  We study the head-on collision of two equal-mass momentarily stationary black
holes, using black hole perturbation theory up to second order. Compared to
first-order results, this significantly improves agreement with numerically
computed waveforms and energy. Much more important, second-order results
correctly indicate the range of validity of perturbation theory. This use of
second-order, to provide ``error bars,'' makes perturbation theory a viable
tool for providing benchmarks for numerical relativity in more generic
collisions and, in some range of collision parameters, for supplying waveform
templates for gravitational wave detection.
