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Eur. Phys. J. C (2020) 80:677https://doi.org/10.1140/epjc/s10052-020-8253-7
Regular Article - Theoretical Physics
Stability analysis for cosmological models in f (T, B) gravity
Geovanny A. Rave Franco 1,a, Celia Escamilla-Rivera1,b , Jackson Levi Said2,3,c
1 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Circuito Exterior C.U., A.P. 70-543, 04510 Mexico, D.F., Mexico2 Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta3 Department of Physics, University of Malta, Msida MSD 2080, Malta
Received: 2 June 2020 / Accepted: 16 July 2020 / Published online: 27 July 2020© The Author(s) 2020
Abstract In this paper we study cosmological solutions ofthe f (T, B) gravity using dynamical system analyses. Forthis purpose, we consider cosmological viable functions off (T, B) that are capable of reproducing the dynamics of theUniverse. We present three specific models of f (T, B) grav-ity which have a general form of their respective solutions bywriting the equations of motion as an autonomous system.Finally, we study its hyperbolic critical points and generaltrajectories in the phase space of the resulting dynamicalvariables which turn out to be compatible with the currentlate-time observations.
1 Introduction
The �CDM cosmological model is supported by overwhelm-ing observational evidence in describing the evolution of theUniverse at all cosmological scales [1,2] which is achievedby the inclusion of matter beyond the standard model of par-ticle physics. This takes the form of dark matter which acts asa stabilizing agent for galactic structures [3,4] and material-izes as cold dark matter particles, while dark energy is repre-sented by the cosmological constant [5–7] and produces themeasured late-time accelerated expansion [8,9]. However,despite great efforts, internal consistency problems persistwith the cosmological constant [10], as well as a severe lackof direct observations of dark matter particles [11].
On the other hand, the effectiveness of the �CDM modelhas also become an open problem in recent years. At itscore, the �CDM model was convinced to describe Hub-ble data but the so-called H0 tension problem calls this intoquestion where the observational discrepancy between modelindependent measurements [12,13] and predicted [14,15]
a e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] (corresponding author)
values of H0 from the early-Universe appears to be grow-ing [16]. While measurements from the tip of the red giantbranch (TRGB, Carnegie-Chicago Hubble Program) point toa lower H0 tension, the issue may ultimately be resolved byfuture observations which may involve more exotic measur-ing techniques such as the use of gravitational wave astron-omy [17,18] with the LISA mission [19,20]. However, it mayalso be the case that physics beyond general relativity (GR)are at play here.
The use of autonomous differential equations to inves-tigate the cosmological dynamics of modified theories ofgravity has been shown to be a powerful tool in elucidat-ing cosmic evolution within these possible models of grav-ity [21]. These analyses can reveal the underlying stabilityconditions of a theory from which it may be possible to con-strain possible models on theoretical grounds alone. Theo-ries beyond GR come in many different flavours [2,22] wheremany are designed to impact the currently observed late-timecosmology dynamics. By and large, the main trust of thesetheories comes in the form of extended theories of gravity[22–24] which build on GR with correction factors that domi-nate for different phenomena. However, these are collectivelyall based on a common mechanism by which gravitation isexpressed through the Levi-Civita connection, i.e. that grav-ity is communicated by means of the curvature of spacetime[1]. In fact, it is the geometric connection which expressesgravity while the metric tensor quantifies the amount of defor-mation present [25]. This is not the only choice where tor-sion has become an increasingly popular replacement andproduced a number of well-motivated theories [26–28].
Teleparallel Gravity (TG) collectively embodies the classof theories of gravity in which gravity is expressed throughtorsion through the teleparallel connection [29]. This connec-tion is torsion-ful while being curvatureless and satisfying themetricity condition. Naturally, all curvature quantities calcu-lated with this connection will vanish irrespective of metriccomponents. Indeed, the Einstein–Hilbert which is based on
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the Ricci scalar R (over-circles represent quantities calcu-lated with the Levi-Civita connection) vanishes when calcu-lated with the teleparallel connection, i.e. in general R = 0while R �= 0. Moreover, the identical dynamical equationscan be arrived at in TG by replacing the Einstein–Hilbertaction with it’s so-called torsion scalar T . By making thissubstitution, we produce the Teleparallel equivalent of Gen-eral Relativity (TEGR), which differs from GR by a boundaryterm B in its Lagrangian.
