The No-Boundary Measure of the Universe

TL;DR

The no-boundary proposal for homogeneous isotropic closed universes with a cosmological constant and a scalar field with a quadratic potential predicts classical behavior at late times if the scalarField is large enough.

Abstract

We consider the no-boundary proposal for homogeneous isotropic closed universes with a cosmological constant and a scalar field with a quadratic potential. In the semi-classical limit, it predicts classical behavior at late times if the initial scalar field is more than a certain minimum. If the classical late time histories are extended back, they may be singular or bounce at a finite radius. The no-boundary proposal provides a probability measure on the classical solutions which selects inflationary histories but is heavily biased towards small amounts of inflation. This would not be compatible with observations. However we argue that the probability for a homogeneous universe should be multiplied by exp(3N) where N is the number of e-foldings of slow roll inflation to obtain the probability for what we observe in our past light cone. This volume weighting is similar to that in eternal inflation. In a landscape potential, it would predict that the universe would have a large amount of inflation and that it would start in an approximately de Sitter state near a saddle-point of the potential. The universe would then have always been in the semi-classical regime.

Figures (3)

• Figure 1: The values of $I_R$ of the one-parameter set of classical histories predicted by the no-boundary proposal in a quadratic potential minisuperspace model with $\mu=3$ and $\Lambda =.03$. There are no classical histories for $\phi_0$ below a critical value $\phi_0^c$ at about $1.2$. The universe therefore requires a minimum amount of matter to behave classically at late times. A critical value $\phi_0^s$ at about $1.5$ separates large $\phi_0$ histories that bounce at a finite radius when extrapolated back from singular histories for smaller $\phi_0$.
• Figure 2: The no-boundary wave function predicts that all histories that behave classically at late times undergo a period of inflation at early times as shown here by the linear growth of the instantaneous Hubble constant $H$ in five representative $\mu=3$ classical histories.
• Figure 3: To account for the different possible locations in the universe of the Hubble volume that contains our data one ought to multiply the no-boundary amplitudes by a volume factor. In regions of the landscape around a maximum of the potential (left), we expect this to have a significant effect on the probability distribution over $\phi_0$ and hence over $N$ (right). The effect of a classicality constraint is also shown.