Classical bigbounce cosmology: dynamical analysis of a homogeneous and irrotational Weyssenhoff fluid
Abstract
A dynamical analysis of an effective homogeneous and irrotational Weyssenhoff fluid in general relativity is performed using the covariant approach that enables the dynamics of the fluid to be determined without assuming any particular form for the spacetime metric. The spin contributions to the field equations produce a bounce that averts an initial singularity, provided that the spin density exceeds the rate of shear. At later times, when the spin contribution can be neglected, a Weyssenhoff fluid reduces to a standard cosmological fluid in general relativity. Numerical solutions for the time evolution of the generalised scale factor in spatiallycurved models are presented, some of which exhibit eternal oscillatory behaviour without any singularities. In spatiallyflat models, analytical solutions for particular values of the equationofstate parameter are derived. Although the scale factor of a Weyssenhoff fluid generically has a positive temporal curvature near a bounce (i.e. ), it requires unreasonable fine tuning of the equationofstate parameter to produce a sufficiently extended period of inflation to fit the current observational data.
pacs:
98.80.k, 98.80.Jk, 04.20.Cv, ,
1 Introduction
The EinsteinCartan (EC) theory of gravity is an extension of Einstein’s theory of general relativity (GR) that includes the spin properties of matter and their influence on the geometrical structure of spacetime ([1]; see also [2], [3]). In GR, the energymomentum of the matter content is assumed to be the source of curvature of a Riemannian spacetime manifold . In the EC theory, the spin of the matter has been postulated, in addition, to be the source of torsion of a RiemannCartan spacetime manifold [4]. Weyssenhoff and Raabe [5] were the first to study the behaviour of perfect fluids with spin. Obukhov and Korotky extended their work in order to build cosmological models based on the EC theory [6] and showed that by assuming the Frenkel condition^{1}^{1}1Note that the Frenkel condition arises naturally when performing a rigorous variation of the action. It simply means that the spin pseudovector is spacelike in the fluid rest frame. the theory may be described by an effective fluid in GR where the effective stressenergy momentum tensor contains some additional spin terms.
The aim of this publication is twofold. First, we wish to investigate the possibility that the spin contributions for a Weyssenhoff fluid may avert an initial singularity, as first suggested by Trautman [7]. Second, since any realistic cosmological model has to include an inflation phase to fit the current observational data, it is also of particular interest to see if the spin contributions are able to generate a dynamical model endowed with an early inflationary era, as first suggested by Gasperini [8]. Scalars fields can generate inflation, but they have not yet been observed. Therefore, it is of interest to examine possible alternatives, such as a Weyssenhoff fluid. In contrast to the approaches of Trautman [7] and Gasperini [8], our use of the covariant formalism enables us to determine the dynamics of a Weyssenhoff fluid without assuming any particular form for the spacetime metric.
The study of the dynamics of a Weyssenhoff fluid in a covariant approach was initiated by Palle [9]. His work has been revised and extended in our previous publication [10]. The present paper builds on [10] to extend the work carried out first by Trautman [7] in an isotropic spacetime, and Kopczynski [11] and Stewart [12] in an anisotropic spacetime. It also generalises the analysis of the inflationary behaviour of Weyssenhoff fluid models made by Gasperini [8] to anisotropic spacetimes.
In our dynamical analysis, we choose to restrict our study to a spatially homogeneous and irrotational Weyssenhoff fluid. This particular choice, which implies a vanishing vorticity and peculiar acceleration, has been motivated by underlying fundamental physical reasons. For a vanishing vorticity, the fluid flow is hypersurfaceorthogonal, which means that the instantaneous rest spaces defined at each spacetime point should mesh together to form a set of 3surfaces in spacetime [13]. These hypersurfaces, which are surfaces of simultaneity for all the fluid observers, define a global cosmic time coordinate determined by the fluid flow. Moreover, by assuming that any peculiar acceleration vanishes, the cosmic time is then uniquely defined. It is worth mentioning that the absence of vorticity is an involutive property, which means that if it is true initially then it will remain so at later times as shown by Ellis et al [14]. Finally, the assumption that there is no vorticity on all scales implies that the fluid has no global rotation. This is in line with recent Bayesian MCMC analysis of WMAP data performed by Bridges et al. [15]. Their work confirms that a physical Bianchi model, which has a nonvanishing vorticity, is statistically disfavored by the data.
