NFW discontinuity in the convergence

[rouzib]

Problem

NFW + m3 Multipole lensing model exhibits discontinuities in the convergence profile around r ≈ 1.

Example:

Context

Relevant code is at:

caustics/src/caustics/lenses/func/nfw.py

Line 141 in 683ca5a

return torch.where(x > 1, f_pos, torch.where(x < 1, f_neg, torch.zeros_like(x)))

Suggested Fix

Apply TNFW-style patch to smooth out these discontinuities with Taylor expansion.

caustics/src/caustics/lenses/func/tnfw.py

Lines 58 to 66 in 683ca5a

	return torch.where(
	x < 1 - 1e-2,
	torch.arccosh(1 / x_lt1) / (1.0 - x_lt1**2).sqrt(),
	torch.where(
	x > 1 + 1e-2,
	torch.arccos(1 / x_gt1) / (x_gt1**2 - 1.0).sqrt(),
	1 - 2 * (x - 1) / 3 + 7 * (x - 1) ** 2 / 15, # where: x == 1
	),
	)