Casimir, Gravitational and Neutron Tests of Dark Energy
aa r X i v : . [ h e p - ph ] D ec Preprint typeset in JHEP style - HYPER VERSION
Casimir, Gravitational and Neutron Tests of DarkEnergy
Philippe Brax
Institut de Physique Th´eorique, CEA, IPhT, CNRS, URA 2306, F-91191Gif/YvetteCedex, FranceE-mail: [email protected]
Anne-Christine Davis
DAMTP, Centre for Mathematical Sciences, University of Cambridge, CB3 0WA, UKE-mail:
Abstract:
We investigate laboratory tests of dark energy theories which modify gravityin a way generalising the inverse power law chameleon models. We make use of the to-mographic description of such theories which captures f ( R ) models in the large curvaturelimit, the dilaton and the symmetron. We consider their effects in various experimentswhere the presence of a new scalar interaction may be uncovered. More precisely, we focuson the Casimir, Eot-wash and neutron experiments. We show that dilatons, symmetronsand generalised chameleon models are efficiently testable in the laboratory. For generalisedchameleons, we revise their status in the light of forthcoming Casimir experiments likeCANNEX in Amsterdam and show that they are within reach of detection. ontents
1. Introduction 22. Tomographic Models 3
3. Planar Solutions in Modified Gravity 7
4. Laboratory Tests 12
5. Application to Models 14
6. Field Theoretic Consistency 18
7. Constraints and Forecast 23
8. Conclusion 27 – 1 – . Introduction
Laboratory tests [1–3] offer a complementary approach to astrophysical and cosmologicalobservations for dark energy/modified gravity theories [4, 5] involving one scalar field cou-pled to matter in a conformal way [6]. Astrophysical and cosmological probes are sensitiveto a deviation of the equation of state for the dark energy fluid from the standard Λ-CDMmodel and its impact on the background cosmology. They are also affected by new scalarinteractions in the Mpc range and its effects on structure formation at the perturbativelevel [7]. Developments in N-body simulations to study non-linear effects in the growth ofstructures [8] combined with new surveys like Euclid [9] will give constraints on the largescale properties of dark energy and modified gravity scenarios which may reach and improveon the level already attained by solar system tests [10, 11]. They will complement them ina range of scales where gravitational properties have never been extensively studied. Onthe theoretical side, the most general type of scalar models where both dark energy andmodifications of gravity can be envisaged has been recently rediscovered in the form of theHorndeski theories [12, 13]. Similarly, bimetric theories of gravity [14] allow one to con-sider theoretically sound extensions of the original Pauli-Fierz theory of massive gravity.For all these models, the landscape of their possible physical consequences has only beenexplored in some corners where both linear on very large scales and non-linear effects, onsmall scales down to laboratory ones, can be analysed. In this paper, we will present newresults for a subset of scalar field models where the effects of the scalar field are screenedby either the chameleon [15, 16] or the Damour-Polyakov [17] mechanisms on small scales.These tomographic theories [18, 19] generalise the inverse power law chameleons, the f ( R )theories [20], dilatons [21] and symmetrons [22, 23]. We consider laboratory experimentswhich are searching for deviations from the electromagnetic Casimir pressure [2], testingthe existence of extra forces like Eot-wash [1] and measuring the neutron energy levelsin the terrestrial gravitational field [3]. Extensions to atomic and neutron interferometryexperiments [24, 25] can also be considered. Using the tomographic method, we are ableto express the Casimir pressure due to the scalar field, the torque between the plates ofthe Eot-wash experiment and the displacement of the neutron energy levels in a simplemanner [26]. For inverse power law chameleons and symmetrons, our analytical results arecompatible with numerical simulations of the exact experimental setup [27, 28]. They canbe applied to all models described tomographically.We also pay attention to the issue of quantum corrections [29] and calculate the oneloop effects in a homogeneous medium. Such effects can be large in dense matter andcan invalidate the predictions made at the classical level. This is particularly true ofinverse power law chameleons where the quantum corrections in the boundary plates oflaboratory experiments can be large for relatively low couplings to matter. For largervalues of the coupling, one cannot guarantee that the scalar field profile between the platesis maintained as the boundary values for the scalar may have been altered drastically byquantum effects. Fortunately, for chameleons at large enough couplings the homogeneoussolution does not hold anymore and the field forms bubbles at the atomic level [24]. Thesebubbles are quantum stable when quantum corrections inside nuclei are tamed by imposing– 2 – vanishingly small coupling in nuclear matter. In this case, the bubble solution betweenthe boundary plates is not sensitive to quantum corrections and experiments tackling thelarge coupling regime of chameleons such as the next generation of Casimir experimentswill give valuable information on the chameleon’s parameter space .We use our results for f ( R ) models in the large curvature regime and find that theireffects in the laboratory are negligible. For dilatons, we find that the 2006 Eot-wash mea-surements at a distance of 55 µ m give a strong restriction on the cosmological mass of thescalar field, although still two orders of magnitude below the bound from the tests of theequivalence principle by the Lunar Ranging experiment. Symmetrons with cosmologicaleffects cannot be effectively tested in the laboratory although the ones with a phase transi-tion in rather dense media can be. Finally, generalised chameleons with inverse power lawpotentials are found to be very close to being detectable by the next generation of Casimirexperiments [30] as soon as their sensitivity will drop below one pN/cm .In section 2, we introduce the tomographic models. In section 3, we consider planarfield configurations. In section 4, we present the various experimental situations that wewill consider and calculate their observables for various models in section 5. In section 6,we analyse the quantum corrections for tomographic models. In section 7, we deduce thepresent laboratory constraints on tomographic models and the forecasts for inverse powerlaw chameleons. We conclude in section 8.
