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REVIEW 4 major objections 6 minor 1 cited by

Cosmological Instabilities and the Role of Matter Interactions in Dynamical Dark Energy Models

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Low-energy ghosts are tachyons; high-energy ghosts are gradient instabilities in interacting two-field dark energy models.

desk verdict Useful two-field extension of the ghost/tachyon/gradient equivalence, but the 'generic' claim holds only for constant coefficients and the abstract overreaches. read the letter →

arxiv 1908.03212 v2 pith:FDTX5PQ3 submitted 2019-08-08 gr-qc astro-ph.COhep-th

William J. Wolf , Macarena Lagos This is my paper · ORCID
classification gr-qcastro-ph.COhep-th PACS 04.50.Kd95.36.+x98.80.-k
keywords darkenergycosmologicalperturbationsghostinstabilitiesgradienttachyoniccanonicaltransformationstwo-fieldactionsstability
open problems Dark Energy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies the stability of cosmological models in which dark energy is a dynamical field rather than a cosmological constant, focusing on the linear perturbations of a dark-energy field interacting with a matter field. Its central claim is that the three standard instabilities—ghosts (negative kinetic energy), gradient instabilities (negative momentum squared), and tachyons (negative mass squared)—are not independent in a two-interacting-field system. Working with a generic quadratic action, the authors show by canonical transformations that a low-energy ghost is the same physical instability as a tachyon, and a high-energy ghost is the same as a gradient instability; only the mathematical basis changes. The significance is that stability conditions for dark energy models should be imposed on invariant combinations of the action coefficients, not on the sign of any single term in a given field basis. Interactions between the fields generally do not determine whether an instability exists, but they set its growth timescale, and in exceptional two-ghost cases they can even make the system stable.

What carries the argument

The machinery is the linear canonical transformation of the two-field Hamiltonian, $Q_i' = a_i Q_i + a_{i+1} P_i$, $P_i' = b_i Q_i + b_{i+1} P_i$ with unit Poisson brackets, combined with a linear field redefinition chosen to diagonalise the transformed kinetic and mass matrices. The key object being manipulated is the quadratic action of eq. (24), whose off-diagonal term is the antisymmetric coupling $D(\dot\phi_1 \phi_2 - \dot\phi_2 \phi_1)$. Because canonical transformations preserve the physics, a negative coefficient that appears as a ghost in one basis must reappear as a tachyon at low $k$ or a gradient at high $k$ in another basis; the transformation moves the sign rather than creating or destroying it.

What would settle it

Numerically evolve the linear scalar perturbations of a concrete two-field dark energy model (for instance a scalar-tensor theory) at low and high energies in two different field bases; the paper predicts identical growth timescales and the same instability structure in every basis, so finding bases that disagree on whether the system is stable or on the growth rate would refute the equivalence claim.

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Extended reading notes

Core claim

The paper derives, for a system of two interacting scalar perturbations of the form $S = \int d^3k\, dt \left[ \frac{1}{2} K_1 \dot\phi_1^2 - \frac{1}{2} M_1^2 \phi_1^2 + D(\dot\phi_1\phi_2 - \dot\phi_2\phi_1) + \frac{1}{2} K_2 \dot\phi_2^2 - \frac{1}{2} M_2^2 \phi_2^2 \right]$, that a linear canonical transformation followed by a field redefinition can move a negative sign in $K_1$ into the mass coefficient at low energies and into the gradient coefficient at high energies, without changing the equations of motion. Thus a low-energy ghost is equivalent to a tachyon and a high-energy ghost to a gradient instability. Since the transformation preserves the Hamiltonian equations and Poisson brackets, any instability that appears in one basis must appear in all bases; the negative coefficient simply changes location. The paper also shows that when one field is unstable, the interaction term $D$ couples the fields so that both grow, and that $|D|$ controls the instability timescale while the sign of $D$ is irrelevant. In the exceptional case of two ghosts, for restricted parameter values the interaction decouples the positive- and negative-energy sectors, giving stable oscillatory solutions.

Load-bearing premise

The central assumption is that any two-field perturbation theory can be rewritten in a standard diagonal form with one antisymmetric interaction term, and that the coefficients are effectively constant at very low and very high energies; if either fails, the claimed equivalences may not be generic.

