# Friedmann-like universes with weak torsion: a dynamical system approach

## Abstract

We consider Friedmann-like universes with torsion and take a step towards studying their stability. In so doing, we apply dynamical-system techniques to an autonomous system of differential equations, which monitors the evolution of these models via the associated density parameters. Assuming relatively weak torsion, we identify the system’s equilibrium points. These are found to represent homogeneous and isotropic spacetimes with nonzero torsion that undergo accelerated expansion. We then examine the linear stability of the aforementioned fixed points. Our results indicate that Friedmann-like cosmologies with weak torsion are generally stable attractors, either asymptotically or in the Lyapunov sense. In addition, depending on the equation of state of the matter, the equilibrium states can also act as intermediate saddle points, marking a transition from a torsional to a torsion-free universe.

## 1 Introduction

Extensions of general relativity that go beyond the boundaries of the Riemannian geometry, by allowing for an asymmetric affine connection, have a long history in the literature. These studies introduce the possibility of spacetime torsion and its associated new degrees of freedom to the gravitational field (e.g. see [1] for a recent review). Therefore, it comes to no surprise that there are many applications of these theories to cosmology, in an effort to illuminate the role and the potential implications of torsion (as well as those of spin) for the evolution of the universe we live in. The topics of research interest, which have varied over the decades, range from the early universe and its initial singularity, to the large-scale kinematics and the late-time universal acceleration (see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] for a representative though incomplete list).

Allowing for arbitrary torsion, introduces anisotropic degrees of freedom into the host spacetime. As a result, the spatially homogeneous and isotropic Friedmann-Robertson-Walker (FRW) cosmologies can only accommodate specific forms of torsion [13]. Vectorial torsion fields, determined by a single scalar function of time (\(\phi =\phi (t)\) – see Sect. 2.1 here), are generally compatible with the FRW symmetries [14]. The latter study focused on finding exact solutions for torsional Friedmann-like models. These were then combined with the primordial nucleosynthesis measurements to constrain the gravitational effects of torsion. Here, following on the work of [14], we investigate the general qualitative behaviour of homogeneous and isotropic torsional cosmologies. Utilising the above named torsion scalar, \(\phi =\phi (t)\), we parametrise the contribution of the torsion field to the universal expansion and to the total (effective) energy density of the universe. Then, assuming relatively weak torsion, we are able to recast the associated Einstein-Cartan equations into an autonomous dynamical system and identify its critical (fixed) equilibrium points. As expected, these include the familiar torsion-free Friedmann models, with varying 3-curvature and a nonzero cosmological constant. In the presence of (weak) torsion, on the other hand, we find that all the critical points correspond to spatially flat cosmologies and that they all undergo accelerated, de Sitter-like, expansion. Put another way, the torsional equilibrium states identified in this work are flat Friedmann-type universes, which are either \(\varLambda \)-dominated or filled with non-conventional matter (dark energy or phantom). This is expected to change, however, if the weak-torsion assumption is relaxed (see Sect. 3.3 here).

Perturbing the aforementioned fixed points, we employ standard dynamical-system techniques to determine their linear stability and then complete the phase-space portraits of their dynamical evolution. Our results show that, with one exception that leads to a “saddle point”, Friedmann-like cosmologies equipped with a weak torsion field are generally stable attractors, either asymptotically or in the Lyapunov sense. More specifically, the attractors correspond to accelerating universes with (weak) torsion, whereas the saddle point marks the transition from an accelerated torsional cosmology to a torsion-free (also accelerating) model.

The plan of the paper is as follows: In Sect. 2 we introduce the key equations of a torsional Friedmann-type universe. The dimensionless variables, the associated autonomous dynamical systems and their critical points are defined and obtained in Sect. 3. We study the stability of the critical points and identify some subtle issues surrounding equilibrium points with zero eigenvalues in Sect. 4. There, we also provide the phase portraits of our dynamical study. Finally, we summarise our conclusions in Sect. 5.

## 2 Friedmann-like universes with torsion

The spatial homogeneity and isotropy of the Friedmann universes severely restricts the forms of torsion they can accommodate naturally. In particular, the torsion fields allowed in an FRW cosmology must depend only on time and should have vanishing spacelike parts.

### 2.1 The torsion field

^{1}\(u_a\) is a timelike 4-velocity vector (i.e. \(u_au^a=-1\)) and \(h_{ab}=g_{ab}+u_au_b\) is the symmetric spacelike tensor orthogonal to it (i.e. \(h_{ab}=h_{ba}\), \(h_{ab}u^b=0\) and \(h_a{}^a=3\)). We also note that an immediate consequence of (1) and (2) is that \(S_a\) becomes the sole carrier of the torsion effects.