2. Tomographic Models
Inverse power law chameleon models and their generalisations are scalar-tensor theoriesdescribed by the Lagrangian S = Z d x √− g ( R πG N − ( ∂φ ) − V ( φ )) + S m ( ψ, A ( φ ) g µν ) (2.1)where A ( φ ) is an arbitrary function which specifies the coupling between matter fields ψ and the scalar φ . The coupling to matter itself is given by β ( φ ) = m Pl d ln A ( φ ) dφ . (2.2)The most important feature of these models is that the scalar field dynamics are determinedby an effective potential which takes into account the presence of the conserved matterdensity ρ of the environment V eff ( φ ) = V ( φ ) + ( A ( φ ) − ρ. (2.3)When the effective potential acquires a matter dependent minimum φ ( ρ ), for instance when V ( φ ) decreases and A ( φ ) increases, the mass of the scalar at the minimum is also matterdependent m ( ρ ). Scalar-tensor theories whose effective potential V eff ( φ ) admits a densitydependent minimum φ ( ρ ) can all be described parametrically from the sole knowledge ofthe mass function m ( ρ ) and the coupling β ( ρ ) at the minimum of the potential [18, 19]. It– 3 –s often simpler to characterise the functions m ( ρ ) and β ( ρ ) using the time evolution of thematter density of the Universe ρ ( a ) = ρ a (2.4)where a ≤ a = 1. This allows oneto describe characteristic models in a simple way, even in situations like laboratory testswhere no cosmology is involved. The field value is given by φ ( a ) − φ c m Pl = 9Ω m H Z aa c da β ( a ) a m ( a ) , (2.5)where the Hubble rate now is H ∼ − GeV and the matter fraction is Ω m ∼ .
27. Wehave identified the mass as the second derivative m ( a ) = d V eff dφ | φ = φ ( ρ ( a )) (2.6)and the coupling β ( a ) = m Pl d ln Adφ | φ = φ ( ρ ( a )) . (2.7)The potential value is given by V ( a ) − V c = − m H Z aa c da β ( a ) m a m ( a ) . (2.8)This parameterisation allows one to obtain V ( φ ) and A ( φ ) implicitly from m ( a ) and β ( a ). Chameleons with a potential of the type V ( φ ) = Λ + Λ n φ n + . . . (2.9)where n >
0, Λ ∼ − eV is the cosmological vacuum energy now, and the couplingfunction is A ( φ ) = exp( βφm Pl ) , (2.10)can be reconstructed using β ( a ) = β (2.11)and m ( a ) = m a − r (2.12)where r = n +2)2( n +1) . The mass scale m is determined by m n +1)0 = ( n + 1) n +1 n (3 β Ω m H m Pl ) n +2 Λ n (2.13)which gives dimensionally m ∼ β ( n +2) / n +1) ( m Pl H ) n/ n +1) H . (2.14)This implies that inverse chameleon models have a cosmological interaction range 1 /m much shorter than the size of the observable Universe for β & .2 Large curvature f(R) A large class of interesting models of the chameleon type consists of the large curvature f ( R ) models with the action S = Z d x √− g f ( R )16 πG N (2.15)where the function f ( R ) is expanded in the large curvature regime f ( R ) = Λ + R − f R n R n +10 R n . (2.16)Here Λ is the cosmological constant term necessary to lead to the late time accelerationof the Universe and R is the present day curvature. These models can be reconstructedusing the constant β ( a ) = 1 / √ m ( a ) = m ( 4Ω Λ0 + Ω m a − Λ0 + Ω m ) ( n +2) / (2.17)where the mass on large cosmological scale is given by m = H s Λ0 + Ω m ( n + 1) f R , (2.18)and Ω Λ0 ≈ .
73 is the dark energy fraction now [19]. When a ≪ a is a power law m ( a ) ∼ m a − r (2.19)where r = n +2)2 . Another relevant example is the environmentally dependent dilaton [21]. This model isinspired by string theory in the large string coupling limit with an exponentially runawaypotential V ( φ ) = V e − φm Pl (2.20)where V is determined to generate the acceleration of the Universe now and the couplingfunction is A ( φ ) = A m ( φ − φ ⋆ ) . (2.21)These models can be described using the coupling function in the matter dominated era β ( a ) = β a (2.22)where β is related to V and is determined by requiring that φ plays the role of late timedark energy which sets β = Ω Λ0 Ω m ∼ .
7, and the mass function which reads m ( a ) = 3 A H a (2.23)and is proportional to the Hubble rate with the mass on cosmological scales now given by m = √ A H . – 5 – .4 Symmetrons Another example is the symmetron where a scalar field has a quartic potential with anon-vanishing minimum V ( φ ) = V + λ φ − µ φ (2.24)and a coupling function A ( φ ) = 1 + β ⋆ φ ⋆ m Pl φ (2.25)where the transition from the minimum of the effective potential at the origin to a non-zerovalue happens at a = a ⋆ . This is a second order phase transition where the mass vanishes.Defining m ⋆ = √ µ, φ ⋆ = 2 β ⋆ ρ ⋆ m ⋆ m Pl , λ = µ φ ⋆ (2.26)where ρ ⋆ = ρ m a ⋆ , the model can be reconstructed using m ( a ) = m ⋆ r − ( a ⋆ a ) (2.27)and β ( a ) = β ⋆ r − ( a ⋆ a ) (2.28)for a > a ⋆ and β ( a ) = 0 for a < a ⋆ . In dense environment, the field is at the origin whilein a sparser one with a > a ⋆ we have φ = φ ⋆ r − ( a ⋆ a ) . (2.29) The inverse power law chameleons, the dilaton and f(R) models in a dense environmentare all described by power law functions m ( a ) = m a − r , β ( a ) = β a − s (2.30)for different choices of r and s . In fact, all these models can be defined by a potential V ( φ ) = V + ǫ Λ − p φ p (2.31)where V is an arbitrary constant, and p = 2 r − − s r − − s (2.32)as long as (2 r − − s ) >
0. The sign ǫ = ± p < − p = 27 | r − − s | Ω m β H m m ( 2 r − − s m Ω m β H m pl ) p (2.33)– 6 –hich is a function of both m and β . For inverse power law chameleons, it is taken to bethe dark energy scale. The coupling function becomes A ( φ ) = β m Pl φ l M l − (2.34)where the power l is given by l = 2 r − − s r − − s (2.35)and the coupling scale is M − l = Ω m l ( 92 r − − s Ω m β H m pl m ) s r − − s . (2.36)Although very explicit, this field parameterisation of the models is cumbersome. We willmostly use the ( m ( a ) , β ( a )) definition in the following.
