Damping of primordial gravitational waves from generalized sources

It has been shown that a cosmological background with an anisotropic stress tensor, appropriate for a free-streaming thermal neutrino background, can damp primordial gravitational waves after they enter the horizon, and can thus affect the cosmic microwave background $B$-mode polarization signature due to such tensor modes. Here we generalize this result and examine the sensitivity of this effect to nonzero neutrino masses, extra neutrino species, and also a possible relativistic background of axions from axion strings. In particular, additional neutrinos with cosmologically interesting neutrino masses at the $O(1)\mathrm{eV}$ level will noticeably reduce damping compared to massless neutrinos for gravitational wave modes with $k{\tau }_0\approx 100–200$, where ${\tau }_0\approx 2/H_0$ and $H_0$ is the present Hubble parameter, while an axion background would produce a phase-dependent damping distinct from that produced by neutrinos.

Figure 1

The $k$ dependence of the damping of an extra massive neutrino is demonstrated. The plots show that the damping is reduced for gravitational wave modes that enter the horizon as neutrinos are beginning to become nonrelativistic. The damping is also less at later times when the neutrino energy fraction has been reduced due to redshifting. The region in each plot on the left around the first minimum is zoomed in the adjacent plot on the right. For the $k{\tau }_0=1000$, the red and blue lines are overlapping, implying a negligible effect of mass.

Figure 2

The damping effect of extra massless neutrino species is shown. Each neutrino species has 2 degrees of freedom; thus $g_{\nu }=6$, 8, and 10 correspond to 3, 4, and 5 neutrinos.

Figure 3

The effect of axions produced by axionic strings.

Figure 4

The square of the ratio of the time derivative of damped modes to undamped modes, which is useful for calculating the $B$-mode correlation function $C_l^{BB}$.