  The Quantum cosmology with Born-Infeld(B-I) type scalar field is considered.
In the extreme limits of small cosmological scale factor the wave function of
the universe can also be obtained by applying the methods developed by
Hartle-Hawking(H-H) and Vilenkin. H-H wave function predicts that most Probable
cosmological constant $\Lambda$ equals to $\frac{1}{\eta}$($\frac{1}{2\eta}$
equals to the maximum of the kinetic energy of scalar field). It is different
from the original results($\Lambda=0$) in cosmological constant obtained by
Hartle-Hawking. The Vilenkin wave function predicts a nucleating unverse with
largest possible cosmological constant and it is larger than $1/\eta$. The
conclusions have been nicely to reconcile with cosmic inflation. We investigate
the inflation model with B-I type scalar field, and find that $\eta$ depends on
the amplitude of tensor perturbation $\delta_h$, with the form
$\frac{1}{\eta}\simeq \frac{m^2}{12\pi[(\frac{9\delta_{\Phi}^2}{N
\delta_h^2})^2-1]}.$ The vacuum energy in inflation epoch depends on the
tensor-to-scalar ratio $\frac{\delta_h}{\delta_{\Phi}}$. The amplitude of the
tensor perturbation ${\delta_{h}}$ can, in principle, be large enough to be
discovered. However, it is only on the border of detectability in future
experiments. If it has been observed in future, this is very interesting to
determine the vacuum energy in inflation epoch.