It is worth pointing out that Szydlowski and Krawiec [16] have considered an isotropic and homogeneous cosmological model in which a Weyssenhoff fluid is proposed as a potential candidate to describe dark energy at late times. In a subsequent publication [17], the authors showed that it is not disfavoured by SNIa data, but it may be in conflict with CMB and BBN observational constraints. By contrast, in this paper, we consider the full evolutionary history of an, in general, anisotropic universe with a Weyssenhoff fluid as its matter source, concentrating in particular on the ‘early universe’ behaviour when the spin terms are significant. Indeed, at late times, when the spin contributions can be neglected, the Weyssenhoff fluid reduces to a standard cosmological fluid. We thus allow for the presence of a nonzero cosmological constant, in accord with current observational constraints.
In , we give a concise description of a Weyssenhoff fluid using a 1+3 covariant approach outlined in Appendix A. The spatial symmetries and macroscopic spin averaging procedure are discussed in . In , we establish the relevant dynamical relations for a homogeneous and irrotational Weyssenhoff fluid. In , we perform a geodesic singularity analysis for such a fluid. In , we analyse the fluid dynamics. The behaviour of the generalised scale factor of such a fluid in a spatiallycurved models is discussed in and explicit analytical solutions in spatiallyflat models are given in . For the reader’s convenience, certain main results obtained in our earlier work [10] will be repeated in the case of a homogeneous and irrotational Weyssenhoff fluid in and . In this paper, we use the signature. To express our results in the opposite signature used by Ellis [14], the correspondence between physical variables can be found in [10].
2 Weyssenhoff fluid description
2.1 Weyssenhoff fluid phenomenology
In the EC theory, the effect of the spin density tensor is locally to induce torsion in the structure of spacetime. In holonomic coordinates, the torsion tensor is defined as the antisymmetric part of the affine connection ,
(1) 
which vanishes in GR since the connection is assumed to be symmetric in its two lower indices. Note that the tilde denotes an EC geometrical object to differentiate it from an effective GR object.
The Weyssenhoff fluid is a continuous macroscopic medium which is characterised on microscopic scales by the spin of the matter fields. The spin density is described by an antisymmetric tensor,
(2) 
which is the source of torsion,
(3) 
This fluid satisfies the Frenkel condition, which requires the intrinsic spin of a matter field to be spacelike in the rest frame of the fluid,
(4) 
where is the velocity of the fluid element. This condition implies an algebraic coupling between spin and torsion according to,
(5) 
and arises naturally from a rigorous variation of the action as shown by [6]. Thus, the torsion contributions to the EC field equations are entirely described in terms of the spin density. It is also useful to introduce a spindensity scalar defined as,
(6) 
Obukhov and Korotky showed [6] that for a perfect fluid the EC field equations reduce to effective GR Einstein field equations with additional spin terms, and a spin field equation.
The former are found to be,
(7) 
where the effective stress energy momentum tensor of the fluid is given by,
(8) 
with effective energy density and pressure of the form,
(9) 
and the physical energy density and pressure satisfy the equation of state,
(10) 
where , is the cosmological constant and the equation of state parameter.
The spin field equation is given by,
(11) 
2.2 Weyssenhoff fluid description in a 1+3 covariant formalism
The covariant formalism outlined in Appendix A can now be used to perform a more transparent analysis of the Weyssenhoff fluid dynamics. Using a covariant approach in [10], we found that the symmetric stressenergy momentum tensor can be recast as,
(12) 
where is the induced metric on the spatial hypersurface, is the rate of shear tensor and is the spatially projected covariant derivative defined in Appendix A.
Similarly, the spin field equation reduces to,
(13) 
where is the expansion rate.
3 Spatial symmetries and macroscopic spin averaging
Although much of our following discussion will concern cosmological models that are anisotropic, it is of interest to consider the status of a Weyssenhoff fluid as a matter source for homogeneous and isotropic models.
