value for $g$ is small enough for the approximation to make sense. The second equality, written in terms of the dimensionless conversion radius [@Fortin:2018ehg] \[EqnConvRadius\] is valid in the large conversion radius limit since in that limit $\hat{q}_\parallel\to1$ and the integral in the exponential can be trivially computed. Hence, from in the large conversion radius limit the normalized surface-subtracted Stokes parameter $R(\chi_0)$ at infinity is given by \[EqnRapprox\] where the norm of the integral in is a number of order one for our benchmark points.

Since the integral in is valid in the large conversion radius limit, it is possible to approximate it by integrating from the origin to infinity. Indeed, the integral is negligible in the interval $[0,r_0/r_{a\to\gamma}]$ due to the $1/(5t^5)$ term. The integral in can therefore be seen as a function of the product $\Delta_ar_{a\to\gamma}$ only. It can be evaluated approximatively in the large and small $|\Delta_ar_{a\to\gamma}|$ regimes using the steepest descent method or a change of variables respectively, leading to \[EqnRapproxanal\] Fig. \[FigBP\] compares the normalized surface-subtracted Stokes parameter $R$ for pure ALP initial state obtained from the evolution equations (blue) and the approximation (red), showing that they agree for small $g$, including our benchmark value 