### 2.2 The \(\varOmega \)-parameters

### 2.3 The deceleration parameter

## 3 The autonomous system

Starting from the Friedmann equations given in Sect. 2.2 earlier, one can arrive to an autonomous system of dynamical equations describing the phase-space evolution of the FRW-like models in terms of the four \(\varOmega \)-parameters defined in the same section.

### 3.1 The dynamical equations

### 3.2 The case of weak torsion

*Einstein-de Sitter universe*(with \(\varOmega _K=0= \varOmega _{\varLambda }\) and \(w=0\)) has \(q=1/2-\varOmega _{\phi }\simeq 1/2\), due to the weakness of the torsion field. On the other hand, in the presence of (weak) torsion, the

*coasting universe*solution (with \(K=0=\varLambda \) and \(w=-1/3\)) has \(q\simeq -\varOmega _{\phi }/2\ne 0\).

### 3.3 The equilibrium points

**(i)** The “trivial” fixed points, namely \((\varOmega _K,\varOmega _{\varLambda },\varOmega _{\phi })=(0,0,0)\) with \(\varOmega _{\rho }=1\) and \(q=(1+3w)/2\), \((\varOmega _{\rho }, \varOmega _{\varLambda },\varOmega _{\phi })=(0,0,0)\) with \(\varOmega _K=1\) and \(q=0\) and \((\varOmega _{\rho },\varOmega _K,\varOmega _{\phi })=(0,0,0)\) with \(\varOmega _{\varLambda }=1\) and \(q=-1\). The first fixed point corresponds to the familiar (torsionless) *Friedmann universes* with conventional matter, Euclidean spatial geometry and no cosmological constant. The second and the third are the classical *Milne* and *de Sitter* solutions respectively.

**(ii)**Assuming non-zero torsion, we demand that \(\varOmega _{\phi }\ne 0\). This ensures that \(q\simeq -1\ne 0\) always (see Eq. (24)), which in turn implies that \(\varOmega _K=0\) at all times (see (23a)).

^{2}On the other hand, expression (23b) allows for \(\varOmega _{\varLambda }\ne 0\) and therefore for a nonzero cosmological constant. On using the above, relations (6) and (22) combine to give

^{3}In the case of a radiative fluid with \(w=1/3\), the above constraints become \(\varOmega _{\rho }\simeq -\varOmega _{\phi }/4\) and \(\varOmega _{\varLambda }\simeq 1-3\varOmega _{\phi }/4\). For pressure-free matter, that is for \(w=0\), expressions (25) and (26) translate into \(\varOmega _{\rho }\simeq -\varOmega _{\phi }/3\) and \(\varOmega _{\varLambda }\simeq 1-2\varOmega _{\phi }/3\) respectively. In either of the aforementioned cases \(\bar{\varOmega }_{\phi }\) has to be negative to guarantee “ghost”-free matter with \(\bar{\varOmega }_{\rho }\) positive. Also, given that \(|\varOmega _{\phi }|\simeq 4|\chi |\ll 1\), we are always dealing with a \(\varLambda \)-dominated, spatially flat FRW-like universes with small amounts of matter (in the form of radiation or dust respectively) and weak torsion. Put another way, Eqs. (25), (26) describe

*de Sitter*-like universes, which is in agreement with the value of the their deceleration parameters (recall that \(q\simeq -1\), when \(\varOmega _{\phi }\ne 0\)).

**(iii)**Allowing for torsion, but switching the cosmological constant off (i.e. assuming that \(\varOmega _{\phi }\ne 0\) and \(\varOmega _{\varLambda }\equiv 0\)), the fixed point defined by (25), (26) has \(q\simeq -1\), \(\varOmega _K=0\),

## 4 Stability analysis

In dynamical terms, the fixed points identified in the last section may be stable attractors, unstable repulsors, or intermediate saddle points. We can determine the stability of the aforementioned equilibrium configurations by perturbing them and then studying their response.

