Chiral Nelson-Barr models: Quality and cosmology

It was recently shown that domain walls from the spontaneous breaking of $CP$ symmetry are exactly stable, and must be inflated away to recover a viable cosmology. We investigate the phenomenological implications of this result in Nelson-Barr solutions of the strong $CP$ problem. Combined with the upper bound on the scale of spontaneous $CP$ breaking necessary to suppress contributions from dangerous, nonrenormalizable operators to θ, this puts an upper bound on the scale of inflation and the reheating temperature after inflation. Minimal Nelson-Barr models are therefore in tension with thermal leptogenesis, models of large-field inflation, or potential future observations of signals from topological remnants of an unrelated, subsequent phase transition. We study how extending Nelson-Barr models with a new, continuous chiral gauge symmetry can ameliorate this tension by forbidding the dangerous dimension-five operators. In particular, we show that gauging a linear combination of baryon number and hypercharge allows for an economic, anomaly-free extension of the minimal Nelson-Barr model, and discuss the phenomenological implications.