Axion–Photon Conversion in Cluster Magnetic Fields

Can an invisible light particle change the X-ray spectrum of a distant galaxy? A magnetic field lets a photon turn into a light axion and back, so part of the light from a distant galaxy can vanish on its way to us. The lost share changes with photon energy and would print small ripples on an X-ray spectrum. Smooth observed spectra therefore cap how strongly axions could couple to light, and stellar cooling sets a separate cap.

Turn magnetic domains, change gas density and axion mass, and watch photons trade places with axions along a cluster sightline.

Photons that arrive $88.3\,\%$

Photons now axions $11.7\,\%$

Linear polarization at the telescope $11.6\,\%$

Helium-burning lifetime left $100\,\%$

Send X-rays through a magnetized cluster

Photon energy

3\,\mathrm{keV}

E

Axion-photon coupling

3.16\times10^{-12}\,\mathrm{GeV^{-1}}

g_{a\gamma}

Magnetic field across the beam

5\,\mu\mathrm{G}

B_\perp

Electron density of the gas

0.00316\,\mathrm{cm^{-3}}

n_e

Axion mass

1\times10^{-13}\,\mathrm{eV}

m_a

Drag a domain arrow in the scene to turn or strengthen its field. Drag across the spectrum to pick the energy shown above.

This coupling is below every published limit that applies at this mass.

Domains, model terms and checks

Number of magnetic domains

24

N

Length of each domain

8\,\mathrm{kpc}

\ell_B

First-domain terms at the chosen energy, in inverse kiloparsecs:

$\Delta_{a\gamma}$

$\omega_{\rm pl}$

Above the critical energy the mixing no longer depends on energy; far below it, conversion is suppressed.

$E_{\rm crit}=\frac{|m_a^2-\omega_{\rm pl}^2|}{2g_{a\gamma}B_\perp}$

$\ell_{\rm osc}=2\pi/\Delta_{\rm osc}$

Each cell uses an exact unitary transfer, so the total probability stays one up to rounding:

$|P_{\gamma\gamma}+P_{\gamma\to a}-1|$

The export saves the inputs, units, field directions, both curves with their labels, and the path profile.

Model assumptions

This is a simplified model of one sightline, not a fit to any observation. The cluster field is a chain of equal cells. Each cell has one uniform field across the beam, with the same strength and a random or common direction. The gas has one constant electron density. Each cell is solved exactly for photons and one axion-like particle. Photon absorption, the turbulent power spectrum of real fields, radial profiles of density and field, the cosmic background light and detector response are all left out. The dashed reference curve is an ideal formula: the uniform-field two-state result, or the average over random field directions. It is not a reconstruction of the solid simulated curve. The stellar-cooling ruler uses a separate equation for horizontal-branch stars from the Particle Data Group review. The numbered limits are printed values from the cited papers and hold only for their own mass ranges and assumptions. The moving tracers only illustrate the computed fractions.

X-ray photons that survive the cluster, by energy

Solid: this simulated sightlineDashed: ideal mean over random field angles

Vertical axis: fraction of photons that arrive. Both curves come from the model, not from a telescope.

Along the sightline at the chosen energy

P_{\gamma\gamma}

P_{\gamma\to a}

\Pi_L

Dotted lines mark domain boundaries. Polarization grows because only one photon direction mixes in each cell.

Stars give a separate limit

Helium-burning lifetime factor

Horizontal-branch star counts $g_{a\gamma}<6.6\times10^{-11}\,\mathrm{GeV^{-1}}$ $m_a\lesssim1000\,\mathrm{eV}$ 95% confidence; PDG 2025 review 89, eq. (89.16); Ayala et al. 2014, arXiv:1406.6053v2, eq. (3)
CAST solar helioscope $g_{a\gamma}<5.8\times10^{-11}\,\mathrm{GeV^{-1}}$ $m_a\lesssim0.02\,\mathrm{eV}$ 95% confidence; PDG 2025 review 89, eq. (89.20)
Fermi-LAT spectrum of NGC 1275 $g_{a\gamma}<5\times10^{-12}\,\mathrm{GeV^{-1}}$ $5\times10^{-10}\lesssim m_a/\mathrm{eV}\lesssim5\times10^{-9}$ 95% confidence; Ajello et al. 2016, arXiv:1603.06978v1, abstract
Chandra spectrum of NGC 1275 $g_{a\gamma}<7.9\times10^{-13}\,\mathrm{GeV^{-1}}$ $m_a\lesssim1\times10^{-12}\,\mathrm{eV}$ 99.7% confidence; Reynolds et al. 2020, arXiv:1907.05475v4, eq. (5), field model B

Grey dashed markers do not apply at the current mass. Limits are copied from the papers, not computed here.

