Effects of observer peculiar motion on the isotropic background frequency spectrum: From the monopole to higher multipoles

Abstract

Context. The observer peculiar motion produces boosting effects in the anisotropy pattern of the considered background with frequency spectral behaviours related to its frequency spectrum. Aims. We study how the frequency spectrum of the background isotropic monopole emission is modified and transferred to the frequency spectra at higher multipoles, ℓ. We performed the analysis in terms of spherical harmonic expansion up to a certain value of ℓmax, for various models of background radiation, spanning the range between the radio and the far-infrared. Methods. We derived a system of linear equations to obtain the spherical harmonic coefficients and provide the explicit solutions up to ℓmax = 6. These are written as linear combinations of the signals at N = ℓmax + 1 colatitudes. We take advantage of the symmetry property of the associated Legendre polynomials with respect to π/2, which allows for the separation of the system into two subsystems: (1) for ℓ = 0 and even multipoles and (2) for odd multipoles. This improves the accuracy of the solutions with respect to an arbitrary choice of the adopted colatitudes. Results. We applied the method to different types of monopole spectra represented in terms of analytical or semi-analytical functions, that is, four types of distortions of the photon distribution function of the cosmic microwave background and four types of extragalactic background signals superimposed onto the cosmic microwave background’s Planckian spectrum, along with several different combinations of these types. We present our results in terms of the spherical harmonic coefficients and of the relationships between the observed and the intrinsic monopole spectra, as well as in terms of the corresponding all-sky maps and angular power spectra. For certain representative cases, we compare the results of the proposed method with those obtained using more computationally demanding numerical integrations or map generation and inversion. The method is generalized to the case of an average map composed by accumulating data taken with sets of different observer velocities, as is necessary when including the effect of the observer motion relative to the Solar System barycentre. Conclusions. The simplicity and efficiency of the proposed method can significantly alleviate the computational effort required for accurate theoretical predictions and for the analysis of data derived by future projects across a variety of cases of interest. Finally, we discuss the superposition of the cosmic microwave background intrinsic anisotropies and of the effects induced by the observer peculiar motion, exploring the possibility of constraining the intrinsic dipole embedded in the kinematic dipole in the presence of background spectral distortions.