Classical and quantum cosmology of Born-Infeld type models

We discuss Born-Infeld type fields (tachyon fields) in classical and quantum cosmology. We first partly review and partly extend the discussion of the classical solutions and focus in particular on the occurrence of singularities. For quantization, we employ geometrodynamics. In the case of constant potential, we discuss both Wheeler-DeWitt quantization and reduced quantization. We are able to give various solutions and discuss their asymptotics. For the case of general potential, we transform the Wheeler-DeWitt equation to a form where it leads to a difference equation. Such a difference equation was previously found in the quantization of black holes. We give explicit results for the cases of constant potential and inverse squared potential and point out special features possessed by solutions of the difference equation.

Figure 1

Plot of the configuration space trajectory (16) for the tachyon model with constant potential.

Figure 2

Plot of the general solution $a(t)$ for the tachyon model with constant potential.

Figure 3

Flow diagram of the tachyon model with the inverse square potential. Inside the grey shaded region, the universe undergoes a decelerated expansion, while it accelerates outside this region. The parameter $\beta$ is given by $\beta =\frac{1}{9}\sqrt{2+\sqrt{9{\kappa }^4{V}_1^2+4}}$; $T$ is in arbitrary units of time.

Figure 4

Plot of the general solution $a(t)$ for the pseudotachyon model with constant potential.

Figure 5

Plot of configuration space trajectory for the pseudotachyon model with constant potential.

Figure 6

Flow diagram of the pseudotachyon model with the inverse square potential for the case $W_1<\frac{2}{9}$; $T$ is in arbitrary units of time.

Figure 7

Phase space trajectories of the constant potential tachyon field models. All solutions (except the one in the middle) evolve out of a big bang singularity marked by the red star and end in the point $(a=\infty ,p_T)$.

Figure 8

Phase space trajectories of the constant potential pseudotachyon field models. All solutions evolve out of a big bang singularity marked by the red star and end in a big brake singularity at the edge of the grey shaded region, where the Hamiltonian (27) becomes ill defined. In the case of a contracting universe, the trajectories go along the same lines, starting from the big brake towards the big crunch singularity.