Revisiting Constraints on Resonant Axion-Photon Conversions from CMB Spectral Distortions

Bryce Cyr; Jens Chluba; Pranav Bharadwaj Gangrekalve Manoj

REVIEW 3 major objections 5 minor 1 cited by

Revisiting Constraints on Resonant Axion-Photon Conversions from CMB Spectral Distortions

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Resonant CMB photon-to-axion conversion in a ~1 nG primordial magnetic field would imprint a characteristic spectral distortion, and COBE/FIRAS residuals already exclude axion-photon couplings that other experiments have not reached.

desk verdict A careful, honest reanalysis of axion-photon conversion limits from CMB spectral distortions; the corrected atomic-hydrogen plasma mass is the real gem and the constraints hold up, so it deserves serious review. read the letter →

arxiv 2411.13701 v1 pith:4ZN3NKDN submitted 2024-11-20 astro-ph.CO

Bryce Cyr , Jens Chluba , Pranav Bharadwaj Gangrekalve Manoj This is my paper · ORCID

classification astro-ph.CO

keywords axionsaxion-likeparticlesCMBspectraldistortionsresonantphoton-axionconversionprimordialmagneticfieldsphotonplasmamassCOBE/FIRASPIXIE

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the near-perfect blackbody spectrum measured by COBE/FIRAS can act as a laboratory for axions and axion-like particles: when the plasma mass of a CMB photon matches the axion mass during cosmic expansion, the photon can resonantly convert into an axion, leaving a frequency-dependent deficit in the CMB spectrum. The authors improve the photon plasma mass by including neutral hydrogen and helium, derive the spectral template of the resulting distortion, and evolve it through the mu- and y-eras with CosmoTherm. For a large-scale primordial magnetic field near the current CMB upper bound, about 1 nG, they find that COBE/FIRAS residuals already exclude axion-photon couplings in mass regions that other experiments have not reached, and a PIXIE-like experiment would extend the reach by about two orders of magnitude. They also treat the large-distortion regime for the first time, finding that stimulated Compton scattering creates a photon excess at low frequencies that prevents the naively expected sign flip of the distortion. A reader should care because CMB spectral distortions are one of the few probes sensitive to axion masses around $10^{-13}$ to $10^{-4}$ eV, a window that other searches cover poorly if at all.

What carries the argument

The load-bearing object is the resonant conversion condition $m_a \simeq m_\gamma(z,\omega)$, where the effective photon mass is computed from the refractive index of the pre- and post-recombination plasma. The paper uses $m_\gamma^2 = \tilde{\omega}_p^2 [X_e - (\omega/\mathrm{eV})^2 \sum_i \kappa_i X_i]$ with $\kappa_{\mathrm{HI}} \simeq 5.0\times 10^{-3}\,\mathrm{eV}^{-2}$, $\kappa_{\mathrm{HeI}} \simeq 1.8\times 10^{-3}\,\mathrm{eV}^{-2}$, and $\kappa_{\mathrm{HeII}} \simeq 3.1\times 10^{-4}\,\mathrm{eV}^{-2}$, so that high-frequency photons can have a negative effective mass-squared and stop converting. Conversion probabilities are handled with the Landau-Zener form $P = 1 - \exp(-\gamma_{\mathrm{con}} x)$, where $\gamma_{\mathrm{con}} \propto (g_{a\gamma\gamma}B_0)^2$ divided by the derivative of $\ln m_\gamma^2$ at the resonance; this expression determines which photon frequencies convert at which redshifts. The paper also defines the axion distortion template $A(x) = (G_3/(3G_2))G(x) - x n_{bb}(x)$, the photon-number-conserving spectral shape left by the conversion before Comptonization drives it toward a $\mu$-type distortion.

What would settle it

A measurement that places an upper limit $B_0^{\rm rms} \lesssim 10^{-2}$ nG on a $\sim$1 Mpc-scale cosmological magnetic field would falsify the core claim, since the distortion scales roughly as $(g_{a\gamma\gamma} B_0)^2$ and would fall below the COBE/FIRAS threshold. A second decisive test would be a direct axion detection at a coupling and mass inside the region the paper's Fig. 13 rules out under the assumed 1 nG field.

