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Dynamical and Cosmological Aspects of Teleparallel and Extended Teleparallel Gravity

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This thesis claims that modified teleparallel gravity theories can reproduce the observed cosmic sequence—radiation, matter, and accelerating dark-energy eras—as stable critical points of a single dynamical system, with late-time…

desk verdict Competent compilation of already-published dynamical-system papers, but the central viability claim rests on freezing λ=Ḧ/H^3 to isolated power-law trajectories. read the letter →

arxiv 2501.11048 v1 pith:UZMKAZEN submitted 2025-01-19 gr-qc

S. A. Kadam This is my paper · ORCID
classification gr-qc MSC 83D0583F0537C75 PACS 04.50.Kd95.36.+x98.80.-k
keywords modifiedteleparallelgravitydynamicalsystemanalysiscosmicaccelerationdarkenergyboundarytermsGauss-Bonnetscalarfieldcosmologycosmologicalphasespace
topics Dark Energy
open problems Dark Energy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This thesis sets out to show that modified teleparallel gravity—a family of theories that describes gravity through torsion rather than curvature—can account for the full sequence of cosmic phases, from radiation and matter domination to the present accelerated expansion, without invoking a cosmological constant. The author constructs autonomous dynamical systems for several such theories, including scalar-field-coupled models and extensions with boundary and Gauss-Bonnet terms, and identifies critical points whose stability matches the expected epochs of cosmic history. For the models examined, stable late-time attractors with equation-of-state parameter near minus one emerge, and the evolution of density parameters and the deceleration parameter comes close to standard Lambda-CDM behaviour. The thesis concludes that these modified teleparallel models form a viable framework for late-time cosmic acceleration.

What carries the argument

The engine of the analysis is the autonomous dynamical system: the Friedmann and Klein-Gordon equations of each theory are rewritten in dimensionless phase-space variables, and the fixed points of the resulting ordinary differential equations are classified by the eigenvalues of the Jacobian matrix, with center-manifold theory used for non-hyperbolic points. A constant dimensionless parameter $\lambda = \ddot{H}/H^3$ closes the system in the boundary-term chapters, and exponential and power-law potentials fix the scalar-field sector. The distinctive load-bearing objects are the teleparallel boundary term $B$ and the teleparallel Gauss-Bonnet term $T_G$, which enter the Lagrangians as new couplings and reshape the phase space; their presence is what distinguishes these models from ordinary $f(T)$ gravity and allows the de Sitter and scaling solutions the thesis highlights.

What would settle it

Compute $\lambda = \ddot{H}/H^3$ along the best-fit $H(z)$ curves the thesis reports for the $f(T,B)$ and $f(T,T_G)$ models; if $\lambda$ moves appreciably across the radiation-to-dark-energy epochs instead of holding near the values used to find the critical points, such as $8$ for radiation and $9/2$ for matter, the fixed-point analysis does not govern the models' real trajectories.

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Extended reading notes

Core claim

The central claim is that, across the studied families of modified teleparallel gravity—$f(T,\phi)$, $f(T,B)$, $f(T,T_G)$, $f(T,B,T_G,B_G)$, and the general teleparallel scalar-tensor formalism—the cosmological field equations can be rewritten as autonomous dynamical systems whose fixed points reproduce radiation-, matter-, and dark-energy-dominated eras in a single phase space. The radiation and matter critical points are saddles, so the Universe can pass through them, while the dark-energy critical points are stable late-time attractors; de Sitter solutions appear as limiting cases. The thesis reports that for representative parameter choices the models yield present-day density parameters near $\Omega_m \approx 0.3$ and $\Omega_{\mathrm{DE}} \approx 0.7$, a transition from deceleration to acceleration at redshift $z \approx 0.6$, and Hubble and distance-modulus curves compatible with observational fits. If correct, this means the observed cosmic acceleration can be explained by the geometry of torsion with added boundary couplings, without a cosmological constant.

Load-bearing premise

The load-bearing premise is that the dimensionless quantity $\lambda = \ddot{H}/H^3$ stays constant along the cosmological trajectories analysed in the boundary-term chapters, so the dynamical systems are autonomous; if the real Universe's $\lambda$ varies with time, the critical points and stability classifications derived under this assumption need not describe the actual evolution.