Editorial extensions

If this is right

  • Stability conditions for dark energy models should be imposed on basis-invariant combinations of coefficients or on eigenfrequencies, not on the individual signs of $K_i$ and $M_i^2$.
  • If any field in a two-field model has a high-energy ghost or gradient instability, the entire interacting system is unstable, with modes growing at least as $e^{kt}$, so such models are not viable unless a high-energy completion intervenes.
  • At low energies, a negative kinetic term and a negative mass term are equally dangerous: both produce exponential growth on a timescale set by the masses and modified by $|D|$, so the phenomenological condition is on the growth rate, e.g. growth slower than the Hubble rate $H_0$.
  • The interaction term $D$ must be included when constraining dark energy models, because even when it does not decide stability it changes the instability timescale.
  • In the exceptional two-ghost case, for $4D^2 > m_1^2 + m_2^2$ and $(D^2 - m_1^2 - m_2^2)^2 > 4 m_1^2 m_2^2$, the low-energy system is stable despite two negative kinetic terms, because the positive- and negative-energy sectors decouple.

Reading between the lines

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

  • If the equivalence is generic, then the standard practice of quoting 'no ghosts' as an independent health condition in modified-gravity cosmology is redundant: a no-ghost requirement in one basis already encodes no-tachyon and no-gradient conditions, and model builders should report basis-invariant stability criteria instead.
  • The same canonical-transformation argument should extend to more than two fields and to vector or tensor perturbations, because it relies on the structure of the quadratic action rather than on the spin of the fields; this would make the ghost–tachyon–gradient equivalence a general property of linearised cosmological perturbations.
  • A concrete check of the paper's practical message would be to compute, for a given dark energy model, the growth timescale from the eigenfrequencies of eq. (26) and compare it with numerical evolution of the full perturbation equations; agreement would confirm that the interaction term $D$ indeed sets the timescale, while disagreement would point to missing time-dependence in the constant-coeffici
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper studies cosmological models with a dynamical dark energy field and analyzes linear scalar perturbations described by two interacting scalar fields. It classifies instabilities as ghosts, gradient instabilities, and tachyons according to the signs of the kinetic, spatial-gradient, and mass coefficients in the quadratic action. Using linear canonical transformations, the paper argues that low-energy ghosts are equivalent to tachyonic instabilities and high-energy ghosts are equivalent to gradient instabilities. It also claims that interactions between the two fields generally do not determine whether an instability is present, but do affect the instability timescale, with exceptional cases in which two ghosts can interact and lead to stable solutions. The conclusion is that stability conditions in modified gravity dark energy models should be imposed on invariant combinations of coefficients rather than on individual signs.

Significance. If the claimed equivalence were established in full generality, this would be a useful conceptual clarification for dark energy model building, where separate positivity conditions on kinetic and gradient terms are commonly imposed. The single-field section is clear and correct, and the constant-coefficient examples in the two-field section are explicit and internally consistent. The observation that the sign of the antisymmetric coupling D does not change the inertia of the Hamiltonian, while its magnitude can affect growth timescales, is a genuinely useful point. However, the paper's advertised 'generic' result is supported only by constant-coefficient examples with particular transformation choices and with D independent of k. Since the cosmological background makes the coefficients time-dependent, the domain of validity of the central claim remains unclear. The manuscript is therefore a potentially valuable contribution, but it needs either additional analysis for the time-dependent case or a careful restriction of its claims.

