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The Heisenberg Limit at Cosmological Scales

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Abstract

For an observation time equal to the universe age, the Heisenberg principle fixes the value of the smallest measurable mass at \(m_\mathrm{H}=1.35 \times 10^{-69}\) kg and prevents to probe the masslessness for any particle using a balance. The corresponding reduced Compton length to \(m_\mathrm{H}\) is , and represents the length limit beyond which masslessness cannot be proved using a metre ruler. In turns, is equated to the luminosity distance \(d_\mathrm{H}\) which corresponds to a red shift \(z_\mathrm{H}\). When using the Concordance-Model parameters, we get \(d_\mathrm{H} = 8.4\) Gpc and \(z_\mathrm{H}=1.3\). Remarkably, \(d_\mathrm{H}\) falls quite short to the radius of the observable universe. According to this result, tensions in cosmological parameters could be nothing else but due to comparing data inside and beyond \(z_\mathrm{H}\). Finally, in terms of quantum quantities, the expansion constant \(H_0\) reveals to be one order of magnitude above the smallest measurable energy, divided by the Planck constant.

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Notes

  1. The International Astronomical Union has adopted the terminology Hubble-Lemaître constant [12]. We recall the contributions by Humason [13,14,15].

  2. Herein, we use indeterminacy when we address massiveness as opposed to masslessness and uncertainty when we refer to measurement accuracy. Principle is used when referring to the general concept rather than an estimate.

  3. Photons were emitted at decoupling about 370.000 years after the Big Bang; photons by galaxies some hundred million years later.

  4. De Broglie estimated the photon mass below \(10^{-53}\) kg through group velocity dispersion [27, 28]. After one century of painstaking experiments, the Particle Data Group (PDG) [29] sets the upper limit just one order of magnitude below, at \(m < 10^{-18}\) eV c\(^{-2}\) or \(1.8 \times 10^{-54}\) kg in the solar wind, but see [30] for a critical assessment. Recently, limits up to \(3.9\times 10^{-51}\) kg were established estimating dispersion in Fast Radio Bursts (FRBs) [31], more stringent than other attempts [32, 33]; for a review, see [34]. The best laboratory test—based on Coulomb’s law—showed an upper limit of \(1.6 \times 10^{-50}\) kg [35], while a space mission was conceived targeting dispersion in the very-low radio frequencies [36].

  5. Massive gravity was first proposed by Fierz and Pauli [37], and later in different contexts including alternatives to dark matter [38]. For gravitons, the PDG [29] sets the upper limit at \(6 \times 10^{-32}\) eV or \(1.1 \times 10^{-67}\) kg, obtained from gravitational lensing [39].

  6. The Taylor expansion of a generic function f(z) is

    $$f(z)=\sum _{i=0}^\infty {c_i} {z^i,}$$
    (3)

    where \(c_i=f^{(i)}(0)/i!\), whereas the (n, m) Padé approximant of f(z) is defined as rational polynomial by

    $$P_{n,m}(z)=\dfrac{\displaystyle {\sum _{i=0}^{n}a_i z^i}}{1+\displaystyle {\sum _{j=1}^{m}b_j z^j}}.$$
    (4)

  7. The check consists in fitting observational data with the Padé approximants. The Padé-approximant based cosmography reproduces observed distances with analytical models. The latter include a set of cosmological parameters which determine the coefficients of the approximants. In Fig. 1, we have used Concordance-Model parameters (\(\Omega _\mathrm{m} = 0.3\) and \(\Omega _{\Lambda } = 0.7\)), and thereby we have identified the coefficients \(q_0\) and \(j_0\). We could have departed from the Concordance-Model parameters and then obtained other values of the Heisenberg radius, but without relevant consequences for our work.

  8. The radius of the observable universe is usually expressed as comoving distance, 14.3 Gpc, roughly at \(z=1100\). For a flat universe, \(\Omega _\mathrm{k} = 0\), the luminosity distance is \((1+z)\) times the comoving distance, achieving thereby extremely large values for increasing z.

  9. McCrea [49] used the term uncertainty for pointing out the consequences of the finitude of the speed of light on the observability of the universe and did not address the Heisenberg principle in any manner. In [50,51,52], the origin of probability in cosmology, the relations to inflation, the boundaries of knowledge are addressed. The latter issue is closest to our work. Further discussions on the limits of knowledge, possibly in connection with Gödel incompleteness theorem, go well beyond the aims of this paper.

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Acknowledgements

ADAMS acknowledges the Erasmus+ programme for visiting the Università di Napoli, SC the Université d’Orléans and Campus France for the hospitality. SC and MB acknowledge the Istituto Nazionale di Fisica Nucleare (INFN), sezione di Napoli, iniziative specifiche MOONLIGHT2 and QGSKY. The authors thank O. Luongo (Frascati) for discussions. Finally, the authors are indebted to the referee for the detailed comments.

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Authors and Affiliations

  1. Institut Denis Poisson (IDP), UMR 7013, Université d’Orléans (UO) et Université de Tours (UT), Centre National de la Recherche Scientifique (CNRS), Parc de Grandmont, 37200, Tours, France

    Alessandro D. A. M. Spallicci

  2. Laboratoire de Physique et Chimie de l’Environnement et de l’Espace (LPC2E), UMR 7328, Centre National de la Recherche Scientifique (CNRS), Université d’Orléans (UO), Centre National d’Études Spatiales (CNES), 3A Avenue de la Recherche Scientifique, 45071, Orléans, France

    Alessandro D. A. M. Spallicci

  3. UFR Sciences et Techniques, Université d’Orléans (UO), Rue de Chartres, 45100, Orléans, France

    Alessandro D. A. M. Spallicci

  4. Observatoire des Sciences de l’Univers en région Centre (OSUC), UMS 3116, Université d’Orléans (UO), 1A rue de la Férollerie, 45071, Orléans, France

    Alessandro D. A. M. Spallicci

  5. Dipartimento di Fisica Ettore Pancini, Università degli Studi di Napoli, Federico II (UNINA), Complesso Universitario Monte S. Angelo, Via Cinthia 9 Edificio G, 80126, Napoli, Italy

    Micol Benetti & Salvatore Capozziello

  6. Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli, Complesso Universitario Monte S. Angelo, Via Cinthia 9 Edificio G, 80126, Napoli, Italy

    Micol Benetti & Salvatore Capozziello

  7. Scuola Superiore Meridionale, Largo San Marcellino 10, 80138, Napoli, Italy

    Micol Benetti & Salvatore Capozziello

  8. Laboratory for Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR), 40 prospect Lenina, 634050, Tomsk, Russia

    Salvatore Capozziello

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  1. Alessandro D. A. M. Spallicci
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  2. Micol Benetti
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  3. Salvatore Capozziello
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Correspondence to Alessandro D. A. M. Spallicci.

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Spallicci, A.D.A.M., Benetti, M. & Capozziello, S. The Heisenberg Limit at Cosmological Scales. Found Phys 52, 23 (2022). https://doi.org/10.1007/s10701-021-00531-z

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