Astroparticle Physics

236 11 The Cosmic Microwave BackgroundSome of the mathematical formalism of the Laplace seriesis given in Appendix A.2. This expansion is analogous toa Fourier series, where the higher-order terms correspondto higher frequencies. Here, terms at higher l correspond tostructures at smaller angular scales. The same mathematicaltechnique is used in the multipole expansion of the poten-tial from an electric charge distribution. The terminology isborrowed from this example and the terms in the series arereferred to as multipole moments. The l = 0termisthemonopole, l = 1 the dipole, etc.Once one has estimates for the coefficients a lm ,theamplitudeof regular variation with angle can be summarizedby definingmultipole expansionangular power spectrumC l = 12l + 1l∑m=−l|a lm | 2 . (11.12)The 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◦ . (11.13)lThe measuring device will in general only be able to resolveangles down to some minimum value; this determines themaximum measurable l.11.5 The Monopole and Dipole Terms“The universe contains the record of itspast the way that sedimentary layers ofrock contain the geological record of theEarth’s past.”Heinz R. PagelsThe l = 0 term in the expansion of T(θ,φ)gives the temperatureaveraged over all directions. The most accurate de-termination of this value comes from the COBE satellite,monopole termIn the 1970s it was discovered that the temperature ofthe CMB in a particular direction was around 0.1% hotterthan in the opposite direction. This corresponds to a nonzerovalue of the l = 1ordipole term in the Laplace expan-sion. The dipole anisotropy has recently been remeasureddipole term〈T 〉=2.725 ± 0.001 K . (11.14)