Stability of the Einstein static Universe in Einstein–Cartan–Brans–Dicke gravity
Abstract
In the present work we consider the existence and stability of Einstein static ES Universe in Brans–Dicke (BD) theory with non-vanishing spacetime torsion. In this theory, torsion field can be generated by the BD scalar field as well as the intrinsic angular momentum (spin) of matter. Assuming the matter content of the Universe to be a Weyssenhoff fluid, which is a generalization of perfect fluid in general relativity (GR) in order to include the spin effects, we find that there exists a stable ES state for a suitable choice of the model parameters. We analyze the stability of the solution by considering linear homogeneous perturbations and discuss the conditions under which the solution can be stable against these type of perturbations. Moreover, using dynamical system techniques and numerical analysis, the stability regions of the ES Universe are parametrized by the BD coupling parameter and first and second derivatives of the BD scalar field potential, and it is explicitly shown that a large class of stable solutions exists within the respective parameter space. This allows for non-singular emergent cosmological scenarios where the Universe oscillates indefinitely about an initial ES solution and is thus past eternal.
1 Introduction
It is well known that inflation, a short-lived and prompt accelerated cosmic expansion era in the very early Universe, is a successful theory in solving some problems from which the standard hot big bang cosmology suffers. Inflationary cosmology predicts a nearly scale-invariant power spectrum for primordial curvature perturbations, which was confirmed by cosmic microwave background (CMB) observations [1, 2, 3, 4]. In spite of its successes, the inflationary scenario still suffers from the problem of initial singularity of the Universe. Some models have been proposed so far in order to cure this problem, i.e., some mechanisms within the framework of quantum gravity such as pre-big bang [5, 6] and cyclic scenarios [7, 8, 9, 10] in string/M theory. Moreover, it has been shown that bouncing cosmology [11] and emergent Universe (EU) scenario [12, 13], as an alternative to cosmic inflation, can also avoid the big bang singularity. The concept of EU is a very interesting idea in standard cosmology with the aim of searching for singularity free inflationary models. In this model, our Universe has no time-like singularity, it is ever existing and has almost a static behavior (Einstein static (ES) state) in the infinite past (\(t\rightarrow -\infty \)) and then evolves into an inflationary era. The Universe is then originated from the ES state rather than the initial big bang singularity. During the last decades, many authors have proposed ES Universe within different scenarios as it provides a setting in which the Universe is ever existing and large enough so that the spacetime may be treated at classical levels. In 1930, Eddington studied the stability of ES solutions in general relativity (GR) and found that these solutions are unstable against homogeneous and isotropic perturbations [14]. In 1967, Harrison introduced a model of closed Universe stuffed from radiation in the presence of a cosmological constant, which asymptotically coincides with the ES model in the infinite past [15] but the scenario does not exit to an inflationary phase. Later studies carried out by Gibbons [16, 17] and Barrow and his coworkers [18] found that the ES Universe with a perfect fluid is neutrally stable with respect to small inhomogeneous vector and tensor linear perturbations, and against scalar perturbations if the sound speed of perfect fluid fulfills \(c_s^2>1/5\). A similar cosmological scenario was considered by Ellis and Maartens where the possibility of avoiding the big-bang singularity without resorting to a quantum regime for spacetime, was investigated [12]. Work along this line has been performed within different models among which we can quote, a closed Universe with a minimally coupled scalar field \(\phi \) and self-interacting potential \(V(\phi )\) [13], EU filled with exotic matter [19], brane-world scenario [20, 21, 22, 23, 24, 25], Einstein–Gauss–Bonnet theory [26, 27], \(f(\mathsf{R})\) theory [28, 29, 30, 31, 32], \(f(\mathsf{T})\) gravity [33], loop quantum cosmology [35, 36, 37, 38, 39, 40] and other gravity theories [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. Moreover, the study of this subject in non-Riemannian spacetimes has been done and recently, the existence and stability of an ES Universe in the framework of Einstein–Cartan (EC) theory has been investigated in [53]. The author has shown that for a spatially closed Universe filled with a Weyssenhoff spinning fluid there is a stable ES state and the Universe can live at this state past-eternally. However, as this stable state corresponds to a center equilibrium point, the Universe may not naturally evolve from it into an inflationary phase. Further studies and developments on this issue has been done in [54] where it is shown that an emergent scenario that avoids the initial singularity of the Universe can be successfully implemented in EC theory.
