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Universal lower bound on the axion decay constant from free streaming effects

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Axion dark matter made by abundance-enhancing mechanisms generically free-streams, and Lyman-alpha data force the decay constant above 10^14-10^15 GeV for masses around 10^-18 eV.

desk verdict A careful, honest paper that probably gets the right order of magnitude, but the 'universal' bounds rest on a transfer-function calibration that is not fully demonstrated. read the letter →

arxiv 2507.01956 v1 pith:TIAPDCI4 submitted 2025-07-02 astro-ph.CO hep-ph

Keisuke Harigaya , Wayne Hu , Rayne Liu , Huangyu Xiao This is my paper
classification astro-ph.COhep-ph PACS 95.35.+d98.80.-k14.80.Va
keywords axiondarkmattermisalignmentmechanismfreestreamingLyman-alphaforestisocurvatureperturbationsdecayconstantparametricresonancewarm
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 argues that any mechanism that boosts the axion relic abundance beyond standard misalignment generically creates nonzero-momentum axion modes. Those modes free-stream like warm dark matter and carry enhanced isocurvature perturbations, which small-scale structure observations constrain. Using the Lyman-$\alpha$ forest, the paper derives universal lower bounds: $f_a \gtrsim 10^{15}\,\mathrm{GeV}\,(10^{-18}\,\mathrm{eV}/m_a)$ for gradient-mode enhancement and $f_a \gtrsim 10^{14}\,\mathrm{GeV}\,(10^{-18}\,\mathrm{eV}/m_a)$ for delayed coherent oscillations. If the argument is correct, the entire enhanced-abundance region of the axion parameter space is closed for $m_a \gtrsim 10^{-18}\,\mathrm{eV}$, leaving only standard zero-mode misalignment at large $f_a$. The bounds are conservative by construction and packaged in scaling parameters so that improved Lyman-$\alpha$ and isocurvature data will tighten them.

What carries the argument

The central object is the free-streaming length $\lambda_{\mathrm{FS}}$ of the nonzero-momentum axions. For a monochromatic spectrum the suppression of the linear matter power spectrum is described by $T_{\mathrm{rel}}(k)=\sin(\lambda_{\mathrm{FS}} k)/(\lambda_{\mathrm{FS}} k)$; the paper turns the Lyman-$\alpha$ warm-dark-matter bound into $\lambda_{\mathrm{FS}}<0.084\,\mathrm{Mpc}$ through a $k_{3/4}$ matching rule, then rewrites $\lambda_{\mathrm{FS}}$ as a condition on the temperature $T_{\mathrm{GM}}$ at which the axions become nonrelativistic. That temperature feeds the abundance formula and produces the lower bound on $f_a$. The parameter $\alpha_{\mathrm{FS}}$ brackets how much a broad, nonthermal momentum spectrum suppresses structure relative to the monochromatic case, and for delayed oscillations the same logic is applied to axions produced by parametric resonance, whose growth rate follows from a Mathieu equation.

What would settle it

Run a full numerical simulation of a concrete enhancement mechanism, such as kinetic misalignment or parametric resonance, with $f_a$ below the quoted bound and $m_a\gtrsim 10^{-18}\,\mathrm{eV}$, and compute the linear matter transfer function from the actual axion momentum distribution; if its $k_{3/4}$ scale shows no suppression below the Lyman-$\alpha$ sensitivity, the universal bound is falsified.

Watch

Extended reading notes

Core claim

The paper claims that nonzero-momentum axions are not a model-dependent accident but a generic byproduct of abundance enhancement, and that their free streaming yields a model-independent lower bound on the axion decay constant. For gradient modes the bound is $f_{\mathrm{GM}} > 2.36\times 10^{15}\,\mathrm{GeV}\,(\alpha_{\mathrm{FS}}^{3/2}/\alpha_{\mathrm{GM}}^{1/2})(10^{-18}\,\mathrm{eV}/m_a)$; for delayed coherent oscillations it is $f_{\mathrm{DO}} > 5.0\times 10^{13}\,\mathrm{GeV}\,(1/\alpha_{\mathrm{DO}}^{3/2})(10^{-18}\,\mathrm{eV}/m_a)$, with resonance and non-resonance branches given by Eqs. (28)-(29). The argument converts the free-streaming scale bound $\lambda_{\mathrm{FS}}<0.084\,\mathrm{Mpc}$ into a temperature bound $T_{\mathrm{GM}}>15.5\,\alpha_{\mathrm{FS}}\,\mathrm{keV}$ and then through the abundance formula into a bound on $f_a$. An independent isocurvature bound gives $m_a>8.0\times10^{-19}\,\mathrm{eV}\,\alpha_{\mathrm{FS}}\alpha_{\mathrm{iso}}^{1/3}$, closing the low-mass side for gradient mechanisms.

