Dynamical compactification from de Sitter space

TL;DR

This work introduces a dynamical compactification mechanism in which a $D$-dimensional de Sitter space with a $q$-form flux nucleates regions with fewer effective dimensions, stabilized by a radion potential in a dimensionally reduced theory. By analyzing $p+2$-dimensional FRW cosmologies and their horizons, the authors classify a family of solutions including AdS$_{p+2} imes S^q$, dS$_{p+2} imes S^q$, Minkowski vacua, and several non-singular interpolating geometries, as well as singular and extremal p-brane-like configurations. They compute semiclassical nucleation rates via Euclidean instantons for both compactification and interpolation, map out a landscape of vacua across different $Q$, $ extΛ$, and $q$, and discuss the global multiverse structure and potential implications for the cosmological constant problem and inflation. The framework yields a qualitatively different picture from four-dimensional eternal inflation by explicitly populating vacua of varying dimensionality and illustrating possible observational consequences and theoretical challenges, including stability and measure issues. Overall, the paper provides a concrete, calculable route to higher-dimensional cosmologies that dynamically yield lower-dimensional universes with rich vacuum structure.

Abstract

We show that D-dimensional de Sitter space is unstable to the nucleation of non-singular geometries containing spacetime regions with different numbers of macroscopic dimensions, leading to a dynamical mechanism of compactification. These and other solutions to Einstein gravity with flux and a cosmological constant are constructed by performing a dimensional reduction under the assumption of q-dimensional spherical symmetry in the full D-dimensional geometry. In addition to the familiar black holes, black branes, and compactification solutions we identify a number of new geometries, some of which are completely non-singular. The dynamical compactification mechanism populates lower-dimensional vacua very differently from false vacuum eternal inflation, which occurs entirely within the context of four-dimensions. We outline the phenomenology of the nucleation rates, finding that the dimensionality of the vacuum plays a key role and that among vacua of the same dimensionality, the rate is highest for smaller values of the cosmological constant. We consider the cosmological constant problem and propose a novel model of slow-roll inflation that is triggered by the compactification process.

Figures (24)

• Figure 1: A portion of the causal diagram for a Schwarzschild black hole. To the future of the event horizon, $\tau$ is a timelike coordinate, and the $\tau$, $x$ plane (which is shown here) resembles a big-crunch cosmology that begins from a completely non-singular "big-bang" at the event horizon. The value of $R$ at the horizon is determined by the stationary point shown on the potential (filled circle). On subsequent surfaces of constant $\tau$, $R$ evolves in the sketched potential $V$, eventually reaching a singularity as $R$ is pushed to zero. To the past of the event horizon, $\tau \rightarrow i\tau$, and $\tau$ becomes a spacelike coordinate. The value of $R$ on surfaces of constant $\tau$ is determined by its motion in the upside-down potential $-V$. This pushes $R \rightarrow \infty$ as the asymptotically flat region far from the black hole is approached.
• Figure 2: A sketch of the causal structure of a solution that interpolates between asymptotically $D-q$ and $D$-dimensional regions across event horizons. On the left, a $D-q$ dimensional non-singular cosmological spacetime is located behind an event horizon. Outside this horizon is an interpolating region to another event horizon, that encloses an asymptotically $D$-dimensional de Sitter region. The nucleation of such solutions from empty $D$-dimensional de Sitter space represents the dynamical compactification of some number of extra dimensions.
• Figure 3: The radion potential Eq. \ref{['eq:radionpotential']} with $\Lambda = 0$ (left) and $\Lambda > 0$ (right). In each plot, a number of potentials at fixed $\Lambda$ are shown with successively larger values of $Q$ from bottom to top. For $\Lambda = 0$ the potential always has a negative minimum and approaches zero from below as $\phi \rightarrow \infty$, while for $\Lambda > 0$ there can exist a minimum of negative, zero, or positive energy and the potential approaches zero from above at large $\phi$.
• Figure 4: Extrema of the radion potential are located at the intersection of each curve (Eq. \ref{['eq:lambdaQrel']} for $\{p=2, q=2 \}$) with a line of fixed $\Lambda$. There is only one intersection at $\Lambda = 0$ and zero, one, or two intersections at $\Lambda > 0$. When there are two intersections, the first corresponds to a minimum (referred to as $R_{-}$ or $\phi_-$) and the second to a maximum (referred to as $R_{+}$ or $\phi_+$). Note that the location of the first intersection is relatively independent of $\Lambda$.
• Figure 5: A sketch of the solutions for $\Lambda = 0$ with a flat or open FRW ansatz for the $p+2$-dimensional metric. Beginning from stationary points $\dot{\phi}=0$ (filled circles), the evolution is in the potential $V$ for $\tau$ timelike (trajectories above the potential) or $- V$ for $\tau$ spacelike (trajectories below the potential). Singularities along a trajectory are denoted by an asterisk. The compactification solution sits at the potential minimum for all $\tau$. The singular (timelike and spacelike), extremal, and oscillatory solutions all have a region where $\tau$ is spacelike and $\phi \rightarrow \infty$, universally approaching the attractor solution Eq. \ref{['eq:L0asymptoticmetric']}. Continuing across an arbitrary stationary point to a region where $\tau$ is timelike will result in a spacelike singularity. There are special stationary points from which $\phi$ oscillates about the potential minimum (indicated by the double arrows on the oscillatory trajectory). For a flat FRW metric ansatz, there are solutions where $\tau$ is everywhere spacelike, and the field interpolates between the potential minimum and $\phi \rightarrow \pm \infty$. Solutions without a stationary point are always singular (timelike).