Coherent and Stochastic Axion Dark Matter from Thermal Relaxation

Ahmadjon Abdujabbarov; Arzu Cilli; Ertan G\"udekli; Farruh Atamurotov; Phongpichit Channuie; Shahid Hussain Gurmani

REVIEW 3 major objections 5 minor 78 references

Coherent and Stochastic Axion Dark Matter from Thermal Relaxation

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A single momentum-dependent optical depth partitions every axion mode into coherent condensate and thermal occupation, unifying misalignment and stochastic production.

desk verdict A serious linear-response framework for axion thermal production with a real gap between the nonlinear field equation and the per-mode identity; worth refereeing, but the central claim needs more support. read the letter →

arxiv 2608.05538 v1 pith:YGWNNY6R submitted 2026-08-06 hep-ph gr-qchep-th

Shahid Hussain Gurmani , Arzu Cilli , Phongpichit Channuie , Ahmadjon Abdujabbarov , Farruh Atamurotov , Ertan Güdekli This is my paper · ORCID

classification hep-phgr-qchep-th

keywords axiondarkmatterthermalrelaxationstochasticproductionmisalignmentopticaldepthfree-streaminglengthisocurvaturehiddengaugebath

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 an axion in a hot plasma does not simply roll to a minimum; the bath attenuates the coherent field and populates stochastic momentum modes, and a single momentum-dependent optical depth $\tau_x$ governs both. It derives the identity $n_x/n_B(x) + |A_x/A_{x,i}|^2 = 1$, partitioning each mode's power between coherent condensate and Bose occupation before any momentum integral. Because the bath response falls with momentum, the stochastic population is colder than Bose equilibrium, with lower mean momentum and shorter free-streaming length. In a weakly coupled realization the stochastic population yields the observed dark matter abundance while the coherent remnant is tiny, suppressing inherited isocurvature and confining production correlations to microscopic scales. A sympathetic reader would care because the result unifies misalignment and stochastic production into one relation and changes predictions for axion relic composition, velocity, and structure.

What carries the argument

The central object is the mode-by-mode optical depth $\tau_x = \int dN\, \gamma_x(N)$, with $\gamma_x = (\Upsilon_h/H)\, S_s(x/(c_m g_h^2))\, S_t(x/(c_\omega g_h^4))$, where $x=p/T_h$ is the comoving momentum of an axion mode and $S_s$, $S_t$ are the spatial and temporal response functions of the hidden Yang-Mills bath. This optical depth enters a linear Langevin equation for each mode; its homogeneous solution gives coherent decay $e^{-\tau_x/2}$ and its noise source gives Bose occupation, so Eq. (33) is the resulting unit partition between coherent power and stochastic occupation. The paper also constructs an amplitude decay rate $\Gamma_A$ and a nonlinear displacement bound $I(N)$ for the homogeneous angle, but the argument's load-bearing identity is the optical-depth partition.

What would settle it

Run a lattice or analytic computation of the sine-Gordon field in a thermal bath with finite correlation time and compare per-mode $|A_x/A_{x,i}|^2$ with $1 - n_x/n_B(x)$; any measurable deviation from unity at one momentum disproves the partition. Alternatively, in a controlled weak-coupling regime, extract the damping rate from the occupation growth and compare it with the amplitude decay rate.

Watch

Extended reading notes

Core claim

The central claim is that thermal relaxation of an axion field is not a single number, an abundance, but a momentum-resolved partition. The optical depth $\tau_x = \int dN \, \gamma_x(N)$ sets simultaneously the survival of the coherent mode, $|A_x/A_{x,i}|^2 = e^{-\tau_x}$, and the growth of stochastic occupation, $n_x = (1-e^{-\tau_x}) n_B(x)$; hence $n_x/n_B(x) + |A_x/A_{x,i}|^2 = 1$. Because the thermal bath has finite spatial and temporal correlation scales, the optical depth decreases with momentum, so high-momentum modes are suppressed; the relic is therefore subthermal in every increasing kinematic moment, in particular mean momentum and free-streaming length. In the benchmark, the stochastic population gives the observed dark matter abundance with $m_a \approx 2.8 \times 10^{2}$ keV, a coherent fraction $f_{\rm coh} = 1.2 \times 10^{-5}$, and a free-streaming wavenumber $k_{\rm fs} \approx 3 \times 10^{4}$ Mpc$^{-1}$, so dark matter is predominantly stochastic rather than misaligned.