Editorial extensions

If this is right

  • If the central claim holds, modified teleparallel gravity can generate the full radiation-to-matter-to-accelerating-dark-energy sequence from a single Lagrangian, with no cosmological constant input.
  • The stable dark-energy critical points mean the models converge to an accelerating late-time state from a broad set of initial conditions, making the acceleration phase an attractor rather than a fine-tuned choice.
  • The boundary and Gauss-Bonnet terms alter the phase-space structure and produce de Sitter and scaling solutions, offering a dynamical route to the coincidence problem within the scope of the paper.
  • The scalar-field models with exponential and power-law potentials give present-day matter and dark-energy densities, transition redshifts, and equation-of-state values close to those inferred from current observations.
  • The fits to Hubble and Supernovae Ia data in the boundary-coupling chapter indicate that the nonminimal boundary coupling with several potentials can match the observed distance-modulus curve, supporting the models' observational viability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: The viability claim is not yet tested against the full cosmological data set; a natural next step is to confront these models with CMB, baryon-acoustic-oscillation, and large-scale-structure likelihoods, which the thesis does not do.
  • Inference: If the constant-$\lambda$ assumption is relaxed, the fixed-point structure of the boundary-term chapters can change qualitatively; integrating the full field equations numerically along the same trajectories would show which stability classifications survive.
  • Inference: Because the $f(T,B)$ power-law models are described as useful for addressing the $H_0$ tension, the same dynamical-system machinery could be used to check whether the stable attractors shift the inferred Hubble constant, a comparison the thesis leaves implicit.
  • Inference: The same phase-space method could be applied to teleparallel models with direct dark-matter and dark-energy interactions, which the thesis does not investigate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This PhD-thesis manuscript compiles dynamical-system analyses of several modified teleparallel gravity theories: teleparallel-Horndeski scalar-tensor models with power-law kinetic couplings (Ch. 2), f(T,φ) gravity with power-law torsion coupling and exponential/power-law potentials (Ch. 3), f(T,B) and f(T,T_G) gravity with boundary/Gauss-Bonnet terms (Ch. 4), the combined f(T,B,T_G,B_G) formalism (Ch. 5), and nonminimal boundary-coupled quintessence with observational Hubble and supernova comparisons (Ch. 6). The recurring method is to introduce dimensionless variables, form an autonomous system, locate critical points, compute eigenvalues, and identify which points can represent radiation-, matter-, and dark-energy-dominated epochs. The abstract concludes that these models provide a viable framework for early and late-time cosmic acceleration.

Significance. If the dynamical-system results were fully sound, the thesis would give a broad and useful phenomenology of teleparallel gravity modifications, with explicit stability tables, phase portraits, and some comparison to H(z) and distance-modulus data. Strengths of the manuscript include the unusually explicit algebraic presentation of the critical points and eigenvalues, the large number of models treated in a unified notation, and the inclusion of observational evolution plots in Chapter 6. However, the central viability claim is weakened by two load-bearing technical issues: the constant-λ closure in the f(T,B) and f(T,B,T_G,B_G) analyses, and the incomplete stability classification of non-hyperbolic de Sitter points. The present-day values Ω_m≈0.3 and Ω_DE≈0.7 are obtained from hand-picked initial conditions and parameter values rather than from a fit, so they are illustrative consistency checks rather than falsifiable predictions. The thesis is therefore a valuable collection of model studies, but the advertised single-framework explanation of the full radiation→matter→dark-energy sequence is not yet established.