The boundary term in TEGR consolidates the fourth-ordercontributions to many beyond GR theories. By extractingthese contributions into a separate scalar, TEGR will have ameaningful and novel impact on extended theories and pro-duce an impactful difference in the predictions of such the-ories. The most direct result of this fact will be that TG willproduce a much broader plethora of theories in which dynam-ical equations are second order. This is totally different to theseverely limited Lovelock theorem in curvature based theo-ries [30]. In fact, TG can produce a large landscape in whichsecond-order field equations are produced [31,32]. TG alsohas a number of other attractive features such as its likeness toYang-mills theory [26] offering a particle physics perspectiveto the theory, the possibility of it giving a definition to thegravitational energy-momentum tensor [33,34], and that itdoes not necessitate the introduction of a Gibbons–Hawking–York boundary term to produce a well-defined Hamiltonianstructure, among others.
Taking the same reasoning as in f (R) gravity [22–24],TEGR can be straightforwardly generalized to produce f (T )
gravity [35–40]. f (T ) gravity is generally second order dueto the weakened Lovelock theorem in TG and has shownpromise in several key observational tests [27,41–47]. How-ever to fully embrace the TG generalization of f (R) grav-ity, we must consider the f (T, B) generalization of TEGR[48–52,52,53]. In this scenario the second and fourth ordercontributions to f (R) gravity are decoupled while this sub-class becomes a particular limit of the arguments T and B,namely f (R) = f (−T + B). f (T, B) gravity is an inter-esting theory of gravity and has shown promise in terms ofsolar system tests and the weak field regime [51,54,55], aswell as cosmologically both in terms of its theoretical struc-ture [48,48,50,52] and confrontation with observational data[56].
In this work, we explore the structure of f (T, B) grav-ity through the dynamical systems approach in the cosmo-logical context of a homogeneous and isotropic Universeusing the Friedmann–Lemaître–Robertson–Walker metric(FLRW). This kind of system has been used to study higher-order modified teleparallel gravity that add a scalar field φ
depending on the boundary term [50], where the stability con-ditions for a number of exact solutions for an FLRW back-ground are studied for a number of solutions such as de SitterUniverse and ideal gas solutions. In Ref. [57], a number of
important reconstructions were presented together with theirdynamical system evolution. This work is also interestingbecause they compare some of their results with Supernovatype 1a data using a chi-square approach. Another interest-ing approach to determining and studying solutions is that ofusing Noether symmetries Ref. [58], which have [50] showngreat promise in producing new cosmological solutions thatadmit more desirable cosmologies. On the other hand, inRef. [52] the all important cosmological thermodynamicshas been explored as well as the mater perturbations, whichwas complemented by background reconstructions of fur-ther cosmological solutions giving a rich literature of mod-els together with Ref. [59]. Reference [60] then developedthe f (T, B) cosmology energy conditions which can giveimportant information about regions of validity of the mod-els.
In the present study, the cosmic acceleration dynamics isreproduced only by a non-canonical φ that mimics the �
term. In our case, we introduce a f (T, B) dark energy whichis fluid-like in order to obtain a richer population of stabil-ity points that can be constrained by current observationalsurveys. We do this by first introducing briefly the techni-cal details of f (T, B) gravity in Sect. 2 and then discussingits dynamical treatment in Sect. 3. In Sect. 4 we lay out themethodology of the analysis which includes the methods bywhich the analysis is conducted. The f (T, B) gravity dynam-ical analysis is then realized in Sect. 5 where the core resultsfor each of the models is presented. Finally, we close in Sect. 6with a summary of our conclusions. In all that follows, Latinindices are used to refer to Minkowski space coordinates,while Greek indices refer to general manifold coordinates.
2 f (T, B) cosmology
We start by considering a flat homogeneous and isotropicFLRW metric in Cartesian coordinates with an absorbedlapse function (N = 1) as (e.g through Ref. [1])
ds2 = −dt2 + a(t)2(dx2 + dy2 + dz2) , (1)
where a(t) is the scale factor. Also, we choose an arbitrarymapping over f (T, B) → −T + f (T, B), which obeys thediffeomorphism invariance. As shown in [56], the choice ofLagrangian where f (T, B) → −T + f (T, B) represents anarbitrary Lagrangian over the torsion scalar and boundaryterm is diffeomorphism invariant. In this proposal our choiceof tetrad is given by
eaμ = diag(1, a(t), a(t), a(t)) , (2)
which reproduces the metric in Eq. (1) and observes the sym-metries of TG. In this spacetime, the torsion scalar can be
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Eur. Phys. J. C (2020) 80 :677 Page 3 of 12 677
given explicitly as
T = 6H2 , (3)
while the boundary term is given by
B = 6(
3H2 + H)
, (4)
which combine to produce the well known Ricci scalar ofthe flat FLRW metric R = −T + B = 6
(H + 2H2
)(where
again over-circles again represent quantities determined withthe Levi-Civita connection).
