On the fast computation of the observer motion effects induced on monopole frequency spectra for tabulated functions

Methods are studied to compute the boosting effects produced by the observer motion that modifies and transfers to higher l the isotropic monopole frequency spectrum of the cosmic background. Explicit analytical solutions for spherical harmonic coefficients are presented and applied to various background spectra, alleviating computational effort. High l frequency spectra are led by higher order derivatives of the spectrum. Tabulated frequency spectra are computed with a relatively poor frequency resolution in comparison with the Doppler shift, calling for interpolation. They are affected by uncertainties due to intrinsic inaccuracies in modelling, observational data or limited computation accuracy, propagate and increase with the derivative order, possibly preventing a trustworthy computation to higher l and of the observed monopole. We filter the original function and the multipole spectra to derive reliable predictions of the harmonic coefficients. For spectra expressed in Taylor series, we derive explicit solutions for the harmonic coefficients up to l=6 in terms of spectra derivatives. We consider filters and study the quality of these methods on suitable analytical approximations, polluted with simulated noise. We consider the extragalactic sources microwave background from radio loud AGN and the 21cm line superimposed to the CMB. Gaussian pre-filtering coupled to a real space filtering of derivatives allows accurate predictions up to l=6, while log-log polynomial representation gives accurate solutions at any l. Describing the 21 cm model variety is difficult, so it is relevant to relax assumptions. Pre-filtering gives accurate predictions up to l=3-4, while further filtering or boosting amplification/deamplification method improves the results allowing reasonable estimations. The methods can extend the range of realistic background models manageable with a fast computation.