# Friedmann-like universes with torsion

## Abstract

We consider spatially homogeneous and isotropic cosmologies with non-vanishing torsion, which assumes a specific form due to the high symmetry of these universes. Using covariant and metric-based techniques, we derive the torsional versions of the continuity, the Friedmann and the Raychaudhuri equations. These show how torsion can drastically change the standard evolution of the Friedmann models, by playing the role of the spatial curvature or that of the cosmological constant. We find, for example, that torsion alone can lead to exponential expansion and thus make the Einstein–de Sitter universe look like the de Sitter cosmos. Also, by modifying the expansion rate of the early universe, torsion could have affected the primordial abundance of helium-4. We show, in particular, that torsion can *reduce* the production of primordial helium-4, unlike other changes to the standard thermal history of the universe. These theoretical results allow us to impose strong observational bounds on the relative strength of the associated torsion field, confining its ratio to the Hubble rate within the narrow interval (\(-\,0.005813,\,+\,0.019370\)) around zero. Finally, turning to static spacetimes, we demonstrate that there exist torsional analogues of the Einstein static universe with all three types of spatial geometry. These models can be stable when the torsion field and the universe’s spatial curvature have the appropriate profiles.

## 1 Introduction

General relativity advocates a geometrical interpretation of gravity, which ceases being a force and becomes a manifestation of the non-Euclidean geometry of the host spacetime. The theory is founded on the assumption of Riemannian geometry, where deviations from Euclidean flatness are described by the symmetric Levi–Civita connection, namely by the Christoffel symbols. Nevertheless, there is no fundamental theoretical reason, apart perhaps from simplicity, for making such an assumption. Allowing for a general asymmetric affine connection introduces spacetime torsion and therefore new geometrical degrees of freedom into the system, since there is now an independent torsional field in addition to the metric. The literature contains a number of suggestions for experimentally testing gravitational theories with non-zero torsion (see [1, 2, 3, 4, 5, 6, 7] for a representative though incomplete list). As yet, however, there is no experimental or observational evidence to support the existence of spacetime torsion. The main reason is that, typically, the effects of torsion start becoming appreciable at extremely high energy densities. These densities can be achieved only in the deep interior of compact objects, like neutron stars and black holes, or during the very early stages of the universe’s expansion. Such environments are still beyond our experimental capabilities.

Torsion does not naturally fit into highly symmetric spacetimes, like the Friedmann–Robertson–Walker (FRW) models of standard cosmology. Given the spatial homogeneity and isotropy of the latter, the allowed torsion field must satisfy a specific profile [8], which falls into the class of the so-called vectorial torsion fields [9, 10, 11]. Practically speaking, spacetime torsion and the associated matter spin are fully determined by a scalar function that depends only on time. Such choices allow us to construct and study the torsional analogues of the classic Friedmann universes. In the process we show that, despite the presence of torsion, the high symmetry of the FRW host preserves the symmetry of the associated Ricci curvature tensor, which implies that the corresponding Einstein and energy-momentum tensors are symmetric as well. Then, using both 1+3 covariant and metric-based techniques, we present the three key formulae monitoring the evolution of these models, namely the analogues of the Friedmann, the Raychaudhuri, and the continuity equations. These enable us to “quantify” the torsion input to the total effective energy density of the system, by means of an associated \(\varOmega \)-parameter, as well as its contribution to the kinematic variables of the cosmological models in question, namely to the Hubble and to the deceleration parameters.

Phenomenologically speaking, torsion can play the role of spatial curvature and reproduce the effects of a cosmological constant, or those of dark energy. As a result, torsional cosmologies (with or without matter) can experience accelerated expansion. We find, in particular, that torsion can force the Milne and the Einstein–de Sitter universes into a phase of accelerated expansion analogous to that of their de Sitter counterpart. These examples suggest that a torsion-dominated early universe, or a dust-dominated late-time cosmos with torsion, could go through a phase of accelerated expansion without the need of a cosmological constant, the inflaton field, or dark energy. Analogous effects were reported in [12, 13, 14, 15, 16], suggesting that torsional cosmologies might deserve further scrutiny.

