Random domains, polarization and stellar cooling

This chapter assumes the two-state formula and the critical energy from Mixing in a cluster magnetic field and plasma, plus the idea of a density matrix for an incoherent mixture of states.

The cell model and what it leaves out

Split the path into $N$ cells of length $L$. In cell $j$ the transverse field has a fixed strength and a direction $\psi_j$ in the sky plane, drawn uniformly and independently. Inside a cell the two-state formula applies to the polarization along $\psi_j$. The cell's transfer matrix in the fixed basis $(A_x,A_y,a)$ is the single-cell solution rotated by $\psi_j$, and the whole sightline is the product of these matrices.

Real cluster fields are turbulent and spread over a range of scales. Matthews and colleagues found that resolving structure below the coherence length can matter, though for the Perseus limit a Gaussian random field gave only slightly weaker results than cells. Matthews et al., 2022 Treat the cell model as a controlled approximation to such a field.

Deriving the mean over random directions

Unpolarized light is the mixture $\rho=\mathrm{diag}(\tfrac12,\tfrac12,0)$. Let $I_a$ be the axion fraction before a cell and $I_\gamma=1-I_a$ the photon fraction. Two facts follow from averaging over the new angle $\psi_j$, which is independent of everything before it.

First, the photon intensity along the new field direction is on average $I_\gamma/2$, because $\cos^2\psi_j$ and $\sin^2\psi_j$ both average to one half and the $\cos\psi_j\sin\psi_j$ term averages to zero. Second, any leftover coherence between the photon and the axion enters the new cell multiplied by $\cos\psi_j$ or $\sin\psi_j$, so its contribution to the next axion fraction also averages to zero.

What remains is a population update. Axions convert back with probability $P_j$, and parallel photons convert forward with the same probability:

Write $D=\tfrac13-\langle I_a\rangle$. The update becomes $D'=D(1-\tfrac32P_j)$, and starting from $D=\tfrac13$,

For small $P_j$ the product becomes an exponential, $\tfrac13[1-\exp(-\tfrac32NP)]$. That continuum form is eq. (5) of Mirizzi, Raffelt and Serpico, 2007. For few cells the sum $NP$ is recovered, and for many cells the result tends to one third: the light ends up shared evenly among two photon polarizations and one axion.

Under these assumptions, independent uniform angles and fixed cell strengths, the product rule is exact for the mean. We checked it against 4000 Lab sightlines and against 2000 sightlines from a separate Python script with a different random-number generator. Both means agree with the formula within statistical error.

Worked example: one sightline against the mean

Take the default preset: 24 cells of 8 kpc, $g_{a\gamma}=10^{-11.5}\,\mathrm{GeV^{-1}}$, $B_\perp=5\,\mu\mathrm G$, $n_e=10^{-2.5}\,\mathrm{cm^{-3}}$, at 3 keV. One cell converts a parallel photon with $P_j=0.0344$. The short-path estimate $(\Delta_{a\gamma}L)^2=0.0373$ is 8 percent too high, because the plasma term is not negligible. Then

The default sightline, with its particular set of angles, gives an axion fraction of only 0.117 at the same energy. That is less than half the mean, and it is a normal outcome, not an error. Across 0.5 to 10 keV the largest gap between this sightline and the mean is 0.225. A fit to data must therefore compare against many realizations, as Reynolds and colleagues did with 500 per field model.

Why the surviving light is polarized

Inside each cell only the polarization along the field loses intensity. Light that entered unpolarized leaves with more intensity across the field than along it. For one cell the linear polarization degree is $P_j/(2-P_j)$, close to $P_j/2$ when $P_j$ is small. Over many cells the preferred direction wanders with the field, so the net polarization depends on the realization. On the default sightline at 3 keV it is 0.116.

This is why polarization is a separate handle. A source with a known, low intrinsic polarization can limit the coupling through its observed polarization rather than through spectral ripples. Day and Krippendorf presented techniques and projected bounds for doing this with X-ray polarimetry of point sources seen through large magnetic fields. Day and Krippendorf, 2018 The Particle Data Group review reports a limit of $1.7\times10^{-12}\,\mathrm{GeV^{-1}}$ at 95 percent confidence from optical polarization of magnetic white dwarfs, whose fields are measured directly. PDG review of axions, 2025

Stars: production instead of propagation

Inside a star there is no need for a large-scale field. A thermal photon passing a nucleus or an electron can convert into an axion in that particle's electric field. This is the Primakoff process. The axions escape freely and carry energy away.

The stellar argument compares the energy a star releases with the energy axions carry away. The PDG review gives a core-averaged helium-burning energy release of about 80 erg per gram per second for stars on the horizontal branch. Primakoff losses add about $30g_{10}^2$ in the same units, with $g_{10}=g_{a\gamma}/10^{-10}\,\mathrm{GeV^{-1}}$. Faster helium consumption shortens the horizontal-branch lifetime by the factor

Red giants are much less affected. The ratio $R$ of horizontal-branch to red-giant stars in a globular cluster therefore falls as the coupling grows. Ayala and colleagues measured an average $R=1.39\pm0.04$ in 39 clusters. Combining it with stellar models and the helium abundance, they found $g_{a\gamma}<6.6\times10^{-11}\,\mathrm{GeV^{-1}}$ at 95 percent confidence. Ayala et al., 2014 At that coupling the factor above is 0.860: horizontal-branch stars would live about 14 percent shorter.

The two limits share only the coupling constant. The X-ray limit needs a model of the cluster field and holds only for axions lighter than about $10^{-12}$ eV. The stellar limit needs stellar evolution and the helium abundance and holds for masses up to a few keV, where the stellar plasma can still make the particles.

Lab tasks

1. Predict. On the default preset, read the axion fraction at 3 keV and compare it with the dashed mean. Before reseeding, guess whether a new realization will land closer to the mean.
2. Change one condition. Drag one cell's arrow by 90 degrees. Watch the profile chart: the conversion in later cells changes, because the state entering them has changed.
3. Compare. Raise the coupling slider until the dashed mean flattens. Check that it approaches two thirds survival.
4. Explain. Raise the coupling to the horizontal-branch limit marked on the stellar ruler. Explain why the X-ray spectrum at that coupling would be grossly distorted, while the stellar lifetime changes by only 14 percent.

Exercises

1. Identical cells each have $P_j=0.02$. Compute the mean axion fraction for unpolarized light with the product rule and with the naive estimate $NP_j/2$, for 12 cells and for 40 cells.
2. What coupling shortens the horizontal-branch lifetime by 3 percent? Why is the published bound larger than this value even though $R$ is known to about 3 percent?
3. Give one reason why polarization and spectral ripples need not give the same limit for the same cluster.

Try it in the Lab: Turn one domain arrow, then raise the coupling to the first stellar limit.