3. Planar Solutions in Modified Gravity
The experimental setups that we will consider in the following can all be well approximatedby two infinite plates separated by a distance 2 d . In this case, the scalar field satifies theKlein-Gordon equation which reduces to d φdz = dVdφ + β ( φ ) ρ ( z ) m Pl (3.1)where the z axis is perpendicular to the plates with z = 0 on the bottom plate. Thedensity is constant between the plates ρ = ρ b and inside them ρ = ρ c . We also assume that A ( φ ) ∼ φ induced by ρ . This is satisfied for all the models we willstudy and comes from the Big Bang Nucleosynthesis (BBN) constraint on the variation ofparticle masses between BBN and now.It is convenient to change variable from φ ( z ) to a ( z ) where φ ( z ) ≡ φ ( a ( z )) and φ ( a ) isgiven by (2.5). The Klein-Gordon equation becomes d adz + γ ( a )( dadz ) = − am ( a )3 (1 − a H ρ Ω m H ) (3.2)and we have defined the effective Hubble rate H ρ = ρ m (3.3)which is constant inside and outside the plates. The function γ ( a ) is given by γ ( a ) = d ln αda (3.4)where α ( a ) = β ( a ) a m ( a ) . (3.5)– 7 –ar enough inside the plates, a ( z ) converges to a stationary value where the source termvanishes for a = a c and we have ρ c = ρ a c (3.6)where ρ = 3 H Ω m m is the matter density in the Universe now. For plates of commondensities in the ρ c ∼
10 g / cm range, this corresponds to a very small a c ∼ . − . Inthis case we have that a ( z ) → a c deep inside the plates.It is useful to define the dimension-less functions f and g such that β ( a ) = β f ( a ) , m ( a ) = m g ( a ) (3.7)where we normalise f (0) = g (0) = 1. We also introduce the dimension-less space variable u = m z. (3.8)and the dimension-less field S ( z ) = Z aa c da ′ f ( a ′ ) a ′ g ( a ′ ) (3.9)The corresponding effective potential V S ( S ) = − Z a ( S ) a c da f ( a )3 a g ( a ) (1 − a H ρ Ω m H ) (3.10)is such that the Klein-Gordon equation becomes d Sdu = ∂V S ∂S (3.11)Notice that this equation is independent of β , i.e. the field configuration does not dependon the value of the matter coupling now. This is a result which was already obtained forinverse power law chameleons and which is general for all tomographic models. We canuse this to integrate the Klein-Gordon. We will assume that the plates are wide enough that a ( z ) becomes constant deep insidethem. This is tantamount to asking that the models we consider are such that large enoughobjects of high density screen the effects of the scalar field. In particular we shall requirethat the mass m ( a ) becomes large enough inside the plates that the variation of φ occursover a thin sheet close to the surface of the body. These conditions were already appliedin the original chameleon papers.We can now integrate the Klein-Gordon equation. In the plates we have( dSdu ) = V − S ( S ) . (3.12)where a − sign signifies that the potential is defined with the density ρ c for z <
0. Inbetween the plates we have( dSdu ) − ( dSdu ) s = 2( V + S ( S ) − V + S ( S s )) (3.13)– 8 –here S s = S ( a ( | z | = d )) is the boundary value of S and similarly for ( dSdz ) s . The +sign is to remind us that this potential is defined with the density ρ b between the plates.Continuity implies that between the plates we have( dSdu ) s = 2 V − S ( S s ) (3.14)and therefore the solution between the plate satisfies( dSdu ) = 2( V + S ( S ) − ∆ V s ) (3.15)where ∆ V s = V + S ( S s ) − V − S ( S s ). The solution a ( z ) has a maximum at z = d . Theresulting scalar configuration forms a bubble between the plates. The profile of the bubbleis determined for 0 ≤ z ≤ d by the integral m z = Z SS s dS ′ q V + S ( S ) − ∆ V s ) (3.16)and the extremal value of S is given by m d = Z S d S s dS ′ q V + S ( S ) − ∆ V s ) (3.17)as a function of S c and S s . As there is an extremum at z = d for a d = a ( z = d ), we havethat V + S ( S d ) = ∆ V s which implies that V + S ( S ) − V + S ( S d ) = Z a d a ( S ) da f ( a )3 a g ( a ) (1 − a H ρ Ω m H ) . (3.18)Between the plates H ρ Ω m H = a − b leading to V + S ( S ) − V + S ( S d ) = Z a d a ( S ) da f ( a )3 a g ( a ) (1 − a a b ) . (3.19)The last term can be neglected as long as a d ≪ a b . In this case, we have S d ≪ S b and thefield deviates significantly from its value in the absence of both plates. This determinesthe field profile completely and we get our final expression for the profile m ( d − z ) = Z S d S dS ′ q V + S ( S ) − V + S ( S d )) (3.20)where m d = Z S d dS ′ q V + S ( S ) − V + S ( S d )) (3.21)These results are valid as long as the chamber is much larger than the range of the scalarfield in the plates m c d ≫
1, which guarantees that both plates are screened. We alsoassume that m b d ≪ m − b . We will apply these results to the models that wehave presented. – 9 – .3 Power law models We focus on generalised power law models defined by m ( a ) = m a − r , β = β a − s (3.22)which describe inverse power law chameleons, the large curvature limit of f ( R ) models for a ≪ S ( a ) = 1(2 r − s −
3) ( a r − − s − a r − − sc ) (3.23)and the potential in between the plates V + S ( a ) − V + S ( a d ) = 13(2 r − − s ) ( a r − − sd − a r − − s ) − r − − s ) ( a r − − s ( aa b ) − a r − − sd ( a d a b ) )(3.24)We concentrate on cases where a d ≪ a b , allowing us to neglect the terms coming from thedensity between the plates. When the plates are screened, we have that a ( z ) ≫ a c betweenthe plates and therefore V + S ( S ) − V + S ( S d ) = (2 r − − s ) p r − − s ) ( S pd − S p ) (3.25)where p = r − − s r − − s . We restrict our attention to 2 r − − s > a d ∼ ( m d ) /r . (3.26)The conditions a c ≪ a d ≪ a b correspond to m − c ≪ d ≪ m − b which is the range ofdistances between the plates where the approximations we have used apply. When d & m − c ,the field is constant between the plates and equal to φ c . When d & m − b , the influence ofthe two plates becomes negligible and the field converges to its constant value φ b . We mustnow distinguish two cases When p <
0, we are in a situation similar to the case of inverse power law chameleonswhere s = 0 and r = 3( n + 2) / n + 1). We find that S d = K p ( m d ) / (2 − p ) (3.27)and K − p/ p = q (2 r − − s ) p | r − − s | I p with I p = R dx √ x p − . Notice that S d decreases when d increases. The bubble is defined by the integral I p (1 − zd ) = Z S ( z ) Sd dx √ x p − . (3.28)Finally, we can also express the field S ( z ) when z ≪ d close to the first plate as S ( z ) = K p ((1 − p I p m z ) / (2 − p ) (3.29)which generalises the usual chameleon result and only depends on m z and not on d at all.– 10 – .3.2 Generalised f(R) models We consider models where p > f ( R ) models where s = 0and r = 3( n + 2) /