After the above considerations over the geometry, our fieldequations for a universe filled with a perfect fluid are
−3H2 (3 fB + 2 fT ) + 3H fB − 3H fB + 1
2f = κ2ρ , (5)
−(
3H2 + H)
(3 fB + 2 fT ) − 2H fT + f B + 1
2f = −κ2 p , (6)
where ρ and p represent the energy density and pressure of aperfect fluid whose equation of state is p = ωρ, respectively.These modified Friedmann equations show explicitly how alinear boundary contribution to the Lagrangian would actas a boundary term while other contributions of B wouldcontribute nontrivially to the dynamics of these equations.
We can rewrite Eqs. (5,6) by considering the modifiedTEGR components contained in the effective fluid contribu-tions
3H2 = κ2 (ρ + ρeff) , (7)
3H2 + 2H = −κ2 (p + peff) , (8)
where
κ2ρeff = 3H2 (3 fB + 2 fT ) − 3H fB + 3H fB − 1
2f , (9)
κ2 peff = 1
2f −
(3H2 + H
)(3 fB + 2 fT ) − 2H fT + f B .
(10)
The latter equation can be combined to obtain
2H = −κ2 (ρ + p + ρeff + peff) . (11)
The effective fluid represents the modified part of the f (T, B)
Lagrangian which turns out to satisfy the conservation equa-tion
ρeff + 3H (ρeff + peff) = 0 . (12)
For our purpose and in order to construct the dynamicalsystem, the f (T, B) Friedmann equations can be rewrittenas
� + �eff = 1 , (13)
3 + 2
(H ′
H
)= − 3 f
6H2 + 9 fB + 6 fT + 3
(H ′
H
)fB
+2
(H ′
H
)fT + 2 f ′
T −(H ′
H
)f ′B − f ′′
B ,
(14)
where
�eff = 3 fB + 2 fT − f ′B − f
6H2 +(H ′
H
)fB , (15)
which each i denotes the effective density parameter �i =κ2ρi/3H2. The prime (′) denotes derivatives with respect toN = ln a, with a chain rule given by d/dt = H(d/dN ).
With the latter equations we can write the continuity equa-tions for each fluid under the consideration
ρ′ + 3(1 + ω)ρ = 0, (16)
ρ′eff + 3(ρeff + peff) = 0, (17)
where the effective fluid is related with the background cos-mology derived from f (T, B) gravity and ω are related tothe cold dark matter and non-relativistic fluids as matter con-tributions. This set of equations impose a condition over theform of the derivative f ′(T, B).
Using the Friedmann equations in Eqs. (13) and (14), wecan directly write down the effective EoS for our f (T, B)
gravity as [56,61]
ωeff = peffρeff
(18)
= −1 + f B − 3H fB − 2H fT − 2H fT3H2 (3 fB + 2 fT ) − 3H fB + 3H fB − 1
2 f,
(19)
which can also be written as having a redshift dependencesimilar to ω(z).
We can explicitly compute from Eq. (13) a dynamicalequation in terms of the Hubble factor and its derivatives as
6
(H ′H
)fB + 2
(H ′H
)fT +
(H ′2H2 + H ′′
H
)fB − f ′
6H2 = 0 . (20)
Notice that only the last term on the r.h.s contains infor-mation about the specific form of f (T, B) theory (or in itsderivative).
3 f (T, B) dynamical system structure
To construct the dynamical autonomous system for ourf (T, B) cosmological model, we follow the approach out-lined in Refs. [61–63]. As a first step we introduce a setof conveniently specified variables which allow us to rewritethe evolution equations as an autonomous phase system. Thisset of equations will be subject to a generic constraint arisingfrom our modified Friedmann equations in Eqs. (13,14). Forthis system we propose to define the parameter [64]
λ = H
H3 = H ′2
H2 + H ′′
H. (21)
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Notice that this expression depends explicitly on N =ln a(time-dependence). It was discussed in the latter refer-ence that for cases when λ = constant, some cosmologicalsolutions can be recover, e.g. if λ = 0, we can obtain a deSitter/quasi de Sitter universe or if λ = 9/2, a matter domina-tion era can be derived. Since this ansatz shows cosmologicalviable scenarios as analogous to models with barotropic flu-ids, along the rest of this work we are going to consider λ =constant. Following this prescription, we can write our Fried-mann evolution equations in term of dynamical variables
x ≡ fB , y ≡ f ′B , z ≡ H ′
H= H
H2 , w ≡ − f
6H2 . (22)
From the latter definitions and the Friedmann evolution inEq. (13) we can derive the constriction equation from thelatter evolution as
� + 3x + 2 fT − y + w + zx = 1 , (23)
where � is a parameter that depends on the other dynamicalvariables. Finally, we can write the autonomous system forthis theory as
z′ = λ − 2z2 , (24)
x ′ = y, (25)
w′ = −6zx − 2z fT − λx − 2zw , (26)
y′ = 3w + (9 + 3z)x + fT (6 + 2z) + 2 f ′T − zy
−3 − 2z . (27)
To follow the constraint of the system in Eqs. (13–23), weneed to write fT as a dynamical variable or write it in termsof the described variables. This can be done by consideringa specific form of f (T, B) as we will show in Sect. 5.
