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

Abstract

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-Measures Theory, where the non-Riemannian volume element ${\rm Y}{d}^{4}x$ is present in the action. As a result of the model dynamics, an upper bound ${\phi }_0$ of admissible values of the scalar field $\phi$ appears, which sets the position of $\mathcal{B}$ in the form of a spacelike hypersurface ${\rm Y}(x)=0$ with a boundary condition: ${\rm Y}\to 0^{+}$ as $\phi \to {\phi }_0^{-}$. A detailed study has established that if the initial kinetic energy density ${\rho }_{kin}^{(in)}$ prevails over initial gradient energy density ${\rho }_{grad}^{(in)}$ then there is an interval of initial values ${\phi }_{in}^{(min)}\le {\phi }_{in}<{\phi }_0$, where ${\rho }_{kin}^{(in)}$ and ${\rho }_{grad}^{(in)}$ 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 ${\rm Y}\to 0^{-}$, 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 ${\rm Y}=0$. Thus, if ${\phi }_{in}^{(min)}\le {\phi }_{in}<{\phi }_0$ and ${\rho }_{kin}^{(in)}>{\rho }_{grad}^{(in)}$, 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 ${\rm Y}=0$.

the paper was an accidental replacement of arXiv:2209.00378