BLAST: Beyond Limber Angular power Spectra Toolkit. A fast and efficient algorithm for 3x2 pt analysis

The advent of next-generation photometric and spectroscopic surveys is approaching, bringing more data with tighter error bars. As a result, theoretical models will become more complex, incorporating additional parameters, which will increase the dimensionality of the parameter space and make posteriors more challenging to explore. Consequently, the need to improve and speed up our current analysis pipelines will grow. In this work, we focus on the $3\times2\mathrm{pt}$ statistics, a summary statistic that has become increasingly popular in recent years due to its great constraining power. These statistics involve calculating angular two-point correlation functions for the auto- and cross-correlations between galaxy clustering and weak lensing. The corresponding model is determined by integrating the product of the power spectrum and two highly-oscillating Bessel functions over three dimensions, which makes the evaluation particularly challenging. Typically, this difficulty is circumvented by employing the so-called Limber approximation, which is an important source of error. We present $\texttt{Blast.jl}$, an innovative and efficient algorithm for calculating angular power spectra without employing the Limber approximation or assuming a scale-dependent growth rate, based on the use of Chebyshev polynomials. The algorithm is compared with the publicly available beyond-Limber codes, whose performances were recently tested by the Rubin Observatory Legacy Survey of Space and Time Dark Energy Science Collaboration. At similar accuracy, $\texttt{Blast.jl}$ is $\approx 10$-$15 \times$ faster than the winning method of the challenge, also showing excellent scaling with respect to various hyper-parameters. $\texttt{Blast.jl}$ is publicly available on GitHub, and we release a repository where we explain how to use the code.