Note: Relaxation time below jamming

Like other critical phenomena, the jamming transition accompanies the divergence of the relaxation time $\tau$. A recent numerical study of frictionless spherical particles proves that $\tau$ is inversely proportional to the lowest non-zero eigenvalue $\lambda_1$ of the dynamical matrix. In this note, we derive the scaling of $\lambda_1$ below the jamming transition point $\varphi_J$ by solving the linearized dynamical equation. The resultant critical exponent agrees with a previous theoretical result for sheared suspension obtained by applying the virtual work theorem to a simple shear, highlighting the universality of the relaxation dynamics below jamming.