The consistent result of cosmological constant from quantum cosmology and inflation with Born–Infeld scalar field

Abstract

Quantum cosmology with a Born–Infeld (BI) type scalar field is considered. In the extreme limits of a small cosmological scale factor the wave function of the universe can also be obtained by applying the methods developed by Hartle–Hawking (HH) and Vilenkin. The HH wave function approach predicts that the most probable cosmological constant Λ equals $\frac{1}{\eta}$ ( $\frac{1}{2\eta}$ equals the maximum of the kinetic energy of the scalar field). It is different from the original results (Λ=0) for the cosmological constant obtained by Hartle–Hawking. The Vilenkin wave function predicts a nucleating universe with the largest possible cosmological constant, and it is larger than 1/η. The conclusions can nicely be reconciled with cosmic inflation. We investigate the inflation model with the BI type scalar field and find that η depends on the amplitude of the tensor perturbation δh, having the form $\frac{1}{\eta}$ $\simeq$ $\frac{m^2}{12\pi[(\frac{9\delta^2_{\Phi}}{N\delta^2_h})^2-1]}.$ The vacuum energy in the inflation epoch depends on the tensor-to-scalar ratio δh/δΦ. The amplitude of the tensor perturbation δh may, in principle, be large enough to be discovered. However, it is only on the border of detectability in future experiments. If it will have been observed in the future, this will be very interesting as regards determining the vacuum energy in the inflation epoch.