The maximum turnaround radius of non-spherical cosmic structures

Three simple idealised models are studied in order to develop some intuition about the leading effect of non-sphericity on maximum turnaround size $R_{\rm TA,max}$ large scale bound cosmic structures. Two them describe intrinsically axisymmetric static mass distributions whereas other is Kerr-de Sitter metric where axisymmetry generated due rotation structure. In all cases fractional change $\delta R_{\rm TA,max}(\theta)/R^{(0)}_{\rm a given structure, compared spherical one with same $M$, depends polar angle $\theta$ and proportional product relevant eccentricity parameter, times square small quantity. This quantity examples ratio two characteristic length scales, while spinning case it $v_{\rm out}/c$ azimuthal speed outmost members over light. Furthermore, angular average $\langle \delta TA,max}\rangle$ zero cases, negative ${\cal O}(v^2_{\rm out}/c^2)$ for Sitter. Thus, an structure very practically any value parameter. We speculate possible further implications our result turn around radius realistic