Drawing conformal diagrams for a fractal landscape

TL;DR

The paper introduces a new qualitative method for drawing conformal diagrams of complex, fractal-like spacetimes in cosmology, simplifying visualization without explicit metric transformations.

Contribution

A novel lightray-based technique for constructing conformal diagrams applicable to arbitrary 1+1-dimensional spacetimes, including fractal cosmological models.

Findings

Diagrams reveal fractal arrangements of lines in inflating universe models

Method simplifies visualization of complex spacetime structures

Applicable to models with nested bubbles of different geometries

Abstract

Generic models of cosmological inflation and the recently proposed scenarios of a recycling universe and the string theory landscape predict spacetimes whose global geometry is a stochastic, self-similar fractal. To visualize the complicated causal structure of such a universe, one usually draws a conformal (Carter-Penrose) diagram. I develop a new method for drawing conformal diagrams, applicable to arbitrary 1+1-dimensional spacetimes. This method is based on a qualitative analysis of intersecting lightrays and thus avoids the need for explicit transformations of the spacetime metric. To demonstrate the power and simplicity of this method, I present derivations of diagrams for spacetimes of varying complication. I then apply the lightray method to three different models of an eternally inflating universe (scalar-field inflation, recycling universe, and string theory landscape) involving the nucleation of nested asymptotically flat, de Sitter and/or anti-de Sitter bubbles. I show that the resulting diagrams contain a characteristic fractal arrangement of lines.