Effectively as Chalnoth said, Carroll is using the Friedmann equation $$ H^2 = \frac{8\pi G}{3}\rho = \frac{1}{3M_P^2}\rho = H_0^2 \frac{\rho}{\rho_c}. $$ The last equality states a relation between ## H_0, M_P## and ## \rho_c ## as follows $$ H_0 = \frac{1}{M_P}\ \sqrt{\frac{\rho_c}{3}}. $$...