Behaviour of Lagrangian Approximations in Spherical Voids

We study the behaviour of spherical Voids in Lagrangian perturbation theories L(n), of which the Zel’dovich approximation is the lowest order solution L(1). We find that at early times higher order L(n) give an increasingly accurate picture of Void expansion. However at late times particle trajectories in L(2) begin to turnaround and converge leading to the contraction of a Void, a sign of pathological behaviour. By contrast particle trajectories in L(3) are well behaved and this approximation gives results in excellent agreement with the exact top-hat solution as long as the Void is not too underdense. For very underdense Voids, L(3) evacuates the Void much too rapidly leading us to conclude that the Zel’dovich approximation L(1), remains the best approximation to apply to the late time study of Voids. The behavior of high order approximations in spherical voids is typical for asymptotic series and may be generic for Lagrangian perturbation theory.