3.1 Spatial symmetries
To be a suitable candidate for the matter content of such a cosmological model, a Weyssenhoff fluid has to be compatible with the Cosmological Principle. In mathematical terms, a fourdimensional spacetime manifold satisfying this principle is foliated by three dimensional spatial hypersurfaces, which are maximally symmetric and thus invariant under the action of translations and rotations.
Although a Weyssenhoff fluid can be expressed as an effective GR fluid, the dynamical nature of such a fluid is rooted in the EC theory. Thus, the dynamics of such a fluid is determined by the translational and the rotational fields, which are respectively the metric and the torsion . The symmetries require the dynamical fields to be invariant under the action of an infinitesimal isometry. Hence, the Lie derivatives of the dynamical fields have to vanish according to,
(14)  
(15) 
where are the Killing vectors generating the spatial isometries. A maximally symmetric spatial hypersurface admits 6 Killings vectors [18]. The 3 Killing vectors generating the infinitesimal translations are related to homogeneity and the 3 Killing pseudovectors generating the infinitesimal rotations are related to isotropy. They satisfy,
(16)  
(17) 
where is threedimensional LeviCivita tensor.
For a cosmological fluid based on the EC theory, such as a Weyssenhoff fluid, we can consider two different forms of the Cosmological Principle:

the Strong Cosmological Principle (SCP), where the Lie derivatives of the metric and of the torsion have to vanish; and

the Weak Cosmological Principle (WCP), where only the Lie derivatives of the metric have to vanish and no restriction is imposed on the torsion.
The translational Killing equation resulting from the symmetries imposed on the metric yields,
(18) 
which is a wellknow result obtained in GR. Hence, the WCP is identical to the GR Cosmological Principle, which implies that the spacetime geometry is described in terms of an FRW metric.
Using the translational Killing equation , the rotational Killing equation resulting from the symmetries imposed on the metric is found to be,
(19) 
For any maximally symmetric space [18], we can choose respectively a Killing vector to vanish at a given point , and independently, a Killing pseudovector to vanish at a given point according to,
(20)  
(21) 
Hence, the homogeneity and isotropy can be considered separately.
By imposing the homogeneity condition on the rotational Killing equation , the spatial covariant derivative of the torsion tensor has to vanish according to,
(22) 
Hence, torsion can only be a function of cosmic time ,
(23) 
By imposing the isotropy condition on the rotational Killing equation , the torsion tensor has to satisfy the constraint,
(24) 
As shown explicitly in a theorem established by Tsamparilis [19] and mentioned subsequently by Boehmer [20], the homogeneity and isotropy constraints taken together put severe restrictions on the torsion tensor. The only nonvanishing components are found to be,
(25)  
(26) 
where is a scalar function of cosmic time , is a fixed index and is the spatial trace of the torsion tensor defined as
(27) 
We now discuss the application of this general framework to a Weyssenhoff fluid.
3.2 Weyssenhoff fluid with macroscopic spin averaging
The algebraic coupling between the spin density and torsion tensors shows that the spin density of a Weyssenhoff fluid can be related to the torsion as,
(28) 
By substituting the nonvanishing components of the torsion and satisfying the SCP into the expression for the spin density of a Weyssenhoff fluid , it is straightforward to show that the spin density tensor has to vanish,
(29) 
Thus, Tsamparilis claims that a Weyssenhoff fluid is incompatible with the SCP [19]. This conclusion would hold if all the dynamical contributions of the spin density were second rank tensors of the form . However, this is not the case since the dynamics contains spin density squared scalar terms. These scalar terms are invariant under spatial isometries like rotations and translations. Hence, they do satisfy the SCP.
In order for the Weyssenhoff fluid to be compatible with the SCP, the spin density tensorial terms have to vanish leaving the scalar terms unaffected. This can be achieved by making the reasonable physical assumption that, locally, macroscopic spin averaging leads to a vanishing expectation value for the spin density tensor according to,
(30) 
However, this macroscopic spin averaging does not lead to a vanishing expectation value for the spin density squared scalar since this term is a variance term,
(31) 
The macroscopic spatial averaging of the spin density was performed in an isotropic case by Gasperini [8]. It can be extended to an anisotropic case provided that on small macroscopic scales the spin density pseudovectors are assumed to be randomly oriented.
