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REVIEW 3 major objections 6 minor 100 references

Cosmological dynamics of holographic dark energy with non-minimally coupled scalar field

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A model with holographic vacuum energy plus a non-minimally coupled scalar field has one viable history: stiff-fluid start, near-dust middle, stable de Sitter end, requiring small negative coupling $\xi$ and small holographic parameter $c$.

desk verdict Competent phase-space analysis of a new NMC + apparent-horizon-HDE combination, but the holographic part is an identity and the one viable sequence rests on hand-picked initial conditions. read the letter →

arxiv 2507.05273 v1 pith:WBZQP3J5 submitted 2025-07-03 gr-qc astro-ph.CO

Amornthep Tita (IF Naresuan Univ. , NAS Mahidol Univ.) , Burin Gumjudpai (NAS Mahidol Univ.) , Pornrad Srisawad (Dept. of Physics Naresuan Univ.) This is my paper
classification gr-qcastro-ph.CO MSC 83F0583D05 PACS 98.80.Cq
keywords holographicdarkenergynon-minimalcouplingscalarfieldcosmologyapparenthorizoncutoffdynamicalsystemsfixedpointanalysisattractorquadraticpotential
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 considers a flat FRW universe containing dust, a non-minimally coupled scalar field with quadratic potential, and holographic vacuum energy whose cutoff is the apparent horizon. It claims that the combined dark energy from the scalar and the holographic component has a single viable cosmological history: a stiff-fluid-dominated start, a nearly dust-dominated transient era, and a stable accelerating final state with effective equation of state $w_{\rm eff}=-1$. The dynamical-system analysis of nine fixed points shows that physical viability requires a negative non-minimal coupling $\xi$ and a holographic parameter $c$ in $(0,1)$, with both small in magnitude. Because $\xi=0$ makes the autonomous system singular, canonical scalar-field holographic dark energy is only approached, never exactly recovered. If the claim is right, this class of two-ingredient dark energy models can mimic a cosmological constant at late times without containing one.

What carries the argument

The machinery is a four-variable autonomous dynamical system in $y$, $s$, $A$, and $\Omega_k$ (potential term, NMC coupling combination, NMC auxiliary variable, and curvature density parameter), built from the NMC Friedmann and Klein-Gordon equations with effective gravitational constant $G_{\rm eff}(\phi)=G/(1-8\pi G\xi\phi^2)$. The central object is the constraint $\Omega_\Lambda=c^2(1-\Omega_k)$, which follows from the apparent-horizon cutoff $L=(H^2+k/a^2)^{-1/2}$ and fixes the holographic density parameter in terms of curvature; together with $x=s^2/(24\xi A)$ and the Friedmann constraint it reduces the phase space to four dimensions. Stability is decided by the eigenvalues of the $4\times 4$ Jacobian at each fixed point, with point 6 ($w_{\rm eff}=-1$) the only stable late-time attractor under $\xi<0$, and point 7 (stiff fluid) and point 5 (near-dust saddle) forming the early and intermediate stages of the claimed path $7 \to 5 \to 6$.

What would settle it

Redo the autonomous system with the future event horizon or Ricci cutoff and check whether a stable $w_{\rm eff}=-1$ attractor and the sequence $7 \to 5 \to 6$ survive; if the attractor disappears, the result is an artifact of the apparent-horizon choice. Separately, sample initial conditions uniformly in the allowed region $\xi<0$, $0<c<1$ and measure the fraction that reach fixed point 6; if only the hand-chosen $s_0=-10^{-7}$, $A_0=-0.7$ traces the sequence, the claimed cosmic history is not generic.

Watch

Extended reading notes

Core claim

The central claim is that in a flat FRW universe with dust, a non-minimally coupled scalar field with $V(\phi)=V_0\phi^2$, and holographic vacuum energy cut off at the apparent horizon, the physically admissible evolution is the sequence of fixed points $7 \to 5 \to 6$: a stiff-fluid kinetic-dominated start, a transient almost-dust epoch, and a stable dark-energy-dominated state with $w_{\rm eff}=-1$. Stability analysis of the nine fixed points of the four-variable autonomous system shows that physical validity requires $\xi<0$ and $0<c<1$; the limit $\xi\to 0^-$ and $c\to 0^+$ approaches, but does not reach, canonical scalar-field holographic dark energy. Numerical integration confirms that for all allowed parameters $w_{\rm eff}\to -1$ at late times, while larger $|\xi|$ erases the dust era and larger $c$ raises $w_{\rm eff}$ without changing its shape.

