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REVIEW 4 major objections 4 minor 51 references

Long-wavelength amplification of an NLS conserved-charge violation in a one-dimensional Gross-Pitaevskii-Poisson field

T0 review · 4 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read In a 1D self-gravitating boson model, the sound-sector phase reduces conditionally to a KPZ-type equation.

desk verdict The full-text conditional KPZ reduction for 1D GPP is careful and honest, but the abstract is a different paper and the central closure step is assumed, not proved. read the letter →

arxiv 2603.27887 v2 pith:MOEXXFYA submitted 2026-03-29 cond-mat.stat-mech

Rin Takada This is my paper · ORCID
classification cond-mat.stat-mech
keywords Kardar-Parisi-ZhangequationGross-Pitaevskii-Poissonsoundmodescoarse-grainedphaseBurgersself-couplingJeansscaleself-gravitatingdarkmatterKPZuniversalityclass
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's central claim is that in a one-dimensional Gross-Pitaevskii–Poisson toy model of self-gravitating bosonic dark matter, the correct field to compare with the Kardar–Parisi–Zhang (KPZ) fixed point is not the microscopic phase but a branch-resolved coarse-grained phase built from the sound sector. Inside an explicitly defined wavenumber window—above the Jeans scale, where the linear modes are stable, and below the healing-scale cutoff, where quantum pressure is small—self-gravity acts as a weak deformation of local sound dynamics. A weakly nonlinear projection of the hydrodynamic equations onto the sound modes yields a nonvanishing same-chirality Burgers self-coupling of strength 3/2. Assuming one sound branch dominates and that the eliminated degrees of freedom close into local diffusion plus white noise, the dominant branch obeys a KPZ-type equation. The paper states plainly that this is a conditional reduction, not a proof of universal KPZ behavior for dark matter.

What carries the argument

The central mechanism is the exact linear sound-mode decomposition of the self-gravitating system. The modes φ_σ = (u + σ ω_sg(k)/ρ_0 k δρ)/2 diagonalize the linearized dynamics; inside the comparison window their frequencies reduce to the local sound speed c_s, so they become the familiar right/left sound combinations. Projecting the nonlinearities onto these modes produces a same-chirality Burgers coupling λ_{σσσ}=3/2. The conditional reduction then relies on (i) one-branch dominance, so interbranch terms are small, and (ii) a local Markov closure, parametrizing the residual noise as local-diffusion-plus-gradient-white-noise. The comparison window itself is defined operationally by the two

What would settle it

Run a direct numerical simulation of the 1D GPP model with one-branch-initialized data inside the comparison window, and measure the equal-time structure factor of the retained branch in the comoving frame. If the branch amplitude ratio |φ_-|/|φ_+| does not remain small, or if the height fluctuations do not scale as t^{1/3} with the predicted stationary/curved/flat one-point law, the conditional reduction fails. The cleanest single check is the stationary subclass: with Brownian initial height, the variance of Θ_σ should follow the KPZ t^{2/3} law with the predicted stationary KPZ scaling func

Watch

Extended reading notes

Core claim

Inside the comparison window k > k_J, G(k) << 1, k << k_micro, the exact linear modes of the 1D GPP model take the local sound form φ_σ = (u + σ c_s/ρ_0 δρ)/2. Projecting the leading quadratic nonlinearities onto these modes gives a Burgers self-coupling λ_{σσσ} = 3/2 for each branch. Under one-branch dominance and a local Markov closure, the coarse-grained phase Θ_σ whose slope is the dominant branch amplitude satisfies ∂_t Θ_σ = D_σ ∂²_X Θ_σ − (3/4)(∂_X Θ_σ)² − ξ_σ + v_∞σ + Δ_σ, a KPZ-type equation in the comoving frame. The author's conclusion is that this provides the proper comparison field and a controlled regime for an exact fixed-point test, not an unconditional universality statemen

Load-bearing premise

The reduction holds only if, after the subdominant sound branch and short-scale modes are eliminated, the remaining coupling closes into a local form with diffusion plus gradient white noise; this closure is assumed, not derived, and it is effectively the KPZ structure itself.

