Cosmic microwave background: polarization and temperature anisotropies from symmetric structures

I explore the undulatory properties of the cosmic microwave background (CMB) physics. I consider the cases of spherical and cylindrical symmetry of the perturbation source, or seed. Such structures could have been left by high energy symmetries breaking in the early universe. I give suitable analytic expressions for the polarization and temperature linear perturbations from this kind of seeds and I show how to get their appearence on the CMB sky. This treatment highlights the undulatory properties of the CMB. I show with numerical examples how the polarization and temperature perturbations propagate beyond the size of their seeds, reaching the CMB sound horizon at the time considered. Just like the waves from a pebble thrown in a pond, the CMB anisotropy appears as a series of temperature and polarization waves surrounding the seed, extending on the scale of the CMB sound horizon at decoupling, roughly $1^{o}$ in the sky. Each wave is characterized by its own value of the CMB perturbation, with the same mean amplitude of the signal coming from the seed interior. These waves could allow to distinguish relics of the very early universe from point-like astrophysical sources, because of their angular extension and amplitude. Also, the analogy between the polarization and temperature CMB signals offers cross correlation possibilities for the future detection instruments like the Planck Surveyor satellite.