Numerical study of cosmic no-hair conjecture. II. Analysis of initial data

We investigate the initial data for localized gravitational waves in space-time with a cosmological constant $\Lambda$. Using a conformal transformation, we find that the Hamiltonian and momentum constraints in the conformal frame turn out to be the same as those for a vacuum space-time without $\Lambda$. As initial data, we consider Brill's waves in the conformal frame and discuss the trapped and antitrapped surfaces in the physical frame. Just as Brill's wave in asymptotically flat space-time, the gravitational "mass" of our case is positive; however, the waves with a large gravitational mass do not always provide trapped surfaces in contrast with the case of $\Lambda =0\mathrm{}$. The large amount of gravitational waves, hence, does not seem to be an obstacle to the cosmic no-hair conjecture.