major comments (4)
  1. [§IV.A; §III.B, after Eq. (17)] The central claim is stated generically in the abstract and in §V, but the two-field derivation assumes all coefficients and transformation parameters are constant in time. In §IV.A the C coefficients are explicitly assumed to be constants in time, and the assumptions before §IV.B fix D to be constant in time and independent of k; the equations of motion (25) and the solutions (26) are constant-coefficient results. This matters physically because §III.B explicitly notes that a time-dependent canonical transformation can remove a single negative kinetic term and produce a stable solution. In a FLRW background K_i, M_i^2 and D evolve on a Hubble timescale, and time derivatives of the transformation coefficients generate additional terms, as shown in Appendix 2 where \dot{D}_1 appears as a mass mixing term. The paper therefore does not establish the 'generic' equivalence for the time-dependent coefficients that occur in cosmological models. The authors should either supply an adiabatic or otherwise general proof, or explicitly restrict the claim to time-independent coefficients and adjust the abstract and discussion accordingly.
  2. [§IV, Eq. (24)] The derivation relies on the assertion that any two-field quadratic action can be brought to the form (24) with only an antisymmetric coupling D by a field redefinition. This is a nontrivial statement when coefficients are time-dependent, since the simultaneous removal of kinetic and mass mixings introduces time-derivative terms and may change the form of D. The paper cites reference [37] for this reduction, but does not state the precise conditions under which (24) is valid. Because the entire subsequent analysis uses (24), the domain of the claimed equivalence is not fully specified. A derivation, or at least a precise statement of the class of theories for which (24) holds, is needed to support the central claim.
  3. [§IV.B, §IV.C] The examples demonstrate the claimed relations for specific sign patterns and for specific choices of canonical-transformation parameters, but they do not add up to the 'generic' statement in the abstract. For instance, §IV.B.1 says that if the ghost is in K_2 instead of K_1, the same transformation leaves a ghost and produces no tachyon; the paper does not give the general construction that maps an arbitrary sign assignment to a tachyon or gradient. In addition, the high-energy results assume D independent of k, whereas the earlier discussion around Eq. (28) explicitly allows D proportional to k. If D is relevant in the high-energy limit, the transformation used in §IV.B.2 and the resulting identification of a gradient instability are not shown to hold. The authors should either prove the equivalences for the full parameter space or moderate the claim from 'generic' to 'shown in representative examples'.
  4. [§IV.D (Example III)] The claim that a two-ghost system can be stable because, after diagonalization, the positive- and negative-energy oscillators are decoupled is correct within the constant-coefficient linear analysis. However, the physical status of this stability is left unclear: the negative-energy sector is decoupled only in the chosen frame, and any small coupling to the environment or to the other field beyond linear order would make the negative-energy oscillator unstable. Since the paper presents this as one of its main results, it should state more carefully that the stability is a feature of the decoupled linear system in a special basis, and discuss whether it survives the couplings that are inevitably present in a cosmological setting, rather than presenting it as a generic stabilization mechanism.
minor comments (6)
  1. [Appendix A.1, Eq. (A.2)] The lowercase k1 and k2 are used for the kinetic coefficients K1 and K2, which is confusing because k is already the wavenumber; please use K1 and K2 or define the shorthand explicitly.
  2. [§IV.C, Eq. (44)] In the D' coupling term, φ_2 and φ_1 appear without primes; they should be φ'_2 and φ'_1.
  3. [§IV.C, after Eq. (44)] The transformed fields are named φ'_1 and φ'_2, but the low-energy coefficients in Eq. (46) are labelled K'_3, M'_3, K'_4, M'_4; the labelling should be made consistent.
  4. [§IV.B, Eq. (42)] The sentence stating that a ghost in K2 instead of K1 'would have kept one ghost in φ'_3 and would exhibit no tachyons' is confusing, because by relabelling the two fields the same construction would produce a tachyon; please clarify that a different transformation is required or that the result follows by symmetry.
  5. [§II.D, Eq. (9)] The statement that the action (9) can absorb gradient terms into M(t,k) should be made more explicit, since a reader may wonder how a gradient instability is represented in the K,M notation used there.
  6. [§IV, Eq. (26)] The factor (2K1K2)^{-1/2} is imaginary for K1K2<0; the paper should state that the expression is to be interpreted by analytic continuation when discussing sign patterns with negative kinetic coefficients.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ghost/tachyon/gradient equivalences are obtained by explicit canonical transformations from the stated two-field action; the only author-overlap citation ([37]) supplies the generic action form and is not the source of the equivalence.

