REVIEW 4 major objections 4 minor 175 references
Averaging Dynamics of Scalar Field-Matter Interacting Models in Anisotropic Universes: The Locally Rotationally Symmetric Bianchi I Spacetime
T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The paper claims that in a locally rotationally symmetric Bianchi I universe with a scalar field, cold dark matter, and a general interaction term, the Hubble parameter acts as a decaying perturbation parameter, making the time-averaged…
desk verdict The paper's central claim is undermined by a mismatch between the general interaction term and the δ=0 interactions, making the analysis of four models invalid. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central machinery is the Taylor expansion of the Hubble-normalized system (41)–(44) around H=0, followed by periodic time-averaging of the truncated vector field, with f=(ω²−1)^{-1} chosen so that the generalized harmonic potential oscillates with a single frequency. This produces a guiding system in the compact phase space (Σ,Ω), written in the time f′=$H^{{-1}}$d/dt, whose equilibrium points carry the late-time dynamics; the averaging theorem from Ref. [121] is the formal instrument that turns the decaying Hubble function into a controlled error bound. The nine interactions (26)–(34) are all instances of the general interaction (24), and the parameters Γ, Γ¯, and m determine whether a given equilibrium exists and whether it is a sink, source, or saddle.
What would settle it
Integrate the full system (41)–(44) for Interaction 5 at γ=2/3, Γ¯=−0.1 with initial data arbitrarily close to Ω=0, and compare with the guiding system (98)–(99). The averaged model predicts attraction to the singular line Ω=0; if the full solution escapes or converges elsewhere, the claimed asymptotic equivalence fails. A more direct test is to compute the L∞ norm of the truncated vector field on the phase space: any unbounded term such as tan(φ−tω) or 1/Ω would violate the hypotheses of Theorem 2.1 and leave the equivalence unproved.
Extended reading notes
Core claim
On its own terms, the paper's discovery is that for the LRS Bianchi I spacetime with a generalized harmonic potential V(φ)=φ²/2+f(1−cos(φ/f)) and any of the nine interactions derived from Q=Γ(H/H0)^{1−δ}ρ_m^α $ρ_φ^{{1−α−β}}$(ρ_m+ρ_φ)^β ˙φ^δ, the field equations can be expanded about H=0 and then time-averaged to produce a guiding system in the compact phase space (Σ,Ω), with Ω²+Σ²≤1 and Ω≥0. The Hubble parameter is the perturbation parameter: the averaging theorem used by the paper gives an error estimate between the full and averaged solutions that shrinks as H(t*)→0, and the numerical integrations presented for all nine interactions show the two systems approaching the same late-time attractors after the early oscillatory transient is suppressed. The stability analysis of the guiding systems yields the equilibrium points P1–P9, M1,2, and N1 with parameter-dependent stability conditions, and the deceleration parameter evaluated at these points exhibits both decelerated and accelerated regimes, with H(t)=C/t at every equilibrium.
Load-bearing premise
The argument assumes that the truncated, averaged vector fields used for each interaction remain a faithful late-time limit of the full system, meaning the general interaction formula really covers the nine cases studied and the vector fields satisfy the boundedness and Lipschitz conditions required by the averaging theorem.
Editorial extensions
If this is right
- For each of the nine interactions, the late-time attractors of the full Bianchi I system coincide with those of the averaged guiding system, so oscillatory transient behavior can be discarded when asking where the model ends up.
- New equilibrium points appear only for specific parameter ranges: P5 for Interaction 3, P6 for Interaction 4, P7 and P8 for Interaction 5, P9 for Interaction 6, M1,2 for Interaction 7 at γ=2, and N1 for Interaction 9, with stability controlled by Γ¯ or m.
- Accelerated late-time expansion is possible at several equilibria, with explicit parameter windows such as P5 accelerating for Γ¯<2−3γ, P6 accelerating for 1<Γ¯<3, and P9 and N1 accelerating in the intervals listed in Section 5.
- All equilibrium solutions have H(t)=C(γ,Γ)/t, and C can be tuned so that H at the present age matches either the 67.4 or 74 km s⁻¹ Mpc⁻¹ local measurements used in the paper.
- Some parameter choices make the guiding system singular at Ω=0 or Σ=±1, and the paper identifies attracting singular lines as physically unacceptable, restricting the viable parameter region.