By considering a Weyssenhoff fluid in the absence of any peculiar acceleration and by performing a macroscopic spin averaging, we indirectly require the fluid to be homogeneous. This follows from the fact that, in this case, the conservation law of momentum leads to a vanishing spatial derivative of the pressure and energy density. This will be explicitly shown in , and can also be derived from the corresponding dynamical equation for an inhomogeneous Weyssenhoff fluid presented in our previous work [10].
Note that even in the absence of a macroscopic spin averaging, the Weyssenhoff fluid is still compatible with the WCP, which we discuss further in . It is worth mentioning that there is no observational evidence so far which would suggest that we should impose the SCP even though the mathematical symmetries make such a principle mathematically appealing. A true test of whether this principle is applicable would be the demonstration of physically observable differences between this case and the WCP.
4 Dynamics of a homogeneous and irrotational Weyssenhoff fluid
The dynamics of a Weyssenhoff fluid with no peculiar acceleration is entirely determined by the symmetric and spin field equations, , and respectively. The former can be used to determine the Ricci identities and the energy conservation law. The latter simply expresses spin propagation.
One important consequence of the spatial averaging of the spin density is that the stressenergy momentum tensor reduces to an elegant expression given by
(32) 
where the only spin contributions affecting the dynamics are the negative spin squared variance terms entering the definition of the effective energy density and pressure , as expected. These spin squared intrinsic interaction terms are a key feature that distinguishes a Weyssenhoff fluid from a perfect fluid in GR and lead to interesting properties we discuss below.
We have to be careful when performing the macroscopic spin averaging on the dynamical equations. The Ricci identities and conservation laws can be entirely determined from the stressenergy momentum tensor . As we have shown, it is perfectly legitimate to perform a macroscopic spin averaging on the stressenergy momentum tensor before obtaining explicitly the dynamical equations. However, this is not the case for the spin field equations . Performing the macroscopic spin averaging at this stage would make these field equations vanish. To be consistent, we first have to determine the dynamical equations and express them in terms of the spin density scalar before performing the spin averaging.
4.1 Ricci identities
The Ricci identities can firstly be applied to the whole spacetime and secondly to the orthogonal 3space. They yield respectively,
(33)  
(34) 
where the spatial vectors are orthogonal to the worldline, i.e. , and the 3space Riemann tensor is related to the Riemann tensor by
(35) 
The Riemann tensor can be decomposed according to its symmetries as [21],
(36) 
where is the tracefree Weyl tensor, which, in turn, can be split into an ‘electric’ and a ‘magnetic’ part [21] according to,
(37)  
(38) 
The Ricci tensor is then simply obtained by substituting the expression for the effective stress energy momentum tensor into the Einstein field equations ,
(39) 
The Riemann tensor can now be recast in terms of the Ricci tensor , the electric and magnetic parts of the Weyl tensor according to the decomposition in the following way,
(40) 
It follows from the relation that the Riemann tensor on the spatial 3space becomes,
(41) 
The information contained in the Ricci identities can now be extracted by projecting them on different hypersurfaces using the decomposition of the corresponding Riemann tensors and following the same procedure as in our previous publication [10].
The Ricci identities applied to the whole spacetime yield respectively the Raychaudhuri equation and the rate of shear propagation equation,
(42)  
(43) 
The Ricci identities applied to the spatial 3space express the spatial curvature. Their contractions yield the spatial Ricci tensor and scalar respectively,
(44)  
(45) 
The above expression for the curvature scalar is a generalisation of the Friedmann equation.
One must take particular care when deducing the time evolution of the rate of shear from the rate of shear propagation equation . This is due to the fact that the rate of shear coupling term and the tidal force term can not simply be neglected. A better route is to deduce the rate of shear evolution equation from the spatial Ricci curvature tensor as shown explicitly by Ellis [22] and outlined below.