Load-bearing premise

The load-bearing premise is that each momentum mode's coherent amplitude decays at the same optical depth that its Bose occupation grows, even though the real axion obeys a nonlinear sine equation whose damping is derived separately; if nonlinear couplings break that equality, the partition identity fails.

Editorial extensions

If this is right

Standard misalignment is the $\tau_x = 0$ limit of the same relation; partial relaxation and full stochastic production are interpolations, not separate mechanisms.
The relic can be dominated by stochastic modes with a tiny coherent fraction ($f_{\rm coh} \approx 1.2 \times 10^{-5}$), which suppresses inherited coherent isocurvature by $f_{\rm coh}^2$.
The stochastic spectrum is colder than Bose equilibrium: the mean momentum ratio is about 0.12--0.13, the free-streaming length is $\sim 2 \times 10^{-4}$ Mpc, and $k_{\rm fs} \sim 3 \times 10^{4}$ Mpc$^{-1}$, so the population's velocity support is narrow.
Production correlations are confined to microscopic scales ($\sim 10^{24}$ Mpc$^{-1}$), far above $k_{\rm fs}$, so the benchmark leaves no axion minicluster seeds despite local non-Gaussianity.
Hidden radiation entropy modes are constrained: the yield response $R_\xi \approx 4$ translates Planck isocurvature bounds into a bound $\delta \xi_h \lesssim 6.5 \times 10^{-6}$.

Reading between the lines

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

Editorial inference: the same partition identity should apply to any light bosonic scalar coupled to a thermal bath, so the framework can be carried over to dark photons, dilatons, or other axionlike fields without re-deriving the core relation.
Editorial inference: because the produced spectrum is subthermal, translating Lyman-alpha free-streaming constraints into an effective thermal mass would overestimate the allowed mass; the direct velocity moment is the correct observable.
Editorial inference: the sharpest quantitative test is a first-principles (lattice or semiclassical) evaluation of the sine-Gordon field's two-point function in a finite-correlation bath, checking whether $|A_x|^2 + n_x/n_B(x) = 1$ survives nonlinear mode mixing.
Editorial inference: the relation could also be used as a model-building target for warm-inflation or reheating scenarios, where the same optical depth would link the inflaton's condensate decay to its stochastic fluctuation spectrum.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a serious axion cosmology paper that deserves a referee, but the central claim is not yet supported at the level the abstract implies. The unit-sum identity (33) is a correct statement for a linear Langevin mode, and the momentum-ordering inequality (35) is a nice result. The problem is that the axion field is nonlinear (Eq. 17), and no argument is given that the linear per-mode description applies to it in the regime used. The stress-test note has this right.

What is actually new: the momentum-resolved optical depth with finite spatial and temporal response, the partition relation (33) before integration, and the clean ordering theorem that a monotonically decreasing optical depth yields a subthermal spectrum. The paper also does some honest things well: it gives a nonlinear bound on the angular displacement (Eqs. 21-23), tests smooth transition profiles and response shapes, and reports the sensitivity of the yield to the O(1) coefficients. The benchmark is a fit, not a prediction, but the authors do not hide that.

The soft spots, in order of size. First, the gap between the nonlinear homogeneous mode and the linear fluctuations. The paper treats the homogeneous angle with the full sine potential, then writes a linear Langevin equation for each mode with no estimate of the mode-coupling terms (parametric resonance, scattering off the condensate). Without a dimensionless ratio showing that the bath optical depth dominates the self-interaction rates, Eq. (33) is a statement about the model, not yet about the axion field. This is the load-bearing issue. Second, the O(1) transport coefficients (c_m, c_omega, kappa, gamma) are not scanned; the quoted mass and free-streaming length carry an order-one uncertainty. Third, the mass is fitted to the DM abundance, which is fine for a benchmark but weakens the 'produces the dark matter abundance' phrasing. None of these is fatal, but the first needs to be addressed for the central claim to stand.

