Hadron bubble evolution into the quark sea

A solution is presented for the evolution of hadron bubbles which nucleate in the quark sea if there is a first-order quark-hadron phase transition at a temperature $T_c$ on the order of 100 MeV. We make three assumptions: (1) the dominant mechanism for transport of latent heat is radiative, e.g., neutrinos; (2) the distance between nucleation sites is greater than the neutrino mean free path; and (3) the effects of hydrodynamic flow can be neglected. Bubbles nucleate with a characteristic radius $\frac{1 \mathrm{fm}}{\Delta }$, where $\Delta$ is a dimensionless parameter for the undercooling (we take $\Delta \ge {10}^{-4\mathrm{}}$, so that the expansion of the Universe can be neglected). We argue that bubbles grow stably and remain spherical until the radius becomes as large as the neutrino mean free path, $l\simeq 10$ cm. The growth then becomes diffusion limited and the bubbles become unstable to formation of dendrites, or finger-like structures, because latent heat can diffuse away more easily from long fingers than from spheres. We study the nonlinear evolution of structure with a "geometrical model" and argue that the hadron bubbles ultimately look like stringy seaweed. The percolation of seaweed-shaped bubbles can leave behind regions of quark phase that are quite small. In fact, one might expect the typical scale to be $L_Q=l\simeq 10$ cm. Protons can easily diffuse out of such small regions (and neutrons back in). Thus, these instabilities can lead to important modifications of inhomogeneous nucleosynthesis, which requires $L_Q\gtrsim 1$ m.