Reading between the lines
- If the asymptotic-equivalence claim survives for interactions whose effective Q contains H0/H, the same averaging route should extend to the LRS Bianchi III and Kantowski-Sachs geometries, a direction the paper flags as future work.
- The singular lines in the guiding systems suggest a sharper test than the paper's chosen initial conditions: trajectories launched near Ω=0 or Σ=±1 may expose discrepancy windows before H decays, so a systematic scan over the full phase-space boundary would stress the equivalence claim.
- Because the guiding systems are two-dimensional and polynomial up to singular denominators, their full bifurcation diagram in the parameter space (γ, Γ¯, m) could in principle be mapped completely, yielding a sharper observational target than the individual equilibrium-point conditions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper applies the method of averaging to the LRS Bianchi I spacetime with a scalar field, cold dark matter, and a phenomenological interaction Q between the dark sectors. The interaction is defined in Eq. (24) as Q = Γ(H/H0)^{1−δ} ρ_m^α ρ_φ^{1−α−β}(ρ_m+ρ_φ)^β \dotϕ^δ, and nine particular choices (26)–(34) are studied. For each interaction, the authors substitute parameter values into the Hubble-normalized system (41)–(44), expand in Taylor series around H=0, average over the fast oscillations, and classify equilibrium points and their stability. They also evaluate the deceleration parameter at the equilibria and provide numerical comparisons between the original and averaged systems. The central claim is that H(t) acts as a small time-dependent perturbation parameter, so that the averaged system captures the asymptotic behavior of the full interacting system.
Significance. If correct, the paper would extend the authors' earlier averaging analysis to a broad family of interacting dark-sector models in an anisotropic spacetime, providing useful qualitative classifications of equilibria and conditions for accelerated expansion. The manuscript is rich in analytic detail: it gives explicit truncated systems, averaged equations, stability tables, phase portraits, and numerical illustrations for nine interactions, and it identifies new equilibrium points (P5–P9, M1,2, N1). These are genuine strengths. However, the central derivation is undermined by load-bearing inconsistencies: the general interaction (24) does not reproduce four of the nine claimed interactions, Section 4.1 analyzes the wrong parameter set for Interaction 1, and the numerical comparisons in Section 6 do not actually integrate the full system. The significance of the results is therefore contingent on substantial corrections.
major comments (4)
- [Section 3, Eqs. (24) and (27)–(30)] For δ=0, Eq. (24) gives Q = Γ(H/H0) ρ_m^α ρ_φ^{1−α−β}(ρ_m+ρ_φ)^β, not Q = Γ(H0/H) ρ_m^α ρ_φ^{1−α−β}(ρ_m+ρ_φ)^β as written in Interactions 2–5. Since Sections 4.2–4.5 substitute α, β, δ into the system (41)–(44) derived from Eq. (24), the “full” systems (57)–(60), (68)–(71), (79)–(82), and (90)–(93) describe Q ∝ H/H0, not Q ∝ H0/H. This is physically consequential: for Q ∝ H0/H with ρ ∼ H^2 and \dotϕ ∼ H, one has Q/\dotϕ ∼ constant as H → 0, whereas the analyzed systems contain only O(H) and O(H^2) interaction terms. Thus the equilibrium points P5–P8, their stability conditions, and the Section 6 numerical results for Interactions 2–5 are not for the models claimed in (27)–(30).
- [Section 4.1, before Eq. (46)] The text states “Setting α = 1, β = −1, δ = 0 in system (41)–(44)” for Interaction 1, but Eq. (26) defines Interaction 1 as (α, β, δ) = (1, 0, 1). The subsequent truncated system (46)–(49), averaged system (50)–(53), and Table I therefore do not describe the declared interaction Q = Γρ_m\dotϕ; they describe a different δ=0 interaction. Consequently, the claimed equivalence with the non-interacting results of Ref. [116] is not established for the stated Interaction 1.
- [Section 2.1, Theorem 2.1, and Section 4] The averaging theorem quoted as Theorem 2.1 requires ∥f^1∥_{L∞} < ∞ and Lipschitz continuity of f^1 in x. The truncated systems used in Section 4 contain unbounded terms and singular denominators, for example tan(φ−tω) in Eq. (60), the denominator \barΣ^2−1 in Eq. (62), and the denominators \barΩ in Eqs. (73), (95), and (99). The manuscript does not verify the boundedness or Lipschitz hypotheses of Theorem 2.1, nor does it restrict the phase space away from these singularities before applying the theorem. The claimed error estimate and the asymptotic equivalence between the full and averaged systems are therefore not rigorously established for these systems.