Load-bearing premise

The entire argument rests on treating the Lyman-$\alpha$ forest limit on $3.1\,\mathrm{keV}$ warm dark matter as a universal cap on how far axion dark matter may free-stream, via a specific $k_{3/4}$ power-matching rule, and on assuming the broad momentum spectra produced by enhancement mechanisms suppress structure no less than the white-noise-plus-power-law family used to set $\alpha_{\mathrm{FS}}$.

Editorial extensions

If this is right

  • For $m_a \gtrsim 10^{-18}$ eV, gradient-energy enhancement of the axion abundance is allowed only for $f_a$ above roughly $10^{15}$ GeV $(10^{-18}\,\mathrm{eV}/m_a)$, pushing axion couplings to Standard Model particles to small values.
  • Delayed coherent oscillation models face the same structure with $f_a \gtrsim 10^{14}$ GeV $(10^{-18}\,\mathrm{eV}/m_a)$, with the resonance and non-resonance branches given by Eqs. (28)-(29).
  • The standard zero-mode misalignment line at large $f_a$ remains the only open path for axion dark matter in this mass range, because it produces no nonzero-momentum modes and no free-streaming suppression.
  • For gradient mechanisms, the isocurvature bound requires $m_a > 8.0\times 10^{-19}\,\mathrm{eV}\,\alpha_{\mathrm{FS}}\alpha_{\mathrm{iso}}^{1/3}$, closing the ultralight end of the enhanced-abundance window.
  • The scaling parameters $\alpha_{\mathrm{FS}}$, $\alpha_{\mathrm{GM}}$, $\alpha_{\mathrm{iso}}$, and $\alpha_{\mathrm{DO}}$ let future Lyman-alpha and isocurvature observations recalibrate these bounds without changing the argument.

Reading between the lines

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

  • The same free-streaming and isocurvature logic should apply to any ultralight scalar whose relic abundance is boosted by gradient or trapped configurations, not only axions, since it is the momentum distribution rather than the particle identity that drives the suppression.
  • A direct detection of axion dark matter with $f_a$ below these bounds and $m_a \gtrsim 10^{-18}$ eV would require a production mechanism outside the generic gradient/trapping class considered here, thereby sharpening the interpretation of direct-detection searches.
  • The machinery can be inverted: measuring the scale at which the matter power spectrum is suppressed could reconstruct the characteristic comoving momentum and production temperature of axion dark matter, turning the exclusion into a diagnostic.
  • The paper notes an adiabatically decreasing axion mass as a possible loophole; deciding whether that scenario is natural and whether it survives the isocurvature constraints would be a concrete next step.
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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 argues that any mechanism that enhances the axion relic abundance relative to standard vacuum misalignment generically produces nonzero-momentum axion modes. Those modes behave as warm dark matter and generate isocurvature perturbations, leading to observationally testable consequences. Using the Lyman-alpha forest bound on warm dark matter (m_X > 3.1 keV from Ref. [44]) converted to a free-streaming scale via the k_{3/4} matching procedure, the authors derive lower bounds on the axion decay constant f_a for two classes of enhancement: gradient-mode mechanisms (Eq. (14): f_GM > 2.36×10^15 GeV α_FS^{3/2}/α_GM^{1/2} (10^{-18} eV/m_a)) and delayed coherent oscillations (Eqs. (23), (28), (29)). They also derive a combined isocurvature/free-streaming lower bound on m_a (Eq. (18): m_a > 8.0×10^{-19} eV α_FS α_iso^{1/3}). The bounds are expressed in terms of scaling parameters (α_FS, α_GM, α_iso, α_DO, α_PR, θ_i) that can be calibrated in concrete models. The internal arithmetic is consistent: the λ_FS < 0.084 Mpc conversion, the T_GM > 15.5α_FS keV bound, and the resulting f_a bounds all check out under the stated assumptions.

