Analytic Spectra of CMB Anisotropies and Polarization Generated by
  Scalar Perturbations in Synchronous Gauge
  We present a detailed analytical calculation of CMB temperature anisotropies
\alpha_k and polarization \beta_k generated by scalar metric perturbations in
synchronous gauge, parallel to our previous work with RGW as a generating
source. This is realized primarily by an analytic time-integration of
Boltzmann's equation, yielding the closed forms of \alpha_k and \beta_k.
Approximations, such as the tight-coupling approximation for photons a prior to
the recombination and the long wavelength limit for scalar perturbations are
used. The residual gauge modes in scalar perturbations are analyzed and a
proper joining condition of scalar perturbations at the radiation-matter
equality is chosen, ensuring the continuity of energy perturbation.
  The resulting analytic expressions of the multipole moments of polarization
a^E_l, and of temperature anisotropies a^T_l are explicit functions of the
scalar perturbations, recombination time, recombination width, photon free
streaming damping factor, baryon fraction, initial amplitude, primordial scalar
spectral index, and the running index. These results show that a longer
recombination width yields higher amplitudes of polarization on large scales
and more damping on small scales, and that a late recombination time shifts the
peaks of C^{XX'}_l to larger angular scales.
  The analytic spectra C^{XX'}_l agree with the numerical ones and with those
observed by WMAP on large scales (l \lesssim 500), but deviate considerably
from the numerical results on smaller scales, showing the limitations of our
approximate analytic calculations. Several possible improvements are pointed
out for further studies.
