Fermions and gravitational gyrotropy

In conventional general relativity without torsion, high-frequency gravitational waves couple to the chiral number density of spin one-half quanta: the polarization of the waves is rotated by $2\pi N_5{\ell }_{\mathrm{Pl}}^2$, where $N_5$ is the chiral column density and ${\ell }_{\mathrm{Pl}}$ is the Planck length. This means that if a primordial distribution of gravitational waves with E-E or B-B correlations passed through a chiral density of fermions in the very early Universe, an E-B correlation will be generated. This in turn will give rise to E-B and T-B correlations in the cosmic microwave background (CMB). Less obviously but more primitively, the condition Albrecht called “cosmic coherence” would be violated, changing the restrictions on the class of admissible cosmological gravitational waves. This altered class of waves would, generally speaking, probe earlier physics than do the conventional waves; their effects on the CMB would be most pronounced for low ($\lesssim 100$) multipoles. Rough estimates indicate that if the tensor-to-scalar ratio is less than about ${10}^{-2}$, it will be hard to constrain a spatially homogeneous primordial $N_5$ by present data.

Figure 1

Alpha- and beta-mode gravitational-wave transfer functions, computed with a modified version of class, using cosmological data are compatible with Planck and WMAP 2013 results. Top: temperature transfer functions. Middle: E mode transfer functions. Bottom: B mode transfer functions. (All are for $\ell =2$.)

Figure 2

Alpha- and beta-mode contributions to the TT, EE, TE and BB correlations (clockwise from top left), for a tensor-to-scalar ratio $r=0.35$ and cosmological data compatible with Planck and WMAP 2013 results, along with low-multipole data from Planck.

Figure 3

Alpha- and beta-mode contributions to the TE correlations, for a tensor-to-scalar ratio $r=0.35$ and cosmological data compatible with Planck and WMAP 2013 results. For negative correlations, the absolute value is shown but the graph is shaded below. The signs of the alpha- and beta-mode contributions correlate closely.

Figure 4

Alpha- and beta-mode contributions to the EB and TB correlations, for a tensor-to-scalar ratio $r=0.35$ and cosmological data compatible with Planck and WMAP 2013 results. Top left: individual alpha-beta and beta-alpha cross-term contributions to the EB correlations. Top right: total cross-term EB contributions. Bottom left: individual alpha-beta and beta-alpha cross-term contributions to the TB correlations. Bottom right: total cross-term TB contributions.