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arXiv:2502.13846 · Journal of Cosmology and Astroparticle Physics

Violation of Bell inequalities from Cosmic Microwave Background data

Roberto Dale, Jesús M. Gandía, Juan Antonío Morales-Lladosa, Ramón Lapiedra

Published Jul 1, 2025• 2 citations
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Abstract

Abstract In a recent paper [R. Dale. R. Lapiedra, and J. A. Morales-Lladosa, Phys. Rev. D 107 (2023) 023506] a cosmic-like Clauser-Horne-Shimony-Holt (CHSH) inequality was proved for the temperature fluctuations δT of the perturbed Cosmic Microwave Background (CMB), assuming local realism. This inequality can be tested from observational data. In fact, no violation for it has been found from the CMB data provided by the COBE satellite in different sky directions. This is a result which, being negative, is not conclusive in relation to the possible violation of the CHSH inequality in a cosmological context. So, the preceding analysis needs to be extended by using more precise current CMB data, like those provided by WMAP and Planck satellites. This is a task of considerable importance but, technically, much more involved in data treatments. In the present, assuming again local realism behind measurements and observational data, this extension is accomplished. The result is that a sound violation of the CHSH inequality is found, which would mean the failure of the assumed local realism in accordance with the quantum origin of the primordial temperature fluctuations in the framework of standard inflation and the orthodox interpretation of the Quantum Mechanics.

