  We investigate the idea that the universe before the Big Bang is the $CPT$
reflection of the universe after the bang, both classically and quantum
mechanically, so that the universe does {\it not} spontaneously violate $CPT$.
We show how $CPT$ symmetry selects a preferred vacuum state for quantum fields
on a $CPT$-invariant cosmological background spacetime. The universe before the
bang and the universe after the bang may be viewed as a universe/anti-universe
pair, emerging directly into the hot, radiation-dominated era we observe in our
past. This, in turn, leads to a remarkably economical explanation of the
cosmological dark matter. With no additional fields beyond Einstein gravity and
the standard model of particle physics (including right-handed neutrinos), a
$\mathbb{Z}_{2}$ symmetry stabilizes one of the right-handed neutrinos. We
calculate its abundance in detail and show that, in order to match the observed
dark matter density, its mass must be $4.8\times10^{8}~{\rm GeV}$. We obtain
several further predictions, including: (i) that the three light neutrinos are
majorana; (ii) that one of these is exactly massless; and (iii) that, in the
absence of an epoch of cosmic inflation, there should be no primordial,
long-wavelength gravitational waves. We also briefly discuss the natural origin
of the matter-antimatter asymmetry within this picture and possibilities for
explaining the cosmological perturbations.
