A rotating condensate tracking V∝r^n cannot reach kination: exact equation of state w=(n-2)/(n+2)

Abstract

A complex scalar with a conserved charge Q, written in polar form as a rotating condensate of amplitude r, has rotational energy density that would scale as a^{-6} (kination, w = 1) if r were held fixed. On a polynomial potential V ∝ r^n the amplitude does not stay fixed: it tracks the minimum of the effective radial potential formed by V plus the centrifugal barrier. Tracking yields the exact equation of state w = (n − 2)/(n + 2), strictly below unity for every finite polynomial degree n, and equal to the kination limit only as n → ∞. Freeze-out of r (the only escape to w = 1) requires a trans-Planckian amplitude, r ≳ O(1) M_Pl. Numerical integration of the radial field equation through a contracting background reproduces the analytic w to a few parts in 10^5 at n = 2, 4, 6. The result is classical field theory with a conserved charge; it is independent of any particular dark-sector model. It is a negative sector result: a rotating condensate of this class cannot supply the stiff phase often invoked to survive a BKL approach to a crunch.