Non-singular Bouncing Cosmology in $f(R,G,T)$--Quintom model

Farzad Milani

REVIEW 4 major objections 4 minor 104 references

Non-singular Bouncing Cosmology in $f(R,G,T)$--Quintom model

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A non-singular bouncing universe arises from f(R,G,T)–quintom gravity, with the effective equation of state crossing the phantom divide line twice during the bounce.

desk verdict All five numerical models set f_G constant, so the advertised f(R,G,T) mechanism cancels; the paper reduces to a tuned f(R,T)-quintom example with an overstated title. read the letter →

arxiv 2510.00688 v3 pith:MIVQQAJH submitted 2025-10-01 gr-qc

Farzad Milani This is my paper

classification gr-qc PACS 04.50.Kd98.80.Jk95.36.+x98.80.Cq

keywords Bouncingcosmologyf(RGT)gravityQuintommodelPhantomdividecrossingNonsingularuniverseDarkenergyCosmologicalperturbationsModified

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 paper aims to show that modified gravity of the form f(R,G,T)—depending on curvature, the Gauss-Bonnet invariant, and the trace of the matter energy-momentum tensor—combined with a two-field 'quintom' scalar sector can produce a non-singular bouncing universe that crosses the phantom divide line twice, something single-field models do not do. A sympathetic reader would care because the same framework is intended to connect the early-universe bounce to late-time dark energy within a single action. The paper derives field equations, conservation laws, and stability conditions, then numerically reconstructs five concrete f(R,G,T) models, reporting non-singular bounces with positive effective energy density and non-negative sound speed. The caveat is that every worked example uses a constant Gauss-Bonnet coupling, so the G-dependent terms cancel exactly in the FLRW equations; what is actually exercised numerically is the quintom-plus-T-coupling dynamics.

What carries the argument

The central object is the action S = ∫ d⁴x √−g [ (1/2κ²)f(R,G,T) + Ξ(φ,ψ) + L_m ], where the quintom Lagrangian Ξ combines a phantom field and a canonical field with opposite-sign kinetic terms. The mechanism is the FLRW reduction R = 6(Ȟ + 2H²) and G = 24H²(Ȟ + H²), which the paper argues suppresses Ostrogradsky instabilities; the double PDL crossing emerges from the interplay between the phantom/canonical fields and the energy-momentum trace coupling f_T. The named identity is the phantom divide line, ω_eff = −1; a double crossing means ω_eff passes from phantom to quintessence and back again around the bounce.

What would settle it

Take one of the five models, say the linear model, and replace f_G = ξ₂ with a non-constant Gauss-Bonnet coupling, e.g. f_G ∝ G or f_G ∝ R, then re-run the same reconstruction; if no non-singular bounce with c_s² ≥ 0 and GS > 0 survives, the central claim that f(R,G,T)-quintom produces stable double-PDL bounces is falsified. Alternatively, compute the full quadratic action for scalar perturbations without assuming f_G constant: if a ghost appears, the stated stability conclusion fails.

Watch

Extended reading notes

Core claim

On its own terms, the central claim is that an f(R,G,T)-quintom coupling realises a double crossing of the phantom divide line (ω_eff = −1) during a non-singular bounce, while Hamiltonian analysis shows that FLRW symmetry reduces the higher-derivative degrees of freedom so that no Ostrogradsky ghost appears. Five reconstructed models—linear, exponential, power-law, teleparallel-inspired, and non-minimal R^2T coupling—are presented as numerical evidence: the scale factor reaches a minimum, the Hubble parameter changes sign, ρ_eff > 0, c_s^2 ≥ 0, and ω_eff crosses −1 twice for appropriate equations of state. Because all the numerical models set f_G constant, the Gauss-Bonnet contribution cance

Load-bearing premise

The load-bearing assumption is that the five numerical models—all with f_G constant, which makes the Gauss-Bonnet terms cancel identically in FLRW equations—actually demonstrate the f(R,G,T) mechanism; if the bounce and double crossing depend on the G dynamics, the paper's demonstrations do not exercise that part of the theory.

Editorial extensions

If this is right

If correct, the framework offers a unified action that avoids the initial singularity through a bounce and also drives late-time accelerated expansion.
The double PDL crossing provides a distinctive signature that could distinguish this bounce mechanism from single-field bounce models or inflationary scenarios.
The stability conditions GS > 0, FS > 0, and c_s² ≥ 0 are stated as checkable criteria; the paper reports they hold numerically across the five models.
The f_T coupling introduces an energy exchange between matter and geometry that allows temporary violation of the null energy condition, a necessary ingredient for the bounce.
Weyl conformal gravity emerges as a limiting case with f(G) = αG, connecting the framework to a conformally invariant theory.

Reading between the lines

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

A reader may infer that the demonstrated bounce mechanism is actually quintom-plus-f_T dynamics: because f_G is constant in every numerical model, the Gauss-Bonnet part is inert in the FLRW background despite being part of the stated framework.
A natural extension left implicit in the paper is to repeat the reconstructions with a non-constant Gauss-Bonnet coupling, such as f_G ∝ G or f_G ∝ R, to see whether the double PDL crossing and stability persist when the higher-derivative terms are genuinely active.
The paper's finite-time singularities in some models suggest that the bounce region is robust but the global solution may need higher-order corrections or limiting-curvature conditions; this is a testable extension, not a claim the paper makes.
A full parameter-space scan of the five models would clarify whether successful bounces are generic or confined to the tuned potentials and initial velocities used here.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know that the advertised f(R,G,T) mechanism is never actually exercised in this paper. All five numerical models set f_G constant, and the paper itself says the Gauss-Bonnet terms cancel exactly. The numerics reduce to f(R,T)-quintom, so the central claim — a novel f(R,G,T)-quintom double PDL crossing — is not demonstrated.

What the paper does well: it is a workmanlike reconstruction of five simple gravity forms (linear, exponential, power-law, teleparallel-inspired, non-minimal) with a quintom pair, and it shows plots of a(t), H(t), ω_eff, and stability coefficients for tuned parameters. The double PDL crossing appears in some models, and c_s^2 stays non-negative on the displayed trajectories. The Weyl gravity limit is also noted. These are okay examples, but the novelty is overstated: the double PDL crossing already appears in the cited quintom bounce literature (ref. [53]), and the f(R,T)-quintom extension is routine.