Load-bearing premise

The load-bearing premise is that the holographic infrared cutoff is the apparent horizon; if a future event horizon, Ricci, or other cutoff were used instead, the relation $\Omega_\Lambda=c^2(1-\Omega_k)$ and hence the attractor structure would change.

Editorial extensions

If this is right

  • If the central claim is correct, the late-time state of this model is always de Sitter-like, with $w_{\rm eff}\to -1$, regardless of the exact allowed values of $\xi$ and $c$.
  • A dust-dominated era exists only in the corner $\xi\to 0^-$, $c\to 0^+$; the model therefore predicts that stronger non-minimal coupling suppresses the matter era.
  • Because $\xi=0$ is inaccessible, canonical scalar-field holographic dark energy is a limit, not a member, of this model family; any observational test that requires the exact GR scalar limit will see a small residual NMC effect.
  • Larger holographic parameter $c$ raises the effective equation of state; the numerical example with $\xi=-10^{-5}$ and $c=0.8$ has $w_{\rm eff}>-1/3$ today, so such parameter values would rule out accelerated expansion in the model.

Reading between the lines

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

  • The authors leave implicit that, in a near-flat universe, the constraint $\Omega_\Lambda=c^2(1-\Omega_k)$ pins the holographic density to an almost constant value, so the holographic ingredient behaves like a tuned cosmological constant and the dynamical evolution is carried mainly by the scalar field.
  • A natural test the paper does not perform is to repeat the autonomous-system analysis with a future-event-horizon or Ricci cutoff; if the stable $w_{\rm eff}=-1$ attractor disappears, the claimed cosmic history is specific to the apparent-horizon choice.
  • Since the numerical sequence $7 \to 5 \to 6$ is shown for the hand-picked initial conditions $s_0=-10^{-7}$, $A_0=-0.7$, random sampling of initial conditions in the allowed region would establish whether the sequence is generic or a fine-tuned trajectory.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript analyzes a flat FRW universe containing pressureless matter, a non-minimally coupled scalar field with a quadratic potential, and holographic vacuum energy built from the effective gravitational constant and the apparent-horizon cutoff. It introduces dimensionless variables, derives a four-dimensional autonomous system for the variables (y, s, A, Ωk), and reports nine fixed points with their eigenvalues and stability conditions for the quadratic-potential case n=2. The central cosmological claim is that a viable evolution follows the sequence 7→5→6: an early stiff-fluid-dominated phase (point 7), a nearly dust-dominated transient (point 5), and a late-time stable de Sitter-like attractor (point 6), which the authors argue requires ξ<0 and 0<c<1. The paper further claims that the canonical scalar-field case ξ=0 cannot be recovered, only approached as ξ→0⁻ and c→0⁺.

Significance. If the claims hold, this is a competent and reasonably complete dynamical-systems study of a specific two-ingredient dark-energy model, and the explicit eigenvalue tables and the comparison with the non-holographic NMC limit of Sami et al. are useful reference material. The derivation of the autonomous system is coherent, and the paper makes its parameter dependence explicit. Its main physical significance is limited, however, by a structural feature of the chosen cutoff: with the apparent-horizon cutoff the holographic density parameter is a fixed fraction of the critical density, so the holographic sector has no independent dynamics, and in the viable corner c→0⁺ it is negligible. The paper does not ship code or machine-checked proofs; its numerical evidence for the claimed sequence is a single family of hand-picked trajectories, which leaves the genericity of the 7→5→6 path unsupported.

