Reading the Chandra NGC 1275 axion limit

We read the full text of arXiv version 4, not the journal version, and did not compare the two. We have not reproduced the limit. The Lab's percent-ripples preset uses the published coupling with schematic field values, and shows only the size of the effect.

Figure 3: how smooth is the spectrum?

Look at the left panel first. The top half shows the combined first-order spectra from Chandra's two grating arms, HEG in blue and MEG in red, with the best power-law model. The bottom half shows data divided by model. The right panel collects those ratios into histograms, with dotted lines at plus and minus 3 percent.

The fitted model is a power law with photon index $\Gamma=1.890\pm0.009$, absorbed by Galactic gas with column density $1.32\times10^{21}\,\mathrm{cm^{-2}}$. The fit statistic is $C=4956$ for 4874 degrees of freedom. The source was observed for about 490 kiloseconds, and the band used is 1 to 9 keV.

The common misreading is to take the plotted points as the fitted data. The caption says the data are heavily binned for display and fitted unbinned. The 3 percent lines only guide the eye to the size of the scatter.

Equation (2) and the two field models

The electron density is a two-component profile taken from earlier X-ray work. It sets the plasma term at every radius:

The field is built from cells. In model A the field in each cell scales as $B_0[n_e(r)/n_e(0)]^{0.7}$ with $B_0\approx25\,\mu\mathrm G$. Cell sizes are drawn between 3.5 and 10 kpc with probability proportional to $L^{-1.2}$. The central 10 kpc is excluded, so the cells run from 10 kpc to the virial radius, 1.8 Mpc, and configurations are kept only if the cluster's Faraday rotation stays at or below $2000\,\mathrm{rad\,m^{-2}}$. In model B the magnetic pressure is fixed at one hundredth of the gas pressure, the field is about 7.5 microgauss at 25 kpc, and for an isothermal cluster it falls as the square root of the density.

The paper says plainly that neither cell models nor Gaussian random fields are fields realized in nature. The relevant features are the transverse strength and the coherence length along the line of sight. Our own chapter on Mixing in a cluster magnetic field and plasma evaluates the critical energy along the model-B profile. It lies far above 9 keV at the limiting coupling, so mixing in Perseus is weak at that coupling.

Figure 4: one realization, many parameters

Both panels use one representative model-B realization. On the left the mass is fixed at $\log_{10}(m_a/\mathrm{eV})=-12.7$ and the coupling steps from $\log_{10}(g_{a\gamma}/\mathrm{GeV^{-1}})=-11.9$ down to $-12.7$. On the right the coupling is fixed at $-12.1$ and the mass steps from $-12.3$ to $-12.9$.

On the left, the dips deepen as the coupling grows, as expected from weak mixing, where conversion scales roughly as the coupling squared. On the right, a heavier mass suppresses the low-energy end. That is the critical-energy behaviour. Do not read these curves as averages. Each is a single field realization, and its ripple positions are specific to that realization.

Figure 5: field model against field realization

Here one parameter point, $m_a=10^{-12}$ eV and $g_{a\gamma}=10^{-12}\,\mathrm{GeV^{-1}}$, is shown with two realizations from each field model, model A in black and model B in red. Below 4 keV the two models give distortions of similar size, with model A's narrower. Above 4 keV both become more periodic, and model A's distortions grow markedly.

The lesson is that two realizations of the same model can differ as much as two models do. Any limit therefore has to marginalize over realizations, which is the next step.

Equations (3) and (4): marginalizing over the field

The authors generated 500 acceptable field configurations per model and computed survival curves over a grid with $\log_{10}(m_a/\mathrm{eV})$ from $-13.6$ to $-11.1$ and $\log_{10}(g_{a\gamma}/\mathrm{GeV^{-1}})$ from $-13$ to $-10.7$. That makes about 260,000 curves. Each curve multiplies the absorbed power law and is fitted to the unbinned spectra, giving a statistic $C(m_a,g_{a\gamma},i)$ for realization $i$.

The posterior is proportional to $\exp(-C/2)$, normalized over masses, couplings and realizations as in eq. (3). Summing over realizations as in eq. (4) gives the posterior over mass and coupling alone. Priors are flat in $\ln m_a$ and $\ln g_{a\gamma}$. The sum over realizations is how the unknown field enters: a coupling is excluded only if almost no plausible field configuration lets it hide under the data.

