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 $\Upsilon d^4x$ 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 $\Upsilon(x)=0$ with a boundary condition: $\Upsilon\rightarrow 0^+$ as $\varphi\rightarrow\varphi_0^{\,\,\,-}$. 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 $\varphi_{in}^{(min)}\leq \varphi_{in} <\varphi_0$, 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 $\Upsilon\rightarrow 0^-$, 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 $\Upsilon=0$. Thus, if $\varphi_{in}^{(min)}\leq \varphi_{in}<\varphi_0$ 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 $\Upsilon=0$.