Possible relationship between initial conditions for inflation and past geodesic incompleteness of the inflationary spacetime

According to the Borde-Guth-Vilenkin (BGV) theorem an expanding region of spacetime cannot be extended to the past beyond some boundary $\mathcal{B}$. Therefore, the inflationary universe must have had some kind of beginning. However, the BGW theorem says nothing about the boundary conditions on $\mathcal{B}$, or even about its location. Here we present a single-scalar field model of the Two-Measure Theory, where the non-Riemannian volume element $\mathrm{{\rm Y}}{d}^{4}x$ is present in the action. As a result of the model dynamics, an upper bound \varphi_0 of admissible values of the scalar field \varphi appears, which sets the position of $\mathcal{B}$ in the form of a spacelike hypersurface Υ(x)=0 with a boundary condition: Υ\to 0^+ as \varphi\to\varphi_0^-. A detailed study has established that if the initial kinetic energy density ρ_{kin}^{(in)} prevails over initial gradient energy density ρ_{grad}^{(in)} then there is an interval of initial values \varphi_{in}^{(min)}\leq\varphi_{in}<\varphi_0, where ρ_{kin}^{(in)} and ρ_{grad}^{(in)} cannot exceed the potential energy density and hence the initial conditions necessary for the onset of inflation are satisfied. It is shown that under almost all possible left-handed boundary conditions on $\mathcal{B}$, that is where Υ\to 0^-, the metric tensor in the Einstein frame has a jump discontinuity on $\mathcal{B}$, so the Christoffel connection coefficients are not defined on the spacelike hypersurface Υ=0. Thus, if \varphi_{in}^{(min)}\leq \varphi_{in}<\varphi_0 and ρ_{kin}^{(in)}>ρ_{grad}^{(in)}, then there was an inflationary stage in the history of our Universe and the congruence of timelike geodesics cannot be extended to the past beyond the hypersurface Υ=0.