4 Stability methodology
We can study our f (T, B) autonomous system in Eqs. (24,25, 26, 27) by performing stability analyses of the criticalpoints, which can be investigated through linear perturba-tions around their critical values as x = x0 + u, wherex = (x, y, z, w) and u = (δx, δy, δz, δw). The equationsof motion for each of our models can be written as x′ = f(x),which upon linearisation can be given by
u′ = Mu , Mi j = ∂ fi∂x j
∣∣∣∣x∗
, (28)
where M is known as the linearisation matrix [21]. Theeigenvalues indicated by ω of M determine the stability(type) of the critical points, whereas the eigenvectors η ofM indicates the principal directions of the perturbations per-formed at linear level. As it is standard in the stability anal-ysis, if Re(ω) < 0 (Re(ω) > 0) the critical point is called
stable (unstable). More specific types of point will be indi-cated for each f (T, B) scenario.
For this case, we should consider perturbations of the fourdynamical variables (x, y, z, w), keeping in mind that theyare not all completely independent because they are boundtogether by the Friedmann constraint in Eq. (13). This depen-dence would then carry over to perturbative level.
From Eq. (24) we notice that there is not an explicit depen-dency of f (T, B), therefore for the critical points followingthe above prescription x∗ / x = f(x∗) = 0 we require that
z = ±√
λ
2, y = 0 . (29)
For each f (T, B) cosmological case we will present the sta-bility results, where we only consider the eigenvalues of thestability matrix M for each of the critical points and for theperturbations that are compatible with Eq. (13).
5 Dynamical analyses for f (T, B) models
In this work we consider three f (T, B) scenarios. They wereselected in order to obtain cosmological viable cases, in par-ticular the late-time observed cosmic acceleration. The fol-lowing models were studied in detail in Ref. [56], wherecosmological constraints of each of them were found. In thefollowing, we will focus on the corresponding parameter val-ues which adapt to our autonomous system.
5.1 Stability analysis for general Taylor expansion model
The form for this model was presented in Ref. [51] as ageneral Taylor expansion of the f (T, B) Lagrangian, givenby
f (T, B) = f (T0, B0) + fT (T0, B0)(T − T0)
+ fB(T0, B0)(B − B0)
+ 1
2! fT T (T0, B0)(T − T0)2
+ 1
2! fBB(T0, B0)(B − B0)2
+ fT B(T0, B0)(T − T0)(B − B0)
+O(T 3, B3) , (30)
which gives the general Taylor expansion of the f (T, B)
Lagrangian about its Minkowski values for the torsion scalarT and boundary term B. We notice from here that we needto take into account beyond linear approximations since B isa boundary term at linear order. Following the form for theFLRW tetrad in Eq. (2), where locally spacetime appears tobe Minkowski with torsion scalar and boundary term values,we can consider T0 = B0 = 0 . Taking constants called Ai ,
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Eur. Phys. J. C (2020) 80 :677 Page 5 of 12 677
the Lagrangian can be rewritten as
f (T, B) � A0 + A1T + A2T2 + A3B
2 + A4T B , (31)
where the linear boundary term vanishes. We notice fromthis specific form that the first term can be seen as A0 ≈�, therefore we are dealing with a cosmic acceleration as aconsequence of the f (T, B). Thus, the form of this modelcan be written in terms of the dynamical variables as
fT = −(3 + z)x − 2w − A1 , (32)
at linear order in torsion and with A0 = 0, i.e we areswitching-off the cosmological constant. This can be donesince an explicitly time-dependent factor appears and then adifferent approach has to be taken. The critical points for thismodel are
w = −A1, x = A1 − 1
3 ±√
λ2
, (33)
which imply that the constriction evolution equation inEq. (23) is now � = 0. According to these points we cancompute the following eigenvalues for the system
ω1 = −3 ∓√
λ
2, ω2 = −3 ∓ 2
√λ
2,
ω3 = ∓4
√λ
2, ω4 = ±2
√λ
2. (34)
Considering values as λ �= 0, we get that Re(ω3) =−Re(ω4) �= 0, implying that for this system all the criti-cal points are saddle-like for any value of λ. In Fig. 1, weshow different views of the phase space of the dynamicalsystem in Eq. (32) on 2-d surfaces. The solutions for the caseare in agreement with the cosmological constraints found in[56]. According to these results, our critical points behaveas Ai < Ai+1 (which states for these values A0 = 0 andA1 = 1), show a quintessence behaviour and when B dom-inates and z ≈ 1. After that, a �CDM model behaviour isobserved.
