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

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.