Where it falls apart: first, the G-dependence vanishes in the examples; calling this an f(R,G,T) implementation when G cancels is not supportable. Second, the bounce is engineered: Section 6.1 says the potential parameters are tuned to violate the NEC near t=0. That is fitting, not prediction. Third, the Hamiltonian analysis in Appendix D is asserted, not derived: there is no actual constraint algebra for the T-coupled action, and the degeneracy condition (76) is a known f(R,G) criterion whose applicability to f(R,G,T) is not shown. Fourth, the manuscript reports finite-time singularities in some solutions (Type II at t≈±0.2 for the exponential model, t∼±1 for the power-law model) while advertising 'non-singular' bouncing cosmology. And there is no code or data, with some initial conditions given as unexplained square roots. There are also algebraic inconsistencies in the teleparallel model's Friedmann equations; I didn't chase them to the end, but they don't need to be — the first three points already sink the central claim.

For peer review: I would not send this to referees; it is a desk reject. If the authors revised it to drop the f(G) claims and focused on the f(R,T)-quintom construction, it might be a minor contribution, but as it stands the central claim doesn't hold.

Referee Report

4 major / 4 minor

Summary. The paper proposes an f(R,G,T)-quintom framework for non-singular bouncing cosmologies, with a phantom field φ and a canonical field ψ, and claims a novel double crossing of the phantom divide during the bounce. It presents the field equations, an energy-momentum decomposition, conservation conditions, a Weyl-gravity limit, a stability analysis based on a degeneracy condition and a quadratic action for ζ, and numerical reconstructions of five models. The advertised central result is that the f(R,G,T) coupling produces stable non-singular bounces with double PDL crossings, ghost-free conditions, and c_s^2 ≥ 0.

Significance. If established, a rigorous f(R,G,T)-quintom bounce with derived stability and a double PDL crossing would be a useful contribution to the bouncing-cosmology literature. The manuscript contains an extensive set of explicit Friedmann expressions and five model reconstructions, and it correctly identifies some of the known technical conditions (e.g., the f(R,G) degeneracy condition). However, the numerical evidence does not exercise the advertised f(R,G,T) mechanism, and the stability analysis is asserted rather than derived for the action actually used. The result as presented therefore does not support the abstract's central claims.

major comments (4)

[§6.2.1–§6.2.5, Appendix F, Abstract] All five numerical models set f_G = ξ2 = const (see Eqs. (F.4), (F.13), (F.19), (F.28) and the text of §6.2.1–§6.2.2). With f_G constant the Gauss–Bonnet invariant enters only through a topological boundary term, and the simplified Friedmann equations used in the numerics contain no surviving ξ2 terms; the paper itself states this cancellation explicitly (§6.2.1, §6.2.2, §6.2.3). Thus the systems actually integrated are f(R,T)-quintom models, not f(R,G,T)-quintom. The abstract's claim that the f(R,G,T) coupling provides the novel double-PDL mechanism is therefore unsupported by the numerical evidence. A demonstration would require at least one model with non-constant f_G whose G-dependent terms survive in the simplified equations.
[§5.1, Appendix D, Eq. (76)] The ghost-free analysis is not derived for the action used. Eq. (76) is the known degeneracy condition for f(R,G) or DHOST-type theories; the manuscript does not show that this condition is sufficient for the f(R,G,T)-quintom action (2), where the T-coupling introduces additional matter degrees of freedom and the quintom sector contains a phantom field. Appendix D asserts a Hamiltonian counting with '8 constraints' and 4 physical DOF, but no explicit constraint algebra is given. Since stability is a central advertised result, this is a load-bearing gap.
[§5.2, Eqs. (79)–(86)] The stability claim c_s^2 ≥ 0 is checked on the tuned solutions rather than derived. The quadratic-action coefficients G_S and F_S are written down without derivation; the regularization parameter ε is inserted by hand in Eq. (80), and condition (85) is imposed to keep the denominator finite. Verifying inequalities on the same parameter choices that were tuned to produce the bounce (§6.1) does not constitute an independent prediction. In addition, several figures show c_s^2 = 0 at discrete times, which is described as 'momentary freezing' rather than an instability; this requires a quantitative justification.
[§6.1, §7] The procedure is partly circular: §6.1 states that the potential parameters are 'tuned... to ensure that the Null Energy Condition (NEC) is violated dynamically near t = 0, facilitating a successful bounce.' The double PDL crossings displayed in Figs. 2, 6, 10, etc. are properties of these constructed solutions, not predictions extracted from the theory. At minimum, the paper should provide a parameter scan or stability boundary separating bouncing from non-bouncing regions. As it stands, 'parametric control' amounts to fitting the desired outcome.

minor comments (4)

[§2.6, Eq. (38) and following text] The text after Eq. (38) lists 'fGR = ∂²f/∂G∂R, fGG = ∂²f/∂²G, and fGR = ∂²f/∂G∂T'; the last symbol should be fGT. Please correct the notation.
[§6.2.5] The initial conditions for the non-minimal coupling model include ˙ψ(0) = (100.2362362)^{1/2} and similar large values, which appear inconsistent with the 'small but non-zero' and sub-Planckian prescription stated in §6.1. The choice needs explanation or the general prescription needs revision.
[§6.2.2, §6.2.3] The text acknowledges finite-time singularities in the exponential model (t ≈ ±0.2) and the power-law model (t ≈ ±1). This should be reconciled with the title/abstract claims of non-singular bouncing cosmology, or explicitly presented as a limitation of those models.
[Appendix D] The sentence 'The complete derivation proceeds as:' is followed by formulas without the promised derivation. Either supply the steps or rephrase to indicate that the expressions are known results adapted to this model.

Circularity Check

2 steps flagged · score 6.0 of 10

All five numerical models set f_G constant, so the f(R,G,T) double-PDL result reduces by construction to f(R,T)-quintom; the bounce itself is parameter-tuned, making the numerical 'confirmations' fitted inputs rather than independent predictions.

renaming known result [Section 6.2.1, Eqs. (91)-(93); Appendix F; also Sections 6.2.2, 6.2.3, 6.2.4, 6.2.5]
"Notably, the Gauss-Bonnet coupling ξ2 completely disappears from the dynamics because all terms involving ξ2 cancel out exactly when fG = ξ2 is constant."

All five numerical reconstructions in §6 take f_G = ξ2 = constant (Eqs. (91), (97), (108), (122), (130); Appendix F sets ˙fG=¨fG=0). With f_G constant the Gauss-Bonnet contributions cancel identically against the -½ξ2G piece of f/2, as the paper states, so the integrated system is f(R,T)-quintom, not f(R,G,T)-quintom. The abstract calls these 'five f(R,G,T) models' confirming a novel double PDL crossing; that is a renaming of an f(R,T)-quintom result because the advertised G-dependent mechanism is absent by construction.
fitted input called prediction [Section 6.1, 'Initial Conditions and Physical Justification'; used again in §6.2.1-§6.2.5]
"The parameters of the potentials (V0, α, β, g) are then tuned for each model to ensure that the Null Energy Condition (NEC) is violated dynamically ρ +p <0 near t = 0, facilitating a successful bounce under these initial conditions."

