FRW and Bianchi type I cosmology of f-essence

Abstract

F-essence is a generalization of the usual Dirac model with the nonstandard kinetic term. In this paper, we introduce a new model of spinor cosmology containing both Ricci scalar and the non minimally coupled spinor fields in its action. We have investigated the cosmology with both isotropy and anisotropy, where the equations of motion of FRW and Bianchi type-I spacetimes have been derived and solved numerically. Finally the quantization of these models through Wheeler-De Witt (WD) wave function has been discussed.

Appendix

As a double check, one can obtain the above field equations from the Einstein and Dirac equations given by:

(78)

(79)

(80)

(81)

For a homogeneous fermionic field ψ(t), (79) and (80) are equivalent to (79) and (80) respectively. On the other hand, the non-vanishing components of the Einstein tensor for the metric (39) are:

(82)

(83)

(84)

(85)

The components of the energy-momentum tensor for the fermionic field as the matter source can be obtained from the standard definition as:

$$ T_{\mu\nu}=2\frac{\partial L_f}{\partial g^{\mu\nu}}-g_{\mu\nu}L_f,$$

(86)

yielding

(87)

Substituting these results into Einstein equations (78), yields the same equations as (46)–(52). In the case of the FRW metric (3), the equations corresponding to the action (1) can be obtained as:

(88)

(89)

(90)

(91)

(92)

where the kinetic terms, the energy density and the pressure take the forms

(93)

(94)