full rationale

The paper's central claim is not derived by assuming the conclusion. Starting from the generic two-field quadratic action (24), the authors define canonical phase-space variables (30)-(31), construct the Hamiltonian (32), apply a linear canonical transformation (34) with a subsequent field redefinition (39), and compute the transformed Lagrangian (40). The low- and high-k limits (42)-(43) and (46)-(49) then show, by explicit algebra, that a negative kinetic coefficient can be recast as a negative mass coefficient at low k or as a negative gradient coefficient at high k while leaving the equations of motion, eq. (26), unchanged. This is a constructive equivalence, not a renaming or a fitted input: the sign of a coefficient in one field basis is shown not to be invariant under canonical transformations, and the instability is tracked through the preserved equations of motion. One limitation is disclosed by the paper itself: after eq. (17) the authors note that a time-dependent canonical transformation can, in some cases, remove a single negative kinetic term, so the constant-coefficient examples do not exhaust all possibilities. This qualifies the word 'generic' in the abstract, but it is an acknowledged scope restriction rather than circularity. The only author-overlapping citation is ref. [37] (Lagos, Bellini, Noller, Ferreira, Baker), used to justify the general action form (24). That is a published, externally checkable result with several non-authors; it is invoked for the setup, not for the equivalence result, and the canonical-transformation derivation stands independently for any action of the form (24). No fitted parameters are promoted to predictions, and no result is imported from a same-author uniqueness theorem to force the conclusion. The derivation is therefore self-contained with respect to its central claim.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new entities. It relies on the standard assumption that a two-field quadratic action can be diagonalized into the form of eq. (24), and that stability can be read off from the signs of K and M^2 in the low- and high-energy limits.

assumptions (3)
  • domain assumption The quadratic action for linear scalar perturbations of a single dark-energy field plus matter can be written as in eq. (24) with only an antisymmetric D coupling and no kinetic or mass mixing.
    Stated in Section IV: 'we note that we have not included mass or kinetic mixing terms because one can always perform a field redefinition to bring the system in the form of eq. (24).' This relies on reference [37].
  • domain assumption Stability of the system can be assessed by the signs of the coefficients K and M^2 in the action, in the low- and high-energy limits, ignoring intermediate scales and time-dependence of the coefficients.
    The paper explicitly restricts to 'high and low energy instabilities' and treats coefficients as constants in the examples. Time-dependent coefficients are only briefly discussed in Section III.B.
  • standard math Linear canonical transformations preserve the physical content of the theory, so instabilities found in any canonical frame are physical.
    Standard Hamiltonian mechanics; used throughout Sections III and IV.

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Cite this review

Pith. "Pith review of Cosmological Instabilities and the Role of Matter Interactions in Dynamical Dark Energy Models." pith.science (2026). https://pith.science/paper/FDTX5PQ3

@misc{pith2026190803212,
  author       = {Pith},
  title        = {Pith review of: Cosmological Instabilities and the Role of Matter Interactions in Dynamical Dark Energy Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FDTX5PQ3}},
  note         = {Machine review of arXiv:1908.03212}
}
read the original abstract

We consider cosmological models with a dynamical dark energy field, and study the presence of three types of commonly found instabilities, namely ghost (when fields have negative kinetic energy), gradient (negative momentum squared) and tachyon (negative mass squared). In particular, we study the linear scalar perturbations of theories with two interacting scalar fields as a proxy for a dark energy and matter fields, and explicitly show how canonical transformations relate these three types of instabilities with each other. We generically show that low-energy ghosts are equivalent to tachyonic instabilities, and that high-energy ghosts are equivalent to gradient instabilities. Via examples we make evident the fact that whenever one of these fields exhibits an instability then the entire physical system becomes unstable, with an unbounded Hamiltonian. Finally, we discuss the role of interactions between the two fields, and show that whereas most of the time interactions will not determine whether an instability is present or not, they may affect the timescale of the instability. We also find exceptional cases in which the two fields are ghosts and hence the physical system is seemingly unstable, but the presence of interactions actually lead to stable solutions. These results are very important for assessing the viability of dark energy models that may exhibit ghost, gradient or tachyonic modes.

