Vacuum energy, the Casimir effect, and Newton's non-constant

We explore two hypotheses. First, the possibility that the quantum vacuum energy density of the Casimir effect contributes to a (local) gravitational vacuum energy density. Second, the possibility that a change in the gravitational coupling implies a change in the cosmological constant. We parametrize these two possibilities in a covariant framework and show that the next generation of Casimir experiments does have a surprisingly good chance of exploring this parameter space.

This article combines several fields of physics, which are not commonly related.This increases the risk of misunderstanding.To minimize this risk we start with two questions of fundamental physics, which we ask in terms of two phenomenological hypotheses: The cosmological constant was introduced by Einstein [1] to prevent gravitational instability in general relativity (GR).While the constant's history is complex, the acceleration of cosmic expansion [2][3][4] provides us with a clear value [5] Λ 0 = 1.1 × 10 −52 m −2 .We may interpret Λ 0 as an energy density contributing via the Newton coupling G 0 and the speed of light c to the matter part of Einstein's field equations, While the interpretation of Λ 0 in GR is clear, its origin remains elusive.According to quantum field theory, the zeropoint energy [6] of all fields of the standard model and beyond as well as the Higgs phase transition [7] should contribute to the measured value of ρ Λ 0 .Zero-point energies of each particle's quantum field provide an infinite contribution to the quantum energy density ρ Q,0 .Those contributions can be rendered finite ρ Q,0 (κ) by introducing regulators like momentum cutoffs κ.We can express the ratio between the cosmological energy density ρ Λ 0 and the regularized quantum contributions ρ Q,0 (κ) as a dimensionless ratio where m Z and m p are the mass of the Z boson and the Planck mass, defining a cutoff choice at the weak and Planck scale, respectively [8].According to our current understanding, Υ 0 should be of order one (or exactly zero).Instead, the values in Eqn.(2) are small but non-vanishing, leading to a severe finetuning problem that was coined cosmological constant problem (CCP) [8,9].
Resolving the CCP will most likely require physics beyond the present SM and GR framework, which gave motivation for research on possible alternative descriptions of quantum gravity.The list of approaches is long, ranging from standard techniques, such as the functional renormalization group [10][11][12][13], to Planck scale fluctuations [14,15], or holographic interpretations [16,17].
New insight (in particular from experiments) concerning the quantum origin of Λ 0 , is urgently needed.In the following we argue that progress can not only be made by astronomical measurements but also by carefully tuned laboratory experiments.

B. The Casimir effect
A setting, where quantum fluctuations are much better under control (both experimentally and theoretically), is the Casimir effect [18], which remains the only known laboratory manifestation of the quantum vacuum causing a detectable interaction between macroscopic bodies.Here, spatial boundary conditions limit the mode spectrum of vacuum (and thermal) electromagnetic fluctuations.Like in the CCP, one needs to introduce a regulator when calculating the otherwise divergent quantum vacuum energy E Q (κ) in a given volume subject to boundary conditions.Contrary to the CCP, the dependence on the regulator κ can be circumvented by subtracting the corresponding quantum vacuum energy in this volume without boundary conditions E Q,0 (κ) giving the renormalized Casimir energy For the ideal case of two infinitely extended, perfectly conducting, parallel plates at distance a and zero temperature, one obtains for the energy per unit area A [18].Real materials show dispersion and their response to electromagnetic fields typically falls off ∝ ω p /ω −2 for high frequencies ω and a plasma frequency ω p , eventually leading to transparency in the far UV.Spectral dielectric properties, curved geometries, roughness, temperature, dynamical effects, and further experimental details can also be considered [19][20][21] and lead to further corrections to (4).The penetration of electromagnetic modes and their energy into a surface can be described by the effective penetration depth δ p ∼ c/ω p of a material.For metals δ p ∼ 100 nm.
While the theory of the Casimir energy is well understood, computation of the local energy density ρ C poses problems [22,23].The assumed step in the dielectric functions between the gap and the bounding surface leads to divergences ∝ z −4 , depending on the distance z from the (Dirichlet) boundary [24,25].One possible solution is the introduction of soft [24,26,27] or movable [28][29][30][31] walls that smooth out the discontinuity, and thereby eliminate divergences [32].However, lacking a detailed first-principles approach, quantitative predictions strongly depend on the assumptions taken.The same is true for the region inside the walls [33].Independent of these details, however, ρ C can be expected to be large near the boundaries, for which it could potentially contribute to gravitational interactions [34].Up to now, experiments have tested configurations in which the Casimir force exceeds the gravitational interaction by several orders of magnitude.The upcoming Cannex setup [35], however, will for the first time probe a regime, in which Casimir and gravitational interactions are of equal strength.This opens up the exciting possibility of testing gravity in physical conditions with modified vacuum energy density.