The ES model has been also studied in the well-known Brans–Dicke (BD) theory and it is shown that a stable past-eternal static solution can be obtained that eventually enters into an unstable phase where the stability of the solution is broken leading to an inflationary period [55]. The BD theory which is an alternative gravity theory is a natural generalization of GR where the gravitational coupling constant is replaced by a scalar field [56, 57]. This theory in which, gravitational interactions are described by the metric of a Riemannian spacetime and a non-minimally coupled scalar field on that spacetime, is an attempt towards improving GR from the standpoint of Mach’s principle [56, 57, 58]. The BD theory has received much interests as it appears naturally in the models dealing with supergravity, Kaluza–Klein theories and in the low energy limit of string theories [59, 60, 61, 62, 63, 64, 65, 66]. Recently, the modified BD theory has been investigated on a general spacetime manifold with non-vanishing torsion field [67] and it was shown that both the scalar field and spin of fermionic particles have contribution in generating the spacetime torsion field. This theory can be viewed as a sort of unification of BD theory and EC theory, i.e., the simplest Poincare gauge theory of gravity, in the framework of which, the gravitational interactions are described by means of spacetime curvature and torsion with the sources being energy-momentum and spin tensors [68]. It is well known that the field equation for spacetime torsion in EC theory is purely algebraically coupled to the spin tensor of matter and therefore the presence of torsion is restricted by spinning matter distribution, thus the spacetime torsion cannot propagate outside the matter through the vacuum [69, 70, 71]. However, in BD theory with torsion (that hereafter we call it as Einstein–Cartan–Brans–Dicke (ECBD) theory) there exists an interesting possibility where a varying gravitational coupling could acts as a source of spacetime torsion. Thus from the standpoint of ECBD theory, the BD scalar field can play the role of a mediator field i.e., from one side, it participates within the gravitational interactions through its non-minimal coupling to curvature and from another side, it behaves as a source, alongside with the spin of fermionic matter, for spacetime torsion field. During the last years, some authors have studied cosmological as well as astrophysical aspects of ECBD theory such as, static spherically symmetric spacetimes in vacuum where it is shown that torsion field can propagate via the scalar field even if the spin angular momentum is absent [72]. Cosmological models in the framework of ECBD theory has been built and investigated in [73, 74, 75] and higher dimensional extension of ECBD theory has been studied in [76] where it is shown that the electromagnetic field and the scalar field appear during the reduction of five-dimensional action. In the case of extreme phenomena where the regimes of ultra-strong gravity are present e.g., the late stages of the gravitational collapse scenario, the BD scalar field may have some effects on stellar structure and final fate of the collapse process [77, 78, 79, 80]. Moreover, from the standpoint of cosmological implications, it has been shown that the BD scalar field, though may be undetectable at the present epoch, could play an important role in the very early Universe [81, 82, 83] (see also [84] and references therein). It is therefore of interest to study the possibility of existence and stability of ES Universe in ECBD theory, as the very early Universe was a hot soup of the fundamental particles including fermionic fields and thus torsion field could be present due to the spin effects of fermions [69, 70, 71, 85, 86, 87, 88, 89, 90]. In the present work, Motivated by the these considerations, we seek for the stable solutions representing an ES state for the Universe in the framework of ECBD theory. Our paper is then organized as follows: in Sect. 2 we derive the field equations of ECBD theory considering a Weyssenhoff fluid for the matter content of the Universe. In Sect. 3, we give the conditions for the stable ES solution and proceed with analyzing this solution using dynamical system approach in Sect. 4. In Sect. 5, we summarize and discuss our results.
