Detecting Chameleon Dark Energy via Electrostatic Analogy
aa r X i v : . [ a s t r o - ph . C O ] M a y Detecting Chameleon Dark Energy via Electrostatic Analogy
Katherine Jones-Smith and Francesc Ferrer
Physics Department and McDonnell Center for the Space Sciences,Washington University, St Louis, MO 63130, USA
The late-time accelerated expansion of the universe could be caused by a scalar field that isscreened on small scales, as in chameleon or symmetron scenarios. We present an analogy betweensuch scalar fields and electrostatics, which allows calculation of the field profile for general extendedbodies. Interestingly, the field demonstrates a ‘lightning rod’ effect, where it becomes enhancednear the ends of a pointed or elongated object. Drawing from this correspondence, we show thatnon-spherical test bodies immersed in a background field will experience a net torque caused by thescalar field. This effect, with no counterpart in the gravitational case, can be potentially tested infuture experiments.
Among the host of cosmological observations thatthe concordance model, ΛCDM, accounts for [1] thelate-time accelerated expansion of the universe posesone of the most compelling problems in physics. Asinferred from different distance measurements, the en-ergy density of the universe is presently dominated bya dark energy component with strongly negative pres-sure; in ΛCDM this role is played by a cosmologicalconstant Λ. Its extremely small value, at odds withtheoretical expectations, has prompted the explorationof alternative models in which a scalar field φ causesthe cosmological acceleration (see e.g. [2–4]).To obtain dark energy behavior with an evolutionof the energy density, however, the scalar must be ex-tremely light, and generically mediates horizon rangedinteractions. Interestingly, constraints from fifth forceexperiments can be evaded in modified gravity theo-ries, where screening mechanisms (reviewed in [5]) areat play in dense environments. The scalar could thenhave significant couplings with ordinary matter at thefundamental level, but the range of the scalar mediatedforce becomes small enough to conform to terrestrialmeasurements [6]. The force gets further reduced forrelatively large bodies if, as in e.g. chameleon or sym-metron scenarios, it is only sourced over a thin shellunder the surface of the body, thus avoiding conflictwith precision measurements of gravity in the sparserenvironment of the solar system.In this Letter, we construct a mathematical anal-ogy between the chameleon scalar field and the electro-static potential. This analogy is in the same spirit asthe many systems identified in the chapter devoted toelectrostatic analogs in Feynman’s classic text [7], forexample, the uniform illumination of a plane, or theflow of an irrotational fluid past a sphere. The under-lying principle at work here is that the same equationshave the same solutions—electrostatics is not germaneto the analogous system, but rather the phrase ‘elec-trostatic analogy’ serves as proxy for saying that theanalogous system obeys the same differential equations as electrostatics. Of course, not all phenomena froman electrostatic system will have counterparts in theanalogous system and vice versa; nonetheless, as Feyn-man shows, one can fruitfully export intuition and so-lutions from electrostatics to its analogs. We show inthis Letter that under conditions relevant to terrestrialexperiments the chameleon obeys the same equationsas the electrostatic potential. Our central finding isthat the chameleon field outside of elongated bodiessuch as ellipsoids is enhanced relative to the sphericalbodies typically considered. This shape enhancementcan be exploited by experimenters to probe new regionsof chameleon parameter space, even in the experimen-tally unfavorable thin shell regime.Fields featuring screening mechanisms often appearin scalar-tensor modifications of general relativity. Wechoose a chameleon field φ to exemplify our calcula-tions [8, 9], but our results extend to other mod-els, like the symmetron [10–12], which also exhibitthe thin shell. In these models, matter follows thegeodesics of the metric ˜ g µν = A ( φ ) g µν , where φ isa scalar