A Bell experiment during inflation: probing quantum entanglement in tensor fluctuations through correlations of primordial scalar curvature perturbations

Abstract

We propose a method that provides an observational signature of the quantum origin of primordial fluctuations generated during inflation. The method gives a prescription for testing a Bell inequality constructed exclusively from the standard scalar and tensor perturbations of minimal single-field inflation. We consider an inflationary spacetime populated by pairs of gravitons entangled in their polarization states. Third-order interactions between two scalars and one graviton transfer polarization information to the scalar sector through the product of spatial derivatives of scalars with the tensor polarization factors. Rather than performing the full multidimensional momentum integrations, we isolate and compute the tensor polarization structure of the primordial scalar eight-point correlation function. This eight-point correlation function factorizes into the product of four scalar two-point functions associated with opposite (mirrored) momentum configurations in Fourier space. This factorization falls from the fact that the two gravitons are spatially well-separated within the cosmological horizon of inflation, replicating the setup of standard Bell experiments. Through these interactions, we track how non-local correlations between both gravitons from polarization entanglement are imprinted on the scalar sector. We show that, for specific configurations of the scalar momenta after the end of inflation (detailed in the text), this observable can be used to construct a Bell-violating quantity in a way that matches the well-known Clauser-Horne-Shimony-Holt inequality definition. In principle, this offers a route to probe the quantum nature of primordial fluctuations through observables accessible today.

Figures (2)

• Figure 1: Scheme of the elements and process needed for a Bell experiment during inflation. Conformal time is denoted as $\eta$, being the end of inflation at $\eta=0$. Blue arrows represent the quantum-informed processes. Black arrows correspond to classic evolution and transmission of the results of the quantum experiment to a common location, where they can be correlated by the final observer. The red, dashed box denotes that the imprinting of the quantum correlations needed for a Bell violation should happen after the quantum state has components at spatially separated locations A and B, and before the end of inflation. Figure taken from our previous work Tejerina2024.
• Figure 2: Representation of the relevant process of this work. We depict the interaction vertices corresponding to two gravitons $\gamma_A$ and $\gamma_B$, each interacting twice, with two different pairs of scalar fluctuations. All happens within one Hubble patch of radius $H_\Lambda^{-1}$. Graviton $A$ interacts once with two scalar fluctuations with momenta $\mathbf{k}_1$ and $\mathbf{k}_2$, and once with two scalar fluctuations with momenta $-\mathbf{k}_1$ and $-\mathbf{k}_2$, in a region $A$ of the Hubble patch. Graviton B interacts analogously with $\mathbf{k}_3$ and $\mathbf{k}_4$, and $-\mathbf{k}_3$ and $-\mathbf{k}_4$, in a region $B$ that is spatially well separated from $A$. The quantum state of the pair of gravitons is a Bell pair of the form \ref{['eq: Bell state of gravitons']}, describing entanglement in their polarization/spin. For certain choices of momenta of the scalar fluctuations $\mathbf{k}_i$ and $-\mathbf{k}_i$ ($i=1,2,3,4$), there is an imprinting of the entanglement of the gravitons in the 8-point scalar correlation function, as described in Sec. \ref{['subsec: our observable - 8 point func']} and \ref{['subsec: CHSH ineq from 8-point']}. The physical interpretation of the process is further described in Sec. \ref{['subsec: our observable - 8 point func']}.