Joseph E. Lesnefsky

We discuss the question of whether or not inflationary spacetimes can be geodesically complete in the infinite past. Geodesic completeness is a necessary condition for averting an initial singularity during eternal inflation. It is frequently argued that cosmological models which are expanding sufficiently fast (having average Hubble expansion rate $H_{\mathrm{avg}}>0$) must be incomplete in null and timelike past directions. This well-known conjecture relies on specific bounds on the integral of the Hubble parameter over a past-directed timelike or null geodesic. As stated, we show this claim is an open issue. We show that the calculation of $H_{\mathrm{avg}}$ yields a continuum of results for a given spacetime predicated upon the underlying topological assumptions. We present an improved definition for $H_{\mathrm{avg}}$ and introduce an uncountably infinite cohort of cosmological solutions which are geodesically complete despite having $H_{\mathrm{avg}}>0$. We discuss a standardized definition for inflationary spacetimes as well as quantum (semiclassical) cosmological concerns over physically reasonable scale factors.

We discuss the question of whether or not inflationary spacetimes can be geodesically complete in the infinite past. Geodesic completeness is a necessary condition for averting an initial singularity during eternal inflation. It is frequently argued that cosmological models which are expanding sufficiently fast (having average Hubble expansion rate $H_{avg}>0$) must be incomplete in null and timelike past directions. This well-known conjecture relies on specific bounds on the integral of the Hubble parameter over a past-directed timelike or null geodesic. As stated, we show this claim is an open issue. We show that the calculation of $H_{avg}$ yields a continuum of results for a given spacetime predicated upon the underlying topological assumptions. We present an improved definition for $H_{avg}$ and introduce an uncountably infinite cohort of cosmological solutions which are geodesically complete despite having $H_{avg}>0$. We discuss a standardized definition for inflationary spacetimes as well as quantum (semi-classical) cosmological concerns over physically reasonable scale factors.

We optimize the first and second intrinsic hyperpolarizabilities for a 1D piecewise linear potential dressed with Dirac delta functions for $N$ noninteracting electrons. The optimized values fall rapidly for $N>1$, but approach constant values of ${\beta }_{\mathrm{int}}=0.40$, ${\gamma }_{\mathrm{int}}^{+}=0.16$, and ${\gamma }_{\mathrm{int}}^{-}=-0.061$ above $N\gtrsim 8$. These apparent bounds are achieved with only two parameters with more general potentials achieving no better value. In contrast to previous studies, analysis of the Hessian matrices of ${\beta }_{\mathrm{int}}$ and ${\gamma }_{\mathrm{int}}$ taken with respect to these parameters shows that the eigenvectors are well aligned with the basis vectors of the parameter space, indicating that the parameterization was well-chosen. The physical significance of the important parameters is also discussed.