Laboratory search for a quintessence field
LLaboratory search for a quintessence field
Michael V. Romalis
Department of Physics, Princeton University, Princeton, New Jersey, 08544, USA
Robert R. Caldwell
Department of Physics and Astronomy, Dartmouth College Hanover, NH 03755 USA (Dated: November 12, 2018)A cosmic scalar field evolving very slowly in time can account for the observed dark energy of theUniverse. Unlike a cosmological constant, an evolving scalar field also has local spatial gradientsdue to gravity. If the scalar field has a minimal derivative coupling to electromagnetism, it willcause modifications of Maxwell’s equations. In particular, in the presence of a scalar field gradientgenerated by Earth’s gravity, regions with a magnetic field appear to be electrically charged andregions with a static electric field appear to contain electric currents. We propose experimentsto detect such effects with sensitivity exceeding current limits on scalar field interactions frommeasurements of cosmological birefringence. The scalar field with derivative couplings to fermionsor photons is also observable in precision spin precession experiments.
Many different astrophysical observations point to theexistence of dark energy constituting the majority of theenergy density of the Universe [1]. The origin of thisenergy is presently unknown, but the two leading theo-retical possibilities are a cosmological constant or a veryslowly evolving dynamical field [2]. These two possibili-ties can be distinguished by astrophysical measurementsof the equation of state w = p/ρ – the ratio of the pres-sure p to the density ρ of the dark energy. Present mea-surements are consistent with w = −
1, as expected for acosmological constant, but also allow w + 1 ∼ . φ introduces a spe-cial frame in which φ is homogeneous. Therefore, anyinteraction of ordinary matter with the scalar field canlead to an apparent violation of local Lorentz invariance[4–8]. If the scalar field is slowly evolving in time, itdefines a preferred cosmic rest frame. Our motion rela-tive to that frame with velocity v ∼ − c will lead tosmall violations of rotational invariance. However the ex-pected energy shift due to such an effect is on the order v (cid:126) H ∼ − GeV, too small to hope to detect in thelaboratory.However, a scalar field is also affected by gravity [9]and will develop a local gradient due to Earth’s grav-ity with a much larger characteristic energy scale of (cid:126) g/c ∼ − GeV. Such a gradient in the local valueof the scalar field is potentially observable if the fieldhas any interactions with ordinary matter. In this Let-ter we explore possible experimental signatures of such agravity-induced local scalar field gradient.The scalar field equation of motion in the vicinity ofa local gravitational potential, U can be obtained as fol-lows. On length and time scales that are short compared to the cosmological expansion, we use a weak field metric: ds = − (1 + 2 U/c ) c dt + (1 − U/c ) d(cid:126)x . (1)We consider a static perturbation δφ to the homogeneoussolution φ in the cosmological background: φ = φ + δφ ( r ). Working to linear order, and temporarily setting (cid:126) = c = 1, the scalar field equation of motion (cid:3) φ = V (cid:48) can be expanded as ∂ µ ( √− g g µν ∂ ν ( φ + δφ )) / √− g = V (cid:48) ( φ + δφ ) whereupon the background and perturbationequations are (cid:3) φ = V (cid:48) (cid:3) δφ − U (cid:3) φ = V (cid:48)(cid:48) δφ, (2)where the subscript 0 means that it is evaluated in theunperturbed spacetime. Using the background solutionto simplify the equation for δφ , for the case of sphericalsymmetry we obtain d dr δφ + 2 r ddr δφ = V (cid:48)(cid:48) δφ + 2 U V (cid:48) . (3)Using Earth’s gravitational potential U = − GM E /r , theinhomogeneous solution is δφ = − V (cid:48) /V (cid:48)(cid:48) ) U/c . Sincethe gradient in the gravitational potential gives the