Astroparticle Physics

240 11 The Cosmic Microwave Backgrounddensity fluctuationsas sound wavesproper distanceangular diameter distanceA detailed modeling of the density fluctuations in theearly universe predicts a large level of structure on distancescales roughly up to the horizon distance. These fluctuationsare essentially sound waves in the primordial plasma, i.e.,regular pressure variations resulting from the infalling ofmatter into small initial density perturbations. These initialperturbations may have been created at a much earlier time,e.g., at the end of the inflationary epoch.By looking at the angular separation of the temperaturefluctuations, in effect one measures the distance betweenthe density perturbations at the time when the photons wereemitted.To relate the angles to distances, one needs to reviewbriefly the proper distance and angular diameter distance.The proper distance d p at a time t is the length one wouldmeasure if one could somehow stop the Hubble expansionand lay meter sticks end to end between two points. In anexpanding universe one finds that the current proper distance(i.e., at t 0 ) to the surface of last scattering is given by∫ t0dtd p (t ls ) = R(t 0 )t lsR(t) , (11.18)angular variationsof temperature variationsNote that this is the current proper distance to the positionof the photon’s emission, assuming that that place hasbeen carried along with the Hubble expansion. (The particle-horizondistance used above is simply the proper distanceto the source of a photon emitted at t = 0.) If matterdomination is assumed, i.e., R ∼ t 2/3 since t ls , then onegets d p (t ls ) = 3(t 0 − t ls ), which can be approximated byd p (t ls ) ≈ 3t 0 .Now, what one wants to know is the angle subtended bya temperature variation which was separated by a distanceperpendicular to our line of sight of δ = 3t ls when the photonswere emitted. To obtain this one needs to divide δ notby the current proper distance from us to the surface of lastscattering, but rather by the distance that it was to us at thetime when the photons started their journey. This locationhas been carried along with the Hubble expansion and isnow further away by a factor equal to the ratio of the scalefactors, R(t 0 )/R(t ls ). Using (11.7), therefore, one findsθ =δ R(t 0 )d p (t ls ) R(t ls ) = δ (1 + z) , (11.19)d p (t ls )