Exact solutions of primordial gravitational waves

The future detection projects of gravitational waves are expected to have the sensitivity of detecting the primordial gravitational waves, which may be useful to get new insight into the very early universe. It is essential to analyze the evolution equation of the gravitational waves to estimate the present field strength of the primordial gravitational waves. In this paper, we obtain analytic solutions of the gravitational wave equation in the presence of non-relativistic matters as well as the cosmological constant. Although it is difficult to obtain the solution directly, we find that the equation for the square of the amplitudes has a simple polynomial solution. This quantity, which is directly related to the energy density of the gravitational waves, turns out to be useful to construct analytic solutions for the amplitudes by using Weierstrass's elliptic functions.