We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent tau. We investigate the case where tau is an element of (3,4), so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, resealed by n(-(tau-2)/(tau-1)), converge to hitting times of a "thinned" Levy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812-854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1-59]. Our results should be contrasted to the case tau > 4, so that the third moment is finite. There, instead, the sizes of the components resealed by n(-2/3) converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous [Ann. Probab. 25 (1997) 812-854] for the Erdos Renyi random graph and extended to the present setting in Bhamidi, van der Hofstad and van Leeuwaarden [Electron. J. Probab. 15 (2010) 1682-1703] and Turova [(2009) Preprint].
