Fluctuations and Correlations in Causal Set Theory

TL;DR

The paper develops a systematic framework to compute fluctuations and correlations of causal-set quantities, focused on the causal set action in sprinklered spacetimes. It introduces two indicators, the cardinality indicator $\zeta$ and the occupation indicator $\chi$, to encode correlations across overlapping causal intervals, and it classifies correlations into $\zeta$-$\zeta$, $\chi$-$\chi$, and $\zeta$-$\chi$ types with explicit strategies to evaluate them. The fluctuations of the causal set action are expressed as a quadratic form in a correlation matrix $\mathcal{K}_{ij}$, which is decomposed into tractable integrals over continuum manifolds, leading to a small core set of parametrizable integrals. The formalism enables concrete calculations of discreteness-induced effects, with potential applications to Minkowski and cosmological spacetimes, and provides a pathway to connect action fluctuations with Everpresent $\Lambda$ phenomenology and dynamical causal-set models.

Abstract

We study the statistical fluctuations (such as the variance) of causal set quantities, with particular focus on the causal set action. To facilitate calculating such fluctuations, we develop tools to account for correlations between causal intervals with different cardinalities. We present a convenient decomposition of the fluctuations of the causal set action into contributions that depend on different kinds of correlations. This decomposition can be used in causal sets approximated by any spacetime manifold $\mathcal M$. Our work paves the way for investigating a number of interesting discreteness effects, such as certain aspects of the Everpresent $Λ$ cosmological model.

Figures (2)

• Figure 1: Two non-empty regions $\mathcal{R}_1$ and $\mathcal{R}_2$ with non-empty overlap $\widetilde{\mathcal{R}}_b = \mathcal{R}_1\cap\mathcal{R}_2$. The union $\mathcal{R}_1\cup\mathcal{R}_2$ can be decomposed into non-overlapping subregions $\widetilde{\mathcal{R}}_b$, $\widetilde{\mathcal{R}}_a = \mathcal{R}_1\backslash \widetilde{\mathcal{R}}_b$ and $\widetilde{\mathcal{R}}_c = \mathcal{R}_2\backslash \widetilde{\mathcal{R}}_b$.
• Figure 2: Setup of $\mathcal{J}^-(x_1)\cap \mathcal{J}^-(x_2)$ in a $1+1$D Minkowski Diamond.