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Forward citations

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Reference graph

Works this paper leans on

80 extracted references · 41 canonical work pages · cited by 1 Pith paper

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    Low Energy We see that if we start with a low-energy ghost field, for instance if ε t = −1 and ε s = ε m = +1, then the new field φ ′ will have the following kinetic and mass terms: K′ = −1 (a2 1 − a2 2m2) − a2 2k2 , (20) M ′2 = m2 + k2 (a2 1 − a2 2m2) − a2 2k2 . (21) First of all, we notice that in this simple scenario, the only relevant free quantities ar...

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    we see that the three type of instabilities are mathemat- ically different: ghosts come from negative time derivativ es, gradient instabilities come from negative spatial derivat ives, and tachyons from negative non-derivative interactions. Nevertheless, from the three solutions for φ k(t) previously discussed we can see that a model with a physical insta...

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    First, we define the phas e space variables as: Q = φ k, P = ∂ L ∂ ˙φ k = K ˙φ k, (13) and obtain the Hamiltonian density H : H = 1 2 P2 K + 1 2 M2Q2

    and calcu- lating its associated Hamiltonian. First, we define the phas e space variables as: Q = φ k, P = ∂ L ∂ ˙φ k = K ˙φ k, (13) and obtain the Hamiltonian density H : H = 1 2 P2 K + 1 2 M2Q2. (14) Next, we proceed to apply the linear canonical transformati on of eq. ( 11) to this Hamiltonian. Under this transformation the new Hamiltonian density be- c...

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    From this final Lagrangian we can identify new kinetic energy and mass terms, analogous to eq. ( 9): K′ ≡ K A , M ′2 ≡ M2 A + d dt (M2Ka2b2 + a1b1 A ) , (17) where we have made an integration by parts in the term φ ′ k ˙φ ′ k so that the new Lagrangian is given by: L ′ = 1 2 K′ ˙φ ′2 k − 1 2 M ′2φ ′2 k . (18) Given this final result we can analyze the effec...

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    Also, as expected, the timescale of the instability is given by k−1 as the factor of a2 2 goes not affect the equations of motion in this case

    High Energy For the same model as before, with ε t = −1 and ε s = ε m = +1, φ starts with a high-energy ghost instability, but the new action will have a high-energy limit, k → ∞ , such that: K′ → 1 a2 2k2 , M ′2 → −1 a2 2 , (23) and thus we see the new field will have a gradient instability instead of a ghost instability, as long as the canonical tran s- ...

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    These four exponen- tial terms will thus always lead an oscillatory behaviour wi th constant amplitudes

    + 4m2 1m2 2 are both always positive. These four exponen- tial terms will thus always lead an oscillatory behaviour wi th constant amplitudes. We emphasize that, in general, the sign of the interaction term D never affects the stability of the system because it al- ways shows ...

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    We see that for W > 0 and (W 2 − 4m2 1m2

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    and show the previously discussed relationship between the three types of instabilities. A. Canonical Transformations A canonical transformation is a transformation of the Hamiltonian variables from {P, Q} to {P′, Q′} such that physics is invariant. In particular, we will gene...

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    ≥ 0 the solutions in the low-energy regime will be stable and oscillating. In this example, a seemingly unst a- ble model with two ghosts turned out to be stable due to the presence of the interaction D (note that if D = 0 then the solu- tions for a model with two ghosts would...

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    must be satisfied for a canonical transformation: {Q′ 1, P′ 1} = a1b2 − a2b1 = 1, {Q′ 2, P′ 2} = c1d2 − c2d1 = 1. (35) As before, we will carry out the transformation to obtain the new Hamiltonian and explore what relations this shows be- tween the types of instabilities we hav...

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    Therefore, we will need to make our final Lagrangian diagonal before making determinations about the relationship between different types of possible in- stabilities. As before, we then use Hamilton’s equation ˙Q′ 1 = ∂ H ′/∂ P′ 1 and ˙Q′ 2 = ∂ H ′/∂ P′ 2 to obtain the new kine...

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    Now, we proceed to consider specific examples and study the stability of this new Hamiltonian. Before that , we note that this Hamiltonian can be simplified by making in- tegration by parts on the terms ˙ φ ′ 1φ ′ 1 and ˙φ ′ 2φ ′ 2 and reabsorbing them into their corresponding m...