Focus question: Why does the same coupling give a smooth spectrum in one setting and strong ripples in another?
One-sentence intuition: Mixing is strong only when the coupling term beats the mismatch between the photon and axion dispersion terms. In a chain of randomly oriented cells the photon fraction averages toward $2/3$ , but any single sightline keeps energy-dependent ripples around that average.

Core mathematical model

Mixing in one magnetic cell

i\frac{d}{dz}\begin{pmatrix}A_\perp\\A_\parallel\\a\end{pmatrix}=-\begin{pmatrix}\Delta_{\rm pl}&0&0\\0&\Delta_{\rm pl}&\Delta_{a\gamma}\\0&\Delta_{a\gamma}&\Delta_a\end{pmatrix}\begin{pmatrix}A_\perp\\A_\parallel\\a\end{pmatrix}

Only the photon polarization parallel to the transverse field mixes. Here $\Delta_{a\gamma}=g_{a\gamma}B_\perp/2$ , $\Delta_{\rm pl}=-\omega_{\rm pl}^2/(2E)$ and $\Delta_a=-m_a^2/(2E)$ .

Two-state conversion in a uniform field

P_{\gamma\to a}=\left(\frac{2\Delta_{a\gamma}}{\Delta_{\rm osc}}\right)^2\sin^2\!\left(\frac{\Delta_{\rm osc}L}{2}\right),\qquad \Delta_{\rm osc}^2=(\Delta_{\rm pl}-\Delta_a)^2+4\Delta_{a\gamma}^2

This is for photons polarized along the field. Unpolarized light converts half as often. The Lab solves each cell exactly and checks this formula in the uniform case.

Mean over random field directions

\langle P_{\gamma\to a}\rangle_N=\frac{1}{3}\left[1-\prod_{j=1}^{N}\left(1-\frac{3}{2}P_j\right)\right]

Each $P_j$ is the parallel-polarization probability of cell j. Many cells drive the mean toward one third, the equipartition among two photon states and one axion state.

A separate stellar equation

\frac{t_{\rm HB}(g)}{t_{\rm HB}(0)}\approx\frac{80}{80+30\,g_{10}^2},\qquad g_{10}=\frac{g_{a\gamma}}{10^{-10}\,\mathrm{GeV^{-1}}}

Primakoff emission in helium-burning cores competes with nuclear burning. Fewer horizontal-branch stars in globular clusters than this predicts would signal axion cooling. The sightline model is not used here.

Common difficulties

Stronger field is not always more conversion

Typical misconceptionDoubling the field doubles the converted share.

Better mental modelThe probability oscillates with distance. Once a cell holds more than half an oscillation, a stronger field can lower the photon loss at a given energy.

The dashed curve is not a fit

Typical misconceptionThe dashed reference was reconstructed from the simulated curve.

Better mental modelIt is an independent ideal formula. In the tangled field it is the average over all field directions, so one sightline scatters around it.

Run the experiment

01
Watch the default sightline
Leave the tangled cluster running and watch the tracers drift between the two lanes.
What to observe: The photon lane thins and refills cell by cell, and the spectrum below dips unevenly with energy.
02
Turn one domain
Drag one arrow in the middle of the chain through a half turn.
What to observe: The path profile changes from that cell onward, and the polarization at the telescope rotates.
03
Make the field uniform
Choose Oscillation in a uniform field and drag the energy cursor.
What to observe: The solid and dashed curves coincide, because the exact transfer reproduces the two-state formula.
04
Compare with stars
Raise the coupling to the first numbered limit on the stellar ruler.
What to observe: There helium burning is about 14 percent shorter, while the cluster sightline is already close to fully mixed. Under their field assumptions, X-ray spectra reach far weaker couplings than star counts.

Axion–Photon Conversion in Cluster Magnetic Fields

Follow X-ray photons through a magnetized cluster

Core mathematical model

Mixing in one magnetic cell

Two-state conversion in a uniform field

Mean over random field directions

A separate stellar equation

Common difficulties

Stronger field is not always more conversion

The dashed curve is not a fit

Run the experiment

Watch the default sightline

Turn one domain

Make the field uniform

Compare with stars

Related experiments and articles

Articles

From two mixing states to a published limit