The Fourier Formalism¶

The Fourier formalism is described by Marsh et al 2022. Marsh et al. showed that, to leading order in $g_{a\gamma}$, the conversion probability $P_{\gamma_i\to a}$ (with $i\in [x,y]$) can be related to $\Delta_i(z)$, or equivalently, the magnetic field profile along the line of sight $B_i(z)$, using Fourier-like transforms. Although this treatment breaks down when the conversion probabilities exceed 5-10 per cent, it is an extremely useful framework when considering how different magnetic field treatments affect the conversion probability.

In this description, we will focus on a polarized beam using the $x$-component only. The unpolarised survival probability can always be obtained directly from

\[P_{\gamma\gamma} = (1 - P_{\gamma a}) = 1 - \frac{1}{2} \left( P_{\gamma_{x}\rightarrow a} + P_{\gamma_{y}\rightarrow a} \right),\]

ALPro comes with routines for applying the Fourier formalism in the massless and massive regimes. More information can be found under Examples & Tutorials.

Massive ALPs¶

In the massive ALP ($m_a \gg \omega_{\rm pl}$) case and focusing on the $x$-component only, the conversion probability can be written as

\[P_{\gamma_x \to a}(\eta) = {\cal F}_s( \Delta_{x})^2 + {\cal F}_c( \Delta_{x})^2\]

where

\[{\cal F}_c = \int^\infty_0 f(z) \cos(\eta z) dz\]

and

\[{\cal F}_s = \int^\infty_0 f(z) \sin(\eta z) dz\]

denote the cosine and sine transforms using a conjugate variable $\eta=m_a^2/2E$. By applying the Wiener Khinchin theorem, the conversion probability can also be expressed in terms of a cosine transform of the autocorrelation function of the line-of-sight magnetic field, $c_{B_x}$, as

\[P_{\gamma_x \to a}(\eta) = \frac{g_{a\gamma}^2}{2} {\cal F}_c \Big( c_{B_x}(L)\Big).\]

Massless ALPs¶

In the massless case ($m_a \ll \omega_{\rm pl}$), the same formalism applies if we transform to new variables. Specifically, $\Delta_x$ is replaced by the function $G=2 \Delta_x/\omega_{\rm pl}^2$, the line of sight distance coordinate is replaced by a phase factor proportional to the electron column density

\[\varphi=\frac{1}{2}\int_0^z{\rm d}z^\prime\omega_{\rm pl}^2(z^\prime),\]

and the conjugate Fourier variable is $\lambda=1/E$. With these transformations, $\Delta_x$ is replaced by the function $G=2 \Delta_x/\omega_{\rm pl}$ with a coordinate change from $z$ to a phase $\varphi$, and a conjugate variable $\lambda=1/E$. Nevertheless, the basic principle is similar to the massive ALP case, as the conversion probability can still be expressed as a simple transform of a function of the line of sight perpendicular magnetic field.

The Fourier Formalism

Contents

The Fourier Formalism¶

Massive ALPs¶

Massless ALPs¶