  We present an analytic model for the splashback mass function of dark matter
halos, which is parameterized by a single coefficient and constructed in the
framework of the generalized excursion set theory and the self-similar
spherical infall model. The value of the single coefficient that quantifies the
diffusive nature of the splashback boundary is determined at various redshifts
by comparing the model with the numerical results from the Erebos N-body
simulations for the Planck and the WMAP7 cosmologies. Showing that the analytic
model with the best-fit coefficient provides excellent matches to the numerical
results in a wide mass range at all redshifts, we employ the Bayesian
Information Criterion test to confirm that our model is most preferred by the
numerical results to the previous models at almost all redshifts for both of
the cosmologies. It is also found that the diffusion coefficient decreases
almost linearly with redshifts, converging to zero at a certain threshold
redshift, $z_{c}$, whose value significantly differs between the Planck and
WMAP7 cosmologies. Our result implies that the splashback mass function of dark
matter halos at $z\ge z_{c}$ is well described by an universal parameter-free
analytic formula and that $z_{c}$ may have a potential to independently
constrain the initial conditions of the universe.
