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REVIEW 2 major objections 4 minor 66 references

Two-field mimetic gravity revisited and Hamiltonian analysis

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The two-field mimetic gravity model contains two scalar degrees of freedom plus two tensor modes, not the single scalar mode claimed earlier, and with opposite-sign kinetic terms its entropy mode is a ghost.

desk verdict A careful correction of the DOF count in two-field mimetic gravity, with a solid linear-perturbation core and a nonlinear Hamiltonian section that leans too heavily on a cited single-field calculation. read the letter →

arxiv 1909.01248 v4 pith:QXPOT64A submitted 2019-09-03 gr-qc astro-ph.COhep-th

Liuyuan Shen , Yunlong Zheng , Mingzhe Li This is my paper · ORCID
classification gr-qcastro-ph.COhep-th
keywords mimeticgravitydegreesoffreedomHamiltoniananalysiscosmologicalperturbationsadiabaticmodeentropyghostinstabilityshiftsymmetry
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

The paper revisits the two-field mimetic gravity model with shift-symmetric scalars and argues that it contains two scalar degrees of freedom on top of the two tensor modes, for a total of four. This corrects an earlier study that reported only one scalar mode. The claim is established in two independent ways: quadratic perturbation theory around a cosmological background, and a full nonlinear Hamiltonian analysis that counts constraints. The paper also shows that when the two scalar kinetic terms enter the mimetic constraint with opposite signs, the entropy mode is a ghost, so the model is quantum-mechanically unstable.

What carries the argument

The central device is the decomposition of the two scalar fluctuations into an adiabatic mode along the background trajectory and an orthogonal entropy mode, together with the Hamiltonian 3+1 split of the metric. Varying the Lagrange multiplier imposes $A=0$, removing the lapse perturbation and leaving the reduced quadratic Lagrangian from which the equations of motion are derived without substituting $\dot{R}=0$ before varying the curvature perturbation $\mathcal{R}$. At the nonlinear level the machinery is the reduced Hamiltonian density obtained by eliminating $\lambda$, whose smeared Hamiltonian and momentum constraints are claimed to close as first-class constraints; that closure fixes the degree-of-freedom count.

What would settle it

Compute the full Poisson bracket $\{H[M],H[N]\}$ for the matter sector of the two-field reduced Hamiltonian without borrowing the single-field result; the four-degree-of-freedom count stands only if the bracket closes on the constraint surface with no tertiary constraints. Separately, for $c=-1$, check whether the reduced Hamiltonian $\sqrt{\pi_\phi^2-\pi_\psi^2}$ is real on the allowed phase space; if not, the Hamiltonian analysis is incomplete for that branch.

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Extended reading notes

Core claim

The paper claims that the two-field mimetic action does not reduce to a single propagating scalar. Decomposing the two scalars into adiabatic and entropy directions, the quadratic action gives a decoupled entropy mode that propagates with unit sound speed and a curvature perturbation that is frozen, $\dot{R}=0$; both modes contribute phase-space degrees of freedom, so the scalar sector has two DOFs. In the full nonlinear reduced Hamiltonian, obtained after eliminating the Lagrange multiplier, the paper finds eight first-class constraints on a twelve-pair phase space and counts $(2\times12-2\times8)/2=4$ degrees of freedom. In the opposite-sign case $c=-1$, the entropy mode has a negative kinetic term and is a ghost.

Load-bearing premise

The argument assumes that the eight constraints of the reduced nonlinear theory are all first-class, with no hidden secondary constraints, and that the matter-sector Poisson algebra matches the single-field result to which the paper refers.

Editorial extensions

If this is right

  • At the background level the two mimetic fields still behave as pressureless dark matter, with $\rho\propto a^{-3}$.
  • The comoving curvature perturbation is frozen, $\dot{\mathcal{R}}=0$, exactly as in the single-field mimetic theory.
  • The entropy perturbation decouples from the curvature mode at linear order, is massless, and propagates with unit sound speed when $c=1$.
  • For $c=-1$ the entropy mode is a ghost, so the model is quantum-mechanically unstable in that branch.
  • The total degree-of-freedom count is four, one more than the earlier analysis concluded, which changes the counting of initial conditions for cosmological perturbations.

