New Journal of Physics

Quick search Find article

Quick search

All journals This journal only

Find article
Volume number: Issue number (if known): Article or page number:

New J. Phys. 4 (2002) 92
PII: S1367-2630(02)52298-9

Satellite Eötvös test of the weak equivalence principle for zero-point vacuum energy

J W Moffat^1,2 and G T Gillies³

¹ Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
² Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9, Canada
³ School of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22904-4746, USA
Email: john.moffat@utoronto.ca and gtg@virginia.edu

New Journal of Physics 4 (2002) 92.1-92.6
Received 13 August 2002
Published 14 November 2002

Abstract. An Eötvös experiment to test the weak equivalence principle (WEP) for zero-point vacuum energy is proposed using a satellite. Following the suggestion of Ross for a terrestrial experiment of this type, the acceleration of a spherical test mass of aluminium would be compared with that of a similar test mass made from another material. The estimated ratio of the zero-point vacuum energy density inside the aluminium sphere to the rest mass energy density is $\sim 1.6\times 10^{-14}$ , which would allow a 1% resolution of a potential WEP violation observed in a satellite mission test that had a baseline sensitivity to WEP violations of $\sim 10^{-16}$ . An observed violation of the WEP for vacuum energy density would constitute a significant clue as to the origin of the cosmological constant and the source of dark energy, and test a recently proposed resolution of the cosmological constant problem, based on a model of nonlocal quantum gravity and quantum field theory.

A recent proposal to resolve the cosmological constant problem was based on a nonlocal quantum gravity theory and quantum field theory [1,2]. The resolution demands that there exist a fundamental low-energy gravitational energy scale, $\Lambda_{\mathrm{Gvac}}\sim 10^{-3}$ eV, above which the coupling of gravitons to vacuum energy density was suppressed by a nonlocal gravitational form factor in the calculation of standard model vacuum polarization loop graphs. This would require a strong violation of the weak equivalence principle (WEP) for coupling of gravitons to pure vacuum energy loop graphs compared to ordinary matter.

It is generally believed that a solution to the cosmological constant problem could alter in a significant way our understanding of gravitation and particle physics [3]. There is mounting observational evidence [4] that the universe is accelerating and that a form of dark energy exists. A small cosmological constant corresponding to a vacuum energy density $\rho_{\mathrm{vac}}\sim (2.4\times 10^{-3}~{\rm eV})^4$ could explain the acceleration of the universe.

We can define an effective cosmological constant

$\begin{equation} \lambda_{\mathrm{eff}}=\lambda_0+\lambda_{\mathrm{vac}}, \end{equation}$

(1)

where $\lambda_0$ is the `bare' cosmological constant in Einstein's classical field equations, and $\lambda_{\mathrm{vac}}$ is the contribution that arises from the vacuum density $\lambda_{\mathrm{vac}}=8\pi G\rho_{ \rm vac}$ . Already at the standard model electroweak scale $\sim 10^2$ GeV, a calculation of the vacuum density $\rho_{\mathrm{vac}}$ , based on local quantum field theory, results in a discrepancy of order 10⁵⁵ with the observational bound

$\begin{equation} \rho_{\mathrm{vac}} \le 10^{-47}~({\rm GeV})^4. \end{equation}$

(2)

This results in a severe fine-tuning problem of order 10⁵⁵, since the virtual quantum fluctuations giving rise to $\lambda_{\mathrm{vac}}$ must cancel $\lambda_0$ to an unbelievable degree of accuracy. This is the `particle physics' source of the cosmological constant problem.

A model of nonlocal quantum gravity with the action S = S_G + S_M was presented in [1] with the gravitational action^Note1 ( $\kappa^2=32\pi G$ )

$\begin{equation} S_G=-\frac{2}{\kappa^2}\int \mathrm{d}^4x~\sqrt{-g}\{R[g,{\cal G} ^{-1}]+2\lambda_0\}. \end{equation}$

(3)

The matter action S_M for the simple case of a scalar field $\phi$ is given by

$\begin{equation} S_M=\frac{1}{2}\int \mathrm{d}^4x~\sqrt{-g}{\cal G}^{-1}(g^{\mu... ...la_\mu\phi{\cal F}^{-1}\nabla_\nu\phi -m^2\phi{\cal F} ^{-1}\phi). \end{equation}$

