Spatially flat cosmological quantum spacetimes
Christian Gaß, Harold C. Steinacker
TL;DR
This work constructs and analyzes a broad class of spatially flat cosmological quantum spacetimes derived from covariant quantum spacetimes built on the fuzzy hyperboloid. A key feature is the local bundle structure ${\mathcal{M}} \ncong { mathcal{M}}^{1,3} \tilde{\times} S^2$, which endows the model with higher-spin degrees of freedom while preserving an $E(3)$ symmetry for the spatial sections. The semi-classical regime is governed by a matrix d'Alembertian $ox_T$ that relates to a geometric d'Alembertian $ox_G$ via a dilaton $ ho^2$, yielding a metric $G_{\,\mu\nu}$ that can be analyzed in local normal coordinates where higher-spin components decouple along timelike curves. Gauge transformations from the underlying matrix model act as symplectomorphisms, and are shown to induce approximate diffeomorphisms on the cosmological spacetime in regimes where higher-spin contributions are suppressed. The paper also introduces a dynamical background framework with time-dependent scalings $ ilde{T}^0=\alpha(Y^0)T^0$, $ ilde{T}^i=\beta(Y^0)T^i$, exploring distinguished gauges (timelike and covariant) and their implications for the effective FLRW geometry, including cases with constant dilaton and exact vanishing higher-spin components. Finally, it outlines how these $k=0$ quantum spacetimes can be embedded into the full IKKT model, with discussions of one-loop stabilization and couplings to extra dimensions, and discusses the potential for expanding versus shrinking cosmologies within this framework.
Abstract
We recently described a cosmological quantum spacetime of vanishing spatial curvature, which can be considered as background for the IKKT matrix model, assuming that the resulting gauge theory couples weakly. Building on this example, we construct a large class of spatially flat cosmological quantum spacetimes. We also elaborate on various details of their algebraic and semi-classical structure as well as the higher spin modes present in these models. In particular, we introduce the notion of approximate diffeomorphisms on the cosmological quantum spacetimes that stem from gauge transformations of the underlying matrix model, and investigate how different gauges are related in the semi-classical regime by approximate diffeomorphisms. Finally, we briefly outline how the described quantum spacetimes could be incorporated into the full IKKT model.