Creating a Universe from Nothing as an Alternative to the Cosmological Principle
Philip D. Mannheim, Daniel A. Norman, Tianye Liu
TL;DR
The paper shows that a pure-geometric 3-tensor fluctuation around a negatively curved Robertson–Walker background can satisfy $G^{\mu\nu}=0$ and $W^{\mu\nu}=0$ (i.e., $\delta G^{\mu\nu}=0$, $\delta W^{\mu\nu}=0$) while producing a nonzero $\delta C^{\mu\lambda\nu\kappa}$, effectively creating a universe from nothing and offering an alternative to the cosmological principle. It derives gauge-invariant fluctuation equations, identifies tensor-sector solutions with $A_T=\tau^2+3$, and provides explicit temporal modes $E^{(1)}_{ij}, E^{(2)}_{ij}$ (and $E^{(3)}_{ij}, E^{(4)}_{ij}$ for $\Lambda\neq0$) that satisfy $\delta G_{\mu\nu}=0$ and $\delta W_{\mu\nu}=0$, enabling a nontrivial gravitational wave geometry without matter. The authors connect these tensor fluctuations to CMB observables by showing the tensor ISW contribution reduces to $\Delta T/T|_{\text{tensor}} = -\int_{\eta_L}^{\eta_0} d\eta\, \partial_{\eta}E_{\chi\chi}$ and derive $C_\ell$ via tensor-mode addition theorems, with both analytic and bandwidth-limited numerical formulations. Numerical evaluation using conformal-gravity-inspired parameters demonstrates a pronounced $\ell$-dependent fall-off similar to standard ISW shapes in a curvature-dominated regime and suggests potential observables in the CMB $B$-mode polarization. Overall, the work proposes a foundational principle in which negative curvature plus pure gravitational fluctuations can generate a universe from nothing, offering a broader alternative to the cosmological principle and tying into conformal gravity’s success in explaining cosmic acceleration and galactic dynamics without dark matter.
Abstract
In the cosmological Robertson-Walker geometry required of the cosmological principle both the Weyl tensor $C^{μλνκ}$ and the Bach tensor $W^{μν}=[2\nabla_κ\nabla_λ-R_{λκ}]C^{μλνκ}$ vanish. In general, in perturbations around the cosmological background neither of the fluctuating $δC^{μλνκ}$ or $δW^{μν}$ would vanish. However, it is possible for $δW^{μν}$ to vanish even as $δC^{μλνκ}$ 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., $G^{μν}=0$, $δG^{μν}=0$, $W^{μν}=0$, $δW^{μν}=0$, where $G^{μν}$ 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 $T^{μν}$ or $δT^{μν}$ 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.