A Casimir turning point as the beginning of a small flat universe

Abstract

We study a flat three-torus universe whose contraction is turned around by the negative Casimir energy of Standard Model fields. The trace anomaly forbids conformal matter alone to turn an anisotropic universe around, so matter with $w<1/3$, most cheaply a positive potential, must supply the negative trace; with antiperiodic fermions the Standard Model supplies the negative energy, and the universe turns at an edge of 722 reduced Planck lengths. There the Higgs quartic coupling is negative, but zero-point fluctuations of the constant gauge modes confine the Higgs zero mode down to a finite-volume threshold $λ_c\approx-0.012$, below the Standard Model value $-0.0098$. The Higgs survives with probability above $1/2$ for a top-quark pole mass below $172.96$ GeV ($\bar m_t<163.16$ GeV; $α_s(M_Z)=0.1180$), and with probability $\approx0.7$ for current data, given a curvature coupling $ξ_H\gtrsim0.08$ at the scale $\bar M_{\mathrm{P}}/722$. The turning point can hold only about five quanta of the lowest modes, whereas a reheating coupling in a time-symmetric past would produce radiation exceeding the Casimir energy by at least $10^{94}$, so we read the turn as a beginning rather than a bounce. Negative Casimir energy cannot hold the shape of the torus: the shape state can be real at most once, and for a real Gaussian state survival of the shape on both branches with $90\%$ probability requires that moment to lie within $0.35$ Hubble times of the turn. The natural state, real at the turn, freezes the torus into a nearly rectangular box whose sides typically differ by $16$-$18\%$. The microwave background can read this fossil only if the shortest cycle lies between $0.985$ and about $1.04$ last-scattering diameters. A top-quark pole mass above $173.7$ GeV, within reach of FCC-ee, would exclude this beginning (Bayes factor $\lesssim0.03$).