Reading between the lines

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

  • If the four-degree-of-freedom count is right, the earlier one-scalar result likely came from substituting $\dot{\mathcal{R}}=0$ into the action before varying the curvature mode, a step the paper identifies as invalid; analogous premature substitutions may affect other multi-field mimetic models.
  • Adding more mimetic scalars would probably add one entropy mode per scalar while leaving the adiabatic curvature mode frozen, so the instability structure of the single-field theory in the presence of matter would persist.
  • The $c=-1$ ghost appears already at quadratic order around any background satisfying the constraint, so avoiding it would require adding higher-derivative or curvature-coupling terms beyond the minimal shift-symmetric action.
  • Because the reduced Hamiltonian for $c=-1$ contains a square root of $\pi_\phi^2-\pi_\psi^2$, a direct test is whether the physical phase space is restricted to the region where that quantity is nonnegative; if not, the Hamiltonian formulation itself may need revision.
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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

2 major / 4 minor

Summary. The manuscript revisits the two-field mimetic gravity model with shift symmetries, defined by the action (2.9). It performs a linear cosmological perturbation analysis in comoving gauge and finds two scalar degrees of freedom (DOFs) — an adiabatic mode with \dot{R}=0 and a propagating entropy mode — contrary to Ref. [46], which claimed only one. It then attempts a full nonlinear Hamiltonian analysis, asserting eight first-class constraints and four DOFs in total. It also analyzes the c=-1 branch (opposite-sign kinetic terms in the mimetic constraint) and concludes that the entropy mode is always a ghost.

Significance. If the results hold, the paper corrects the DOF count in the two-field mimetic model and identifies a new ghost instability. The linear perturbation derivation in Sec. 3.3 is explicit, and the methodological criticism of Ref. [46] is well founded: substituting \dot{R}=0 into the action before varying is invalid because it is a differential equation, not an algebraic constraint. The Hamiltonian reduction for the adiabatic mode is also internally consistent, and the c=-1 ghost conclusion is a concrete falsifiable prediction at the linear level. However, the nonlinear confirmation is incomplete: the first-class nature of the reduced constraints is asserted rather than demonstrated, and the c=-1 branch introduces a square-root Hamiltonian with possible non-differentiable points. The central claim is defensible on the linear side, but the nonlinear section needs substantial completion before the paper can be accepted.

major comments (2)
  1. [Sec. 4, Eqs. (4.10)-(4.12)] The claim that the eight constraints {\pi_N, \pi_i, H_R, H_i} are all first-class is the load-bearing step for the nonlinear DOF count (2\times12-2\times8)/2 = 4, but the matter-sector Poisson brackets are not computed. In particular, {H_m[M], H_m[N]} in (4.12) is only asserted to equal D_m[h^{ij}(M\partial_jN-N\partial_jM)], with the calculation deferred to Ref. [36]. That reference treats the single-field mimetic model and does not cover the two-field matter Hamiltonian (4.7) or the c=-1 branch. Without this calculation, the first-class status of the reduced Hamiltonian constraint is unestablished, and the claimed nonlinear confirmation of the four-DOF result is not supported.
  2. [Sec. 4, Eq. (4.7)] The reduction that eliminates \lambda and \pi_\lambda treats the pair (\pi_\lambda, \Phi_2) as second-class, but the subsequent constraint algebra (4.10)-(4.12) is computed with ordinary Poisson brackets. In a system with second-class constraints, the reduced Hamiltonian (4.7) must be used with Dirac brackets; the naive Poisson bracket algebra is not sufficient to establish that H_R and H_i are first-class. The paper should either compute the Dirac brackets or demonstrate that they coincide with the Poisson brackets on the relevant variables. In addition, for c=-1 the argument of the square root in (4.7) vanishes on a locus (e.g., where the spatial kinetic terms cancel), making the Hamiltonian non-differentiable there; the paper does not discuss whether the constraint algebra closes on that locus, which is directly relevant to the c=-1 ghost claim in Sec. 5.
minor comments (4)
  1. [Sec. 3.3, Eqs. (3.25)-(3.28)] The linear Hamiltonian reduction for the adiabatic mode imposes two constraints, \pi_B\approx0 and \pi_R+2a^3k^2B\approx0, and concludes that this sector contributes one DOF. This is correct, but only because the two constraints are second-class (their Poisson bracket is -2a^3k^2, nonzero for k\neq0). Adding this one-line check would make the counting explicit and avoid a possible misreading as first-class constraints.
  2. [Sec. 3.3, below Eq. (3.17)] The notation "\delta \dot{s}^2" is ambiguous; it should be written as \delta\dot{s}^2 or \dot{\delta s}^2 to avoid confusion between the square of the time derivative and the time derivative of the squared perturbation.
  3. [Sec. 4, before Eq. (4.8)] The phrase "the rest four primary constraints" is imprecise: \pi_N is one constraint and \pi_i are three, and the secondary constraints H_R and H_i arise from requiring consistency of these primary constraints. Please rephrase for clarity.
  4. [Sec. 2, footnote 3 and Sec. 1, paragraph 1] There are minor typos: "orignial" in Sec. 1 should be "original," and "inlcude" in footnote 3 should be "include." Also, Eq. (2.12) appears to have a missing parenthesis in the derivative term.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the two-field DOF count and ghost conclusion are derived from the paper's own perturbation action and Hamiltonian constraint algebra.