major comments (3)
  1. [II, Eqs. (10)–(17)] The apparent-horizon cutoff makes the holographic sector a fixed fraction of the total density. Substituting Eq. (11) into Eq. (10) and using the definitions (13) yields Eq. (17), ΩΛ = c²(1−Ωk); in the flat case studied in Sections IV and V this reduces to ΩΛ = c² identically. Hence ΩΛ carries no independent dynamics, and the split between scalar and holographic dark energy at the late-time attractor is fixed by the parameter c rather than by the evolution. Moreover, the viable corner identified by the authors is ξ→0⁻ and c→0⁺, where ΩΛ→0; the holographic ingredient is therefore negligible in exactly the regime that produces the dust era. This undercuts the abstract's two-ingredient dark-energy claim. The authors should either adopt a cutoff with independent dynamics (e.g., future event horizon, Ricci, or Granda-Oliveros) or explicitly reframe the conclusions as a model with a fixed fractional vacuum component plus an NMC scalar field.
  2. [IV–V, Figs. 2–3] The claimed viable sequence 7→5→6 is exhibited for a single hand-selected family of initial conditions: s0 = −10⁻⁷, A0 = −0.7, Ωm0 = 0.3233, Ωk0 = −0.0004, with y0 fixed by Eq. (19). Points 7 and 5 are saddles, so in the four-dimensional autonomous system a trajectory that visits both must lie on the intersection of the unstable manifold of point 7 and the stable manifold of point 5. A single numerical orbit does not establish that the near-dust era or the subsequent attraction to point 6 is representative of the model. Please provide a basin-of-attraction calculation, unstable-manifold shooting, or a measure of initial conditions on the constraint surface that reach point 6 through a near-dust phase; otherwise the abstract statement that viable evolution follows 7→5→6 is not supported.
  3. [III, Eq. (18), and VI] The exclusion of ξ=0 is a coordinate artifact of the chosen variables rather than a dynamical obstruction of the theory. Equation (18), x = s²/(24ξA), is indeterminate at ξ=0 because the variables x, s, and A are not independent coordinates in that limit; a canonical scalar holographic model at ξ=0 is a separate, well-defined autonomous system. The paper acknowledges this in Section VI, but the abstract's statement that the model 'cannot completely recover the canonical scalar case' should be phrased as a limitation of the variable choice, not as a property of the physical model.
minor comments (6)
  1. [II, Eq. (2)] The potential term in the action appears with a plus sign, +V(φ), which is inconsistent with the standard canonical-scalar convention and with the Friedmann equation (7) used later; it should be −V(φ) for the metric signature (−,+,+,+).
  2. [II, Eq. (6)] The Klein-Gordon equation is written as ∇μ∇νφ − ξRφ = 0, which is a two-index tensor equation rather than the scalar equation used subsequently; it should read □φ − ξRφ = 0 or ∇μ∇^μφ − ξRφ = 0.
  3. [III, after Eq. (13)] The statement 's = 0 implies ξ = 0' is not correct in general: s = 16πG_eff ξ φ φ̇/H also vanishes when φ = 0 or φ̇ = 0. The canonical-scalar limit should be discussed in terms of the combination ξφ², not s alone.
  4. [Throughout] The symbol c is used both for the speed of light in ℏ = c = 1 and for the dimensionless holographic parameter 0 ≤ c < 1. This is a recurring source of possible confusion and should be disambiguated, for instance by renaming the holographic parameter c_h.
  5. [V, Fig. 3] The text states that for any allowed values of ξ and c, w_eff approaches −1 at late times, but the right panel shows that for ξ = −10⁻⁵ and c = 0.8 the present-day w_eff is above −1/3. The statement should clarify that the late-time approach does not imply current acceleration for all parameter choices, and that 'allowed' excludes the parameter region preferred by the model's own viability criteria.
  6. [Table II, points 7–8] The stability conditions for points 7 and 8 contain an unbalanced parenthesis, e.g. '(−17 + 12c² − √(1 + 24c²))/[48(c² − 2) < ξ', which should be rewritten as an unambiguous interval.

Circularity Check

2 steps flagged · score 4.0 of 10

Partial structural circularity: the holographic component is fixed by the cutoff identity and the canonical ξ=0 limit is excluded by the chosen variables, but the NMC attractor analysis itself is self-contained.

  1. self definitional [Section III, equation (17)]
    "According to the energy density of HDE in equation (12), there is a relationship between the HDE density parameter and the spatial curvature term. This results in the constraint ΩΛ = c2(1 − Ωk). This relation is independent of the NMC coupling."

    With L = 1/sqrt(H^2 + k/a^2) and ρΛ = 3c^2/(8πG_eff L^2), the definition of ΩΛ gives ΩΛ = c^2(1 − Ωk) identically. In the flat case this means the holographic vacuum energy is a fixed fraction c^2 of the critical density at every epoch. The late-time dark-energy contribution ΩΛc = c^2 at fixed point 6 is therefore fixed by the choice of cutoff, not derived from the field dynamics. The claim that dark energy is contributed by both the scalar field and the holographic vacuum energy is thus partly a bookkeeping identity.

  2. self definitional [Section III, equation (18), and Sections IV/VI]
    "x = s2/(24ξA), where ξ ≠ 0, otherwise indeterminate. … Since zero NMC coupling, ξ = 0, is not allowed in the autonomous system, the model can not completely recover canonical scalar field case."