field. Due to this conformal coupling theeffective potential that appears in the Klein-Gordonequation includes a piece that depends on ρ m thedensity of matter, V eff ( φ ) = V ( φ ) + A ( φ ) ρ m . In thechameleon family of models V ( φ ) is monotonically de-creasing while A ( φ ) is monotonically increasing [8, 9].With this choice the effective potential has a minimum;both the value of the field at the minimum and themass of the field, m φ = ∂ V eff /∂φ , depend on ρ m .Thin shells do not exist for all couplings and poten-tials [13], so from now on we make the usual choice A ( φ ) = e βφ/M pl . Replacing the dimensionless constant β by α ( φ ∞ ) ≡ M pl d log A/ d φ , where M pl = 1 / √ πG ,our results also apply to generic conformal couplings.For static configurations of the field in a weak grav-itational background, the chameleon obeys the Klein-Gordon equation ∇ φ = ∂V eff /∂φ. (1)In general this is an unwieldy non-linear differentialequation, but in some regimes, approximations allowfor its easy solution. For example, the chameleonfield outside a sphere (radius R c , density ρ c , Newto-nian gravitational potential Φ c ) immersed in a uni-form background medium (density ρ ∞ < ρ c ) is well-approximated by a Yukawa profile [8, 9]. There aretwo versions of the Yukawa profile, depending on thedensity contrast between the body and its environ-ment [8, 9]. If the density contrast is slight, the bodydoes not disturb the surroundings too much, and thefield is only slightly perturbed from the value whichminimizes V eff corresponding the density ρ ∞ ( we de-note this value φ ∞ ). This is known as the thick shellregime; see [9] for details. If a chameleon field with thecanonical gravitational-strength coupling β ∼ V eff appropriate tothe density ρ c ; we call this value φ c . It only starts to‘see’ the exterior density as r approaches R c , and onlyover the course of a thin shell of material (thickness∆ R c ) just underneath the surface of the body does thefield begin to vary. Once outside the body, the field iswell-described by a Yukawa profile φ ( r ) ≈ (cid:18) − β πM P l (cid:19) (cid:18) R c R c (cid:19) M c e − m ∞ ( r − R c ) r + φ ∞ , (2)where ∆ R c R c ≡ φ ∞ − φ c βM P l Φ c ≪ /r for all intents and purposes: φ = φ ∞ + ( φ c − φ ∞ ) R c r . (4)which is the same as Eq. (2) as the reader can check.To summarize, in the thin shell regime, φ is constantthroughout its core. Only a thin shell of material justunderneath the surface contributes to the exterior field, where the profile is technically Yukawa. Given the den-sities and distances relevant to a terrestrial experiment,we can ignore the exponent, yielding simply a 1 /r be-havior.But we can also obtain this 1 /r behavior in a simplerway: a massive Yukawa profile results from makinga second order Taylor expansion of the field aroundits minimum, reducing Eq. 1 to ∇ φ ≈ m φ . If wefind later we can ignore the mass, we may as well justsolve Laplace’s equation, ∇ φ = 0 to determine thechameleon field exterior to the body. The solution toLaplace’s equation in spherical coordinates is simply φ = A + B/r where
A, B are determined by boundaryconditions. Very far from the sphere we should have φ → φ ∞ , and at r = R c we assume φ = φ c , hence A = φ ∞ and B = ( φ c − φ ∞ ) R c . Thus we recoverEq. (4).Note the collective behavior of the chameleon fieldfor a thin shelled sphere is precisely the same as the be-havior of the electrostatic potential ψ for a conductingsphere: inside the sphere both ψ and φ are constant,and outside the sphere both fields obey Laplace’s equa-tion. In a conductor there is a thin layer of chargeresiding on the surface that sources the electric fieldoutside the sphere; similarly inside the thin shell re-gion the chameleon field Eq. (1) may be approximatedas Poisson’s equation, with ∇ φ = βρ c /M P l .Thus there is an analogy between the chameleon fieldfor thin shelled objects and the electrostatic potentialof conducting objects, by the principle that the samedifferential equations have the same solutions.Let us examine more closely the charged surfacelayer. We know how to interpret this in an electro-static context– we say there is a surface layer of electriccharge σ , which is related