Blowing in the Dark Matter Wind

Interactions between dark matter and ordinary matter will transfer momentum, and therefore give rise to a force on ordinary matter due to the dark matter `wind.' We show that this force can be maximal in a realistic model of dark matter, meaning that an order-1 fraction of the dark matter momentum incident on a target of ordinary matter is reflected. The model consists of light ($m_\phi \lsim \text{eV}$) scalar dark matter with an effective interaction $\phi^2 \bar{\psi}\psi$, where $\psi$ is an electron or nucleon field. If the coupling is repulsive and sufficiently strong, the field $\phi$ is excluded from ordinary matter, analogous to the Meissner effect for photons in a superconductor. We show that there is a large region of parameter space that is compatible with existing constraints, where the force is large enough to be detected by existing force probes, such as satellite tests of the equivalence principle and torsion balance experiments. However, shielding of the dark matter by ordinary matter prevents existing experiments from being sensitive to the dark matter force. We show that precise measurements of spacecraft trajectories proposed to test long distance modifications of gravity are sensitive to this force for a wide range of parameters.


Introduction
Understanding the microscopic nature of dark matter is one of the major open problems in particle physics and cosmology.Dark matter has been detected only through its gravitational interactions, but additional interactions are needed in order to explain its relic abundance, giving hope that we can find additional signals of dark matter in the laboratory or in astrophysical or cosmological observations.The allowed dark matter masses and interactions span a vast parameter space, and it is important to carry out experimental searches for as wide a range of models as possible.
In this paper, we consider a new probe of dark matter: the force on ordinary matter due to an incident flux of dark matter-the dark matter 'wind.' Due to the motion of the solar system in the galaxy relative to the dark matter halo, the local dark matter is moving relative to the solar system with an average speed v DM ∼ 300 km/s ∼ 10 −3 c.We can estimate the maximal force that could be exerted by the dark matter wind by assuming that the full flux of dark matter on an object is reflected by the object.If the dark matter interacted strongly with a macroscopic target with cross-sectional area A, the force exerted by the dark matter on the target is given by where ρ DM ∼ 0.3 GeV/cm 3 is the dark matter density near the earth.The corresponding acceleration of a solid sphere of radius R and density ρ is (1.2) These accelerations are large enough to be detected using sensitive low-frequency force probes, such as those used to test the equivalence principle [1][2][3].On the other hand, this interaction also means that any surrounding ordinary matter (including the earth's atmosphere) can partially or completely shield the force probes from the dark matter.We will see that this effect makes existing experiments insensitive to the dark matter force.We will comment on the possibility of future experiments that can detect the effect.
The interactions of dark matter with ordinary matter are strongly constrained by direct dark matter searches, as well as cosmological and astrophysical constraints.Nonetheless, we will show that there is a model of dark matter where the force on ordinary matter can be maximal, and which is compatible with all these constraints.In this model, dark matter is described by a light scalar field ϕ with an effective coupling where ψ is an electron or nucleon field.The absence of a Yukawa coupling ϕ ψψ can be explained by a Z 2 symmetry under which ϕ → −ϕ.In a region containing matter, we have ⟨ ψψ⟩ = n ψ in the non-relativistic limit, where n ψ is the number density of ψ particles1 .Therefore, this interaction gives an additional contribution to the dark matter mass inside matter: (1.4) We will consider the case f ψ > 0, so that ∆m2 ϕ > 0. Inside matter, the relation between the energy and momentum of a ϕ particle is modified: If ∆m 2 ϕ is sufficiently large, then the propagation of dark matter is suppressed inside matter.For a ϕ particle in vacuum with momentum k incident on a region of ordinary matter, energy conservation gives and we see that k 2 matt < 0 for ∆m 2 ϕ > k 2 vac .If this is satisfied, then the matter region is classically forbidden, and the propagation of the dark matter is exponentially suppressed.This is a coherent scattering effect that is important when the density of scatterers is large compared to the de Broglie wavelength of the dark matter.It is analogous to the Meissner effect for photons in a superconductor, so we call this the 'dark matter Meissner effect.' The shielding of the field ϕ in this model by ordinary matter was previously discussed in [4,5].However, they did not discuss the force from the dark matter wind that is the focus of the present work.The dark matter force on planets in this model was considered in [6], but there result included a large coherent enhancement factor that we believe is not present.
We will show that there is a large region of the parameter space of this model where the dark matter Meissner effect takes place, which is compatible with existing laboratory and astrophysical/cosmological constraints.This opens the exciting possibility that it may be possible to directly measure the force exerted by dark matter on ordinary matter.We emphasize that here we are discussing the average force due to the motion of the dark matter, rather than the oscillatory force due to coherent ϕ oscillations with frequency ω = m ϕ , which has been previously considered in the literature [7,8].We expect that we can approximate the wind force as time-independent if we average over time scales longer than the coherence time of these oscillations, which is of order . (1.7) The strong interaction between ordinary matter and dark matter makes the detection of the dark matter wind force possible, but it also means that ordinary matter can act as a shield for the dark matter wind.Because of this shielding effect, we are not able to identify any existing force experiments that are sensitive to this force.We will give estimates for the size of the force, and discuss some possible future experiments that may be sensitive to it.Further work is needed to design an experiment that is sensitive to this force.This paper is organized as follows.In §2, we discuss the physics of the dark matter Meissner effect.In §3 we review existing constraints on the model.In §4 we give estimates for the size of the force on potential experimental targets.Our conclusions are given in §5.Appendices give additional details of our calculations, and discuss UV completions and fine tuning.