Editorial extensions

If this is right

  • The KPZ question for self-gravitating BECDM should be posed with the branch-resolved coarse-grained phase as the field, not the raw unwrapped phase.
  • The exact curved, flat, and stationary KPZ one-point data become testable benchmarks for the GPP toy model within the comparison window.
  • A failure of the reduction would show up as a violation of the KPZ t^{1/3} scaling or as a nonlocal effective noise, not as a failure of the sound-mode identification.
  • The paper's dictionary from microscopic initial data to KPZ subclasses gives a concrete recipe for setting up initial conditions in numerical tests.
  • Since the reduction is conditional, establishing a full fixed-point stability analysis remains the open step before any universality claim.

Reading between the lines

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

  • If the conditional reduction survives direct numerical simulation, the same sound-mode projection could be applied to other weakly perturbed integrable quantum gases to look for KPZ-like effective dynamics in their phase sectors.
  • The local Markov closure is the piece most likely to break; a test that measures the effective noise correlation function of the retained branch would either validate the white-noise ansatz or force a nonlocal extension.
  • The manuscript includes an abstract describing a separate result on long-wavelength amplification of an NLS conserved-charge violation (A₃ = O(L)); that claim is not developed in the body and would need its own evidence chain if it is intended as part of the same contribution.
  • A practical extension is to derive the effective diffusion constant D_σ and noise amplitude from the microscopic parameters via a controlled closure, rather than leaving them as free coefficients.
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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 / 4 minor

Summary. The paper studies a one-dimensional Gross-Pitaevskii-Poisson (GPP) toy model as a possible microscopic setting for comparing dark-matter wave dynamics with the (1+1)-dimensional KPZ universality class. The main analytic thread is: exact diagonalization of the linearized (δρ, u) system gives sound modes with Jeans-modified Bogoliubov dispersion; an operational comparison window k>kJ, G(k)≪1, k≪k_cut is defined; projecting the weakly nonlinear hydrodynamic equations onto the local sound sector yields a nonvanishing same-chirality Burgers self-coupling λσσσ=3/2; under one-branch dominance and a local Markov closure, the branch-resolved coarse-grained phase Θσ is claimed to reduce conditionally to a KPZ-type equation, Eq. (168). Section V then maps initial geometries to the curved, flat, and stationary KPZ subclasses and lists exact fixed-point benchmarks. The paper repeatedly and explicitly states that the result is conditional and does not establish KPZ universality for BECDM.

Significance. If the conditional reduction were to be made rigorous or numerically supported, the paper would provide a concrete bridge from a deterministic self-gravitating bosonic field model to exact KPZ fixed-point data, and would correctly identify the branch-resolved coarse-grained phase rather than the raw microscopic phase as the meaningful comparison field. The manuscript has real strengths: the exact linear sound-mode analysis, the operational definition of the comparison window with explicit existence bounds in Appendix B, and the clean, checkable projection algebra in Appendix C that yields the λ=3/2 self-coupling without fitting. The paper is also commendably explicit about what is not proved. However, the decisive step — the local Markov closure and the white-noise representation — is assumed rather than derived, and the effective parameters Dσ and Δσ are left undetermined. The significance is therefore that of a well-posed, partially justified proposal rather than an established reduction.

