Cosmic microwave background: Polarization and temperature anisotropies from symmetric structures

Perturbations in the cosmic microwave background (CMB) are generated by primordial inhomogeneities. I consider the case of CMB anisotropies from one single ordered perturbation source, or seed, existing well before decoupling between matter and radiation. Such structures could have been left by high energy symmetries breaking in the early universe. I focus on the cases of spherical and cylindrical symmetry of the seed. I give general analytic expressions for the polarization and temperature linear perturbations, factoring out of the Fourier integral the dependence on the photon propagation direction and on the geometric coordinates describing the seed. I show how the CMB perturbations manifestly reflect the symmetries of their seeds. In particular, polarization is uniquely linked to the shape of the source because of its tensorial nature. CMB anisotropies are obtained with a line of sight integration. They are a function of the position and orientation of the seed along the photons path. 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, CMB anisotropy from a seed intersecting the last scattering surface 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 deg 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; as expected for a linear structure with size $L<~H^{-1}$ and density contrast $\delta$ at decoupling, the temperature anisotropy is $\delta T/T\simeq \delta {(L/H}^{-1}{)}^2,$ roughly ten times stronger than the polarization. These waves could allow one to distinguish relics from high energy processes of the early universe from pointlike astrophysical sources, because of their angular extension and amplitude. Also, the marked analogy between polarization and temperature signals offers cross correlation possibilities for the future detection instruments. It would be interesting to detect these signals in the next 10 arc min CMB map provided by the Planck Surveyor satellite experiment.