The bounce—and the NEC violation that drives it—is the physical outcome being 'confirmed' in Section 6 and the Abstract. Here the parameters are explicitly tuned to make ρ+p<0 at t=0, i.e., the success criterion is input by construction, and the same tuned solutions are then reported as numerical evidence for non-singular bounces and PDL crossings. The claimed prediction is therefore a fitted output of the parameter choice, not an independent result.

full rationale

Most of the formal variation and conservation algebra in Sections 2, 5 and Appendices A-C is algebraic and self-contained; there is no evident circularity there. The circular content is concentrated in the numerical confirmation claim. First, the paper repeatedly acknowledges that in every one of its five reconstructions f_G is constant and the Gauss-Bonnet terms cancel exactly (e.g., §6.2.1, §6.2.3, Appendix F). Once fG=const., the FLRW equations explicitly reduce to the f(R,T)-quintom system, so the abstract's central assertion that the numerics confirm an f(R,G,T)-induced double PDL crossing is a label over an f(R,T) result, not a test of the advertised G mechanism. This is a reduction by construction of the central 'prediction'. Second, the initial-field and potential parameters are chosen so that ρ+p<0 near t=0, i.e., the bounce condition is enforced by the parameter choice; using those same solutions to claim confirmation of the bounce/PDL is fitting labeled as prediction. The c_s²≥0 checks are consistency checks on those tuned solutions, not independent predictions. The ghost-free criterion (76) is an external DHOST condition and is automatically satisfied for the fG=const. cases actually solved, so it does not independently validate the T-dependent action; I do not count this as a separate circular step. The self-citations [44],[45] are used to correct prior component expressions and are not load-bearing for the central claim. Taking all this together, the advertised f(R,G,T) mechanism is not exercised in the quantitative evidence, and the bounce behavior is parameter-enforced: partial circularity with score 6.

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

The model depends on many tuned free parameters (potential couplings, initial conditions, ε) and on the assumption that linear G is inert, which together mean the advertised f(R,G,T) mechanism is never actually exercised. No new physical entities are introduced beyond the standard quintom fields.

free parameters (5)

potential parameters V0, α, β, g = Vary per model: V0=0.25, α=2, β=1 (linear); V0=2.5, n=2, α=1, g=0.1 (exponential); V0=25, α=-0.01, g=0.1 (non-minimal);
Section 6.1 states 'The parameters of the potentials (V0,α,β,g) are then tuned for each model to ensure that the Null Energy Condition (NEC) is violated dynamically ρ+p<0 near t=0.'
coupling constants ξ1, ξ2, ξ3 = ξ1=ξ2=ξ3=1 in linear/power-law; ξ2=ξ3=1 in exponential/teleparallel/non-minimal
Chosen to satisfy perturbativity and numerical stability (Section 6.2); no observational or theoretical derivation.
initial field values φ(0), ψ(0), φ̇(0), ψ̇(0) = e.g., φ(0)=-0.05, ψ(0)=0.05, φ̇(0)=0.1, ψ̇(0)=-0.1; non-minimal uses ψ̇(0)=sqrt(100.2362362), sqrt(66.72589056), or 0.01
Section 6.1 says 'The specific numerical values (±0.05, ±0.1) are chosen for numerical convenience to produce clear graphical results' and the non-minimal values are given without justification.
regularization parameter ε = not specified
Introduced in Eq. (80) to make f_T(ρ+p)/(2H^2+ε) finite at the bounce; no physical interpretation or numerical value is given.
exponents n, m and scale R0, b = e.g., n=2, b=1 in exponential; n≥2, m≥2 in power-law
Ad hoc model parameters chosen for the example solutions; no scan or justification beyond 'to meet several physical criteria'.

assumptions (5)

domain assumption Flat FLRW metric with the given symmetry is the only background considered.
The entire stability and ghost-freedom discussion is limited to FLRW symmetry (Section 5), which the paper claims suppresses higher-derivative terms.
standard math Linear Gauss-Bonnet term (f_G = const.) is a topological boundary term with no dynamics, so G cancels in the Friedmann equations.
Used throughout Section 6; all five numerical models have f_G=ξ2=const. and the paper repeatedly states 'the Gauss-Bonnet terms cancel exactly'.
ad hoc to paper Degeneracy condition det(f_RR f_RG; f_GR f_GG)=0 is sufficient for ghost freedom in f(R,G,T).
Quoted from DHOST literature [21]; not derived for the full f(R,G,T) action with T-dependence. Appendix D does not compute the relevant Hessian or constraint algebra.
domain assumption Matter is a perfect fluid with L_m=p and standard conservation when consistency conditions are imposed.
Used to derive Θ_μν and conservation equations in Sections 2.5-2.6 and Appendix A; the fluid Lagrangian choice is assumed.
ad hoc to paper Quintom potentials (exponential, quadratic+coupling) are chosen to enable PDL crossings.
No derivation from fundamental theory; potential forms are selected to make the bounce work.

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Cite this review

Pith. "Pith review of Non-singular Bouncing Cosmology in $f(R,G,T)$--Quintom model." pith.science (2026). https://pith.science/paper/MIVQQAJH

@misc{pith2026251000688,
  author       = {Pith},
  title        = {Pith review of: Non-singular Bouncing Cosmology in $f(R,G,T)$--Quintom model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MIVQQAJH}},
  note         = {Machine review of arXiv:2510.00688}
}

abstract

We present a unified framework for non-singular bouncing cosmologies in modified gravity, combining $f(R,G,T)$ geometry with quintom scalar dynamics in a flat FLRW universe. While single-field models achieve phantom divide line (PDL) crossing and stable bounces, our $f(R,G,T)$-quintom coupling provides a novel implementation of a \textit{double} PDL crossing of $\omega_{\text{eff}}$ during the bounce. We address stability concerns through Hamiltonian analysis, showing that FLRW symmetry constraints suppress Ostrogradsky instabilities by reducing higher-derivative terms to metric invariant. The scalar field equation of motion is explicitly derived, confirming cancellation of pathological modes. Numerical reconstruction of five $f(R,G,T)$ models confirms non-singular bounces with $\rho_{\text{eff}}>0$ and $c_s^2 \geq 0$, alongside parametric control over energy condition violations. Our work extends prior studies by: (1) unifying early-time bounce dynamics with late-time dark energy, (2) demonstrating a novel double-PDL crossing signature compatible with FLRW stability, and (3) establishing explicit ghost-free criteria for higher-derivative terms.

Figures

Figures reproduced from arXiv: 2510.00688 by the authors.

