In galaxy cluster weak lensing, a significant systematic issue is cluster member dilution also known as boost factors. The idea is that if some cluster galaxies are misidentified as source (background) galaxies, then your weak lensing signal is diluted due to the fact that the cluster member galaxy won't be sheared or magnified. Traditionally, one calculates or estimates this correction and "boosts" the data vector. This boost factor is radially dependent, since you will tend to misidentify cluster members close to the cluster more than those farther out. Mathematically this looks like
\Delta\Sigma_{\rm corrected}(R) = (1-f_{\rm cl})^{-1}(R)\Delta\Sigma(R)
where f_{\rm cl} is the fraction of cluster members misidentified as being source galaxies. For shorthand, we write \mathcal{B} = (1-f_{\rm cl})^{-1}. This module provides multiple models for \mathcal{B}.
In McClintock et al. (in prep.) we model the boost factor with an NFW model:
\mathcal{B}(R) = 1+B_0\frac{1-F(x)}{x^2-1}
where x=R/R_s and
F(x) = \Biggl\lbrace
\begin{eqnarray}
\frac{\tan^{-1}\sqrt{x^2-1}}{\sqrt{x^2-1}} : x > 1\\
1 : x = 1\\
\frac{\tanh^{-1}\sqrt{1-x^2}}{\sqrt{1-x^2}} : x < 1.
\end{eqnarray}
Parameters that need to be specified by the user are B_0 and the scale radius R_s. To use this, you would do:
from cluster_toolkit import boostfactors import numpy as np R = np.logspace(-2, 3, 100) #Mpc/h comoving B0 = 0.1 #Typical value Rs = 1.0 #Mpc/h comoving; typical value B = boostfactors.boost_nfw_at_R(R, B0, Rs)
In Melchior et al. we used a power law for the boost factor.
\mathcal{B} = 1 + B_0\left(\frac{R}{R_s}\right)^\alpha
Here, the input parameters are B_0, the scale radius R_s, and the exponent \alpha. This is also available in this module:
from cluster_toolkit import boostfactors import numpy as np R = np.logspace(-2, 3, 100) #Mpc/h comoving B0 = 0.1 #Typical value Rs = 1.0 #Mpc/h comoving; typical value alpha = -1.0 #arbitrary B = boostfactors.boost_powerlaw_at_R(R, B0, Rs, alpha)
This figure shows the NFW boost factor model:
This figure shows how the boost factor changes the \Delta\Sigma(R) profile:
