DYNAMICAL SYSTEM ANALYSIS OF DARK ENERGY MODELS IN SCALAR COUPLED METRIC-TORSION THEORIES

Dynamical System Analysis of Dark Energy Models in Scalar Coupled Metric-Torsion Theories

Arshdeep Singh Bhatia and Sourav Sur Department of Physics & Astrophysics
University of Delhi
New Delhi - 110 007, India
arshdeepsb@gmail.com; asbhatia@physics.du.ac.in
sourav.sur@gmail.com; sourav@physics.du.ac.in
Abstract

We study the phase space dynamics of cosmological models in the theoretical formulations of non-minimal metric-torsion couplings with a scalar field, and investigate in particular the critical points which yield stable solutions exhibiting cosmic acceleration driven by the dark energy. The latter is so defined that it effectively has no direct interaction with the cosmological fluid, although in an equivalent scalar-tensor cosmological setup the scalar field interacts with the fluid (which we consider to be the pressureless dust). Determining the conditions for the existence of the stable critical points we check their physical viability in both Einstein and Jordan frames. We also verify that in either of these frames, the evolution of the universe at the corresponding stable points matches with that given by the respective exact solutions we have found in an earlier work (arXiv:1611.00654 [gr-qc]). We not only examine the regions of physical relevance in the phase space when the coupling parameter is varied, but also demonstrate the evolution profiles of the cosmological parameters of interest along fiducial trajectories in the effectively non-interacting scenarios, in both Einstein and Jordan frames.

dark energy theory; alternative theories of gravity; torsion; scalar tensor gravity; phase plane analysis.

1 Introduction

Dynamical stability is a major requirement for cosmological solutions representing dark energy (DE) that supposedly drives the late-time cosmic acceleration [1]. While the question as to how the DE evolves has been contemplated by a plethora of theoretical surmises and conjectures [2, 3, 4], observations have mostly been in favour of a non-dynamical DE, reminiscent of a cosmological constant , at low to moderately high redshifts [5, 6, 7]. However, some scope is there to look for (albeit mild) deviations from the concordant CDM model, comprising of and cold dark matter (CDM) as the dominant constituents of the universe [8, 9, 10]. In fact, the dynamical aspects of the DE are always worth examining, for a sufficiently longer span of evolution, tracing back from deep in the past, till extrapolating to high blueshifts in the future [3, 4, 11]. The theoretical motivation for this is obvious, in view of the well-known fine tuning and coincidence problems affecting the CDM cosmology [8, 9].

Extensive searches for the dynamical DE, within the standard Friedmann-Robertson-Walker (FRW) framework, have mostly accounted for the scalar field candidates, such as quintessence, k-essence, tachyon, dilaton, chameleon, etc. [12, 13, 14, 15, 16], which have had many intriguing features [2, 3]. However, in recent years the focus has shifted to a purely geometric characterization of the DE in the so-called modified gravity theories [17] of e.g. the type [18], where is the Riemannian curvature scalar. Such theories can also be mapped to scalar-tensor theories [19, 20, 21, 22], and hence give rise to interacting (or unified) dark energy–matter scenarios [23] under conformal transformations. One’s perception though, of a ‘geometrical’ DE, is not limited to the formulations in the Riemannian space-time only. We may equally well look into the cosmologies emerging from the rather conventional extensions of General Relativity (GR), such as that formulated in the four-dimensional Riemann-Cartan () space-time with torsion — an antisymmetric tensor field that generalizes the Levi-Civita connections in GR [24, 25, 26, 27, 28, 29, 30]. Torsion is often considered as a geometric entity that provides a classical background for quantized spinning matter, and is therefore an inherent part of a fundamental (quantum gravitational) theory, such as string theory [28, 31, 32]. A completely antisymmetric torsion can have its source in the closed string massless Kalb-Ramond mode [33, 34], with interesting implications in cosmology and astrophysics [35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Among other torsion scenarios of interest in the cosmological context, most notable are those based on the teleparallel theories [45], extended gravity theories [46], Poincaré gauge theory of gravity [47, 48, 49], etc.

We in this paper turn our attention to the formalism of a metric-scalar-torsion (MST) theory developed in an earlier work (henceforth ‘paper I’) [50]. Such a theory deals with the Lagrangian non-minimally coupled to a scalar field (of presumably primordial origin), in a way that no uniqueness problem arises [28, 50, 51]. Now, in the standard cosmological framework, the torsion degrees of freedom get restricted by the FRW metric structure [30]. Also since acts as the source of the trace mode of torsion (via the corresponding equation of motion), we effectively have a scalar-tensor equivalent MST setup. The pseudo-trace mode of torsion can give rise to a mass term for , via suitable augmentation of the effective action with say, some higher order torsion terms [50]. Considering further a pressureless dust-like cosmological matter, viable DE solutions in Einstein and Jordan frames have been worked out analytically in paper I [50], keeping the cosmological parameters within the corresponding error estimates for the CDM model from recent observations. However, there remains the important question:

would these (and possibly a few other) solutions, persist over time (i.e. stable), once subjected to fluctuations in the solution space (or the phase space)?”