5.2 Stability analysis for power law model
If we consider a Lagrangian of separated power law stylemodels for the torsion and boundary scalars, we can write amodel like [50]
f (T, B) = b0Bk + t0T
m . (35)
This is an interesting model since it was already been shownin Ref. [65] that for m < 0 the Friedmann equations will beeffected mostly in the accelerating late-time universe whilefor m > 0 this impact will take place for the early universe,assuming no input from the boundary contribution. By incor-porating the boundary term, this analysis will reveal an effectof B on the combined evolution within f (T, B) cosmology.
The form for this model can be written in terms of the dynam-ical variables as:
fT = −mw − m
k(3 + z)x . (36)
The critical points for this scenario are
w = k − m
m(1 − k), x = − k
m
(m − 1
k − 1
) 1
3 ±√
λ2
. (37)
For this case, the constriction (23) gives again � = 0.According to Eq. (36), we can analyse independently the
positive and negative roots with z = ±√
λ2 as follows.
5.2.1 Analysis for the positive branch
Critical points. For this case, the eigenvalues derived fromthe stability matrix are
ω1 = −√2√
λ − 3 , (38)
ω2 = −2√
2√
λ , (39)
ω3 = − 1
4k
(√α − β + γ + k
(√2√
λ(1 − 2m) − 6)
+2(√
2√
λ + 6)m
), (40)
ω4 = 1
4k
(√α − β + γ + k
(√2√
λ(2m − 1) + 6)
−2(√
2√
λ + 6)m
), (41)
where
α = 2k2(λ(2m + 1)2 + 6
√2√
λ(6m − 1) + 18)
, (42)
β = 8km(λ + 6
√2√
λ(m + 1) + 2λm + 18)
, (43)
γ = 8(λ + 6
√2√
λ + 18)m2 , (44)
which under the conditions Re(ω1), Re(ω2) < 0 for anyvalue of λ we get saddle/attractors points.
• Case Re(ω3) = 0. For the condition Re(ω3) = 0 thecritical regions are1
1.
k = 1 ∧ 0 < m <1
2∧ 0 < λ < 72m2 − 72m + 18 ,
(45)
2.
0 <m <1
2∧ 2m < k ≤ 1 ∧ λ
1 From this point, along the text we refer to the symbol ∨ as or, ∧ asand.
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Fig. 1 Different views of the phase space of the dynamical system inEq. (32) for λ �= 0. The system was reduced to a 2-d surface represent-ing the different perspectives of its constraint. The arrows represent the
direction of the velocity field and the trajectories reveal their stabilityproperties as described for this model
= 18(k − 2m)2
(−2km + k + 2m)2 , (46)
• For the condition Re(ω4) = 0 the critical regions are
1.
k =1 ∧[(
0 < m <1
2∧ λ > 72m2 − 72m + 18
)
∨(m ≥ 1
2∧ λ > 0
)], (47)
2.
0 <m <1
2∧ 2m < k ≤ 1 ∧ λ
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Eur. Phys. J. C (2020) 80 :677 Page 7 of 12 677
= 18(k − 2m)2
(−2km + k + 2m)2 , (48)
• Attractor regions. These cases can happen under the fol-lowing conditions:
1.
0 < m ≤ 1
2∧ 0 < k < 1 ∧ λ > 18 , (49)
2.
m >1
2∧ 0 < k < 1 ∧ λ > 0 , (50)
Properties:
• If m > k then, w < −1/3 (b0 and c0 fixed as positive).• If m < k then, we get �CDM. (b0 and c0 fixed as posi-
tive).• If b0 < t0 and vice versa, we get a crossover over the
phantom divided-line (w = −1).• We recover �CDM and late cosmic acceleration.