C. Parametrizing question H Q↔Λ
Despite their suggestively analogous names, it is not clear whether the cosmological vacuum energy density ρ Λ 0 and the quantum energy density ρ C are related at all, i.e. if the CCP actually exists.To test possible relations between the two densities, we formulate the following working hypothesis: The net cosmological energy density ρ Λ is a function of the bare cosmological energy density ρ Λ 0 and a quantum contribution ρ C .For small ρ C this relation can be linearized The free parameter of this hypothesis is α.Since ρ C arises from (electromagnetic) vacuum fluctuations, which should fully contribute to a total energy density, even values such |α| = 1, are reasonable, leading to the CCP.This is, for example, the case in well-known cosmological models [36].In contrast, topological arguments of loop diagrams [37] might suggest that α is strongly suppressed, or zero.Thus, a laboratory test of the hypothesis (5) would give valuable information in this respect.Note that ρ Λ 0 might change on cosmological time-scales.However, for the time-scales involved in an experimental setup, such a dependence is negligible.

D. Scale-dependent gravitational couplings: H Λ↔G
The gravitational couplings G(k) = G( x) and Λ(k) = Λ( x) have to be generalized to SD quantities as well.Such couplings can be dealt with in a theoretical framework, known as SD gravity.[38] SD couplings are common in the effective field theory approach to quantum gravity [10][11][12][13]39].In these approaches, the functional form of G(k) and Λ(k) is determined by renormalization group equations, analogous to most other quantum field theories [40,41].The usual known values of these couplings in a weak curvature expansion are then associated with the asymptotic values Even though a uniquely accepted running of gravitational couplings is not available, we can extract useful information from the SD of gravitational couplings in the close vicinity of the deep infrared (IR).In this regime one can expect power-law running of the gravitational couplings [42].Thus, we can expand the SD gravitational couplings around k 0 = 0 [43,44] 7) Here, C i parametrize the first effects of running couplings, when we depart from the classical IR limit.This expansion is in the same spirit at the spirit of the effective field approach presented in [45].Even quite different approaches to quantum gravity (e.g.[46]) can, in principle, be mapped to Eqns.(7,8), resulting in different values for these parameters.Depending on the theory the C i can be small or large [47].Up to now, no experimental limits exist.To extract quantitative predictions from SD couplings like Eqns.(7,8), we need to set the RG scale k in terms of physical parameters '( x, a, . . .)' of a given observation (or experiment) like energy, mass, or plate distance k → k( x, a, . . .).This crucial procedure is known as scale-setting.There exists a large variety of scalesetting methods in conventional quantum field theory [48][49][50][51] and in quantum gravity [52][53][54][55][56][57][58][59][60][61].The applicability of these methods depends strongly on the physical and experimental context.Scale-setting also plays an important role in the attempts to solve the cosmological constant problem [62] (see also [42,60,63,64]).Note, that in its "Critique of the Asymptotic Safety Program" Donoghue raises the question, whether SD couplings are a universal concept (meaning that C i are universal constants) or whether this running is "only" subjective for a particular observable, in which case the constants C i are only applicable for a specific observable [65].Either way, universal or subjective, obtaining observable input for these parameters is a novelty.Independent of which quantum gravity model gives rise to Eqns.(7,8), or which scale-setting method is used, after the scale-setting the SD couplings can be written as local quantities.Thus, the Einstein field equations only remain consistent if they are generalized for SD couplings [52].These read where Note that already at this point it is possible to establish a relation between H Q↔Λ and H Λ↔G .The seemingly innocent relation (5) has a notable consequence.Since the Casimir energy density is a function of external dimensionful quantities and local coordintes x, the same must be true for the resulting ρ Λ .This dependence on local and global scales k ≡ k( x) renders ρ Λ (k) = ρ Λ ( x) a scale-dependent (SD) quantity.Consequently, the definition in terms of the gravitational couplings (1) has to be generalized to allow for SD III. RESULTS

