CMB polarization as a tool to constrain the optical properties of the Universe

We present a novel formalism to describe the $in$ $vacuo$ conversion between polarization states of propagating radiation, also known as generalized Faraday effect (GFE), in a cosmological context. Thinking of GFE as a potential tracer of new, isotropy- and/or parity-violating physics, we apply our formalism to the cosmic microwave background (CMB) polarized anisotropy power spectra, providing a simple framework to easily compute their observed modifications. In so doing, we re-interpret previously known results, namely the $in$ $vacuo$ rotation of the linear polarization plane of CMB photons (or cosmic birefringence) but also point out that GFE could lead to the partial conversion of linear into circular polarization. We notice that GFE can be seen as an effect of light propagating in an anisotropic and/or chiral medium (a "dark crystal") and recast its parameters as the components of an effective "cosmic susceptibility tensor". For a wave number-independent susceptibility tensor, this allows us to set an observational bound on a GFE-induced CMB circularly polarized power spectrum, or $VV$, at $C_{\ell}^{VV} < 2 \times 10^{-5} \mu K^2$ (95 \% C.L.), at its peak $\ell\simeq 370$, which is some 3 orders of magnitude better than presently available direct $VV$ measurements. We argue that, unless dramatic technological improvements will arise in direct $V$-modes measurements, cosmic variance-limited linear polarization surveys expected within this decade should provide, as a byproduct, superior bounds on GFE-induced circular polarization of the CMB.