Watch

Extended reading notes

Core claim

The central claim is that resonant CMB photon-to-axion conversion, computed with an updated plasma mass and a full spectral-distortion treatment, produces competitive new constraints on the axion-photon coupling $g_{a\gamma\gamma}$. For axion masses roughly $10^{-13}$ to $10^{-4}$ eV and a comoving magnetic field of order 1 nG, the paper finds that COBE/FIRAS data explore regions of parameter space not yet accessed by other experimental probes, on both the high- and low-mass ends, with a PIXIE-type experiment improving the reach by about two orders of magnitude. This improvement is attributed to three changes: the refractive index of neutral atomic hydrogen is used instead of a molecular-hydrogen value, helium contributions are included, and the full shape of the distortion rather than a total-energy proxy is compared to data. For multiple conversions at low masses, the analytic conversion probability can change sign near $10^{-12}$ to $10^{-11}$ eV, but the CosmoTherm solutions keep the final distortion negative because photon condensation at low frequencies supplies the extra photons. The resulting constraints and forecasts appear in the paper's Figs. 11 and 13.

Load-bearing premise

The load-bearing premise is that a large-scale primordial magnetic field with present-day amplitude near 1 nG, coherence length near 1 Mpc, and redshift scaling $B(z)=B_0(1+z)^2$ exists; if the field is absent or weaker than about $10^{-2}$ nG today, the claimed unexplored parameter-space regions from COBE/FIRAS disappear.

Editorial extensions

If this is right

COBE/FIRAS residuals already exclude some $g_{a\gamma\gamma}$ values near $m_a \sim 10^{-13}$ to $10^{-4}$ eV that other laboratory and astrophysical searches do not reach, if a ~1 nG large-scale magnetic field exists.
A PIXIE-like experiment would improve the $g_{a\gamma\gamma}$ reach by about two orders of magnitude, allowing a present-day field strength near $10^{-4}$ nG to produce a detectable distortion.
The analytic distortion template and energy estimate reproduce the CosmoTherm constraints well, and the full numerical shape tightens the PIXIE forecast by roughly 30%.
Multiple conversions for $m_a \lesssim 10^{-10}$ eV make the distortion sign and shape sensitive to the details of recombination and reionization, so spectral-distortion data becomes a probe of those epochs as well.
In the large-distortion regime, strong axion conversions do not produce the sign-flipped positive distortion one might expect; stimulated Compton scattering and low-frequency photon destruction keep the net distortion negative, weakening constraints relative to naive estimates.

Reading between the lines

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

Editorial inference: the corrected atomic-hydrogen and helium refractive indices should also change predictions for axion dark matter converting into photons; the helium-induced conversion spikes in the Wien tail would be a source of signal rather than a negligible correction in the $\Omega_a \simeq \Omega_{\rm dm}$ scenario.
Editorial inference: the new template $A(x)$ is a generic signature of high-frequency photon removal, so future spectral-distortion fits could include it as an independent component alongside $\mu$ and $y$; because its null lies near the $\mu$-distortion null, multi-band data will be needed to separate them.
Editorial inference: the limits in Fig. 13 are conditioned on the assumed $B_0$; a null PIXIE measurement would tighten $g_{a\gamma\gamma}$ only for a fixed field, so the analytic framework could be used to present joint exclusions on $(m_a, g_{a\gamma\gamma}, B_0)$ rather than separate curves.

