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 three-tensor gravitational wave fluctuating around a background with a constant negative three-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 the $B$ mode polarization tensor 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 the connection of bandwidth limited functions to signal processing theory, a treatment that is new to the literature. 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. These positive energy graviton modes can then excite negative energy particles out of the vacuum into positive energy states, to thus produce “something” with $\delta W_{\mu \nu }\ne 0$. As well as using the Bach tensor as a diagnostic, we consider dynamics based on it.

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• Cosmic microwave background
• Cosmology
• Relativistic aspects of cosmology