full rationale

The central claims are derived from the paper's own equations. The linear perturbation analysis starts from the ADM action (3.15), obtains the quadratic Lagrangian (3.18)-(3.20), derives the constraint A = 0 from variation with respect to δλ, and then obtains the equations of motion (3.24). The DOF count is not a fitted or imported quantity; it follows from the canonical Hamiltonian (3.25)-(3.28) and from the full nonlinear constraint counting in Sec. 4. The disagreement with Ref. [46] is a substantive reanalysis: the authors argue that ẚ = 0 should not be substituted into the Lagrangian before varying with respect to R, and this is a calculational judgment rather than a circular definition. The c = -1 ghost conclusion is read directly from the sign in Eq. (3.19), L_δs ∝ c (δṁ^2 - k^2 δs^2/a^2), so it is an output of the computation, not an input. The only external result relied on is the single-field mimetic Hamiltonian constraint algebra of Ref. [36], which is cited for a computational shortcut in the bracket calculations; it is not by the present authors and does not already contain the two-field DOF count. Any potential weaknesses, such as the incompletely demonstrated matter-sector bracket in Eq. (4.12) or the non-global reality of the reduced Hamiltonian for c = -1, are correctness and rigor concerns, not circularity. Accordingly, no circular step is identified.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the equivalence between the metric-redefinition and Lagrange-multiplier formulations, the shift-symmetry restriction, the γ = 0 reduction, and the unproven constraint algebra for the two-field Hamiltonian. No new entities are introduced. The only discrete parameter is c, which controls the sign of one kinetic term.

free parameters (2)
  • c = ±1 (discrete sign choice)
    Relative sign of the two kinetic terms in the mimetic constraint X + cY = 1; not fitted to data, but the ghost-instability claim applies specifically to c = -1.
  • α, β (absorbed) = 1 and ±1 after field redefinition
    Constants in the conformal factor A = α X~ + β Y~; absorbed into the scalar fields by φ→φ/√|α|, ψ→ψ/√|β|, so no continuous numerical value remains.
assumptions (4)
  • domain assumption The Lagrange-multiplier action (2.9) is equivalent to the original non-invertible conformal transformation (2.6) with constraint X + cY = 1.
    The paper uses Eq. (2.9) for all subsequent analysis; the equivalence to the metric redefinition is stated, not proven in this paper.
  • domain assumption Shift symmetry on both fields, so α and β are constants.
    Assumed in Sec. 2 and inherited from Ref. [46]; the general non-shift-symmetric case is left for future work.
  • domain assumption Without loss of generality γ = 0 in A = α X~ + β Y~ + 2γ Z~; the cross term can be removed by a linear transformation of the field space.
    Stated in Sec. 2 without proof; if this diagonalization fails for the Lorentzian-signature case c = -1, the constraint and DOF count could be different.
  • domain assumption The Poisson bracket algebra of the reduced Hamiltonian constraints follows the single-field results of Ref. [36]; all commutators vanish on the constraint surface.
    Sec. 4 gives only the structure of the commutators and refers to Ref. [36] for details; the two-field square-root Hamiltonian is not fully checked for the c = -1 case.

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Pith. "Pith review of Two-field mimetic gravity revisited and Hamiltonian analysis." pith.science (2026). https://pith.science/paper/QXPOT64A

@misc{pith2026190901248,
  author       = {Pith},
  title        = {Pith review of: Two-field mimetic gravity revisited and Hamiltonian analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXPOT64A}},
  note         = {Machine review of arXiv:1909.01248}
}
read the original abstract

We revisit the two-field mimetic gravity model with shift symmetries recently proposed in the literature, especially the problems of degrees of freedom and stabilities. We first study the model at the linear cosmological perturbation level by quadratic Lagrangian and Hamiltonian formulations. We show that there are actually two (instead of one) scalar degrees of freedom in this model in addition to two tensor modes. We then push on the study to the full non-linear level in terms of the Hamiltonian analysis, and confirm our result from the linear perturbation theory. We also consider the case where the kinetic terms of the two mimetic scalar fields have opposite signs in the constraint equation. We point out that in this case the model always suffers from the ghost instability problem.

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