Significance. If the results hold, they would exclude a large portion of the axion parameter space for m_a ≳ 10^{-18} eV, specifically the regions where enhanced abundance mechanisms such as kinetic misalignment, acoustic misalignment, and trapping/delayed oscillation operate. This would be a powerful, complementary constraint to direct searches and astrophysical probes. A notable strength is that the authors provide explicit scaling parameters so that the bounds can be updated with improved observations and calibrated for specific models. The paper is also careful to state its assumptions (e.g., the conservative choice of quantum seeds for parametric resonance, the treatment of g_* at low temperatures). However, the title and abstract claim 'universal' lower bounds, and the actual numerical reach depends on ranges of the scaling parameters that are not derived from first principles in this manuscript.

major comments (4)
  1. [§III.B, Eqs. (11)-(14)] The universal character of the gradient-mode bound rests on the assertion that the white-noise-plus-power-law spectra with n=4,∞ bracket the actual spectra of kinetic misalignment, acoustic misalignment, and parametric resonance, giving 0.8<α_FS<4. This is not demonstrated in the paper; it is attributed to Ref. [34], which is by the same group. Since Eq. (14) scales as α_FS^{3/2}, an unmodeled spectrum with a broader low-momentum core or a different high-momentum tail could move the bound outside the quoted factor range. Please provide explicit examples, simulations, or analytic arguments showing that the listed mechanisms fall within this envelope, or soften the 'universal' claim accordingly.
  2. [§IV.A, Eq. (22)] The parameter α_DO is introduced as O(1), but the fiducial value α_DO=3 and the 'very conservative' range 1≤α_DO≤100 are not derived from the trapping mechanisms. Because Eq. (23) scales as α_DO^{-3/2}, the abstract's headline bound f_a ≳ 10^{14} GeV (10^{-18} eV/m_a) corresponds to the fiducial choice, not to the conservative edge of the stated range. The paper should either derive a physically motivated range for α_DO or explicitly state that the delayed-oscillation bound is illustrative rather than universal.
  3. [§IV.B, Eqs. (24)-(26)] The parametric resonance analysis is linear: it neglects cosmic expansion on the oscillation time scale, assumes a sinusoidal zero mode, and uses quantum fluctuation seeds. For the delayed-oscillation mechanism with θ_i near π, the zero-mode oscillation is anharmonic and the small-amplitude limit may not apply. Since the with-resonance bound Eq. (28) provides the strongest delayed-oscillation constraint, this assumption is load-bearing. Please justify the linear approximation over the parameter range where Eq. (28) is used, or provide a non-linear estimate showing that the result is robust.
  4. [§III.B, Eq. (11)] The conversion of the Lyman-alpha bound m_X > 3.1 keV into λ_FS < 0.084 Mpc relies on the k_{3/4} matching prescription of Ref. [38], which was validated for thermal WDM and zero-mode axions. The paper then applies this calibration to broad non-thermal spectra, absorbing the shape uncertainty into α_FS. The correctness of this application is not demonstrated. A direct test of the k_{3/4} procedure on the actual non-thermal spectra (e.g., from lattice simulations of kinetic misalignment) would substantially strengthen the universality claim.
minor comments (6)
  1. [Abstract] The notation 'fa ≳ 10^{15} GeV (10^{-18}eV/ma)' is dimensionally ambiguous; please write 'f_a ≥ 10^{15} GeV × (10^{-18} eV / m_a)' or use equivalent wording.
  2. [§III.B, text after Eq. (13)] The sentence 'For the range of power laws n=4,∞, 0.8<α_FS<4 approximately' is ambiguous: it should specify which power law corresponds to which end of the α_FS range.
  3. [Figure 2] The caption does not explain the meaning of the shaded region or the dashed lines (e.g., f_MM^{-1}, f_MT^{-1}); please expand the caption to make the figure self-contained.
  4. [References] Several references lack complete publication data, e.g., Refs. [8], [11], [19], [24], [25], [34], [35], [36], [37], [40], [41], [42], [45], [49]; please complete the journal, volume, page, and year information.
  5. [§III.C, Eq. (15)] The symbols A_s and k_CMB are used without explicit definitions; please state that these are the Planck normalization A_s=2.054×10^{-9} at k_CMB=0.05 Mpc^{-1}.
  6. [§IV.B, Eq. (24)] The subscript 'DO' on a_DO is inconsistent in a few places (sometimes written as a_{DO}, sometimes as a_{DO} with different spacing); please ensure uniform notation.