    The statement that the canonical ξ = 0 case cannot be recovered follows directly from the chosen variable x = s^2/(24ξA), which is singular at ξ = 0 by construction. The paper itself concedes that the variables must be redefined for ξ = 0, so the non-recoverability is an artifact of the coordinate choice rather than an independent physical result of the field equations. Presenting this as a conclusion of the model is definitional circularity, even though the paper is transparent about the reason.

full rationale

The paper is largely a self-contained dynamical-systems analysis of a specified Jordan-frame action with a non-minimally coupled scalar field and a holographic vacuum term. The fixed-point table, eigenvalues, and stability conditions are derived from the autonomous system through linearization; the late-time attractor point 6 with w_eff = −1 and its stability for ξ < 0 are genuine mathematical consequences of the model, not fitted outputs. No parameter is fitted to a subset of data and then renamed as a prediction: the initial densities are taken from DESI+CMB+Union3 and y0 is fixed by the Friedmann constraint, which is standard procedure. The self-citation [63] (Tsujikawa-Gumjudpai) is used only to motivate the parameter range ξ > −7×10^−3 and is not load-bearing for the attractor calculation. However, two claims presented as results are built into the paper's definitions. First, ΩΛ = c^2(1 − Ωk) is an identity following from the apparent-horizon cutoff, so the holographic component's late-time role is fixed by construction rather than dynamically predicted. Second, the statement that the canonical ξ = 0 limit cannot be recovered is a direct consequence of the variable x = s^2/(24ξA) being singular at ξ = 0; the authors acknowledge this but still list it as a finding. The claimed sequence 7 → 5 → 6 is exhibited only for hand-chosen initial conditions without a basin-of-attraction analysis, which weakens the generality of the claim but is not itself circularity. Overall, the circularity is partial and structural, confined to the holographic-bookkeeping identity and the coordinate-singularity claim, while the NMC attractor analysis itself retains independent content.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

V0 (potential amplitude) cancels out of the autonomous system through the variable y, so it is not a free parameter. Observational inputs Ωm0 = 0.3233 and Ωk0 = -0.0004 are taken from DESI+CMB+Union3, not fitted. The four entries above are the genuine degrees of freedom the central claim depends on: two model parameters plus two hand-picked initial conditions. The apparent-horizon cutoff (axiom 3) is the main structural assumption and directly produces the ΩΛ = c²(1 - Ωk) identity.

free parameters (4)
  • ξ (NMC coupling) = negative and small; numerics use -1e-6 to -1e-2; physical window -7e-3 < ξ < 0 via ref [63]
    Theory parameter; stability and the non-negative kinetic term force ξ < 0, and the dust era forces ξ → 0⁻.
  • c (holographic parameter) = 0 < c < 1; c ≪ 1 preferred; numerics use c = 0.1, 0.6, 0.8
    Enters the ρΛ definition; dust and stiff eras require c → 0⁺, while larger c raises w_eff and can suppress acceleration.
  • s0 (initial value of NMC variable s) = -1e-7
    Chosen by hand in section V to place the trajectory on the claimed 7 → 5 → 6 route; s < 0 is required for an expanding universe with ξ < 0.
  • A0 (initial value of NMC variable A) = -0.7
    Chosen by hand in section V; A is confined to (-1, 0) for ξ < 0, and y0 is then fixed by the Friedmann constraint.
assumptions (6)
  • domain assumption NMC action (2) with G4 = (1/2)[(8πG)⁻¹ - ξφ²] and V(φ) = V0φ²
    Defines the theory under study; the full fixed-point table is computed only for n = 2 (quadratic potential).
  • standard math CKN bound gives ρΛ = 3c²/8πG L² (eq. 4)
    Adopted from Cohen-Kaplan-Nelson 1999; the holographic density formula is the starting point of the HDE sector.
  • domain assumption Apparent horizon chosen as IR cutoff, L = 1/sqrt(H² + k/a²) (eq. 11)
    This choice yields the identity ΩΛ = c²(1 - Ωk) (eq. 17), making the holographic component a fixed fraction of the total density; it is load-bearing for the claim that both ingredients drive dark energy.
  • ad hoc to paper ξ ≠ 0 is required because x = s²/(24ξA) (eq. 18) is indeterminate at ξ = 0
    The authors acknowledge this; the claimed non-recoverability of the canonical ξ = 0 limit is an artifact of the variable choice, not a physical barrier.
  • domain assumption Flat universe (Ωk ≈ 0) assumed for the viable sequence
    Justified by the DESI+CMB+Union3 curvature bound (Ωk0 = -0.0004 ± 0.0019) cited in section IV.
  • domain assumption Perturbation constraint ξ > -7.0e-3 from ref [63]
    Borrowed external constraint used to define the physical window for the quadratic potential; ref [63] shares an author with this paper but is an independent published result.