major comments (4)
  1. [§IV.E, §IV.F, §IV.G; Eqs. (149), (153), (168), (177)] The local Markov closure is the load-bearing step, and it is only postulated. Rσ in Eq. (148) is an exact deterministic remainder; replacing it by Dσ∂²Xφσ − ∂Xξσ in Eq. (149), and then assuming ξσ is space-time white noise in Eq. (177), imposes the KPZ form rather than deriving it. The GPP equations (38)–(39) are deterministic, so an effective stochastic forcing must arise from eliminated degrees of freedom through a specified mechanism (mixing, random initial data, timescale separation, etc.). No such mechanism or controlled limit is supplied. The paper explicitly labels this assumption unproved in §IV.G, but it is precisely the step that turns Burgers self-coupling into a stochastic KPZ equation. As it stands, Eq. (168) is a restatement of the closure assumption, not a consequence of the microscopic dynamics. I would need either a controlled derivation of the closure or a numerical tes
  2. [§IV.E, §IV.G(i); Eqs. (140)–(141), (146)] One-branch dominance is assumed throughout the window, but it is only an initial-time condition. The cross terms in Eqs. (140)–(141) generate the opposite branch from a dominant one at second order, so |φ−σ|≪|φσ| is not preserved automatically. The estimate in Eq. (146) is a schematic instantaneous ratio; it does not control the duration over which dominance persists. The KPZ fixed-point comparison in §V.C requires t→∞, and the paper gives no argument that dominance or the Burgers form survives on that time scale. This is not a minor caveat: without a decoupling estimate, the one-branch Burgers equation (148) cannot be integrated to the long-time benchmarks.
  3. [§IV.B, §V.C; comparison window versus KPZ scaling limit] The comparison window is a finite band: k>kJ with G(k)≪1 and k<k_cut. The exact KPZ scaling limits in Eq. (182) are taken with X̂=X/(κσt^{2/3}) fixed as t→∞, which probes wavevectors k∼t^{−2/3}→0. This is outside the operational window, whose lower edge kJ is positive. The paper does not explain how a model with a finite lower cutoff can produce the infrared KPZ fixed point; self-gravity is not screened below kJ, and no limit removing the Jeans instability is discussed. Thus the connection between the finite-band conditional reduction and the F2/F1/F0 benchmarks is not established. This gap concerns the very possibility of using the proposed dictionary as a fixed-point test.
  4. [§V.B, §V.E; Dσ and Δσ] The benchmark dictionary is incomplete in a way that limits its testability. The comparison amplitudes Γσ and κσ in Eq. (180) depend on Dσ and Δσ, but the paper never derives or constrains these effective parameters from the microscopic g, ρ0, κ, or the coarse-graining kernel. Only the initial Brownian amplitude A_initσ is specified in §V.E. Without Dσ and Δσ, one cannot compute the predicted Airy scaling or Tracy–Widom/Baik–Rains fluctuations from GPP initial data. If Dσ and Δσ are treated as free fitting parameters, the claim that the reduction can be tested against exact KPZ benchmarks is considerably weakened and should be stated as such.
minor comments (4)
  1. [Abstract vs. full text] The abstract at the head of the manuscript describes a different set of results: a conserved-charge Q3 violation, the identity Q̇3=κF3, the coefficient A3=Var(F3)/Var(Q3)=O(L), and Monte Carlo checks for 16≤L≤512. None of these quantities or simulations appears in the body, whose title is 'Conditional KPZ reduction in a one-dimensional model of bosonic dark matter'. The body does not define Q3 or F3. The abstract must be rewritten to describe the actual paper; as it stands it prevents the reader from knowing what is being claimed.
  2. [§V.D, around Eq. (219)] There is a typesetting error: 'Fw(s, t) = = = =⇒' contains repeated equals signs.
  3. [Reference [23]] The reference 'A. Borodin, P. L. Ferrari, M. Prähofer, et al.' is incomplete; it should list the full author list, presumably including H. Spohn, or use the standard journal citation.
  4. [§V.C, Table I] The variance of the curved subclass is labeled μ_c,2; the subscript notation is not introduced. A one-line definition would improve readability.

Circularity Check

1 steps flagged · score 4.0 of 10

Conditional KPZ reduction is partly self-definitional: the local Markov closure in Eq. (149) supplies the diffusive and gradient-noise terms that reappear as the claimed KPZ equation (168).

  1. self definitional [Sec. IV.E Eq. (149); Sec. IV.F Eq. (168); Sec. IV.G assumption (ii)]
    "By a local Markov closure we mean the following working assumption: after eliminating the subdominant branch and the short-scale degrees of freedom, the retained long-wavelength field is described by an effective stochastic equation that is local in X and t... We therefore parametrize the relevant local part of the remainder as Rσ =Dσ∂²xϕσ −∂xξσ +∆(ϕ)σ."

    The closure (149) is chosen to be a diffusion term plus a gradient noise. After one integration in Sec. IV.F, these are precisely the Dσ∂²XΘσ and −ξσ terms that make Eq. (168) a KPZ-type equation rather than an inviscid Burgers equation. The 3/4 nonlinear coefficient is genuinely obtained from the sound-mode projection, but the stochastic/diffusive part of the claimed reduction is the closure assumption restated. Section IV.G concedes this: '(ii) that the coarse-graining procedure closes into a local Markov form, with effective diffusion and noise.' Thus part of the derived equation is equal to an input assumption, though the paper honestly labels it as conditional.

full rationale

The paper contains no parameter fitting called prediction and no self-citation chain: the author's own prior work is not used as load-bearing evidence. The exact linear sound modes, Jeans scale, comparison-window bounds, and the same-chirality Burgers coupling λσσσ=3/2 are derived algebraically from the GPP equations. The abstract's Q3/F3 statement is likewise an exact identity plus an externally checkable Monte Carlo scaling claim, not a circular fit. The only near-circular element is the local Markov/white-noise closure: Eq. (149) and Eq. (177) assume the diffusive and white-noise terms that then appear in the final KPZ-type equation (168)/(173). Because the paper explicitly lists this as an unproved assumption (Sec. IV.G) and the nonlinear coefficient is independently derived, the reduction is not fully circular; it is a conditional statement whose stochastic content is assumed. Score 4 reflects this partial self-definitional structure without treating the honest labeling as deception.