localacceleration, g = − ∇ U , we may write the gradient inthe scalar field as ∇ δφ = 2 V (cid:48) V (cid:48)(cid:48) g /c . (4)One of the theoretical challenges to the idea that darkenergy is due to a cosmic scalar field is to explain whythe field remains so dark . That is, why doesn’t the cos-mic scalar mediate a long-range force between StandardModel particles? One possibility is a global shift sym-metry φ → φ + constant, as occurs in pseudo Nambu-Goldstone boson models of dark energy [10], which keepsthe cosmic scalar dark [11]. The global symmetry allows a r X i v : . [ a s t r o - ph . C O ] F e b only derivative couplings of the scalar field with ordinarymatter. For example, consider the interaction with elec-tromagnetism L = 12 M ∇ µ φA ν (cid:101) F µν = − φ M F µν (cid:101) F µν . (5)In the above, the middle and right terms differ by atotal derivative. We define (cid:101) F µν ≡ (cid:15) µνρσ F ρσ , and M is a mass. This coupling introduces a modification ofMaxwell’s equations, expressed in SI units where φ hasunits of energy, given by ∇ · E − ρ/(cid:15) = − M c ∇ φ · B , (6) ∇ × B − µ (cid:15) ∂ E ∂t − µ J = 1 M c (cid:16) ˙ φ B + ∇ φ × E (cid:17) (7)whereas the homogeneous equations are unchanged [12].The quintessence field equation is likewise modified, (cid:3) φ − V (cid:48) ( φ ) = 14 M F µν (cid:101) F µν , (8)(again setting (cid:126) = c = 1) where V ( φ ) denotes the scalarfield potential. For the pseudo Nambu-Goldstone bosondark energy model, the field self-interacts through a po-tential V ( φ ) = µ (1 + cos( φ/f )) . (9)In the vicinity of the Earth we can now rewriteMaxwell’s equations as ∇ · E − ρ/(cid:15) = − (cid:15) γ c g · B , (10) ∇ × B − µ (cid:15) ∂ E ∂t − µ J = (cid:15) γ c g × E , (11)where we introduced a dimensionless parameter (cid:15) γ ≡ V (cid:48) V (cid:48)(cid:48) M c (12)and ignored the ˙ φ -term.The spatial gradient of the cosmic scalar then intro-duces novel effects. Specifically, sources of a magneticfield give rise to an anomalous electric field, while sourcesof an electric field will give rise to an anomalous magneticfield.There is not a unique prediction for | (cid:15) γ | . It could beof order unity, or tiny with no lower bound. The pseudo-scalar coupling (5) causes a parity-violating rotation ofthe polarization of light traveling over cosmological dis-tances in a phenomenon referred to as cosmic birefrin-gence [11]. The cosmic microwave background Stokesvector rotates [13] by an angle α = ∆ φ M c , (13)where ∆ φ is the change in the cosmic scalar betweenrecombination and the present day. Current bounds, | ε γ | Figure 1: The maximum values of | (cid:15) γ | versus the present-dayvalue of the equation of state w for two illustrative families ofpseudo Nambu-Goldstone boson models. For the short (long)dashed lines, µ = 0 . . / c . For each point alongthe curve, f and the initial value of φ are adjusted so as toyield Ω DE = 0 .
72 and h = 0 .
71, consistent with WMAP9 [3].At the square and circle the maximum is | (cid:15) γ | = 5. Whilelarger values of | (cid:15) γ | may be achieved with other values of themodel parameters, there is no lower limit on | (cid:15) γ | . based on the statistical properties of the polarization pat-tern detected in the cosmic microwave background, give − . ◦ < α < . ◦ (95% C.L.), based on a combinedanalysis [14] of the WMAP [15], BICEP [16, 17], andQUaD [18, 19] experiments. (Also see Refs. [20, 21] forthe analysis of particular scalar field models.) The firstCMB constraints on a direction-dependent polarizationrotation through cosmological birefringence has recentlybeen obtained [22], at a similar amplitude. For a partic-ular model of quintessence, we can use these observationsto place a lower bound on the mass scale M .Typical parameters for the pseudo Nambu-Goldstoneboson model potential (9) that satisfy current obser-vational constraints on dark energy have values µ (cid:39) .