Circularity Check

2 steps flagged · score 4.0 of 10

The free-streaming bounds are anchored in external Lyman-alpha data (mX > 3.1 keV and the k3/4 matching prescription), so the core derivation is not circular; the principal self-citation burden is the alpha_FS spectral envelope imported from the authors' own Ref. [34], which sets the numerical scale of every headline bound.

  1. ansatz smuggled in via citation [Section III.B (Free Streaming Bound), paragraph after Eq. (13); envelope enters Eqs. (14), (18), (27)-(29)]
    "To account for different spectra of momenta we can introduce a scaling parameter αFS and finally convert to temperature TGM > 15.5αFS keV. Following Ref. [34], we consider a range of spectra defined by white noise field fluctuations with a constant power spectrum for q < q∗ and a q−n power law fall off for q > q∗. For the range of power laws n = 4, ∞, 0.8 < αFS < 4 approximately."

    Every headline free-streaming bound — fGM (Eq. 14), the delayed-oscillation bounds (Eqs. 27-29), and the mass bound (Eq. 18) — scales with αFS (or inherits it through TDO). The claimed 'universal, model-independent' result therefore rests on the assertion that the momentum spectra of all enhancement mechanisms (kinetic misalignment, acoustic misalignment, parametric resonance, domain walls) are bracketed by the white-noise-plus-q^{-n} family with αFS in [0.8,4]. That family is introduced in this paper by definition ('we consider a range of spectra defined by...'), and the numerical envelope is attributed to Ref. [34], which is the authors' own prior work (R. Liu, W. Hu, H. Xiao, PRD 111, 023535).

  2. self citation load bearing [Section III.B, Eqs. (6)-(8) and the k3/4-matching paragraph leading to Eq. (11)]
    "Integrating the free streaming through to the matter dominated epoch, this scale is [34], ... For a monochromatic spectrum of a single momentum q∗, the suppression of the linear matter power spectrum due to free streaming can be characterized by a relative transfer function [33, 34] Trel(k) = sin(λFSk)/(λFSk). ... We adopt the prescription from Ref. [38] ... by matching the scale k3/4 ... This prescription matches direct tests of WDM mass constraints and zero mode axion mass constraints."

    The externally anchored WDM mass bound (mX > 3.1 keV, Ref. [44]) reaches the axion temperature bound TGM > 15.5 keV — the input for every headline fa bound — only through the monochromatic transfer function Trel(k) = sin(λFS k)/(λFS k) and its k3/4 matching. The transfer-function machinery is jointly cited to the independent Ref. [33] and the same-author Ref. [34], so the self-citation is partly corroborated; however, the claim that the k3/4 prescription is validated for 'zero mode axion mass constraints' (a case not present in the external Refs. [38,44]) and the specific λFS normalization used in Eq. (11) trace to the authors' own line of work.

