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.
