Improved Master for the LSS: Fast and accurate analysis of the two point power spectra and correlation functions

Abstract: We review the methodology for measurements of two point functions of the cosmological observables, both power spectra and correlation functions. For pseudo-$C_\ell$ estimators, we will argue that the window weighted overdensity field can yield more optimal measurements as the window acts as an inverse noise weight, an effect that becomes more important for surveys with a variable selection function. We then discuss the impact of approximations made in the {\sc Master} algorithm and suggest improvements, the {\sc $i$Master} algorithm, that uses the theoretical model to give unbiased results for arbitrarily complex windows provided that the model satisfies weak accuracy conditions. The methodology of {\sc $i$Master} algorithm is also generalized to the correlation functions to reconstruct the binned power spectra, for E/B mode separation, or to properly convolve the correlation functions to account for the scale cuts in the Fourier space model. We also show that the errors in the window estimation lead to both additive and multiplicative effects on the over density field. Multiplicative effects show up as mis-estimated window power and are typically higher order but can be important at the few percent precision requirements. Accurate estimation of window power can be required up to scales of $\sim 2\ell_\text{max}$ or larger. Misestimation of the window power leads to biases in the measured power spectra which scale as ${\delta C_\ell}\sim M^W_{\ell\ell'}\delta W_{\ell'}$, where the $M^W_{\ell\ell'}$ scales as $\sim(2\ell+1)C_\ell$ leading to effects that can be important at high $\ell$. While the notation in this paper is geared towards photometric galaxy surveys, the discussion is equally applicable to spectroscopic galaxy, intensity mapping and CMB surveys.