2. We find that the midpoint value between the plate is S d = ˜ K p ( m d ) / (2 − p ) (3.30)where ˜ K − p/ p = q (2 r − − s ) p | r − − s | J p with J p = R dx √ − x p . The bubble is defined by the integral J p (1 − zd ) = Z S ( z ) Sd dx √ − x p (3.31)and the field S ( z ) when z ≪ d close to the first plate is S ( z ) S d = J p zd . (3.32)The field profile is linear in this case contrary to the generalised chameleon behaviour. The dilaton can be described by the low density part of its potential where we have chosen φ ⋆ = 0 and r = 3 / , s = − p = 1, i.e. the linear approximation an exponentialpotential in the corresponding range of field values. This allows one to get exact expressionsfor the profile. The midpoint value between the plate is S d = ( m d ) . (3.34)The bubble is defined by the integral2(1 − zd ) = Z S ( z ) Sd dx √ − x (3.35)and the field S ( z ) becomes S ( z ) = S d (1 − (1 − zd ) ) (3.36)and for small z S ( z ) S d = 2 zd (3.37)The field profile is also linear contrary to the generalised chameleon behaviour.– 11 – .4 Symmetron In the symmetron case, the field is at the origin deep inside the plates. Between the platesthe field varies according to S ( a ) = S ⋆ r − ( a ⋆ a ) (3.38)where S ⋆ = 23 a ⋆ (3.39)and the potential becomes V + S ( S ) − V + S ( S d ) = 118 a ⋆ ((1 − S S ⋆ ) − (1 − S d S ⋆ ) + 2(1 − S b S ⋆ )( S S ⋆ − S d S ⋆ ))) (3.40)corresponding to a quadratic potential close to the origin. Defining x = S/S ⋆ and x d = S d /S ⋆ and the new variable x = q − (1 − x d ) cosh θ (3.41)where cosh θ d = 1 / (1 − x d ), the maximal value S d is given by m ⋆ d = Z θ d dθ sinh θ q (sinh θ + 2 cosh θ b (1 − cosh θ ))(1 − cosh θ cosh θ d ) (3.42)The symmetron has a non-vanishing profile in between the two plates only when m b d islarger than a critical value m b d c obtained by taking θ d to 0. We find that m b d c = π √ m b = m ⋆ r − a ⋆ a b and we have assumed that a b > a ⋆ , allowing the symmetron toprobe the symmetry breaking part of its potential between the plates. In the case when m b d ≪ S d = 0. Obviously in thiscase we have S ( z ) = 0.
4. Laboratory Tests
We will concentrate on the Casimir effect [2] induced by the presence of the scalar fieldcoupled to the plates and having a bubble profile between the boundary plates. Let us firstrewrite the field equation inside and outside the plates d φdz = ∂V eff ( φ ) ∂φ (4.1)from which we get the boundary value( dφdz ) s = 2( V eff ( φ d ) − V eff ( φ s )) = 2( V eff ( φ s ) − V eff ( φ c )) (4.2)– 12 –nd using the explicit expression of V eff ( φ ) we have A ( φ s ) = V ( φ c ) − V ( φ d ) + ρ c A ( φ c ) − ρ b A ( φ d ) ρ c − ρ b (4.3)for the value of the field on the boundaries.The Casimir force F φ on one of the plates of surface area A is simply obtained byintegrating F φ A = − Z D + dd dxρ c dA ( φ ) dx (4.4)for a constant density plate of width D . We obtain the pressure F φ A = − ρ c ( A ( φ c ) − A ( φ s )) (4.5)In the case where ρ c ≫ ρ b , this expression simplifies and we get F φ A = V ( φ c ) − V ( φ d ) + ρ b ( A ( φ c ) − A ( φ d )) (4.6)In the absence of a second plate, there is a vacuum pressure due to the scalar field where wereplace φ d → φ b where φ b is the minimum of the effective potential for a density ρ = ρ b . Ina real experiment where the plates have a large but finite width, the vacuum pressure fromthe outside of the chamber on the plates would cancel leaving the plate in equilibrium if itwere not for the presence of the second plate which offsets the pressure on the inner sideof the plate. As a result, the vacuum pressure must be removed and the effective pressurefelt by one plate is ∆ F φ A = V eff ( φ b ) − V eff ( φ d ) (4.7)corresponding to the difference between the effective potential in vacuum compared to thevalue it takes in between the plates. This can be expressed as∆ F φ A = − m β H m m Z a b a d da f ( a ) a g ( a ) (1 − a a b ) (4.8)where ρ b = ρ a b . This proves that the scalar field adds an extra attracting pressure betweenthe plate as the integrand is always positive. It is convenient to rewrite this expression interms of V S ( S ): ∆ F φ A = − m β H m m ( V + S ( S d ) − V + S ( S b )) . (4.9)The value of S d depends on the masses m c and m b . When m c d & m b d ≪
1, thefield has a non trivial profile between the plates and we have calculated S d in the previoussection. When m c d .
1, the field is constant between the plates and S d = S c . Finallywhen the plates are not screened and m c D . D is the width of the plates, we have S d = S b and no Casimir pressure is present. We will use these results in the next section.– 13 – .2 The Eotwash experiment The search for the presence of new interactions by the Eotwash experiment [1] involves twoplates separated by a distance D in which holes of radii r h have been drilled regularly ona circle. The two plates rotate with respect to each other. The gravitational and scalarinteractions induce a torque on the plates which depends on the potential energy of theconfiguration. The potential energy is obtained by calculation the amount of work requiredto approach one plate from infinity [28, 31]. Defining by A ( θ ) the surface area of the twoplates which face each other (this is not the whole surface area because of the presence ofthe holes), a good approximation to the torque expressed as the derivative of the potentialenergy of the configuration with respect to the rotation angle θ is given by T ∼ a θ Z d max D dx ( ∆ F φ A ( x )) . (4.10)where a θ = dAdθ depends on the experiment. When the Casimir pressure due to the scalarfield decreases fast enough with d , the upper bound d max can be taken to be infinite. Whenthis is not the case, the upper bound is the maximal distance below which the scalar forceis not suppressed by the Yukawa fall-off. We will discuss the value of d max for the differentmodels that we have considered in the following section. Neutrons in empty space between two mirrors have quantized energy levels in the terrestrialgravitational field [3,26]. The scalar field induced a shift in the energy levels of the neutrondue to the change in the potential energy V ( z ) = m n gz + m n ( A ( φ ( z )) −
1) (4.11)close to the lower mirror. The correction term is given by δV ( z ) = 9Ω m β m n H m Z a ( S ) a c da f ( a ) a g ( a ) (4.12)where a ( S ) depends on z . This leads to a shift in the energy levels given by δE n = < ψ n | δV ( z ) | ψ n > (4.13)where | ψ n > is the n-th Airy level of the neutron. We will evaluate this shift in the nextsection.