to the external field gradientvia ∂ψ/∂n = σ , where n is the direction normal to thesurface (and we have taken ǫ = 1). Similarly we canwrite ∂φ/∂n = ̺δ where ̺ = βρ c /M P l is the volumedensity of ‘chameleon charge’ and δ is the thicknessof the layer over which this chameleon charge is dis-tributed. For the sphere, setting ∂φ/∂n = ( βρ c /M P l ) δ ,and solving for δ we find δ = ( φ ∞ − φ c ) R c βM P l Φ c , (5)which is identical to the thickness of the shell ∆ R c found in a completely independent way by Khoury andWeltman, Eq. (3). This suggests we interpret βρ c /M P l as the volume density of ‘chameleon charge’. Doingso independently reproduces the thickness of the shellderived in [9], and is consistent with Eq. (1) reducing toPoisson’s equation inside the shell region with the RHSgiven by βρ c /M P l . Note that this chameleon chargerepresents the material within the body that interactswith the chameleon field outside, and that it is confinedto the shell/surface layer. Note further that the thethick shell regime does not extrapolate from the thinshell regime increasing the thickness of the shell (infact it doesnt extrapolate at all).Given that the thin shell effect arises via the densitycontrast and boundary conditions, it stands to reasonthat although the effect was first derived for a sphere, ceteris paribus a less symmetric shape would still pos-sess a thin shell. We can use the analogy to deter-mine the chameleon profile for less symmetric shapes;in this Letter we work with ellipsoids to illustrate theinteresting shape-dependent effects we have identified.Ellipsoids also have the merit that they can also becompared with spherical results in the limit that theeccentricity ε → ξ, η, ϕ ); the surface of an ellipsoid has radialcoordinate ξ = ξ ; furthermore ξ = 1 /ε . η measuresthe latitude, with the poles at η = ± η = 0. It is convenient to introduce an equiv-alent radius R e such that the volume of the ellipsoidis πR e . We first consider the chameleon field pro-duced by an ellipsoid of arbitrary material, assumingonly that it possesses a thin shell. Its interior is at aconstant value determined by the density ρ c , and in theexterior it is the solution to Laplace’s equation. Therelevant solution to Laplace’s equation in these coordi-nates is obtained from [14]; φ = φ ∞ + ( φ c − φ ∞ ) Q ( ξ ) Q ( ξ ) (6)where Q ( ξ ) = ln[( ξ + 1) / ( ξ − / r ≫ a where a is the interfocal distance of the ellipsoid and r is the radial spherical coordinate, the chameleon profilecan be written φ = φ ∞ − f ( ξ )( φ ∞ − φ c ) R e r (7)where f ( ξ ) = 2[ ξ ( ξ − / ξ + 1) / ( ξ − , (8)and we have set a = 2 R e / [ ξ ( ξ − / .In comparing to the case of the sphere (Eq. 4) wesee the ellipsoid has a shape enhancement f ( ξ ) > m is given by [9] F = − m ∇ ( βφ/M P l ) . (9)Using the chameleon profile outside the sphere, Eq. 4,there will be unmitigated suppression of the force viathe thin shell factor. However, if we were to use anellipsoidal source instead of a spherical one, we woulduse Eq. 7 instead of Eq. 4 in calculating the force. Theellipsoidal profile still possesses the thin shell suppres-sion factor, but the suppression effect can leveraged bythe enhancement f ( ξ ).We refer to this mitigation of the suppression fac-tor as a ’lightning rod’ effect because in electromag-netism the electric field at the polar region of an elon-gated object is enhanced relative to the polar regionof a sphere, though we stress once again that electro-magnetic phenomena are not germane to this set up.The enhancement arises as a feature of the elongatedellipsoid, which has a preferred axis (its major axis)which the sphere lacks. The reader can confirm thatas ε →
