Can the jet steepen the light curves of gamma-ray burst afterglows?

Beaming of relativistic ejecta in gamma-ray bursts (GRBs) has been postulated by many authors in order to reduce the total GRB energy; thus it is very important to look for the observational evidence of beaming. Rhoads has pointed out recently that the dynamics of the blast wave, which is formed when the beamed ejecta sweep the external medium, will be significantly modified by the sideways expansion caused by the increased swept-up matter. He has claimed that shortly after the bulk Lorentz factor (Gamma) of the blast wave drops below the inverse of the initial opening angle (theta(0)) of the beamed ejecta, there will be a sharp break in the afterglow light curves. However, some other authors have performed numerical calculations and shown that the break of the light curve is weaker and much smoother than the one analytically predicted. In this paper we reanalyze the dynamical evolution of the jet blast wave and calculate the jet emission analytically; we find that the sharp break predicted by Rhoads will actually not exist, and for most cases the afterglow light curve will be almost unaffected by sideways expansion unless the beaming angle is extremely small. We demonstrate that only when theta(0), < 0.1 may the afterglow light curves be steepened by sideways expansion, and in fact there cannot be two breaks as claimed before. We have also constructed a simple numerical code to verify our conclusion.