Answering this requires an in-depth analysis of the MST-cosmological dynamics in both Einstein and Jordan frames. Our objective in this paper is to carry out such an analysis, by constructing (from cosmological equations in either cases) the autonomous system of equations in terms of suitable phase space variables.

For simplicity, we take into account only the two dominant components of the universe, viz. the dust and the scalar field , whence the phase space is a two-dimensional (i.e. a phase plane). However, instead of working with and its mass , throughout the analysis we resort to a torsion scalar and a torsion constant , so as to have a clear understanding of torsion’s effect on the dynamics. Whereas is equal to the time-integral of the norm of the torsion trace vector (as defined in the original Jordan frame), is given by the norm of the pseudo-trace vector of torsion (modulo some numerical factor).

We follow the standard methodology based on linear perturbation theory [2, 11] to determine the critical points (CPs) in the phase plane and their characteristic type and nature. This is essential, since each CP represents an equilibrium state of the system (the universe) in the asymptotic limit (, where is the cosmological scale factor). Now, a given cosmological solution is considered stable if it transpires to the dynamical evolution of the universe at a stable CP. However, as is common in a plethora of contexts in the literature [2, 11, 52, 53, 54], there are instances of more than one CPs existing in a certain parametric domain, i.e. the range of values of a system parameter (e.g. a coupling parameter). The same is the situation we find here, in both Einstein and Jordan frames, for certain domains of our MST-coupling parameter . This compels us to analyse the dynamical evolution at each individual CP and figure out the appropriate one(s) in the respective (Einstein or Jordan) frame. Additionally, we have to determine the parametric domain(s) in which a stable CP supports solution(s) that exhibit cosmic acceleration in the asymptotic limit. We do so in both Einstein and Jordan frames, and hence show that the corresponding exact solutions found analytically in paper are indeed stable. Numerically solving the autonomous equations, subject to appropriate initial conditions, we work out a host of relevant trajectories in the phase plane, in order to examine the overall dynamics of the system and its constituents leading up to the CPs. Also, for certain fiducial settings, we demonstrate the evolution of cosmological parameters of interest, such as the effective DE density and equation of state (EoS) parameters, and , along the corresponding trajectories.

We carry out the dynamical analysis first in the Einstein frame, in which the cosmological equations are rather simple and have resemblance with those for quintessence. There are however two major differences. Firstly, the torsion scalar interacts with the (apriori dust-like) fluid, thus affecting the dynamics of both. As such, it is not possible to make a direct comparison of the MST-cosmological parameters with those estimated (from observations) for known models, such as CDM. It is rather convenient to resort to the scenario in which the critical density of the universe, , is decomposed into two effective non-interacting components, viz. the dust-like matter and a left-over, supposedly the DE [50]. In such a scenario, the physical relevance of the existent CPs in the phase plane is implicated by the eventual extinction of the matter sector, irrespective of the initial conditions. Secondly, it is not desirable to have the chosen phase space variables depending explicitly on the system parameter . Otherwise their calibration would keep on changing with , different domains of which are assigned for the existence of the CPs and(or) their physical relevance. Therefore, if instead of we choose to work with a redefined (quintessence-like) field which absorbs in it, the physically admissible regions for the trajectories in the phase plane would get altered in shape and size [55].

Repeating the analysis in the Jordan frame is straightforward, but cumbersome because of an explicit -dependence of the gravitational coupling factor , where is the generalization of the Newton’s constant. There are some interesting consequences of this though, culminating from two legitimate standpoints. In principle, we may resort to one of the following: (a) a conventional scenario in which the critical density varies with and is not conserved, although the matter density is conserved [11], and (b) an effective scenario in which the critical density , defined as in a minimally coupled theory, is the sum of the densities of the dust and a left-over (supposedly the DE), which are individually conserved [50]. Now, since it is the same Jordan frame MST setup looked from different perspectives, the general outcomes of the dynamical analysis remain the same in both the scenarios, viz. the same number of CPs of the same type and nature. One difference is there though — in the conventional scenario, the physically admissible regions are confined within two similar curves (conic sections) in the phase plane, whereas only the outer curves are there in the effective scenario, for the same values of the effective Brans-Dicke parameter ()111We choose to take as the Jordan frame system parameter, different domains of which ascribe to the existent and(or) the physical relevant CPs. The limiting values set on from extensive studies [56, 57], provide an independent credibility check of the cosmological solutions [50].. The cosmological dynamics in the effective scenario, although by-and-large similar to that in the Einstein frame, has an intriguing feature, viz. the existence of a stable CP that supports solutions which not only exhibit cosmic acceleration in the asymptotic limit, but also a super-accelerating or phantom regime in course of their evolution. One such stable solution is actually that found in paper I, for which the phantom barrier crossing takes place at an epoch in the near past, whereafter the phantom regime continues eternally [50].