(4)

Here, ${\cal G}$ and ${\cal F}$ are nonlocal regularizing, entire functions and $\nabla_\mu$ is the covariant derivative with respect to the metric $g_{\mu\nu}$ . We choose the covariant functions

$\begin{equation} {\cal G}(x)=\exp[-{\cal D}(x)/\Lambda_G^2], \end{equation}$	(5)
$\begin{equation} {\cal F}(x)=\exp[-({\cal D}(x)+m^2)/\Lambda_M^2)], \end{equation}$	(6)

where ${\cal D}\equiv\nabla_\mu\nabla^\mu$ , and $\Lambda_G$ and $\Lambda_M$ are gravitational and matter energy scales, respectively.

A calculation of the first-order vacuum polarization tensor $\Pi^{\mathrm{Gvac}}_{\mu\nu\rho\sigma}$ for the coupling of gravitons to a loop consisting of photons yields for the vacuum energy density

$\begin{equation} \rho_{\mathrm{vac}}\sim M^2_{\mathrm{PL}}\Pi^{\mathrm{Gvac}}(p)\sim\Lambda_{ \rm Gvac}^4. \end{equation}$

(7)

If we choose $\Lambda_{\mathrm{Gvac}}\le 10^{-3}$ eV, then the quantum correction to the bare cosmological constant $\lambda_0$ is suppressed sufficiently to satisfy the observational bound on $\lambda$ , and it is protected from large unstable radiative corrections.

This provides a solution to the cosmological constant problem at the energy level of the standard model and possible higher energy extensions of the standard model. The universal fixed gravitational scale $\Lambda_{\mathrm{Gvac}}$ corresponds to the fundamental length $\ell_{\mathrm{Gvac}}\le 1$ mm at which virtual gravitational radiative corrections to pure vacuum energy are cut off. However, it is postulated in the model that gravitons coupled to ordinary matter have a form factor that is controlled by the energy scale, $\Lambda_{GM}=\Lambda_M\sim 1-10$ TeV, which avoids any measurable violation of the WEP for gravitons coupled to ordinary matter and guarantees that the calculations of standard model diagrams agree with experiment. This would lead to a strong violation of the WEP for coupling of low energy gravitons to pure zero-point vacuum energy compared to the coupling to ordinary matter.

To explore the possibility that the cosmological constant arises from zero-point energy, Ross [5] has suggested a novel scenario in which the violation of the WEP that would result from such an effect might be observed in an Eötvös experiment. In his proposal, the acceleration of a spherical test mass of aluminium would be compared with that of a similar test mass made from another material (e.g. a metal like copper or silver). He chose aluminium because it has a relatively sharp transition from reflectance to absorption of electromagnetic waves at photon energies of approximately 15.5 eV. His analysis indicated that the magnitude of the missing zero-point energy density inside aluminium is given by

$\begin{equation} {\cal E}=\frac{4\pi }{(\hbar c)^{3}}\int_{0}^{E_{\max}}E^{3}~\mathrm{d} E, \end{equation}$

(8)

where E_max is the energy at which aluminium becomes transparent. For E_max = 15.5 eV, one obtains ${\cal E} =2.37\times 10^{19}~{\rm eV~cm}^{-3}$ . The rest mass energy density is 1.52 × 10³³ eV, so he found that the ratio of the zero-point energy density inside the aluminium sphere to the rest mass energy density is 1.6 × 10^-14. Therefore, if comparisons were made between test masses of aluminium and, e.g., copper, a violation of WEP at approximately this level should be observed if the zero-point energy does not couple to the gravitational field. Such an experiment would be a direct test of the role that a purely quantum mechanical effect plays in general relativity [6].

In a long series of elegant experiments with rotating torsion balances, the Eöt-Wash Group has searched for composition dependence in the gravitational force via tests of the universality of free fall. In terms of the standard Eötvös parameter $\eta$ , they have reached sensitivities of $\eta \sim 1.1\times 10^{-12}$ in comparisons of the accelerations of Be and Al/Cu test masses [7] and, more recently, have resolved differential accelerations of approximately 1.0 × 10^-14 cm s^-2 in experiments with other masses [8]. Drop-tower experiments now under way in Germany [9] have as their goal testing WEP at sensitivities of $\eta \sim 1\times 10^{-13}$ , and Unnikrishnan [10] describes a methodology under study at the Tata Institute of Fundamental Research in India wherein torsion balance experiments aiming at sensitivities of $\eta \sim 1\times 10^{-14}$ are being developed.

