General relativity has predicted the existence of gravitational waves (GW), which are waves of the distortions of space-time with two degrees of polarization and the propagation speed of light. Alternative theories predict more polarizations, up to a maximum of six, and possible frequency dependent propagation speed from the light speed. The polarization and dispersion properties of GWs shed light on the spin and mass information of gravitons. Although GWs have not been directly detected yet,

more »

... eir amplitude upper-bounds has been addressed by research using different types of detectors. For example, the amplitude upper-bounds for the stochastic background derived from pulsar timing observations have already become astrophysically interesting. The present paper reviews proposals to test the gravity theories in the radiation regime by observing GWs using pulsar timing arrays. We also present the estimation for the upper-bounds on the amplitude of alternative modes for the stochastic background of GW.