5.2.2 Analysis for the negative branch
Critical points. For this case, the eigenvalues derived fromthe stability matrix are given by
ω1 = √2√
λ − 3 , (51)
ω2 = 2√
2√
λ , (52)
ω3 = 1
4k
⎛⎝−
√γ k2
4m2 + η
⎞⎠ + ζ , (53)
ω4 = 1
4k
⎛⎝
√γ k2
4m2 + η
⎞⎠ + ζ. (54)
with
ζ = 1
4k
(k
(√2√
λ(1 − 2m) + 6)
+ 2(√
2√
λ − 6)m
), (55)
η = 8m2(λ(k − 1)2 + 6
√2√
λ(k − 1) + 18)
−8mk(
3√
2√
λ(3k − 2) − kλ + λ + 18)
. (56)
Notice that according to the values of ω1 and ω2, the criticalpoint associated to the negative root case corresponds to ascenario where the universe has a contraction (accelerated)phase λ < 2(λ > 2), respectively, which represents a saddlepoint. On the other hand, according to the value of ω1, wenotice that if 0 < λ < 9
2 , the critical point is hyperbolic andsaddle-type. To simplify the analysed regions, we considercases where ω1, or ω2, only have a non-vanishing real part.
The next case to explore will be with a vanished real part(which correspond to a non-hyperbolic case).
• Conditions with Re(ω3) = 0. These regions are
A:
0 < m < 1 ∧ ((0 < k < 1 ∧ λ = 18)
∨(k = 1 ∧ λ > 72m2 − 72m + 18
)
∨(k > 1 ∧ λ = 18)) , (57)
B:
k =1 ∧[(
m ≥ 1 ∧ λ ≥ 72m2 − 72m + 18)
∨(
3
4< m < 1 ∧ λ = 72m2 − 72m + 18
)],
(58)
C:
0 < k < 1 ∧(m ≥ 1 ∧ λ = 18(k − 2m)2
(−2km + k + 2m)2
),
(59)
D:
1 < k < 2
∧(
3k
2k + 2< m ≤ 1 ∧ λ = 18(k − 2m)2
(−2km + k + 2m)2
),
(60)
E:
k = 2 ∧ m = 1 ∧ 9
2< λ ≤ 18 , (61)
F:
k > 2 ∧ 1 ≤ m <3k
2k + 2∧ λ
= 18(k − 2m)2
(−2km + k + 2m)2 , (62)
G:
k < 0 ∧ [(m > 1 ∧ λ = 18) ∨ (1 ≤ m
≤ 3
2∧ λ = 18(k − 2m)2
(−2km + k + 2m)2
)], (63)
H:
m >3
2∧ − 2m
2m − 3< k < 0 ∧ λ
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677 Page 8 of 12 Eur. Phys. J. C (2020) 80 :677
= 18(k − 2m)2
(−2km + k + 2m)2 , (64)
• Condition Re(ω4) = 0. These regions are:
A:
k = 1 ∧ 3
4< m ≤ 1 ∧ 9
2< λ
≤ 72m2 − 72m + 18 , (65)
B:
[m ≥ 1 ∧ 0 < k < 1∧(
λ = 18 ∨ λ = 18(k − 2m)2
(−2km + k + 2m)2
)]
∨(m>1 ∧ k=1 ∧ 9
2< λ<72m2 − 72m + 18
)
∨ (k > 1 ∧ λ = 18) , (66)
C:
1 < k < 2 ∧[(
3k
2k + 2< m < 1 ∧ λ
= 18(k − 2m)2
(−2km + k + 2m)2
)∨ (m = 1 ∧ λ = 18)
],
(67)
D:
k = 2 ∧ m = 1 ∧ 9
2< λ ≤ 18 , (68)
E:
k > 2 ∧(
1 ≤ m <3k
2k + 2∧ λ
= 18(k − 2m)2
(−2km + k + 2m)2
), (69)
F:
k < 0 ∧[(0 < m < 1 ∧ λ = 18) ∨
(1 ≤ m≤3
2∧ λ
= 18(k − 2m)2
(−2km + k + 2m)2
)], (70)
G:
m >3
2∧ − 2m
2m − 3< k < 0 ∧ λ
= 18(k − 2m)2
(−2km + k + 2m)2 . (71)
• Saddle regions. These regions are determine by the con-dition Re(ω3), Re(ω4) > 0, therefore
A:
0 < m <1
2∧ [(0 < k < 1 ∧ λ > 18)
∨(k > 1 ∧ 9
2< λ < 18
)], (72)
B:
m =1
2∧ [(0 < k < 1 ∧ λ > 18)
∨(k > 1 ∧ 9
2< λ < 18
)], (73)
C:
1
2< m < 1 ∧ 0 < k < 1 ∧ λ > 18 , (74)
D:
m ≥1 ∧[(
0 < k <2m
1 + 2m∧ λ > 18
)
∨(k >
2m
1 + 2m∧ λ > 18
)], (75)
E:
1 < m <3
2∧ k > − 2m
2m − 3∧ 9
2< λ
<18(k − 2m)2
(−2km + k + 2m)2 , (76)
F:
k < 0 ∧[(
0 < m < 1 ∧ 9
2< λ < 18
)
∧(m > 1 ∧ 9
2< λ <
18(k − 2m)2
(k + 2m − 2km)2
)].