2 Field equations of ECBD theory
3 Static Universe in ECBD
The corresponding r–m relation corresponding to some given potentials
Potential \(V(\Phi )\) | r | m | r–m relation |
---|---|---|---|
\(\Phi ^{\alpha }\) | \(\alpha \) | \(\alpha -1\) | \(m=r-1\) |
\(\lambda \Phi ^{\alpha }+\eta \Phi ^{\beta }\) | \(\frac{\lambda (\alpha -\beta ) \Phi ^{\alpha }}{\lambda \Phi ^{\alpha }+\eta \Phi ^{\beta }}+\beta \) | \(\frac{\alpha \lambda (\alpha -\beta ) \Phi ^{\alpha }}{\alpha \lambda \Phi ^{\alpha }+\beta \eta \Phi ^{\beta }}+\beta -1\) | \(m=-\frac{\alpha \beta }{r}+\alpha +\beta -1\) |
\(\Phi ^{\alpha } \log ^{\beta }(\lambda \Phi )\) | \(\frac{\beta }{\log (\lambda \Phi )}+\alpha \) | \(\frac{\alpha }{\alpha \log (\lambda \Phi )+\beta } +\frac{\beta -1}{\log (\lambda \Phi )}+\alpha -1\) | \(m=-\frac{(r-\alpha )^2}{\beta r}+r-1\) |
\(\Phi ^{\alpha } \exp \left( \lambda \Phi ^{\beta }\right) \) | \(\alpha +\beta \lambda \Phi ^{\beta }\) | \(\beta \left( \frac{\alpha \beta }{\beta \lambda \Phi ^{\beta }+\alpha }+\lambda \Phi ^{\beta }\right) +\beta +\alpha -1\) | \(m=\frac{\alpha \beta }{r}+r-(\beta +1)\) |
4 Stability of ECBD theory through the dynamical system approach
Point | Coordinates \((\mathsf{X,Y,Q,Z})\) |
---|---|
\(\mathsf{P}_{1}\) | \(\left( 0,0,\frac{(50-51 \gamma )}{3 (13 \gamma +4 (\gamma -2) r)},\frac{4 (3 \gamma +(2-3 \gamma ) r)}{3 (13 \gamma +4 (\gamma -2) r)}\right) \) |
\(\mathsf{P}_{2}\) | \(\left( 0,-i,0,0\right) \) |
\(\mathsf{P}_{3}\) | \(\left( 0,i,0,0\right) \) |
\(\mathsf{P}_{4}\) | \(\left( -\frac{1}{3\sqrt{2 (1 + \omega ) + r(4 - r) }},\frac{r-2}{3\sqrt{2 (1 + \omega ) + r(4 - r) }},\frac{4(3+\omega )}{3 [(-\,4 + r) r - 2 (1 + \omega )]},0\right) \) |
\(\mathsf{P}_{5}\) | \(\left( \frac{1}{3\sqrt{2 (1 + \omega ) + r(4 - r)}},\frac{2-r}{3\sqrt{2 (1 + \omega ) + r(4 - r)}},\frac{4(3+\omega )}{3 [(-\,4 + r) r - 2 (1 + \omega )]},0\right) \) |
\(\mathsf{P}_{6}\) | \(\left( -\frac{\sqrt{51 \gamma -50}}{9\sqrt{2(2-\gamma ) (3 + \omega )}}i,0,0,\frac{8 (3 \gamma -\,4) }{27 (\gamma -2)}\right) \) |
\(\mathsf{P}_{7}\) | \(\left( \frac{\sqrt{51 \gamma -50}}{9\sqrt{2(2-\gamma ) (3 + \omega )}}i,0,0,\frac{8 (3 \gamma -4) }{27 (\gamma -2)}\right) \) |
\(\mathsf{P}_{8}\) | \(\left( \frac{(2-3 \gamma )}{3 \sqrt{\gamma (3 \gamma -8) (2 \omega +3)+8 (\omega +1)}},\frac{(4-3 \gamma )}{\sqrt{\gamma (3 \gamma -8) (2 \omega +3)+8 (\omega +1)}},0,0\right) \) |
\(\mathsf{P}_{9}\) | \(\left( \frac{(3 \gamma -2 )}{3 \sqrt{\gamma (3 \gamma -8) (2 \omega +3)+8 (\omega +1)}},\frac{(3 \gamma -4)}{\sqrt{\gamma (3 \gamma -8) (2 \omega +3)+8 (\omega +1)}},0,0\right) \) |
As it is not possible to visualize the full 4D phase-space constructed by \((\mathsf{X,Y,Z,Q})\) variables, we proceed to pursue the dynamical evolution of the system in 3D and 2D subspaces of the full phase-space. Figure 3 presents 2D and 3D slices of full phase-space for a Universe dominated by a dust fluid. We observe that, given the initial data set, the Universe experiences fluctuations around the center equilibrium point. The 3D simulation gives us better visualizing of the dynamical evolution of the Universe within the subspace of 4D phase-space. We observe that though the amplitude of departures from stability grows for a limited time interval in some regions of the 3D phase-space, the system comes back to the neighborhood of the center fixed point as the time passes and it keeps behaving in this way for later times. In Figs. 4 and 5 we have plotted numerical integration of the system (49)–(52) for other EoS parameters, where we observe that the overall behavior of the system is same as the dust case but with different orbits. In Figs. 6, 7 and 8 we have sketched the dynamical evolution of the scale factor and BD scalar field where it is seen that these quantities undergo oscillations around their equilibrium values. The scale factor remains finite and nonzero and also the change from an expanding regime to a contracting one occurs smoothly, see the inset diagrams in Figs. 6, 7 and 8. This behavior of the scale factor signals that the Kretschmann invariant \((\mathsf{K}=12[(\ddot{a}/a)^2+(\dot{a}/a)^4])\) behaves regularly during the dynamical evolution of the Universe, hence, the ES state lives within a nonsingular stable regime, past-eternally. One may also argue that the perturbation modes never damp out, nor do they grow up in time but instead, they continue to exist (with oscillating and non-growing amplitude) as the Universe evolves. If the perturbation modes go to zero as the time passes, the Universe may undergo an unstoppable collapse process and thus a spacetime singularity may be the end-state of the Universe. In case these modes grow up, inevitable expansion of the Universe could occur so that the BD scalar field can go to a vanishing value which would correspond to an infinitely strong gravitational coupling. Hence, we require that the perturbations remain and evolve within the system in order that they could act as a balancing effect and possibly prevent the Universe from instantaneous collapse or expansion, at least, as long as the Universe stays in an static state. However, the Universe have to eventually leave the static state and enters to an inflationary phase. A suitable mechanism