Tachyon driven quantum cosmology in string theory

Recently an effective action of the SD$p$-brane decaying process in string theory has been proposed. This effective description involves the Tachyon driven matter coupled to bosonic ten-dimensional Type II supergravity. Here the Hamiltonian formulation of this system is given. Exact solutions for the corresponding quantum theory by solving the Wheeler-deWitt equation in the late-time limit of the rolling tachyon are found. The energy spectrum and the probability densities for several values of $p$ are shown and their possible interpretation is discussed. In the process the effects of electromagnetic fields are also incorporated and it is shown that in this case the interpretation of tachyon regarded as “matter clock” is modified.

Figure 1

The figure shows the probability density $|\Psi {|}^2$ for the “physical” case of a SD3-brane ($p=3$) (with $\mu =0.1$ and $\nu =0.7$) and its variation with respect to the radii $a_{\perp }$ and $a_{\parallel }$. The maxima of the quantum solution $\Psi$ determines a trajectory in the $a_{\perp }-a_{\parallel }$ plane. These maxima satisfy an inverse relation between both radii as shown in Eq. (26).

Figure 2

The probability density $|\Psi {|}^2$ (also with $\mu =0.1$ and $\nu =0.7$) for one extreme case with $p=1$. The solution shows that for this case, the maxima of $|\Psi {|}^2$ determines an evolution which is almost projected on the axis $a_{\perp }$, that is, for large values of $a_{\perp }$ it is almost independent on $a_{\parallel }$.

Figure 3

The figure corresponds with the other extreme case with $p=8$ which shows that the probability density $|\Psi {|}^2$ is independent on $a_{\perp }$. This corresponds with a wave function $\Psi$ describing an evolution whose maxima are localized around fixed values of $a_{\parallel }$.