- [Section 6.2, Figures 30–47] The numerical comparisons labeled “original system” are actually integrations of the Taylor-truncated systems: for example, the caption of Figure 30 refers to system (46)–(49) while the text cites the full system (41)–(44), and Figures 32–47 likewise cite the truncated systems (58)–(60), (68)–(71), (79)–(82), and (90)–(93). These numerics therefore compare the averaged system only with the truncated H-expansion, not with the full system (41)–(44). As a result, the observed late-time agreement does not test or validate the paper's central claim that the averaged system captures the asymptotic behavior of the full interacting system.
minor comments (4)
- [Throughout] The manuscript contains numerous typographical errors, including “Aditionally”, “de results”, “for the prurpose”, “stars at q = 2”, and a duplicated phrase “It always exists It always exists”; a thorough copyedit is needed.
- [Section 4.6, Figures 13–14] The captions of Figures 13 and 14 state m = 0.5 and m = −0.5, while the text says the figures are plotted with m = 0.1 and m = −0.1; these should be reconciled.
- [Section 5.1, Eqs. (150)–(171)] The expressions H(t) = C/t are obtained by solving \dot H/H = const and setting the integration constant to zero; this choice is not derived from the model's initial conditions, and the tuning of γ and \barΓ in Figures 22–23 to match observed H0 values should be described as parameter fitting rather than a prediction.
- [Section 2.1, after Eq. (21)] The claim that f^0(x,t;ω) can always be made to vanish by tuning the frequency ω is stated without demonstration; a brief justification or reference would help the reader assess the reduction to the standard form (18).
Circularity Check
No significant circularity: the asymptotic equivalence is supported by an external averaging theorem and the paper's own numerical comparisons, not by fitted parameters or self-referential definitions.
full rationale
The paper's derivation chain runs from the interaction ansatz (24) through the field equations (39), the Hubble-normalized system (41)-(44), a Taylor expansion about H=0, periodic averaging, and the guiding systems such as (54)-(55). The equilibrium points and stability conditions are computed algebraically from the averaged vector fields; no parameter is fitted to those equilibria or to the claimed asymptotic agreement. The asymptotic-equivalence claim rests on Theorem 2.1 of Ref. [121], which is an external result, and it is checked numerically in Section 6 for every interaction. The self-citations [115-119] are methodological (how to set up the standard form and the truncation) and are corroborated by the paper's own numerical integrations, so they are not load-bearing in a circular way. The main defensible concern is internal consistency rather than circularity: for delta=0, Eq. (24) yields the factor H/H0, but Interactions 2-5 in Eqs. (27)-(30) are written with H0/H, meaning the full systems analyzed in Sections 4.2-4.5 are not the stated interactions. That is a correctness/modeling error, but it does not make any result equivalent to its input by construction, so it does not raise the circularity score.
Assumptions & free parameters
free parameters (5)
- alpha, beta, delta (interaction exponents) =
Nine chosen sets listed in Eqs. (26)-(34)
- Gamma (coupling constant) =
Not fitted; rescaled to Gamma-bar or m in some interactions
- gamma (barotropic index) =
0 <= gamma <= 2
- omega (oscillator frequency) =
omega > 1; e.g., omega = sqrt(2) in Fig. 21
- H0 (reference Hubble scale) =
Not fitted; set to 1 in some expressions
assumptions (6)
- domain assumption Averaging theorem (Theorem 2.1 and Proposition 1 from Ref. [121])
- domain assumption Field equations (39) for LRS Bianchi I with interaction
- domain assumption Generalized harmonic potential V(phi) = phi^2/2 + f(1 - cos(phi/f)) with f = (omega^2 - 1)^(-1)
- ad hoc to paper Interaction term Q of the form (24)
- domain assumption H is strictly decreasing to zero and the system can be Taylor-expanded around H = 0
- domain assumption The zeroth-order term f^0 in Eq. (21) can be tuned to zero by choosing omega
Cite this review
Pith. "Pith review of Averaging Dynamics of Scalar Field-Matter Interacting Models in Anisotropic Universes: The Locally Rotationally Symmetric Bianchi I Spacetime." pith.science (2026). https://pith.science/paper/U32VRGVO
@misc{pith2026250801876,
author = {Pith},
title = {Pith review of: Averaging Dynamics of Scalar Field-Matter Interacting Models in Anisotropic Universes: The Locally Rotationally Symmetric Bianchi I Spacetime},
year = {2026},
howpublished = {\url{https://pith.science/paper/U32VRGVO}},
note = {Machine review of arXiv:2508.01876}