5. Application to Models
We can now use the results of the previous section to calculate the effect of the scalarfield, and its Casimir energy. We focus on the case where the plates are screened as in theabsence of screening, no Casimir pressure is generated by the scalar field. In the case of– 14 –ower models with p <
0, the field value in the presence of the plates is much smaller thanthe one in their absence S d ≪ S b and we get V + S ( S d ) − V + ( S b ) = (2 r − − s ) p | r − − s | K pp ( m d ) p/ (2 − p ) (5.1)corresponding to a Casimir Pressure∆ F φ A = Λ ( p p B ( , − p ) Λ d ) p − p (5.2)where B ( ., . ) is the Euler B function. This generalises the inverse power chameleon casewhere p = − n [32]. Notice that the Casimir pressure only depends on the scale Λ and thedistance d . When Λ is taken to be the dark energy scale, this Casimir pressure is withinreach of the next generation of Casimir experiments. We will present new forecasts at theend the paper.For power law models with p >
0, the contribution of S b = a r − − sb / (2 r − − s ) cannotbe neglected anymore. In this case, we find a Casimir pressure∆ F φ A = 27Ω m β H m m (2 r − − s ) p | r − − s | ( ˜ K pp ( m d ) p/ (2 − p ) − S pb r − − s ) . (5.3)We define the distance d ⋆ where the a d ⋆ = a b , i.e. when the profile is that the scalarfield begins to feel the effect of the matter density between the plates and its resultingsuppression effect: ( m d ⋆ ) p/ − p = 3 S pb (2 r − − s ) ˜ K pp . (5.4)We then find that ∆ F φ A = 81Ω m β H m m (2 r − − s ) p S pb | r − − s | (( dd ⋆ ) p −
1) (5.5)As long as d ≪ d ⋆ , the distance dependence becomes negligible and the pressure constant∆ F φ A = − m β H m m a r − − sb | r − − s | (2 r − − s ) (5.6)as p < r > m b d ≪
1, we havethat S d = 0 (5.7)and the Casimir pressure is given by a constant∆ F φ A = −
92 Ω m β H m a ⋆ m (5.8)We can rewrite this result as ∆ F φ A = − µ λ (5.9)which is the height of the symmetron potential.– 15 – .2 Gravitational experiment We can use the previous result on the Casimir pressure to infer the torque on the rotatingplates in the Eotwash experiment in the screened case. Let us first focus on power lawmodels. When p <
0, the Casimir pressure falls off at infinity and two cases must beenvisaged. When p < −
2, the fall is fast enough that no dependence on d max is of relevanceand T θ = a θ − pp + 2 ( p p B ( , − p ) ) p − p Λ (Λ d ) ( p +2) / (2 − p ) (5.10)When − < p <
0, the torque is sensitive to the long distance behaviour of the Casimirpressure which becomes negligible when d = d ⋆ where a d ⋆ = a b , i.e. we take d max = d ⋆ ,implying that T θ = a θ − pp + 2 ( p p B ( , − p ) ) p − p Λ [(Λ d ) ( p +2) / (2 − p ) − (Λ d ⋆ ) ( p +2) / (2 − p ) ] . (5.11)which is independent of d as long as d ≪ d ⋆ where we have here( m d ⋆ ) p/ (2 − p ) = 3 S pb (2 r − − s ) K pp . (5.12)Notice that the torque depends on the combinations Λ d and Λ d ⋆ , i.e. it probes distancesof the order of the inverse dark energy scale which is about Λ − ∼ µ m. Its order ofmagnitude is then around a θ Λ which is very close to the bound found by Eot-wash.For power law models with p >
0, the situation is similar to the case − < p < T θ = a θ r − − s Ω m β H m m S pb d ⋆ [ 2 − pp + 2 (1 − ( dd ⋆ ) ( p +2) / (2 − p ) ) − (1 − dd ⋆ )] . (5.13)As 0 < p < T θ = − a θ r − − s pp + 2 Ω m β H m m S pb d ⋆ (5.14)which is a function of m and β .For the symmetrons, the Casimir force is independent of the distance as long as thefield vanishes between the plates. This is true as long as d < d c = π √ m b , hence the torqueis given by T θ = − a θ µ d c λ (5.15)which depends on µ and λ . The dependence on the coupling strength β ⋆ only appearswhen the electrostatic shielding between the plates is taken into account.– 16 – igure 1: The phase diagram of symmetron models for µ = Λ = 2 . − eV as an examplewhere d < d c and the field vanishes between the plates. The Eot-wash experiment is sensitive tosymmetrons for small values of λ to the left of the vertical line. For values of M ⋆ larger than thetop horizontal line, the symmetron is in its vacuum phase and no constraints apply. Below thebottom curve the symmetron is not excluded while it is excluded between the bottom curve andthe horizontal line. For power law models we have that δV ( z ) = 9Ω m β m n H m r − − s ((2 r − − s ) S ( z )) p . (5.16)When p <
0, we generalise the inverse power law chameleons and we find δV ( z ) = β m n m Pl Λ( 2 − p √ z ) / (2 − p ) (5.17)which has been thoroughly studied [24].Focusing on the models with p > δV ( z ) = 9Ω m β m n H m r − − s ((2 r − − s ) J p S d ) r − − s r − − s ( zd ) r − − s r − − s (5.18)The shift in the energy levels is then given by δE n = 3Ω m β m n H m r − − s α n,r ((2 r − − s ) J p S d ) r − − s r − − s ( z d ) r − − s r − − s (5.19)where the numbers α n,r = < ψ n | ( zz ) r − − s r − − s | ψ n > (5.20)– 17 –re of order one, and z = (cid:16) m n g (cid:17) / . In practice, the distance between the plates isadjusted to select different energy levels and therefore z ∼ d . The correction to the energylevels has a dependence on δE n ∼ β m n H m ( m d ) (2 r − − s ) /r (5.21)Contrary to p <
0, the result depends on d , i.e. on the details of the experimental setup.It also depends on the ratio of the distance between the two plates and the size of thepresent horizon. For the present day sensitivities at the 10 − eV, the deviation δE n is notobservable for both f ( R ) models and dilatons. Finally, for cosmological symmetrons wehave that S ( z ) = 0 and no deviation of the energy levels is expected.