Looking for observational signatures of torsion, we find that the latter can affect the outcome of primordial nucleosynthesis, since it changes the expansion rate of the universe. This can be used to put observational constraints on the allowed torsion fields. Here, we are able to calculate the torsion effect on the amount of helium-4 produced during primordial nucleosynthesis. Combining our theoretical result with the currently allowed range of the primordial helium-4 abundance, leads to a very strong constraint on the strength of the associated torsion field. In particular, the relative torsion contribution to the volume expansion of the universe is found to lie within the narrow interval (\(-\,0.005813,\,+\,0.019370\)) around zero.

Finally, we turn our attention to static spacetimes with torsion. In particular, we study the structure of the torsional analogue of the Einstein-static universe and investigate its linear stability. We find that there exist static models with all three types of spatial geometry, that is Euclidean, spherical or hyperbolic. Our last step is to use standard perturbative techniques to test the linear stability of these new spacetimes. We show that static solutions with positive curvature are unstable, while those with zero or negative 3-curvature can achieve stable configurations.

## 2 Spacetimes with torsion

Riemannian geometry demands the symmetry of the affine connection, thus ensuring torsion-free spaces. Nevertheless, by treating torsion as an independent geometrical field, in addition to the metric, one extends the possibilities to the so-called Riemann–Cartan spaces.

### 2.1 Torsion and contortion

^{1}

### 2.2 Field equations and Bianchi identities

### 2.3 Kinematics

^{2}The volume scalar monitors the convergence/divergence of the worldlines tangent to the \(u_a\)-field, while the shear and the vorticity describe kinematic anisotropies and rotation respectively. Finally, non-zero 4-acceleration implies that the aforementioned worldlines are not autoparallel curves.

## 3 FRW-like models with torsion

Spatially homogeneous and isotropic, Friedmann-like, spacetimes cannot naturally accommodate any arbitrary form of torsion. In what follows, we will investigate the implications of such highly symmetric torsion fields for the evolution of the cosmological spacetime.

### 3.1 The torsion field

### 3.2 Conservation laws

Applying relations (12) and (13) to the second of the twice-contracted Bianchi identities (see Eq. (8) in Sect. 2.2), it is straightforward to show that the right-hand side of the latter relation vanishes. This ensures that \(G_{[ab]}=0\), which in turn guarantees that \(R_{[ab]}=0\) and \(T_{[ab]}=0\) as well. Consequently, in spatially homogeneous and isotropic spacetimes, the Ricci and the energy-momentum tensors retain their familiar (Riemannian) symmetry despite the presence of torsion.^{3}

*p*are its energy density and isotropic pressure respectively. Then, starting from (16), we obtain

### 3.3 The Raychaudhuri equation

*H*being the associated Hubble parameter.

^{4}According to (20), the divergence/convergence of worldlines in FRW-type cosmologies with torsion is not solely determined by the scale-factor evolution. Substituting (20) into Eq. (18), we obtain

### 3.4 The Friedmann equations

### 3.5 Characteristic solutions

The analytical solutions presented in this section are examples aiming at demonstrating the versatility of the torsion effects upon the FRW-like host. In order to maintain the analytical nature of our treatment, as well as for reasons of mathematical simplicity and physical transparency, we will do so by assuming a simple time-invariant torsion field (i.e. one with \(\phi =\phi _0=\) constant).^{5} Nevertheless, our formalism can be readily extended to time-varying torsion as well.

#### 3.5.1 Vacuum and torsion-dominated solutions

^{6}When \(\rho =0=p=\varLambda \), \(K=-\,1\) and \(\phi \ne 0\), we obtain what one might call the torsion analogue of the classical Milne universe. Then, Eq. (24) factorises as \(({\dot{a}}/a+2\phi +1/a) ({\dot{a}}/a+2\phi -1/a)=0\), giving

Given that in empty models torsion cannot be associated with the spin of the matter, it would have to be treated as an independent, generic spacetime feature (the same also holds for the curvature). Nevertheless, to first approximation, our vacuum solutions also govern the evolution of a Friedmann-like universe with matter, provided that torsion dominates over the matter (i.e. for \(\varOmega _{\phi }\gg \varOmega _{\rho }\) – see Eq. (26) in Sect. 3.4). Put another way, the solutions obtained in this section can be seen as limiting cases of low-density FRW-type universes with non-zero torsion.

