Measurability of Vacuum Fluctuations and Dark Energy

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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 magnitude 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 typically exploit macroscopic quantum effects in superconductors. From our measurability assumption we obtain a formula for the observable dark energy density in the universe 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 superconductors. Our central hypothesis implies that vacuum fluctuations at very high frequencies do not contribute to the cosmological constant. In that sense it puts a cosmological 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 energy 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. relevance for other mesoscopic systems as well [29]. The spectrum in eq. (4) is identical (up to a factor 4/R) to The central hypothesis is testable experimentally, e.g. the measured spectrum in Josephson junctions. It arises in the experiment by Warburton, Barber and Blamire out of the fluctuation dissipation theorem [16, 30, 31] in [28] currently under way. If the central hypothesis is a universal way and it formally describes a quantum me- true, then the vacuum fluctuations producing the mea- chanical oscillator of frequency ω. The linear term in ω sured spectra in the Josephson junction are gravitation- describes the zeropoint energy of this quantum mechani- ally active (i.e. contributing to dark energy density), and cal oscillator, whereas the second term describes thermal hence there will be a cutoff near νc in the measured spec- states of this oscillator. There is vacuum energy associ- trum. Otherwise the dark energy density of the universe ated with the zeropoint term and we may identify it with is exceeded. The converse is also true: If the cutoff is the source of dark energy (for more details on the theoret- not observed, the observed vacuum fluctuations in the ical background, see Section 5). For the vacuum fluctua- Josephson junction cannot be gravitationally active. In tion noise term 1 ¯hω to be measurable, it must be not too this case the central hypothesis is false. 2 small relative to the thermal noise term ¯hω/(ehω/kT¯ −1). We now demonstrate how, under the assumption that For low frequencies the thermal noise dominates, for large the central hypothesis is true, the cosmological constant frequencies the quantum noise. The frequency ω where problem is solved and at the same time the correct order 0 both terms have the same size is given by of magnitude of dark energy density in the universe is predicted. kT ω = ln 3. (5) Recall that quantum field theories predict a divergent 0 ¯h (infinite) amount of vacuum energy given by Now suppose we want to measure vacuum fluctuation ∞ 1 d3k 2j ~ 2 2 spectra at very large frequencies ω. From eq. (5) we can ρˆvac = (−1) (2j + 1) 3 k + m (3) 2 Z−∞ (2π) q achieve higher frequencies by choosing to do our measurements at higher temperatures since increasing the tem- in units whereh ¯ = c = 1. Here ~k is the momentum perature increases the frequency ω0. As ω0 increases, the associated with a vacuum fluctuation, m is the mass of vacuum fluctuation term dominates relative to thermal the particle under consideration, and j is the spin. The noise for all frequencies ω>ω0. central hypothesis now immediately explains why most There is however a practical limit to this procedure. of the vacuum energy in eq. (3) is not gravitationally Namely, if the temperature T becomes too high, then a active: In most cases there will be no suitable detec- superconducting state will no longer exist. Since super- tor to measure the vacuum fluctuations under considera- conducting devices such as Josephson junctions appear tion. And what is not measurable is also not observable. to be the only experimentally feasible devices to measure Only in very rare cases will there be such a detector, and high frequency quantum noise spectra (see the arguments only in these cases the corresponding vacuum energy is in section 5), this means that there is a maximum fre- measurable and thus physically relevant. According to quency ωc above which a superconducting detector is no our hypothesis, the detectable part of vacuum energy is longer functional and the quantum fluctuation spectrum gravitationally active and responsible for the current ac- becomes unmeasurable. This critical frequency is given celerated expansion of the universe. by For strong and electro-weak interactions it is unlikely that a suitable macroscopic detector exists that can mea- ¯hωc ∼ kTc, (6) sure the corresponding vacuum spectra. The only candi- where T is the largest possible critical temperature of