**Figure 2.** FIG. 2: Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

**Figure 3.** FIG. 3: (a) Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

Figures from the paper (18 more)

**Figure 4.** Figure 4: FIG. 4: Stability analysis for the linear coupling model ( [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]

**Figure 5.** Figure 5: FIG. 5: Evolution of (a) scale factor [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]

**Figure 6.** Figure 6: FIG. 6: Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]

**Figure 7.** Figure 7: FIG. 7: (a) Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

**Figure 8.** Figure 8: FIG. 8: Stability analysis for the exponential model ( [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]

**Figure 9.** Figure 9: FIG. 9: Evolution of (a) scale factor [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]

**Figure 10.** Figure 10: FIG. 10: Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p033_10.png]

**Figure 11.** Figure 11: FIG. 11: Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p033_11.png]

**Figure 12.** Figure 12: FIG. 12: (a) Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p034_12.png]

**Figure 13.** Figure 13: FIG. 13: Stability analysis for the power-law model ( [PITH_FULL_IMAGE:figures/full_fig_p035_13.png]

**Figure 14.** Figure 14: FIG. 14: Evolution of (a) scale factor [PITH_FULL_IMAGE:figures/full_fig_p036_14.png]

**Figure 15.** Figure 15: FIG. 15: Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p037_15.png]

**Figure 16.** Figure 16: FIG. 16: (a) Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p038_16.png]

**Figure 17.** Figure 17: FIG. 17: Stability analysis for the teleparallel model ( [PITH_FULL_IMAGE:figures/full_fig_p040_17.png]

**Figure 18.** Figure 18: FIG. 18: Scale factor evolution showing (a) comparative bou [PITH_FULL_IMAGE:figures/full_fig_p041_18.png]

**Figure 19.** Figure 19: FIG. 19: Hubble parameter evolution showing (a) comparativ [PITH_FULL_IMAGE:figures/full_fig_p042_19.png]

**Figure 20.** Figure 20: FIG. 20: Effective EoS showing (a) comparative evolution for d [PITH_FULL_IMAGE:figures/full_fig_p043_20.png]

**Figure 21.** Figure 21: FIG. 21: Stability analysis for the non-minimal coupling mo [PITH_FULL_IMAGE:figures/full_fig_p044_21.png]

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Reference graph

Works this paper leans on

104 extracted references · 3 linked inside Pith

[53]

P. D. Mannheim. Are galactic rotation curves really ﬂat ? Astrophys. J. , 479(2):659–664, 1997. 69

1997
[1]

General Bounce Capability: The model supports bouncing solutions for a range of ω values; the ω = −1 case is shown as a representative example. 36
[2]

Bounce Dynamics: • Scale factor a(t) reaches minimum at t = 0 without singularity • Hubble parameter H(t) changes sign smoothly, indicating torsion-mediated bounce • Similar to linear model behavior but with modiﬁed late-time evolution
[3]

This suggests fundamen tally diﬀerent dynamics between the dark energy-driven bounce phase and subsequent r adiation-dominated expan- sion

exhibits inverted behavior with a(t) showing a maximum at t = 0, while H(t) still changes sign but reﬂects the decelerating expansion typical of radiation domination. This suggests fundamen tally diﬀerent dynamics between the dark energy-driven bounce phase and subsequent r adiation-dominated expan- sion. The eﬀective equation of state (Fig. 2 and 3) show...
[4]

3a exhibits no PDL crossing, with ωeﬀ remaining ﬁrmly in the quintessence region ( ωeﬀ > −1), consistent with standard cosmology

shown in Fig. 3a exhibits no PDL crossing, with ωeﬀ remaining ﬁrmly in the quintessence region ( ωeﬀ > −1), consistent with standard cosmology. The comparative view in Fig. 3b clearly demonstrates the se diﬀerent behaviors. These results demonstrate that: (1) successful bouncing cosm ology requires dark energy domination (ω ≤ − 1/3), (2) the phantom divid...
[5]

(B.48) So, by comparing with the equations of motion (11) and (12), we hav e ∇µT (Ξ) µν = 0

Covariant Derivative of T(Ξ): By using eq.(A.25), one obtains ∇µT (Ξ) µν = − ˙φ ( ¨φ + 3H ˙φ −V,φ ) + ˙ψ ( ¨ψ + 3H ˙ψ +V,ψ ) . (B.48) So, by comparing with the equations of motion (11) and (12), we hav e ∇µT (Ξ) µν = 0. (B.49) Appendix C: Substitution of FLR W Expressions into the Covariant Divergence
[6]

Each of these models oﬀers unique perspectives on the evolution of the universe, emphasizing t he intricate relationship between gravitational interactions and cosmic expansion

DYNAMICS OF BOUNCING COSMOLOGICAL MODELS This section investigates the dynamics and implications of various cos mological models, including the Linear [6], Exponential Function of Curvature [48], Pow er-Law [49], Modiﬁed Teleparallel Gravity [50], and Non-Minimal Coupling [51] models. Each of these models oﬀers unique perspectives on the evolution of the ...
[7]

Torsion-Dominated Phase: The quadratic torsion term T 2 creates eﬀective energy components: ρtors = −ξ3 2 (5ρ2 − 14ρp − 3p2), p tors = −ξ3 2 (9p2 − 6ρp +ρ2) (125)

Show all 104 references

[8]

(a) ωeﬀ for ω = −1 (black) (b) ωeﬀ for ω = −1/3 (red) FIG

Energy Condition Violation : The torsion term modiﬁes the Null Energy Condition: ρ +p +ξ3(2ρ2 − 5ρp + 3p2) ≥ 0 (126) allowing a temporary NEC violation during bounce. (a) ωeﬀ for ω = −1 (black) (b) ωeﬀ for ω = −1/3 (red) FIG. 15: Eﬀective equation of state ωeﬀ(t) showing (a) p...
[9]

Forω = −1: 37 • Remains in phantom regime ( ωeﬀ< −1) throughout −0.3<t< 0.3 • Reaches maximum ωeﬀ at t = 0 (bounce point) • Transitions to ωeﬀ> 0 for |t|> 0.3 (matter/radiation-like)
[10]

16: (a) Eﬀective equation of state ωeﬀ(t) for ω = 1/3 (blue line) and (b) composite view comparing all cases: ω = −1 (black), ω = −1/3 (red), and ω = 1/3 (blue)

Forω = −1/3: • Shows quintessence behavior ( ωeﬀ> −1) in −0.5<t< 0.5 • Crosses PDL twice at t ∼ ±25 (phantom-quintessence transitions) • Demonstrates torsion-mediated vacuum stability near bounce (a) ωeﬀ for ω = 1/3 (blue) (b) Comparative evolution FIG. 16: (a) Eﬀective equati...
[11]