A homogeneous Weyssenhoff fluid satisfies the spatial curvature identity,
(46) 
Hence, by substituting this identity into the expression for the spatial Ricci tensor , the propagation equation for the rate of shear is found to be,
(47) 
This tensorial expression can be recast in terms of a scalar relation involving the rate of shear scalar according to,
(48) 
4.2 Conservation laws
The effective energy conservation and momentum conservation laws are obtained by projecting the conservation equation of the effective stress energy momentum tensor ,
(49) 
respectively along the worldline and on the orthogonal spatial hypersurface according to,
(50)  
(51) 
It is worth mentioning that the momentum conservation law expresses the homogeneity of the Weyssenhoff fluid. This is due to the fact that according to this law, the energy density, the pressure and the spin density of the fluid have to be a function of cosmic time only. Hence, the torsion tensor has also to be a function of cosmic time only, which is the homogeneity requirement . This is only the case for a Weyssenhoff fluid with no peculiar acceleration on which a macroscopic spin averaging has been performed, as otherwise the momentum conservation law would contain additional terms.
4.3 Spin propagation relation
The spin conversation law results from twice projecting the antisymmetric field equations onto the hypersurface orthogonal to the worldline,
(52) 
This tensorial expression can be recast in terms of a scalar relation involving the spindensity scalar in according to,
(53) 
This expression implies that the spin density is inversely proportional to the volume of the fluid. Note that although the tensorial expression vanishes due to the macroscopic spin averaging , the scalar expression still applies because it is related to the spin variance .
The effective energy conservation equation can now be recast in terms of the true (i.e. not effective) energy density and pressure of the fluid by substituting the spin propagation equation ,
(54) 
4.4 Comparision with previous results
Let us compare our results with the conclusions reached by Lu and Cheng [23] for an isotropic Weyssenhoff fluid without any macroscopic spin averaging as shown in Appendix A of their publication.
In an isotropic spacetime, the dynamics of a Weyssenhoff fluid, without a macroscopic spin averaging, is greatly simplified as we now briefly explain. The projection of the effective Einstein field equations along the worldline and on the orthogonal spatial hypersurfaces, yields the following constraint,
(55) 
It arises from the fact that, in an isotropic case, the timespace components of the Ricci tensor vanish. From the expression for the stressenergy momentum tensor , it is clear that the constraint implies a vanishing spin divergence,
(56) 
Moreover, the isotropy constraint implies a vanishing rate of shear (i.e. ). Thus, in this case, the effective stress energy momentum tensor without the macroscopic spin averaging reduces to the elegant expression obtained by performing the macroscopic spin averaging.
Hence, for a Weyssenhoff fluid and isotropic spacetime, our results can be compared to those of Lu and Cheng [23]. The results of our analysis do not agree with the conclusions outlined in [23]. First, they argue that the isotropic Friedmann equation implies that the spin density has to be a function of time only, with which we agree. Then, they claim that this stands in contradiction with the fact that the spin density has also to be a function of space in order to satisfy the projection constraint , which we dispute. The projection constraint simply implies a vanishing orthogonal projection of the spin divergence on the spatial hypersurface , which is perfectly compatible with the spin density being a function of time only. Hence, contrary to their claim, a Weyssenhoff fluid model seems to be perfectly consistent with an isotropic spacetime (i.e. obeying the WCP), even without spin averaging.
5 Geodesic singularity analysis
For a homogeneous and irrotational Weyssenhoff fluid satisfying the macroscopic spin averaging condition, the fluid congruence is geodesic. To study the behaviour of such a fluid congruence near a singularity, we use the 1+3 covariant formalism, which applies on local as well as on global scales for a homogeneous fluid model.
In order for singularities in the timelike geodesic congruence to occur, the Raychaudhuri equation has to satisfy the condition,
(57) 
near the singularity, as we now explain. First, we recast the singularity condition in terms of the inverse expansion rate as,
(58) 
After integrating with respect to cosmic time , we find,
(59) 
where and is some arbitrary cosmic time near the singularity. Thus, if (), the model describes a fluid evolving on a spatially expanding (collapsing) hypersurface at . According to the integrated singularity condition , must vanish within a finite past (future) time interval with respect to . Thus a geodesic singularity, defined by , occurs at .
The homogeneity requirement allows us to define up to a constant factor a generalised scale factor according to,
(60) 
In a covariant approach, is generally a locally defined variable. If the model is homogeneous, however, can be globally defined and interpreted as a cosmological scale factor.