(77)
In Fig.(2) we show the dynamical phase space for thismodel.
5.3 Stability analysis for mixed power law model
In order to reproduce several important power law scale fac-tors relevant for several cosmological epochs, in Ref. [50] aform of f (T, B) given by
f (T, B) = f0BkTm, (78)
was presented, where the second and fourth order contribu-tions will now be mixed, and f0, k,m are arbitrary constants.
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Fig. 2 Different views of the phase space of the dynamical system inEq. (36) for λ �= 0. The system was reduced to a 2-d surface represent-
ing the different perspectives of its constraint with x = − km
(m−1k−1
)1
3+z .
The arrows represent the direction of the velocity field and the trajec-tories reveal their stability properties as described for this model
We can recover GR limit when the index powers vanish, i.e.when k = 0 = m. For this case, the model can be written interms of the dynamical variables through
fT = −mw. (79)
In comparison to the latter f (T, B) scenarios, this case hasthe following particularity where
x = fB = k f0Bk−1Tm = k
Bf = k
6(3H2 + H)f
= f
6H2
k
3 + HH2
= − wk
3 + z, (80)
from which we can notice that x is not an independent vari-able of the dynamical system. In the same way, when y = x ′we obtain directly that y = y(w, z). With these conditions,
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the autonomous system for this case can be reduced to a 2-ddynamical phase space
z′=λ − 2z2 , w′=w
[6z(k + m − 1)+λk+2z2(m − 1)
3 + z
].
(81)
The critical point of the latter system are
z = ±√
λ
2, and w = 0. (82)
Under these values, the constriction of the system is givenby � = 1. Again, we can consider the two roots as follow:
5.3.1 Critical points
1. Positive branch. This case are determinate by the condi-tion m > 0, with eigenvalues
ω1 = −2√
2√
λ , (83)
ω2 = (k + m − 1)6√
λ2 + λ
3 +√
λ2
. (84)
If λ > 0, we obtain that the eigenvalues are real for anyvalue of λ, m and k. On the other hand, ω2 = 0 if k =1−m, which represent a non-hyperbolic point. We obtainan attractor point if k < 1−m, and saddle-like otherwise.
2. Negative branch. The eigenvalues for this case are
ω1 = 2√
2√
λ , (85)
ω2 = (k + m − 1)−6
√λ2 + λ
3 −√
λ2
. (86)
Again, both values are real. The critical point is repulsor-like if k < 1−m and λ > 0∧λ �= 18. We obtain a saddlepoint if k > 1 − m and λ > 0 ∧ λ �= 18. For m > k orm < k we get a phantom-like EoS (w < −1).
This result reinforces our work in Ref. [56] where wefind that for mixed power law models, that the equation ofstate does cross the phantom line (ω = −1) but preservesthe quintessence behaviour up to high redshifts where themodel limits to �CDM. For lower redshifts, both m > k andm < k scenarios mimick phantom energy. In fact, these twoscenarios correspond to the critical points that we find in theabove analysis.
6 Discussion
Dynamical systems can reveal a lot of information about thecosmology of theories beyond GR, which may be difficultto study at background or using direct cosmological pertur-bation theory. In this work, we analysed dynamical systemswithin the f (T, B) gravity context which was first studied inRefs. [57,66,67] in a context of scalar field in order to studythe implications on inflation solutions. TG offers an avenueto constructing theories which exhibit torsion rather thancurvature by exchanging the Levi-Civita connection with itsteleparallel connection analog. This produces a wide rangeof potential cosmological models since TG is naturally lowerorder and so produces novel manifestations of gravity in addi-tion to those constructed through extensions to GR [22–24].
f (T, B) gravity is an interesting context within which tostudy dynamical systems since it is one of the rare higherorder theories in TG that occurs naturally. Indeed, in Sect. 3we outline our strategy in terms of which dynamical variableswill produce a suitable dynamical analysis of cosmologicalsystems. f (T, B) gravity acts as a TG generalisation of f (R)
gravity in that the second and fourth order contributions aredecoupled from one another through the torsion scalar Tand boundary term B with coincidence only for cases wheref (T, B) = f (−T +B) = f (R). The effect of this point alsoplays out in the dynamical systems analysis where we alsomust take the dynamical variable defined λ in Eq. (21) whichis directly analogous to the approach taken in [64]. Indeed,the analysis where this parameter was probed against possibleconstant values was studied in [68]. A feature we explore inthis work through the methodology outlined in Sect. 4.