A. Weak modifications of the Netwon potential
Now, we explore the Weak Gravitational curvature and Weak SD limit (WG-WSD) of (9).For this purpose, we isolate the Ricci curvature tensor on the left-hand side 12) The WG-WSD is achieved by an expansion in formally small deviations from the flat Minkowsi background.The line element is expanded with the parameter ǫ Φ and the SD gravitational coupling is expanded with the parameter ǫ G .For both bookkeeping parameters (ǫ Φ , ǫ G ) we consider where (Φ, Ψ, Ξ) are small deformations (potentials) of the flat metric and ∆G ≪ G 0 is the SD correction to the gravitational coupling G 0 .Further, for non-relativistic matter at rest with energy density ρ M = ρ M (r, θ, φ), the stress energy tensor in spherical coordinates is With this and the expansion (13), the time-time component of Eqn. ( 12) reads where a global factor of −ǫ Φ has been canceled.This expansion is valid and physically reasonable in a regime where ǫ Φ > ǫ G > ǫ 2 Φ .Note that we also expect contributions ∼ ∆G(k)ρ M to Eqn. (15).Such contributions do exist, but they are much smaller than the leading contributions shown in Eqn.(15).Next, we drop the formal expansion parameters, keeping in mind the smallness of the ∆G contribution.For all practical purposes in the context of Casimir experiments, the cosmological term can be neglected with respect to the other terms.Thus, the IR correction to the gravitational coupling is given in terms of the local Casimir energy density ∆G(k) = ∆G(ρ C ( x)) ≡ ∆G( x).Now, the Poisson equation ( 15) can be solved by the usual Green's function method where V 1 is the region of the gravitational source.Here, we defined the apparent gravitational energy density When we insert the result ( 16) into the geodesic equation for a test particle with position x µ , we find, for the spatial components, in the non-relativistic limit In order to relate this acceleration to a force in Newton's second law, we have two mass densities at our disposal.The first one is the 'original' mass density ρ M /c 2 , corresponding to the mass we find in the absence of ∆G: The second is the apparent gravitational mass density (17), which extends the first one by the influence of the local vacuum energy density, and thus is the one that has to be used when calculating the gravitational force caused by one object on another, F 12 = − F 21 .To leading order in (ǫ G , ǫ Φ ), the force, sensed by an extended body with volume V 2 is given by The crucial difference between Eqn. (19) and the usual expression for the gravitational force between extended bodies is the distinction between ρM and ρ M , which only appears for ∇ 2 ∆G 0.

B. Density scale-setting: linking H Q↔Λ and H Λ↔G
To relate an effective action to real observables, one has to choose the RG scale in terms of variables that describe the system under consideration.Since these variables, are in many cases local.Thus, the scale-setting can imply the breaking of local symmetries, unless one takes particular care throughout the scale-setting procedure.Below, we show how a scale-setting based on the definition of density and a manifestly covariant scale-setting, both leading to the same type of expression.
Since we are interested in the leading corrections to the asymptotic limit (6), we insert Eqns.( 7) and ( 8) into the definition (11) and expand to first order in Here, the second term is the aforementioned correction to the asymptotic definition (1).By construction, Eqn. ( 20) is equal to ρ Λ defined in Eqn.(5).Thus, by subtracting Eqns.( 5) and (20), we obtain the unique scale-setting that is consistent with the working hypothesis in Eqn.(5), Reinserting this into the IR expansion (7), we find that the gravitational coupling inherits a weak dependence on the electromagnetic Casimir energy density Comparing Eqn.(22) with the weak SD expansion in Eqn. ( 13), we can identify This correction has to be inserted into Eqn.( 16) when calculating the modified gravitational potential or the induced force between two objects according to Eqn. (19).The three phenomenological parameters in the following discussion will be (α, C 1 , C 3 ).The density induced scale-setting combines the expansions (7, 8), which are consistent with covariant equations, with the relation (5), which is explicitly not covariant.This is OK as long as one understands (5) as a matter contribution to the equations of motion.There is, however, a way to formulate the spirit of equation ( 5) in a covariant language.This will be discussed in the next subsection.