}
read the original abstract
We consider an anisotropic cosmological model based on the locally rotational Bianchi I spacetime, incorporating a scalar field and a non-zero cosmological interaction term. The framework of averaging theory is employed to study the associated non-linear differential equations. Through a qualitative analysis of the gravitational field equations, we obtain valuable insights into the structure of the solution space for the anisotropic scalar field model with a generalized harmonic potential. The interaction between the scalar field and matter is described by a general expression that depends on the Hubble parameter, the time derivative of the scalar field, and the energy densities of cold dark matter and dark energy. This formulation involves real parameters that modulate the interaction, as well as a coupling constant with the dimensions of the Hubble parameter. We show that the Hubble parameter serves as a time-dependent perturbation parameter, controlling the discrepancy between the full system and its time-averaged counterpart. As this parameter decreases, both systems converge to the same asymptotic behaviour. This enables the suppression of oscillatory effects, significantly simplifying the dynamical analysis. Finally, we identify conditions on the interaction parameters that ensure the regularity of the system's evolution by preventing the emergence of singularities.
Reference graph
Works this paper leans on
- [116]
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[1]
It always exists and it verifies Ωm = 1
P1 = (0 , 0) with eigenvalues { 3(γ−2) 2 , 3(γ−1) 2 }. It always exists and it verifies Ωm = 1 . It describes a matter-dominated flat FLRW solution. The point is i) sink for 0 ≤ γ <1, ii) saddle for 1 < γ≤ 2, iii) non-hyperbolic for γ = 1, 2. 12
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[3]
Moreover, the evo- lution equation for H, given by the Raychaudhuri equation, decouples
INTERACTING ANISOTROPIC SCALAR FIELD COSMOLOGY For a scalar field theory with exponential potential Hubble-normalized quantities are used. Moreover, the evo- lution equation for H, given by the Raychaudhuri equation, decouples. Then, one can work in reduced phase space. The equilibrium points typically give the asymptotic of the remaining reduced system, ...
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[4]
This means that for each of the interactions (26)-(34), we obtain a different dynamical system when the corresponding values of α, β, δare substituted into (41)-(44)
A VERAGING DYNAMICS OF INTERACTIONS For the study of the anisotropic cosmological field equations we follow the approach established before in [115–118]. This means that for each of the interactions (26)-(34), we obtain a different dynamical system when the corresponding values of α, β, δare substituted into (41)-(44). Each of these systems is studied by ...
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[5]
They always exist and both represent an anisotropic Bianchi I vacuum solution
P3,4 = ( ±1, 0) with eigenvalues { 3 2 , −3(γ − 2)}. They always exist and both represent an anisotropic Bianchi I vacuum solution. They are i) sources for 0 ≤ γ <2, ii) non-hyperbolic for γ = 2. For this first interaction, the results are very similar to those obtained for the non-interacting model investigated in [116] because in the averaging process, ...
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[6]
It always exists and we verify that Ωm = 1
P1 = (0, 0), with eigenvalues n 3(γ−2) 2 , 1 2 ¯Γ + 3γ − 3 o . It always exists and we verify that Ωm = 1. It describes a matter-dominated flat FLRW solution. The point is i) sink for 0 ≤ γ <2, ¯Γ < −3(γ − 1), ii) saddle for 0 ≤ γ <2, ¯Γ > −3(γ − 1), iii) non-hyperbolic for ¯Γ = −3(γ − 1) or γ = 2
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[7]
It always exists It always exists and represents a scalar field dominated flat FLRW solution
P2 = (0, 1), with eigenvalues − 3 2 , −¯Γ − 3γ + 3 . It always exists It always exists and represents a scalar field dominated flat FLRW solution. The point is i) sink for 0 ≤ γ ≤ 2, ¯Γ > −3(γ − 1), ii) saddle for 0 ≤ γ ≤ 2, ¯Γ < −3(γ − 1), iii) non-hyperbolic for ¯Γ = −3(γ − 1)
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[8]
These points are anisotropic vacuum Bianchi I solutions
P3,4 = ( ±1, 0) with eigenvalues {0, 0}. These points are anisotropic vacuum Bianchi I solutions. These points make the numerators of the system vanish, but they also make the denominator of the equation for Ω vanish. To study their stability, the limit of the eigenvalues is considered when Σ → ±1. For that prurpose, we evaluate the Jacobian matrix in Ω =...