full rationale

The derivation chain producing the headline numbers is anchored in external observation: mX > 3.1 keV comes from Ref. [44] (Villasenor et al., no author overlap), the k3/4-matching prescription from Ref. [38] (Irsic et al., no author overlap), and the isocurvature normalization fiso < 3e-3 (Ref. [40]) is also observationally derived from Lyman-alpha data. The axion abundance formulas (Eqs. (2)-(5), (20)) and the parametric-resonance analysis of Eqs. (24)-(26) are derived in the paper from the cosine potential, not imported. The same-author citations (Refs. [34], [40], [37]) supply the transfer-function shape, the alpha_FS envelope, and part of the isocurvature modeling. Of these, the alpha_FS envelope is the most load-bearing: it is attributed to Ref. [34] (three of the four present authors), the spectra family is introduced by ansatz rather than derived from the production mechanisms, and Eq. (14) scales as alpha_FS^{3/2}, so the numerical 'universal' bound inherits an unverified same-author assumption. Mitigating factors: the envelope is openly parameterized rather than fitted to the target data; conservative fiducial choices (alpha_FS = 1) are stated; the paper explicitly reports the resulting factor uncertainties (0.7-8); the monochromatic transfer function is jointly sourced to the independent Ref. [33]; the k1/2-matching check shows numerical robustness; and footnote 5 openly flags a loophole (adiabatic mass decrease) that can evade the bounds. No claim reduces by construction to a fitted parameter renamed as a prediction, and the central fa bounds are genuinely determined by external WDM and isocurvature limits. The score of 4 reflects the load-bearing same-author alpha_FS envelope and the self-sourced zero-mode validation of k3/4 matching, without treating the well-anchored external chain as circular.

Assumptions & free parameters 6 free parameters · 9 assumptions · 0 invented entities

The central claim rests on the axion cosine-potential Lagrangian with a mass that is constant in Secs. II-IV (Sec. V relaxes this), on the calibration of the Lyman-alpha WDM limit into a free-streaming scale through a transfer-function shape from Refs. [33,34] (same group) with the Ref. [38] matching prescription, and on a white-noise ansatz for the isocurvature spectrum normalized by αiso. Five O(1) scaling parameters (αFS, αGM, αiso, αDO, αPR) with author-chosen fiducial values, plus an optimized θi in the delayed-oscillation case, carry factors of 0.3-10 in the final bounds. The parametric-resonance efficiency is estimated from a linearized Mathieu treatment with quantum-fluctuation seeds. No new particles, forces, or dimensions are introduced.

