Critical gravitational collapse of a massive complex scalar field

We study the critical collapse of a massive complex scalar field coupled minimally to gravity. Taking as initial data a simple Gaussian pulse with a shape similar to the harmonic ansatz for boson stars, we obtain critical collapse of type I and II when varying the Gaussian width sigma. For sigma <= 0.5 we find collapse of type II with a critical exponent gamma = 0.38 +/- 0.01 and an echoing period Delta = 3.4 +/- 0.1. These values are very similar to the well-known results for a real massless scalar field. On the other hand, for sigma >= 2.5 we obtain the collapse of type I. In this case we find that the critical solutions turn out to be an unstable boson stars in the ground state: all the data obtained from our simulations can be contrasted with the characteristic values for unstable boson stars and their corresponding Lyapunov exponents.