  There is mounting observational evidence that the expansion of our universe
is undergoing an acceleration. A dark energy component has usually been invoked
as the most feasible mechanism for the acceleration. However, it is desirable
to explore alternative possibilities motivated by particle physics before
adopting such an untested entity. In this work, we focus our attention on an
acceleration mechanism: one arising from gravitational leakage into extra
dimensions. We confront this scenario with high-$z$ type Ia supernovae compiled
by Tonry et al. (2003) and recent measurements of the X-ray gas mass fractions
in clusters of galaxies published by Allen et al. (2002,2003). A combination of
the two databases gives at a 99% confidence level that
$\Omega_m=0.29^{+0.04}_{-0.02}$, $\Omega_{rc}=0.21^{+0.08}_{-0.08}$, and
$\Omega_k=-0.36^{+0.31}_{-0.35}$, indicating a closed universe. We then
constrain the model using the test of the turnaround redshift, $z_{q=0}$, at
which the universe switches from deceleration to acceleration. We show that, in
order to explain that acceleration happened earlier than $z_{q=0} = 0.6$ within
the framework of gravitational leakage into extra dimensions, a low matter
density, $\Omega_m < 0.27$, or a closed universe is necessary.
