Extension of the Bjorken energy density formula of the initial state for relativistic heavy ion collisions

For relativistic heavy ion collisions, the Bjorken formula is very useful for estimating the initial energy density once an initial time to is specified. However, it cannot be trusted at low energies, e.g., well below root(s)(NN )approximate to 50 GeV for central Au+Au collisions, when to is smaller than the finite time it takes for the two nuclei to cross each other. Here I extend the Bjorken formula by including the finite time duration of the initial energy production. Analytical solutions for the formed energy density in the central space-time-rapidity region are derived for several time profiles. Compared to the Bjorken formula at low energies, the maximum energy density reached is much lower, increases much faster with the collision energy, and is much less sensitive to the uncertainty of the formation time, while the energy density time evolution is much longer. Comparisons with results from a multiphase transport confirm the key features of these solutions. The effect of the finite longitudinal width of the initial energy production, which is neglected in the analytical results, is investigated with the transport model and shown to be small. This work thus provides a general model for the initial energy production of relativistic heavy ion collisions that is also valid at low energies.