6. Field Theoretic Consistency
So far we have treated the field theoretical models as classical field theories. Most of thetheories we have discussed have unusual features such as inverse power law potentials andnon integer powers of the field. In this section we will describe their properties usingthe language of effective field theories when they are embedded in an environment witha uniform energy density. In this case, the tomographic field theories have a well-definedminimum of the effective potential around which one can expand the potential in pertur-bation. This will allow us to discuss their validity and the quantum corrections which canbe easily calculated at the one loop order.
The tomographic models in the presence of a constant density environment are field theorieswith a potential described by V eff = − m β H m Z a ( φ ) a c da f ( a )3 a g ( a ) (1 − a H ρ Ω m H ) (6.1)where a ( φ ) has to be computed using the mapping φ ( a ) = 9Ω m β H m Pl m Z a ( φ ) a c da f ( a )3 a g ( a ) . (6.2)These theories can be expanded around a background field value φ corresponding to a valueof a = ¯ a in an infinite series which defines the tree level Lagrangian of an effective theory L = 12 ( ∂δφ ) + ∞ X i =0 λ i i ! δφ i (6.3)where we have λ i = d i V eff dφ p | φ = ¯ φ . This can be easily reexpressed as λ i = ( m m β H m Pl ) i − ˜ λ i where we identify the dimension-less coupling ˜ λ i = d i V S dS i | S = ¯ S and ¯ S = R ¯ aa c da f ( a )3 a g ( a ) . Theinfinite series can be rewritten as L = 12 ( ∂δφ ) + ∞ X i =0 ( ˜ λ i i ! )( m m β H m ) δφ i ˜Λ i − (6.4)– 18 – igure 2: Chameleons are such that the plates are not screened below the bottom curve (brown)and quantum effects are strong for values of the coupling larger than the top curve (green). Themiddle curve (red) is the limit below which the field is constant between the plates. with ˜Λ = m β H m Pl m . The terms with i < i = 4 is the marginal interaction with adimension-less coupling constant while the i > i = 4 at low energy, below a cut-off scalethat we will determine. It will also tell us when the perturbative expansion makes senseand no strong coupling issue arises.Practically we have ˜ λ = − f ( a )( a − − a − c ) (6.5)and ˜ λ i = a g ( a ) f ( a ) d ˜ λ i − da (6.6)recursively. Explicitly we find that the second coupling is˜ λ = g ( a ) − d ln fda g ( a ) a ( a − − a − c ) (6.7)We are considering the effective field theory when the matter density is ρ c and we areexpanding around the vacuum value a = a c . In this case ˜ λ = g . To go further, we shallassume that around a c , the mass function has a power law dependence g ( a ) = a − r and thecoupling function f ( a ) = a − s . In this case we find that˜ λ i ∼ a − r +6 − s ( a − r + s ) i − (6.8)– 19 –mplying that the infinite series can be written as L = 12 ( ∂δφ ) + ∞ X i =0 ( κ i i ! ) δφ i Λ i − (6.9)where the coupling constants are obtained to be κ i ∼ m ( ¯ φ ) m β ( ¯ φ ) ρ c (6.10)and the cut-off is Λ = β ( ¯ φ ) ρ c m ( ¯ φ ) m Pl . (6.11)This is the final form of the effective action for tomographic models. First of all, wefind that the tomographic models viewed as effective field theories are only defined at lowenergy below the cut-off Λ . The smallest cut-off scales is obtained in dense matter whereΛ ∼ a r − − sc β H m m pl . For cosmologically relevant models where m & H , this is avery small scale much smaller than the scale of the standard model of particle physics.Hence all the tomographic models in dense matter have a very low cut-off scale. Similarly,the tomographic models can be strongly coupled in dense matter when m ( ¯ φ ) & ( β ( ¯ φ ) ρ c m Pl ) / . (6.12)In this case, the effective field theory is not well-defined and one cannot consider thattomographic models describe the behaviour of massive particles with self-interactions atan energy below the cut-off scale Λ . This does not imply that the tomographic models arenot well defined themselves. It simply means that one cannot describe them as an effectivefield theory around a constant background field ¯ φ . For these cases, perturbation theoryfails. Quantum corrections and the quantum stability of the tomographic models are very im-portant to guarantee that the results we have obtained at tree level for the laboratoryexperiments such as the Casimir effect stand when quantum effects are taken into account.The quantum effects of the scalar field δφ can be calculated when the effective field the-ory is not strongly coupled and when the scalar field has a mass below the cut-off scaleΛ . Fortunately, the latter is equivalent here to requiring that the model is not stronglycoupled. In this case, the one loop correction to the potential is δV = m ( ¯ φ )32 π ln m ( ¯ φ )Λ (6.13)Around the vacuum value a = a c , the effective potential V eff ( ¯ φ ) = − m β H m Z a ( ¯ φ ) a c da f ( a )3 a g ( a ) (1 − a H ρ Ω m H ) + m ( ¯ φ )32 π ln m ( ¯ φ )Λ (6.14)– 20 –as a new minimum which is shifted a c → a c + δa where δaa c = r π m c m β c ρ c ln m c Λ (6.15)The quantum corrections are negligible when [29] m c ≤ ( 6 π √ β c ρ c √ rm Pl ) / (6.16)which is similar to the requirement that the theory must not be strongly coupled. Hencewe have found that tomographic models which are not strongly coupled at tree level donot suffer from any quantum instability.The quantum corrections are larger in dense media. In the cosmological vacuum, theabsence of quantum correction is guaranteed when m H . ( β m pl H ) / (6.17)For theories with β &
1, the right hand side is of order 10 , hence for cosmological modelswhere m /H & the quantum corrections are always negligible on cosmological scales.For dense media, quantum corrections can play a major role. The failure to calculate quantum corrections and the strong coupling issue can be resolvedin certain cases when a constant background field ¯ φ is not an appropriate description ofthe vacuum structure of the model. This is in particular the case for inverse power lawchameleons. In [24, 33, 34], it was shown that for large values of β , the nuclei in each atomof the dense medium become screened when φ c ≤ β m Pl Φ n (6.18)where Φ n = m n πm Pl2 R n whilst m n and R n are the mass and the radius of the nucleus.In this case, the homogeneous solution where φ = φ c inside the dense body is not validanymore. The scalar field becomes inhomogeneous and forms bubbles centered at eachatom and similar to the bubble solution between two plates but on atomic scales. Outsidea radius R ⋆ ∼ ¯ D ( R n ¯ D ) ( n +1) / (2 n +1) the solution for R ⋆ ≪ r ≪ R grows like a bubble in φ ( r ) ∼ Λ( n +2 √ Λ r ) / ( n +2) before reaching a maximum φ D = ( √
