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This article constructs flat-sky approximations in a controlled way in thecontext of the cosmic microwave background observations for the computation ofboth spectra and bispectra. For angular spectra, it is explicitly shown thatthere exists a whole family of flat-sky approximations of similar accuracy forwhich the expression and amplitude of next to leading order terms can beexplicitly computed. It is noted that in this context two limiting cases can beencountered for which the expressions can be further simplified. Theycorrespond to cases where either the sources are localized in a narrow region(thin-shell approximation) or are slowly varying over a large distance (whichleads to the so-called Limber approximation). Applying this to the calculationof the spectra it is shown that, as long as the late integrated Sachs-Wolfecontribution is neglected, the flat-sky approximation at leading order isaccurate at 1% level for any multipole. Generalization of this constructionscheme to the bispectra led to the introduction of an alternative descriptionof the bispectra for which the flat-sky approximation is well controlled. Thisis not the case for the usual description of the bispectrum in terms of reducedbispectrum for which a flat-sky approximation is proposed but thenext-to-leading order terms of which remain obscure.

CMB spectra and bispectra calculations: making the flat-sky approximation rigorous