By means of exact calculations, the complete recoil spectrum of a suddenly-perturbed N-electron Fermi sea is evaluated. Its threshold shape is compared with the theory of Nozieres and de Dominicis (1969) and Doniach and Sunjic (1970) which is believed to be asymptotically valid for small excitation energies and N to infinity . For N approximately=10, the N= infinity threshold behaviour is apparent. However, the distance from threshold over which the asymptotic threshold shape remains universally valid for all final-state interactions is limited to a small percentage of the Fermi energy. The present work provides a simple method for computing complete X-ray photoemission spectral (XPS) lineshapes for independent-electron metals. It also offers potential explanations of: (i) the incompatibilities between XPS threshold exponents and exponents extracted from other data; (ii) the occurrence of nearly-symmetric XPS lines; and (iii) the lack of mirror symmetry of absorption and emission edges.