Show all 175 references
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[9]
P2 = (0 , 1), with eigenvalues −3, −2 ¯Γ + 3γ − 3 . It always exists, it represents a scalar field domi- nated solution and is i) sink for 0 ≤ γ ≤ 2, ¯Γ > −3(γ − 1), ii) saddle for 0 ≤ γ ≤ 2, ¯Γ < −3(γ − 1), iii) non-hyperbolic for ¯Γ = −3(γ − 1)
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[11]
P5 = (0, q ¯Γ 3(1−γ) ) with eigenvalues ¯Γ + 3γ − 6, 2 ¯Γ + 3γ − 3 . This point exist for 0 ≤ γ <1, 0 ≤ ¯Γ ≤ 3 − 3γ or 1 < γ≤ 2, 3 − 3γ ≤ ¯Γ ≤ 0 and is i) sink for 0 ≤ γ <1, 0 ≤ ¯Γ < 3 − 3γ, ii) saddle for 1 < γ <2, 3 − 3γ <¯Γ ≤ 0 or γ = 2, −3 < ¯Γ < 0 iii) non-hyperbolic for ...
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[12]
It always exists and is i) sink for 0 ≤ γ <2, ¯Γ < −3(γ − 1), ii) saddle for 0 ≤ γ <2, ¯Γ > −3(γ − 1), iii) non-hyperbolic for ¯Γ = −3(γ − 1) or γ = 2
P1 = (0, 0) with eigenvalues n 3(γ−2) 2 , 1 2 ¯Γ + 3γ − 3 o . It always exists and is i) sink for 0 ≤ γ <2, ¯Γ < −3(γ − 1), ii) saddle for 0 ≤ γ <2, ¯Γ > −3(γ − 1), iii) non-hyperbolic for ¯Γ = −3(γ − 1) or γ = 2
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[13]
They always exist and are i) sources for 0 ≤ γ <2, ¯Γ > −3, ii) saddles for 0 ≤ γ <2, ¯Γ < −3, iii) non-hyperbolic for ¯Γ = −3 or γ = 2
P3,4 = (±1, 0) with eigenvalues 6 − 3γ, 1 2 ¯Γ + 3 . They always exist and are i) sources for 0 ≤ γ <2, ¯Γ > −3, ii) saddles for 0 ≤ γ <2, ¯Γ < −3, iii) non-hyperbolic for ¯Γ = −3 or γ = 2
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[14]
This point verifies Ωm = ¯Γ 3−3γ
P6 = 0, q 1 + ¯Γ 3(γ−1) , with eigenvalues 1 2 −¯Γ − 3 , −¯Γ − 3γ + 3 . This point verifies Ωm = ¯Γ 3−3γ . It exist for 0 ≤ γ <1, 0 ≤ ¯Γ < 3 − 3γ or 1 < γ≤ 2, 3 − 3γ <¯Γ ≤ 0. The point is i) sink for 1 < γ≤ 2, −3(γ − 1) < ¯Γ ≤ 0, 22 ii) saddle for 0 ≤ γ <1, 0 ≤ ¯Γ < −3(γ − 1)....
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[15]
They belong to the singular line ¯Ω = 0 and are non-hyperbolic
P3,4 = (±1, 0), with eigenvalues {0, 0}. They belong to the singular line ¯Ω = 0 and are non-hyperbolic
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[16]
It has Ωm = 1 6 3 − √ 9γ+12¯Γ−9√γ−1 and it exists exists for 0 ≤ γ < 1, 0 ≤ ¯Γ ≤ 1 4 (3 − 3γ) or 1 < γ ≤ 2, 1 4 (3 − 3γ) ≤ ¯Γ ≤ 0
P7 = 0, rq 3+ 12¯Γ γ−1 +9 √ 6 , with eigenvalues 1 2 3γ − √ 3 p γ − 1 q 3γ + 4¯Γ − 3 − 9 , −2 √ 3 p γ − 1 q 3γ + 4¯Γ − 3 . It has Ωm = 1 6 3 − √ 9γ+12¯Γ−9√γ−1 and it exists exists for 0 ≤ γ < 1, 0 ≤ ¯Γ ≤ 1 4 (3 − 3γ) or 1 < γ ≤ 2, 1 4 (3 − 3γ) ≤ ¯Γ ≤ 0. The point is i) ...