Bounce Mechanism : a(t) ≈amin [ 1 + t2 2τ 2 ] , τ −2 = ξ3 2 (5ρ2 c − 14ρcpc − 3p2 c) − 12ξ2H 2(H 2 + ˙H) (127) where ρc,pc are critical values at bounce
[12]

Phase Transitions: The torsion coupling creates eﬀective potentials: Veﬀ(φ,ψ ) = 1 2mp(φ2 +ψ2) − 2 3φ2ψ +ξ3T 2(φ,ψ ) (128)
[13]

Figure 17 shows the results for the ω = −1 case

Singularity Avoidance : Rmax ∼ξ−1/2 3 (Finite curvature at bounce) (129) Stability Veriﬁcation: Using the teleparallel-inspired model, we numerically verify the stability criteria. Figure 17 shows the results for the ω = −1 case. The teleparallel model satisﬁes GS(t)> 0 for al...
[14]

Energy-Dependent Gravitational Constant : Geﬀ = GN 1 + 2ξ3RT ≈GN [1 − 2ξ3RT ] (135) becomes matter-density dependent, modifying gravitational inte ractions in high-energy regimes
[15]

Curvature-Induced Pressure: The coupling generates an eﬀective pressure through curvature-matter interaction: peﬀ =p + ξ3R2 κ2 ( ¨RT + 2 ˙R ˙T +R ¨T + 2H ˙RT + 2HR ˙T ) (136) This provides an additional repulsive force during cosmic contractio n
[16]

The EoS evolution reveals:

Bounce Mechanism : For radiation-dominated universe ( ω = 1/3), the coupling terms dominate when: ξ3R2T ∼ρrad ⇒amin ∼ ( ρrad,0 ξ3R2 0T0 ) 1/4 (137) determining the minimum scale factor before bounce occurrence. The EoS evolution reveals:
[17]

19: Hubble parameter evolution showing (a) comparativ e dynamics for dark energy cases and (b) radiation-dominated case with characteristic H(t) proﬁle and sign-change at bounce

Forω = 1/3: 41 (a) ω = −1 (black) vs ω = −1/3 (red) (b) ω = 1/3 (blue) FIG. 19: Hubble parameter evolution showing (a) comparativ e dynamics for dark energy cases and (b) radiation-dominated case with characteristic H(t) proﬁle and sign-change at bounce. The red curve ( ˙ψ(0) ...
[18]

This is physically signiﬁcant because the R2T coupling term introduces corrections that scale with both curvature and energy density

Forω = −1 and ω = −1/3: • Equilibrium points without PDL crossing • Eﬀective potential minimum at t = 0: Veﬀ ≈V0e−0.01φ + 1 2m2 pψ2 + 0.1φψ +ξ3R2T (138) The non-minimal coupling model exhibits a particularly interesting fea ture: its bounce dynamics are most eﬃcient and robust...
[19]

Radiation domination provides most stable bounce framewor k: The equation of state ω = 1/3 yields symmetric energy conditions and avoids anisotropic instabilitie s during the contraction phase, creating optimal conditions for a no n-singular bounce
[20]

Non-minimal coupling generates eﬀective UV cutoﬀ : LUV ∼ξ3R2T ≈ξ3(⊔ ⊓R)2 (139) 44 This introduces a natural regularization scale that prevents curv ature singularities, acting as a gravitational analogue of higher-derivative terms in eﬀective ﬁ eld theory
[21]

Late-time behavior controlled by curvature-matter interp lay: The R2T cou- pling becomes negligible at low curvatures, ensuring standard ΛCDM e volution while pro- viding a dynamical dark energy component through curvature-ma tter energy exchange
[22]

SUMMAR Y AND CONCLUSION This work has developed a comprehensive framework for nonsingula r bouncing cosmolo- gies within modiﬁed f (R,G,T ) gravity coupled to a quintom scalar ﬁeld model. Our inves- tigation successfully addresses the challenge of unifying early-time bounce dy...
[23]

Uniﬁed Cosmological Description: We have demonstrated that the synthesis of f (R,G,T ) geometry with quintom scalar ﬁelds provides a powerful mechanism for describing a complete cosmic history. The model naturally incorporates a non- singular bounce at early times, followed by...
[24]

Novel Phantom Divide Phenomenology: A distinctive prediction of our frame- work is the double crossing of the phantom divide line ( ωeﬀ = −1) during the bounce phase. This feature emerges from the dynamic interplay between the curv ature-matter coupling and the two-ﬁeld quinto...
[25]

The derived conditions for ghos t avoidance (GS > 0) and gradient stability ( FS > 0) ensure theoretical consistency throughout the cosmic evolut ion

Theoretical Consistency and Stability: Through rigorous Hamiltonian analysis in Section 5, we conﬁrmed the theory contains the correct number o f physical degrees of freedom without Ostrogradsky instabilities. The derived conditions for ghos t avoidance (GS > 0) and gradient s...
[26]

(B.1) So, by using eq

Covariant Derivative of T : The covariant derivative of the trace of energy-momentum tenso r T , as a scalar, is ex- pressed as: ∇µT = ∂µT. (B.1) So, by using eq. (24) for the time component ( µ = 0), we have ∇0T =∂0T = ˙ρ − 3 ˙p. (B.2) For the spatial components ( µ =i = 1, 2...
[27]

(B.5) Considering the Bianchi identity within the framework of general re lativity, we ﬁnd that ∇ν□fR − □ ∇νfR = −Rµν∇µfR

Covariant Derivative of T(R): Using Equation (A.22), we derive: κ2∇µT (R) µν = ∇µ ( fRRµν − 1 2gµνf +gµν□fR − ∇µ∇νfR ) = ( ∇µfR)Rµν +fR∇µRµν − 1 2fR∇µ(gµνR) − 1 2fG∇µ(gµνG) − 1 2fT ∇µ(gµνT ) + ∇ν□fR − □ ∇νfR. (B.5) Considering the Bianchi identity within the framework of gener...
[28]

(A.23), term by term

Covariant Derivative of T(G): To calculate ∇µT (G) µν , we need to apply covariant derivative on eq. (A.23), term by term. • The First Term: We start with the expression for T (G) 1µν , as the ﬁrst term of the Gauss-Bonnet energy- momentum tensor, eq.(A.23), T (G) 1µν = 2 κ2R ...
[29]

(B.44) Next, we take the covariant derivative: ∇µT (T ) µν = ∇µ ( −fT κ2 ( T (m) µν + Θµν ) ) (B.45) = − 1 κ2 (( T (m) µν + Θµν ) ∇µfT +fT ∇µ(T (m) µν + Θµν) )