major comments (4)
  1. [§4.3, Eqs. (4.10) and surrounding text] The paragraph preceding Eq. (4.10) states that λ=Ḧ/H^3 is 'treated as a constant throughout the analysis.' For a power-law scale factor a∝t^p one has λ=2/p^2, so the values λ=8 (radiation) and λ=9/2 (matter) correspond exactly to isolated power-law solutions. During a physical transition between these epochs, λ is time-dependent, and no evolution equation for λ is supplied for the f(T,B) system or for the f(T,B,T_G,B_G) system in Chapter 5 that builds on the same assumption. Consequently, the critical points and stability tables in §4.3 describe a restricted system with λ frozen, not the actual phase flow of the cosmological model. The f(T,T_G) mixed-power-law model in §4.5.1 is less affected because λ is solved from the variables in Eq. (4.28), but the summary claim treats all models as one viable framework, and for the f(T,B) and f(T,B,T_G,B_G) analyses the constant-λ closure is load-bearing.
  2. [§4.3.1, point C4, and §4.3.2, point P4 (Tables 4.2, 4.6 and Figs. 4.3, 4.6)] The de Sitter critical points C4 and P4 have eigenvalues {0,0,0,-4}, so linear stability theory is inconclusive. The text states that the central-manifold condition is not satisfied and therefore the method is not applied, but then concludes from 2D phase portraits that the point is an attractor. A two-dimensional projection of a four-dimensional flow cannot establish asymptotic stability; at most it suggests attraction in the projected directions. Without a valid center-manifold reduction, a Lyapunov function, or a rigorous normally-hyperbolic argument, the late-time attractor status of these points is unproven, and this status is central to the claimed dark-energy epoch.
  3. [§3.3.1 and §3.3.2, Figs. 3.3-3.4 and 3.7-3.8] The presented present-day values Ω_m≈0.3, Ω_DE≈0.7, q0≈-0.61, and the transition redshifts are obtained by evolving the autonomous system from a specific hand-picked initial condition (e.g., x0=10^-8.89, y0=10^-2.89, u0=10^-5.96, ρ0=10^-0.9) with chosen parameters β=-0.2, σ=-0.30, λ=-0.2. No parameter estimation, likelihood, or goodness-of-fit is performed, so statements that these values 'agree with Planck' or are 'compatible with ΛCDM' are consistency checks rather than tests of the model. This does not invalidate the dynamical-system analysis, but it should be reframed in the abstract and conclusions as an illustrative compatibility exercise, not as evidence that the model is observationally preferred.
  4. [Chapters 2-5 (general method)] The thesis repeatedly infers a viable cosmic history from the coexistence of a stable dark-energy point and saddle radiation/matter points in the same parameter range. Strictly, one must also exhibit a trajectory (heteroclinic orbit or explicit numerical evolution) that connects the radiation saddle to the matter saddle and then to the dark-energy attractor. The 2D phase portraits and the single evolution plots from selected initial conditions do not establish this sequence for the full higher-dimensional systems. For the central claim of a single model describing radiation→matter→dark-energy in sequence, at least one representative heteroclinic chain or a full-dimension numerical trajectory with stated initial conditions is needed.
minor comments (6)
  1. [Abstract] The sentence 'Teleparallel gravity, is an alternative to General Relativity, explains gravitation through torsion' contains a misplaced comma; it should read 'Teleparallel gravity, an alternative to General Relativity, explains gravitation through torsion.'
  2. [Eq. (1.15)] The tetrad for flat FLRW spacetime is written as (1,a(t),a(t),a(t)), which is ambiguous; it should be written as the diagonal tetrad diag(1,a(t),a(t),a(t)) to avoid confusion with a four-vector.
  3. [§2.3.1 and §2.3.2] There are missing closing parentheses in references to 'action equation Eq. (2.11' and similar; please proofread the equation cross-references.
  4. [Throughout] The cosmological model name is written inconsistently as FLRW, FLRW, and 'FLR W'; one standard form should be used throughout.
  5. [§4.3.2 heading] The subsection heading 'Radiaiton-dominated Critical points' contains a typo: it should be 'Radiation-dominated Critical points.'
  6. [Figure captions, Chapter 2] Some phase-portrait captions list parameter values such as τ, ζ, δ without defining them in the caption or immediately preceding text; please clarify each symbol when it first appears in a figure.

Circularity Check

2 steps flagged · score 6.0 of 10

The claimed radiation/matter/de-Sitter epoch sequence is put in by hand through the constant-λ closure, and the quoted present-day density parameters are fixed by chosen initial conditions rather than predicted.

  1. fitted input called prediction [Chapter 4, §4.3, before Eq. (4.10); Tables 4.1–4.4 (f(T,B)) and Tables 4.5–4.8 (power-law model)]
    "To express the autonomous dynamical system, we define the parameter λ = Ḧ/H^3 [39,93] and is treated as a constant throughout the analysis. To note the value of the parameter λ = 8, 9/2, connects with the radiation, matter dominated phase, respectively, whereas for DE, it depends on the dynamical variables X and Y."

    For a power-law scale factor a(t) ∝ t^p one has λ = Ḧ/H^3 = 2/p^2. The values quoted are exactly the power-law epochs: radiation p = 1/2 gives λ = 8, matter p = 2/3 gives λ = 9/2, and de Sitter gives λ = 0. The critical points C1/P1, C2/P2 and C4/P4 are then required to exist precisely at λ = 8, λ = 9/2 and λ = 0, so the 'radiation-dominated', 'matter-dominated' and 'de Sitter' epochs are not derived from the field equations; they are inserted as the assumed constant value of λ. No evolution equation for λ is supplied, so the claimed sequence from radiation to matter to dark energy is a property of the closure assumption, not of the full f(T,B) dynamics. The same constant-λ construction is carried into the f(T,B,T_G,B_G) analysis of Chapter 5.