#### 3.5.2 Solutions with matter

^{7}

The solutions presented in Sects. 3.5.1 and 3.5.2 are characteristic of the versatile and the occasionally nontrivial implications of the torsion field, even when the latter takes the very restricted form imposed by the high symmetry of the Friedmann-like host.

## 4 Observational bounds on cosmic torsion

The literature contains a number of proposals for observational tests of torsion, the majority of which work within the realm of our solar system [1, 2, 3, 4, 5, 6, 7]. Here, we will attempt to put cosmological bounds on the torsion field, by exploiting the fact that it “gravitates” and therefore modifies the expansion dynamics of the host universe.

### 4.1 Steady-state torsion

^{8}The former relation shows that torsion changes the Hubble-flow rate, which means that it can affect physical interactions that are sensitive to the pace of the cosmic expansion, like the Big-Bang Nucleosynthesis (BBN) of helium-4 for example (see Sect. 4.2 next).

### 4.2 BBN bounds on torsion

*reduce*the freeze-out temperature. Then, the neutron-to-proton ratio (\({\mathcal {N}}=n/p\)) will freeze-in at

*lower*temperatures and the residual helium-4 abundance will

*decrease*compared to that in the standard (torsion-free) Friedmann universe. The slowing of the expansion rate allows neutrons and protons to remain in non-relativistic kinetic equilibrium down to lower temperatures, with fewer neutrons per proton surviving before the equilibrium is broken at \(T_{fr}\).

The above torsion effect provides a very rare (if not unique) example of a modified early-universe model with a reduced helium-4 abundance. All other common modifications (i.e. extra light neutrino species, magnetic fields, anisotropies, Brans–Dicke fields, etc) lead to higher freeze-out temperatures. This increases the frozen-in *n* / *p* ratio and therefore enhances the residual abundance of helium-4. In the presence of torsion this happens when \(\lambda <0\).

## 5 Static spacetimes with torsion

The extra degrees of freedom that torsion introduces are expected to relax some of the standard constraints associated with static spacetimes. We will therefore now turn our attention to the study of static (homogeneous and isotropic) models with torsion.

### 5.1 The Einstein-static analogue

Static spacetimes with non-zero torsion have been studied in the past, assuming matter in the form of the Weyssenhoff fluid [28]. The latter, however, is incompatible with the high symmetry of the FRW-like models and the Einstein static universe as well. For this reason, an unpolarised spin field was adopted, with a spin tensor that averages to zero (e.g. see [29, 30, 31]). Here, instead, we address the FRW-compatibility issue by adopting a form for the torsion/spin fields that is compatible with the spatial isotropy and homogeneity of the Friedmannian spacetimes (see Eqs. (12) and (15) in Sect. 3.1 earlier).

### 5.2 Stability of the static model

Our stability analysis assumed homogeneous linear perturbations, similar to those employed by Eddington in his classic study of Einstein’s static world [33]. This implies that the stable configurations reported here may prove unstable when inhomogeneous perturbations of all the three possible types (i.e. scalar, vector and tensor) are accounted for (see [34] for such a linear-stability analysis on the classic Einstein-static spacetime).^{9}

## 6 Discussion

Allowing for an asymmetric affine connection provides the simplest classical extension of general relativity, by incorporating the effects of spacetime torsion into the theory. The latter can then be use to study a variety of theoretical problems, ranging from singularity theorems and cosmology, to supergravity and quantum gravity (e.g. see [40] and references therein). However, torsion and spin are generally incompatible with the high symmetry of the FRW cosmologies, which means that one needs to consider torsion/spin fields that preserve both the homogeneity and the isotropy of these universes. Here, we have addressed this issue by adopting a specific profile for the torsion tensor that belongs to the class of the vectorial torsion fields and it is monitored by a single scalar function of time [8, 11]. Nevertheless, even this highly constraint form of torsion was found capable of drastically altering the standard evolution of the classic FRW cosmologies.