2Λ ¯ DI n ) / ( n +2) (6.19)where 2 ¯ D is the interatomic distance. Apart from the steep increase around each atom,the solution is of average φ D which is very different from the homogeneous solution. Infact we have the relation between φ D and the maximal value between the boundary platesof the experiments that we have considered φ D = ( ¯ Dd ) / ( n +2) φ d . (6.20)– 21 –mplying that φ D ≪ φ d . This boundary value is small enough to guarantee the existenceof the bubble solution between the plates .For such values of β , the quantum corrections to the chameleon potential inside thenuclei can be large and therefore not calculable. In this case this implies that the bubblestructure between atoms may be destroyed by those quantum corrections inside the nucleusif the quantum corrected minimum inside the nuclei is much larger than its tree level value.Unfortunately, the behaviour of inverse power law models in nuclear matter goes beyondthe domain of validity of such models which we have assumed to be valid for densitiessmaller or equal to the ones during BBN, i.e. a few g / cm . At much larger densities, themodels have to altered to guarantee that the quantum corrections are negligible. This canbe achieved by appropriately modifying the matter m ( a ) and coupling function β ( a ) attree level for a . a BBN . For instance an interpolation to a symmetron-like behaviour athigh density m ( a ) = m a − r r − ( a ⋆ a ) , β ( a ) = β r − ( a ⋆ a ) (6.21)with a ⋆ < a BBN and where the field would be stuck to a vanishing value in the regime a < a ⋆ would annul the effects of the quantum corrections. The details are left for futurework. For the laboratory experiments that we have considered where two dense plates are sepa-rated by a gas, the quantum corrections are relevant for models of the generalised chameleontype as inside the plates the quantum corrections can be large. When the quantum cor-rections are not negligible, the minimum of the effective potential can be shifted by a largeamount or even disappear. On the other hand, the field equations between the plates arenot affected by the quantum corrections. Hence the only role played by the potentially largequantum corrections is to modify the boundary values of the field and its first derivativeon the plates.The derivation of the Casimir pressure (4.7) is not affected much, implying that theonly effect of the quantum corrections is to shift the boundary value φ s and therefore tomodify the value φ d between the plates. Using (6.15) as an order of magnitude estimate ofthe value a q of the new minimum a q ∼ a c m c m β c ρ c . (6.22)Therefore the structure of the bubble between the plate is preserved as long as a q ≪ a d .When this is not the case, the boundary value is affected too much to guarantee that abubble solution still exists between the plates and therefore the classical results are largelyaffected. For even larger values of β , the cloud of electrons in the plate serves as the source for the scalar fieldwhich becomes homogeneous again. For chameleons, this happens for very large couplings which are alreadyexcluded [35]. – 22 –t higher values of the coupling, chameleon models are described by an inhomogeneoussolution at the atomic level. Again this solution is highly sensitive to the quantum stabilityof chameleons in nuclear matter. As long as the coupling in this dense environment is smallenough, which requires us to modify the models for density higher than the BBN ones, thebubbles formed at the atomic level are quantum stable and the bubble solution betweenthe boundary plates is preserved. In this case, the experiments probing the large couplinglimit of chameleon models are immune from quantum corrections.
7. Constraints and Forecast
The most stringent experimental constraint on the intrinsic value of the Casimir pressurehas been obtained with a distance 2 d = 746 nm between the two boundary plates andreads | ∆ F φ A | ≤ .
35 mPa where we have 1mPa = 1 .
44 10 Λ [36]. The experiment hasbeen performed with a pressure of 10 − Torr between the plates of width D = 0 . ρ b = 6 . − GeV and a b ∼ · − . The plate density is of the orderof ρ c = 10 g . cm − . For the 2006 Eot-wash experiment [37], we consider the bound obtainedfor a separation between the plates of 2 d = 55 µ m is | T | ≤ a θ Λ T (7.1)where Λ T = 0 .
35Λ [31]. The pressure was lower than in the Casimir experiment cor-responding 10 − T and a b ∼ . − . We must also modify the torque that we havecalculated in order to take into account the effects of a thin electrostatic shielding sheet ofwidth d s = 10 µ m between the plates in the Eotwash experiment. This reduces the observedtorque which becomes T obs = e − m c d s T θ (7.2)When the mass in dense media is very large, this imposes a strong reduction of the signal.We combine these bounds as an illustration for f ( R ) models in the large curvaturelimit, dilatons, symmetrons and inverse power law chameleons. For f ( R ) models, theCasimir bound on m /H & − for n & .
01 is irrelevant compared to the usual boundfrom solar system tests m /H & . For values of m satisfying the solar system bound,the effect of the electrostatic shielding is so large that f ( R ) models cannot be tested bythese experiments as m c d s ≫
1. For dilatons with β ∼ .
7, the electrostatic shielding isefficient when m /H & , implying that no constraint can be obtained for such largemasses. For masses m /H & , quantum corrections become large in the plates. Theplates are screened provided m /H & · . Below this value, the field between theplates is equal to the one in the plates. This does not change the leading order expressionfor the torque and we find that the Eot-wash experiment requires that m /H &
55. Thisis less stringent than the solar system tests coming from the Lunar Ranging experiment m /H & M ⋆ = φ ⋆ m pl β ⋆ , (7.3)– 23 – igure 3: The top curve (green) represents the Eot-wash limit. It is in the quantum correctedpart of the phase diagram of chameleons which corresponds to β being larger than values on themiddle curve (red) where quantum effects start being relevant. The lower curve (blue) representsthe limit below which the plates are not screened and the torque vanishes. This implies thatchameleons are not excluded for very low couplings. The ascending curve (brown) is the Eot-washlimit when the field between the plates is constant and equal to the values inside the plates for m c d .
