In observational cosmology people often measure and model cross-correlations between different tracer maps. There are generally two ways to measure cross-correlations:
- in real space (two-point function):
$$ \xi^{uv}(\theta) \equiv \left\langle u(\boldsymbol{\theta'}) v(\boldsymbol{\theta'}+\boldsymbol{\theta}) \right\rangle_{\boldsymbol{\theta}'} $$
- in harmonic space (angular power spectrum)
$$ \xi^{uv}(\theta)=\int d \ell \frac{\ell}{2 \pi} C_{\ell}^{uv} J_{0}(\ell \theta) $$
I'm wondering in the sense of measurements, modelling, systematics, etc, what are the pros and cons of these two methods?