The singularity condition can now be recast in terms of the scale factor and reduces to,
(61) 
One must also require the scale factor to obey the consistency condition, which requires the expansion rate squared to be positively defined at all times according to,
(62) 
To determine explicitly these two conditions , the Friedmann and Raychaudhuri equations are recast in terms of the scale factor, using respectively the expressions for the Ricci and stressenergymomentum tensor as,
(63)  
(64) 
Using the Friedmann and the Raychaudhuri equations, the consistency and singularity conditions can respectively be explicitly expressed as,
(65)  
(66) 
The scaling of the energy density , the spin density squared and the rate of shear squared can be deduced respectively from the energy conservation law , the spin propagation equation and the rate of shear propagation equation by recasting the expansion rate in terms of the scale factor according to,
(67)  
(68)  
(69) 
Note that the bar corresponds to an arbitrary event (defined by a cosmic time ), subject only to the condition .
Furthermore, the spatial Ricci scalar is the Gaussian curvature of the spatial hypersurface, which scales according to,
(70) 
and the cosmological constant has by definition no scale dependence,
(71) 
Let us now assume the existence of singularities in the timelike geodesic congruence for a homogeneous and irrotational Weyssenhoff fluid. By comparing the scaling relations for the spatial Ricci scalar and the cosmological constant with those obtained for the spin density squared and the rate of shear squared , we see that in the limit where the model tends towards a singularity (i.e. ), the contribution due to curvature and the cosmological constant is negligible. Hence, for a Weyssenhoff fluid with a physically reasonable equationofstate parameter (i.e. ), the consistency and singularity conditions merge into a single condition according to,
(72) 
Moreover, we can recast this condition in terms of the scale dependence . In the limit where the model tends towards a singularity, the condition becomes,
(73) 
Provided the equationofstate parameter , the singularity condition can only hold if the rate of shear squared is larger than the spin squared (i.e. ). Hence, in the opposite case, where the macroscopic spin density squared of the Weyssenhoff fluid is larger than the fluid anisotropies according to,
(74) 
there will be no singularity on any scale. This is a generalisation of the result established independently for a Bianchi I metric by Kopczynski [11] and Stewart and Hajieck [12].
Our singularity analysis is based on the assumption that the Weyssenhoff fluid flow lines are geodesics, which implies that the macroscopic fluid (i.e. with spin averaging) has to be homogeneous. A key question is whether this still holds in presence of small inhomogeneities. According to Ellis [24], the HawkingPenrose singularity theorems apply not only to homogeneous models but also to approximately homogeneous models with local pressure inhomogeneities. By analogy, if there is no singularity for geodesics fluid flow lines, singularities may still be averted provided the real fluid flow lines can be described as small perturbations around geodesics.
In following sections, we will assume that the spinshear condition holds, which guarantees the absence of singularities for homogeneous models.
6 Dynamical evolution: general considerations
For a homogeneous fluid, the Gaussian curvature depends only on the scale factor according to,
(75) 
where is the normalised curvature parameter.
To analyse the dynamics of a homogeneous and irrotational Weyssenhoff fluid, let us first recast explicitly the Friedmann and Raychaudhuri equations in terms of the physical quantities using the expression for the Gaussian curvature according to,
(76)  
(77) 
We will now discuss in more details the geodesic singularities presented in , drawing out more fully the geometrical and physical applications.
6.1 Geometric interpretation of the solutions
As outlined above, at stages of the dynamical evolution for which the scale factor is small, a Weyssenhoff fluid with an equationofstate parameter is dominated by the spin density and rate of shear contributions. This follows from the scaling properties of the energy of the spin density and of the rate of shear . Provided the spinshear condition is satisfied, there can be no singularity (), because the negative sign of the spin squared terms in the RHS of the Friedmann equation would imply the existence of an imaginary rate of expansion, which is physically unacceptable () as discussed before in . For physical consistency, the RHS of the Friedmann equation has to be positively defined at all times,
(78) 
which clearly excludes the presence of a singularity provided . The physical interpretation is that as one goes backwards in cosmic time from the present epoch the spin contributions to the field equations dominate and produce a bounce, which we may take to occur at , that avoids an initial singularity (i.e. ). Since this model contains no initial singularity, the temporal evolution of the model, governed by the Friedmann and Raychaudhuri equations, extends symmetrically to the negative part of the time arrow. In order to satisfy the time symmetry requirement and avoid a kink in the time evolution of the scale factor at , the expansion rate at the bounce has to vanish, , and the temporal curvature of the scale factor has to be finite. Thus, the scale factor goes through an extremum at the bounce ^{2}^{2}2Note that, throughout the paper, a zero subscript denotes the value of a quantity at the bounce (i.e. ) and not at the present epoch.. The energy density at the bounce, , is determined by the limit where the consistency requirement becomes an equality,
(79) 
where and denote respectively the spin energy density and the rate of shear evaluated at the bounce. Note that this particular choice for the energy density at has been made in order for the expansion rate to vanish at the bounce. This can be shown explicitly by evaluating the Friedmann equation at the bounce using the expression for the energy density .
