  If the source of the current accelerating expansion of the universe is a
positive cosmological constant, Banks and Fischler argued that there exists an
upper limit of the total number of e-foldings of inflation. We further
elaborate on the upper limit in the senses of viewing the cosmological horizon
as the boundary of a cavity and of the holographic D-bound in a de Sitter
space. Assuming a simple evolution model of inflation, we obtain an expression
of the upper limit in terms of the cosmological constant, the energy density of
inflaton when the inflation starts, the energy density as the inflation ends,
and reheating temperature. We discuss how the upper limit is modified in the
different evolution models of the universe. The holographic D-bound gives more
high upper limit than the entropy threshold in the cavity. For the most
extremal case where the initial energy density of inflation is as high as the
Planck energy, and the reheating temperature is as low as the energy scale of
nucleosynthesis, the D-bound gives the upper limit as 146 and the entropy
threshold as 122. For reasonable assumption in the simplest cosmology, the
holographic D-bound leads to a value about 85, while the cavity model gives a
value around 65 for the upper limit, which is close to the value in order to
solve the flatness problem and horizon problem in the hot big bang cosmology.
