Fast computation of the observer motion effects induced on monopole frequency spectra for tabulated functions
Authors: Trombetti T. , Burigana C. , Tucci M. , Toffolatti L. 
Published: January 2024

Context. Various methods have been studied to compute the boosting effects produced by the observer peculiar motion, which modifies and transfers the isotropic monopole frequency spectrum of the cosmic background radiation to higher multipoles. Explicit analytical solutions for the spherical harmonic expansion coefficients were already presented and applied to different types of background spectrum, strongly alleviating the computational effort needed for accurate theoretical predictions. The frequency spectra at higher multipoles are inherently led by higher-order derivatives of the monopole spectrum. Provided that it can be well described by analytic or semi-analytic functions, the computation of its transfer is not affected by numerical instabilities when evaluated at the required level of numerical accuracy. Instead, monopole frequency spectra described by tabulated functions are computed with a relatively poor frequency resolution in comparison with the Doppler shift, which necessitates interpolation of the tabular representation. The spectra are also affected by uncertainties related to intrinsic inaccuracies in the modelling or in the related observational data as well as to limited accuracy in their numerical computation. These uncertainties propagate and increase with the derivative order, possibly preventing the trustworthy computation of the transfer to higher multipoles and of the observed monopole.
Aims. We study methods to filter the original function or its derivatives and the multipole spectra, to mitigate numerical instabilities, and to derive reliable predictions of the harmonic coefficients for different cosmic background models.
Methods. From the analytical solutions, and assuming that the monopole spectrum can be expanded in Taylor’s series, we derive explicit expressions for the harmonic coefficients up to the multipole ℓmax = 6 in terms of monopole spectrum derivatives. We then consider different low-pass filters: prefiltering in Fourier space of the tabular representation; filtering in both real and Fourier space of the numerical derivatives; interpolation approaches; and a dedicated method based on amplification and deamplification of the boosted signal. We study the quality of these methods when applied to suitable analytical approximations of the tabulated functions, possibly polluted with simulated noise. These methods are then applied to the tabulations.
Results. We consider two very different types of monopole spectra superimposed to the cosmic microwave background: the (smooth) extragalactic source microwave background signal from radio-loud active galactic nuclei and the (feature-rich) redshifted 21 cm line, and present our results in terms of spherical harmonic coefficients. The direct prediction of these coefficients can be noisy at ℓ > 1 or, depending on the uncertainty level, even at ℓ ≤ 1. Without assuming a functional form for the extragalactic background spectrum, the Gaussian prefiltering coupled to the sequential real-space filtering of derivatives allows us to derive accurate predictions up to ℓ ∼ 6, while a log–log polynomial representation, which is appropriate over several decades, gives accurate solutions at any ℓ. Instead, it is difficult to characterise the 21 cm line model variety, and so it is relevant to work without assumptions about the underlying function. Typically, the prefiltering provides accurate predictions up to ℓ ≃ 3 or 4, while the further sequential filtering of the derivatives or the boosting amplification and deamplification method improves the results up to ℓ = 4, while also allowing reasonable estimations of the spectrum at higher ℓ.
Conclusions. The proposed methods can significantly extend the range of realistic cosmic background models manageable with a fast computation, beyond the cases characterised a priori by analytical or semi-analytical functions. These methods require only an affordable increase in computation time compared to the direct calculation via simple interpolation.