1. The triangular region between these three curves (red,blue,brown) is excluded by theEotwash experiment. For large values of n &
20, no constraints from Eot-wash apply. Notice thatthe top part of the exclusion zone is in the quantum corrected region of the parameter space andtherefore cannot be trusted. the mass of the scalar field when the density is much larger than the critical density ρ ⋆ isgiven by m ( a ) = µ ( a ⋆ a c ) / (7.4)The Eot-wash bound can be expressed asΛ T ≥ π µ λ e − ( a⋆ac ) / µd s (7.5)which can be written as M ⋆ ≤ √ ρ c d s ln( πµ λ Λ T ) (7.6)as long as 4 λ Λ T ≤ πµ . The torque calculation that we have presented applies only when a ⋆ ≥ a c where a ⋆ = ( ρ µ M ⋆ ) / . (7.7)For large values of M ⋆ , the symmetron is always in its vacuum phase and there is notorque between the plates. The combined constraints can be seen in Figure 1 for µ = Λ.For large values of λ , no constraints are obtained while at small λ , values of M ⋆ are bounded– 24 – igure 4: For large values of β higher than the top curve (green), the chameleons are in theirstrongly coupled phase and the quantum corrections which are present for values larger than thelower curve (red) are not important anymore. from above. This part of the parameter space can be tested by laboratory experimentssimply because a ⋆ ≤ a b . Cosmologically interesting models where µ ∼ H and a ⋆ ≥ − corresponding to a phase transition in the recent Universe cannot be probed by currentexperiments as a b is too small. For these models, the field value in all the laboratoryenvironment is equal to zero, implying that no Casimir effect or torque is present. When a b > a ⋆ , the parameter space is constrained by β ⋆ ≤ π a ⋆ m Λ T µ ρ e ( a⋆ac ) / µd s (7.8)and for µ ∼ H and small values of a ⋆ , the Eot-wash experiment implies a tight constrainton β ⋆ . − .Finally, for inverse power law chameleons the phenomenology is richer. We focus hereon the Eot-wash experiment. At small coupling β , the boundary plates are not screened.The Eot-wash bound is satisfied for values of β larger than a lower bound which dependson n . For n = 1, we must have β & β thequantum corrections are not under control in the plates and the bubble solution between theplates is not calculable anymore, see Figure 2. In fact as we show in Figure 3 the quantumbound is always lower than the Eot-wash upper bound for n .
20 implying that presentday experimental results are not compatible with the absence of quantum corrections in thepart of the parameter space between the top (green) and middle (red) curves of Figure 3.For n &
20, the Eot-wash result does not constrain chameleons due to the large suppressionby the electrostatic shield. For larger values of β & , chameleons are in their strongcoupling phase where bubbles appear at the atomic scale. This guarantees that the bubblebetween the plates is present and the Eot-wash bound applies, i.e. the strong coupling– 25 – igure 5: The distance below which new Casimir experiments with plates of surface areas 1 cm would feel a scalar Casimir force of respectively from top to bottom 0 . . d = 10 µ m, the new Casimir experiments with a sensitivity of 0 . models are allowed. Finally, for low values of β , the plates are separated by a distancelower than the distance d screen defined by d screen = m − c and the field between the platesbecomes equal to φ c . This modifies the expression of the torque T obs = a θ e − m c d s [ Z d screen d ( V ( φ c ) − V ( φ b )) dx + Z d ⋆ d screen ∆ F φ A dx ] (7.9)where the Casimir pressure ∆ F φ A is given by (5.2). The first term goes to zero as β → n . For n >
2, the integral is dominated by x ∼ d screen which increases as β → β ≪ n > n ≤
2, the integral is dominated by its upper boundand increases with it. The torque remains large for small β . This is valid as long as m c D &
1. When this is not the case anymore, the torque vanishes and the chameleons arenot constrained by the Eot-wash experiment (see Figure 3).
Casimir tests are particularly relevant for inverse power law chameleon models and theirgeneralisations. In the regime where the boundary plates are screened, and in particularfor very large values of β , the Casimir pressure due to the chameleons is independent of β .The next generation of Casimir experiments such as CANNEX in Amsterdam will test theelectromagnetic Casimir effect at distances larger than 2 d & µ m, with a sensitivity whichcould reach 0 . A = 1cm . At large coupling, the Casimirforce between the plates due to the scalar field is independent of β and depends only on– 26 – . In figure 4 we show, for such a setting, the distance between the plates for which thescalar Casimir force reaches the requires sensitivity. In particular, we find that at the 0 . d = 10 µ m.Even with a precision of 0 . n .
8. Conclusion
We have considered laboratory tests of tomographic models such as f ( R ) in the large cur-vature regime, inverse power law chameleons, dilatons and symmetrons. We have givensimple analytical expressions for the Casimir pressure due to the scalar, the torque betweentwo rotating plates in the Eot-wash experiment, and the shift of the energy levels of theneutron in the terrestrial gravitational field. We have analysed in detail the behaviour of f ( R ) models, inverse power law chameleons, dilatons and symmetrons in these experiments.We have also analysed the quantum corrections which can be particularly large for inversepower law chameleon models. We have shown that these quantum corrections are undercontrol when the tomographic models are at small coupling in a perturbative expansionaround the density dependent minimum of their effective potential. At strong coupling,and for models such as inverse chameleon models, the homogeneous approximation whichhas been used to calculate the quantum correction fails. The scalar field becomes inhomo-geneous at the atomic level and as long as the coupling to nuclear matter is assumed to bevanishingly small, the quantum corrections are tamed at large coupling for matter densitieslower than the BBN one. In this strong coupling phase, inverse power law chameleons arewithin reach of the next generation of Casimir experiments.We would like to thank C. Burrage and A. Upadhye for comments and suggestions.We are grateful to A. Amalsi and R. Sedmik for discussions and sharing information aboutthe CANNEX experiment, and to G. Pignol for lively discussions about the dilaton and theneutron experiments. P.B. acknowledges partial support from the European Union FP7ITN INVISIBLES (Marie Curie Actions, PITN- GA-2011- 289442) and from the AgenceNationale de la Recherche under contract ANR 2010 BLANC 0413 01. ACD acknowledgespartial support from STFC under grants ST/L000385/1 and ST/L000636/1. References [1] E. G. Adelberger, B. R. Heckel and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. (2003) 77[hep-ph/0307284].[2] S. K. Lamoreaux, Phys. Rev. Lett. (1997) 5 [Erratum-ibid. (1998) 5475].[3] V. V. Nesvizhevsky, H. G. Borner, A. M. Gagarski, A. K. Petoukhov, G. A. Petrov,H. Abele, S. Baessler and G. Divkovic et al. , Phys. Rev. D (2003) 102002[hep-ph/0306198].[4] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D (2006) 1753[hep-th/0603057]. – 27 –
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