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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

Read this one. It's a careful, honest reanalysis of resonant axion-photon conversion limits from CMB spectral distortions, and it holds up better than most papers in this space. The genuinely new bits are the corrected atomic-hydrogen refractive index coefficient (the old literature value comes from molecular hydrogen, H2, not HI; they show the coefficient drops from ~7.7e-3 to ~5.0e-3 eV^-2), the inclusion of helium in the photon plasma mass, and the explicit axion distortion template A(x). They validate the analytic small-distortion framework with CosmoTherm and find the full-shape analysis tightens the COBE/FIRAS and PIXIE constraints by up to ~30% relative to the energy-density-only estimate. The broad agreement with Mirizzi, Redondo, and Sigl (2009) is a good consistency check. The central constraints are built on clean logic: Landau-Zener conversion probability, no fitted normalization, external COBE/FIRAS and Planck data. I did not find a load-bearing error.

The soft spots are real but not fatal. The 'previously unexplored regions' claim in the abstract and Fig. 13 is conditional on a ~1 nG comoving primordial magnetic field at ~1 Mpc coherence. That's a Planck upper bound, not a detection, and the authors say the limits degrade roughly linearly with B0, becoming suboptimal below ~1e-2 nG. They flag it, but the visibility of that caveat matters. The large-distortion regime is only partially treated: the numerical run develops a photon shock at low frequencies and they stop at scattering y~0.3, and they note Landau-Zener corrections may matter at gamma_con ~1. That's fine because the Neff bound keeps them out of that regime, but the 'first treatment' is more of a first look. No code or data release accompanies the numerical constraint curves, and the FIRAS likelihood itself is deferred to the companion dark-photon paper. Minor: the paper leans on that companion, Ref [45], for machinery, which is legitimate but makes this paper less self-contained.

Bottom line: this is a solid refinement of an existing constraint program, with one genuinely useful correction (the HI vs H2 coefficient) and a credible numerical upgrade. Send it to a serious referee. I'd bring it to reading group and would cite it for the plasma mass treatment.

Referee Report

3 major / 5 minor

Summary. The paper revisits constraints on resonant CMB photon-to-axion conversions from CMB spectral distortions. It improves the modeling of the photon plasma mass by adding neutral hydrogen and helium contributions to the free-electron term, derives an analytic spectral template for the axion distortion, and uses the CosmoTherm code to evolve the distortion through the thermalization epochs. The resulting limits are compared with COBE/FIRAS data and with a PIXIE-like forecast, and are mapped into the gaγγ–ma plane for several assumed primordial magnetic field amplitudes. The paper also explores the large-distortion regime, where a significant fraction of CMB photons are converted.

Significance. If the central derivation holds, the paper provides a useful update to spectral-distortion constraints on axion-photon couplings, with quantitative improvements from the full spectral template rather than energy-density-only limits, and a physically better motivated plasma-mass treatment. The analytic framework in Eqs. (17)–(21) is transparent and cross-checked against CosmoTherm, the broad agreement with Mirizzi et al. is a valuable consistency check, and the PIXIE forecasts give a concrete, falsifiable target. The authors are also explicit that the constraints are conditional on the existence of a large-scale primordial magnetic field near the current Planck upper bound, and they show how the reach degrades with B0. The main caveats are associated with the normalization convention for the magnetic-field fudge factor, the self-consistency of the low-mass constraints with the recombination history, and the completeness of the large-distortion treatment.

major comments (3)

[Sec. V B and Eq. (3b)] The angle-averaging argument in footnote 12 gives ⟨B_T²⟩ = (1/3) B0², so the rms transverse field entering Eq. (3b) linearly is B_T = B0/√3, not f = 1/3 as stated in the text. Since κ is linear in B_T and the conversion signal scales as κ², using f = 1/3 in Fig. 13 normalizes the gaγγ constraints differently from the stated convention by a factor √3 (the limits are weaker by √3 for fixed B0). Please correct the definition of f or recompute Fig. 13, and state the convention unambiguously.
[Sec. II B and Sec. V (Figs. 1 and 11)] The low-mass constraints (ma ≲ few × 10⁻¹⁰ eV) are computed with the standard CosmoRec recombination history and assume that the conversion does not alter that history. At the FIRAS 95% limit in this mass range, Eq. (21) gives |Δργ/ργ| ≈ 6 × 10⁻⁵, corresponding to |ϵρ| ≈ 10⁻³ at the conversion redshift; this is not obviously negligible for the ionization balance and would feed back into Xe and hence into mγ and the resonance locations. Please quantify this feedback or restrict the mass range over which the constraints are quoted.
[Sec. IV A and Sec. V A (Figs. 4–6, 12)] The treatment of the large-distortion regime is presented as complete ("we treat for the first time the large-distortion regime"), but the CosmoTherm integrations are stopped at a scattering y-parameter of about 0.3 because a photon shock renders the numerical treatment insufficient (Sec. IV A), and the Landau-Zener expression is used up to γcon ≃ 1 despite caveats from Refs. [48–50]. The conclusion that no sign flip occurs and the interpretation of Fig. 12 therefore rest on an incomplete numerical evolution. Please either complete the evolution with a method that handles the shock or soften the claims to describe an exploratory treatment of the onset of the large-distortion regime.