This paper is organized as follows: starting with a general description of the MST formalism in §2, we write down the scalar-tensor equivalent actions in both Jordan and Einstein frames, in terms of the torsion scalar . Considering first the Einstein frame MST-cosmological setup in §3, we proceed as follows: (i) work out (in §3.1) the equations of motion for the effectively non-interacting dust and DE sectors, (ii) construct from them (in §3.2) the autonomous system of equations and examine the domains of existence and(or) physical relevance of the CPs and also their type and nature, (iii) study (in §3.3) the dynamical evolution of the universe at each of the CPs and examine the stability of viable DE solutions such as the one found in paper I, and finally (iv) obtain the phase plane trajectories (in §3.4) by numerically solving the autonomous equations for appropriate sets of initial conditions, and hence illustrate the evolution of the DE density and EoS parameters along a fiducial trajectory that leads up to a stable CP. Almost the same chronology is maintained while repeating the dynamical analysis in §4 for the MST-cosmological setup in the Jordan frame. Some characteristic differences are there of course, compared to the Einstein frame analysis. Accordingly, a detailed account of, for e.g. the shape and size of the physically admissible region(s) with the variation in the system parameter , is given. Also the phantom barrier crossing in the effective Jordan frame scenario is illustrated clearly, by working out the evolution along trajectories for fiducial settings corresponding to two different values of . We conclude in §5 with a summary and a discussion on some implications and possible extensions.

We use the same notations and conventions as in paper I, viz. the metric signature throughout is , units are chosen so that the speed of light , and the determinant of the metric tensor is denoted by .

2 The general MST formalism in the cosmological setup

Let us discuss the basic tenets of the metric-scalar-torsion (MST) formalism, viz. that of scalar field couplings to the four-dimensional Riemann-Cartan () Lagrangian [50]. The space-time is characterized by an asymmetric affine connection: , which incorporates the third-rank torsion tensor defined as: . Essentially, the Riemannian () covariant derivative in GR (defined via the symmetric Levi-Cevita connections ) is replaced with that () defined via , preserving the metricity condition .

The torsion tensor can be decomposed into three irreducible modes, viz. the trace , the pseudo-trace and the (pseudo-)tracefree tensorial residue , whence the analogue of the Ricci scalar curvature is

(2.1)

Accordingly, the free Lagrangian has a purely algebraic dependence on torsion222Note that the term in is merely a total divergence (or, a boundary term).. Now, while coupling a scalar field to , one encounters the well-known problem of non-uniqueness of the resulting action under the minimal coupling scheme () [28, 58]. A simple (and convenient) way to avoid this is to assume a non-minimal term , so that upto a total divergence the action is [50]

(2.2)

where is a dimensionless coupling constant, is the scalar field potential, and is the Lagrangian density for other matter fields in the theory. Eq. (2), dubbed as the ‘MST action’ [50], leads to the equation of motion , which implies that the scalar field acts a source of the trace mode of torsion. Moreover, in order to preserve the FRW metric structure in a standard cosmological setup, one requires the tensor mode of torsion to vanish altogether, and the vector modes and to have only their temporal components existent [30, 59]. Also since the torsion field is generally taken to be massless [24, 32], one expects the scalar field source , of its trace mode to be massless as well. However, the pseudo-trace mode of torsion, , may effectively lead to a scalar field potential , where is a mass parameter for . Such a possibility arises from a suitable augmentation of the MST action (2) with say, some higher order torsion term(s) such as , whence one gets via the corresponding equation of motion333Note that, a mass term for may also result from a norm-fixing constraint on [50], similar to that in the vector-tensor gravity theories of Einstein-æther type [60, 61, 62], or in the mimetic gravity theories [63, 64, 65]. However, such an analogy has no specific physical motivation. [50]. The effective MST action then assumes the form of a scalar-tensor action in the Jordan frame [50]:

(2.3)

where and is the value444The parameter is of course taken to be positive definite, as otherwise the underlying quantum gravitational theory would be unbounded from below. of at the present epoch , such that the running gravitational coupling parameter has its present-day value , the Newton’s constant.