date where we know that a suitable macroscopic detector c any superconductor. Currently, the largest T known for exists is the electromagnetic interaction. Photonic vac- c high-T superconductors is approximately T = 139 K uum fluctuations induce measurable spectra in supercon- c c [32]. ducting devices, as experimentally confirmed by Koch et In fact, technically feasible solutions of superconduct- al. [15] up to frequencies of 0.6 THz. For photons, m =0 ing materials that are used in practice to build Josephson and the integration over all ~k in eq. (3) is just an integra- junctions have lower Tc. For example, the well-known tion over all frequencies ν since E = ~k2 + m2 = |~k| = YBCO materials have a maximum critical temperature hν =¯hω. p Tc of 93 K [23]. To optimize a Josephson quantum noise 4 detector one needs to avoid quasiparticle currents, which vacuum fluctuations are no longer measurable, by our means that the junction should operate at a tempera- central hypothesis they also can no longer have a large- ′ ture Tc well below Tc. Let us choose as a rough estimate scale gravitational effect. ′ ′ ′ Tc ≈ 80 K. One obtains νc ∼ kTc/h ≈ 1.7 THz. It is From this perspective, the dark energy density of the encouraging that this value is so close to that of eq. (2), universe is given by integrating the energy density over thus providing us with the correct order of magnitude of all measurable vacuum fluctuations, to obtain measurable dark energy density if the central hypothesis νc 8πν2 1 is true. ρdark = 3 hνdν (11) Z0 c 2 πh = ν4 (12) III. ESTIMATE OF OBSERVABLE DARK c3 c ENERGY DENSITY ln4(1+2η) (kT )4 = c . (13) 8π2 ¯h3c3 To get a better estimate of the proportionality constant in eq. (6), let us more carefully analyze when an experi- This result can be considered an analogue of the Stefan- ment based on Josephson junctions will be able to resolve Boltzmann law quantum noise. In the data in Fig. 1, the overall precision π2 (kT )4 ρrad = (14) by which the power spectrum can be measured at a given 15 ¯h3c3 frequency in the Koch experiment is of the order 10-40%. for radiation energy density. The difference is that in Other experiments based on SQUIDs yield fluctuations of eq. (13) the temperature is the largest possible critical similar order of magnitude in the measured power spec- temperature Tc of superconductors, and the proportion- 1 4 trum [33]. The quantum noise term 2 ¯hω is measurable ln (1+2η) −5 −3 as soon as it is larger than the standard deviation of the ality constant 8π2 ≈ 1.4 × 10 → 1.5 × 10 is much smaller than for the Stefan-Boltzmann law (i.e. fluctuations of the measured spectrum. That is to say, 2 π compared with 15 ≃ 0.66). Using Tc = 139 K and the 1 ¯hω 3 ¯hω >η · (7) known dark energy density ρdark = (3.9 ± 0.4) GeV/m 2 ehω/kT¯ − 1 [25, 26, 27] we may write eq. (13) as 4 where we estimate η ≃ 0.1 → 0.4 from the fluctuations (kTc) ρdark = σ (15) of the Koch data in Fig. 1. Condition (7) can be written ¯h3c3 as where the dimensionless constant σ is given by ¯hω 4 > ln(1 + 2η). (8) ln (1+2η) 3 3 ρdark −3 kT σ = 2 =¯h c 4 = (1.46 ± 0.15) × 10 . 8π (kTc) We thus obtain from eq. (6) the critical frequency νc be- (16) yond which measurements of the spectrum become im- possible as IV. FURTHER CONSTRAINTS ON DARK ln(1 + 2η) kT ENERGY DENSITY ν ≈ c . (9) c 2π ¯h The explanation of why vacuum fluctuations produce The largest critical temperature of a high-T supercon- a measurable spectrum of noise in resistively shunted ductors achieved so far is approximately Tc = 139K [32]. Josephson junctions is based on two fundamental effects, Our result yields, with Tc = 139K and η ≃ 0.1 → 0.4, the fluctuation dissipation theorem [16, 17, 20, 30, 31]

12 and the AC Josephson effect [22, 23, 24]. The Josephson νc ≈ ln(1 + 2η) × 2.89 × 10 Hz ≃ 0.5 → 1.7THz. (10) effect requires the existence of a superconducting state, and in this way we were led to an estimate on observable Recall that the dark energy density measured in as- (measurable) dark energy density in the previous sec- tronomical observations is correctly reproduced for νc = 12 tion. The fluctuation dissipation theorem requires that 1.69 × 10 Hz [11]. It is interesting that our argument the system under consideration is close to thermal equi- based on measurability of vacuum fluctuations predicts librium. We now show that the latter condition also pro- the correct order of magnitude of dark energy density in vides an upper bound on observable (measurable) dark the universe. This is especially hearting since the cosmo- energy density. In other words, both effects imply that logical constant problem is usually plagued by estimates 120 the cosmological constant