minor comments (5)

[Sec. II A, after Eq. (3)] The sentence "This assumption is not crucial when considering considering axion masses" contains a duplicated word; please edit.
[Sec. II B, Eq. (15)] The index set "i = e, HI, HeI, HeII" is not consistent with the displayed term Xe having no κi; please clarify that κe ≡ 1 and define Xp explicitly.
[Fig. 13] The ordinate label "f ga" should be "f gaγγ" and the f convention should be stated in the caption, especially in light of the normalization issue raised above.
[Sec. IV C] The multiple-conversion implementation replaces δ(z − zcon,i) with a Gaussian of width Δz/z ≈ 10⁻²; a sentence justifying that this width is small compared with the thermalization timescale would improve reproducibility.
[General] The paper would benefit from a data/code availability statement, since the FIRAS likelihood is described only by reference to Ref. [45].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the axion distortion template is derived from first principles and compared to external data without fitted normalization.

full rationale

The derivation chain is self-contained. The conversion probability is the Landau-Zener expression, Eq. (2), with gamma_con computed from microphysical inputs (gaγγ, B0, ma, H(z), and the plasma-mass derivative). The plasma mass, Eq. (15), is assembled from atomic-physics refractive-index coefficients (kappa_HI, kappa_HeI, kappa_HeII) and the recombination history from CosmoRec; none of these coefficients is fitted to the COBE/FIRAS spectral residuals used for the constraints. The initial distortion is derived analytically in Eqs. (19)-(21) and Eqs. (30)-(31), giving the axion shape A(x); the same shape is then evolved with CosmoTherm, whose solutions are compared with the analytic result and with the external COBE/FIRAS and PIXIE sensitivities in Fig. 11. No fitted normalization enters the signal template: the constraint is obtained by comparing the predicted |Delta rho / rho|_dis with the external limit |Delta rho / rho|_dis <~ 6e-5. The reliance on Ref. [45] is for shared numerical/likelihood machinery developed for the companion dark-photon study; it is code-based and does not smuggle in the axion conclusion, and the paper explicitly re-derives the axion-specific spectral shape. The assumption of a ~1 nG large-scale primordial magnetic field is an external input (Planck upper bound), not a parameter fitted to the distortion, and the paper states the constraints scale roughly linearly with B0 and weaken for B0 <~ 1e-2 nG (Sec. V B, Fig. 13). Acknowledged limitations (photon-shock numerics for y >~ 0.3, Landau-Zener corrections near gamma_con ~ 1, neglected band averaging) are honest caveats, not circular steps. No uniqueness theorem, ansatz-by-citation, or fitted-input-as-prediction pattern is present.