While it is not yet clear what ultimate sensitivities might be reached by terrestrial experiments of these types, it is generally accepted that significant gains in sensitivity will be made by space-based WEP experiments, and several of them are presently in various stages of planning and development. Among these are the STEP satellite [11], the MICROSCOPE experiment [12], the Galileo Galelei (GG) mission [13] and Project SEE [14]. The target sensitivities of each of them are the following: STEP, $\eta \sim 10^{-18}$ ; MICROSCOPE, $\eta \sim 10^{-15}$ ; GG, $\eta \sim 10^{-17}$ ; SEE, $\eta \sim 10^{-16}$ . The first three missions are in advanced stages of planning and hardware testing, and are expected to be launched over the next few years. Project SEE is still undergoing rigorous conceptual evaluation and is not yet a scheduled mission. However, one of the significant points of interest about it is the relative simplicity of the test mass configuration: one large spherical mass and one small one undergo a three-body interaction with the Earth during the orbit of the capsule containing them [15]. Analysis of the small mass motion then yields measures of WEP violation, the absolute value of the Newtonian gravitational constant and its time variation. Because it is still an early stage endeavour, it might be possible to incorporate into the mission schedule a test of WEP at the SEE target sensitivity using aluminium and copper test masses, in order to examine the zero-point energy coupling. This would avoid the need to design, fund and carry out a separate independent mission, which would be very difficult to justify because of the large costs. Further details of the SEE mission are given in the review by Sanders and Gillies [16].

The website http://www.phys.utk.edu/see/ provides a description of the fundamental design features of the proposed SEE mission. It is meant to be a multi-functional platform for gravitational physics experiments, providing the capability to measure the strength of the gravitational interaction between a pair of test bodies co-orbiting the Earth in a drag-free environment. The nearly identical circular orbits of the test bodies in the field of the Earth result in gravitationally governed relative motions between them of the type predicted first by George Darwin in 1897 and exhibited in the celestial mechanics of two of the moons of Saturn [16]. When one of the masses is much smaller than the other, an analysis of the mechanics of the interaction reveals how measurements of the relative motions can lead to a determination of the absolute value of the Newtonian gravitational constant and its time-rate of change, as well as limits on violations of the WEP. As with any high precision measurement system of this type, great care must be taken to circumvent the problems arising from competing effects, noise and other sources of experimental uncertainty. The satellite itself provides the first line of defence, with its capsule consisting of a series of concentric cylindrical shells that serve as radiation baffles and which provide the drag-free containment for the test masses. The orbital configuration will be a sun-synchronous trajectory that keeps the capsule illuminated at all times and thus much closer to a state of constant thermal equilibrium. The large and small test masses would be nominally 500 kg and 100 g respectively, and small masses of different materials could be orbited simultaneously to make differential measurements of the type needed to search for a WEP violation. The details of the optimal orbital parameters, composition of the error budget and thermo-mechanics of the capsule are still under development.

If the SEE mission were able to accommodate a test of the type proposed here, the level at which the WEP violation is predicted to occur, $\sim 1.6\times 10^{-14}$ , means that a nearly 1% resolution of it would be possible within the context of the SEE interaction. In principle, this degree of resolution would not only provide a definitive statement of the presence of such an effect if it appeared at that level, but it would also allow for enough sensitivity to observe weaker manifestations of variants of the nominal prediction. The key issue in any case becomes one of maximizing the signal size and, as always, minimizing the consequences of competing effects. To accomplish the former task, a careful re-analysis of potential test mass materials should be carried out to ensure that those with maximum estimated difference in zero-point energy density are selected. This is an important point, particularly in light of possible complexities that arise in calculating the exact values of the Casimir effect cut-offs for spherical mass configurations [17].

