Minimal noise in non-quantized gravity

Abstract

An elementary prediction of the quantization of the gravitational field is that the Newtonian interaction can entangle pairs of massive objects. Conversely, in models of gravity in which the field is not quantized, the gravitational interaction necessarily comes with some level of noise, i.e., non-reversibility. Here, we give a systematic classification of all possible such models consistent with the basic requirements that the non-relativistic limit is Galilean invariant and reproduces the Newtonian interaction on average. We demonstrate that for any such model to be non-entangling, a quantifiable, minimal amount of noise must be injected into any experimental system. Thus, measuring gravitating systems at noise levels below this threshold would be equivalent to demonstrating that Newtonian gravity is entangling. As concrete examples, we analyze our general predictions in a number of experimental setups, and test it on the classical-quantum gravity models of Oppenheim et al., as well as on a recent model of Newtonian gravity as an entropic force.

Figures (2)

• Figure 1: Schematic experimental setups. Left: two mechanical oscillators placed at distance $d$, used to measure their mutual gravitational interaction Westphal:2020okxAgafonova:2024evrManley:2026evd. Fluctuations in the force are schematically depicted by the dashed masses. Center: a pair of free-falling masses, approximated as two-state systems $\ket{L,R}$ and initially superposed into two locations, following Bose:2017nin. Right: a hybrid system consisting of a mechanical source mass coupled to a two-state system, such as an atom interferometer near a large source mass kovachy2015quantumasenbaum2017phaseCarney:2021yfwoverstreet2022observation.
• Figure 2: The parameter space of classical-quantum gravity. Here we show the two-dimensional space of free parameters $D_0$, $D_2$ that define CQ gravity. The lower left region in gray violates the basic tradeoff relation $D_0 D_2 \geq 1$ and is thus not part of the parameter space. The top right dark gray region is already excluded because it would produce more noise [see Eq. \ref{['eq:cq_forcenoise']}] than observed in the LISA Pathfinder (a device of linear size $\ell \approx 46~{\rm mm}$), which operated with acceleration noise $\sqrt{S_{aa}} \lesssim 10^{-15}~{\rm m/s^2}/\sqrt{\rm Hz}$armano2024depth. The dashed region shows the part of this parameter space would be ruled out by comparing the force noise predicted by Eq. \ref{['eq:cq_forcenoise']} to a future acceleration measurement at the general non-entangling threshold $S_{aa} \approx 10^{-18}~{\rm m/s^2}/\sqrt{\rm Hz}$ [see Eq. \ref{['eq:SFF_thresh']}]. To render this plot, we have not assumed that CQ is non-entangling; if we had, then the measurement at the $10^{-18}$ level would be sufficient to completely rule out the model.