is small. of vacuum energy too large by a factor 10 [5, 6]. Our The fluctuation dissipation theorem predicts a current proposition for the resolution of the cosmological con- (I) spectrum (A2/Hz) in the shunt resistor given by stant problem is simply that for frequencies higher than ν a macroscopic detector no longer exists to measure ¯hω − c S (ω)=2¯hω coth Re Z 1(ω), (17) the spectrum of vacuum fluctuations. As soon as the I 2kT 5 and a voltage (V ) spectrum (V2/Hz) given by If we make the same estimate for the dark energy density, the corresponding temperature of a black body with ¯hω the same energy density as dark energy density is T ≃ 30 SV (ω)=2¯hω coth Re Z(ω), (18) 2kT K, so with some suitable cooling this neither disturbs thermal equilibrium, nor disturbs a junction operating where Z(ω) is the impedance. As said before, the deriva- in a superconducting state. tion of the fluctuation dissipation theorem is based on Note that the experiments that successfully confirm the assumption of thermal equilibrium in the resistor, or the QED predictions of the Casimir effect (see, e.g., [34]) that the system is at least near thermal equilibrium. just provide measurements of the Casimir force, they do The noise spectra measured with Josephson junctions not provide us with a measured spectrum of vacuum fluc- correspond to real currents of real electrons, rather than tuations, as required by the central hypothesis. If the virtual fluctuations, and hence these currents will gener- central hypothesis is correct, it is clear that the vacuum ate heat in the shunting resistor. The dissipated power fluctuations that formally (i.e. by entering into the QED is P = I × V , which again provides a reason why there calculations) influence the Casimir effect are gravitation- must be an upper cutoff in the measurable spectrum. ally inactive since their frequency is much larger than νc Namely, if the frequency becomes too high, then the as- (see also [35]). sociated heat production through the power dissipation will become so large that it takes the system away from thermal equilibrium. Consequently, the fluctuation dis- V. THEORETICAL BACKGROUND sipation theorem would no longer be valid. To illustrate this, assume there is a physical device that A. The fluctuation dissipation theorem and dark could measure the vacuum noise spectra to frequencies energy much higher than νc, say of the same order of magnitude as relevant for the Casimir effect. Consider an arbitrary observable x(t) of a quantum Consider two Casimir plates separated by a distance L. mechanical system and consider the quantum mechanical The wavelengths of vacuum fluctuations that will fit into expectation of the time derivative of x, denoted by hx˙(t)i. the cavity formed by the plates must satisfy λ ≤ L but Let F (t) be an external force. If linear response theory since λ = c/ν this means that the minimum frequency is applicable we may write of vacuum fluctuations associated with the Casimir effect t must satisfy ′ ′ ′ hx˙(t)i = G(t − t )F (t )dt , (21) c Z−∞ νmin ≥ . L where the function G is the ‘generalized conductance’. Experimental measurements of the Casimir effect are Its Fourier transform is given by made with L ≃ O(0.1 µ) → O(1 µ) [34], so assume ∞ −6 14 iωt L = 1 µ = 10 m to give νmin ≃ 3 × 10 Hz, some G(ω)= dte G(t). (22) Z0 200 times greater than the cutoff frequency νc in eq. (2). If the predicted cutoff frequency in the Josephson junc- The fluctuation dissipation theorem [16, 17, 20, 30, 31] tion were increased to νmin, this would imply an energy yields a very general relation between the power spec- density of about trum Sx˙ of the stochastic processx ˙(t) and the real part of G(ω): 4 νmin 4 8 3 ρdark ≃ 3.9 × 2 × 10 GeV/m (19) ¯hω νc S (ω) = 2¯hω coth Re G(ω) (23) x˙ 2kT = 6.2 × 109 GeV/m3 (20) 1 ¯hω = ¯hω + · 4Re G(ω)(24) which is dissipated in the shunt resistor. Suppose that 2 exp(¯hω/kT ) − 1 this energy density came from a black body described by The function in square brackets is universal, it does not the Stefan-Boltzmann law, i.e. depend on details of the quantum system considered, in 2 4 particular on its Hamiltonian H. However, G(ω) is sys- π k 4 9 T =6.2 × 10 GeV/m3. tem dependent. The universal function in the square 15¯h3c3 brackets can be physically associated with the mean en- What would the corresponding temperature be? Quite ergy of a quantum mechanical oscillator of frequency ω at 1 high, namely T ≃ 6000 K. It is clear that the Joseph- temperature T . Its ground state energy is given by 2 ¯hω. son experiment could not operate with a heat source at Note that this oscillator has nothing to do with the origi- that temperature, and thermal equilibrium