Astroparticle Physics

11.4 CMB Anisotropies 235Once the photons and matter decoupled, the photonssimply continue unimpeded to the present day. One can definea surface of last scattering as the sphere centered aboutus with a radius equal to the mean distance to the last placewhere the CMB photons scattered. To a good approximationthis is equal to the distance to where decoupling took placeand the time of last scattering is essentially the same as t dec .So, when the CMB is detected, one is probing the conditionsin the universe at a time of approximately 380 000 years afterthe Big Bang.surface of last scatteringtransparent universe11.4 CMB Anisotropies“As physics advances farther and fartherevery day and develops new axioms,it will require fresh assistancefrom mathematics.”Francis BaconThe initial measurements of the CMB temperature by Penziasand Wilson indicated that the temperature was independentof direction, i.e., that the radiation was isotropic, towithin an accuracy of around 10%. More precise measurementseventually revealed that the temperature is about onepart in one thousand hotter in one particular direction of thesky than in the opposite. This is called the dipole anisotropyand is interpreted as being caused by the motion of the Earththrough the CMB. Then in 1992 the COBE satellite foundanisotropies at smaller angular separations at a level of onepart in 10 5 . These small variations in temperature have recentlybeen measured down to angles of several tenths of adegree by several groups, including the WMAP 1 satellite,from which one can extract a wealth of information aboutthe early universe.In order to study the CMB anisotropies one begins witha measurement of the CMB temperature as a function of direction,i.e., T(θ,φ),whereθ and φ are spherical coordinates,i.e., polar and azimuthal angle, respectively. As withany function of direction, it can be expanded in sphericalharmonic functions Y lm (θ, φ) (a Laplace series),dipole anisotropydiscoveryof small-angle anisotropiesLaplace series∞∑ l∑T(θ,φ)= a lm Y lm (θ, φ) . (11.11)l=0 m=−l1 WMAP – Wilkinson Microwave Anisotropy Probe