Assumptions & free parameters 5 free parameters · 6 assumptions · 2 invented entities

The derived content is the Burgers coupling λ=3/2 in the sound sector; everything beyond it (diffusion, noise, drift, initial Brownian amplitude) is posited by the closure rather than computed. The comparison-window parameters δq, δg are hand-set. This is why the KPZ reduction is conditional.

free parameters (5)
  • Dσ (effective diffusion in reduced KPZ equation) = unspecified
    Coefficient of ∂²_Xφσ in the local Markov closure, Eq. (149); not derived from microscopic constants.
  • ξσ noise / Δσ amplitude = unspecified
    Effective stochastic forcing in Eq. (149); later assumed white with amplitude Δσ in Eq. (177).
  • v∞,σ (uniform drift) = unspecified
    Spatially uniform integration constant in Eq. (159); value not fixed by the model.
  • δq, δg (window smallness parameters) = positive, ≪1, unspecified
    Operationally define the comparison window in Sec. IV.B; chosen by hand.
  • A_init,σ (initial Brownian amplitude) = depends on initial two-point function
    Sets stationary-subclass normalization in Sec. V.E; not fixed by the dynamics.
assumptions (6)
  • ad hoc to paper One-branch dominance |φ−σ|≪|φσ| throughout the comparison window
    Assumed in Eq. (147); paper lists it as unproved in Sec. IV.G.
  • ad hoc to paper Local Markov closure: after eliminating subdominant branch and short scales, remainder is local with form Dσ∂²Xφσ−∂Xξσ
    Eq. (149); this step supplies the KPZ diffusive and noise terms.
  • ad hoc to paper Effective forcing is space-time white noise
    Eq. (177); assumed only to enter standard KPZ normalization, acknowledged in Sec. V.B.
  • domain assumption Coarse-graining kernel KΛ is normalized, local, and low-pass with sharp cutoff properties
    Sec. IV.A, Eqs. (80)-(81); detailed microscopic UV cutoff not modeled.
  • domain assumption 1D GPP toy model with periodic box and subtracted-mean Poisson term is a valid arena for BECDM/KPZ questions
    Sec. III.A; paper says it is a minimal model, not literal Newtonian cosmology.
  • standard math Exact KPZ one-point laws and processes (TW, Airy, Baik–Rains, Ferrari–Spohn g_sc) are correct as cited
    Section V relies on [23,24,26-30] without rederivation.
invented entities (2)
  • Branch-resolved coarse-grained phase Θσ
    purpose: KPZ height candidate defined by φσ=∂XΘσ
    Constructed from microscopic fields via projection and low-pass coarse graining; not independently observable.
  • Effective stochastic forcing ξσ
    purpose: Represents eliminated subdominant branch and short-scale degrees of freedom
    Its statistics are assumed white (Eq. 177); no falsifiable prediction tied to microscopic data.

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Cite this review

Pith. "Pith review of Long-wavelength amplification of an NLS conserved-charge violation in a one-dimensional Gross-Pitaevskii-Poisson field." pith.science (2026). https://pith.science/paper/MOEXXFYA

@misc{pith2026260327887,
  author       = {Pith},
  title        = {Pith review of: Long-wavelength amplification of an NLS conserved-charge violation in a one-dimensional Gross-Pitaevskii-Poisson field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MOEXXFYA}},
  note         = {Machine review of arXiv:2603.27887}
}
abstract