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Pith. "Pith review of Cosmological dynamics of holographic dark energy with non-minimally coupled scalar field." pith.science (2026). https://pith.science/paper/WBZQP3J5

@misc{pith2026250705273,
  author       = {Pith},
  title        = {Pith review of: Cosmological dynamics of holographic dark energy with non-minimally coupled scalar field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WBZQP3J5}},
  note         = {Machine review of arXiv:2507.05273}
}
abstract

In this study, we consider FRW universe filled with matter, non-minimally coupling (NMC) scalar field under $V(\phi) = V_{0}\phi^{2}$ potential and holographic vacuum energy. Dark energy is contributed from both holographic vacuum energy and the NMC scalar field. NMC effective gravitational constant $G_\text{eff}(\phi)$, is naturally defined at the action level. Therefore, the gravitational constant in the holographic vacuum density is an effective one, i.e. $ \rho_{\Lambda} = {3c^{2}}/{8\pi G_{\text{eff}}L^{2}}\,. $ Apparent horizon is chosen as IR holographic cutoff scale as it is a trapped null surface. There are nine fixed points in this dynamical system with four independent dimensionless parameters. We consider flat case and find that viable cosmological evolution follows the sequence: an initial stiff-fluid-dominated phase, transitioning through a nearly dust-dominated era, and eventually reaching a stable dark energy-dominating state. Stability analysis requires that $\xi <0$ and $0 < c < 1$ for the theory to be physically valid. Since zero NMC coupling, $\xi=0$, is not allowed in the autonomous system, the model can not completely recover canonical scalar field case. That is to say, as $\xi \rightarrow 0^-$ and $c \rightarrow 0^+$, the model can only approach the canonical scalar case but can not completely recover it. To approach dust or stiff fluid dominations, both magnitudes of the NMC coupling and the holographic parameter must be small. Numerical integration shows that for any allowed values of $\xi$ and $c$, $w_\text{eff}$ approaches $-1$ at late times. Increasing of $c$ does not change shape of the $w_{\rm eff}$, but larger $c$ increases $w_\text{eff}$. As $\xi$ becomes stronger, dust era gradually disappears. Good behaviors of the dynamics require $-1 \ll \xi <0$ and $0 < c \ll 1$.

Figures

Figures reproduced from arXiv: 2507.05273 by the authors.

Figure 1
Figure 1. FIG. 1: Parametric plot between [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Phase portrait illustrates the evolution of a flat uni [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The figure shows evolution of the effective equation of [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The figure shows evolution of the NMC scalar field densi [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The figure shows evolution of variables [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

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Works this paper leans on

100 extracted references · 67 canonical work pages

  1. [1]

    The effective equation of state parameter at this point is indeterminate due to the zero deno minator in equation (27)

    Fixed point 1 Fixed point 1 corresponds to a completely spatial curvature-domin ated point, Ω kc = 1. The effective equation of state parameter at this point is indeterminate due to the zero deno minator in equation (27). The eigenvalues of the Jacobian matrix M (equation (29)) are given by, λ1 = 0, λ 2 = 2, λ 3 = −1, λ 4 = −2. (30) Since the eigenvalues i...

  2. [2]

    Fixed point 2 Fixed point 2 corresponds to a spatial curvature-dominated stat e, where the effective equation of state parameter is weff = −1/3. The eigenvalues of the Jacobian matrix are given by, λ1 = 1, λ 2 = 2, λ3 = − [ 4ξ ( c2 − 4 ) + c2 + 96ξ2] − √ c4[8ξ(194ξ − 23) + 1] + 32 c2ξ(6ξ − 1)(116ξ − 13) + 256ξ(9ξ − 1)(1 − 6ξ)2 32ξ(−1 + c2 + 6ξ) , λ4 = − [ 4...

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    The equation of state parameter for this point is weff = −1/3 for all ξ

    Fixed point 3 Fixed point 3 corresponds to a spatial curvature-dominated epoc h, Ω kc = 1, when ξ = (1 − c2)/6. The equation of state parameter for this point is weff = −1/3 for all ξ. The eigenvalues for this point are given by λ1 = 2, λ 2 = 4(−1 + c2 + 6ξ) c2 , λ 3 = 3c2 + 24ξ − 4 c2 , λ 4 = 6c4ξ + 4c2(6ξ − 1) + 4(1 − 6ξ)2 3c4ξ , (32) where c ⁄= 0 and ξ ...

