Gravitational waves in a closed spacetime via deviation equation

Within the closed universe, we obtain the amplitude and frequency of gravitational waves in the terms of discrete wave numbers, wave propagation time, and cosmological constant using the deviation equation in the first-order perturbed metric. We demonstrate that the cosmological constant effect on GWs is only seen in the early universe. Also, by considering the time evolution of a gravitational wave in a closed spacetime, we investigate its effect on a circle of nearby massless particles, which will be compared with this case in the flat spacetime. Expanding the universe has effective damping on GWs; thus, we suggest it can be used as a tool to characterize the large-scale curvature of the universe