Let us now define a dimensionless scalar field as:

(2.4)

so that at , , and the effective gravitational coupling factor is given by

(2.5)

The Jordan frame action (2.3) is then expressed as

(2.6)

where and are two dimensionful constants, given by

(2.7)

One may note that the field and the constant facilitates a clear understanding of the roles of the individual torsion modes in the MST-cosmological dynamics. If one makes interpretations in terms of torsion parameters chosen as the norms of the trace and pseudo-trace vector modes of torsion [50], then it is easy to see that

(2.8)

So, we always have , and simply in a cosmological space-time described by the spatially flat FRW metric, viz. diag, where is the comoving time and is the scale factor. Henceforth, we shall appropriately refer to as the ‘torsion scalar’ and as the ‘torsion constant’.

Now, as is usual in a scalar-tensor equivalent theory, the equations of motion are much simpler in the Einstein frame than in the Jordan frame [20, 21]. The Einstein frame MST-action can be obtained from Eq. (2.6), under the conformal transformation [50]:

(2.9)

where is a dimensionful constant, is the Einstein frame metric determinant, is the corresponding curvature scalar, and is the corresponding matter Lagrangian density. Despite their mathematical equivalence, the Einstein and Jordan frames in general have different outcomes of physical measurements. The reason is obviously the gravitational coupling factor, which varies in one frame and not in the other. In fact, there is a longstanding debate as to which of these frames is actually of physical relevance [20, 21]. For completeness therefore, we shall subsequently carry out the dynamical analysis for our MST-cosmological formalism in both the Einstein and Jordan frames, taking one or the other to be physically relevant. We shall set up first the corresponding (Einstein or Jordan frame) cosmological equations, for the two system constituents, viz. the torsion scalar and the cosmological matter in the form of a pressureless (non-relativistic) dust. Defining suitable variables for the corresponding two-dimensional phase space (or the phase plane), we shall then construct the autonomous system of equations and look for stable solutions representing an effective DE evolution. Since the dust feels the effect of torsion555That is, the dust-like fluid and the torsion scalar have a mutual interaction, which results from either the conformal transformation or the varying gravitational coupling factor ., we shall resort to an effectively non-interacting picture in the respective (Einstein or Jordan) frame.

3 Phase plane analysis in the Einstein frame

Let us consider, in this section, the Einstein frame to be suitable for physical observations, and drop for brevity the hats over all quantities defined in this frame. We shall however continue with the expressions (2.8) for the norms and , as defined in the Jordan frame, in order to keep track of the individual terms of our original MST-action (2). We have therefore the relationships:

(3.1)

in the Einstein frame, with being the corresponding comoving time coordinate.

3.1 Cosmological equations and the effective scenario

The Friedmann and Raychaudhuri equations, obtained from the action (2.9), are

(3.2)

where is the Hubble parameter corresponding to the Einstein frame scale factor (the overhead dot ), is the energy density of the fluid matter, whereas and are respectively the field energy density and pressure:

(3.3)

The corresponding energy-momentum conservation relation, given by

(3.4)

imply that the cosmological fluid does not retain its ‘dust’ interpretation in the Einstein frame, as its energy density depends explicitly on the torsion scalar :

(3.5)

where is the present-day value of .

Now, it is easy to see that under a field redefinition , the above Eqs. (3.2)–(3.5) correspond to those in an interacting system of an apriori pressureless cosmological fluid and a quintessence scalar field with an exponential potential [12]. Such a correspondence is however misleading, since the dynamical analysis of the system in terms of the torsion scalar has a radical difference with that in terms of the redefined field (see the discussion in the next subsection). In other words, working with not only pinpoints the dynamical effects of torsion on the evolution of the universe, but also leads to results different from those of the dynamical analysis for the standard scalar-tensor cosmologies in the Einstein frame (which are essentially the systems of interacting quintessence and cosmological matter) [11]. Moreover, Eq. (3.5) suggests that the expression for the critical (or total) density of the universe, viz. , is barely of any use when it comes to making a comparison with the parametric estimations of well-known models, such as CDM, from physical observations. It is rather convenient to express [50]

(3.6)

where is an effective (dust-like) matter density and is a surplus density (which we consider to be due to the DE). These are given respectively as

(3.7)

The Friedmann equation can then be recast as

(3.8)

where and are the actual and the effective matter density parameters respectively, is the density parameter for the field , whereas is that for the DE. Identifying the DE pressure as , we also have the DE conservation relation (obtained using Eqs. (3.4)):

(3.9)

Eqs. (3.7)–(3.9) govern the dynamical evolution of the system. One may in principle look for their outright solutions, for e.g. by guessing suitable solution ansatze, and then examine the physical viability of those solutions [50]. A rather general alternative is to construct an autonomous system of first order coupled differential equations, out of Eqs. (3.7)–(3.9), and look for such a system the plausible real roots and their stability against small fluctuations in the solution space (or the phase space).