  2. fitted input called prediction [Chapter 3, §3.3.1 and §3.3.2, text around Figs. 3.3 and 3.7]
    "From the evolution plots of standard density parameters, we observe that at present, Ωm ≈ 0.3, which agrees with the Planck observation results [20]. The dominant presence of DE at the present epoch is quite visible from the derived value of the DE density parameter ΩDE ≈ 0.7 [89]. ... for the initial conditions x0 = 10^-8.89, y0 = 10^-2.89, u0 = 10^-5.96, ρ0 = 10^-0.9 for model 3.3.1."

    The autonomous system for (x,y,u,ρ) is deterministic: the entire trajectory, and hence the present-day values Ωm ≈ 0.3 and ΩDE ≈ 0.7, is fixed once the initial conditions are chosen. The thesis presents these values as a successful outcome ('derived value') but they are, on inspection, outputs of the stated initial condition vector, not predictions of the modified-gravity model. Matching ΛCDM/Planck values is therefore a tuning of initial data rather than an independent test. The same applies to the transition redshift values quoted from the same integrations.

full rationale

The thesis contains a large amount of standard dynamical-system work: deriving autonomous systems from the Friedmann and Klein-Gordon equations, computing critical points, and classifying stability by eigenvalues. That core analysis is self-contained and not circular. The circularity is concentrated in the interpretive layer. First, the f(T,B) analysis (Chapter 4) closes the system by declaring λ = Ḧ/H^3 constant, and then identifies radiation, matter and de Sitter critical points by imposing λ = 8, λ = 9/2 and λ = 0 respectively. Since these numbers are exactly the values of λ for the corresponding power-law solutions, the 'epochs' are equivalent to the input assumption rather than emergent from the dynamics. The f(T,T_G) mixed-power-law model partially improves on this by solving λ from the phase-space variables in Eq. (4.28), and the paper notes the singular k = 0,1 limits honestly, so the f(T,T_G) part is less exposed. Second, the present-day cosmological parameters quoted as successful outcomes are fixed by chosen initial conditions in the numerical integrations; no independent, parameter-free prediction of Ωm and ΩDE is demonstrated. The thesis also makes normal use of the author's own prior publications for the λ variable and for model forms; I do not treat that as load-bearing circularity beyond the constant-λ issue. The acknowledged non-hyperbolic stability gaps and the failed central-manifold conditions are limitations rather than circular steps. Overall, the central viability claim is partially circular: the epoch sequence is built into the constant-λ closure, and the quoted present-day agreement is fitted through initial conditions. A score of 6 reflects this partial reduction without claiming that every result in the thesis is definitionally true.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central results rest on a chain of model choices: specific Lagrangian forms, constant λ, and hand-picked coupling parameters. The thesis does not provide an independent mechanism from which these choices follow, so they are counted as free parameters or ad hoc assumptions.

free parameters (6)
  • α (power-law index in kinetic couplings X^α T and X^α I2) = general α, plus α=1,2 chosen by hand
    Chapter 2 actions in Eqs. (2.11) and (2.37) depend on α; stability conditions require α>0 and the value is not derived.
  • λ (potential slope in V=V0 e^{-κτφ}) = chosen as λ=√(2/9) etc. in phase portraits
    The exponential potential yields constant λ, but the specific values used to illustrate stability are hand-picked from the allowed ranges.
  • σ (coupling slope in F=F0 e^{-ηκφ}) = -0.30 in Chapter 3 plots
    Exponential coupling function parameter; stability conditions are expressed in terms of σ and the plotted value is chosen by hand.
  • β (power in G(T)=α(-T)^β) = -0.2 or 1.1 in different models
    Power-law torsion coupling in f(T,φ); stability conditions vary with β and the values are selected to give stable attractors.
  • ξ, α, ζ, p, m, k (boundary-term and Gauss-Bonnet coupling constants) = e.g., ξ=-2.4, ζ=1.0001, p=-1, k=0.029, m=0.5
    These constants in f(T,B)= -T + ξT + αB log B, f(T,B)=ζT+β(-B)^p, and f(T,T_G)=f0 T^k T_G^m are chosen to obtain stable accelerating critical points.
  • initial conditions for evolution plots = e.g., x0=10^-8.89, y0=10^-2.89, u0=10^-5.96, ρ0=10^-0.9
    The density-parameter evolution and present-day values (Ωm≈0.3, ΩDE≈0.7) depend on the chosen initial conditions, not on independent constraints.
assumptions (5)
  • domain assumption FLRW metric and perfect-fluid energy-momentum tensor with matter EoS ω_m=0 and radiation EoS ω_r=1/3
    Used throughout all chapters to derive the Friedmann equations and the dynamical systems.
  • domain assumption Weitzenböck gauge with vanishing spin connection for the tetrad
    Chapter 1, Sec 1.6, tetrad in Eq. (1.15), consistent with the teleparallel formalism.
  • ad hoc to paper λ=Ḧ/H^3 is treated as a constant in the autonomous systems
    Chapter 4, Sec 4.3 states λ is treated as a constant; this restricts trajectories to constant-jerk solutions without justification.
  • ad hoc to paper The chosen Lagrangian functions (logarithmic boundary term, power-law models, exponential/power-law potentials) are representative of the viable parameter space
    These forms are selected for integrability and the existence of critical points; no reconstruction from observations is provided.
  • standard math The equivalence between f(R) and f(T,B) when f(T,B)=f(-T+B) is accepted
    Chapter 1, Sec 1.6.2, used to relate the modified teleparallel models to known f(R) results.

