Beyond Zel'dovich-Type Approximations in Gravitational Instability Theory --- Pad\'e Prescription in Spheroidal Collapse ---

Among several analytic approximations for the growth of density fluctuations in the expanding Universe, Zel’dovich approximation in Lagrangian coordinate scheme is known to be unusually accurate even in mildly non-linear regime. This approximation is very similar to the Pad'e approximation in appearance. We first establish, however, that these two are actually different and independent approximations with each other by using a model of spheroidal mass collapse. Then we propose Pad'e-prescribed Zel’dovich-type approximations and demonstrate, within this model, that they are much accurate than any other known nonlinear approximations.