Old phase remnants in first-order phase transitions

First-order phase transitions (FOPTs) are usually described by the nucleation and expansion of new phase bubbles in the old phase background. While the dynamics of new phase bubbles have been extensively studied, a comprehensive treatment of the shrinking old phase remnants remained undeveloped. We present a novel formalism for remnant statistics in FOPTs and perform the first calculations of their distribution. By shifting to the reverse time description, we identify the shrinking remnants with expanding old phase bubbles, allowing a quantitative evolution and determination of the population statistics. Our results not only provide essential input for cosmological FOPT-induced soliton/primordial black hole formation scenarios, but can also be readily applied to generic FOPTs.