state (contracting universe) onto the outgoing one (expanding). Known properties of the limit allow to explicitly determine the scattering matrix and thus to relate the properties of the incoming and outgoing states. The procedure is explained in detail in Section \[sec:scatter\] through an application to the cases $(i)$ and $(ii)$ listed above.

The paper is organized as follows: First, in Sec. \[sec:frame\] we briefly introduce the main aspects of the LQC framework used to characterize considered models, as well as its WDW analog. Next, in Sec. \[sec:limit\] we analyze in detail the exact WDW limit of LQC states, which is next used in Sec. \[sec:scatter\] to construct the scattering picture mentioned in the previous paragraph. That picture is then employed in Sec. \[sec:triangle\] to the analysis of the dispersions of the incoming/outgoing WDW states, which allows in turn to derive the triangle inequalities relating them. We also show there (Sec. \[sec:tr-rel\]), that the dispersions of the outgoing/incoming components of the WDW limit equal the dispersions of a genuine LQC state in the asymptotic future and past, thus extending the inequalities to these quantities. We conclude with Section \[sec:concl\] with the discussion of the main results as well as the possibilities 