Figure 6 and equation (5): the limit

The top panel is model A and the bottom panel model B. The heavy black line bounds the excluded region at 99.7 percent, shaded light blue. Yellow shading marks the 95 percent allowed region, and red dots are best-fit points. The figure also shows earlier limits from M87 (Marsh et al. 2017), from SN1987A (Payez et al. 2015) and from Fermi, plus the projected IAXO sensitivity.

For $m_a<10^{-12}$ eV the paper gives $g_{a\gamma}<6.3\times10^{-13}\,\mathrm{GeV^{-1}}$ for model A and $7.9\times10^{-13}\,\mathrm{GeV^{-1}}$ for model B at 99.7 percent. Equation (5) states the model-B value as the main result.

The easiest misreading is the yellow island. At 95 percent, model B prefers $g_{a\gamma}\approx4$ to $8\times10^{-13}\,\mathrm{GeV^{-1}}$, driven by a modest dip at 1.2 keV and a broad hump at 3 to 4 keV. The authors attribute these to plausible residual calibration errors in the grating effective area, and so they report the result as an upper limit only.

Four kinds of statement

Published result. The 99.7 percent limits for models A and B in Reynolds et al. 2020 are refereed results. So are the horizontal-branch limit $g_{a\gamma}<6.6\times10^{-11}\,\mathrm{GeV^{-1}}$ of Ayala et al., 2014 and the Fermi-LAT limit of $5\times10^{-12}\,\mathrm{GeV^{-1}}$ for 0.5 to 5 neV from six years of data on the same galaxy, Ajello et al., 2016. We read only the abstract of the Fermi paper. The PDG review of axions, 2025 collects these and quotes the X-ray bound as $8\times10^{-13}\,\mathrm{GeV^{-1}}$.

Preprint. The texts read here are arXiv versions. Version 4 of Reynolds et al. still carries the header "Submitted to ApJ". The Matthews et al. reanalysis was read in its arXiv version 1, marked as accepted to the same journal. Any wording change in the final journal versions has not been checked.

Model prediction. The IAXO line in Fig. 6 is a projected sensitivity, not a measurement. So are the X-ray polarimetry bounds of Day and Krippendorf, 2018, whose abstract presents techniques and projected bounds; we read only the abstract. Every survival curve in the paper, and every curve in the Lab, is a model prediction for an assumed field.

Open question. How much does the limit depend on the field? Matthews et al., 2022 reanalysed the same spectra and found limits weaker by 0.1 to 0.3 dex below about $10^{-13}$ eV for other field models. They concluded that model B is robust if the gas-to-magnetic pressure ratio is about 100, and that the field's strength and structure remain uncertain. Whether the 1.2 keV dip is purely calibration is also open.

How the other channels differ

Absorption. In X-rays, absorption is part of the baseline model: Galactic gas dims low energies smoothly, and Matthews et al. found the limits insensitive to adding a partially covering absorber. In gamma rays the role reverses. Very high-energy photons are absorbed by pair production on extragalactic background light, and photons that travel part of the way as axions could avoid it. The PDG review calls that possibility inconclusive at present.

Turbulent fields. Every cluster limit inherits the field model. The PDG review notes that gamma-ray limits rely on magnetic fields not yet confirmed by direct observations in those directions.

Polarization. Only light polarized along the field converts, so polarization carries independent information. The PDG review quotes $1.7\times10^{-12}\,\mathrm{GeV^{-1}}$ at 95 percent from optical polarization of magnetic white dwarfs, whose fields are measured through Zeeman splitting.

Stellar cooling. Stars need no external field at all. They limit the same coupling by comparing nuclear energy release with axion losses. That limit holds up to masses of a few keV and is about 80 times looser than the Chandra limit.

Checklist for the next limit you read

1. Which spectrum, which band and which baseline model, including absorption?
2. Which field model, and what observations constrain it?
3. How many field realizations, and is the limit marginalized over them?
4. Which mass range, which confidence level and which priors?
5. Is any preferred region discussed, and what instrumental effect could produce it?
6. Which version of the text did you read, and which statements are projections rather than measurements?

Try it in the Lab: Choose Percent-level ripples and compare the solid curve with the dashed mean.