  Collisionless shock acceleration, which transfers localized particle energies
to non-thermal energetic particles via electromagnetic potential, is ubiquitous
in space plasma. We investigate dynamics of collisionless electrostatic shocks
that appear at interface of two plasma slabs with different pressures using
one-dimensional particle-in-cell (PIC) simulations and find that the shock
structure transforms to a double-layer structure at the high density gradient.
The threshold condition of the structure transformation is identified as
density ratio of the two plasma slabs $\Gamma$ $\sim 40$ regardless of the
temperature ratio between them. We then update the collisionless shock model
that takes into account density expansion effects caused by a rarefaction wave
to improve the prediction of the critical Mach numbers. The new critical Mach
numbers are benchmarked by PIC simulations for a wide range of $\Gamma$.
Furthermore, we introduce a semi-analytical approach to forecast the shock
velocity just from the initial conditions based on a new concept of the
accelerated fraction $\alpha$.