Using \(1+3\) covariant and metric-based techniques, we derived the associated continuity, Friedmann and Raychaudhuri equations. These allowed us to quantify the relative strength of the torsion effects by means of an associated \(\varOmega \)-parameter. A number of new possibilities emerged. We found that torsion can play the role of the spatial curvature and mimic the effects of the cosmological constant, depending on the specifics of the scenario in hand. The orientation of the torsion vector, relative to the fundamental 4-velocity field, was a decisive factor, since it determines whether torsion will tend to decelerate or accelerate the expansion of the host spacetime. Empty spacetimes with zero 3-curvature, no cosmological constant and non-zero torsion are not necessarily static, but can experience exponential expansion (see also [22] for similar results). The introduction of spatial curvature, or matter, did not seem to change the aforementioned picture. So, in the presence of torsion, the Einstein–de Sitter universe can experience exponential de Sitter-like inflation.^{10} All these findings raise the possibility that universes with non-zero torsion might have gone through an early (or a late) phase of accelerated expansion without requiring a cosmological constant, an inflaton field, or some sort of dark energy.

Looking for potentially observable cosmological signatures of torsion, we considered its effects on primordial nucleosynthesis. We found that torsion can increase, as well as decrease, the production of helium-4, by changing the expansion rate of the universe at the time of primordial nucleosynthesis. Using our solution for a radiation-dominated Friedmann universe, we were able to calculated the expected abundance of helium-4 from primordial nucleosynthesis when torsion is present. Combining these results with the observationally allowed range of the helium-4 abundance, we were able to impose strong constraints on the relative strength of the torsion field.

Our study also found that there exist Einstein-static universes with torsion that are not closed, but can have all three types of spatial curvature. Unlike the classic (torsion-free) Einstein model, for appropriate choices of the torsion field and of the spatial curvature, these static universes can be stable against linear scalar perturbations even for pressureless (dust) matter. Overall, despite the restrictions imposed by their high symmetry, FRW-like universes with torsion exhibit a rich phenomenology that could distinguish them from their general-relativistic counterparts.

## Footnotes

- 1.
In the literature the definitions of the torsion and the contortion tensors vary. In this study, we have adopted the conventions of [17], which follow those of [12, 13], though in the latter the metric signature is (\(+,-,-,-\)). Also note that the tildas will always indicate purely Riemannian (torsion free) variables.

- 2.
Overdots indicate temporal derivatives (along the timelike \(u_{a}\)-field). For instance \(A_{a}={\dot{u}}_{a}= u^{b}\nabla _{b}u_{a}\) by definition. Spatial derivatives (orthogonal to \(u_{a}\)), on the other hand, are denoted by the covariant operator \(\mathrm {D}_{a}=h_{a}{}^{b}\nabla _{b}\). Therefore, \(\varTheta =\mathrm {D}^{a}u_{a}=h^{ab}\nabla _{b}u_{a}\), \(\sigma _{ab}= \mathrm {D}_{\langle b}u_{a\rangle }=h_{\langle b}{}^{d} h_{a\rangle }{}^{c}\nabla _{d}u_{c}\), etc [20, 21]. Also, round brackets denote symmetrisation and square antisymmetrisation, while angled ones indicate the symmetric and trace-free part of second rank tensors (e.g. \(\sigma _{ab}= \mathrm {D}_{\langle b}u_{a\rangle }=\mathrm {D}_{(b}u_{a)}- (\mathrm {D}^{c}u_{c}/3)h_{ab}\) by construction).

- 3.
In order to show the symmetry of the Ricci and energy-momentum tensors in FRW-like models with torsion, one needs to remember that 4-velocity split (see Eq. (10) in Sect. 2.3) reduces to \(\nabla _{b}u_{a}= (\varTheta /3)h_{ab}\) and that \(\nabla _{a}\phi =-{\dot{\phi }}u_{a}\) (since \(\mathrm {D}_{a}\phi =0\) by default) in these highly symmetric spacetimes.

- 4.
Following (20), the dimensionless ratio \(\phi /H\) measures the “relative strength”of the torsion effects.

