABLE]

From

[TABLE]

where $\left[a,b\right]=ab-ba$ one finds the following expressions for spinor affine connections:

[TABLE]

In (8) $\gamma_{\beta}=e_{\beta}^{(b)}\bar{\gamma}_{b}$ and $\gamma^{\alpha}=e^{\alpha}_{(a)}\bar{\gamma}^{a}$ are the Dirac matrices in curve space-time and $e^{\alpha}_{(a)}$ and $e_{\beta}^{(b)}$ are the tetrad vectors.

We consider the case when the spinor field depends on $t$ only. The spinor field equations in this case read

[TABLE]

where we denote $F_{S}=dF/dS.$ From (10) one easily finds

[TABLE]

The energy-momentum tensor of the spinor field

[TABLE]

in our case gives the following nontrivial components SpinBInm

[TABLE]

Taking into account that in our case, $\Box S=\ddot{S}+3\frac{\dot{a}}{a}\dot{S}$, in view of

[TABLE]

for the metric (6) from (4) we find

[TABLE]

In a recent paper SpinBInm it was shown that if instead of ordinary scalar we deal with $S=\bar{\psi}\psi$ as component by component, then for the second derivative in (14a) we get an additional term, namely $\bar{\psi}\Gamma_{\nu}\partial_{\mu}\psi-\partial_{\mu}\bar{\psi}\Gamma_{\nu}\psi$. In our case $\psi$ is a function of $t$, moreover $\Gamma_{0}=0$ and $\Gamma_{i}=(\dot{a}/2)\bar{\gamma}^{i}\bar{\gamma}^{0}$. On account of that we can write $\bar{\psi}\Gamma_{\nu}\partial_{\mu}\psi-\partial_{\mu}\bar{\psi}\Gamma_{\nu}\psi=\frac{\dot{a}}{2}\left(\bar{\psi}\bar{\gamma}^{i}\bar{\gamma}^{0}\dot{\psi}-\dot{\bar{\psi}}\bar{\gamma}^{i}\bar{\gamma}^{0}\psi\right).$ Further multiplying (10a) by $\bar{\psi}\bar{\gamma}^{i}$ from the left and (10b) by $\bar{\gamma}^{i}\psi$ from the right and adding them we find $\left(\bar{\psi}\bar{\gamma}^{i}\bar{\gamma}^{0}\dot{\psi}-\dot{\bar{\psi}}\bar{\gamma}^{i}\bar{\gamma}^{0}\psi\right)\equiv 0.$ Thus in this case we can deal with $S$ as an ordinary scalar.

In view of (11) from (15b) we find

[TABLE]

Further from (11) we find that $\ddot{S}=-3\frac{\ddot{a}}{a}S+12\frac{{\dot{a}}^{2}}{a^{2}}S$. Then on account of (16) we rewrite (15a) as

[TABLE]

in view of (11) which can be written as

[TABLE]

III Numerical analysis

In what follows we solve this equation numerically. For simplicity we set $m=1$, $\kappa_{1}=1$ and $C_{0}=1$. We consider two cases, one with non-minimal coupling, another with minimal coupling, so that the role of non-minimal coupling becomes clear.

Case 1 In this case we consider the non-minimal coupling with nonlinear term (plotted in solid blue line) setting $\lambda_{1}=1,\,\lambda=1$.

Case 2 As a second case we consider nonlinear spinor field with minimal coupling setting $\lambda_{1}=0,\,\lambda=1$ (plotted in dot red line).

The initial value $a(0)$ was chosen in such a way that the initial value of $\dot{a}(0)$ that was determined from (16) remains real. As it was mentioned earlier, the nonlinear spinor field can simulate different types of dark energy. Here we consider different types of nonlinearity and compare the results for there different cases.

Dust

Let us begin with linear spinor field. Setting $\lambda=0$ from (13) we find $T_{0}^{0}=mS$ and $T_{1}^{1}=T_{2}^{2}=T_{3}^{3}=0$. It means the linear spinor field behaves like dust. In Fig. 1 the behavior or scale factor $a$ is plotted for non-minimal and minimal coupling. As one sees, non-minimal coupling in this case leads to th e rapid expansion of the Universe.

Radiation

Let us first consider the case when the Universe is filled with radiation. In this case the spinor field nonlinearity is given by Saha2018ECAA

[TABLE]

The corresponding solution is given in Fig. 2. Here the blue solid line stand for non-minimal coupling with nonlinear spinor field, and red dot line stands for minimal coupling with nonlinear spinor field. Like in the previous case here too we see that the non-minimal coupling plays significant role in the evolution of the Universe and leads to its rapid expansion.

Quintessence

Let us consider the spinor field nonlinearity which is responsible for quintessence. In this case the spinor field nonlinearity can be given by Saha2018ECAA

[TABLE]

Let us set $W=-1/2$. The solution to the equation (18) is plotted in the Fig. 3. Here we see that the nonlinear term plays the key role in the evolution of the Universe. The presence of non-minimality is hardly distinguishable.

Chaplygin Gas

Another choice of spinor field nonlinearity could be the one that describes a Chaplygin gas. As it was shown in Saha2018ECAA spinor field nonlinearity in this case takes the form

[TABLE]

with $A>0$ and $0<\alpha\leq 1$. Inserting it into (18) and setting $A=1$ and $\alpha=0.5$ we have solved the equation is question numerically. The result is illustrated in Fig.  4. As in case of quintessence, here too the prime role belongs to the spinor field nonlinearity.

Modified Quintessence

The discovery of late time acceleration gives rise a number of problems. One of the problems is the eternal acceleration. To avoid this a modified quintessence was proposed. In this case the spinor field nonlinearity takes the form Saha2018ECAA

[TABLE]

where $\varepsilon_{\rm cr}$ is come constant. We set $W=-1/2$ and $\varepsilon_{\rm cr}=0.01$ Then the solution to the equation (18) takes the from drawn in Fig. 5. We again see that the evolution of the Universe is dominated by the dark energy given by the spinor field nonlinearity.

Modified Chaplygin Gas

We also consider the case when the dark energy is the combination of quintessence and Chaplygin gas. In this case the spinor field nonlinearity takes the form Saha2018ECAA

[TABLE]

with $W<-1/3$ and $A>0$. We have taken $W=-1/2$ and $\alpha=2.$ The corresponding solution is illustrated in the Fig. 23. Like other previous cases spinor field nonlinearity plays principal role in the evolution of the Universe.

IV conclusion

Since in a FRW Universe the non-diagonal components of the energy-momentum tensor of the spinor field do not exist the spinor field does not impose any additional restriction on the geometry of the Universe as it takes place for the anisotropic cosmological models. This is true for both cases with minimal and non-minimal coupling. If the spinor field nonlinearity behaves like an ordinary matter, e.g., radiation, the presence of non-minimality becomes significant and in this case the non-minimal coupling leads to the rapid rapid expansion of the Universe, whereas if the spinor field nonlinearity describes a dark energy, the evolution of the Universe is totally dominated by it and the presence of non-minimality remains almost unnoticeable.

**Acknowledgments

**This work is supported in part by a joint Romanian-JINR, Dubna Research Project, Order no.396/27.05.2019 p-71.