Another concern would be one of interpreting the meaning of an unequivocal positive result. The task would be one of establishing, for instance, whether or not the signal might alternatively be due to the revelation of the presence of a long sought new weak force [18]. Here again, selecting the proper experimental strategy within the context of theoretical guidance would play the key role. Moreover, while the prediction of the WEP violation in [5] is based on a straightforward analysis of the vacuum energy density deficit in aluminium relative to that present in other materials, a careful reconfirmation of that interpretation will be important to establish that the level of the violation is indeed what is claimed. A significant reduction of the strength of the expected violation brought on by any reanalysis of the coupling of gravity to the difference in vacuum energy densities in the test masses might place the measurement below the reach of the SEE Mission's sensitivity. Lastly, if a satellite experiment were successful in revealing the predicted effect, physics would be faced with one additional concern: the need to repeat and independently confirm a finding of this significance. Perhaps completely new technologies for terrestrial experiments could eventually be called on to meet any such future need.

As a point of historical interest, we note that one of the earliest cases where vacuum polarization effects were discussed within the context of a gravitational physics experiment was in Long's interpretation of his non-null result for a breakdown in the inverse square law of Newtonian gravity [19]. He claimed to find a scale dependence in the Newtonian gravitational constant, G, and argued that a certain type of vacuum polarization process could explain his positive results. He further argued that the results consistent with no breakdown that were obtained in the null experiments of others [20] did not invalidate his findings of a variation in G with inter-mass spacing, because of the lack of a polarizing gravitational field in such experiments. Chen [21] extended the argument and contended that vacuum polarization could even resolve the differences between the null and the non-null results via a very weak shielding mechanism that could arise from it, but the very stringent limits on gravitational shielding phenomena [22,23] derived from recent experiments block that possibility. The present consensus is that the Newtonian inverse square law has been proven to be valid over the ranges through which it has been tested experimentally. The new laboratory searches for possible violations of it now focus on the sub-millimetric regime, and are driven largely by predictions for short range variations in gravity that arise in extra-dimensional theories [24,25].

We call for a satellite test of the WEP that is designed to determine if the cosmological constant arises from zero-point energy, and whether the WEP is violated for coupling of gravitons to pure vacuum energy compared to ordinary matter. An observed violation of the WEP would serve as experimental support for the conjectured resolution of the cosmological constant problem, based on a model of nonlocal quantum gravity and quantum field theory [1,2].

Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Council of Canada. GTG was supported in part by a NASA subcontract from the University of Tennessee to the University of Virginia, and he thanks D K Ross, A J Sanders and C S Unnikrishnan for interesting discussions.

References

[1]: Moffat J W 2002 The cosmological constant problem and nonlocal quantum gravity MRST Mtg (Perimeter Institute for Theoretical Physics, Waterloo, Ontario, 2002)

Moffat J W 2002 Proc. 18th IAP Coll. on Observational and Theoretical Results on the Accelerating Universe (Paris, 2002) at press
[2]: Moffat J W 2001 Preprint hep-ph/0102088 v2
Preprint
[3]: Weinberg S 1989 The cosmological constant problem Rev. Mod. Phys. 61 1
CrossRef MathSciNet
Straumann N 2002 Preprint astro-ph/0203330
Preprint
Peebles P J E and Ratra B 2002 Preprint astro-ph/0207347
Preprint
Ellwanger U 2002 Preprint hep-ph/0203252
Preprint
[4]: Perlmutter S et al 1997 Astrophys. J. 483 565 (Preprint astro-ph/9608192)
IOPscience Preprint
Riess A G et al 1998 Astron. J. 116 1009 (Preprint astro-ph/9805201)
IOPscience Preprint
Garnavich P M et al 1998 Astrophys. J. 509 74 ( Preprint astro-ph/9806396)
IOPscience Preprint
Perlmutter S et al 1999 Astrophys. J. 517 565 ( Preprint astro-ph/9812133)
IOPscience Preprint
Riess A G et al 2002 Astrophys. J. at press

(Riess A G et al 2001 (Preprint astro-ph/0104455))
Preprint
[5]: Ross D K 1999 Nuovo Cimento B 114 1073
[6]: Gillies G T and Unnikrishnan C S 2002 Experimental perspectives on the interplay between quantum and gravity physics Advances in the Interplay between Quantum and Gravity Physics ed P G Bergmann and V de Sabbata (Dordrecht: Kluwer) pp 103-21