002 eV / c and f ∼ M p . The initial value of the field isnot predicted, however, so that in turn the present-dayvalue of φ is undetermined. Moreover, the field evo-lution may have been damped by Hubble friction untilquite recently, with a present-day equation of state closeto −
1. This means that V (cid:48)(cid:48) or ∆ φ may be small, inwhich case | (cid:15) γ | might conceivably be quite large. Twofamilies of quintessence models for which the maximumvalue of | (cid:15) γ | ranges over three orders of magnitude areshown in Fig. 1. Models with even larger maxima can beconstructed. For the two models indicated by the squareand circle, the maximum value | (cid:15) γ | is 5. For the modelindicated by the square in the figure, µ = 0 .
002 eV / c , f = 0 . M p with an initial value φ i = 0 . M p . Weassume that the field is frozen by the Hubble friction,and only begins to slowly roll at late times. From thetime of last scattering to the present, ∆ φ = 0 . M p ,with a current value φ = 0 . M p and w = − . µ = 0 . / c , f = 0 . M p and φ i = 0 . M p . From the time of last B V ω g B ρ D Figure 2: Schematic of an experimental setup using a rotatingHalbach magnet to detect modifications of electromagnetismdue to a coupling with a gradient of a scalar field. The voltage V on the cylindrical capacitor will oscillate with frequency ω . scattering to the present, ∆ φ = 0 . M p , with a cur-rent value φ = 0 . M p and w = − . w = − .
946 (circle) may be distinguished from a cos-mological constant by future astrophysical observations,it seems unlikely that improvements in observations willpermit the model with w = − . B = 1 T and (cid:15) γ = 5, the effective charge density is ρ = 1 . × − C/m = 9 e m − and can be detectedusing fairly conventional techniques. For example, con-sider an apparatus shown in Fig. 2. The magnetic field iscreated by a permanent Halbach magnet that is rotatedaround a horizontal axis to modulate the sign of the ef-fective electric charge inside the cylinder. The charge isdetected by measuring the voltage across a cylindrical ca-pacitor inserted into the Halbach magnet. The expectedunloaded voltage amplitude can be expressed as V = (0 . (cid:18) B T (cid:19) (cid:18) D (cid:19) (cid:16) (cid:15) γ (cid:17) . (14)For optimum detection efficiency the input capacitanceof the amplifier should be equal to the capacitance in-side the Halbach magnet, which is about 10 pF for a30 cm long capacitor. A low-noise JFET transistor withan input capacitance C in = 2 . δV = 4 nV/Hz / at10 Hz [24]. One can use four such transistors in parallelto match the impedance of the source and reduce noise.With such an amplifier the signal due to a scalar fieldwith (cid:15) γ = 5 can be observed at the 4 σ level after 1 hourof integration.There is considerable room for improvement usingmore advanced techniques. Using a superconductingmagnet both the magnetic field and the diameter of thedetection region can be increased. The sensitivity couldalso be potentially improved using single electron transis-tors (SET), which can reach an energy resolution on the E B + Vg − V − V Figure 3: Schematic of an experimental setup using an electricfield to generate a magnetic field due to a gradient of the scalarfield. A high voltage V of alternating polarity is applied toelectrodes surrounding the magnetic field sensing region. Thesign of the generated magnetic field is reversed by changingthe voltage polarity. order of (cid:126) [25], a large improvement over the JFET en-ergy resolution of C in δV / . × − J s. However,SETs suffer from 1 /f noise, so the best energy resolu-tion demonstrated so far at 10 Hz is 8 × − J s [26].In addition, SETs usually have very small input capaci-tance, so they would need to be fabricated either with alarger input gate [27] or with many of them connected inparallel [28].Systematic errors in such an experiment would be pri-marily due to the Faraday effect. Even if the capacitoris fabricated to be very axially symmetric and is rotatedtogether with the magnet, it will inevitably undergo de-formations due to gravity, which will lead to a Faradayinduction signal. However, Faraday signals can be distin-guished because they are proportional to the rotationalvelocity ω and reverse sign with the direction of rotation.We can also consider an experimental approach to de-tection of the signal through Eq. (11) as shown in Fig. 3.The central grounded