# Spin precession in anisotropic cosmologies

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## Abstract

We consider the precession of a Dirac particle spin in some anisotropic Bianchi universes. This effect is present already in the Bianchi-I universe. We discuss in some detail the geodesics and the spin precession for both the Kasner and the Heckmann–Schucking solutions. In the Bianchi-IX universe the spin precession acquires the chaotic character due to the stochasticity of the oscillatory approach to the cosmological singularity. The related helicity flip of fermions in the very early universe may produce the sterile particles contributing to dark matter.

### Keywords

Early Universe Equivalence Principle Sterile Neutrino Dirac Particle Spin Precession## 1 Introduction

In almost all the applications of mathematical cosmology to the elaboration of observational data the isotropic Friedmann cosmological models are used. However, in the very early universe, the effects of anisotropies could be essential. As is well known, the most simple and well studied anisotropic cosmological models are the spatially homogeneous Bianchi models (see e.g. [1, 2]). Remarkably, already in Bianchi models one can observe such interesting and important phenomenon as the oscillatory approach to the cosmological singularity [3, 4, 5]. However, to the best of our knowledge, the behavior of quantum particles in the Bianchi universes has not been studied in detail. We think that the filling of this gap can be of interest not only from the theoretical point of view, but that it may also reveal some interesting physical effects in the very early universe.

Especially promising can be the study of the motion of Dirac particles (quarks and leptons) in gravitational fields. While this study has a rather long history [6, 7], some essential progress was made in a recent series of papers [8, 9, 10, 11, 12]. In particular, the general expressions, characterizing the spin motion in rather general gravitational fields were elaborated in paper [12]. Here we apply this formalism to the study of the behavior of quantum particles with spin in some Bianchi universes. We found a novel effect of anisotropy induced spin precession, revealed already in the simplest case of the Bianchi-I universe. We consider in some details the geodesics and the spin precession in Bianchi-I universes, putting special emphasis on the Kasner [13] and the Heckmann–Schucking [14] solutions. It is interesting also from the point of view of the study of cosmic jets, which was undertaken in Ref. [15].

Then we consider the precession in the Bianchi-IX universe. Here, first of all, two qualitatively different contributions to the angular velocity are present and, second, the oscillatory approach to the singularity [3, 4] implies the stochasticity of the changes of the direction of the precession axis.

We also consider the possible physical consequences of these effects in the very early universe, including the appearance of effective magnetic field. The similar precession effects are also present for classical rotators due to the equivalence principle and might be manifested in the structure formation in the very early universe.

The equivalence principle implied also the helicity flip which is of special interest for massive Dirac neutrinos. The neutrinos produced as active ones are becoming sterile due to gravity-induced helicity flip and may contribute to fermionic dark matter. The structure of the paper is as follows: in the second section we briefly describe the precession of the Dirac particle in gravitational field; in the third section we give the general formulas for geodesics and spin rotation in the Bianchi-I universes, and, in particular, in the empty Bianchi-I universes evolving, following the Kasner solution; Sect. 4 is devoted to the Heckmann–Schucking solution for a Bianchi-I universe filled with a dust-like matter; in the fifth section we consider the precession of spin in a Bianchi-IX universe; in the concluding section we discuss possible physical applications of described effects and give a short outlook of the future directions of investigations.

## 2 The precession of the Dirac particle in a gravitational field

*m*and its momentum \(p_a\) by introducing the velocity \(v_a\). Thus, the precession velocities are

## 3 The evolution of a spinning particle in the Kasner universe

Obviously, this effect can be essential in the early universe, i.e. at the very small values of the proper cosmic time *t*.

*u*[17]:

First of all, let us find the velocities of a particle, moving in a Kasner universe, resolving the geodesic Eqs. [15, 18].

*t*is

*D*is an integration constant. For the particles in the rest frame (\(C_1=C_2=C_3=0\)) the proper time \(\tau \) coincides with the coordinate one

*t*, hence, \(D=1\) and

*t*is big enough to make the term \(C^1/t^{p_1}\) dominating in the expression (26). (Let us note, however, that the value of the time parameter is still not as big to make the Kasner regime invalid and to have a transition to the isotropization due to the presence of matter, which will be described in the next section.) A similar limit was considered in Ref. [15], where it was related to the possible production of cosmic jets. In this context, the component velocity of a particle oriented along the axis of contracting dimension tends to the velocity of light. What is the behavior of the angular momentum of the jet? It is easy to see that in this case the factor \(\gamma \) behaves as

*t*is

Let us recall that due to the equivalence principle, the macroscopic angular momentum is evolving like spin. So, the angular momenta of jets are changing very slowly.

It is curious that even in the vicinity of the cosmological singularity of the Kasner universe, the angular momentum remains quite stable and does not exhibit any singular behavior. Perhaps, one can say that, in a way, the rotation possesses some smoothing effect.