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Pith. "Pith review of Dynamical and Cosmological Aspects of Teleparallel and Extended Teleparallel Gravity." pith.science (2026). https://pith.science/paper/UZMKAZEN

@misc{pith2026250111048,
  author       = {Pith},
  title        = {Pith review of: Dynamical and Cosmological Aspects of Teleparallel and Extended Teleparallel Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UZMKAZEN}},
  note         = {Machine review of arXiv:2501.11048}
}
abstract

This thesis investigates modified teleparallel gravity models with a scalar field and teleparallel boundary terms, focusing on their cosmological implications for late-time cosmic acceleration. Teleparallel gravity, is an alternative to General Relativity, explains gravitation through torsion. The study presents the teleparallel analog of the Horndeski theory and its dynamical system approach, analyzing the newly developed $f(T,\phi)$ gravity model with two potential functions. It examines the phase space, conditions for different cosmological epochs, and the transition from early to late-time cosmic evolution. The inclusion of boundary terms, such as the teleparallel boundary term $B$ and the Gauss-Bonnet term $T_G$, enhances gravitational interactions. The modified teleparallel gravity models, such as $f(T, B)$, $f(T, T_G)$, and $f(T, B, T_G, B_G)$, are explored in terms of their stability and cosmological scenarios, successfully describing the accelerated expansion and late-time attractors. The thesis assesses the impact of a non-canonical scalar field coupled to the boundary term $B$ on the Universe's evolution, comparing findings from Hubble data and Supernovae Ia data. Overall, the analysis suggests that modified teleparallel gravity models effectively explain early to late-time cosmic acceleration, revealing the strong dynamical foundation and offering a versatile framework for addressing challenges in cosmic evolution.

Figures

Figures reproduced from arXiv: 2501.11048 by the authors.