The core results of the work are presented in Sect. 5where the models (that are cosmological viable at back-ground level) are analysed. We start by probing the generalTaylor expansion model in Eq. (30) where the arbitrary func-tion is expanded about the Minkowski values of the scalararguments up to quadratic order (due to the linear form of Bbeing a boundary term). Here we find the critical points inEq. (33) with system eigenvalues in Eq. (34). We find that forany nonvanishing constant value of λ, all the critical valuesare all saddle points. An interesting feature of this investi-gation is that the constraints found are consistent with Ref.[56] showing consistency in its confrontation with observa-tions. The evolution for the various parameter combinationis shown in Fig. 1.
Afterward, we then study the power law model in Sect. 5.2which considers a power law form for both scalar contrib-utors. In this case, the determining factor is the z variablewhich depends only on derivatives of the Hubble parameteras defined in Eq. (22). In this scenario, we find either attrac-tors or saddle points for the positive branch and repellentor saddle points for the negative branch which is shown inFig. 2.
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Fig. 3 Different views of the phase space of the dynamical system in Eq. (79). The system was reduced to a 2-d surface representing differentperspectives. The arrows represent the direction of the velocity field and the trajectories reveal their stability properties as described for this model
A similar picture also emerges for the mixed model inves-tigated in Sect. 5.3. Here, we again find the same variable tobe the determining factor in the behaviour of the dynamics ofthe system. On the other hand, the system turns out to be rela-tively straightforward to analysis with clear cut results whichtally with the general results of the power law model. Theseare shown in Fig. 3. The above results can be linked in a morestraightforward manner if we consider directly the form forthe Equation of State (EoS). According to the generic EoSreported in [56] and using our dynamical results, we obtainthat for the first two cases (Taylor and power law model)
ωeff = −1 ∓ 2
3
√λ
2. (87)
Meanwhile, for the mixed power law model2
ωeffω→0−−−→ −1 ∓ 2
3(k + m)
√λ
2. (88)
Notice that we recover �CDM in the GR limit. In bothEoS scenarios, we recover a �CDM model when λ van-ishes, this can happen when we obtain de Sitter solutions asH = constant. Notice that we can rewrite the z-variable fromEq. (22) using the definition of the second cosmographicparameter, the deceleration parameter (q = −aa/a2) asH = −H2(q+1), therefore z = −(q+1). In terms of λ, this
latter parameter can be written as q = −1 ∓√
λ2 , which for
H = constant we obtain that q = −1. Also, we can rewritethe ansatz for λ given in Eq. (21) in terms of the third cosmo-graphic parameter, the jerk (
...a /aH3) as j = H
H3 − 3q − 2,
which again in terms of λ is j = λ ∓ 3√
λ2 + 1. Notice that
when λ = 0, we recover the standard value j = 1.Another important feature of the analysis in this work is
the role of couplings between the torsion tensor T and theboundary term B. As discussed in the introduction, these rep-resent the second order and fourth order contributions to thefield equations respectively, and in a particular combination,
2 We consider an approximate solution to avoid divergence in the crit-ical points obtained for this case.
forming f (R) = f (−T + B) gravity. In f (R) gravity, thesecouplings do appear but in a very prescribed format. In thepresent case we allow for more novel models to develop. Inparticular in cases 1 and 3 of Sect. 5, the coupling term inthe Lagrangian plays an impactful role in the dynamics thatensue. This is an interesting property that should be investi-gated further.
From the results obtained with this proposal, we notice thatit will be interesting to study the behaviour of other f (T, B)
gravity models, which together with their confrontation withobservational data, may open an avenue for producing otherviable cosmological scenarios. Furthermore, the role of avarying λ is also an important future work which may betterexpose the dynamical behaviour of f (T, B) gravity, sincefrom the Taylor and Power Law cases we will require anon-autonomous system with λ �= const. Another impor-tant higher-order extension to TG is f (T, TG) which mayalso have interesting properties. This study will be reportedelsewhere.
Acknowledgements CE-R acknowledges the Royal AstronomicalSociety as FRAS 10147 and PAPIIT Project IA100220. CE-R and JLSwould like to acknowledge networking support by the COST ActionCA18108. JLS would also like to acknowledge funding support fromCosmology@MALTA which is supported by the University of Malta.
DataAvailability Statement This manuscript has no associated data orthe data will not be deposited. [Authors’ comment: This is a theoreticalwork based on exploring the stability of f (T, B) gravity. For this reason,no associated data forms part of this work.]
Open Access This article is licensed under a Creative Commons Attri-bution 4.0 International License, which permits use, sharing, adaptation,distribution and reproduction in any medium or format, as long as yougive appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changeswere made. The images or other third party material in this articleare included in the article’s Creative Commons licence, unless indi-cated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permit-ted use, you will need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecomm
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