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Measurability of Vacuum Fluctuations and Dark Energy

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Measurability of Vacuum Fluctuations and Dark Energy Measurability of vacuum fluctuations and dark energy Christian Beck School of Mathematical Sciences Queen Mary, University of London Mile End Road, London E1 4NS, UK∗ Michael C. Mackey Centre for Nonlinear Dynamics in Physiology and Medicine Departments of Physiology, Physics and Mathematics McGill University, Montreal, Quebec, Canada† (Dated: February 5, 2008) Vacuum fluctuations of the electromagnetic field induce current fluctuations in resistively shunted Josephson junctions that are measurable in terms of a physically relevant power spectrum. In this paper we investigate under which conditions vacuum fluctuations can be gravitationally active, thus contributing to the dark energy density of the universe. Our central hypothesis is that vacuum fluctuations are gravitationally active if and only if they are measurable in terms of a physical power spectrum in a suitable macroscopic or mesoscopic detector. This hypothesis is consistent with the observed dark energy density in the universe and offers a resolution of the cosmological constant problem. Using this hypothesis we show that the observable vacuum energy density ρvac in the universe is related to the largest possible critical temperature Tc of superconductors through 4 (kTc) −3 vac · ρ = σ h¯3c3 , where σ is a small constant of the order 10 . This relation can be regarded as an analog of the Stefan-Boltzmann law for dark energy. Our hypothesis is testable in Josephson junctions where we predict there should be a cutoff in the measured spectrum at 1.7 THz if the hypothesis is true. PACS numbers: 95.36.+x, 74.81.Fa, 03.70.+k I. INTRODUCTION by many orders of magnitude if gravitational coupling is taken into account. While the currently observed dark energy density in the universe is of the order m4 , where A fundamental problem relevant to quantum field the- ν mν is a typical neutrino mass scale, quantum field the- ories, cosmology, and statistical mechanics is whether ories predict that there is an infinite vacuum energy vacuum fluctuations are ‘real’ in the sense that the density, since the corresponding integrals diverge. As- corresponding vacuum energy has a gravitational effect suming a cutoff on the order of the Planck scale, one [1, 2, 3, 4]. Quantum electrodynamic (QED) explana- still has a vacuum energy density too large by a factor tions of phenomena such as the Casimir effect, the Lamb 10120. This is the famous cosmological constant problem shift, van der Waals forces, spontaneous emission from [5, 6, 7, 8, 9, 10]. atoms, etc. provide indirect evidence for the existence of The relevant question would thus seem to be not vacuum fluctuations. However, Jaffe [1] emphasizes that whether vacuum fluctuations exist (they certainly exist the Casimir effect, which is generally believed to pro- as a useful theoretical tool), but under which conditions vide stringent evidence for the existence of QED vacuum arXiv:astro-ph/0605418v3 11 Dec 2006 they have a physical reality in the sense that they pro- fluctuations, can be explained without any reference to duce a directly measurable spectrum of fluctuations in vacuum fluctuations. Indeed it seems that for most QED macroscopic or mesoscopic detectors which could have a phenomena vacuum fluctuations are a useful mathemat- gravitational effect [11, 12, 13, 14]. With respect to this ical tool to theoretically describe these effects, but there question, a very interesting experiment was performed by are formulations such as Schwinger’s source theory [3] Koch, van Harlingen and Clarke [15] in 1982. Koch et al. that completely avoid any reference to vacuum fluctua- experimentally measured the spectral density of the noise tions. current in a resistively shunted Josephson junction and It is well known that the amount of vacuum energy showed that the data were described by the theoretically formally predicted by quantum field theories is too large derived spectrum [16, 17, 18, 19, 20, 21] 2¯hω ¯hω S (ω) = coth I R 2kT ∗ Electronic address: [email protected]; 4 1 ¯hω URL: http://www.maths.qmul.ac.uk/∼beck = ¯hω + , (1) †Electronic address: [email protected]; R 2 exp(¯hω/kT ) − 1 URL: http://www.cnd.mcgill.ca/people mackey.html; also: Mathematical Institute, University of Oxford, 24-29 St Giles’, where R is the shunting resistance, ω = 2πν is the fre- Oxford OX1 3LB, UK quency, h is Planck’s constant, k is Boltzmann’s con- 2 stant, and T is the temperature. The first term in this capable of reaching the cosmologically interesting fre- equation is independent of temperature and increases quency νc [28]. linearly with the frequency ν. This term is induced The central hypothesis that we explore in this paper is by zero-point fluctuations of the electromagnetic field, that vacuum fluctuations are gravitationally active (and