NumCosmoXcorKernelRadialKDepGrowth

Scale-dependent growth across a smooth transition in $k$.

\begin{equation} g(\chi,k) = \exp\left[-\alpha\,\sigma(k)\, \frac{\chi_\mathrm{ref} - \chi}{\chi_\mathrm{ref}}\right] , \qquad \sigma(k) = \frac{(k/k_t)^2}{1 + (k/k_t)^2} . \end{equation}

Below $k_t$ the growth is scale-free, $g \to 1$; above it the effect saturates at $\alpha$ and accumulates along the line of sight, in the manner of free-streaming or a modified growth rate. It is an emulation with the right structure, closed form and exactly evaluable, not a fit to any particular model: $\alpha = 0$ recovers the scale-independent case exactly.

$g = 1$ at $\chi_\mathrm{ref}$ and departs from it on both sides — suppression at smaller $\chi$, enhancement at larger. Setting $\chi_\mathrm{ref}$ at or beyond the outer edge of the window therefore makes it a pure suppression across the whole support. The factor is deliberately left monotone and unclamped: bounding it would introduce a kink, and a kink inside an integration domain is exactly what these kernels exist to avoid.