Astroparticle Physics

388 A Mathematical AppendixLaplace series for thetemperature variationsfinite series: practical limitTo quantify the temperature variations of the CMB, theLaplace series can be used to describeof the CMB T (θ, φ) = T(θ,φ)−〈T 〉= ∑ l∑a lm Y lm (θ, φ) ,l≥1 m=−l(A.16)where 〈T 〉 is the temperature averaged over all directions.Here the sum starts at l = 1, not l = 0, since by constructionthe l = 0 term gives the average temperature, whichhas been subtracted off. In some references one expandsT /〈T 〉 rather than T . This gives the equivalent informationbut with the coefficients simply differing from those in(A.16) by a factor of 〈T 〉.In practice one determines the coefficients a lm up tosome l max by means of a statistical parameter estimationtechnique such as the method of maximum likelihood. Thisprocedure will use as input the measured temperatures andinformation about their accuracy to determine estimates forthe coefficients a lm and their uncertainties.Once one has estimates for the coefficients a lm , one cansummarize the amplitude of regular variation with angle bydefiningC l = 12l + 1l∑m=−l|a lm | 2 . (A.17)angular power spectrumThe set of numbers C l is called the angular power spectrum.The value of C l represents the level of structure found at anangular separationθ = 180◦l. (A.18)The measuring device will in general only be able to resolveangles down to some minimum value; this determines themaximum measurable l.