Can weak self-gravity be treated as a small, generic perturbation of the integrable one-dimensional cubic nonlinear Schr\"odinger equation? We test this question by following the first nontrivial local NLS charge, $Q_3$. In the Gross-Pitaevskii-Poisson model it obeys the exact identity $\dot Q_3=\kappa F_3$, where $F_3$ is the instantaneous gravitational force on the charge. We quantify the initial violation through the normalized mean-square displacement of $Q_3$. Its curvature coefficient, $A_3={\rm Var}(F_3)/{\rm Var}(Q_3)$, measures the root-mean-square initial speed of the charge in units of the charge's own ensemble width. For a UV-regulated, linearly stable Gaussian initial-data preparation, Wick contraction gives ${\rm Var}(Q_3)=O(L)$ and ${\rm Var}(F_3)=O(L^2)$, hence $A_3=O(L)$. Direct Monte Carlo evaluation of the full nonlinear observables confirms these powers for $16\leq L\leq 512$. The linear growth of $A_3$ means that admitting longer wavelengths makes nominally weak gravity increasingly effective at displacing the NLS charge. Along the Jeans-stable sequence $\kappa_L\propto L^{-2}$, the associated short-time curvature scale is $L^{3/2}$ instead of the $L^2$ benchmark of an infrared-uniform perturbation. This is an equal-time result, not a relaxation-rate or KPZ measurement. The local sound sector contains the Burgers self-coupling required for KPZ, but unscreened gravity also generates a relevant inverse-gradient coupling and, at fixed strength, a Jeans band. A dynamical connection to KPZ therefore requires a projected charge-relaxation calculation and a branch-resolved structure-factor test.

Figures

Figures reproduced from arXiv: 2603.27887 by the authors.

Figure 1
Figure 1. FIG. 1. Representative TASEP initial conditions. (a) step [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison of the cumulative distribution functions [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of the probability densities [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

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Reference graph

Works this paper leans on

51 extracted references · 6 linked inside Pith

  1. [1]

    Derivation of the hydrodynamic equations without self-gravity For completeness, we first summarize the elementary derivation of the hydrodynamic equations used in Sec. II. Let ψ(x, t) =R(x, t) eiθ(x,t), R := √ρ, u :=∂ xθ.(A1) Then ∂tψ= ∂tR+ iR ∂tθ eiθ,(A2) ∂xψ= ∂xR+ iRu eiθ,(A3) ∂2 xψ= ∂2 xR−Ru 2 + 2i(∂xR)u+ iR ∂xu eiθ.(A4) Substituting these expressions ...

  2. [2]

    Derivation of the background decomposition without self-gravity We next summarize the exact rewriting used in Sec. II. Around a uniform backgroundρ 0 >0, write ρ=ρ 0 +δρ, ˜h :=θ+gρ 0t, u=∂ x˜h.(A14) Substituting this into the continuity equation gives ∂t(ρ0 +δρ) +∂ x (ρ0 +δρ)∂ x˜h = 0,(A15) hence ∂tδρ+ρ 0∂2 x˜h=−∂ x δρ ∂x˜h =: Sρ.(A16) For the phase equat...

  3. [3]

    Derivation of the hydrodynamic equations with self-gravity We now summarize the corresponding derivation for the self-gravitating model of Sec. III. Starting from i∂tψ=− 1 2 ∂2 xψ+g|ψ| 2ψ+ Φψ, ∂ 2 xΦ =κ(ρ−¯ρ), (A21) we again write ψ(x, t) =R(x, t) eiθ(x,t), R= √ρ, u=∂ xθ.(A22) The derivative formulas for∂ tψand∂ 2 xψare the same as above. Substituting the...

  4. [4]

    Linearization and Jeans-modified dispersion with self-gravity Around a uniform backgroundρ 0 >0, we write ρ=ρ 0+δρ,Φ =δΦ, ˜θ :=θ+gρ 0t, u=∂ x ˜θ.(A28) 21 Since the background density is subtracted in the Poisson equation, the uniform background satisfies Φ 0 = 0. Keeping only terms linear inδρ,δΦ, andu, one obtains ∂tδρ+ρ 0∂2 x ˜θ= 0,(A29) ∂t ˜θ+g δρ+δΦ− ...

  5. [5]

    W eakly nonlinear reorganization with self-gravity Keeping the same linear part fixed and moving the nonlinear terms to the right-hand side, one may rewrite the self-gravitating equations as ∂tδρ+ρ 0∂2 x ˜θ=−∂ x δρ ∂x ˜θ =: Sρ,(A45) ∂t ˜θ+g δρ+δΦ− 1 4ρ0 ∂2 xδρ =− 1 2 ∂x ˜θ 2 + ∆Qnl[δρ] =: Sθ,(A46) ∂2 xδΦ−κδρ= 0.(A47) In Fourier space this becomes ∂tδρk −ρ...

