Measure problem in cosmology

The Hamiltonian structure of general relativity provides a natural canonical measure on the space of all classical universes, i.e., the multiverse. We review this construction and show how one can visualize the measure in terms of a “magnetic flux” of solutions through phase space. Previous studies identified a divergence in the measure, which we observe to be due to the dilatation invariance of flat Friedmann-Lemaitre-Robertson-Walker universes. We show that the divergence is removed if we identify universes which are so flat they cannot be observationally distinguished. The resulting measure is independent of time and of the choice of coordinates on the space of fields. We further show that, for some quantities of interest, the measure is very insensitive to the details of how the identification is made. One such quantity is the probability of inflation in simple scalar field models. We find that, according to our implementation of the canonical measure, the probability for $N$ e-folds of inflation in single-field, slow-roll models is suppressed by of order $\exp (-3N)$ and we discuss the implications of this result.

Figure 1

Set of classical trajectories for the inflationary model with $V=\frac{1}{2}{m}^2{\varphi }^2$ and $k=0$, in the coordinates ($\varphi$, $\dot{\varphi }$, $\lambda$), where $\lambda \equiv \ln a$. The upper panel shows the projection onto the ($\varphi$, $\dot{\varphi }$) plane, and the lower panel shows the projection onto the ($\varphi$, $\lambda$) plane. The dashed lines indicate the projection of the measure surface $S$, which takes the form of an elliptical cylinder and which each trajectory crosses once. The parameters used were $m^2=0.05$, $H_S=0.1$.

Figure 2

The set of trajectories in the ($\varphi$, $H$) plane for $V(\varphi )=\frac{1}{2}{m}^2{\varphi }^2$ and $k=0$, and the same parameters as in Fig. 1. The measure surface $S$ is taken at $H=0.1$, and the trajectories plotted are equally spaced in $\varphi$ on that surface. Only the trajectories with positive $\dot{\varphi }$ on the measure surface are shown: those with negative $\dot{\varphi }$ are obtained by mirror reflection.

Figure 3

Illustration of the tuning needed to create the inflationary solution as one follows $\varphi$ backward in time from a measure surface $H=H_S$. Trajectories which $\varphi$ is too large on the measure surface encounter the classical turning point $H=\sqrt{V(\varphi )/3}$, turn around and end up kinetic dominated. Trajectories in which the initial $\varphi$ is too small fall behind the inflationary solution and also head towards kinetic domination.

Figure 4

Same as Fig. 2, but for an open universe with ${\Omega }_k=0.25$ on the measure surface $H=H_S$.

Figure 5

Same as Fig. 2, but for a closed universe with ${\Omega }_k=-0.25$ on the measure surface $H=H_S$.