### 4.1 Perturbing the equilibrium points

^{4}

### 4.2 Stability of fixed points with \(\bar{\varOmega }_{\phi }=0\)

### 4.3 Stability of fixed points with \(\bar{\varOmega }_{\phi }\ne 0\) and \(\bar{\varOmega }_{\varLambda }\ne 0\)

#### 4.3.1 The case of \(\omega _K=0\)

#### 4.3.2 The case of \(\omega _K\ne 0\)

Allowing for curvature perturbations and assuming radiative matter, the linear system is given by (45). Then, the characteristic polynomial is \(\mathcal {P}_1(\lambda )= -\lambda (\lambda +2)(\lambda +4-3\bar{\varOmega }_{\phi }/2)\), with eigenvalues \(\lambda _1=0\), \(\lambda _2=-2\) and \(\lambda _3=-4+3\bar{\varOmega }_{\phi }/2<0\) (since \(|\bar{\varOmega }_{\phi }|\ll 1\)). When dealing with non-relativistic (pressure-free) matter, the linear system is (46) and the characteristic polynomial reads \(\mathcal {P}_2(\lambda )= -\lambda (\lambda +2)(\lambda +3-\bar{\varOmega }_{\phi })\). Here, the eigenvalues are \(\lambda _1=0\), \(\lambda _2=-2\) and \(\lambda _3=-3+\bar{\varOmega }_{\phi }<0\) (given that \(|\bar{\varOmega }_{\phi }|\ll 1\)).

### 4.4 Stability of fixed points with \(\bar{\varOmega }_{\phi }\ne 0\) and \(\bar{\varOmega }_{\varLambda }=0\)

^{5}

## 5 Discussion

Dynamical system techniques have been extensively used to study the qualitative evolution of a wide range of cosmological solutions (e.g. see [22, 23] for reviews). Among others, there have been applications to cosmologies with torsion (see [24] and references therein.), Nevertheless, to the best of our knowledge, qualitative methods have not been used to study cosmological models based on pure Einstein-Cartan gravity, which is the simplest classical extension of general relativity. Here, we have attempted a step in this direction, by employing dynamical systems to study Friedmann-like universes with torsion. Before proceeding, however, one should bear in mind that the high symmetry of the FRW spacetimes imposes severe constraints on the form of the allowed torsion field [13]. Therefore, following on the work of [14], we have considered vectorial torsion, determined by a single scalar function of time (\(\phi =\phi (t)\)). We also introduced an effective density parameter (\(\varOmega _{\phi }\)) to measure the torsion contribution to the total (effective) energy density of the universe. Then, assuming weak torsion (namely setting \(\varOmega _{\phi }\ll 1\)), we wrote the associated Einstein-Cartan equations as an autonomous system of differential equations.

The torsional equilibrium states of the aforementioned system corresponded to accelerating universes with zero spatial curvature. These were either \(\varLambda \)-dominated cosmologies, or they were filled with non-conventional matter, which satisfied a dark-energy/phantom equation of state. The stability analysis of these fixed points showed that they are all stable attractors, either asymptotically or à la Lyapunov, with the exception of the phantom-dominated solution. The latter was found to act as an intermediate saddle point, marking the transition from an accelerated torsional Friedmann-like universe to its (also accelerating) torsion-free counterpart.

Our results so far have been obtained under the assumption of weak torsion, with the dimensionless parameters \(\phi /H\) and \(\varOmega _{\phi }=-4\phi /H\) restricted to values considerably smaller than unity. Relaxing these constraints, while remaining within the FRW framework, should add extra degrees of freedom to the solutions. The associated equilibrium points, in particular, are very likely to exhibit a richer and more versatile behaviour (like that reported in [14] for example) and they should not necessarily identify themselves with accelerating, spatially flat spacetimes (see footnote 2 in Sect. 3.3). However, in order to accommodate strong torsion fields to our analysis, one needs (among others) an evolution law for the torsion scalar. We will return to the investigation of the strong-torsion regime in our future work, as it goes beyond the scope of the present paper.

## Footnotes

- 1.
- 2.
The constraints \(q\simeq -1\) and \(\varOmega _K=0\) are a direct consequence of our weak torsion assumption. This is reflected in Eq. (13), which ensures that the aforementioned conditions do not apply for a general torsion field. In that case, however, one needs an evolution equation for the the torsion scalar (\(\phi \)), in order to proceed.

- 3.
Following (22), the value \(w=-1\) of the barotropic index is also incompatible with our assumption that \(\varOmega _{\phi }\ne 0\), which meant that \(q=-1\) and \(\varOmega _K=0\).

- 4.
Hereafter, overbars will always indicate variables evaluated at the equilibrium points.

- 5.
Keeping \(\bar{\varOmega }_{\varLambda }=0\), but allowing for nonzero perturbations (i.e. assuming that \(\omega _{\varLambda }\ne 0\)), we find that \(\omega _{\varLambda }^{\prime }=0\) (see Eq. (39) in Sect. 4.1). Incorporating this equation to (59) adds a zero eigenvalue to the system. This alters the nature of the stability, which is no longer asymptotic but of the Lyapunov type.

## Notes

### Acknowledgements

JDB was supported by the Science and Technology Facilities Council (STFC) of the UK. CGT acknowledges support from a visiting fellowship by Clare Hall and visitor support by DAMTP at the University of Cambridge, where the main part of this work was conducted.

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