3.2 Autonomous equations and the critical points

Defining the phase space variables as

(3.10)

we obtain from the above cosmological equations, the autonomous equations:

(3.11)
(3.12)

where is the number of e-foldings, and one also has the constraint

(3.13)

Inverting the definition of the variable in Eq. (3.10), we express

(3.14)

Eqs. (3.6), (3.7) and (3.13) imply that the effective matter density parameter is

(3.15)

Moreover, the total pressure being , we have from Eqs. (3.6), (3.7) and (3.10) the total equation of state (EoS) parameter of the system given by

(3.16)

i.e. the EoS parameter for the DE is

(3.17)

since , where , by Eq. (3.8).

It is worth noting here that the autonomous system of equations (3.11)–(3.13) is symmetric under the interchange , which means that we can restrict our analysis to the region of the phase plane without loss of generality. Moreover, as mentioned in the previous subsection, the MST-cosmological equations (3.2) and (3.3) correspond to those for quintessence, under the redefinition . Such a correspondence may not in general be reflected in the dynamical analysis though, when the (dimensionless) phase space variables are defined using . Actually, in comparison to the standard (and even the interacting) quintessence scenarios, the MST setup has the intriguing aspect of the coupling parameter playing a potentially active role in determining the viable cosmologies. So it is imperative to allow for a discrete alteration of the value of in the dynamical analysis. However, for simplicity we may keep the other parameter in the theory, viz. , to remain fixed. Now in such a situation, while working with instead of , we cannot use the above definitions (3.10) of the phase space variables and , with a mere substitution of by therein. The reason is that being proportional to , such a substitution would mean and explicitly dependent on . Therefore their calibration would change when the value of is changed, thus giving rise to an ambiguity in the analysis. So, in terms of one has to define altogether different phase space variables, under the demand that they need to be free from any explicit -dependence666Their implicit dependence on is not a worry though, as that would not affect their calibration. Hence, we would have a different set of autonomous equations which may lead to a different dynamics of the same system, if we resort to the redefined field , instead of persisting with the torsion scalar .

Now, the objective of analysing autonomous equations, say and , is to determine the critical points, or the equilibrium solutions, and consequently examine the type and nature of such solutions, i.e. their stability in the two-dimensional phase space formed by and . Here, and are given functions of and for a particular system, for e.g. the right hand sides of Eqs. (3.11) and (3.12). By definition, a critical point (CP) is assigned coordinates at which and vanish. Now, for small changes about , we have the following eigenvalue equation in the linear perturbation theory [2]:

(3.18)

The eigenvalues and of the matrix , i.e. their type (real, imaginary or complex) and their sign, determine whether the solutions for and have modes that are exponentially growing or decaying with , and as such assert the type and the nature of the CPs. Specifically, there are the following cases:

  1. Real and : (a) If they are both negative, then as , which means that all phase space trajectories in the vicinity would terminate at the CP , i.e. the latter acts as a stable nodal sink, or an attractor. (b) If they are both positive, then the perturbations build up over time (or ), taking the trajectories away from the CP , which therefore acts as an unstable nodal source. (c) If they are of opposite sign, then the CP is a saddle point, i.e. it acts as an attractor along a particular direction (viz. the attractor axis) and as an unstable point along the direction normal to that. In general, the trajectories may tend towards the saddle point, but eventually get repelled away. Only for the trajectories that start from somewhere on the attractor axis, proceed along that, and manage to reach the saddle point, the latter acts as an attractor. Otherwise, the saddle point is unstable in nature.

  2. Imaginary and : The trajectories would in general describe an ellipse, having at its center the CP , which is stable and is called a center.

  3. Complex and : Depending on whether their real parts are negative or positive, the trajectories would spiral towards or away from the CP , rendering the latter to be a stable spiral sink or an unstable spiral source.

In the case of vanishing and , the type and nature of a CP are indeterministic if we persist with the linear perturbation theory, which actually breaks down [2].

For our autonomous set of equations (3.11)–(3.13), the above methodology implies the existence of five distinct CPs (discounting for the multiplicity of course). The coordinates of these CPs and the domains of their existence and physical relevance (given by appropriate range of values of the parameter ) are listed in Table 3.2.

\tbl

Einstein frame critical points and the parametric domains of their existence and physical relevance. \topruleCP: Location        Parametric domains of in the phase plane existence physical relevance \colrule: \colrule: \colrule: \colrule: \colrule: \botrule

The eigenvalues and of the linear perturbation matrix at each CP, and also the type and nature of the CPs are shown in Table 3.2.