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    in L ′, which will allow us to easily study the stability of the system by simply looking the the signs of the diagonal kinetic and mass terms. While we could do this using only the canonical trans- formation we have defined, we introduce an additional field redefinition at this ...

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    (39) After making the field redefinition, we will later choose α 1 and α 2 such that they diagonalize the kinetic and mass ma- trices thereby eliminating problematic mixing terms. Firs t we will show that this reproduces the results of the single field ex- ample from the beginnin...

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    This sta- bility analysis is done, as in the previous section, by looki ng at the sign of the kinetic and mass coefficients in the low and high-energy regimes. 9

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    (42) Here x = ( m4 1 + 2m2 1(4D2 + m2

    Low Energy In the low-energy regime, k → 0, we find the following ki- netic and mass coefficients: K′ 3 → x [ √ z2 + 16D2m2 1 − z ] 16D2c2 1m2 1 , M ′2 3 → x [ y − √ y2 − 16D2m2 2 ] 16D2c2 1 , K′ 4 → x [ √ z2 + 16D2m2 1 + z ] 16D2c2 1m2 1 , M ′2 4 → −x [ √ y2 − 16D2m2 2 + y ] 16...

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    Also, we have defined y = (4D2 + m2 1 + m2

    + (4D2 − m2 2)2)1/2, and is al- ways real. Also, we have defined y = (4D2 + m2 1 + m2

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    and z = (−4D2 + m2 1 + m2 2), in order to easily determine if there is a minus sign in the kinetic and mass terms, and thus if there is an instability. We find that M ′2 4 < 0, and hence we trans- form a low-energy ghost into a tachyon while all of the other coefficients in this...

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    This example generalizes the results found for a single field in the previous section

    Thus, we can clearly see that the entire system is unstable, and instabilities ar e the same physical entity regardless of the explicit term they ap - pear in within the Lagrangian, and regardless of the field in which they appear within an interacting system. This example gene...

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    (43) Here again the canonical transformation has allowed us to move a minus sign in the kinetic term of the action into the mass term of the action

    High Energy In the limit k → ∞ we obtain the following kinetic and mass terms in the transformed action: K′ 3 → 1 c2 1 , M ′2 3 → k2 c2 1 , K′ 4 → k2 D2c2 1 , M ′2 4 → − k4 D2c2 1 . (43) Here again the canonical transformation has allowed us to move a minus sign in the kinetic...

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    (46) Here we see that depending on the relative size of D2 and m2 1,2, the negative signs will appear in different terms

    Low Energy In the low-energy regime, k → 0, we find the following ki- netic and mass coefficients: K′ 3 → − 1 a2 2 ( D2 − m2 1 ), M ′2 3 → m2 2 a2 2 ( D2 − m2 2 ), K′ 4 → − 1 c2 2 ( D2 − m2 2 ), M ′2 4 → m2 1 c2 2 ( D2 − m2 1 ). (46) Here we see that depending on the relative si...

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    If D2 is larger than one of the masses but not the other then we would have a mix of ghosts and tachyons that would likewise correspond to unstable solutions in eq

    to be dominated by real expo- nentials. If D2 is larger than one of the masses but not the other then we would have a mix of ghosts and tachyons that would likewise correspond to unstable solutions in eq. ( 29). On the other hand, if 4 D2 > (m2 1 + m2

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    and (D2 − m2 1 − m2 2)2 > 4m2 1m2 2, which are the same conditions as W > 0 and W 2 − 4m2 1m2 2 > 0 mentioned previously, there will still be two ghosts in the action; however, this will then correspond to t he low energy limit of eq. (

  20. [26]

    with stable, oscillating solutions. 10

  21. [27]

    In this case, we see that the two initial ghosts appear now as two gradients

    High Energy In the limit k → ∞ we obtain the following kinetic and mass terms in the transformed action: K′ 3 → 1 a2 2k2 , M ′2 3 → − 1 a2 2 , (47) K′ 4 → 1 c2 2k2 , M ′2 4 → − 1 c2 2 . In this case, we see that the two initial ghosts appear now as two gradients. Similarly to ...