C. Covariant scale-setting: H Q↔Λ and H Λ↔G unified
For this we give the effective action of SD gravity coupled to matter.In the low curvature expansion it can be written as Here, L m (φ, k) is the scale-dependent effective Lagrangian of Standard Model fields φ.For the Casimir effect, the relevant field is the electromagnetic U(1) gauge field.In the spirit of the expansion (7,8) in the gravitational sector, we can also expand the electromagnetic Lagrange density 8) Here, the densities of the effective electromagnetic Lagrangian are Here, we defined the usual (one-loop) prefactors [66] Now, the expansions (25, 26) replace the hypothesis (5) and the coefficients α i take the role of the parameter α.To avoid confusion with the notation, remember that α EM is the electromagnetic coupling constant.With In terms of the action, the cosmological constant problem becomes the question "how and to which extend quantum modes of L m (φ, k) contribute to Λ(k)".This question is now parametrized in terms of α i .Varying (24) with respect to the metric field g µν gives rise to the field equations (9).It is not clear whether general covariance is broken by quantum gravity [67], but it is certainly a feature, we would like to perserve.The general covariance of the system, even after the scale-setting, can be assured by the variational scale-setting prescription [55], which complements (9) with the condition Inserting the IR expansions (7,8,25) into the covariant scalesetting condition, we can solve (28) for the optimal scale In the second line we have neglected the ∼ R and higher order ∼ (C 2 , C 4 ) contributions.Interestingly, the only local contribution to this optimal scale, comes from the matter Lagrangian L m,1 (φ).For the purpose of the Casimir experiment, we are not interested in the dynamics of the electromagnetic field itself.Therefore, the electromagnetic Lagrangian enters the equations only in terms of a local background value.This background value is obtained from a summation over all modes consistent with the Casimir boundary conditions Here, we have used the fact, that in the Casimir setting the magnetic modes do not contribute B 2 = B • E = 0, while the summation over all electric modes gives E 2 /2 = ρ C (x).Note that using this summation, subject to the boundary conditions, breaks Lorentz invariance.This is, however, not a problem since it is a direct consequence of the boundary conditions of the experimental setup.With this, the local background value of the optimal scale is In this relation, we have summarized all constant terms into k 2 0 .This constant k 2 0 will not contribute to the equations of motion.The value (31) can then be used when solving the field equations ( 9), or their non-relativistic approximation (19).For (17) we need ∇ 2 ∆G.With ( 31) and ( 7) we find Now, we can compare the result of this the formal covariant scale-setting (32) with the result of the density-driven scalesetting (23).It turns out that in the end, the local part of the covariant scale-setting corresponds exactly to the density scalesetting (21) if one relabels the proportionality constant of our hypothesis This is a remarkable result.It shows that by assuming scaledependence of all couplings (7, 8, and 25) in the effective action, combined with the covariant scale-setting (28) gives the same result as the combination of the hypotheses H Q↔Λ and H Λ↔G .The hypotheses can thus be covariantly unified into a single hypothesis of a scale-dependent Lagrangian with covariant scale-setting (28).Thus the two hypotheses look independent, but they are essentially the same concept.The result (23) is the product of a covariant scale-setting procedure (28) combined with covariant field equations ( 9) which are then applied to the non-covariant boundary conditions of the Casimir problem.Such a covariant method might also be applicable to different systems, e.g. with a mild time-evolution (running) of the cosmological vacuum energy, which has been explored in [68,69].

