Mixing in a cluster magnetic field and plasma

The previous chapter, Photon and axion as two mixing states, derived the two-state probability. This chapter puts cluster numbers into it.

The plasma frequency of intracluster gas

Free electrons respond to a passing wave and shift its dispersion. For X-rays far above the plasma frequency the photon behaves as if it had mass $\omega_{\rm pl}$, with

Here $n_e$ is the electron number density, $m_e$ the electron mass and $\alpha$ the fine-structure constant. Evaluated with CODATA constants,

Meyer, Horns and Raue, 2013 print $3.69\times10^{-11}$ for the coefficient. Ours is 0.6 percent larger. We kept the value computed from the constants and recorded the difference instead of matching the rounded one.

Cluster gas has densities of roughly $10^{-3}$ to $10^{-1}\,\mathrm{cm^{-3}}$, so $\omega_{\rm pl}$ is a few times $10^{-12}$ eV. At $n_e=10^{-2.5}\,\mathrm{cm^{-3}}$ it is $2.09\times10^{-12}$ eV, and the photon term at 1 keV is $\Delta_{\rm pl}=-0.341\,\mathrm{kpc^{-1}}$. It scales as density over energy.

One ratio decides the regime

The amplitude factor from the two-state formula can be rewritten with a single ratio. The mismatch is $\Delta_{\rm pl}-\Delta_a=(m_a^2-\omega_{\rm pl}^2)/(2E)$ and twice the mixing is $g_{a\gamma}B_\perp$. Their ratio is $E_{\rm crit}/E$, with

Substituting gives an exact form for the largest possible conversion:

Far above $E_{\rm crit}$ the factor approaches one. Far below it, it falls as $(E/E_{\rm crit})^2$. The same critical energy appears as eq. (6) of Meyer, Horns and Raue, written as $2.5$ GeV times the mass difference in square nanoelectronvolts, divided by the coupling in units of $10^{-11}\,\mathrm{GeV^{-1}}$ and the field in microgauss. Recomputing their prefactor from the constants gives 2.56 GeV, so their 2.5 is a rounded value.

Three cases from the Lab presets

Default tangled field. With $g_{a\gamma}=10^{-11.5}\,\mathrm{GeV^{-1}}$, $B_\perp=5\,\mu\mathrm G$, $n_e=10^{-2.5}\,\mathrm{cm^{-3}}$ and $m_a=10^{-13}$ eV, the axion mass is negligible next to the plasma frequency. The formula gives $E_{\rm crit}=7.04$ keV. Meyer's rounded prefactor gives 6.88 keV, a 2 percent difference that comes entirely from the rounding. Across the 1 to 9 keV band the amplitude factor climbs from about 0.02 to above 0.6, so conversion grows with energy.

Mass matches the plasma. Set $m_a=\omega_{\rm pl}=2.09\times10^{-12}$ eV. The diagonal entries are equal, the amplitude factor is one at every energy and $\Delta_{\rm osc}=2\Delta_{a\gamma}$. With $g_{a\gamma}=10^{-12}\,\mathrm{GeV^{-1}}$, $B_\perp=5\,\mu\mathrm G$ and a uniform 96 kpc path,

Unpolarized survival is 0.776, the same at 0.5 keV as at 10 keV. In a real cluster the density changes along the path, so a resonance like this would hold only in a thin shell where $\omega_{\rm pl}$ happens to equal $m_a$.

Heavier axion. Raise the mass to $10^{-11}$ eV with $g_{a\gamma}=10^{-11}\,\mathrm{GeV^{-1}}$. Now the axion term dominates and $E_{\rm crit}=49.0$ keV. The amplitude factor is $4.2\times10^{-4}$ at 1 keV and $1.7\times10^{-2}$ at 6.5 keV, a forty-fold rise. The spectrum is untouched at low energy and dips at high energy. A low-energy cutoff of this kind is how X-ray searches lose sensitivity as the mass grows.

The Perseus profile at the published limit

Reynolds and colleagues describe the Perseus electron density with a two-component profile from earlier X-ray work, their eq. (2):

In their field model B the magnetic pressure is a fixed fraction of the gas pressure, the field is about 7.5 microgauss at 25 kpc, and treating the cluster as isothermal makes the field fall as the square root of the density. Reynolds et al., 2020 Leave out their random cell structure and evaluate only that radial scaling at the coupling they exclude, $g_{a\gamma}=7.9\times10^{-13}\,\mathrm{GeV^{-1}}$:

Every critical energy sits far above the Chandra band. At this coupling Perseus mixes X-rays only weakly: a single coherent cell converts well under one percent. A percent-level signal builds up only because the photons cross many cells. At 5 keV the oscillation length runs from about 8 kpc near the centre to about 74 kpc at 200 kpc, close to the cell sizes, so each cell's contribution changes phase with energy. That is where the ripples come from.

The Lab does not apply a telescope's energy resolution. Ripples narrower than the instrument resolution would be partly smoothed in a real spectrum, and a real analysis has to pass the model through the instrument response.

Lab tasks

1. Predict. On the default preset, open the advanced panel and read the critical energy. Say whether a 2 keV photon or an 8 keV photon converts more on average.
2. Change one condition. Raise the density slider by one decade. Predict how the critical energy changes, then check.
3. Compare. Choose the plasma-resonance preset and drag the energy cursor across the whole band. The solid curve should stay flat.
4. Explain. Choose the heavier-axion preset. Find roughly where the dashed mean first drops below 0.95. It happens near 3.6 keV. Explain why this is more than ten times below the critical energy: each of the 24 cells adds a small conversion, and the small amounts accumulate.

Exercises

1. A cluster has $n_e=10^{-3}\,\mathrm{cm^{-3}}$ and $B_\perp=2\,\mu\mathrm G$. For a very light axion and $g_{a\gamma}=10^{-12}\,\mathrm{GeV^{-1}}$, estimate $E_{\rm crit}$.
2. At fixed coupling and field, how does $E_{\rm crit}$ change if the density doubles? If instead the axion mass rises well above the plasma frequency and then doubles?
3. Why does the resonant case lose its energy dependence even though $\Delta_{\rm pl}$ itself still depends on energy?

Try it in the Lab: Compare Mass matches the plasma with Heavier axion, low-energy cutoff, and read the critical energy in the advanced panel.