In phase estimation algorithms, we have $U|\psi\rangle = e^{2\pi i\theta}|\psi\rangle$, where $|\psi\rangle$ is an eigenvector and $ e^{2\pi i\theta}$ is the corresponding eigenvalue. Since $U$ represents the time-evolution of the quantum system, suppose the original Hamiltonian has an eigenvalue $E_1$, then after certain time-evolution, the value becomes $e^{-iE_1t}$, I wonder to simulate the time-evolution in the algorithm, how can we determine the time $t$?
Should we let $-iE_1t = 2\pi i\theta$ to solve for $t$? Also, given that $\theta\in[0,1]$, to make phase estimation less confusing, can we scale the Hamiltonian being simulated to make its eigenvalues within $[0,2\pi]$?
Thanks!!