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

The central derivation has one external environmental assumption (a primordial magnetic field near the Planck bound) and one modeling parameter (the fudge factor f). The axion itself is a prior theoretical entity, not introduced by this paper. The atomic kappa coefficients are physical inputs from oscillator strengths, not fitted to the distortion data.

free parameters (2)

Primordial magnetic field amplitude B0rms = 1 nG benchmark (also 0.1 and 0.01 nG)
The conversion rate gamma_con is proportional to B0rms, and the 'unexplored parameter space' claim in Fig. 13 is shown for B0rms near the Planck upper bound. The constraints weaken roughly linearly with B0rms and become suboptimal below about 10^-2 nG.
Fudge factor f relating transverse to total B-field = 1/3 from simple angle-averaging, retained free
Sec. V B introduces f via Brms,T = f B0rms and argues pre-recombination scattering complicates the simple value; Fig. 13 therefore constrains f times gaγγ rather than gaγγ alone.

assumptions (4)

domain assumption Landau-Zener formula P = 1 - exp(-gamma_con x) with gamma_con from Eq. (3a) governs resonant photon-axion conversion.
Adopted from Refs. [46-49]; requires B-field coherence length larger than the oscillation length (Eq. 4) and neglects phase interference under the condition of Eq. (6).
domain assumption The photon thermal mass is m_gamma^2 = -omega^2(n^2-1) with refractive contributions from free electrons, HI, HeI and HeII (Eq. 15).
Standard plasma dispersion; the paper neglects damping near resonances, continuum transitions, collisional corrections, and magnetic-field corrections, as stated in Sec. II B.
domain assumption A large-scale primordial magnetic field with B(z) = B0(1+z)^2 and coherence length ~1 Mpc exists at the required epochs.
Required for conversion; B0rms near 1 nG is an upper limit from Planck, not a detection, and the paper notes constraints degrade for smaller fields (Sec. V B).
domain assumption Standard recombination and thermalization histories from CosmoRec and CosmoTherm, with LCDM expansion, describe the background.
Used to obtain Xe, XHI, XHeI, XHeII and to evolve distortions; the paper assumes no back-reaction of conversion on recombination history (Sec. II B).

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Cite this review

Pith. "Pith review of Revisiting Constraints on Resonant Axion-Photon Conversions from CMB Spectral Distortions." pith.science (2026). https://pith.science/paper/4ZN3NKDN

@misc{pith2026241113701,
  author       = {Pith},
  title        = {Pith review of: Revisiting Constraints on Resonant Axion-Photon Conversions from CMB Spectral Distortions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ZN3NKDN}},
  note         = {Machine review of arXiv:2411.13701}
}

read the original abstract

Axions and axion-like particles (ALPs) remain highly motivated extensions to the standard model due to their ability to address open questions such as the relic abundance of dark matter and the strong CP problem. Axions are also capable of undergoing a resonant mixing with photons when the masses of the two fields are roughly equal, producing a wide array of phenomenological consequences. Here, we revisit constraints coming from conversions of the cosmic microwave background (CMB) into axions, which will induce a distortion to the frequency spectrum of the background photons. We introduce a more detailed description for the modeling of the plasma mass of the photon, showcasing how the inclusion of Helium recombination can alter the conversion probability for photons in the Wien tail. Our results include an updated analytic framework, which allows us to define the precise spectral shape of the axion distortion, as well as a numeric component which utilized the code \texttt{CosmoTherm} to fully characterize the distortion, providing a slight increase in the constraining power over the analytics. We also treat for the first time the large-distortion regime for resonant axion-photon conversions. Under the assumption of large-scale primordial magnetic fields near the limit obtained from CMB observations, we find that spectral distortions can probe previously unexplored regions of the axion parameter space.

Figures

Figures reproduced from arXiv: 2411.13701 by the authors.

**Figure 2.** Dependence of ∆ργ/ργ [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Normalized distortion spectra M∗ (x) = 1.401 M(x), Y ∗ (x) = Y (x)/4 and A ∗ (x) = −4.3354 A(x) [with 4.3354 = 1/0.23066]. The axion distortion has a new shape, with a null close to that of the µ-distortion. Note that A(x) has a negative sign in comparison to the µ and y distortions. with βM = 2.1923. Heuristically, G(x), Y (x), and M(x) describe the spectral shape of temperature shifts, y-distortions, and µ-distor… view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Illustration of the initial spectrum for [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 6.** Figure 6: Illustration of distortion evolution for [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Final distortion with respect to a blackbody for [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 9.** Figure 9: Differential conversion probability for ma = 10−13 eV and ϵ = 10−3 and varying frequency. The conversion redshifts can be easily understood with [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 8.** Figure 8: Conversion probabilities for ϵ = 10−3 and various axion masses. High-frequency structure becomes visible for ma ≲ 10−10 eV. At axion masses ma ≲ 10−12 eV, the conversion probability significantly increases again at high frequencies. Although a large spike is seen aro…

