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 ℬ. Therefore, the inflationary universe must have had some kind of beginning. However, the BGW theorem says nothing about the boundary conditions on 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 Υ d⁴x is present in the action. As a result of the model dynamics, an upper bound φ₀ of admissible values of the scalar field φ appears, which sets the position of ℬ  in the form of a spacelike hypersurface Υ(x) = 0 with a boundary condition:  Υ → 0⁺ as φ → φ₀^-. 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  φ_in^(min) ≤ φ_in < φ₀, 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 ℬ, that is where Υ → 0^-, the metric tensor in the Einstein frame has a jump discontinuity on ℬ, so the Christoffel connection coefficients are not defined on the spacelike hypersurface Υ = 0. Thus, if  φ_in^(min) ≤ φ_in < φ₀ 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.