Damping of tensor mode in spatially closed cosmology

We derive an integro-differential equation for propagation of cosmological gravitation waves in spatially closed cosmology whereas the traceless transverse tensor part of the anisotropic stress tensor is free-streaming neutrinos (including antineutrinos), which have been traveling essentially without collision since temperature dropped below about $10^{10}\mathrm{K}$. We studied the short wavelengths and long wavelengths of gravitational waves (GWs) that enter the horizon in closed spacetime. The solution shows that the anisotropic stress reduces the squared amplitude by 76% for wavelengths that enter the horizon during radiation-dominated phase and this reduction is less for the wavelength that enters the horizon at later times. At the end we compare the results to the flat case and then we investigate the dependence of the evolution of GWs on cosmological parameters.

Figure 1

In the radiation-dominated era when the perturbation enters the horizon, the free-streaming neutrinos in closed cosmology (solid line) have a greater effect than the flat case (dashed line) on the damping of gravitational waves.