Covariant Derivative of T(T): We begin with the expression for the energy-momentum tensor T (T ) µν : T (T ) µν = −fT κ2 ( T (m) µν + Θµν ) (B.42) where T (m) µν = (ρ +p)uµuν +pgµν (B.43) and Θµν =gαβ∂T (m) αβ ∂gµν . (B.44) Next, we take the covariant derivative: ∇µT (T ) µν =...
[30]

(C.7) The left-hand side of (C.1) for ν = 0 gives: κ2∇µT (R) µ0 = 12fRH ˙H + 3fR ¨H − ˙f 2 (C.8) where the total derivative ˙f is: ˙f =fR ˙R +fG ˙G +fT ˙T

FLR W Computation for ∇µT (R) µν For the FLR W metric, the conservation equation for the Ricci sect or becomes: ∇µT (R) µν = − 1 2κ2gµν (fG∇µG +fT ∇µT ), (C.1) Considering the temporal component ( ν = 0) and using FLR W expressions: R = 6( ˙H + 2H 2), (C.2) G = 24H 2( ˙H +H 2)...
[31]

FLR W Computation for ∇µT (G) µν Following the computation of ∇µT (R) µν , we now derive the FLR W expression for the Gauss- Bonnet sector. Starting from the general expression: ∇µT (G) µν = 1 κ2 ( RfG − 2(∇α∇αfG) + 4Rαβ∇α∇βfG ) ∇νR + 4 κ2 ( Rµρνλ∇µ∇λ∇ρfG −Rνρ∇ρ(∇α∇αfG) ) − 4 ...
[32]

Canonical Momentum Expressions The Hamiltonian analysis begins with the conjugate momentum calculat ions following standard results in modiﬁed gravity theories [46, 47]. The complete derivation proceeds as: πij = δL δ ˙gij = √−g 2κ2 [ fR δR δ ˙gij +fG δG δ ˙gij +fT δT δ ˙gij ]...
[33]

Matter Coupling Term: The fT term vanishes because: δT δ ˙gij = 0 as the matter stress-energy tensor T depends on gij but not on ˙gij [40]

Key Simpliﬁcations in Momentum Expressions I. Matter Coupling Term: The fT term vanishes because: δT δ ˙gij = 0 as the matter stress-energy tensor T depends on gij but not on ˙gij [40]. II. Gauss-Bonnet Decomposition: The full variation of G contains: δG δ ˙gij = 4(KikKj k −KK...
[34]

Constraint Structure and Degree of F reedom Counting Primary constraints: Φ0 =π0 − δL δ ˙N ≈ 0 (Lapse function) (D.6) Φi =πi − δL δ ˙Ni ≈ 0 (Shift vectors) (D.7) Secondary constraints: Four additional constraints emerge from time conservation of Φµ, corresponding to diﬀeomorph...
[35]

The key condition is: det  fRR fRG fGR fGG   = 0 (D.10) For linear Gauss-Bonnet ( fG = const.), this holds automatically since fGG = fRG = 0

Ostrogradsky Instability Avoidance The higher-derivative terms in f (R,G,T ) could introduce Ostrogradsky instabilities un- less degeneracy conditions are met. The key condition is: det  fRR fRG fGR fGG   = 0 (D.10) For linear Gauss-Bonnet ( fG = const.), this holds automa...
[36]

Physical Interpretation • The ﬁrst term in (D.3) represents usual gravitational momentum f rom Einstein-Hilbert gravity [55] • The second term encodes Gauss-Bonnet modiﬁcations, which vanis h when fG = const. (total derivative) [46] but become dynamical when fG depends on othe...
[37]

Perturbation V ariables in FLR W Metric The general perturbed FLR W metric contains four scalar perturbation functions following standard cosmological perturbation theory [56–58]: • Φ(t, x) - Newtonian potential: – Governs time-time component of metric perturbations – Represen...
[38]

Gauge Fixing to Newtonian Gauge The general perturbed metric [56]: ds2 = −(1 + 2Φ)dt2 + 2a∂iBdxidt +a2[(1 − 2Ψ)δij + 2∂i∂jE]dxidxj (E.1) can be simpliﬁed to Newtonian (longitudinal) gauge [56, 58]: ds2 = −(1 + 2Φ)dt2 +a2(1 − 2Ψ)δijdxidxj (E.2) through gauge transformations: St...
[39]

Appendix F: Derivation of Modiﬁed F riedmann Equations

Physical Interpretation of Newtonian Gauge The simpliﬁed metric describes [60, 61]: • Time dilation : Governed by Φ (Newtonian potential analog) 64 • Spatial curvature : Governed by Ψ (volume changes) • Removed degrees : – B eliminated (no frame-dragging) – E eliminated (no an...
[40]

Exponential F unction of Curvature Forf (R,G,T ) = R − 2Λe−(R0/R)n +ξ2G +ξ3T , we compute the partial derivatives: fR = 1 − 2nΛue−u, f G =ξ2, f T =ξ3 (F.1) ˙fR = −2nΛue−u [ n Ru − n + 1 R ] ˙R, (F.2) ¨fR = 2n2Λue−u [ (1 −u) ( ¨R R − (n + 1) ˙R2 R2 ) +nu ˙R2 R2 (2 −u) ] (F.3) ˙...
[41]

Power-Law Modiﬁed Gravity Forf (R,G,T ) = R +ξ1Rn +ξ2G +ξ3Tm, we compute the partial derivatives: fR = 1 + nξ1Rn−1, f G =ξ2, f T =mξ3Tm−1 (F.11) ˙fR =n(n − 1)ξ1Rn−2 ˙R, ¨fR =n(n − 1)ξ1Rn−3[(n − 2) ˙R2 +R ¨R] (F.12) ˙fG = 0, ¨fG = 0 (F.13) Substituting into the general eﬀective...
[42]

Modiﬁed T eleparallel Gravity Cosmology For the speciﬁc form f (R,G,T ) = R +ξ2G +ξ3T 2, we compute the partial derivatives: fR = 1, f G =ξ2, f T = 2ξ3T (F.18) ˙fR = 0, ¨fR = 0, ˙fG = 0, ¨fG = 0 (F.19) Substituting into the general eﬀective Friedmann equations (62) a nd (63), ...
[43]

Non-Minimal Curvature-Matter Coupling Forf (R,G,T ) = R +ξ2G +ξ3R2T , we compute the partial derivatives: fR = 1 + 2ξ3RT, f G =ξ2, f T =ξ3R2 (F.26) ˙fR = 2ξ3( ˙RT +R ˙T ), ¨fR = 2ξ3( ¨RT + 2 ˙R ˙T +R ¨T ) (F.27) ˙fG = 0, ¨fG = 0 (F.28) 68 Substituting into the general eﬀective...
[44]

A. G. Riess et al. Observational evidence from supernova e for an accelerating universe and a cosmological constant. Astron. J. , 116(3):1009–1038, 1998

