Anisotropies in the Cosmic Microwave Background: An Analytic Approach

We introduce a simple yet powerful {\it analytic} method which obtains the structure of cosmic microwave background anisotropies to better than 5-10\% in temperature fluctuations on {\it all} scales. It is applicable to {\it any} model in which the potential fluctuations at recombination are both linear and known. Moreover, it recovers and explains the presence of the ``Doppler peaks'' at degree scales as {\it driven} acoustic oscillations of the photon-baryon fluid. We treat in detail such subtleties as the time dependence of the gravitational driving force, anisotropic stress from the neutrino quadrupole, and damping during the recombination process, again all from an analytic standpoint. We apply this formalism to the standard cold dark matter model to gain physical insight into the anisotropies, including the dependence of the peak locations and heights on cosmological parameters such as $\Omega_b$ and $h$, as well as model parameters such as the ionization history. Damping due to the finite thickness of the last scattering surface and photon diffusion are further more shown to be identical. In addition to being a powerful probe into the nature of anisotropies, this treatment can be used in place of the standard Boltzmann code where 5-10\% accuracy in temperature fluctuations is satisfactory and/or speed is essential. Equally importantly, it can be used as a portable standard by which numerical codes can be tested and compared.