  6. [6]

    The conditionG(k)⩽δ g and the lower edge of the window From Appendix A, the Jeans wavenumber is k2 J = 2 p c4s +ρ 0κ−c 2 s .(B6) On the other hand, the gravity measure is defined by G(k) := ρ0κ c2sk2 +k 4/4 .(B7) To require that self-gravity remain a weak deformation of the local sound sector, we impose G(k)⩽δ g,0< δ g ≪1.(B8) Since all quantities are pos...

  7. [7]

    Quantum-pressure bound and the upper edge of the window To require that the quantum-pressure correction re- main small relative to the local sound term, we define εq(k) := k2 4c2s (B17) and impose εq(k)⩽δ q,0< δ q ≪1.(B18) This is equivalent to k2 4c2s ⩽δ q ⇒k 2 ⩽4c 2 sδq.(B19) Hence, k2 ⩽k 2 q (δq), k 2 q (δq) := 4c2 sδq,(B20) and therefore kq(δq) = 2cs ...

  8. [8]

    (B14) and (B20), the general condition be- comes 2 r c4s + ρ0κ δg −c 2 s <min{k 2 micro,4c 2 sδq}.(B25)

    Operational window and the general existence condition The operational comparison window is kmin(δg)< k < kmax := min{kmicro, kq(δq)}.(B22) A window exists if and only if kmin(δg)< kmax.(B23) Since all relevant quantities are positive, this is equivalent to k2 min(δg)< k2 max.(B24) Using Eqs. (B14) and (B20), the general condition be- comes 2 r c4s + ρ0κ ...

Show all 51 references
  1. [9]

    (B24) becomes k2 min(δg)<4c 2 sδq.(B28) Substituting Eq

    Regime I: the upper edge is set by the local-sound condition Suppose that the sound bound is the active upper cut- off, namely kq(δq)⩽k micro.(B26) Then kmax =k q(δq) = 2cs p δq,(B27) 23 and Eq. (B24) becomes k2 min(δg)<4c 2 sδq.(B28) Substituting Eq. (B14), 2 r c4s + ρ0κ δg −...

  2. [10]

    (B23) becomes k2 min(δg)< k2 micro.(B39) Substituting Eq

    Regime II: the upper edge is set by the microscopic ultraviolet cutoff Suppose instead that the microscopic cutoff is the ac- tive upper bound, namely kmicro < kq(δq).(B37) Then kmax =k micro,(B38) and Eq. (B23) becomes k2 min(δg)< k2 micro.(B39) Substituting Eq. (B14), 2 r c4...

  3. [11]

    Using Eq

    Dimensionless form It is convenient to introduce the dimensionless gravity ratio µ := ρ0κ c4s = κ g2ρ0 ,(B45) where, the second equality follows fromc 2 s =gρ 0. Using Eq. (B45), we can rewrite Eq. (B6) as k2 J = 2 p c4s +ρ 0κ−c 2 s = 2c2 s r 1 + ρ0κ c4s −1 = 2c2 s p 1 +µ−1 .(...

  4. [12]

    (B1) or Eq

    Summary The comparison window is defined by the simultaneous requirements k⩾k min(δg), k⩽k q(δq), k⩽k micro,(B51) with k2 J = 2 p c4s +ρ 0κ−c 2 s ,(B52) k2 min(δg) = 2 r c4s + ρ0κ δg −c 2 s ,(B53) k2 q (δq) = 4c2 sδq.(B54) A nonempty window exists if and only if kmin(δg)<min{k...

  5. [13]

    Leading weakly nonlinear system in the comparison window We begin from the weakly nonlinear equations ∂tδρ+ρ 0∂xu=−∂ x(δρ u),(C2) ∂tu+g ∂xδρ+∂ xδΦ− 1 4ρ0 ∂3 xδρ =−u ∂xu+∂ x∆Qnl[δρ].(C3) Inside the comparison window, the local sound sector is characterized by εq(k)≪1, G(k)≪1.(C...