1998
[45]

Perlmutter et al

S. Perlmutter et al. Measurements of Ω and Λ from 42 high-r edshift supernovae. Astrophys. J., 517(2):565–586, 1999

1999
[46]

H. V. Peiris et al. First-year Wilkinson Microwave Aniso tropy Probe (WMAP)* observations: Implications for inﬂation. Astrophys. J. Suppl. Ser. , 148(1):213–231, 2003

2003
[47]

D. N. Spergel et al. Three-year Wilkinson Microwave Anis otropy Probe (WMAP) observations: Implications for cosmology. Astrophys. J. Suppl. Ser. , 170(2):377–408, 2007

2007
[48]

Aghanim et al

N. Aghanim et al. Planck 2018 results-I. overview and the cosmological legacy of Planck. Astron. Astrophys., 641:A1, 2020

2018
[49]

Nojiri and S

S. Nojiri and S. D. Odintsov. Uniﬁed cosmic history in mod iﬁed gravity: From F (R) theory to lorentz non-invariant models. Phys. Rep. , 505(2-4):59–144, 2011

2011
[50]

Faraoni, E

V. Faraoni, E. Gunzig, and P. Nardone. Conformal transfo rmations in classical gravitational theories and in cosmology. arXiv preprint gr-qc/9811047 , 1998

1998 arXiv
[51]

H. Ess´ en. General relativity as a conformally invariant scalar gauge ﬁeld theory. Int. J. Theor. Phys., 29(2):183–187, 1990

1990
[52]

M. V. Takook and M. R. Tanhayi. Linear weyl gravity in de si tter universe. JHEP, 2010(12): 1–15, 2010

2010
[54]

R. P. Feynman et al. Feynman Lectures on Gravitation. Adv. Book Program. Addison-Wesley, Reading, Massachusetts, 1st edition, 1995. ISBN 0-201-627 34-5

1995
[55]

Hu and I

W. Hu and I. Sawicki. Models of f (R) cosmic acceleration that evade solar system tests. Phys. Rev. D , 76(6):064004, 2007

2007
[56]

Deﬀayet et al

C. Deﬀayet et al. Imperfect dark energy from kinetic grav ity braiding. JCAP, 2010(10):026, 2010

2010
[57]

Qiu et al

T. Qiu et al. Bouncing galileon cosmologies. JCAP, 2011(10):036, 2011

2011
[58]

Lin et al

C. Lin et al. A matter bounce by means of ghost condensati on. JCAP, 2011(04):019, 2011

2011
[59]

Nojiri and S

S. Nojiri and S. D. Odintsov. Modiﬁed Gauss–Bonnet theo ry as gravitational alternative for dark energy. Phys. Lett. B , 631(1-2):1–6, 2005

2005
[60]

Bamba et al

K. Bamba et al. The future of the universe in modiﬁed grav itational theories: Approaching a ﬁnite-time future singularity. JCAP, 2008(10):045, 2008

2008
[61]

Feng et al

B. Feng et al. Dark energy constraints from the cosmic ag e and supernova. Phys. Lett. B , 607 (1-2):35–41, 2005

2005
[62]

Ye and Y.-S

G. Ye and Y.-S. Piao. Bounce in general relativity and hi gher-order derivative operators. Phys. Rev. D , 99(8):084019, 2019

2019
[63]

Cai et al

Y.-F. Cai et al. Towards a nonsingular bouncing cosmolo gy. JCAP, 2012(08):020, 2012

2012
[64]

Langlois

D. Langlois. Degenerate higher-order scalar-tensor ( dhost) theories. arXiv:1707.03625, 2017

2017 arXiv
[65]

A. Vikman. Can dark energy evolve to the phantom? Phys. Rev. D , 71(2):023515, 2005

2005
[66]

Armendariz-Picon et al

C. Armendariz-Picon et al. Dynamical solution to the pr oblem of a small cosmological constant and late-time cosmic acceleration. Phys. Rev. Lett. , 85(21):4438, 2000

2000
[67]

Khoury et al

J. Khoury et al. From big crunch to big bang. Phys. Rev. D , 65(8):086007, 2002

2002
[68]

Elitzur et al

S. Elitzur et al. From big bang to big crunch and beyond. JHEP, 2002(06):017, 2002

2002
[69]

Sadeghi et al

J. Sadeghi et al. Bouncing universe with the non-minima lly coupled scalar ﬁeld and its reconstruction. Mod. Phys. Lett. A , 24(29):2363–2376, 2009

2009
[70]

Bamba et al

K. Bamba et al. Bounce cosmology from F (R) gravity and F (R) bigravity. J. Cosmol. Astropart. Phys., 2014(01):008, 2014

2014
[71]

D. A. Eliezer and R. P. Woodard. Instability of higher-d iﬀerence initial-value theories. Phys. Rev. D , 40(2):465, 1989

1989
[72]

R. P. Woodard. Avoiding dark energy with 1 /R modiﬁcations of gravity. In E. J. Lerner and J. B. Almeida, editors, The Invisible Universe: Dark Matter and Dark Energy , volume 720 of 70 Lect. Notes Phys. , pages 403–433. Springer, 2007

2007
[73]

P Woodard

R. P Woodard. The theorem of ostrogradsky. arXiv preprint arXiv:1506.02210 , 2015

2015 arXiv
[74]

T. A. Smecker and R. F. G. Wyse. Type Ia supernovae: Const raints on baryonic dark matter. In AIP Conf. Proc. , volume 222, pages 467–470. AIP, 1991

1991
[75]

Ruiz-Lapuente et al

P. Ruiz-Lapuente et al. Type Ia supernova scenarios and the hubble sequence. Astrophys. J. , 447(2):L69, 1995

1995
[76]

Nomoto et al

K. Nomoto et al. Type Ia supernovae: Their origin and pos sible applications in cosmology. Science, 276(5317):1378–1382, 1997

1997
[77]

D. J. Eisenstein et al. Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies. Astrophys. J. , 633(2):560, 2005

2005
[78]

Ratra and P

B. Ratra and P. J. E. Peebles. Cosmological consequence s of a rolling homogeneous scalar ﬁeld. Phys. Rev. D , 37(12):3406, 1988

1988
[79]

R. R. Caldwell. A phantom menace? cosmological consequ ences of a dark energy component with super-negative equation of state. Phys. Lett. B , 545(1-2):23–29, 2002

2002
[80]

A. Sen. Tachyon matter. JHEP, 2002(07):065, 2002

2002
[81]

Gasperini et al

M. Gasperini et al. Quintessence as a runaway dilaton. Phys. Rev. D , 65(2):023508, 2001

2001
[82]

