https://hal-insu.archives-ouvertes.fr/insu-03646063Bernardeau, FrancisFrancisBernardeauIPHT - Institut de Physique Théorique - UMR CNRS 3681 - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - Université Paris-Saclay - CNRS - Centre National de la Recherche ScientifiqueCNRS - Centre National de la Recherche ScientifiquePitrou, CyrilCyrilPitrouUzan, Jean-PhilippeJean-PhilippeUzanIAP - Institut d'Astrophysique de Paris - INSU - CNRS - Institut national des sciences de l'Univers - SU - Sorbonne Université - CNRS - Centre National de la Recherche ScientifiqueCMB spectra and bispectra calculations: making the flat-sky approximation rigorousHAL CCSD2011Astrophysics - Cosmology and Nongalactic AstrophysicsGeneral Relativity and Quantum Cosmology[SDU] Sciences of the Universe [physics]Gestionnaire, Hal Sorbonne Université2022-04-19 16:27:272022-04-21 03:14:452022-04-19 16:27:27enJournal articles10.1088/1475-7516/2011/02/0151This article constructs flat-sky approximations in a controlled way in the context of the cosmic microwave background observations for the computation of both spectra and bispectra. For angular spectra, it is explicitly shown that there exists a whole family of flat-sky approximations of similar accuracy for which the expression and amplitude of next to leading order terms can be explicitly computed. It is noted that in this context two limiting cases can be encountered for which the expressions can be further simplified. They correspond to cases where either the sources are localized in a narrow region (thin-shell approximation) or are slowly varying over a large distance (which leads to the so-called Limber approximation). Applying this to the calculation of the spectra it is shown that, as long as the late integrated Sachs-Wolfe contribution is neglected, the flat-sky approximation at leading order is accurate at 1% level for any multipole. Generalization of this construction scheme to the bispectra led to the introduction of an alternative description of the bispectra for which the flat-sky approximation is well controlled. This is not the case for the usual description of the bispectrum in terms of reduced bispectrum for which a flat-sky approximation is proposed but the next-to-leading order terms of which remain obscure.