Quantitative expressions or the curve in various cases are derived in below. Before doing so, however, it is worth noting that qualitatively, the general shape of the curve for a Weyssenhoff fluid is closely related to the temporal curvature of the scale factor , which is explicitly given by the Raychaudhuri equation , and also to the range of values for , which is determined by the consistency condition on the Friedmann equation .
In this section, let us discuss one particular class of Weyssenhoff fluid models for which the cosmological constant is small (and positively defined),
(80) 
and the curvature is also small
(81) 
The two constraints and on the class of models imply that the sign of the temporal curvature of the scale factor depends only on the value of the equationofstate parameter , which yields three different cases.
In the first case, where , the RHS of the Raychaudhuri equation implies that the temporal curvature of the scale factor is positively defined at all times,
(82) 
The positive sign of implies that the scale factor is minimal at the bounce and the model is perpetually inflating (for ).
In the second case, where , by comparing the consistency requirement with the Raychaudhuri equation , the temporal curvature of the scale factor is found to be negatively defined at all times,
(83) 
Note that for a model with an equationofstate parameter , we reach the same conclusion as for a fluid with an equation of state parameter , which is that the model has a timesymmetric evolution and bounces at . The negative sign of implies that the scale factor is maximal at the bounce and is deflating (for ) until it eventually collapses.
In the third case, where , the symmetric time evolution of the scale factor can be split into five parts. Firstly, for a small cosmic time, i.e. where the value of depends on the scale parameter the sign of the temporal curvature of the scale factor is positive. This corresponds to the spin dominated phase. Secondly, for a specific cosmic time, i.e. , the temporal curvature of the scale factor vanishes as the time evolution of the scale factor reaches an inflection point. Then, for a larger cosmic time, i.e. , the temporal curvature of the scale factor has the opposite sign until it reaches the second inflection point . This corresponds to the matter dominated phase. Finally, for large cosmic time, i.e. , the sign of the temporal curvature of the scale factor becomes positive again. This corresponds to the cosmological constant dominated phase. The behaviour of in terms of cosmic time is summarised as follows,
(84)  
(85)  
(86)  
(87)  
(88) 
In the first and second cases, the results obtained for the symmetric time evolution of the scale factor are interesting mathematical solutions, but they are inconsistent with current cosmological observations. In order to satisfy the current cosmological data, the positively defined time evolution of the model has to inflate, i.e. , at early time (), and produce a sufficient amount of inflation. At later time (), the energy density of the fluid dominates the dynamics and acts like a brake on the expansion .
During the spindominated phase, the contribution due to the cosmological constant can be safely neglected and the positive temporal curvature of the scale factor leads to an inflation phase. The inflatability condition, , may be deduced from the Raychaudhuri equation according to,
(89) 
This inflation phase ends when this inequality is no longer satisfied, which corresponds to the inflection point of the temporal evolution of the scale factor, i.e. . Hence, at the end of inflation the density is given by,
(90) 
The temporal evolution of this model for a positively defined time is characterised by a maximal physical energy density coinciding with the start of an inflation phase ending when the energy density reaches the density threshold . At the end of inflation, the model enters a matter dominated phase. During this stage, the Weyssenhoff fluid model reduces asymptotically to the cosmological solution obtained for a perfect fluid in GR in the limit where the cosmic time is sufficiently large , which eventually leads to a cosmological constant dominated phase for .
