Vacuum structure and the arrow of time

We find ourselves in an extended era of entropy production. Unlike most other observations, the arrow of time is usually regarded as a constraint on initial conditions. I argue, however, that it primarily constrains the vacuum structure of the theory. I exhibit simple scalar field potentials in which low-entropy initial conditions are not necessary, or not sufficient, for an arrow of time to arise. In particular, in some models an arrow of time obtains even if the initial vacuum has arbitrarily large entropy. In such examples, the dynamical resolution of the arrow of time problem arises from properties of the vacuum structure similar to those that allow the string landscape to solve the cosmological constant problem without producing an empty universe, particularly its high dimensionality and the large difference in vacuum energy between neighboring vacua.

Figure 1

The observed arrow of time is defined as the entropy produced in our past light cone (blue, $t_f$) since some early time that we can confidently extrapolate to (red, $t_i$): $\Delta S=S(t_f)-S(t_i)$. With $t_i$ equal to the era of big bang nucleosynthesis, one finds $\Delta S\sim {10}^{103}$.

Figure 2

A landscape with one metastable de Sitter vacuum $A$ and a terminal vacuum $T$. The vertical axis indicates both the effective potential and, schematically, the free energy: Ordinary observers, who see a long arrow of time, are drawn higher up than Boltzmann brains. The dominant histories are indicated by chains of arrows. (a) With ${\Gamma }_A>{\Gamma }_{\mathrm{BB},A}$ and initial conditions (IC) similar to our own universe, most observers are ordinary observers (OO) who perceive an arrow of time. (b) If initial conditions are high entropy, then the most likely observers are Boltzmann brains. (c) If vacuum $A$ decays too slowly $({\Gamma }_A<{\Gamma }_{\mathrm{BB},A})$, then many more Boltzmann brains than ordinary observers are produced.

Figure 3

Two landscapes with four vacua each. In both models, the universe starts out in vacuum $C$, and the cosmological constant and low-energy physics are the same in identically labeled vacua. In model (a), the most likely way to produce observers is by a rare fluctuation to a Boltzmann brain state of vacuum $B$; no arrow of time is predicted. In model (b), the only way to produce observers is by first fluctuating up to vacuum $A$; a large arrow of time is predicted even though initial conditions have arbitrarily large entropy. The string landscape shares crucial features of this second model.