Creating a Universe from Nothing as an Alternative to the Cosmological Principle

TL;DR

This paper constructs an exact, matter-free gravitational model with negative curvature that can be created from nothing, offering an alternative to the cosmological principle and explaining cosmic microwave background anisotropies.

Contribution

It introduces a novel geometric model of a universe emerging from nothing, characterized by a specific gravitational wave solution with no matter fields and vanishing Bach tensor.

Findings

Model is exactly solvable and purely geometric.

Contributes to understanding CMB anisotropies and polarization.

Provides an analytic method for spatial mode analysis.

Abstract

In the cosmological Robertson-Walker geometry required of the cosmological principle both the Weyl tensor $C^{\mu\lambda\nu\kappa}$ and the Bach tensor $W^{\mu\nu}=[2\nabla_{\kappa}\nabla_{\lambda}-R_{\lambda\kappa}]C^{\mu\lambda\nu\kappa}$ vanish. In general, in perturbations around the cosmological background neither of the fluctuating $\delta C^{\mu\lambda\nu\kappa}$ or $\delta W^{\mu\nu}$ would vanish. However, it is possible for $\delta W^{\mu\nu}$ to vanish even as $\delta C^{\mu\lambda\nu\kappa}$ does not. In this paper we construct an explicit model in which this is the case. The model consists of a 3-tensor gravitational wave fluctuating around a background with a constant negative 3-curvature. The model is exactly solvable and consists purely of geometric quantities with no matter fields at all (i.e., $G^{\mu\nu}=0$, $\delta G^{\mu\nu}=0$, $W^{\mu\nu}=0$, $\delta W^{\mu\nu}=0$, where $G^{\mu\nu}$ is the Einstein tensor). The model can thus be created out of nothing, with creating a universe from nothing thus being an alternative principle to the cosmological principle. The fluctuating gravitational wave contributes to the temperature anisotropy in the cosmic microwave background and its $B$ mode polarization in a calculable manner, one for which we provide a simple analytic way of treating spatial modes that is based on the use of a spatial mode addition theorem. In addition, we provide a treatment of the anisotropy that is based on properties of bandwidth limited functions. Classically by ``nothing" we mean that there are no $T^{\mu\nu}$ or $\delta T^{\mu\nu}$ matter field terms. Quantum-mechanically by ``nothing" we mean that all fields other than the gravitational field are in a negative energy mode vacuum state, with the only occupied positive energy modes being graviton modes. As well as use the Bach tensor as a diagnostic, we consider dynamics based on it.