  Our observed Universe has a very strong arrow of time rooted in its low
entropy starting point. This low entropy start can be related to various
"tuning puzzles" about the early state of the Universe. Here we explore the
relationship between the arrow of time and the emergence of classical from
quantum in the hopes of ultimately gaining insights into cosmological initial
conditions. Our focus is on einselection, the process whereby interactions with
an environment select preferred states for a quantum system. This process plays
an essential role in the emergence of classical from quantum. Studies of
einselection have so far been limited to cases that exhibit an arrow of time.
Here we study the ability of equilibrium systems to exhibit einselection -- and
investigate whether detailed balance prevents this -- motivated by the question
of whether classicality requires an arrow of time. We present calculations in
the adapted Caldeira-Leggett model which demonstrate that einselection can
indeed take place in equilibrium systems, and show how this phenomenon is tied
to histories which express an arrow of time, despite the global equilibrium. We
discuss some interesting implications of our results for cosmology and
cosmological initial conditions. We are intrigued and a bit surprised by the
role the consistent histories formalism has ended up playing in our analysis.