DOI
10.1088/1475-7516/2025/07/044

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Abstract In a recent paper [R. Dale. R. Lapiedra, and J. A. Morales-Lladosa, Phys. Rev. D 107 (2023) 023506] a cosmic-like Clauser-Horne-Shimony-Holt (CHSH) inequality was proved for the temperature fluctuations δT of the perturbed Cosmic Microwave Background (CMB), assuming local realism. This inequality can be tested from observational data. In fact, no violation for it has been found from the CMB data provided by the COBE satellite in different sky directions. This is a result which, being negative, is not conclusive in relation to the possible violation of the CHSH inequality in a cosmological context. So, the preceding analysis needs to be extended by using more precise current CMB data, like those provided by WMAP and Planck satellites. This is a task of considerable importance but, technically, much more involved in data treatments. In the present, assuming again local realism behind measurements and observational data, this extension is accomplished. The result is that a sound violation of the CHSH inequality is found, which would mean the failure of the assumed local realism in accordance with the quantum origin of the primordial temperature fluctuations in the framework of standard inflation and the orthodox interpretation of the Quantum Mechanics.
The temperature measurements, TTT, of the perturbed cosmic microwave background, performed by the cosmic background explorer satellite (COBE), are considered. A dichotomist function, $f=\ifmmode\pm\else\textpm\fi{}1$, is defined such that f=+1f=+1f=+1 if $\ensuremath{\delta}T>0$ and $f=\ensuremath{-}1$ if $\ensuremath{\delta}T<0$, where $\ensuremath{\delta}T$ stands for the fluctuation of TTT. Then, it is assumed that behind the appearance of these fluctuations there is local realism. Under this assumption, some specific Clauser-Horne-Shimony-Holt (CHSH) inequalities are proved for these fluctuation temperatures measured by COBE in the different sky directions. The calculation of these inequalities from the actual temperature measurements shows that these inequalities are not violated. This result cannot be anticipated by calculating the commutators of the cosmic density quantum operators. This must be remarked here since, in the case of a system of two entangled spin 12\frac{1}{2}21​ particles, its CHSH inequalities violation can be inferred from the nonvanishing value of the corresponding spin measurement commutators. The above nonviolation of the observed cosmic CHSH inequalities is compatible with the existence of local realism behind the cosmic measurement results. Nevertheless, assuming again local realism, some new cosmic CHSH inequalities can be derived for the case of the WMAP measurements whose accuracy is better than the one of the above considered COBE measurements. More specifically, in the WMAP case, some significant cross correlations between the temperature and polarization maps are detected, and the new cosmic CHSH inequalities are the ones built with these cross correlations. Now, the occasional violation of these CHSH inequalities would mean the failure of the assumed local realism in accordance with the quantum origin of the primordial temperature and polarization fluctuations in the framework of standard inflation.
A number of papers have suggested that it is inappropriate to combine data from different experiments when undertaking experimental tests of Bell's inequalities. It has been suggested that a correct analysis, using a single probability space, leads to inequalities which are not violated by experiment. If correct, this would be contrary to the normal interpretation of such experimental data. However, in this note, a generalised Clauser-Horne-Shimony-Holt (CHSH) inequality is derived for a system of four experiments constructed on a single probability space which combines the data from the four experiments. It is shown that this leads to the standard CHSH inequality which is normally used to interpret experimental data. Thus the commonly accepted conclusion that experimental violations of Bell's inequality imply that local realistic models are inconsistent with the predictions of quantum mechanics has not been challenged.
A finite non-classical framework for qubit physics is described that challenges the conclusion that the Bell Inequality has been shown to have been violated experimentally, even approximately. This framework postulates the primacy of a fractal-like ‘invariant set’ geometry I U in cosmological state space, on which the universe evolves deterministically and causally, and from which space-time and the laws of physics in space-time are emergent. Consistent with the assumed primacy of I U , a non-Euclidean (and hence non-classical) metric g p is defined in cosmological state space. Here, p is a large but finite integer (whose inverse may reflect the weakness of gravity). Points that do not lie on I U are necessarily g p -distant from points that do. g p is related to the p-adic metric of number theory. Using number-theoretic properties of spherical triangles, the Clauser-Horne-Shimony-Holt (CHSH) inequality, whose violation would rule out local realism, is shown to be undefined in this framework. Moreover, the CHSH-like inequalities violated experimentally are shown to be g p -distant from the CHSH inequality. This result fails in the singular limit p = ∞ , at which g p is Euclidean and the corresponding model classical. Although Invariant Set Theory is deterministic and locally causal, it is not conspiratorial and does not compromise experimenter free will. The relationship between Invariant Set Theory, Bohmian Theory, The Cellular Automaton Interpretation of Quantum Theory and p-adic Quantum Theory is discussed.
According to the theory of cosmic inflation, the large scale structures observed in our Universe (galaxies, clusters of galaxies, Cosmic Background Microwave—CMB—anisotropy…) are of quantum mechanical origin. They are nothing but vacuum fluctuations, stretched to cosmological scales by the cosmic expansion and amplified by gravitational instability. At the end of inflation, these perturbations are placed in a two-mode squeezed state with the strongest squeezing ever produced in Nature (much larger than anything that can be made in the laboratory on Earth). This article studies whether astrophysical observations could unambiguously reveal this quantum origin by borrowing ideas from quantum information theory. It is argued that some of the tools needed to carry out this task have been discussed long ago by J. Bell in a, so far, largely unrecognized contribution. A detailled study of his paper and of the criticisms that have been put forward against his work is presented. Although J. Bell could not have realized it when he wrote his letter since the quantum state of cosmological perturbations was not yet fully characterized at that time, it is also shown that Cosmology and cosmic inflation represent the most interesting frameworks to apply the concepts he investigated. This confirms that cosmic inflation is not only a successful paradigm to understand the early Universe. It is also the only situation in Physics where one crucially needs General Relativity and Quantum Mechanics to derive the predictions of a theory and, where, at the same time, we have high-accuracy data to test these predictions, making inflation a playground of utmost importance to discuss foundational issues in Quantum Mechanics.
One of the cornerstones of inflationary cosmology is that primordial density fluctuations have a quantum mechanical origin. However, most physicists consider that such quantum mechanical effects disappear in CMB data due to decoherence. In this conference report, we show that the violation of Bell inequalities in an initial state of our universe increases exponentially with the number of modes to measure in inflation. This indicates that some evidence that our universe has a quantum mechanical origin may survive in CMB data, even if quantum entanglement decays exponentially afterward due to decoherence.
The Bell-CHSH (Clauser-Horne-Shimony-Holt) inequality in the vacuum state of a relativistic scalar quantum field is analyzed. Using Weyl operators built with smeared fields localized in the Rindler wedges, the Bell-CHSH inequality is expressed in terms of the Lorentz invariant inner products of test functions. A numerical framework for these inner products is devised. Causality is also explicitly checked by a numerical evaluation of the Pauli-Jordan function. Violations of the Bell-CHSH inequality are reported for different values of the particle mass parameter. Published by the American Physical Society 2024