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[17]
It has Ωm = 1 6 √ 9γ+12¯Γ−9√γ−1 + 3 , it exists for 0 ≤ γ <1, 0 ≤ ¯Γ ≤ 1 4 (3 − 3γ) or 1 < γ≤ 2, 1 4 (3 − 3γ) ≤ ¯Γ ≤ 0
P8 = 0, r 3− q 12¯Γ γ−1 +9 √ 6 with eigenvalues 1 2 3γ + √ 3 p γ − 1 q 3γ + 4¯Γ − 3 − 9 , 2 √ 3 p γ − 1 q 3γ + 4¯Γ − 3 . It has Ωm = 1 6 √ 9γ+12¯Γ−9√γ−1 + 3 , it exists for 0 ≤ γ <1, 0 ≤ ¯Γ ≤ 1 4 (3 − 3γ) or 1 < γ≤ 2, 1 4 (3 − 3γ) ≤ ¯Γ ≤ 0. The point is a i) sink for 0 ...
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[20]
It exist for i) m <0, 1 ≤ γ ≤ 2 or ii) m = 0, 0 ≤ γ <1 or iii) m = 0, 1 < γ≤ 2 or iv) m >0, 0 ≤ γ ≤ 1
P9 = 0, q 1−γ 1+m−γ , with eigenvalues n 3 − 3γ, 3(γ+(γ−2)m−1) 2(−γ+m+1) o . It exist for i) m <0, 1 ≤ γ ≤ 2 or ii) m = 0, 0 ≤ γ <1 or iii) m = 0, 1 < γ≤ 2 or iv) m >0, 0 ≤ γ ≤ 1. Additionally, we verify that Ωm = m 1+m−γ . The point is a i) sink for 1 < γ≤ 2, m >0 or 1 < γ≤ 2...
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[21]
It always exists and it verifies Ωm = 1
P1 = (0, 0) with eigenvalues n 3(γ−2) 2 , 3 2 (γ + m − 1) o . It always exists and it verifies Ωm = 1. It describes a matter-dominated flat FLRW solution. The point is a i) sink for m ≤ −1, 0 ≤ γ <2 or −1 < m <1, 0 ≤ γ <1 − m, ii) saddle for −1 < m≤ 1, 1 − m < γ <2 or m >1, 0 ...
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[22]
It always exists and represents a scalar field domi- nated solution
P2 = (0 , 1) with eigenvalues − 3 2 , −3(γ + m − 1) . It always exists and represents a scalar field domi- nated solution. The point always exists and is a i) sink for −1 < m≤ 1, 1 − m < γ≤ 2 or m >1, 0 ≤ γ ≤ 2, ii) saddle for m <−1, 0 ≤ γ ≤ 2 or −1 ≤ m <1, 0 ≤ γ <1 − m, iii) ...
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[23]
They always exist and both represent an anisotropic Bianchi I vacuum solution
P3,4 = ( ±1, 0) with eigenvalues { 3 2 , −3(γ − 2)}. They always exist and both represent an anisotropic Bianchi I vacuum solution. They are i) sources for 0 ≤ γ <2, ii) non-hyperbolic for γ = 2
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[24]
The points M1,2 = ± q γ+m−1 m , q 2−γ m only exist for γ = 2 and m ≤ −1
A special case arises when γ = 2. The points M1,2 = ± q γ+m−1 m , q 2−γ m only exist for γ = 2 and m ≤ −1. But in this case, the points are M1,2 = ± q m+1 m , 0 and they belong to the set of equilibrium points defined by the line ¯Ω = 0. The eigenvalues for this normally hyper...
