  In loop quantum cosmology (LQC) the big bang singularity is generically
resolved by a big bounce. This feature holds even when modified quantization
prescriptions of the Hamiltonian constraint are used such as in mLQC-I and
mLQC-II. While the later describes an effective description qualitatively
similar to that of standard LQC, the former describes an asymmetric evolution
with an emergent Planckian de-Sitter pre-bounce phase even in the absence of a
potential. We consider the potential relation of these canonically quantized
non-singular models with effective actions based on a geometric description. We
find a 3-parameter family of metric-affine $f(\mathcal{R})$ theories which
accurately approximate the effective dynamics of LQC and mLQC-II in all regimes
and mLQC-I in the post-bounce phase. Two of the parameters are fixed by
enforcing equivalence at the bounce, and the background evolution of the
relevant observables can be fitted with only one free parameter. It is seen
that the non-perturbative effects of these loop cosmologies are universally
encoded by a logarithmic correction that only depends on the bounce curvature
of the model. In addition, we find that the best fit value of the free
parameter can be very approximately written in terms of fundamental parameters
of the underlying quantum description for the three models. The values of the
best fits can be written in terms of the bounce density in a simple manner, and
the values for each model are related to one another by a proportionality
relation involving only the Barbero-Immirzi parameter.