- 5.
It goes without saying that the same solutions also (approximately) hold in the case of slowly varying torsion.

- 6.
Vacuum torsional spacetimes with no cosmological constant and spherical spatial geometry do not exist in our scheme. Indeed, in such an environment Eq. (24) recasts into \(({\dot{a}}/a+2\phi )^2=-\,1/a^2\), which is impossible.

- 7.We can also extract graduated inflation [26], by setting \(\phi \propto t^{-1}\). Indeed, substituted into Eq. (24), this choice leads to$$\begin{aligned} a= a_0\left( {\frac{t}{t_0}}\right) ^{-2\phi _0t_0}\mathrm {e}^{\pm \sqrt{{\frac{\kappa \rho _0t_0^2}{3(2\phi _0t_0+1)^2}}} \left( {\frac{t}{t_0}}\right) ^{2\phi _0t_0+1}}. \end{aligned}$$(38)
- 8.
When \(\lambda =-\,1/2\) we have \(\varOmega _{\phi }=1\) (see Eq. (26)), corresponding to a purely torsional (empty) FRW-like universe.

- 9.
- 10.
These results were obtained after assuming a time-invariant torsion field. Nevertheless, our formalism can be readily extended to include time-varying torsion (e.g. see footnote 7 and Sect. 4.1), which is what we intent to do in future work. In that case, one might also have to go beyond the analytical treatment and use numerical techniques as well.

## Notes

### Acknowledgements

We would like to thank Peter Stichel for drawing our attention to reference [8]. CGT acknowledges support from a visiting fellowship by Clare Hall College and visitor support by DAMTP at the University of Cambridge. JDB was supported by the Science and Technology Facilities Council (STFC) of the UK.