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[25]
It always exists and it verifies Ωm = 1
P1 = (0 , 0) with eigenvalues { 3(γ−2) 2 , 3(γ−1) 2 }. It always exists and it verifies Ωm = 1 . It describes a matter-dominated flat FLRW solution. The point is i) sink for 0 ≤ γ <1, ii) saddle for 1 < γ≤ 2, iii) non-hyperbolic for γ = 1, 2
-
[26]
It always exists and represents a scalar field dominated flat FLRW solution
P2 = (0, 1) with eigenvalues {− 3 2 , −3(γ − 1)}. It always exists and represents a scalar field dominated flat FLRW solution. The point is i) saddle for 0 ≤ γ <1, ii) sink for 1 < γ≤ 2, iii) non-hyperbolic for γ = 1
-
[27]
They always exist and both represent an anisotropic Bianchi I vacuum solution
P3,4 = ( ±1, 0) with eigenvalues { 3 2 , −3(γ − 2)}. They always exist and both represent an anisotropic Bianchi I vacuum solution. They are i) sources for 0 ≤ γ <2, 34 ii) non-hyperbolic for γ = 2. We note that the equilibrium points for interaction 8 (33) are the same as tho...
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[28]
It always exists and it verifies Ωm = 1
P1 = (0 , 0), with eigenvalues { 3(γ−2) 2 , 3(γ−1) 2 }. It always exists and it verifies Ωm = 1 . It describes a matter-dominated flat FLRW solution. The point is i) sink for 0 ≤ γ <1, ii) saddle for 1 < γ≤ 2, iii) non-hyperbolic for γ = 1, 2
-
[29]
It always exists and represents a scalar field domi- nated solution
P2 = (0, 1), with eigenvalues − 3 2 , −3(γ + m − 1) . It always exists and represents a scalar field domi- nated solution. The point always exists and is a i) sink for −1 < m≤ 1, 1 − m < γ≤ 2 or m >1, 0 ≤ γ ≤ 2, ii) saddle for m <−1, 0 ≤ γ ≤ 2 or −1 ≤ m <1, 0 ≤ γ <1 − m, iii) ...
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[30]
They always exist, to study their stability we substitute ¯Ω = 0 in the Jacobian matrix and then take the limit as Σ → ±1
P3,4 = (±1, 0). They always exist, to study their stability we substitute ¯Ω = 0 in the Jacobian matrix and then take the limit as Σ → ±1. The eigenvalues are 3 2 (4 − 2γ), 3 2 . They are i) sources for 0 ≤ γ <2, ii) non-hyperbolic for γ = 2
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[31]
The point exists for m > 0, 0 ≤ γ ≤ 1 or m <0, 1 ≤ γ ≤ 2 and is a i) sink for 0 ≤ γ <1, −γ2+2γ−1 γ−2 < m <1 − γ or 1 < γ≤ 2, m <1 − γ, ii) saddle for i
N1 = 0, q 1−γ m , with eigenvalues 3((γ−1)2+(γ−2)m) 2m , − 3(γ−1)(γ+m−1) m . The point exists for m > 0, 0 ≤ γ ≤ 1 or m <0, 1 ≤ γ ≤ 2 and is a i) sink for 0 ≤ γ <1, −γ2+2γ−1 γ−2 < m <1 − γ or 1 < γ≤ 2, m <1 − γ, ii) saddle for i. 0 ≤ γ <1, 0 < m <−γ2+2γ−1 γ−2 or ii. 0 ≤ γ <1, ...
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[32]
The definition of the deceleration parameter q is as follows q := −1 − ˙H(t) H 2(t)
DECELERATION PARAMETER FOR THE EQUILIBRIUM POINTS In this section, we investigate the dynamical behaviour of the deceleration parameter. The definition of the deceleration parameter q is as follows q := −1 − ˙H(t) H 2(t) . (145) For the models under consideration, equation (14...
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[33]
It describes acceleration for 0 ≤ γ < 2 3
For P1 = (0, 0), the values are qseries = qaveraged = −1 + 3γ 2 . It describes acceleration for 0 ≤ γ < 2 3 . Additionally, we can obtain an expression for H by solving the following differential equation ˙H H = − 3γ 2 . Setting the integration constant to zero, the Hubble fun...
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[34]
(152) In Figure 21 we see that qseries oscillates between deceleration and acceleration
For P2 = (0, 1), the values are qseries = 1 2 + 3 2 cos(2(φ − tω)), (151) qaveraged = 1 2 . (152) In Figure 21 we see that qseries oscillates between deceleration and acceleration. On the other hand, qaveraged is a constant decelerated value. We verify that for qaveraged the H...