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    Fixed point 4 Fixed point 4 corresponds to weff = −1. The eigenvalues are given by, λ1 = −2, λ 2 = 3, λ3 = 3c2(4ξ − 1) − √ 3ξ [c4ξ (48ξ2 + 104ξ − 21) + 6c2 (128ξ3 + 40ξ2 − 28ξ + 3) + 2(6ξ − 1)(3 − 16ξ)2] 4(−1 + c2 + 6ξ) λ4 = 3c2(4ξ − 1) + √ 3ξ [c4ξ (48ξ2 + 104ξ − 21) + 6c2 (128ξ3 + 40ξ2 − 28ξ + 3) + 2(6ξ − 1)(3 − 16ξ)2] 4(−1 + c2 + 6ξ) . (33) Since there i...

  5. [5]

    Fixed point 5 The eigenvalues of this fixed point depend on ξ only when ξ ⁄= 0. These are given by λ1 = 4ξ 1 − 4ξ , λ 2 = 1 − 8ξ 1 − 4ξ , λ 3 = 3 − 16ξ 1 − 4ξ , λ 4 = 3c2(1 − 4ξ)2 − 96ξ2 + 34ξ − 3 2(4ξ − 1)(−1 + c2 + 6ξ) , (34) where ξ ⁄= 1/4 and ξ ⁄= (1 − c2)/6. Since xc must be positive and this corresponds to ξ < 0, the eigenvalues are either positive o...

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    The stability therefore depends only on ξ and this can be considered in three cases

    Fixed point 6 Eigenvalues of the Jacobian matrix at this point are given by λ1 = −4ξ(1 + c2) 4ξ − 1 , λ 2 = −2, λ 3 = 3 − 4ξ(4 + c2) 4ξ − 1 , λ 4 = 3c2 − 2c2ξ(8 + c2) − 96ξ2 + 34ξ − 3 (4ξ − 1)(−1 + c2 + 6ξ) , (39) where ξ ⁄= 1 /4 and ξ ⁄= (1 − c2)/6. The stability therefore depends only on ξ and this can be considered in three cases. First, for 0 < ξ < 1/...

  7. [7]

    (40) To avoid divergence in the eigenvalues, the coupling parameter must satisfy ξ ⁄= (1 − c2)/6

    Fixed point 7 Eigenvalues of this fixed point are given by λ1 = 12ξ − 2 √ 6ξ(−1 + c2 + 6ξ), λ2 = c2(3 − 12ξ) + (6ξ − 1) [ 2 √ 6ξ(−1 + c2 + 6ξ) − 12ξ + 3 ] −1 + c2 + 6ξ , λ3 = 2c2 [ √ 6ξ(−1 + c2 + 6ξ) − 12ξ + 2 ] + 2(6ξ − 1) [ 2 √ 6ξ(−1 + c2 + 6ξ) − 12ξ + 2 ] −1 + c2 + 6ξ , λ4 = 2c2 [ √ 6ξ(−1 + c2 + 6ξ) − 12ξ + 3 ] + 2(6ξ − 1) [ 2 √ 6ξ(−1 + c2 + 6ξ) − 12ξ +...

  8. [8]

    (43) and the equation of state is weff = c2 ( 3 − 24ξ − 2 √ 6ξ(−1 + c2 + 6ξ) ) + (−1 + 6ξ) ( 3 − 24ξ − 4 √ 6ξ(−1 + c2 + 6ξ) ) 3(−1 + c2 + 6ξ)

    Fixed point 8 Eigenvalues of this fixed point are given by λ1 = 12ξ + 2 √ 6ξ(−1 + c2 + 6ξ), 11 λ2 = c2(3 − 12ξ) + (6ξ − 1) [ −2 √ 6ξ(−1 + c2 + 6ξ) − 12ξ + 3 ] −1 + c2 + 6ξ , λ3 = 2c2 [ − √ 6ξ(−1 + c2 + 6ξ) − 12ξ + 2 ] + 2(6ξ − 1) [ −2 √ 6ξ(−1 + c2 + 6ξ) − 12ξ + 2 ] −1 + c2 + 6ξ , λ4 = 2c2 [ − √ 6ξ(−1 + c2 + 6ξ) − 12ξ + 3 ] + 2(6ξ − 1) [ −2 √ 6ξ(−1 + c2 + 6...

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