free parameters (6)
  • αFS = 1 (fiducial); range 0.8-4
    Spectral shape factor for free streaming in Eq. (13); converts the monochromatic sin(λFS k)/λFS k result to white-noise spectra with a q^-n falloff (Refs. [33,34]). The fiducial value shifts the fa bounds by factors 0.7-8 across the stated range.
  • αGM = 1 (fiducial); ≤1 by construction
    Ratio of the actual axion number density to na = ma fa²/2 at the nonrelativistic transition, Eq. (5). Acoustic misalignment can saturate αGM ≈ 1; other mechanisms give smaller values, strengthening the bound as αGM^(1/2).
  • αiso = 1 (fiducial); range 0.1-100
    White-noise normalization of the isocurvature power spectrum, Eq. (15), related to δ²rms. Load-bearing for the mass bound (18) and the upper fa bound (19); not derived from the microphysics of any listed mechanism.
  • αDO = 3 (fiducial); range 1-100
    O(1) factor in the missing-dark-matter horizon constraint, Eqs. (21)-(22), giving TDO > 1.19 keV/αDO. The stated range changes the delayed-oscillation bounds by factors 0.5-10.
  • αPR = 1 (fiducial)
    Resonance band factor in q* ≈ αPR θi ma aDO/2, Eq. (25); larger if resonantly produced axions upscatter. Strengthens the resonance free-streaming bound as αPR^-1, Eq. (28).
  • θi = Optimized to evade free streaming; e.g. αPR θi ~ 0.1-1 in Eqs. (30)-(31)
    Initial misalignment angle. In the delayed-oscillation analysis it is chosen at 'the most optimal value to evade the free-streaming bound' (Sec. IV B) to keep the bounds model-independent; specific models predict it (Refs. [22,23]). Also sets the standard line fMM via αMM(θi).
assumptions (9)
  • domain assumption Axion field described by the cosine-potential Lagrangian L = (1/2)(∂φ)^2 - ma² fa² (1 - cos(φ/fa)), with the axion mass treated as constant in Secs. II-IV
    Eq. (1). The cosine shape sets up the misalignment and Mathieu equations; the constant-mass assumption is relaxed only schematically in Sec. V.
  • domain assumption Monochromatic nonzero axion modes free-stream with transfer function Trel(k) = sin(λFS k)/(λFS k); the Lyman-alpha WDM bound is converted into λFS < 0.084 Mpc via the k3/4 matching prescription of Ref. [38]
    Sec. III B, Eqs. (6)-(11). The transfer function comes from Refs. [33,34] (two of the present authors); the matching prescription from Ref. [38] is validated there against WDM mass constraints and zero-mode axion mass constraints.
  • domain assumption The isocurvature power spectrum of gradient axion modes is white noise for k ≪ q*, normalized by f²iso = (αiso/As)(kCMB/(aGM ma))³
    Eq. (15), following Ref. [40] (an author overlap). αiso ≈ δ²rms with fiducial 1 and range 0.1-100; the white-noise extrapolation to kCMB is asserted by causality and not derived from any specific mechanism.
  • ad hoc to paper Parametric resonance in the delayed-oscillation case is treated linearly: sinusoidal zero mode, no cosmic expansion, Mathieu growth rate Γ ≈ θi² ma/16, with fluctuation seeds given by quantum fluctuations
    Eqs. (24)-(26). The quantum-seed assumption sets the ln(fa/ma) factor in the resonance condition (26), which controls the delayed-oscillation bounds (28)-(29); no numerical validation is provided.
  • ad hoc to paper Dark matter absent for T > TDO produces a step in the matter power spectrum constrained by the Lyman-alpha k3/4 scale, giving TDO > 1.19 keV/αDO
    Eqs. (21)-(22). The relation k3/4 > aDO H(TDO) αDO is asserted as 'generically expect'; the quantitative coefficient 1.19 keV and the αDO = O(1) factor are not derived or cited.
  • domain assumption Free-streaming and isocurvature bounds are treated as independent constraints
    Footnote 2. The paper notes that when the two bounds are comparable, the joint constraint changes shape and is not computed; the presented exclusion assumes one bound dominates.
  • domain assumption Resonantly produced axions at q* ≈ (αPR/2) θi ma aDO are nonrelativistic at aDO
    Footnote 3. The paper states relativistic axions would only strengthen the bound, but the nonrelativistic assumption sets Eq. (27) and hence the delayed-oscillation free-streaming bounds.
  • domain assumption The external WDM bound mX > 3.1 keV (Ref. [44]) with ΩX h² ≈ 0.12 applies to the axion free-streaming case through the matching prescription
    Sec. III B. This external Lyman-alpha constraint is the observational anchor of the entire free-streaming argument; the translated value λFS < 0.084 Mpc feeds every headline bound.
  • domain assumption Entropic and energy degrees of freedom satisfy g⋆S(TGM) ≈ g⋆S(T0) for the keV-scale temperatures involved in the bounds
    Used in evaluating Eqs. (5) and (16)-(17); the paper states this is appropriate for evaluating the lower bound (Sec. III B-C).

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Pith. "Pith review of Universal lower bound on the axion decay constant from free streaming effects." pith.science (2026). https://pith.science/paper/TIAPDCI4

@misc{pith2026250701956,
  author       = {Pith},
  title        = {Pith review of: Universal lower bound on the axion decay constant from free streaming effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TIAPDCI4}},
  note         = {Machine review of arXiv:2507.01956}
}
abstract

We show that enhancement of the axion relic abundance compared to the standard misalignment contribution generically leads to the production of nonzero momentum axion modes, resulting in warm dark matter behavior and enhanced isocurvature perturbations. It leads to universal constraints on the axion parameter space that are independent of detailed model assumptions and cosmological history. For models enhancing relic abundance with gradient axion modes, observations of the Lyman-$\alpha$ forest impose a lower bound on the axion decay constant, $f_a \gtrsim 10^{15} {\rm GeV}\,(10^{-18}{\rm eV}/m_a)$, from the free-streaming effect. For models relying on the delay of coherent axion oscillations, we obtain a slightly weaker bound, $f_a \gtrsim 10^{14} {\rm GeV}\,(10^{-18}{\rm eV}/m_a)$. We make relatively conservative choices to establish these universal bounds but also provide scaling parameters that can be calibrated for stronger constraints in concrete models and updated as observations improve.

Figures

Figures reproduced from arXiv: 2507.01956 by the authors.

Figure 1
Figure 1. FIG. 1. In the standard misalignment mechanism, the axion [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Constraints on [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Bounds on gradient modes and delayed oscillations [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.