which could provide a setting in order that the Universe departs from the static regime is a slight change within the EoS parameter so that this parameter turns out to be time dependent temporarily and under this change within the EoS parameter, static equilibrium can be broken and the Universe could have chance to escape the static state and eventually enters an inflationary era. The study of how perturbations could affect such a transition may not be an easy task but one could intuitively imagine that the perturbations which have had dynamical evolution along with the Universe could possibly assist it to escape the static regime and begins its inflationary expansion. In order to have an exit to inflation scenario, we assume that the EoS parameter turns out to be time dependent with functionality \(\gamma (\eta )=\gamma _0+\gamma _1(1-\mathrm{exp}(\alpha \eta /\eta _0))\) and then solve the system of differential equations (49)–(52) numerically. The results are shown in Fig. 9 where we see that after oscillations around its static value, the scale factor begins to increase (with no return) in a short time period allowing thus the Universe to enter an inflationary regime. Meanwhile, having oscillations around its static value, the BD scalar field also starts to decrease and finally settles down to a non-vanishing constant value and this value can be set, using a suitable system of units, to the gravitational coupling constant. It is worth mentioning that since in addition to the spin effects, the BD scalar field could also act as a source of torsion field, then the ECBD theory is reduced to the EC theory after the BD scalar field comes to a rest at a constant value but the torsion field may not be totally vanished as the spin effects are still present.
5 Concluding remarks
The study of early Universe physics has been a hot topic in the fields of cosmology and astronomy. Over the past decades, a huge amount of research works have been done in this arena, which have extended our knowledge of the origin and evolution of the Universe. One of the most beautiful and popular cosmological models describing a non-singular state of the early Universe is the ES model that the study of which has been extensively carried out in the literature. In the present work we investigated the existence and stability of the ES Universe in the context of ECBD theory and showed that the corresponding ES solution is stable in the sense of dynamically corresponding to a center equilibrium point. Moreover, The solution is stable against small homogeneous perturbations in the sense that these perturbations prompt the scale factor and BD scalar field to oscillate about their static values so that the Universe undergoes small departures from its stable phase in the from of contractions and expansions. Since the effects of spin contribution to the field equation (18) show themselves as negative pressure, one may argue that the spacetime torsion generated by spin contribution, induces gravitational repulsion in fermionic matter at extremely high densities and tends to destabilize the ES state via this repulsive effect and finally breaks down the stability. Examples of cosmological as well as astrophysical nonsingular scenarios have shown that the initial singularity of the Universe or the singularity as the end-state of a gravitational collapse process can be remedied as a result of the repulsive effects due to spin [104, 105, 106, 107, 108, 109]. However, we observed that even if this repulsive effect is considered within the BD theory the static stable state for the Universe could exist but with different conditions on model parameters in comparison to the BD theory without torsion and spin effects [55]. The conditions on stability as discussed in the present model put some restrictions on the values of EoS and BD parameters along with a dimensionless parameter which is related to the scalar field potential and its derivative. Hence, it is easy to figure out that the nature of the fixed point crucially depends on model parameters so that this dependence could be helpful for providing an exit to inflation scenario, in which, assuming a slowly varying EoS parameter for a short time interval, the Universe that has been living in a stable past-eternal static state (a center equilibrium point) could eventually enter into a phase where the stability of the solution is broken leading to an inflationary era. It is not however far fetched to ideate the possibility of having a time dependent EoS parameter for a short period of time due to disturbances that the Universe were experiencing. Another possibility is that, as the BD coupling parameter bears the ratio of the scalar to tensor couplings to matter in such a way that the larger the values of this parameter the smaller the scalar field effects, one can consider a running BD coupling parameter in the sense that it gets smaller values, therefore significant contribution of the scalar field at the early stages of the Universe, while evolving to larger values at present epoch [110, 111].
Notes
Acknowledgements
This work has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM) under research project No. 1/6025–70.
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