Figure 2.1
Figure 2.1. 2D phase portrait for model 2.3 for general α. For the critical points L and M, the stability conditions depend on χ, which represents u co-ordinate, and the phase plots are analysed in the xy-axis plane; hence critical points L and M may show saddle point behaviour. From the phase diagram, it can be observed that critical points, H, I, D, and E trajectories are attracted towards the critical point, hence describing… view at source ↗
Figure 2.2
Figure 2.2. 2D phase portrait for model 2.3, α = 1. 2.3.2 Case B: α = 2 In this case, we have analysed cosmological implications by using a dynamical system approach for particular value α = 2 in action Eq. (2.11). In this case, the set of dynamical variables to obtain an autonomous dynamical system can be defined as follows, x = κϕ˙ √ 6H , y = κ √ V √ 3H , u = 5 2 κ 2ϕ˙4 , ρ = κ √ρr √ 3H , λ = −V ′ (ϕ) κV (ϕ) , Γ = V (ϕ)V ′′(ϕ… view at source ↗
Figure 2.3
Figure 2.3. 2D phase portrait for model 2.3, α = 2. The upper left plot in [PITH_FULL_IMAGE:figures/full_fig_p055_2_3.png] view at source ↗
Figures from the paper (48 more)
Figure 2.4
Figure 2.4. Figure 2.4: 2D phase portrait for model 2.4, for general α [PITH_FULL_IMAGE:figures/full_fig_p059_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: 2D phase portrait for model 2.4, for general α. In above [PITH_FULL_IMAGE:figures/full_fig_p060_2_5.png]
Figure 2.6
Figure 2.6. Figure 2.6: 2D phase portrait for model 2.4, α = 1. Here, the left phase portrait is plotted for the parametric values u = 0, ρ = 0 and λ = q 2 9 and the right side plot parametric values are x = 0, ρ = 0, σ = 1 4 for right side figure. The phase space plots for the dynamical sy…
Figure 2.7
Figure 2.7. Figure 2.7: 2D phase portrait for model 2.4 α = 1. 2.4.2 Case B: α = 2 In this case, we have discussed dynamical system analysis for model 2.4, α = 2. The evolution expressions can be obtained by limiting Eqs. (2.38) to (2.40) using α = 2. The dynamical variables to obtain an au…
Figure 2.8
Figure 2.8. Figure 2.8: 2D phase portrait for model 2.4, α = 2. Here in [PITH_FULL_IMAGE:figures/full_fig_p064_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: 2D phase portrait for model 2.4, α = 2. Critical points xc yc uc ρc Existence condition q ωtot A 0 0 1 0 - 2 3 1 9 B 1 0 0 0 - 2 1 C −1 0 0 0 - 2 1 D σ √ σ 2 + 1 −2σ 2 0 λ = 0, σ 3 − 3σ ̸= 0 −1 −1 E σ − √ σ 2 + 1 −2σ 2 0 λ = 0, σ 3 − 3σ ̸= 0 −1 −1 F q3 2 λ q 3 2 q 1 …
Figure 3.1
Figure 3.1. Figure 3.1: 2D phase portraits for model 3.3.1. In above [PITH_FULL_IMAGE:figures/full_fig_p075_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: 2D and 3D region plots for model 3.3.1 . Ωr Ωm ΩDE -2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 Log10(1+z) q -2 0 2 4 6 -1.0 -0.5 0.0 0.5 1.0 Log10(1+z) [PITH_FULL_IMAGE:figures/full_fig_p076_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: Density parameters and deceleration parameter for model 3.3.1 [PITH_FULL_IMAGE:figures/full_fig_p076_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: The EoS for DE (ωDE) and total EoS (ωtot) parameter for model 3.3.1. In Figs. 3.3, 3.4 we have for radiation (Ωr), matter (Ωm), DE (ΩDE) in redshift. Right Panel–Deceleration parameter (q) in redshift for β = −0.2, σ = −0.30, λ = −0.2 for the initial conditions x0 = …
Figure 3.5
Figure 3.5. Figure 3.5: 2D phase portrait for model 3.3.2. The [PITH_FULL_IMAGE:figures/full_fig_p080_3_5.png]
Figure 3.6
Figure 3.6. Figure 3.6: 2D region plot for model 3.3.2. Ωr Ωm ΩDE -2 0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 Log10(1+z) q -2 0 2 4 6 -1.0 -0.5 0.0 0.5 1.0 Log10(1+z) [PITH_FULL_IMAGE:figures/full_fig_p081_3_6.png]
Figure 3.7
Figure 3.7. Figure 3.7: Density parameter and deceleration parameter for model 3.3.2. ωDE ωtot -2 0 2 4 6 -1.0 -0.5 0.0 0.5 Log10(1+z) [PITH_FULL_IMAGE:figures/full_fig_p081_3_7.png]
Figure 3.8