Gillies G T and Unnikrishnan C S 2002 Quantum physics-motivated measurement and interpretation of the Newtonian gravitational constant Advances in the Interplay between Quantum and Gravity Physics ed P G Bergmann and V de Sabbata (Dordrecht: Kluwer) pp 123-31
[7]: Su Y et al 1994 Phys. Rev. D 50 3614
CrossRef
[8]: Baeßler S et al 1999 Phys. Rev. Lett. 83 3585
CrossRef
[9]: Vodel W et al 2001 IEEE Trans. Appl. Supercond. 11 1379
CrossRef
[10]: Unnikrishnan C S 1994 Class. Quantum Grav. 11 A195
IOPscience
[11]: Mester J et al 2001 Class. Quantum Grav. 18 2475
IOPscience
[12]: Touboul P et al 2002 Acta Astronaut. 50 443
CrossRef
[13]: Nobili A M et al 2000 Class. Quantum Grav. 17 2347
IOPscience
[14]: Sanders A J et al 2000 Class. Quantum Grav. 17 2331
IOPscience
[15]: Sanders A J and Deeds W E 1992 Phys. Rev. D 46 489
CrossRef
[16]: Sanders A J and Gillies G T 1996 Riv. Nuovo Cimento 19 1
CrossRef
[17]: Hagen C R 2000 Phys. Rev. D 61 065005
CrossRef
Hagen C R 2001 Eur. Phys. J. C 19 677
CrossRef
[18]: Fischbach E et al 1986 Phys. Rev. Lett. 56 3
CrossRef PubMed
[19]: Long D R 1976 Nature 260 417
CrossRef
Long D R 1980 Nuovo Cimento B 55 252
CrossRef
Long D R 1981 Nuovo Cimento B 62 130
CrossRef
[20]: Spero R et al 1980 Phys. Rev. Lett. 44 1645
CrossRef
[21]: Chen S G 1989 Nuovo Cimento B 104 611
CrossRef
[22]: Unnikrishnan C S and Gillies G T 2000 Phys. Rev. D 61 101101
CrossRef
[23]: Unnikrishnan C S, Mohapatra A K and Gillies G T 2001 Phys. Rev. D 63 062002
CrossRef
[24]: Arkani-Hamad N, Dimopoulus S and Dvali G 2000 Phys. Rev. Lett. 84 586
CrossRef PubMed
[25]: Ritter R C and Gillies G T 2002 Laboratory searches for effects beyond classical gravity Advances in the Interplay between Quantum and Gravity Physics ed P G Bergmann and V de Sabbata (Dordrecht: Kluwer) pp 339-52

Notes

Note1: A paper is in preparation in which more complete details of the model will be provided.

Your last 10 viewed

Satellite Eötvös test of the weak equivalence principle for zero-point vacuum energy
J W Moffat and G T Gillies 2002 New J. Phys. 4 92
A brain–computer interface using electrocorticographic signals in humans
Eric C Leuthardt et al 2004 J. Neural Eng. 1 63
The Woman Who Knew Too Much: Alice Stewart and the Secrets of Radiation
Richard Wakeford 2000 J. Radiol. Prot. 20 475
Brownian motion on a sphere: distribution of solid angles
M M G Krishna et al 2000 J. Phys. A: Math. Gen. 33 5965
A New Measurement of the Bulk Flow of X-Ray Luminous Clusters of Galaxies
A. Kashlinsky et al. 2010 ApJ 712 L81
Application of the density matrix renormalization group method to finite temperatures and two-dimensional systems
Naokazu Shibata 2003 J. Phys. A: Math. Gen. 36 R381
The gravitational-wave signature of core-collapse supernovae
Christian D Ott 2009 Class. Quantum Grav. 26 063001
Chromospheric Activity in the Late A- and Early F-Type Stars of Open Clusters. I. Hyades, Praesepe, and Coma
Brian L. Rachford 1998 ApJ 505 255
Dewetting films: bifurcations and concentrations
A L Bertozzi et al 2001 Nonlinearity 14 1569
Bounds on general entropy measures
Dominic W Berry and Barry C Sanders 2003 J. Phys. A: Math. Gen. 36 12255

IOPscience

New Journal of Physics

New Journal of Physics