Reheating surface volume renormalization and eternal inflation under broken scale invariance

Recently, Cotler, Ivo, and Maldacena have shown that under scale invariance, eternal inflation is interpreted as a failure of the reheating surface volume renormalization due to an ill-defined scaling dimension. Meanwhile, observations suggest that scale invariance is slightly broken, which is parametrized by deviation of the spectral index from unity, varepsilon=n_s-1. Motivated by this, we study how the scale invariance breaking effect modifies the no-eternal inflation condition as the renormalizability of the reheating surface volume. For this purpose, we adopt two independent approaches: investigating the behavior of the normal ordered reheating surface volume operator under a shift of the curvature perturbation, and analyzing the coarse-graining structure to obtain the scaling dimension. Both approaches commonly reveal that as long as varepsilon is negative, which is consistent with observations, the fractal structure of the reheating surface volume breaks down at macroscopic scale, moderating the no-eternal inflation condition.