which produces measurable noise currents in the Joseph- hence contribute to the dark energy density of the uni- son junction [19]. The second term is due to ordinary verse) if and only if they are measurable (in the form of Bose-Einstein statistics and describes thermal noise. The a spectral density) in a suitable macroscopic or meso- experimental data of [15], reproduced in Figure 1, con- scopic detector. We will show that this basic hypothe- firm the form (1) of the spectrum up to a frequency of sis: a) provides a possible solution to the cosmological approximately ν ≃ 6 × 1011 Hz. constant problem; b) predicts the correct order of mag- nitude of dark energy currently observed in the universe; and c) is testable in future laboratory experiments based on Josephson junctions. We also argue that the optimal detector for measurable quantum noise spectra will typ- ically exploit macroscopic quantum effects in supercon- ductors. From our measurability assumption we obtain a formula for the observable dark energy density in the uni- verse that is a kind of analogue of the Stefan-Boltzmann law for vacuum energy, but in which the temperature T is not a free parameter but rather given by the largest possible critical temperature Tc of high-Tc superconduc- tors. Our central hypothesis implies that vacuum fluctua- tions at very high frequencies do not contribute to the cosmological constant. In that sense it puts a cosmolog- ical limitation on general relativity. However, it should be clear that all models of dark energy in the universe require new physics in one way or another [7]. The advan- FIG. 1: Spectral density of current noise as measured in [15] for two different temperatures. The solid line is the prediction tage of our approach is that it is experimentally testable. of eq. (1), while the dashed line neglects the zero-point term. This paper is organized as follows. In Section II we explain the central hypothesis of this paper and discuss The data in Fig. 1 represent a physically measured spec- how it can help to avoid the cosmological constant prob- trum induced by vacuum fluctuations. The spectrum lem. In Section III we show that assuming the central is measurable due to subtle nonlinear mixing effects in hypothesis is true one obtains the correct order of mag- Josephson junctions. These mixing effects, which have nitude of dark energy density in the universe. Section IV nothing to do with either the Casimir effect or van der shows how the near-equilibrium condition of the fluctu- Waals forces, are a consequence of the AC Josephson ef- ation dissipation theorem can be used to obtain further fect [22, 23, 24]. constraints on the dark energy density. Finally, in Sec- tion V we provide some theoretical background for our In [11] we suggested an extension of the Koch exper- measurability approach. We discuss the fluctuation dissi- iment to higher frequencies. We based this suggestion on the observation that if the vacuum energy associated pation theorem and its potential relation to dark energy, as well as the AC Josephson effect and its relation to with the measured zero-point fluctuations in Fig. 1 is measurability of vacuum fluctuation spectra. gravitationally active (in the sense that the vacuum en- ergy is the source of dark energy), then there must be a cutoff in the measured spectrum at a critical frequency νc. Otherwise the corresponding vacuum energy density II. MEASURABILITY AND GRAVITATIONAL would exceed the currently measured dark energy density ACTIVITY OF VACUUM FLUCTUATIONS [25, 26, 27] of the universe. The relevant cutoff frequency was predicted in [11] to be given by Let us once again state the following central hypoth- 12 esis: νc ≃ (1.69 ± 0.05) × 10 Hz, (2) only 3 times larger than the largest frequency reached in Vacuum fluctuations are gravitationally ac- the 1982 Koch et al. experiment [15]. tive if and only if they are measurable in terms In this paper we discuss the measurability of vacuum of a physically relevant power spectrum in a fluctuations as inspired by the Koch et al. experiment. macroscopic or mesoscopic detector We are motivated by the fact that this experiment will now be repeated with new types of Josephson junctions It is important to make our usage of terms precise. 3 1. By ‘vacuum fluctuations that are gravitationally These detectors of photonic vacuum fluctuations will active’ we mean those that contribute to the cur- no longer function if the frequency ν becomes too large. rently measured dark energy density of the uni- To see this, consider the spectrum verse. 1 ¯hω S(ω)= ¯hω + (4) 2. By ‘measurable vacuum fluctuations’ we mean 2 exp(¯hω/kT ) − 1 those that induce a measurable quantum noise power spectrum in a suitable macroscopic or meso- occurring in the square brackets of eq. (1). This spectrum scopic detector, for example in a resistively shunted is at the root of the problem considered here and it is of Josephson junction.
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