Bottom line: the paper is a real contribution with a clean formal core, but it is not finished. A good referee could help the authors close the nonlinearity gap and quantify the coefficient dependence. The reading group would find the ordering argument interesting, but I would not cite the central result until the gap is closed.

Referee Report

3 major / 5 minor

Summary. The paper proposes a thermal-relaxation mechanism for axion dark matter in which a hidden Yang-Mills bath both damps a coherent axion-like field and populates stochastic momentum modes. The central result is Eq. (33), a mode-by-mode identity stating that the fractional Bose occupation plus the surviving coherent power sum to one, with both controlled by a single momentum-dependent optical depth. The authors derive this identity within a linear Langevin model, use monotonicity of the response to show that the produced population is kinematically colder than Bose equilibrium, and construct a benchmark hidden-sector model whose stochastic modes match the dark-matter abundance. They also compute free-streaming lengths, density power spectra, skewness, and isocurvature transfer, concluding that production correlations remain microscopic and that the relic is a subthermal stochastic population rather than a rescaled thermal distribution.

Significance. If Eq. (33) applied to the real axion field, the paper would provide an attractive organizing principle that unifies misalignment, partial relaxation, and stochastic production into a single momentum-resolved statement, with the monotonicity argument in Eq. (35) giving robust kinematic restrictions that are independent of many ultraviolet details. The paper is commendable for reporting an explicit benchmark with dimensionless validity ratios, for providing analytic derivations in the appendix, and for making concrete, checkable statements about free-streaming scales and hidden entropy perturbations. The main limitation is that the central identity is demonstrated only for a linear Langevin mode and its connection to the nonlinear dynamics of Eq. (17) is not established; the present significance is therefore at the level of a model result rather than a general field-theoretic statement.

major comments (3)

[§IV and Appendix A, Eq. (33)] The central mode-partition identity is derived from the linear Langevin equation (A13) and the linearized rate equation (A16), but the actual field evolution is given by the nonlinear Eq. (17), with q(N)=3+dlnH/dN+Υh/H and the restoring term μ² sinϑ. In the linearized treatment the coherent amplitude decays as e^{-τx/2} with γx from Eq. (30), which depends only on Υh/H times response factors. In the nonlinear treatment, the homogeneous amplitude is damped by ΓA from Eq. (19), which depends on both q and μ (for example ΓA≃μ²/q in the strongly overdamped regime). The two rates are not shown to be equal or even proportional, and no dimensionless ratio in Table I bounds the self-interaction mode couplings (such as ϑ₀→ϑ_k+ϑ_{-k}) relative to γx. Because Eq. (33) is stated as a property of every mode before momentum integration, the derivation needs either to follow from Eq. (17) or to be accompanied by an explicit approximation regime with controlled error.
[§V, Eqs. (38)–(39)] The abstract and Section V state that the benchmark 'produces the dark matter abundance', but as implemented this is an abundance-matching fit rather than a prediction: Eq. (39) defines m_a^DM by equating Y_a to the observed ρ_DM/s_0, and the benchmark value m_a=2.83×10² keV is the output of that matching. The paper should state this distinction explicitly, and the substantive claims should be framed as the composition and momentum distribution for a specified axion mass rather than as a prediction of the axion mass itself.
[§IV, Eq. (28) vs Eq. (33)] Equation (33) is asserted for every mode, but the kinetic equation (29) is restricted to the quasiparticle sector x≥x_q defined in Eq. (28), while modes below x_q are explicitly excluded and treated as part of the long-wavelength coherent field in Section III. The paper never reconciles the coherent survival computed from ΓA in Eqs. (19)–(20) with the coherent amplitude e^{-τx/2} appearing in Eq. (33); moreover, the benchmark coherent fraction in Eq. (41) is computed from Eq. (40), not from the partition identity. The domain of validity of Eq. (33) should be stated precisely, and the relationship between the two coherent-survival calculations should be established.