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    (48) Now x = ( −2m2 1 ( 4D2 + m2 2 ) + ( m2 2 − 4D2)2 + m4 1 )1/2 , y = (4D2 − m2 1 + m2

    Low Energy In the low energy limit, k → 0, we find the following kinetic and mass coefficients: K′ 3 → x [ √ z2 − 16D2m2 1 + z ] 16D2c2 1m2 1 , M ′2 3 → x [ y − √ y2 − 16D2m2 2 ] 16D2c2 1 , K′ 4 → − x [ z − √ z2 − 16D2m2 1 ] 16D2c2 1m2 1 , M ′2 4 → − x [ √ y2 − 16D2m2 2 + y ] 16...

  23. [29]

    However, in eq

    are exceptional as we find that most of the time the size of |D| does not determine whether an instability is present or not. However, in eq. ( 28) we do find that the timescale of the instability does depend on D, as long as it is relevant on the given energy regime. Therefore,...

  24. [30]

    and z = (4D2 + m2 1 − m2 2). Notice that x is now real for restricted values of the coefficients D, m1 and m2 and again must be so in order to have a valid transformation because the fields in our initial system are real. These re- stricted values correspond to W 2 − 4m2 1m2 2 >...

  25. [31]

    (49) In this case, we find that the high-energy limit gives us an unstable solution with two ghosts

    High Energy In the limit k → ∞ we obtain the following kinetic and mass terms in the transformed action: K′ 3 → − 1 c2 1 , M ′2 3 → k2 c2 1 , K′ 4 → − 1 c2 1 , M ′2 4 → k2 c2 1 . (49) In this case, we find that the high-energy limit gives us an unstable solution with two ghosts...

  26. [32]

    and use our transformation eq. ( 34) to obtain the new Hamiltonian which has the following form: H ′ = Γ 1P′2 1 + Γ 2P′ 1Q′ 1 + Γ 3Q′2 1 + Γ 4P′2 2 + Γ 5P′ 2Q′ 2 + Γ 6Q′2 2 + Γ 7Q′ 1P′ 2 + Γ 8P′ 1Q′ 2 + Γ 9Q′ 1Q′ 2 + Γ 10P′ 1P′ 2, (36) where the Γ s are terms comprised of coef...

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    when D is not relevant and thus go as ekt. V . DISCUSSION There are a number of cosmological models in the literature that have a dynamical evolution of dark energy, and provide alternatives to the standard cosmological constant paradi gm. These models typically describe dark ...

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    After- wards, we study the sign of the kinetic K1,2(t, k) and mass m1,2(t, k) terms in the low and high-energy regimes, while studying also the behaviour of the interaction term D

    (this has been explicitly done in [33]). After- wards, we study the sign of the kinetic K1,2(t, k) and mass m1,2(t, k) terms in the low and high-energy regimes, while studying also the behaviour of the interaction term D. From the result obtained in [33], we find that in the hi...

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    (A.1) A ≡ a2 2 ( c2 2 ( D2 + k2m2 1 )( D2 + k1m2 2 ) + c2 1k1m2 1 ) + 2a1a2c1c2D2 + a2 1 ( c2 2k2m2 2 + c2 1 )

    Transformed Lagrangian The transformed Lagrangian for two fields is given by: L ′ = C1 2A ˙φ ′ 1 2 + C2 A ˙φ ′ 1φ ′ 1 − C3 2A φ ′2 1 + C4 2A ˙φ ′ 2 2 + C5 A ˙φ ′ 2φ ′ 2 − C6 2A φ ′2 2 + C7 A ˙φ ′ 1 ˙φ ′ 2 + C8 A ˙φ ′ 1φ ′ 2 + C9 A ˙φ ′ 2φ ′ 1 + C10 A φ ′ 1φ ′ 2. (A.1) A ≡ a2 2 ...

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    (A.4) They can be integrated by parts to bring them into an antisym- metric form as desired for the study of stability

    Interaction terms Consider terms in our Lagrangian of the following: ∫ d4x [ D1 ˙φ 1φ 2 + D2 ˙φ 2φ 1 ] . (A.4) They can be integrated by parts to bring them into an antisym- metric form as desired for the study of stability. We rewrite these terms as: ∫ d4x [ − ˙D1 φ 1φ 2 − D1...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.