\tbl

Eigenvalues of the linear perturbation matrix at the critical points, and the type and nature of these points in the Einstein frame. \topruleCP Eigenvalues of at CP For parametric range: CP type (nature) \colrule : Nodal Source (Unstable) \colrule : Nodal Source (Unstable) : Indeterministic : Nodal Sink (Stable) \colrule : Saddle point (Unstable) : Indeterministic \colrule : Nodal Sink (Stable) : Indeterministic \colrule : Saddle point (Unstable) : Indeterministic \botrule

The criterion for their physical relevance actually follows from Eq. (3.15) for the effective matter density parameter and the constraint (3.13). The presence of the exponential factor in Eq. (3.15), where , implies that may keep on evolving with time (or ) even after the system reaches a CP. So the condition for a physically realistic matter density, viz. , may get violated at some epoch, leaving the corresponding cosmological model unphysical at that CP. The exception(s) though is(are) the scenario(s) in which in the asymptotic limit. From Eq. (3.13) we find the corresponding (physically relevant) CP(s) to be on the circumference of an ellipse, whose center is at the origin of the phase plane:

(3.19)

The region enclosed by this ellipse (let us call it ) is therefore the physically admissible region in the phase plane. The identification of this region should be emphasized as the key difference between the dynamical system analysis here for the Einstein frame MST-cosmology, and that for the standard (quintessence-type) scalar field models in which the energy densities due to the field and the cosmological fluid matter are individually conserved. See below the comparison in the next subsection.

3.3 Dynamical evolution of the universe at each critical point

For viable cosmologies, particularly from the perspective of the late-time cosmic acceleration (or of the DE), it is necessary to identify the supportive CP(s). As the criterion (3.19) of the physical relevance of the CPs follows from the argument that (i.e. ) asymptotically, the cosmological solutions at each of the CPs are given entirely by the energy component with the density parameter . Although we are choosing to call it the ’DE’, it is yet to be verified whether this component actually complies with an accelerated expansion of the universe at late times (preceded by a regime of a decelerated one). Let us therefore look into the characteristics of the CPs and the dynamical aspects of the universe at each CP:

CP : Exists for all values of the parameter and is situated at the intersection of the ellipse (3.19) and the negative -axis. It acts an unstable nodal source, since the eigenvalues and of the linear perturbation matrix are real and positive. The equilibrium state of the solutions, implicated by this CP and given by the DE with density parameter , is of an extremely decelerated expansion of the universe, because the EoS parameter of the DE is , i.e. the DE behaves like a steep fluid.

CP : Exists for all values of and is at the intersection of the ellipse (3.19) and the positive -axis. It acts as an unstable nodal source when and as a stable nodal sink when . The solutions at this CP, in either case, are given by a steep fluid-like DE, exhibiting an extremely decelerated expansion of the universe. For however, the type and nature of this CP are indeterministic, as both the eigenvalues and of vanish.

CP : Exists for all values of and lies on the -axis, but inside the ellipse (3.19) except for . It acts as an unstable saddle point (of no physical relevance though) for , whereas for it coincides with , whence its type and nature are indeterministic. .

CP : Exists whenever and is constrained to lie on the circumference of the ellipse (3.19). Whereas for it coincides with and (i.e. its type and nature are indeterministic), for it acts as a stable nodal sink and leads to a DE dominated accelerated expansion of the universe if further (as can be checked easily777From Eq. (3.16) we see that at this CP , viz. , the total EoS parameter for the system is . So, the acceleration condition implies .). This is in fact the only CP (among the five) that can support solution(s) in presence of a non-vanishing potential term for the torsion scalar , and hence the viable DE model(s).

CP : Exists whenever , but is neither on the -axis nor on the ellipse (3.19), unless for , whence it coincides with and (i.e. its type and nature are indeterministic). For it acts as an unstable saddle point (which is physically irrelevant as well).