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[35]
The Hubble function is H(t) = 1 3t
For P3,4 = (±1, 0), the values are qseries = qaveraged = 2, they describe deceleration. The Hubble function is H(t) = 1 3t
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[36]
(154) We focus on qaveraged, it describes acceleration for ¯Γ < 2 − 3γ
For P5 = (0, q ¯Γ 3(1−γ) ), the values are qseries = −1 + ¯Γ 2 + 3γ 2 − ¯Γ cos(2(φ − tω)) 2(γ − 1) , (153) qaveraged = −1 + ¯Γ 2 + 3γ 2 . (154) We focus on qaveraged, it describes acceleration for ¯Γ < 2 − 3γ. The Hubble function is H(t) = 2 t ¯Γ + 3γ . (155) This function sat...
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(157) We focus on qaveraged, it describes acceleration for ¯Γ > 1
For P6 = 0, q 1 + ¯Γ 3(γ−1) , the values are qseries = 1 2 + ¯Γ(1 − γ + cos(2(φ − tω))) 2(γ − 1) + 3 2 cos(2(φ − tω)), (156) qaveraged = 1 2 − ¯Γ 2 . (157) We focus on qaveraged, it describes acceleration for ¯Γ > 1. The Hubble function is H(t) = 2 t 3 − ¯Γ . (158) This functi...
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(160) We focus on qaveraged
For P7 = 0, rq 3+ 12¯Γ γ−1 +9 √ 6 , the values are qseries = − 1 4 + 3γ 4 − 1 4 √ 3 p γ − 1 q 3γ + 4¯Γ − 3 (159) + 3√γ − 1 + p 9γ + 12¯Γ − 9 cos(2(φ − tω)) 4√γ − 1 , qaveraged = − 1 4 + 3γ 4 − 1 4 √ 3 p γ − 1 q 3γ + 4¯Γ − 3. (160) We focus on qaveraged. This quantity re...
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[39]
(163) We focus on qaveraged
For P8 = 0, r 3− q 12¯Γ γ−1 +9 √ 6 , the values are qseries = − 1 4 + 3γ 4 + 1 4 √ 3 p γ − 1 q 3γ + 4¯Γ − 3 (162) + 3√γ − 1 − p 9γ + 12¯Γ − 9 cos(2(φ − tω)) 4√γ − 1 , qaveraged = − 1 4 + 3γ 4 + 1 4 √ 3 p γ − 1 q 3γ + 4¯Γ − 3. (163) We focus on qaveraged. This quantity r...
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[40]
(166) We focus on qaveraged
For P9 = 0, √1−γ√m−γ+1 , the values are qseries = −γ + 3γm − 2m + 1 −2γ + 2m + 2 − 3(γ − 1) cos(2(φ − tω)) 2(−γ + m + 1) , (165) qaveraged = −γ + 3γm − 2m + 1 −2γ + 2m + 2 . (166) We focus on qaveraged. It describes acceleration for • 0 ≤ γ < 2 3 , m < γ− 1 or • 0 ≤ γ < 2 3 , ...
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[41]
They describe acceleration for 0 ≤ γ <2, 1 4 (3γ − 6) < m <0
For M1,2 = ± q m+1 m , 0 , the values are qseries = qaveraged = −3γ+4m+6 2m . They describe acceleration for 0 ≤ γ <2, 1 4 (3γ − 6) < m <0. The Hubble function is H(t) = 2m t(−3γ + 6m + 6), (168) which is positive for • m <γ−2 2 or • m >0. In this case, qaveraged < 0 and H(t) ...
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(170) We focus on qaveraged
For N1 = 0, q 1−γ m , the values are qseries = 3(γ − 1)2 + (3γ − 2)m 2m − 3(γ − 1) cos(2(φ − tω)) 2m , (169) qaveraged = 3(γ − 1)2 + (3γ − 2)m 2m . (170) We focus on qaveraged. It describes acceleration for • γ = 2 3 , m <0 or • 0 ≤ γ < 2 3 , m <0 or • 0 ≤ γ < 2 3 , m >−3γ2+6γ...
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[43]
StiffnessSwitching
NUMERICAL SIMULATIONS In this section, we present the numerical solutions of both thefull and averaged systems corresponding to each of the interaction models under consideration. The computations were performed using Wolfram Mathematica[135], under a student license. The nume...
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CONCLUSION In this research, the Locally Rotationally Symmetric Bianchi I spacetime with scalar field a nonzero matter component, with interaction was investigated. This analysis extends the previous studie [116] by introducing the interactiong function Q that characterizes th...
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