# What are the pros and cons of angular cross-correlation in real space and harmonic space?

In observational cosmology people often measure and model cross-correlations between different tracer maps. There are generally two ways to measure cross-correlations:

1. in real space (two-point function):

$$\xi^{uv}(\theta) \equiv \left\langle u(\boldsymbol{\theta'}) v(\boldsymbol{\theta'}+\boldsymbol{\theta}) \right\rangle_{\boldsymbol{\theta}'}$$

1. 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?

• For observational data I wonder if there are challenges to getting a power spectrum in situations where an incomplete (i.e. <4𝜋 sr) map is available. Hard truncations introduces strong artifacts and while apodization helps it isn't necessarily perfect. Then again I'm guessing that it would be easier to handle incomplete maps with correlations but I don't know. – uhoh Nov 11 '20 at 7:52
• Incomplete sky coverage introduces mode-mixing in C_l's. There are plenty of tools that can solve this problem, for example, PolSpice and NaMaster. But I agree if you use real-space correlation function you don't have to worry about that. – Astrolumos Nov 12 '20 at 0:02