Let us now make a comparison with the (saddle and stable) CPs found in the dynamical analysis for the standard scenario of (non-interacting) dust and quintessence scalar field with an exponential potential , where is some numerical factor [2, 11]. Among the saddle point(s) that could exist, depending on the value of , there is one always at the origin of the phase plane. This point supports solutions which require to get obliterated asymptotically, leaving the dust as the sole constituent of the universe. Among the stable point(s) that could exist, there is one that supports solutions which, depending on , are given in the asymptotic limit either entirely by a non-dynamic DE component (i.e. a cosmological constant ) or by a CDM configuration with the densities of and the dust of the same order of magnitude. In contrast, the analysis of our Einstein frame MST-cosmological dynamics leads to, depending on the value of the parameter , the saddle points () and (), and the stable point () or (). However, we additionally have the criterion (3.19) that permits only those solutions for which the cosmological matter gets obliterated asymptotically. None of the saddle points ( and ) satisfies this criterion though. Nevertheless, as demonstrated below, the CP that lies in the region enclosed by the ellipse (3.19), has significance in funnelling physical trajectories towards the stable point . The latter is of course the only CP which supports solutions exhibiting the cosmic acceleration in the asymptotic limit, and that too for restricted up to a maximum value . So the DE models in the Einstein frame MST setup are viable only for a fixed parametric range . This also suggests (from Table 3.2) that only four CPs are of practical importance, viz. two unstable points and at the intersections of the -axis and the ellipse (3.19), a saddle point on the -axis and inside this ellipse, and a stable point on the circumference of the ellipse.

As to the status of the exact solution we have found in paper I by explicitly solving the Einstein frame MST-cosmological equations [50], note that:

(i) The small parametric bound , for the viability of an almost CDM-like DE model described by such a solution [50], is compatible with the rather loose bound we have obtained here from the dynamical analysis.

(ii) The universe described by such a solution must transpire to the dynamics of the DE and the cosmological matter at the stable point . This can be seen by working out that at , i.e. at , the torsion scalar is given by , whence the expression for the Hubble parameter:

(3.20)

is precisely the same as the asymptotic (i.e. the limiting) form of that we have had in paper I, while deriving the exact solution in the Einstein frame (see section 4.1 therein). The stability of such a solution is thus established.

Now, for a clear understanding of the qualitative aspects of the evolution of the universe leading up to the stable point , let us refer back for convenience, to the original decomposition of the critical density into the densities of the (interacting) torsion scalar and the cosmological fluid in the Einstein frame. Note the following:

I. Any point on the -axis of the phase plane implicates a non-dynamic (i.e. , since ), whose contribution to the total energy density of the universe, via the potential , has a fixed value. The all-important torsion mode is therefore the pseudo-trace , which is assumed to give rise to the potential, that acts a cosmological constant (with the corresponding EoS parameter ). Hence the overall configuration for a point on the -axis is that of CDM. Moreover, the constraint (3.13) implies that the further such a point is from the origin, the greater is the contribution of torsion to the energy content of the universe (subject always to the condition , however). The CDM trajectory hence shows the evolution along the -axis, upto the point , i.e. the apex of the elliptic boundary (3.19) of the physically admissible region in the phase plane.

II. A point anywhere except on the -axis of the phase plane represents a system configuration in which the torsion scalar is dynamical. The extent of such dynamics is determined by the magnitude of , or equivalently by the contribution of the trace mode of torsion to the total energy content of the universe. However, the viability of a DE evolution depends on how dominant is the potential , and hence the torsion pseudo-trace , over the dynamical mode . In other words, the latter has to be quite subdued, which is commensurable with the smallness of the parameter . In fact, it is easy to see that for , the stable point located at the boundary (3.19) of supports cosmologies which can be summed up as small deviations from CDM. A DE model of such sort has been the one studied earlier [50], in which statistical bounds on are found by demanding that the value of, say, the Hubble constant has to be within the error limits of that predicted for CDM from physical observations. Although these bounds are important from the observational perspective, in order to see the overall qualitative nature of stable cosmologies represented by it suffices one to resort to the general (model-independent) upper limits — the coarse one, viz. , for the stable point to exist in the first place, and the tighter one, viz. , in order that a phase of accelerated expansion of the universe is supported by .

3.4 Numerical solutions of the autonomous equations

The limitation of the equilibrium solutions of the autonomous equations (or the critical points in the phase plane) is that they do not provide any quantitative information as to what the state of a system has been prior to reaching them. To overcome this (atleast partially), we require to find particular solutions of the autonomous equations. Doing so analytically is however a formidable proposition for the coupled set (3.11)–(3.13). We therefore resort to numerical techniques for a given range of initial values of the variables and . Each set of numerical solutions and traces out a trajectory representing the system’s evolution in the phase plane, right from the point of origin (in accord with the initial conditions) till the termination at one of the CPs. This also enables us to see the variation (with ) of any explicit function of or(and) , over the lifetime of a phase plane trajectory. Hence, in a cosmology resulting from the chosen initial conditions, we can in principle plot the quantities of interest, such as , and , over a significantly large span of time (or ).

(a)
(b)
(c)
(d)
(e)
Figure 6: Einstein frame phase portraits for different values of the parameter . The dots represent the critical points, arrows mark the direction of time-evolution along trajectories and the dashed curve demarcates the permissible region for cosmologies with non-negative effective matter density.

