Figure 1

Evolution of the scale factor cutoff during structure formation. The constant scale factor time surface ${\Sigma }_1$ lies in the early, approximately homogeneous universe and coincides with a surface of constant FRW time $\tau$. As density perturbations grow, some geodesics decouple from the Hubble flow, stop expanding, and become trapped in collapsed regions such as galaxies. If the scale factor cutoff exceeds the largest scale factor ever reached along such geodesics, then the rule of De Simone et al. requires that their entire future evolution be included. (A similar result is obtained if the local formulation of the scale factor measure is used as a general definition; see Sec. 5d.) Therefore, the later cutoff surface ${\Sigma }_2$ no longer agrees everywhere with a constant FRW time surface; it includes the entire future of collapsed regions (gray/green), which show up as spikes since the figure is drawn in comoving coordinates.Reuse & Permissions

Figure 2

The light shaded (green) slices are surfaces of constant scale factor time, ${\Sigma }_{\eta }$; they have fixed comoving size but increasing physical volume. Fat geodesics (dark shaded, purple) have fixed physical width and thus decreasing comoving size. If the initial slice is chosen in the attractor regime, the fat geodesics define a representative finite sample of the total four-volume. Thus, the results of the scale factor measure can be reproduced by following a single geodesic of fixed width, starting in the longest-lived metastable vacuum.Reuse & Permissions