**Figure 10.** Figure 10: Final distortion for ϵ = 10−3 and varying masses. The complicated conversion structure leads complex spectral responses. Note that we scaled the distortions for ma = 10−9 eV, ma = 10−10 eV and ma = 10−11 eV to make them more comparable. We also note that we expect som…

**Figure 11.** Figure 11: CMB spectral distortion limits (95% c.l.) from [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: Dependence of effective distortion energy density [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 11.** Figure 11: We note that this assumption does not seriously [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]

**Figure 13.** Figure 13: Upper limits on gaγγ from CMB spectral distortions for different values of the large scale B-field. The factor f represents a fudge factor discussed in the main text. The “Other constraints” contour does not necessarily depend on f. A breakdown of these constraints c…

**Figure 14.** Figure 14: Constraints on the model parameters valid for gen [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]

**Figure 12.** Figure 12: The numerics, however, indicate that large dis [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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Determination of the conversion redshifts Given the expression for the effective photon mass, we can numerically determine the conversion redshifts (zcon) from the condition m2 a ≈ m2 γ. Since the terms ∝ ω2 can dominate over the plasma contribution, it is first useful to determine critical frequency ωf = Xe κHI XHI + κHeI XHeI + κHeII XHeII 1/2 eV (16) a...

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µ-era Assuming a single conversion of photons into dark pho- tons, the final µ-parameter can be reliably estimated by [68, 69] µcon ≃ 3 κ " ∆ργ ργ con − 4 3 ∆Nγ Nγ con # Jµ(zcon), (17) where κ ≃ 2.1419 [i.e., 3 /κ ≈ 1.4007] is a dimen- sionless numerical factor, ϵρ = ∆ ργ/ργ con and ϵN = ∆Nγ/Nγ con respectively denote the fractional change of the photon e...

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[45], the overall effective energy release can be used to model constraints in the y-era as well as in the transition to the µ-era

y-era and µ/y-transition As explained in Chluba et al. [45], the overall effective energy release can be used to model constraints in the y-era as well as in the transition to the µ-era. This is simply given by ∆ργ ργ dis ≈ " ∆ργ ργ con − 4 3 ∆Nγ Nγ con # Jbb(zcon) ≈ −0.2307 γcon Jbb(zcon), (21) where we used the approximations from above. Given an axion ...

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Initial spectrum For the large distortion regime we have to more carefully consider the initial condition [45], as it is possible for the background temperature to exhibit large jumps during a conversion process. One has the following relationships [70, 71] ∆Tin TCMB = − (1 + ϵρ)1/4 − 1 (1 + ϵρ)1/4 ≈ −ϵρ 4 ϵCMB = − ϵρ 1 + ϵρ ≈ −ϵρ, where in the rightmost ...

1923

[6] [5]

In the small distortion limit, the typical spectral dis- tortion that is initially formed is given by A(x), almost independent9 of the value of γcon ≲ 0.01. For axion mod- els with the same value for γcon, differences in the final distortion shape and constraints can thus only arise from the variation of the axion mass, which directly controls the convers...

[7] [6]

Other constraints

Late distortion solutions To confirm the validity of the analytic treatment, we implemented the multiple conversion scenario in CosmoTherm. For this we assume that the total change in the CMB temperature is negligible and that we can simply set Tin ≈ TCMB. The effect of this is that all timescales for photon interactions are slightly modified at a level o...

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WMAP-Collaboration, ApJS 192, 18 (2011), arXiv:1001.4538 [astro-ph.CO]

arXiv 2011