The forgotten V

Anybody who ever read a book or some canonical introductory references on CMB polarization has found a small paragraph about the formal definition of the Stokes parameter Q, U, and V. From the combination of the Q and U parameters of the CMB signal it is straightforward to define the well known polarization field P=Q + iU which is now probably the most desired observable in cosmology. The poor V has always been left in the corner and in most of the paper I have ever read up to now is always neglected or claimed to be simply zero. As a (maybe) good student I always found myself thinking “Why is V damped so directly?”.

In the last year a few papers were published providing some further detail about the circular polarization of the CMB i.e. about the so-called V modes.

It's nice to see that a primordial circular polarization is not such an exotic phenomenon.

The V parameter is invariant for a rotation of the reference frame so, if present, it contributes to the brightness temperature. If the primordial plasma is even weakly magnetized and its magnetic field can be taken as sufficiently homogeneous (i.e. its inohomogeneity scale is fo the order of the Hubble radius) and we dealt with adiabatic scalar perturbations (see this paper for the contribution of tensor perturbation), a non vanishing V is produced.

The VT and VV power spectra are computed solving Boltzmann equations with the famous line of sight approach. VE and VB power spectra are much smaller and not quoted. The peculiar aspect of the V field is nevertheless its dependence on the frequency channel which is a direct consequence of the dependence of the source functions in the Boltzmann equations on the plasma and Larmor frequency. The lower the frequency, the higher the signal. The result is such that for a tensor to scalar ration of 0.1 and and a frequency channel of namely 30Ghz and a primordial magnetic field magnitude set by the actual constraints derived by CMB osbervations ( less then 10nG) the VT correlation signal could be as high as B modes on large scales, while the VV is much lower everywhere. Nevertheless, considering frequencies below the GHz, the VT signal could be 6 or 7 orders of magnitude larger then a putative B modes signal from gravitational lensing. Not as negligible as it seemed...

In a second (more technical) paper an opposite approach is also implemented: the magnetic field is seen not as a “ circular polarizer” starting from unpolarized CMB initial condition while as a tool to measure primordial polarization. We can in fact claim that the plasma doesn't posses a specified linear polarization without forbidding however the presence of a primordial circular polarization. If the plasma is not magnetized, the circular polarization evolves independently but, conversely, if it is magnetized the circular polarization can affect temperature and thus also E mode polarization anisotropies power spectra. Bounds on V polarization could in principle be set analyzing T and E power spectra computed with a proper set of equation. The paper describes then how and when this option can be taken into account with respect to a direct detection strategy of the circular polarization in order to obtain the best constraints on the intensity of an uniform primordial magnetic field.

Though the ideas is really interesting, unfortunately foregrounds are the Sword of Damocles and can forbid any progress in the desired direction.

In any case the first mandatory step should be updating the very old limits on the circular polarization set in the early Eighties at the VLA...