## 4 Geodesics and jets in the Heckmann–Schucking universe

In Ref. [15] an interesting possibility of production of jets in the Kasner spacetime was considered. Such a possibility is connected with the fact that at the expansion the velocity of test particles in the contracting direction is growing, tending to that of light [see Eq. (27)]. It is particularly interesting to study such e phenomenon in a more realistic Heckmann–Schucking model, which represents the Bianchi-I universe filled with dust [14]. Note that this solution can easily be generalized for the case when the stiff matter and the cosmological constant are also present [19, 20].

*a*,

*b*, and

*c*for a Bianchi-I universe (11) as

*R*(

*t*) can be written in the form

*R*(

*t*). The former, in turn satisfy the equations

Let us note that we cannot make a naive transition from the Heckmann–Schucking solution (52) to the Kasner solution (16) by requiring that \(M \rightarrow 0\). It is connected with the fact that the integrals for the anisotropy functions (51) logarithmically diverge at \(t \rightarrow \infty \) if \(M=0\). Thus, the scale \(t_0=4\sqrt{S}/3M\) loses sense and one should introduce another time scale to make it convergent. This phenomenon can be considered as a particular example of the automodelity of the second order [21]. Under the automodelity of the second order one means the nonexistence of a finite limit of some observable when a particular parameter is tending to zero or to infinity. Instead, one has a power dependence on this parameter. We would like to stress that this power is similar to anomalous dimension in the renormalization group approach to quantum field theory [22].

*a*(

*t*) has a non-monotonic behavior if, as usual, we choose \(p_1 < 0\). Indeed, this factor is infinitely large at the singularity at \(t = 0\), then it begin decreasing arriving the point of minimal contraction at the moment

*a*is

The plot of the function *f*(*u*) is presented in Fig. 1. It decreases monotonically from \(f(1)\approx 1.9\) until \(f(\infty )=1\).

## 5 Precession in a Bianchi-IX universe

*a*,

*b*, and

*c*are some functions of time, as usual. Its inverse matrix \(W^c_{\ \hat{b}}\) is

Thus, we have seen that the “gravitomagnetic” velocity in the Bianchi-IX universe is the same as in Bianchi-I universe, however, in the Bianchi-IX universe there is also the “gravitoelectric” precession. The presence of this term (64) is connected with the presence of a spatial curvature in the Bianchi-IX universe, in contrast to the Bianchi-I universe. It is connected with the fact that the anholonomy coefficients are non-vanishing in the Bianchi-IX universe.

*u*[17] as

*u*becomes less than 1. In this case the change of the “Kasner era” occurs [1, 3, 4]. This change is described by the following formula:

After the change of the Kasner era all the components of the velocity \(\mathbf {\Omega }_{(2)}\) just change the sign, preserving the absolute values, as follows immediately from (67).

## 6 Discussion and outlook

We have seen that the precession of a Dirac particle spin exists already in the Bianchi-I universe. Interestingly, the Kasner indices play the role similar to the moments of inertia in the Euler equation for the rigid body precession. In the Bianchi-IX universe the precession acquires the chaotic character due to the stochasticity of the oscillatory approach to the cosmological singularity [3, 4, 5]. Remarkably, the formulas for the changes of the precession direction are nicely expressible in terms of the Lifshitz–Khalatnikov parameter *u*.

What physical consequences could it have for the very early universe?

^{1}the gravity-induced helicity flip may turn them to sterile neutrinos which remain in this state after the universe becomes isotropic and contribute to fermionic dark matter. As soon as the rotation period is defined by the age of the universe in the anisotropic phase, the amounts of sterile and active neutrinos at the end of this phase are of the same order:

The anisotropic metrics, in the case of some scale parameters being much smaller than others, may provide the model of transitions between spaces of different (effective) dimension [27, 28, 29]. The spin dynamics in that case is manifesting the interesting effects [30]. Other interesting directions of investigation could be connected with the study of the spin precession in Bianchi II universes, in the generalized Melvin cosmologies in the presence of electromagnetic fields [31] and in the double Kasner universes [26]. We hope to study these topics in detail in future publications.

## Footnotes

- 1.
It is definitely so in the Standard Model and we will not consider its extensions in which sterile Dirac neutrinos may be produced.

## Notes

### Acknowledgments

The authors are grateful to I.M. Khalatnikov, Yu.N. Obukhov and A.J. Silenko for useful discussions. The work of A.K. was partially supported by the RFBR through the Grant No. 14-02-00894 while the work of O.T. was partially supported through the Grant No. 14-01-00647. O.T. is grateful to the Sezione di INFN in Bologna and to the Dipartimento di Fisica e Astronomia dell’Università di Bologna for kind hospitality during his visits to Bologna in the Falls of 2014 and 2015.

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