  6. [14]

    Change of variables to local sound modes We now define the local sound fields by ϕ± := 1 2 u± cs ρ0 δρ .(C9) Solving foruandδρ, we immediately obtain the inverse relations u=ϕ + +ϕ −, δρ= ρ0 cs ϕ+ −ϕ − .(C10)

  7. [15]

    (C10) into Eq

    Projection of the continuity equation Substituting Eq. (C10) into Eq. (C7), we find ∂t ρ0 cs (ϕ+ −ϕ −) +ρ 0∂x(ϕ+ +ϕ −) =−∂ x ρ0 cs (ϕ+ −ϕ −)(ϕ+ +ϕ −) .(C11) Multiplying through byc s/ρ0 yields ∂tϕ− −∂ tϕ+ −c s∂xϕ+ −c s∂xϕ− =∂ x(ϕ2 + −ϕ 2 −),(C12) or equivalently (∂t +c s∂x)ϕ+ ...

  8. [16]

    (C10) into Eq

    Projection of the Euler equation Next, substituting Eq. (C10) into Eq. (C8), we obtain ∂t(ϕ+ +ϕ −) + c2 s ρ0 ∂x ρ0 cs (ϕ+ −ϕ −) =−(ϕ + +ϕ −)(∂xϕ+ +∂ xϕ−).(C16) Since c2 s ρ0 ∂x ρ0 cs (ϕ+ −ϕ −) =c s∂x(ϕ+ −ϕ −),(C17) this becomes (∂t +c s∂x)ϕ+ + (∂t −c s∂x)ϕ− =−(ϕ + +ϕ −)(∂xϕ+ +...

  9. [17]

    (C15) and (C19)

    Isolation of the two chiral branches To isolate the equation forϕ +, we add Eqs. (C15) and (C19). Dividing by 2, we obtain ∂tϕ+ +c s∂xϕ+ =− 3 2 ϕ+∂xϕ+ − 1 2 ϕ+∂xϕ− − 1 2 ϕ−∂xϕ+ + 1 2 ϕ−∂xϕ−. (C20) To isolate the equation forϕ −, we subtract Eq. (C15) from Eq. (C19). Dividing b...

  10. [18]

    Reading off the coupling coefficients It is useful to write Eqs. (C20) and (C21) schematically as ∂tϕσ +σc s∂xϕσ =− X α,β=± λσαβ ϕα∂xϕβ.(C22) Then the equation forϕ + immediately gives λ+++ = 3 2 , λ ++− = 1 2 , λ+−+ = 1 2 , λ +−− =− 1 2 , (C23) while the equation forϕ − gives...

  11. [19]

    One-branch form and the Burgers equation Suppose now that one branch dominates: |ϕ−σ| ≪ |ϕσ|.(C27) Then Eqs. (C20) and (C21) reduce schematically to ∂tϕσ +σc s∂xϕσ + 3 2 ϕσ∂xϕσ =R σ,(C28) where the remainderR σ contains all subleading terms, namely Rσ =R cross σ +R q σ +R g σ,...

  12. [20]

    (C34) into Eq

    Integration to the KPZ-type phase equation Finally, define the branch-resolved coarse-grained phase by φσ =∂ X Θσ.(C34) Substituting Eq. (C34) into Eq. (C33), we obtain ∂X ∂tΘσ + 3 2 (∂X Θσ)(∂2 X Θσ) =D σ∂3 X Θσ −∂ X ξσ + ∆(φ) σ . (C35) Using (∂X Θσ)(∂2 X Θσ) =∂ X 1 2 (∂X Θσ)2...

  13. [21]

    Thus, the conditional KPZ reduction of Sec

    Summary The derivation presented here shows explicitly that: (i) the local sound variablesϕ ± diagonalize the leading linear sound sector; (ii) the weakly nonlinear projection yields nonvanish- ing same-chirality self-couplings λ+++ =λ −−− = 3 2 ; (C40) (iii) under one-branch ...

  14. [22]

    Bertone, D

    G. Bertone, D. Hooper, and J. Silk, Physics Reports 405, 279 (2005)

    2005

  15. [23]

    L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Phys. Rev. D95, 043541 (2017)

    2017

  16. [24]

    W. Hu, R. Barkana, and A. Gruzinov, Phys. Rev. Lett. 85, 1158 (2000)

    2000

  17. [25]

    P. J. E. Peebles, The Astrophysical Journal534, L127 (2000)

    2000

  18. [26]

    Su´ arez, V

    A. Su´ arez, V. H. Robles, and T. Matos, inAccelerated Cosmic Expansion, edited by C. Moreno Gonz´ alez, J. E. Madriz Aguilar, and L. M. Reyes Barrera (Springer, Cham, 2014) pp. 107–142

    2014

  19. [27]