6.2 Amount of inflation
The amount of inflation is measured by the number of efolds, which is determined using the scaling of the energy density , the initial and final energy densities, and found to be,
(91) 
Using the scaling relations obtained for the spin density squared and the rate of shear squared , the initial and final values of these quantities are found to be related by the number of efolds according to,
(92)  
(93) 
By recasting the initial values of the spin density squared and rate of shear squared in terms of their final values according to and respectively, the expression for the number of efolds reduces to an elegant expression,
(94) 
and is shown in . It worth mentioning that the amount of inflation is independent of the rate of shear or the spin density of the fluid. Let us mention that Bianchi models based on a Weyssenhoff fluid have been studied previously by Lu and Cheng [23]. However, the authors did not try to estimate the amount of inflation in their analysis.
The only way to have achieve a substantial number of efolds is by requiring an equation of state of the form
(95) 
which corresponds to no standard fluid and has therefore no acceptable physical basis. This conclusion has already been reached by Gasperini [8] in the isotropic case. We have showed that the same result still holds in the anisotropic case.
It is interesting to note that a cosmic string fluid has an equationofstate parameter . A hybrid Weyssenhoff fluid made for example of fermionic matter cosmic strings [25] and matter fields where the cosmic strings contribution dominates the dynamics at the era of interest has an equationofstate parameter of the form where the value of the fine tuning parameter depends crucially on the ratio between the cosmic string and the matter fields densities. Although such a fluid is a candidate to obtain an inflation phase at an early positively defined time (i.e. just after the bounce), it does not reduce to the cosmological standard model at later times when the spin contribution can be safely neglected. This is due to the fact that the density of the cosmic strings contribution scales as . Hence, if the cosmic strings contribution dominates the behaviour of the cosmic fluid for an early positively defined time, it will do so at all times.
However, this problem may potentially be overcome by assuming that, at early times, the cosmic strings decay into the matter fields of the standard model leading to a reheating phase. It would be worth further investigating this possibility.
The fine tuning parameter has a magnitude that is related to the number of efolds according to,
(96) 
To obtain, for example, an inflationary phase with efolds which is a characteristic range of values for current parameter estimations the equation of state has to be very fine tuned such that . It is worth noting that this is a similar order of magnitude to the factor relating the ratio of the cosmological constant predicted by summing the zero point energy of the Standard Model fields up to the Planck cutoff to that inferred from cosmological observations, although this is almost certainly just a numerical coincidence.
7 Quantitative dynamical evolution of spatiallycurved models
Our general approach allows one to investigate models with nonzero spatial curvature and a cosmological constant. In general, it is not possible to find analytical solutions for the time evolution of the scale factor. However, the behaviour of the solutions can be analysed by integrating the dynamical equations numerically. The analysis and plots of the time evolution of the scale factor in spatiallycurved models are presented below.
7.1 Solutions in presence of a cosmological constant
The dynamics of a homogeneous and anisotropic Weyssenhoff fluid in a spatiallycurved model in presence of a cosmological constant relies on the Fridemann and Raychaudhuri equations. Using the scaling relation obtained for the energy density , for the spin density , and for rate of shear , the Friedmann and Raychaudhuri equations can be recast respectively as,
(97)  
(98) 
where for , is the scale factor, the energy density, the spin density and the rate of shear.
For convenience, we introduce six dimensionless parameters defined as,
(99)  
(100)  
(101)  
(102)  
(103)  
(104) 
which are the scale factor parameter , the cosmic time parameter , the rate of shear squared parameter and the spin density squared parameter , the curvature parameter , the cosmological constant parameter . Note that and depend on , whereas , , and are constant, defined in terms of quantities at the bounce .
The consistency condition at the bounce can be recast in terms of dimensionless parameters as,
(105) 
Using , the Friedmann and Raychaudhuri equations can also be recast respectively in terms of the dimensionless parameters according to,
(106)  
(107) 
where a prime denotes a derivative with respect to the rescaled cosmic time parameter . It worth emphasizing that the dynamics of a homogeneous Weyssenhoff fluid does not depend explicitly on the rate of shear parameter squared . This is due to the fact that the rate of shear scales like the spin density , and follows explicitly from the fact that the