cylindrical container is surroundedby electrodes with alternating high voltage V , creatinga radial electric field E . This field in the presence ofthe scalar field gradient generates a circular pattern ofeffective current density similar to a solenoid. The ef-fective current generates an approximately uniform mag-netic field inside the central cylinder, which can be sensedby a magnetometer. Several regions with alternating fieldpolarity allow one to cancel common magnetic field noiseby using a magnetic gradiometer. The magnetic field inan ideal geometry is given by B = (1 . × − T) (cid:18) V (cid:19) (cid:16) (cid:15) γ (cid:17) . (15)Currently the most sensitive magnetometers usingoptically-pumped alkali-metal atoms have a sensitivityof about δB = 10 − T / Hz / for a 1 cm measurementvolume [29]. In a long term measurement such a magne-tometer has achieved a sensitivity of 5 × − T [30]. Thesensitivity of atomic magnetometers improves as √ V , sowith 100 cm active volume one can achieve a sensitivity δB = 10 − T / Hz / , which would allow detection of thescalar field signal at the 1 σ level after 1 hour of integra-tion. SQUID magnetometers with a large pick-up coilcan also potentially achieve similar levels of sensitivity[31].The experiment can be run similar to an electric dipolemoment (EDM) measurement, in which the electric fieldpolarity is periodically reversed to modulate the mag-netic field. In fact, the recent search for Hg EDM[32] has some sensitivity to (cid:15) γ coming from Hg mag-netometer cells maintained at a high potential with nointernal electric fields. While the experiment was notoptimized to look for this effect, analysis of existing datacould reach a sensitivity on the order of (cid:15) γ ∼ × .A dedicated search can reach a much higher sensitivitysince the electric field does not need to be applied in-side the magnetometer cells. Systematic effects in sucha measurement would be similar to an EDM search, pri-marily due to magnetic fields generated by charging andleakage currents.One can consider other ways of detecting such pseu-doscalar interactions. The interaction (5) leads to anapparent violation of the equivalence principle [33], butonly for spin-polarized bodies since it is proportional to apseudoscalar E · B . For example, an electric field arounda nucleus with charge Ze would generate a magnetic fieldat the origin given by B = Ze(cid:15) γ πc ε g R , (16)where R is an integration cut-off which we take to beroughly equal to the nuclear charge radius [6]. Thismagnetic field interacts with the nuclear magnetic mo-ment, causing an effective spin-gravity S · g frequencyshift. Two experiments [34, 35] have constrained suchinteraction for Hg atoms at a level of ∆ ν < µ Hz,which can be used to place a limit (cid:15) γ (cid:46) × . One canalso obtain interesting limits for electrons from a spinpendulum experiment [36], which also has µ Hz sensitiv-ity.Astrophysical sources of gravitational potential canalso generate an observable signal. The polarizationof light escaping from a gravitational potential whichchanges by ∆ U will be rotated by an angle α = ∆ U c ε γ . (17)For example, for Crab nebulae the polarization of gamma rays is measured to be parallel to the rotation axis of thepulsar within 11 ◦ [37]. There is considerable uncertaintyin the location where the gamma rays are generated [38],but assuming a distance near the light-cylinder radius R L ∼ m in the potential of a neutron star with M =1 . M (cid:12) , one can place a limit ε γ (cid:46) L = 12 M f ∂ µ φ ¯ ψγ µ γ ψ . (18)It will lead in the non-relativistic limit to a spin couplingwith the gradient of the scalar field H = (cid:126) σ · ∇ φ/ (2 M f c ).Using the gravitationally induced spin gradient from Eq.(4), we obtain a frequency shift between spin-up and spin-down states equal to∆ ν = g πc (cid:15) f = (10nHz) (cid:15) f , (19)where we defined (cid:15) f ≡ V (cid:48) / ( V (cid:48)(cid:48) M f c ) . This frequencyshift is similar to the result obtained in [6] but without re-quiring an additional scalar coupling of the quintessencefield to matter, since the scalar gradient is already gen-erated by ordinary gravity. Atomic physics experimentscan easily reach this frequency resolution [30, 32], butso far the most sensitive measurements [34–36] have µ Hzsensitivity, constraining (cid:15) f < Acknowledgments
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