minor comments (5)

[§II] The error-function and compact-cubic transition profiles mentioned after Eq. (11) are never defined; Figure 5 shows their effect, but the functional forms and their parameter values should be given so that the results are reproducible.
[Appendix A] The appendix states that 'the numerical material accompanying the article' documents the independent checks, but no repository, DOI, or link is provided; please include a deposition identifier.
[Throughout] Several displays contain formatting artifacts, for example the inequality in Eq. (22) and the subscript in Eq. (A4); the final typeset version should be checked carefully.
[Table I] Table I lists Ti/M⋆ and ξh,i but not the actual initial hidden-bath temperature; since Figure 2 quotes specific Th values, the table should include Ti and Th,i for completeness.
[Eq. (34)] The measure d ln x x³ n_B(x) is used in Eqs. (34) and (35), but it is not defined as an explicit momentum average; stating the measure explicitly before Eq. (34) would avoid confusion with an average over the produced distribution.

Circularity Check

0 steps flagged · score 1.0 of 10

No substantial circularity: the central mode-partition identity is a derived consequence of the stated linear Langevin model, and the abundance-matching mass is explicitly fitted rather than presented as a prediction.

full rationale

I find no circular step that reduces a prediction to its inputs. The central identity, Eq. (33), follows in Appendix A by solving the linear Langevin equation (A13); the solution gives nx = (1 - e^{-tau_x}) nB(x) and |Ax/Ax,i|^2 = e^{-tau_x}, so the unit sum is an algebraic consequence of using the same optical depth tau_x for both coherent decay and noise-driven occupation. This is a bookkeeping identity of the model, not a fitted result and not a relabeled input: tau_x is computed from the bath response in Eqs. (10) and (30), with the noise normalization fixed by the fluctuation-dissipation relation. The late mass in Eq. (39) is defined by matching the observed relic density, but the paper explicitly says it 'reproduces the observed abundance' rather than claiming an independent prediction; the subthermal spectrum, kinematic ordering, and free-streaming bounds follow from response monotonicity and are not fit parameters. No load-bearing self-citation appears. The only author-overlapping citation I identify is Ref. [47] (with coauthor F. Atamurotov), which supports a peripheral statement about dilaton-coupled reheating and prolonged inflation; the thermal-curvature and open-system steps are cited to independent sources (e.g., Refs. [3,4,9,48,49]) and are rederived in the appendices. The nonlinear-sine concern raised in the skeptic brief is a regime-validity question about linearizing Eq. (17) to obtain the Langevin mode equation (A13), not a circularity of the derivation chain.

Assumptions & free parameters 11 free parameters · 7 assumptions · 3 invented entities

The central benchmark rests on about a dozen hand-chosen parameters and several unverified microphysical assumptions. The mass is fitted to the DM abundance. The main physical simplification is the linear Langevin description with monotonic finite-range response.

free parameters (11)

lambda_1 = 6.00
Scalar quartic coupling in Eq. (2). Chosen by hand for the benchmark; sets the thermal mass via Eq. (5).
lambda_Hchi = 2.00e-3
Higgs portal coupling in Eq. (6). Chosen to keep the scalar multiplet in equilibrium (Eq. 7).
m_chi = 1.00e4 GeV
Scalar multiplet mass; sets F2(y) and the disappearance of thermal curvature.
M_star = 1.25e17 GeV
UV cutoff in Eq. (2); controls the size of the thermal and threshold contributions.
f_a = 2.50e8 GeV
Axion decay constant; sets field normalization and yields.
f_h = 2.50e8 GeV
Hidden axion coupling scale in Eq. (9); sets dissipation strength.
xi_h,i = 0.25
Initial hidden-to-visible temperature ratio; controls hidden bath temperature and yield.
N_chi = 16
Number of scalar components; enhances thermal mass in Eq. (5).
m_a = 2.83e2 keV (abundance matching gives 2.75-2.91e2 keV)
Late axion mass. Fixed by matching Omega_DM via Eq. (39); this is a fit, not a prediction.
O(1) transport coefficients (Ng, kappa, gamma, c_m, c_omega) = not specified
Eqs. (10) and (26) contain unspecified dimensionless constants for the hidden gauge response and correlation lengths. The yield and spectrum depend on their values.
T_h,* and Delta_h = not specified
Hidden Higgs transition temperature and width in Eq. (11). The paper claims insensitivity to profile shape, but the transition time affects the final optical depth.