0, the spherical results are obtained. An ex-periment would likely probe the near field close to thesharp tip of a dense body rather than the asymptoticfield discussed above. We will return to the analysis ofrealistic force experiments in future work.We now turn to another shape enhancement effectdemonstrated by the chameleon field. We begin bycalculating the chameleon force on an extended bodythat cannot be treated as a test mass. It follows fromEq. 9 that the force on an extended body with density ρ c and volume element dV be given by F = Z vol dV βρ c M P l ∇ φ. (10)In the thin shell regime φ = φ c in the core so it isonly necessary to integrate over the shell. As notedthe field obeys Poisson’s equation in this region. Wedenote the thickness of the shell as ∆ R , and take z tobe the coordinate along the local normal to the surfaceˆ n . Assuming the gradient of the field vanishes at z = 0where the shell meets the core, we obtain φ = φ c + 12 βρ c M P l z . (11)Substituting eq (11) into eq (10) yields F net = Z da Z ∆ R dz (cid:18) βρ c M P (cid:19) z ˆ n = 12 Z da (cid:18) βρ c ∆ RM P l (cid:19) ˆ n . (12)Using βρ c ∆ R/M
P l = ∂φ/∂n we can write this as F net = 12 Z da (cid:18) ∂φ∂n (cid:19) ˆ n . (13)Note that this is the same expression for the force on aconductor in an electrostatics context, if we take ǫ = 1and replace φ by the electrostatic potential ψ . Us-ing Eq.(13) one can calculate the chameleon force ona spherically symmetric extended body and show thatit only differs from that of the test mass by the thinshell factor. At first this may seem surprising, becauseit says a test mass experiences greater force than theextended object of the same mass. But the entire testmass experiences the chameleon field gradient whereasin the extended body, only the material in the shellsees a field gradient, so the suppression makes sense.In graduating from a test mass to an extended sphere,the body ‘acquires’ a thin shell.So the chameleon field exerts a force on an extendedbody given by Eq. 13. It follows that if this force acts ata distance r it will result in a torque, given by τ = r × F : τ = 12 Z da ( ∂φ/∂n ) r × ˆ n . (14)Note that an extended body can experience a torque,where a true test mass cannot–treated as a point par-ticle it has no radial extent and thus lacks a leverarm at which the force can act to produce a torque.Combining this result with our earlier finding that thechameleon field produced by an ellipsoid is enhancedin the polar regions, we suspect an ellipsoid immersedin a chameleon field will experience non-zero torque,due wholly to the chameleon field. To determine thiswe need to pull together several results. Khoury andWeltman [9] argue that the ambient chameleon field in-side a vacuum chamber has a uniform gradient, whosemagnitude χ = |∇ φ | they estimate. Introducing a testmass would not disrupt this field configuration butsince we are considering an extended source we seekthe solution to Laplace’s equation in which the am-bient chameleon field adjusts to the presence of thisextended body, which is also a source of chameleonfield. Morse and Feshbach [14] provide the solutionto Laplace’s equation for a conducting ellipsoid whoseinterior is held at a constant potential and which isimmersed in a uniform electric field that makes an an-gle γ with the ellipsoid’s major axis; by analogy thechameleon field for a thin shelled ellipsoid immersed in a chameleon field with uniform gradient is given bythe same expression. The resulting expression for thechameleon field is φ + φ where φ is given by Eq. 6)and φ = χ R e [ ξ ( ξ − / { cos γ η (cid:20) ξ Q ( ξ ) Q ( ξ ) − ξ (cid:21) + sin γ cos ϕ p − η " p ξ − Q ( ξ ) Q ( ξ ) − p ξ − } (15)Here Q ( ξ ) = ξQ ( ξ ) − Q ( ξ ) = p ξ − ξ/ ( ξ − − Q ( ξ )].For this profile, the chameleon torque on the ellipsoidhas one non-vanishing component τ y = πR e χ sin γ cos γ g ( ξ ) (16)where the shape dependent factor g ( ξ ) = 2 / − ξ ) Q ( ξ ) Q ( ξ ) Q ( ξ ) ξ / ( ξ − / . (17)Note that the torque vanishes in the spherical limit ξ → ∞ . It is remarkable that the chameleon fieldproduces a torque because a gravitational field withuniform gradient would not produce a torque on anellipsoid[16]. Before estimating the magnitude of thetorque we point out several important features. τ isindependent of β and ρ ; this is analogous to the cor-responding electrostatic torque on a conductor beingindependent of the total charge. The insensitivity ofterrestrial experiments to β is well-known [6] and un-fortunate from the stand point of detection. However,any value of β & β would be ruled out. To estimate the magnitude of thetorque we follow [9] and use V ( φ ) = M /φ , R vac ∼
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