Eternal inflation: The inside story

Motivated by the lessons of black hole complementarity, we develop a causal patch description of eternal inflation. We argue that an observer cannot ascribe a semiclassical geometry to regions outside his horizon, because the large-scale metric is governed by the fluctuations of quantum fields. In order to identify what is within the horizon, it is necessary to understand the late time asymptotics. Any given worldline will eventually exit from eternal inflation into a terminal vacuum. If the cosmological constant is negative, the universe crunches. If it is zero, then we find that the observer’s fate depends on the mechanism of eternal inflation. Worldlines emerging from an eternal inflation phase driven by thermal fluctuations end in a singularity. By contrast, if eternal inflation ends by bubble nucleation, the observer can emerge into an asymptotic, locally flat region. As evidence that bubble collisions preserve this property, we present an exact solution describing the collision of two bubbles.

Figure 1

A conformal diagram for the global geometry in FVEI. The hats represent the future infinity of open FRW universes with vanishing cosmological constant. Regions with negative cosmological constant end in a big crunch singularity (squiggly lines). The part of the universe which remains in the inflationary phase has zero comoving volume but infinite physical volume. The thick diamond shows an example of a causally connected region accessible to a single observer. The local description developed here is confined to such regions.

Figure 2

On the left, a spin is measured after the apparatus has entered a black hole, leaving the outside observer ignorant of the outcome. On the right, the measurement takes place outside the black hole, beyond the causal past of an infalling observer. Either way, the observer cannot ascribe a definite outcome to the measurement. If the geometry depends on this outcome, it cannot be treated semiclassically, since it would involve a quantum superposition of macroscopically distinct metrics.

Figure 3

Examples of causal horizons in cosmology: A closed FRW universe (left) and de Sitter space (right). The observer is on the left edge of each diagram. She cannot access the result of the measurement in the (white) region outside her causal past.

Figure 4

The slow-roll potential $V=\frac{1}{2}{m}^2{\varphi }^2$. The SREI regime is $\varphi >m^{-1/2}$.

Figure 5

A potential with a plateau supporting eternal inflation and a relatively short-lived classical slow roll.

Figure 6

Even a weak perturbation will eventually go nonlinear, but there is plenty of time to avoid getting trapped in the collapse. This conformal diagram shows an exact solution constructed with the help of Birkhoff’s theorem. The blue region is a small portion of a closed universe (the polar cap of the three-sphere). Because its curvature radius is much larger than its size, it represents only a slightly overdense region embedded in a flat universe. The world volume of the unperturbed dust particles is shown in yellow. The white region is vacuous and is a portion of the Schwarzschild solution. It develops as the overdense region decouples from the ambient expansion. The perturbation is first fully seen at ${\eta }_{\mathrm{see}}$. It goes nonlinear much later, at ${\eta }_{\mathrm{turn}}$. Only at ${\eta }_{\mathrm{doom}}$ does the inert observer enter a black hole.

Figure 7

A strong perturbation catches the observer unaware (compare to Fig. 1). This conformal diagram shows another exact solution, but this time we keep half of the three-sphere of the closed universe (blue). This represents a strong perturbation because its size is comparable to its curvature radius. The observer can see the overdense region only as he is already falling into the black hole it forms.

Figure 8

The Coleman-De Luccia bounce geometry, as an embedding in Minkowski space. The flat piece has zero cosmological constant, while the curved piece has positive cosmological constant. A domain wall separates the two regions.

Figure 9

A constant time slice of the two-bubble solution. The spatial geometry consists of two 3-hyperboloids glued across a wall of dust to each other. Here the 3-hyperboloids are pictured, with one dimension suppressed, in the Poincaré disk representation. The two disks are to be glued along the dust wall (heavy line), and the shaded regions are to be thrown away.

Figure 10

At late time, the wall of dust moves to the center of the hyperboloids. In this limit, the geometry reverts to one hyperboloid. Physically, this occurs because the dust has diluted and no longer causes a discontinuity in the geometry across the wall. Again, the shaded regions are to be thrown away and the two sides glued along the heavy line.

Figure 11

A typical observer sees a boosted version of the previous figure, with the dust wall asymptotically moving away at a finite velocity.

Figure 12

The hypersurface where inflationary modes freeze out is spacelike. Hence, frozen modes can be measured only after inflation has ended.