What does $C^{XY}_{\ell}$ mean when we weasure $a_{\ell m}$ in the sky? In cosmology context, we have the general formula for the angular power spectrum $C_{\ell}$ :
$$C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\operatorname{Var}\left(a_{l m}\right)\quad(1)$$
In practise, we have to measure the $a_{\ell m}$ indirectly with the fluctuations of temperature by the relation :
$$a_{l m}=\int_{0}^{2 \pi} d \varphi \int_{0}^{\pi} \sin \theta d \theta \delta T(\theta, \varphi) Y_{l}^{m *}(\vec{\theta})\quad(2)$$
which is understable and allows to know how we measure the $a_{\ell m}$.
Now, how can I handle the following quantity with 2 probes X and Y (for example X represents a spectroscopic probe and a the photometric probe) :
$$\hat{C}_{\ell}^{X Y}=\frac{1}{2 \ell+1} \sum_{m} a_{\ell m}^{X}\left(a_{\ell m}^{Y}\right)^{*}\quad(3)$$

*

*It seems to be a mix between the 2 observales but are the $a_{\ell m}^{X}$ and $\left(a_{\ell m}^{Y}\right)^{*}$ are measured like in the first case in eq(1) with eq(2) ?

For example, I don't know how to measure :
$$\left(a_{\ell m}^{Y}\right)^{*}$$
Does $\hat{C}_{\ell}^{X Y}$ correspond to a "cross" angular power spectrum ? I mean physically, how to interpret it ?


*If we have a high statistics, I mean if we have a lot of $a_{\ell m}$ measures, i.e with a Shoit noise which is quasi-null, could we consider all $a_{\ell m}$ (with a high $\ell_{max}$ in the sum definition) as a constant and a same quantity for all $\ell$ and $m$ ?

In this case, that would imply that we fix the temperature fluctuations for a given probe ($X$ or $Y$, on all the full sky for CMB, wouldn't it ?
Any clarifications are welcome.
 A: 
It seems to be a mix between the 2 observales but are the ℓ and (ℓ)∗ are measured like in the first case in eq(1) with eq(2) ?

Yes, for example say $X$ was temperature, then $a_{\ell m}^X$ would be exactly the same as your Eq 2. If $Y$ was a different probe, say lensing, then you could compute $a_{\ell m}^Y$ using the same formula you gave for $a_{\ell m}$, except you would replace the temperature fluctuation $\delta T(\theta, \varphi)$ with $\delta L(\theta, \varphi)$, where $\delta L$ is some measure of lensing (eg, magnification). To compute $(a_{\ell m}^Y)^\star$, you just compute $a_{\ell m}^Y$ as described in the previousu sentence and then take the complex conjugate.

If we have a high statistics, I mean if we have a lot of ℓ measures, i.e with a Shoit noise which is quasi-null, could we consider all ℓ (with a high ℓ in the sum definition) as a constant and a same quantity for all ℓ and  ?

Let's say you want to measure the $C_\ell$ for a single value of $\ell$. Measuring $a_{\ell m}$ for each value of $m$ gives you an independent measurement; so the sum defining the $C_\ell$ is essentially an average of $2\ell + 1$ independent measurements. So, for large $\ell$, you have more independent measurements, and therefore the statistical uncertainty on $C_\ell$ is smaller.
Finally, you can think of $C_\ell^{AB}$ for a given $\ell$ as a covariance matrix, with indices $A$ and $B$. For $A=B=X$, $C_\ell^{XX}$ gives you have the variance of $X$ (for a given $\ell$). For $A=B=Y$, $C_\ell^{YY}$ gives you the variance of $Y$ (for a given $\ell$). For $A=X$ and $B=Y$, $C_\ell^{XY}$ gives you the covariance of $X$ and $Y$, which is a measure of how much $X$ is correlated with $Y$ (for a given $\ell$).