would be de- nal Hamiltonian H of the quantum system under consid- stroyed long before. This simple argument once again eration. Also note that the fluctuation dissipation theo- shows that there must be a cutoff in the measurable spec- rem is valid for arbitrary Hamiltonians H, H need not de- trum at frequencies much smaller than νmin. scribe a harmonic oscillator at all but can be a much more 6 complicated Hamiltonian function. Nevertheless, the To briefly explain this effect, remember that a Joseph- 1 universal function Huni = 2 ¯hω +¯hω/(exp(¯hω/kT ) − 1) son junction consists of two superconductors with an in- that occurs in the square brackets of eq. (24) can always sulator sandwiched in between. In the Ginzburg-Landau be formally interpreted as the mean energy of a harmonic theory, each superconductor is described by a complex oscillator. wave function, whose absolute value squared yields the Our physical interpretation is to identify the zeropoint density of superconducting electrons. Denote the phase energy of Huni as the source of dark energy. This os- difference between the two wave functions of the two su- cillator is universally present everywhere, but typically perconductors by ϕ(t). Josephson [22] made the remark- manifests measurable effects only in dissipative media. able prediction that at zero external voltage a supercon- 1 Note that the term 2 ¯hω, which describes the zeropoint ductive current given by energy of this oscillator, is invariant under transforma- tions H → H + const of the system Hamiltonian H. Is = Ic sin ϕ (25) This is obvious, since the fluctuation dissipation theorem is valid for any Hamiltonian H. In other words, the flows between the two superconducting electrodes. Here Ic is the maximum superconducting current the junction zeropoint energy of our dark energy oscillator Huni is an invariant, it is invariant under renormalization of the can support. Moreover, he predicted that if a voltage system Hamiltonian H. This is a strong hint that the difference V is maintained across the junction, then the 1 phase difference ϕ evolves according to corresponding vacuum energy 2 ¯hω is indeed a physically observable quantity. It is likely to have gravitational rel- 2eV evance since it is invariant under arbitrary re-definitions ϕ˙ = , (26) ¯h of the system Hamiltonian H. A further interpretation (as suggested in the classical i.e. the current in eq. (25) thus becomes an oscillating papers and textbooks on the subject [16, 17, 19]) is to current with amplitude Ic and frequency associate the term 1 ¯hω with the zeropoint fluctuations of 2 2eV the electromagnetic field. We thus arrive at the follow- ν = . (27) ing, in our opinion, most plausible interpretation of the h fluctuation dissipation theorem: The term 1 ¯hω describes 2 This frequency is the well-known Josephson frequency, the gravitationally active part of the vacuum fluctuations and the corresponding effect is called the AC Josephson of the electromagnetic field. effect. The quantum energy hν given by eq. (27) can be Once again let us emphasize that we consider the uni- interpreted as the energy change of a Cooper pair that is versal function Huni occuring in the fluctuation dissipa- transferred across the junction. The AC Josephson effect tion theorem as a potential source of dark energy, rather is a very general and universal effect that always occurs than the system dependent Hamiltonian H. In recent whenever two superconducting electrodes are connected papers [37, 38], the role of H and Huni is confused. The by a weak link. authors re-derive the well-known fact that the fluctua- The AC Josephson effect connects quantum mechan- tion dissipation theorem is valid for arbitrary Hamiltoni- ics (i.e. differences of phases of macroscopic wave func- ans H, in particular for those where an arbitrary additive tions) with measurable classical quantities (currents or constant is added to H. However, their argument relates voltages). A Josephson junction can be regarded as a to the system Hamiltonian H and not to Huni and is perfect voltage-to-frequency converter, satisfying the re- thus not applicable to our model of dark energy based lation 2eV = hν. For distinct DC voltages, it is also a on Huni. In particular, the considerations in [37, 38] perfect frequency-to-voltage converter. This inverse AC provide no insight into the physical interpretation of the Josephson effect is, for example, used to maintain the SI vacuum energy associated with Huni, which is invariant unit Volt. and universal. In the experiments of Koch et al. [15] the quantum noise in the shunt resistor is mixed down at the Joseph- son frequency 2eV/h to produce measurable