Angpow: a software for the fast computation of accurate tomographic
  power spectra
  The statistical distribution of galaxies is a powerful probe to constrain
cosmological models and gravity. In particular the matter power spectrum $P(k)$
brings information about the cosmological distance evolution and the galaxy
clustering together. However the building of $P(k)$ from galaxy catalogues
needs a cosmological model to convert angles on the sky and redshifts into
distances, which leads to difficulties when comparing data with predicted
$P(k)$ from other cosmological models, and for photometric surveys like LSST.
The angular power spectrum $C_\ell(z_1,z_2)$ between two bins located at
redshift $z_1$ and $z_2$ contains the same information than the matter power
spectrum, is free from any cosmological assumption, but the prediction of
$C_\ell(z_1,z_2)$ from $P(k)$ is a costly computation when performed exactly.
The Angpow software aims at computing quickly and accurately the auto
($z_1=z_2$) and cross ($z_1 \neq z_2$) angular power spectra between redshift
bins. We describe the developed algorithm, based on developments on the
Chebyshev polynomial basis and on the Clenshaw-Curtis quadrature method. We
validate the results with other codes, and benchmark the performance. Angpow is
flexible and can handle any user defined power spectra, transfer functions, and
redshift selection windows. The code is fast enough to be embedded inside
programs exploring large cosmological parameter spaces through the
$C_\ell(z_1,z_2)$ comparison with data. We emphasize that the Limber's
approximation, often used to fasten the computation, gives wrong $C_\ell$
values for cross-correlations.
