Reduced shear correction for DeltaSigma model

[combet]

When we measure DeltaSigma from the data, we basically compute g*Sigma_c, where g is the reduced shear and not the shear gamma (g = gamma/(1-kappa)). However, the model definition of DeltaSigma is DeltaSigma = gamma*Sigma_c.

Far from the halo center, where kappa<<1, g = gamma and the DeltaSigma model (that we have currently in CLMM) that corresponds to gamma*Sigma_c is sufficient. However, when kappa starts to be "large" (R<~1 Mpc), kappa becomes sufficiently large that we need to account for it. In practice, this means multiplying DeltaSigma as defined in CLMM's theory modules by

(See McClintock (2019))

Below the the difference between the two for a 1e15 Msun cluster at z=0.5 and a source at z=2.

We should provide a function that implements the correction.

[hsinfan1996]

Does that mean users need to provide z_src to compute Sigma_c in the correction factor?

[combet]

Yes, this becomes dependent on the source redshift (either individual or distribution).

[combet]

@hsinfan1996, @m-aguena - I was going through open issues to see if anything outstanding needed addressing for v2.0 and I came upon this old one... I don't think we did anything about it, right?

[hsinfan1996]

@combet Where is the DeltaSigma calculation you are referring to in the code?

[combet]

So the DeltaSigma in the theory module corresponds to shear * Sigma_crit. We actually define the shear from DeltaSigma in theory (e.g. parent_class.py, l. 306)

def _eval_tangential_shear_core(self, r_proj, z_cl, z_src):
        delta_sigma = self.eval_excess_surface_density(r_proj, z_cl)
        sigma_c = self.cosmo.eval_sigma_crit(z_cl, z_src)
        return delta_sigma / sigma_c

But when we compute DeltaSigma from the data, what we do is reduce_shear * Sigma_crit, as the data only gives access to the reduced shear. So in the inner regions, I think the model DeltaSigma will not be the right model for the data and this may biased the mass estimation in a fit. So I think the user should have the option to compute the correction factor to apply to the model before fitting. But then, yes, the model depends on the data through the redshifts...

Not sure if I'm making sense 😅

[hsinfan1996]

I see. So basically there should be a version of excess surface density that is corrected by 1/(1-kappa).

[combet]

Actually, thinking about it again it may be a non-problem.

It's really dataops that may be "misleading", as it's multiplying the tan and cross components by sigmacrit when the flag is_deltasigma=True. If the tan and cross components are shear (say in simulations where this can be known), then yes, the result will be DeltaSigma and match the definition in theory. If these components are reduced shear (the case in real life), then the result will be not quite DeltaSigma in the inner regions. But maybe it's up to the user to know what they are doing and we can simply expand a bit the docstring of the is_deltasigma flag to make that point?

[hsinfan1996]

The docstrings for the function compute_tangential_and_cross_components in dataops mention the concern, but I am not sure if this is enough. We can definitely port this part to the argument is_deltasigma and maybe galaxycluster.py as well.

CLMM/clmm/dataops/__init__.py

Lines 71 to 79 in 157bdb0

	Finally, if the critical surface density (:math:`\Sigma_\text{crit}`) is provided, an estimate
	of the excess surface density :math:`\widehat{\Delta\Sigma}` is obtained from
	.. math::
	\widehat{\Delta\Sigma_{t,x}} = g_{t,x} \times \Sigma_\text{crit}(z_l, z_{\text{src}})
	If :math:`g_{t,x}` correspond to the shear, the above expression is an accurate. However, if
	:math:`g_{t,x}` correspond to ellipticities or reduced shear, this expression only gives an
	estimate :math:`\widehat{\Delta\Sigma_{t,x}}`, valid only in the weak lensing regime.

[combet]

Ah, thanks, I completely missed the warning. Yes, might be worth moving it to the is_deltasigma level in case people don't read further than that.