D. First estimate of the experimental reach
Now we discuss the perspective for experimental tests of the hypothesis (5).The three phenomenological parameters in this discussion are {α, C 1 , C 3 }.From the results in Eqns.(19 and (23)) it is clear that any Casimir experiment, which is sensitive enough to also measure the gravitational attraction between the two plates [35], will be suited to resolve, to some extend, the difference between ρ M and ρM .This will then allow to determine, or constrain α, C 1 , and C 3 .To obtain a glimpse on the experimental relevance of this effect, it is necessary to make further assumptions about experimental details.For the force between the plates, we have to integrate in Eqn.(19) over the volume of the two plates.Novel corrections to this force arise from non-vanishing ρ C (z) inside the plate material, while the functional form of this energy density is irrelevant for this purpose.When entering the plates, the vacuum energy density is assumed to drop exponentially from a starting value ρ C,id to zero Here, ρ C,id = σ C,id /a, is estimated by the average of the ideallized Casimir energy density (4).Since Eqn. ( 34) is a bold simplification of a still unknown, but likely more complicated functional form of ρ C (z) [25-27, 32, 33], we defer a detailed numerical integration of Eqn.(19).Instead, we revisit the WG-WSD assumption of Eqn.(13) to get a feeling of what we could expect to find for the phenomenological parameters.By inspecting the result (19), it is reasonable to assume that the integrated energy density of the matter material ρ M is bigger than the correction due to the integral over SD of the gravitational coupling.Thus, Using Eqns.( 23) and ( 4), and realistic exemplary values of (z = a/2, a = 10 −5 m, ρ M /c 2 = ρ gold /c 2 = 19.3g/cm 3 /c 2 , δ p = 10 −7 m, D = 0.01 m) in the above expression, we find that ???ben is here Note that δ p and the boundary value ρ C,id refer to the assumed exponential attenuation model in Eqn.(34), which is subject to significant theoretical and experimental uncertainty.Nevertheless, the skin depth was investigated quantitatively in experiments [70].Moreover, since our assumption of a constant ρ C between the plates and an exponential fall-off within the boundary underestimates the true volume where ρM 0 (i.e.where the SD modification of the gravitational force is sourced), the bound (36) is a worst case estimate.This implies a very strong impact on the parameter space {α, C 1 , C 3 }.A more realistic modeling of the experimental situation in this, and other configurations, such as the measurement of the potential Φ(z) [cf.Eqn. ( 16)] with test particles, will be part of our future projects.
IV. FINALLY

A. Discussion
We have shown how to obtain novel experimental insight into the possible connection between the quantum vacuum energy and the energy density corresponding to the cosmological coupling Λ.
The sensitivity in Eqn. ( 36) is, despite of its large uncertainty due to the simple assumed model for attenuation inside the material, overwhelmingly strong when we compare them with standard quantum gravity corrections to the Newtonian potential [71].These leading corrections are typically suppressed by the extremely small factors r S /a or λ 2 p /a 2 , where r S and λ p are the Schwarzschild radius and the Planck length, respectively, and a is a typical length scale.Small corrections imply that a phenomenological pre-factor α (name chosen analogous to our α) of such corrections would be allowed to be very large The reason for the discrepancy between the usual expectation Eqn.(37) and the strength of our result Eqn. ( 36) is fourfold: i) Comparable order of ρ Λ 0 and ρ C .
ii) The SD quantities ρ Λ ↔ Λ(k) ↔ G(k) are linked through the hypothesis H Q↔Λ and H Λ↔G .
iii) ∆t µν contribution to the equations of motion.iv) ∆t µν enhanced for small skin depth.
None of these four aspects are considered in the usual estimate [71].Note that i-iv are not independent ad-hoc assumptions, but are rather natural consequences of the hypothesis (5).
We will interpret the result (36) in inverted order of the items above.iv) Eqn.(36) has to be interpreted with care since Eqn.(34) has large theoretical uncertainties.
iii) The result (36) would have to be recalculated if the ∆t µν term was absent from the modified field equations ( 9), or if there would be additional non-minimal terms.
i) Finally, except for the scenarios iv)-ii), there remains the possibility that Eqn.(36) provides an opportunity to experimentally test the relation between the quantum and cosmological energy densities, and thereby to possibly gain insight on the CCP (2).
What can an experimental sensitivity of α ≈ 10 −30 teach us about the CCP (2)?The CCP arises from the ambition to predict ρ Λ in terms of ρ Q such that ρ Λ = ρ Λ (ρ Q ).Without loss of generality we can write this ambition in factorized form as In terms of Eqn.(38) the CCP is the statement that Υ 0 is an extremely small number (2).The fact that Υ 0 is a small number, as measured in cosmology without additional Casimir energy contribution (i.e.ρ Q = ρ Q,0 ), does not imply that it is constant.It could be a function Υ = Υ(ρ Q ).According to the definition (3), the quantum vacuum energy density ρ Q , in turn, is for sure changing with additional small Casimir contributions and hence Υ 0 = Υ(ρ Q,0 ).We define the dependence of the CCP on changes in the quantum energy density (39) in terms of the logarithmic derivative Inserting Eqns.( 2) and ( 5) into Eqn.(40) allows us to relate the observable α to Υ 0 and Υ ′ 0 via This relation clearly states that experimental insight on α coming from Casimir experiments can, without knowing Υ ′ 0 from some theory, not give unambiguous information on the CCP.This may come as no surprise, as Casimir experiments probe a difference in ρ Q according to Eqn. (39), while the CCP is caused by the absolute values of ρ Λ (and ρ Q ).If such theoretical connection Υ ′ can be established, however, direct experimental investigations of the CCP by means of force metrology would be possible.