## References

- 1.R.T. Hammond, Rep. Prog. Phys.
**65**, 599 (2002)ADSCrossRefGoogle Scholar - 2.Y. Mao, M. Tegmark, A.H. Guth, S. Cabi, Phys. Rev. D
**76**, 104029 (2007)ADSCrossRefGoogle Scholar - 3.V.A. Kostelecky, N. Russell, J. Tasson, Phys. Rev. Lett.
**100**, 111102 (2008)ADSCrossRefGoogle Scholar - 4.R. March, G. Bellettini, R. Tauraso, S. Dell’Angello, Phys. Rev. D
**83**, 104008 (2011)ADSCrossRefGoogle Scholar - 5.F.W. Hehl, Y.N. Obukhov, D. Puetzfeld, Phys. Lett. A
**377**, 1775 (2013)ADSMathSciNetCrossRefGoogle Scholar - 6.D. Puetzfeld, Y.N. Obukhov, Int. J. Mod. Phys. D
**23**, 1442004 (2014)ADSCrossRefGoogle Scholar - 7.R.-H. Lin, X.H. Zhai, X.Z. Li, Eur. Phys. J. C
**77**, 504 (2017)ADSCrossRefGoogle Scholar - 8.M. Tsamparlis, Phys. Rev. D
**24**, 1451 (1981)ADSMathSciNetCrossRefGoogle Scholar - 9.G.J. Olmo, Int. J. Mod. Phys. D
**20**, 413 (2011)ADSMathSciNetCrossRefGoogle Scholar - 10.S. Capozzielo, R. Cianci, C. Stornaiolo, S. Vignolo, Phys. Scr.
**78**, 065010 (2008)ADSCrossRefGoogle Scholar - 11.Jimenez J. Beltran, T.S. Koivisto, Phys. Lett. B
**756**, 400 (2016)ADSCrossRefGoogle Scholar - 12.N.J. Poplawski, Phys. Lett. B
**694**, 181 (2010)ADSMathSciNetCrossRefGoogle Scholar - 13.N.J. Poplawski, Astron. Rev.
**8**, 108 (2013)ADSCrossRefGoogle Scholar - 14.A.N. Ivanov, M. Wellenzohn, Astrophys. J.
**829**, 47 (2016)ADSCrossRefGoogle Scholar - 15.S. Akhshabi, E. Qorani, F. Khajenabi, Europhys. Lett.
**119**, 29002 (2017)ADSCrossRefGoogle Scholar - 16.R. Banerjee, S. Chakraborty, P. Mukherjee, Phys. Rev. D
**98**, 083506 (2018)ADSCrossRefGoogle Scholar - 17.K. Pasmatsiou, C.G. Tsagas, J.D. Barrow, Phys. Rev. D
**95**, 104007 (2017)ADSMathSciNetCrossRefGoogle Scholar - 18.F.W. Hehl, P. von der Heyde, G.D. Kerlick, Rev. Mod. Phys.
**48**, 373 (1976)ADSCrossRefGoogle Scholar - 19.F.W. Hehl, Y.N. Obukhov, Ann. Fond. Broglie
**32**, 157 (2007)Google Scholar - 20.C.G. Tsagas, A. Challinor, R. Maartens, Phys. Rep.
**465**, 61 (2008)ADSMathSciNetCrossRefGoogle Scholar - 21.G.F.R. Ellis, R. Maartens, M.A.H. MacCallum,
*Relativistic Cosmology*(Cambridge University Press, Cambridge, 2012)CrossRefGoogle Scholar - 22.D. Iosifidis, C.G. Tsagas, A.C. Petkou, Phys. Rev. D
**98**, 104037 (2018)ADSCrossRefGoogle Scholar - 23.H. Nariai, Prog. Theor. Phys.
**40**, 48 (1969)Google Scholar - 24.C. Mathiazhagen, V.B. Johri, Class. Quantum Gravity
**1**, L29 (1984)ADSCrossRefGoogle Scholar - 25.J.D. Barrow, K.-I. Maeda, Nucl. Phys. B
**341**, 294 (1990)ADSCrossRefGoogle Scholar - 26.J.D. Barrow, Phys. Lett. B
**235**, 40 (1990)ADSMathSciNetCrossRefGoogle Scholar - 27.R.H. Cyburt, B.D. Fields, K.A. Olive, T.-H. Yeh, Rev. Mod. Phys.
**88**, 015004 (2016)ADSCrossRefGoogle Scholar - 28.J. Weyssenhoff, J. Raade, Acta Phys. Polon.
**9**, 7 (1947)Google Scholar - 29.B. Kuchowicz, Gen. Relativ. Gravit.
**9**, 511 (1978)ADSMathSciNetCrossRefGoogle Scholar - 30.M. Gasperini, Phys. Rev. Lett.
**56**, 2873 (1986)ADSCrossRefGoogle Scholar - 31.K. Atazadeh, JCAP
**06**, 020 (2014)ADSMathSciNetCrossRefGoogle Scholar - 32.C.G. Böhmer, Class. Quantum Gravity
**21**, 1119 (2004)ADSCrossRefGoogle Scholar - 33.A.S. Eddington, Mon. Not. R. Astron. Soc.
**90**, 668 (1930)ADSCrossRefGoogle Scholar - 34.J.D. Barrow, G.F.R. Ellis, R. Maartens, C.G. Tsagas, Class. Quantum Gravity
**20**, L155 (2003)ADSCrossRefGoogle Scholar - 35.E.R. Harrison, Rev. Mod. Phys.
**39**, 862 (1967)ADSCrossRefGoogle Scholar - 36.G.W. Gibbons, Nucl. Phys. B
**292**, 784 (1987)ADSCrossRefGoogle Scholar - 37.G.W. Gibbons, Nucl. Phys. B
**310**, 636 (1988)ADSCrossRefGoogle Scholar - 38.J.D. Barrow, K. Yamamoto, Phys. Rev. D
**85**, 083505 (2012)ADSCrossRefGoogle Scholar - 39.J.D. Barrow, C.G. Tsagas, Class. Quantum Gravity
**26**, 195003 (2009)ADSCrossRefGoogle Scholar - 40.M. Blagojevic, F.W. Hehl,
*Gauge Theories and Gravitation*(Imperial College Press, London, 2013)CrossRefGoogle Scholar - 41.J.V. Narlikar,
*Introduction to Cosmology*(Jones and Bartlett, Boston, 1983)zbMATHGoogle Scholar

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