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