B. Clarifying comments
• The result isn't Lorentz invariant, is this contradictory or inconsistent?One has to distinguish at which level the fundamental symmetries like Lorentz invariance are broken.While breaking this symmetry at the level of the equations of motion is typically considered conflictive, a breaking at the level of solution and its source terms is totally OK.Our results are derived from the Lorentz invariant system of equations ( 9) and (28).Breaking of Lorentz symmetry occurs only at two points.First, when going to the Newtonian limit and second by considering a particular background configuration for the optimal scale (30)."No, there is no inconsiteny with Lorentz symmetry, since the breaking only occurs at the choice of the source term of the physical system under consideration." • The result violates general covariance, is this contradictory or inconsistent?
The theory (24) and its equations ( 9) and ( 28) are invariant under general coordinate transformations."Like explained in for the previous comment, a symmetry breaking at the level of (approximate) solutions is not in conflict with the consistency of the theory.
• Why should the two hypotheses H Q↔Λ and H Λ↔G be combined?
The two hypotheses seem like a combination of independent assumptions.The question arises, why should such assumptions be combined.However, as shown in subsection III C, H Q↔Λ and H Λ↔G have the same origin."The two hypotheses are just to faces of a single concept, namely covariant scale-dependence with covariant scale-setting." • What if α 1 = 0? It could happen, that the leading scale-dependence parameters of some of the expansions are actually exactly zero (e.g.α 1 ).In this case, the expansions have to be continued to the subleading terms such as for example α 2 , α 3 ."We have to consider subleading coefficients."

C. Conclusion
We explored the hypothesis (5) that the cosmological energy density ρ Λ is influenced by changes in the quantum energy density in terms of the Casimir vacuum energy density ρ C (H Q↔Λ ).The local nature of ρ C made it then inevitable to introduce SD to the gravitational couplings.This SD of the vacuum energy density in the gravitational sector, is then related to the frequently used link between the value of the gravitational coupling and the cosmological coupling, which is our second hypothesis (H Λ↔G ).In subsection III C we showed, that the same result can be obtained from the concept of universal covariant SD.SD, when minimally combined with diffeomorphism invariance, then led in the WG-WSD limit to a modification of the gravitational potential (16).
This means, that experiments, which are sensitive to both the gravitational force (19) and the Casimir force, have the potential to actually test the hypotheses.The scope of such tests is exemplified in the bound (36).Naturally, this inequality depends on the parameters of both hypothesis, namely α for H Q↔Λ and C i for H Λ↔G .Eventually, our results may lead to new insight on the CCP (41).

H
Q↔Λ : Is the quantum vacuum energy density linked to the cosmological energy density?H Λ↔G : Does a change of the gravitational coupling G imply a change in the cosmological coupling Λ and vice versa?II.INTRODUCTION A. Vacuum energy and the cosmological constant