Figure 3.8. Figure 3.8: The EoS for DE (ωDE) and total EoS (ωtot) parameter for model 3.3.2. 3.4 Conclusion In this work, we have studied the dynamical system analysis for the recently developed scalar￾torsion f(T, ϕ) gravity [58–60] formalism. Two well-motivated non-minimally coupling func…
Figure 4.1
Figure 4.1. Figure 4.1: 2D plot for stability and acceleration region for critical point C3 for model 4.3.1 [PITH_FULL_IMAGE:figures/full_fig_p089_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: 2D phase portrait for model 4.3.1 . C4 -10 -5 0 5 10 -10 -5 0 5 10 V W [PITH_FULL_IMAGE:figures/full_fig_p090_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: 2D phase portrait for model 4.3.1 . The above 2D phase plots are for λ = 0.3, ξ = −2.4, α = 1.3. In [PITH_FULL_IMAGE:figures/full_fig_p090_4_3.png]
Figure 4.4
Figure 4.4. Figure 4.4: Density parameters for model 4.3.1. The initial conditions used to plot Figs. 4.4, 4.5 are X = −1.2 × 102.2 , Y = 2.2 × 10−3.4 , V = 1.02 × 102.6 , W = 4.5 × 10−8.1 , λ = 0.3, ξ = −2.4, α = 1.3. q -10 -5 0 5 10 -1.3870 -1.3865 -1.3860 -1.3855 N = log(a) ωDE ωtot -10 …
Figure 4.5
Figure 4.5. Figure 4.5: Deceleration parameter for Model 4.3.1. 4.3.2 Power law model We consider, ˜f(T, B) = ζT + β(−B) p , (4.13) where ˜f2(B) = β(−B) p , is nonlinear and capable of studying observational tests for the theory by referring to the most recent SN-Ia data [83]. This is a pro…
Figure 4.6
Figure 4.6. Figure 4.6: 2D phase portrait for the dynamical system for model 4.3.2 [PITH_FULL_IMAGE:figures/full_fig_p094_4_6.png]
Figure 4.7
Figure 4.7. Figure 4.7: Stability and acceleration of the Universe for model 4.3.2. Ωm Ωr ΩDE -2 -1 0 1 2 3 4 5 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 N = log(a) [PITH_FULL_IMAGE:figures/full_fig_p095_4_7.png]
Figure 4.8
Figure 4.8. Figure 4.8: Evolution of the density parameters for model 4.3.2. q -10 -5 0 5 10 -1.3870 -1.3865 -1.3860 -1.3855 N = log(a) ωDE ωtot -4 -2 0 2 4 -4 -2 0 2 4 N = log(a) [PITH_FULL_IMAGE:figures/full_fig_p095_4_8.png]
Figure 4.9
Figure 4.9. Figure 4.9: Deceleration (q) and EoS parameters ωDE, ωtot for model 4.3.2. The initial conditions for [PITH_FULL_IMAGE:figures/full_fig_p095_4_9.png]
Figure 4.10
Figure 4.10. Figure 4.10: 2D phase portrait with k = 0.029, m = 0.5 for model 4.5.1 [PITH_FULL_IMAGE:figures/full_fig_p101_4_10.png]
Figure 4.11
Figure 4.11. Figure 4.11: Region plot for critical points A3 and A4 at x4 = − 1 3 for model 4.5.1. In [PITH_FULL_IMAGE:figures/full_fig_p102_4_11.png]
Figure 4.12
Figure 4.12. Figure 4.12: 3D phase portrait with k = 0.029, m = 0.5 for model 4.5.1. The 3D phase portrait presented in [PITH_FULL_IMAGE:figures/full_fig_p102_4_12.png]
Figure 4.13
Figure 4.13. Figure 4.13: Evolution of the density parameters for model 4.5.1. q -15 -10 -5 0 5 10 15 20 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 N=log(a) ωDE wtot. -10 0 10 20 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 N=log(a) [PITH_FULL_IMAGE:figures/full_fig_p103_4_13.png]
Figure 4.14
Figure 4.14. Figure 4.14: Deceleration and EoS parameter for model 4.5.1. The initial conditions for above plots are X = −10−1 , Y = 10−5 , Z = 10−10, V = 10−11 , k = 0.029, m = 0.5. The evolution plot for the standard density parameter is plotted in [PITH_FULL_IMAGE:figures/full_fig_p103_4…
Figure 4.15
Figure 4.15. Figure 4.15: 2D phase portrait for model 4.5.2. • DE-dominated critical points: In critical points B3, B4, the value of parameter λ and the parameter Z is depend on ϵ1 and ϵ2. The critical point B3 and at B4 may describe current cosmic acceleration, respectively, at ϵ1 > −1 and …
Figure 4.16
Figure 4.16. Figure 4.16: Region plots for critical points B3 and B4 showing stability and acceleration for model 4.5.2. We wish to mention here that the fixed points B1 and B2 exist only for the value of λ as 8 and 9 2 , respectively. For the choice of the variables of the model, all three …
Figure 4.17
Figure 4.17. Figure 4.17: Evolution of the density parameters for model 4.5.2. The evolution of density parameters Ωr, Ωm, ΩDE have been presented in [PITH_FULL_IMAGE:figures/full_fig_p107_4_17.png]