assumptions (7)

standard math The scalar multiplet variance is given by the free thermal integral Eq. (A1) with Gaussian contraction Eq. (A2).
Used to derive thermal mass Eq. (5).
domain assumption The hidden gauge bath's low-frequency topological response has the form Upsilon_h = kappa gamma (N_g alpha_h)^5 T_h^3 / f_h^2 with an O(1) prefactor (Eq. 10).
Perturbative gauge transport result, not derived in the paper; central to the optical depth.
ad hoc to paper The bath correlators are exponentially localized in time and space with scales set by gauge couplings (Eqs. 24-26).
Chosen finite range profiles; the result that the spectrum is subthermal follows from assuming these are monotonically decreasing.
domain assumption The local transport approximation epsilon_c << 1 holds (Eq. 27).
Needed for the rate equation (29) to apply.
ad hoc to paper The quasiparticle sector x >= x_q separates from the long-wavelength coherent field (Eq. 28).
This horizon split is chosen to avoid conflating isocurvature with stochastic occupation; it is not derived from first principles.
standard math The fluctuation-dissipation theorem fixes the noise normalization from the dissipative response (Refs [51,52]).
Sets n_B(x) as the equilibrium occupation in Eq. (29).
domain assumption Planck constraints on isocurvature and Lyman-alpha free streaming are used as external data.
Used to place bounds in Section VI; external observational input.

invented entities (3)

Hidden Yang-Mills bath (G_h, alpha_h)
purpose: Provides dissipative response Upsilon_h and stochastic forcing that populates axion modes; Eq. (9).
No observed particles or forces; benchmark is photophobic and outside detection ranges. Only cosmological constraints such as Delta_N_eff and isocurvature apply.
Scalar multiplet chi_A with N_chi real components
purpose: Generates the transient thermal curvature in Eq. (2).
Introduced to realize the mechanism; not observed.
Hidden Higgs condensate that gives hidden gauge bosons mass near T_h,*
purpose: Suppresses topological response after production, freezing the distribution.
Included to end the interaction; transition parameters are unspecified.

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Cite this review

Pith. "Pith review of Coherent and Stochastic Axion Dark Matter from Thermal Relaxation." pith.science (2026). https://pith.science/paper/YGWNNY6R

@misc{pith2026260805538,
  author       = {Pith},
  title        = {Pith review of: Coherent and Stochastic Axion Dark Matter from Thermal Relaxation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YGWNNY6R}},
  note         = {Machine review of arXiv:2608.05538}
}

read the original abstract

An axion driven toward a thermal minimum need not remain a coherent condensate. We derive the coupled attenuation of the field mean and production of stochastic momentum modes in an expanding plasma. A single momentum dependent optical depth partitions every mode between coherent power and Bose occupation, placing standard misalignment, partial relaxation, and stochastic production in one dynamical relation. Finite spatial and temporal response suppresses high momentum production, so the stochastic relic has lower mean momentum and a shorter free streaming length than an equilibrium population at fixed mass and bath temperature. A weakly coupled realization produces the dark matter abundance, reduces inherited coherent isocurvature, and confines production correlations to scales removed by collisionless propagation. Thermal relaxation therefore determines the composition, momentum distribution, and transported structure of axion dark matter.

Figures

Figures reproduced from arXiv: 2608.05538 by the authors.

**Figure 2.** FIG. 2. Momentum dependent stochastic production. The left panel shows the occupation relative to the Bose distribution at [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. Abundance composition and momentum compression. The left panel compares the stochastic abundance with the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Microscopic production and gravitational smoothing. The left panel compares the CMB pivot scale, the high redshift [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Finite response and transition profile tests. The left panel compares the exponential response with the Lorentzian [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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