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REVIEW 4 major objections 4 minor 104 references

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.

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2026-08-04 13:23 UTC pith:MIVQQAJH

load-bearing objection 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. the 4 major comments →

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

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

Farzad Milani This is my paper
classification gr-qc PACS 04.50.Kd98.80.Jk95.36.+x98.80.Cq
keywords Bouncing cosmologyf(R,G,T) gravityQuintom modelPhantom divide crossingNonsingular universeDark energyCosmological perturbationsModified gravity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

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.

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

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.

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.

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 this falsifier. Get emailed when new claim-graph text bears on it.

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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.

Where Pith is reading between the lines

These are 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.

Editorial analysis

A structured set of objections, weighed in public.

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

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)
  1. [§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.
  2. [§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.
  3. [§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.
  4. [§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)
  1. [§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.
  2. [§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.
  3. [§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.
  4. [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

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.

specific steps
  1. 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.

  2. 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.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 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'.
axioms (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.

pith-pipeline@v1.3.0-alltime-deepseek · 39686 in / 15261 out tokens · 407485 ms · 2026-08-04T13:23:38.881756+00:00 · methodology

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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}
}
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read the original 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 Farzad Milani.

Figure 1
Figure 1. Figure 1: FIG. 1: Evolution of (a) scale factor [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a) Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Stability analysis for the linear coupling model ( [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Evolution of (a) scale factor [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: (a) Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Stability analysis for the exponential model ( [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Evolution of (a) scale factor [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: (a) Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p034_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Stability analysis for the power-law model ( [PITH_FULL_IMAGE:figures/full_fig_p035_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Evolution of (a) scale factor [PITH_FULL_IMAGE:figures/full_fig_p036_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p037_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: (a) Effective equation of state [PITH_FULL_IMAGE:figures/full_fig_p038_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Stability analysis for the teleparallel model ( [PITH_FULL_IMAGE:figures/full_fig_p040_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Scale factor evolution showing (a) comparative bou [PITH_FULL_IMAGE:figures/full_fig_p041_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Hubble parameter evolution showing (a) comparativ [PITH_FULL_IMAGE:figures/full_fig_p042_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Effective EoS showing (a) comparative evolution for d [PITH_FULL_IMAGE:figures/full_fig_p043_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: Stability analysis for the non-minimal coupling mo [PITH_FULL_IMAGE:figures/full_fig_p044_21.png] view at source ↗

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Reference graph

Works this paper leans on

104 extracted references · 3 linked inside Pith

  1. [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. [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. [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. [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. [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. [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. [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)

  8. [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) persistent phantom phase for ω = −1 and (b) phase transitions for ω = −1/3. The P...

  9. [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. [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 equation of state ωeff(t) for ω = 1/3 (blue line) and (b) composite view comparing all ...

  11. [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. [12]

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

  13. [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 all t, avoiding ghost pathologies. The sound speed remains real with c2 s(t) ≥ 0, ...

  14. [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. [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. [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. [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) = (100.236)1/2) shows strongest bounce dynamics. • Successful bounce with two PD...

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    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 in the radiation-dominated er a (ω = 1/3). This is physically significant becaus...

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    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

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    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

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    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

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    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 dynamics with late-time accelerated expansion in a single theoretical framework, w...

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    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 a smooth transition to the standard expansion his tory, culminating in dark ene...

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    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 quintom dynamics, representing a unique observa tional signature that differentiates ou...

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    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 stability ( FS > 0) ensure theoretical consistency throughout the cosmic evolut i...

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    (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, 3), we take into account the spatial indices, for a T just dependent on proper...

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    (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 general re lativity, we find that ∇ν□fR − □ ∇νfR = −Rµν∇µfR. Therefore, κ2∇µT (R) µν =...

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    (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 (RµνfG +gµν□fG − ∇µ∇νfG). (B.8) Applying the covariant derivative ∇µ using the p...

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    (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 = ∇µ ( −fT κ2 ( T (m) µν + Θµν ) ) (B.45) = − 1 κ2 (( T (m) µν + Θµν ) ∇µfT +fT ∇...

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    (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), (C.3) T = −ρ + 3p, (C.4) we compute the derivatives: ˙R = 6( ¨H + 4H ˙H), (C.5...

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    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 κ2fG ( Rρλ∇ρRνλ + 1 2Rρλαβ∇νRρλαβ ) + 4 κ2 (fGR∇µR +fGG∇µG +fGT ∇µT ) ( 1 2Rρλξ ...

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    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 ] (D.1) = √−g 2κ2 [ fR(Kij −Kgij) + 4fG(KikKj k −KKij + R 2Kij − ✘✘✘✘✘✘✘✘✘✘✘ ✘1 2...

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    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 −KKij) + terms proportional to R and K 2 The cancellations occur because: • The R-d...

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    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 diffeomorphism invariance. Degrees of freedom counting using Dirac’s formula: Total DOF = 1...

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    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 automatically since fGG = fRG = 0. The FLR W symmetry provides additional protection b...

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    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 other fields [42] • All other terms cancel or become boundary terms in closed univers...

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    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 – Represents gravitational potential in Newtonian limit – Affects particle energies ( E ≈ √...

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    Appendix F: Derivation of Modified F riedmann Equations

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