Figure 4.18
Figure 4.18. Figure 4.18: Deceleration and EoS parameter for model 4.5.2. The initial conditions in this case are X = 10−1.3 , Z = 0.02 × 101.2 , V = 0.02 × 10−2.5 , W = 0.0021 × 10−3.7 , m = 0.67, k = 0.785. 4.6 Conclusion The general dynamical system, which is dependent on the form of ˜f(T…
Figure 5.1
Figure 5.1. Figure 5.1: 2D phase portrait for the dynamical system, for model 5.3.1. In [PITH_FULL_IMAGE:figures/full_fig_p118_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: Behaviour of density parameters for matter, radiation, and DE, q for Model 5.3.1. 5.3.2 Sum of separated power law (particular case) The second form considered for F(T, B, TG, BG) is a particular form of F(T, B, TG, BG) = t0T m + b0Bn + g0T k G, which has been succes…
Figure 5.3
Figure 5.3. Figure 5.3: 2D phase portrait for model 5.3.2. In [PITH_FULL_IMAGE:figures/full_fig_p123_5_3.png]
Figure 5.4
Figure 5.4. Figure 5.4: Behaviour of density parameters for matter and DE, q for model 5.3.2 [PITH_FULL_IMAGE:figures/full_fig_p123_5_4.png]
Figure 6.1
Figure 6.1. Figure 6.1: Evolution of EoS and standard density parameters for Model 6.2.1. In this case, the plots are plotted for the initial conditions are xC = 10−8.89, yC = 10−2.89, uC = 10−5.96, ρC = 10−0.75, λ = −0.01, α = −5.2. -2 0 2 4 6 -2 -1 0 1 2 Log10(1+z) ω ωDE ωtot ωΛCDM -1 0 1…
Figure 6.2
Figure 6.2. Figure 6.2: Evolution of Hubble and deceleration parameters for model 6.2.1. The plot shows the contribution amount of DM Ωm ≈ 0.3 and DE ΩDE ≈ 0.7 density parameters. The time of matter-radiation equality is around z ≈ 3387 and is denoted with a pointed arrow in [PITH_FULL_IMA…
Figure 6.3
Figure 6.3. Figure 6.3: Plot of the observed distance modulus function µ(z) and the predicted ΛCDM model distance modulus function µΛCDM(z) for model 6.2.1. In the 2D phase space portrait shown in [PITH_FULL_IMAGE:figures/full_fig_p133_6_3.png]
Figure 6.4
Figure 6.4. Figure 6.4: 2D phase space for model 6.2.1. The dynamical variables x and y are used to plot the phase portrait. The critical points AR and EDE are connected by the trajectory showing a red line. The phase space trajectories indicate that the solution transitions from a saddle c…
Figure 6.5
Figure 6.5. Figure 6.5: Evolution of EoS and standard density parameters for model 6.2.2 [PITH_FULL_IMAGE:figures/full_fig_p136_6_5.png]
Figure 6.6
Figure 6.6. Figure 6.6: Evolution of Hubble and deceleration parameters for model 6.2.2. 0.0 0.5 1.0 1.5 2.0 2.5 32 34 36 38 40 42 44 46 z μ ( z ) μ(z) μ(ΛCDM)(z) [PITH_FULL_IMAGE:figures/full_fig_p137_6_6.png]
Figure 6.7
Figure 6.7. Figure 6.7: Plot of the observed distance modulus function µ(z) and the predicted ΛCDM model distance modulus function µΛCDM(z) for Model 6.2.2. In the 2D phase space portrait represented in [PITH_FULL_IMAGE:figures/full_fig_p137_6_7.png]
Figure 6.8
Figure 6.8. Figure 6.8: 2D phase space for model 6.2.2 [PITH_FULL_IMAGE:figures/full_fig_p137_6_8.png]
Figure 6.9
Figure 6.9. Figure 6.9: depicts the behavior of the EoS parameter ωtot, which starts from 1 3 for radiation, approaches 0 during the DM-dominated period, and ultimately tends to −1. Both ωΛCDM and ωDE approach −1 at late times, with the current value of ωDE being equal to −1 at z = 0. The d…
Figure 6.10
Figure 6.10. Figure 6.10: Evolution of Hubble and deceleration parameters for model 6.2.3. 0.0 0.5 1.0 1.5 2.0 2.5 32 34 36 38 40 42 44 46 z μ ( z ) μ(z) μ(ΛCDM)(z) [PITH_FULL_IMAGE:figures/full_fig_p141_6_10.png]
Figure 6.11
Figure 6.11. Figure 6.11: Plot of the observed distance modulus function µ(z) and the predicted ΛCDM model distance modulus function µΛCDM(z) for model 6.2.3. In [PITH_FULL_IMAGE:figures/full_fig_p141_6_11.png]
Figure 6.12
Figure 6.12. Figure 6.12: 2D phase space for model 6.2.3 [PITH_FULL_IMAGE:figures/full_fig_p141_6_12.png]

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Reviewed August 10, 2026 · model on record in the stance chip above.