  In $\Lambda$CDM cosmology, structure formation is halted shortly after dark
energy dominates the mass/energy budget of the Universe. A manifestation of
this effect is that in such a cosmology the turnaround radius has an upper
bound. Recently, a new, local, test for the existence of dark energy in the
form of a cosmological constant was proposed based on this turnaround bound.
Before designing an experiment that, through high-precision determination of
masses and turnaround radii, will challenge $\Lambda$CDM cosmology, we have to
answer two important questions: first, when turnaround-scale structures are
predicted to be close enough to their maximum size, so that a possible
violation of the bound may be observable. Second, which is the best mass scale
to target for possible violations of the bound. Using the Press-Schechter
formalism, we find that turnaround structures have in practice already stopped
forming, and consequently, the turnaround radius of structures must be very
close to the maximum value today. We also find that the mass scale of $\sim
10^{13} M_\odot$ characterizes turnaround structures that start to form in a
statistically important number density today. This mass scale also separates
turnaround structures with different cosmological evolution: smaller structures
are no longer readjusting their mass distribution inside the turnaround scale,
they asymptotically approach their ultimate abundance from higher values, and
they are common enough to have, at some epoch, experienced major mergers with
structures of comparable mass; larger structures exhibit the opposite behavior.
We call this mass scale the transitional mass scale and we argue that it is the
optimal for the purpose outlined above. As a corollary result, we explain the
different accretion behavior of small and larger structures observed in already
conducted numerical simulations.
