Dark Sector Ringing at the Bottom of the Well

Abstract

The strong-CP problem is read in the geometry of the Euler circle. The QCD vacuum angle theta is a point on a circle, periodic modulo 2pi, and CP conjugation is a reflection across the real axis, theta to -theta, so CP is conserved only at theta = 0 and pi. Measurement places the physical angle within 10^-10 of zero, an unexplained smallness for a quantity that is naturally of order one. The Peccei-Quinn mechanism promotes the static angle to a dynamical field, the axion, which obeys a damped-pendulum equation in the expanding universe: Hubble friction holds it frozen, the QCD-induced mass releases it, and it spirals into the CP-conserving bottom of the well with its swing amplitude tapering as a^(-3/2). The Vafa-Witten theorem fixes that bottom at theta = 0. The residual oscillation left behind is cold dark matter, its abundance set by the initial misalignment. Cast as a phasor on the circle, the axion angle is the damped counterpart of the undamped neutrino phase: one circle, one CP reflection, two radial fates forked by a single dissipation term, a constant-radius cylinder where unitarity forbids friction and a collapsing funnel where Hubble friction acts. The same field shows both faces of the dark sector, frozen it acts as dark energy (w about -1) and ringing it acts as matter (w about 0); the condensate is also the lowest-entropy form of dark matter, and its misalignment is an imbalance spent along the arrow of cosmic time, the entropy residing in the expanding horizon rather than in the field. These readings connect the construction to the horizon-entropy and dark-sector accounts of the Euler Cosmos series. The account is a geometric retelling of established physics, not a new dynamics or a prediction.