    Schive, T

    H.-Y. Schive, T. Chiueh, and T. Broadhurst, Nature Physics10, 496 (2014)

    2014

  20. [28]

    D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett.121, 151301 (2018)

    2018

  21. [29]

    Chavanis, Phys

    P.-H. Chavanis, Phys. Rev. D84, 063518 (2011)

    2011

  22. [30]

    Dominguez, D

    A. Dominguez, D. Hochberg, J. M. Martin-Garcia, J. Perez-Mercader, and L. S. Schulman, Astron. Astro- phys.344, 27 (1999), arXiv:astro-ph/9901208

    1999 arXiv

  23. [31]

    Gaite and A

    J. Gaite and A. Dom ´ ınguez, Journal of Physics A: Math- ematical and Theoretical40, 6849 (2007)

    2007

  24. [32]

    Kulkarni, D

    M. Kulkarni, D. A. Huse, and H. Spohn, Phys. Rev. A 92, 043612 (2015)

    2015

  25. [33]

    Spohn, Journal of Statistical Physics154, 1191 (2014)

    H. Spohn, Journal of Statistical Physics154, 1191 (2014)

    2014

  26. [34]

    van Beijeren, Phys

    H. van Beijeren, Phys. Rev. Lett.108, 180601 (2012)

    2012

  27. [35]

    Shukla, Phys

    S. Shukla, Phys. Rev. Fluids10, 114601 (2025)

    2025

  28. [36]

    Carusotto and C

    I. Carusotto and C. Ciuti, Rev. Mod. Phys.85, 299 (2013)

    2013

  29. [37]

    Altman, L

    E. Altman, L. M. Sieberer, L. Chen, S. Diehl, and J. Toner, Phys. Rev. X5, 011017 (2015)

    2015

  30. [38]

    Fontaineet al., Nature608, 687 (2022)

    Q. Fontaineet al., Nature608, 687 (2022)

    2022

  31. [39]

    Kardar, G

    M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. 56, 889 (1986)

    1986

  32. [40]

    The kardar-parisi-zhang equation and uni- versality class,

    I. Corwin, “The kardar-parisi-zhang equation and uni- versality class,” (2011), arXiv:1106.1596 [math.PR]

    2011 arXiv

  33. [41]

    K. A. Takeuchi, Physica A: Statistical Mechanics and its Applications504, 77 (2018), lecture Notes of the 14th International Summer School on Fundamental Problems in Statistical Physics

    2018

  34. [42]

    K. A. Takeuchi,Physics of the Kardar-Parisi-Zhang Uni- versality Class, Kihon Hosoku kara Yomitoku Butsuri- gaku Saizensen, Vol. 38 (Kyoritsu Shuppan, Tokyo, 2026) [in Japanese]

    2026

  35. [43]

    Transversal fluctuations for in- creasing subsequences on the plane,

    K. Johansson, “Transversal fluctuations for in- creasing subsequences on the plane,” (1999), arXiv:math/9910146 [math.PR]

    1999 arXiv

  36. [44]

    Borodin, P

    A. Borodin, P. L. Ferrari, M. Pr¨ ahofer,et al., J. Stat. Phys.129, 1055 (2007)

    2007

  37. [45]

    Ferrari and H

    P. Ferrari and H. Spohn, Commun. Math. Phys.265, 1 (2006)

    2006

  38. [46]

    L. He, L. M. Sieberer, and S. Diehl, Phys. Rev. Lett. 118, 085301 (2017)

    2017

  39. [47]

    Johansson, Communications in Mathematical Physics 209, 437 (2000)

    K. Johansson, Communications in Mathematical Physics 209, 437 (2000)

    2000

  40. [48]

    Pr¨ ahofer and H

    M. Pr¨ ahofer and H. Spohn, Phys. Rev. Lett.84, 4882 (2000)

    2000

  41. [49]

    C. A. Tracy and H. Widom, Commun. Math. Phys.159, 151 (1994), arXiv:hep-th/9211141

    1994 arXiv

  42. [50]

    C. A. Tracy and H. Widom, Commun. Math. Phys.177, 727 (1996), arXiv:solv-int/9509007

    1996 arXiv

  43. [51]

    Limiting distributions for a polynuclear growth model with external sources,

    J. Baik and E. Rains, “Limiting distributions for a polynuclear growth model with external sources,” (2000), arXiv:math/0003130 [math.PR]

    2000 arXiv

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