Figs. 6(a) – (e) show the trajectories the system follows in the plane to reach a CP for different initial values and for the parametric settings and . As mentioned in the last subsection, there are actually four CPs to be taken into account, viz. those in the physically admissible region , or on its elliptic boundary (3.19). These CPs are superimposed for comparison in each of the Figs. 6(a)–(e).

Our main interest however, from the point of view of the accelerating cosmologies, is in the Figs. 6(a), (b) and (c), corresponding to the settings and respectively. Figs. 6(d) and (e), corresponding to and respectively, are only for the sake of completeness in the illustration. We see that the trajectories originating somewhere in the elliptically bounded region of the phase plane, excluding the -axis, terminate at the stable point . On the other hand, the trajectories which originate at a point on the -axis, tend to terminate at the saddle point . In fact, the trajectories originating anywhere in except the -axis are deflected in a direction parallel to the -axis towards , which deflects them in a direction vertically above it, i.e. towards situated at . In other words, the saddle point has the effect of funnelling trajectories towards the stable point . Of course, the latter exhibits its own attractive nature as well. The -axis (i.e. ) is the stable axis for and the line is its unstable axis. As is increased from a small value (say, ), the area of the region increases, with the decrease in the eccentricity of its elliptic boundary. Accordingly, the saddle point and the stable point shift away from the -axis, and so do the unstable points and . The shift continues till , whence the CPs and tend to coincide with the CP (not shown in the Figs. 6(a) – (e)), which approaches the elliptic boundary of the region from outside. Further increase in the value of (beyond ) would take and outside the physical realm, whereas and would cease to exist.

Let us now examine the evolution of some cosmological parameters of interest, and the torsion parameters, along a fiducial trajectory corresponding to a particular setting, say . As to the initial conditions for this fiducial trajectory, we may conveniently set them at the present epoch (), i.e. by appropriately choosing the values of and . One obvious choice is that in line with the exact solution obtained in paper I, whose stability we have already established in this paper. Such a solution corresponds to taking the ansatz , whence it follows from Eq. (3.10) that constant . Moreover, since (or by the definition (3.14)), we have from Eq. (3.15) . Considering now the fiducial values:

(i) (which is roughly the observational prediction [5, 6, 7] for most of the model-independent and model-dependent DE parametrizations), and

(ii) (which is the order of magnitude estimation [50] for the above ansatz, using the WMAP and Planck results [5, 7]),

we have the initial conditions

(3.21)
(a) Density parameters vs.
(b) EoS parameters vs.
Figure 9: Evolution of (a) the density parameters and , and (b) EoS parameters and , over the trajectory with initial conditions for , .

Fig. 9(a) shows the evolution of the density parameters and over the fiducial trajectory (with the above initial conditions) for a fairly wide range of e-foldings 888considering that corresponds to a redshift .. As expected (in view of the small value of ), these parameters vary with similar to their CDM analogues. There are some subtleties however. Note that the curves in Fig. 9(a) are asymmetric about . Actually, as we go from the present () to the future regime (), and rapidly approach a near-saturation to the values and respectively. That is, the DE tends to dominate completely over the dust-like matter even in the not-so-distant future, which is quite identical to the case in CDM. On the other hand, as we go back in the past (), and first approach each other rapidly, reach an equality point, then diverge with the same rapidity, attain extremum values, and finally approach each other once again (albeit very slowly) further back in the past. This is of course dissimilar to what happens for CDM, and its root cause can be traced to the original interaction between the torsion scalar and the cosmological fluid. Although this interaction gets obscured in the effective picture, it leaves its imprint on the density profiles. In fact, the dissimilarity with CDM is also evident from the evolution patterns of the EoS parameters, viz. and corresponding to the dynamical DE and the system respectively, over the fiducial trajectory. These are shown in Fig. 9(b), for the same range of , viz. .

As to the evolution of the torsion scalar , and that of the torsion parameters and , over the fiducial trajectory, first note that since is constant throughout, Eq. (3.14) implies . Therefore, , i.e. varies linearly with , as shown in Fig. 12(a). Using Eqs. (3.1) and (3.10) we can now work out the functional forms of the parameters and , and plot them after appropriate dimensional scaling. We choose to scale with the Hubble parameter , in order to have a direct measure of the effect of the trace mode of torsion on the cosmological evolution. However, being a constant, it is imperative to consider its ratio with and see how that evolves with , for the fiducial setting. The expressions of and are found to be

(3.22)
(a) Torsion scalar and trace parameter vs.
(b) Torsion pseudo-trace to trace ratio vs.
Figure 12: Evolution of (a) the torsion scalar and the trace parameter (in units of the Hubble parameter ), and (b) the ratio of the torsion pseudo-trace and trace parameters,