Wei et al

H. Wei et al. Hessence: A new view of quintom dark energy. Class. Quantum Gravity , 22(16): 3189, 2005

2005
[83]

Harko et al

T. Harko et al. f (R,T ) gravity. Phys. Rev. D , 84:024020, 2011

2011
[84]

Charmousis and A

C. Charmousis and A. Padilla. The instability of vacuum in f (G) gravity. JHEP, 2008:028, 2008

2008
[85]

Nojiri and S

S. Nojiri and S. D. Odintsov. Modiﬁed gauss-bonnet theo ry as gravitational alternative for dark energy. Phys. Lett. B , 631:1–6, 2005

2005
[86]

Lilley and P

M. Lilley and P. Peter. Bouncing alternatives to inﬂati on. Comptes Rendus Physique , 16(10): 1038–1047, 2015

2015
[87]

Ghanaatian and F

M. Ghanaatian and F. Milani. Bouncing universe with the modiﬁed gravity coupled to the weyl tensor. Gen. Relativ. Gravit. , 46(9):1789, 2014

2014
[88]

Ghanaatian et al

M. Ghanaatian et al. Bouncing theory in the modiﬁed weyl gravity and its stability conditions. Ann. Phys. , 397:458–473, 2018

2018
[89]

De Felice and S

A. De Felice and S. Tsujikawa. f (R) theories. Living Rev. Relativ. , 13(1):3, 2010

2010
[90]

T. P. Sotiriou and V. Faraoni. f (R) theories of gravity. Rev. Mod. Phys. , 82:451–497, 2010. 71

2010
[91]

Elizalde et al

E. Elizalde et al. Oscillations of the f (r) dark energy i n the accelerating universe. Eur. Phys. J. C , 72(2):1843, 2012

2012
[92]

Bamba et al

K. Bamba et al. Dark energy cosmology: the equivalent de scription via diﬀerent theoretical models and cosmography tests. Astrophys. Space Sci. , 342(1):155–228, 2012

2012
[93]

Cai et al

Y.-F. Cai et al. f (T ) teleparallel gravity and cosmology. Rep. Prog. Phys. , 79(10):106901, 2016

2016
[94]

Bertolami et al

O. Bertolami et al. Extra force in f (R) modiﬁed theories of gravity. Phys. Rev. D , 75(10): 104016, 2007

2007
[95]

Bamba et al

K. Bamba et al. Crossing of the phantom divide in modiﬁed gravity. Phys. Rev. D , 79(8): 083014, 2009

2009
[96]

Cai and other

Y.-F. Cai and other. Bouncing universe with quintom mat ter. JHEP, 2007(10):071, 2007

2007
[97]

Arnowitt, S

R. Arnowitt, S. Deser, and C. W. Misner. The dynamics of g eneral relativity, gravitation: An introduction to current research. Chap, 7:227–265, 1962

1962
[98]

C. W. Misner, K. S. Thorne, and J. A. Wheeler. Gravitation. W. H. Freeman, 1973

1973
[99]

V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger. Theory of cosmological perturba- tions, volume 215. 1992

1992
[100]

Kodama and M

H. Kodama and M. Sasaki. Cosmological perturbation the ory. Prog. Theor. Phys. Suppl. , 78: 1–166, 1984

1984
[101]

Ma and E

C.-P. Ma and E. Bertschinger. Cosmological perturbati on theory in the synchronous and conformal newtonian gauges. Astrophys. J. , 455:7, 1995

1995
[102]

J. M. Bardeen. Gauge-invariant cosmological perturba tions. Phys. Rev. D , 22:1882–1905, 1980

1905
[103]

Weinberg

S. Weinberg. Cosmology. Oxford Univ. Press, 2008

2008
[104]

Dodelson

S. Dodelson. Modern Cosmology. Academic Press, 2003. 72

2003

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [53]

P. D. Mannheim. Are galactic rotation curves really ﬂat ? Astrophys. J. , 479(2):659–664, 1997. 69

1997

[2] [1]

General Bounce Capability: The model supports bouncing solutions for a range of ω values; the ω = −1 case is shown as a representative example. 36

[3] [2]

Bounce Dynamics: • Scale factor a(t) reaches minimum at t = 0 without singularity • Hubble parameter H(t) changes sign smoothly, indicating torsion-mediated bounce • Similar to linear model behavior but with modiﬁed late-time evolution

[4] [3]

This suggests fundamen tally diﬀerent dynamics between the dark energy-driven bounce phase and subsequent r adiation-dominated expan- sion

exhibits inverted behavior with a(t) showing a maximum at t = 0, while H(t) still changes sign but reﬂects the decelerating expansion typical of radiation domination. This suggests fundamen tally diﬀerent dynamics between the dark energy-driven bounce phase and subsequent r adiation-dominated expan- sion. The eﬀective equation of state (Fig. 2 and 3) show...

[5] [4]

3a exhibits no PDL crossing, with ωeﬀ remaining ﬁrmly in the quintessence region ( ωeﬀ > −1), consistent with standard cosmology

shown in Fig. 3a exhibits no PDL crossing, with ωeﬀ remaining ﬁrmly in the quintessence region ( ωeﬀ > −1), consistent with standard cosmology. The comparative view in Fig. 3b clearly demonstrates the se diﬀerent behaviors. These results demonstrate that: (1) successful bouncing cosm ology requires dark energy domination (ω ≤ − 1/3), (2) the phantom divid...

[6] [5]

(B.48) So, by comparing with the equations of motion (11) and (12), we hav e ∇µT (Ξ) µν = 0

Covariant Derivative of T(Ξ): By using eq.(A.25), one obtains ∇µT (Ξ) µν = − ˙φ ( ¨φ + 3H ˙φ −V,φ ) + ˙ψ ( ¨ψ + 3H ˙ψ +V,ψ ) . (B.48) So, by comparing with the equations of motion (11) and (12), we hav e ∇µT (Ξ) µν = 0. (B.49) Appendix C: Substitution of FLR W Expressions into the Covariant Divergence

[7] [6]

Each of these models oﬀers unique perspectives on the evolution of the universe, emphasizing t he intricate relationship between gravitational interactions and cosmic expansion

DYNAMICS OF BOUNCING COSMOLOGICAL MODELS This section investigates the dynamics and implications of various cos mological models, including the Linear [6], Exponential Function of Curvature [48], Pow er-Law [49], Modiﬁed Teleparallel Gravity [50], and Non-Minimal Coupling [51] models. Each of these models oﬀers unique perspectives on the evolution of the ...

[8] [7]

Torsion-Dominated Phase: The quadratic torsion term T 2 creates eﬀective energy components: ρtors = −ξ3 2 (5ρ2 − 14ρp − 3p2), p tors = −ξ3 2 (9p2 − 6ρp +ρ2) (125)