Germanium material based on band gap engineering has aroused great interest for the CMOS-compatible optoelectronic integrated circuits due to its quasi-direct band gap structure. While many technologies have been conquered for germanium light, optimization is the bottleneck due to the excessive threshold current density, low luminescence efficiency and unstable problem in the laser device. The proper understanding of inter-valley scattering mechanisms between direct and indirect valleys in germanium is of paramount importance in view of the optimization of Ge as optical gain medium. The paper focuses on the inter-valley scattering mechanisms in strained Ge in theory based on a time-dependent Hamiltonian describing the electron-phonon interaction. The impacts of temperature and strain on the inter-valley scattering between direct and indirect valleys are discussed quantificationally. For the electrons in direct valley, emitting inter-valley phonon scattering is the dominant mechanism for momentum and energy relaxation of electrons both at the low and room temperature, and they are more likely to be scattered by inter-valley phonons to the L valleys with lower energy. For the electrons in L valleys, inter-valley scattering is important only for electrons with sufficient energy to scatter into the direct valley, which can happen in germanium devices under high electric field. Numerical results also indicate that enhanced indirect-to-direct inter-valley scattering and reduced direct-to-indirect inter-valley scattering are reliable by introducing tensile strain in Ge material at room temperature. The results offer fundamental understanding of phonon engineering for further optimization of the germanium light sources.