voltage fluc- B. The AC Josephson effect and measurability of tuations. The measurement frequency in these exper- vacuum fluctuations iments is usually much smaller than the Josephson frequency. However, due to the specific nonlinear properties The fluctuation dissipation theorem quite generally ex- of Josephson junctions, the measured voltage fluctuations plains the existence of quantum noise in resistors, but on are influenced by quantum fluctuations at the Josephson its own it does not suffice to make the power spectrum frequency and its harmonics [18, 21]. In this way quan- of quantum fluctuations measurable in an experiment. tum fluctuations in the THz regime become experimen- Putting a voltmeter directly into the dissipative medium tally accessible. The frequency variable ν in Fig. 1 is would not prove to be a feasible method to measure the experimentally varied by applying different DC voltages zeropoint spectrum at high frequencies. Rather, we need across the junction, thus making direct use of formula a more sophisticated method, and the AC Josephson ef- (27). Josephson oscillations are clearly necessary to do fect [22, 23, 24] satisfies this requirement. these types of measurements. 7

The AC Josephson effect has been experimentally ob- Virtual photons as a useful tool for the theoretician are served up to Josephson frequencies in the low-THz re- still allowed to persist at higher frequencies. These can gion. The energy gap in cuprates limits the maximum then still be used as a theoretical tool to do calculations value of the Josephson frequency to ν ∼ 15 THz, but in for the Casimir effect, Lamb shift, spontaneous emissions practice one seems to be able to only reach the 2 THz of atoms etc. in just the same way as before, keeping in region [36]. If the central hypothesis is true, then the mind that in many cases, e.g. for the Casimir effect, they largest attainable Josephson frequency also constrains are not needed at all to explain the effect [1]. the dark energy density of the universe. The central hypothesis merely implies that for photonic vacuum fluctuations the measurability in the form of a physically relevant spectrum ceases to exist for VI. CONCLUSIONS AND OUTLOOK ν > νc = 1.7 THz, for the reasons we have given above. This is connected with a kind of phase transition for the In this paper we have proposed a possible solution to gravitational activity of the virtual photons at ν = νc. the cosmological constant problem, based on the central It is indeed plausible that vacuum fluctuations are only hypothesis that vacuum fluctuations are only gravitation- gravitationally active if they are measurable in the form ally active if they are measurable in the form of a noise of a frequency spectrum in a macroscopic or mesoscopic spectrum in suitable macroscopic or mesoscopic detec- detector, as stated by our central hypothesis. How else tors. From this assumption a universal low-energy cutoff should or could these vacuum fluctuations push galaxies frequency νc consistent with the currently observed dark apart and accelerate the expansion of the universe if the energy density in the universe is predicted in a rather effect is not measurable with a macroscopic detector? As natural way. One obtains νc ∼ kTc/h, where Tc is the shown above the central hypothesis leads to the correct largest possible critical temperature of high-Tc supercon- dark energy density in the universe, and the cosmological ductors. This means νc is in the THz region. Moreover, constant problem is avoided. The validity or non-validity the dark energy density of the universe is most naturally of the central hypothesis will soon be tested [28]. identified as ordinary electromagnetic vacuum energy of virtual photons with frequency ν < νc, and given by a kind of analogue of the Stefan-Boltzmann law for dark Acknowledgments 4 3 3 energy, ρdark ∼ (kTc) /(¯h c ). Suppose the universal cutoff νc =1.7 THz correspond- This work was supported by the Engineering and Phys- ing to the dark energy density is observed in the exper- ical Sciences Research Council (EPSRC, UK), the Natu- iment planned by Warburton et al. [28]. Would this in- ral Sciences and Engineering Research Council (NSERC, validate successful QED predictions for other effects such Canada) and the Mathematics of Information Technol- as the Casimir effect [34, 35] or Lamb shift? ogy and Complex Systems (MITACS, Canada). This re- We think the answer is ‘no’. Once again we emphasize search was carried out while MCM was visiting the Math- that the cutoff is predicted for the measurable spectrum. ematics Institute of the University of Oxford.

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