The Dark Matter Meissner Effect
In this section we discuss the dark matter Meissner effect in more detail.We begin with some simple estimates for the the parameter regime where the effect takes place and can lead to a maximal force on a target.We then explain the methods used to perform precise calculations of the force, and give some parametric estimates for various limiting cases.

Estimates
In the presence of matter, coherent scattering effects can be important if the density of scatterers is sufficiently high.For example, for the earth's atmosphere, the ratio between the DM de Broglie wavelength and the distance among two atoms is roughly given by λdB n where λdB = 1/m ϕ v ϕ is the reduced de Broglie wavelength of the dark matter particles in vacuum and n atm is the number density of nucleons near the earth's surface.As long as this ratio is large, we can approximate the matter as continuous.Inside matter, we then have where n ψ is the number density of ψ particles (nucleons or electrons) in matter. 2 Therefore, as discussed in the introduction, dark matter will be excluded from regions of sufficiently high density if ∆m 2 ϕ > k 2 vac , where ∆m 2 ϕ is the matter contribution to the dark matter mass.This condition is satisfied for This can be written more intuitively as λdB > ∼ L skin . (2.4) If this condition is satisfied, then the propagation of dark matter into the target has an exponential suppression ∼ e −d/L skin , where d is the distance into the target and3 is the skin depth associated with the dark matter Meissner effect.In order for the force to be maximal, for a target of linear size R we also require so that the exponential suppression causes an order-1 fraction of the dark matter to be reflected from the target.If Eqs. (2.3) and (2.6) are both satisfied, we expect the force to be maximal, so that the estimate Eq. (1.2) holds.
Assuming we are interested in experimental targets with the density of ordinary matter (∼ g/cm 3 ) and with size ∼ 1-10 cm, the region of parameters that we can hope to probe is roughly (see Fig. 4) (2.7) The upper bounds on m ϕ and 1/f come from the requirement that the force be maximal; the lower bounds come from constraints on the model from nucleosynthesis and supernova cooling.These constraints will be reviewed in §3.

Quantitative Calculations
We now discuss how to perform quantitative calculations of the force due to the dark matter wind.We consider a monochromatic wave of dark matter incident on a target localized at the origin.The force on the target can be computed using one of two approximations.In the first, we approximate the dark matter as a classical field and find the classical scattering solution for the field in the presence of the target.We can then compute the force on the target by computing the momentum transferred to the target by the field.The classical field approximation is valid for m ϕ ≪ 10 eV, which is satisfied in most (but not all) of the phenomenologically interesting parameter space of our model.In the second approximation, we treat the incident dark matter wave as a superposition of ϕ particles, and compute the scattering probability of these particles from the target using non-relativistic quantum scattering theory.We can then compute the force by adding up the momentum transferred to the target by each scattering particle.In Appendix A, we show that these two methods give identical results in the non-relativistic limit.
We begin with the classical field picture.We assume that far from the target the solution is a plane wave in the +z direction: where is the frequency in vacuum.To compute the force, we consider a steady state solution of the form ϕ(r, t) = Re e −iωt ψ(r) . (2.10) The equations of motion of the field in the presence of the target give which we can rewrite as where This is the time-independent non-relativistic Schrödinger equation that describes scattering solutions for a ϕ particle with potential V eff (r).We are interested in the non-relativistic case (k ≪ m ϕ ), but it is interesting that we obtain the non-relativistic Schrödinger equation even if the system is relativistic.More importantly for us, this allows us to use standard results from quantum mechanical scattering theory in the classical calculation, and will be used to show that the quantum and classical calculations predict the same time-averaged force.
We illustrate this by consider a potential with spherical symmetry.We expand the wavefunction in spherical Bessel functions and Legendre polynomials (see Appendix A): Here S ℓ is the quantum-mechanical S-matrix element for the partial wave with angular momentum ℓ, given by where δ ℓ is the phase shift for the partial wave ℓ in the effective potential Eq. (2.13).The result is that the time-averaged force on a target is given by where is the density of dark matter.Note that the force vanishes in the no-scattering limit S ℓ → 1, as it must.The derivation of this result is given in Appendix A for both the classical and quantum pictures.Formulas for S ℓ for various geometries are given in the appendix: see Eq. (A.3) together with Eq. (A.22) (for a solid sphere), Eq. (A.25) (for a hollow sphere).
We now use this result to understand how the force depends on the three length scales in the problem: the reduced de Broglie wavelength λdB , the skin depth L skin = (f /n ψ ) 1/2 , and the size of the target R.We first consider the limit ( The first inequality is equivalent to V eff ≫ k 2 /2m ϕ , the strong potential regime.Because λdB ≪ R, we are in the classical geometric limit where the we can compute the force assuming that the dark matter is made of classical particles that scatter elastically from the target.This is the case where the naive picture used to derive the estimate Eq. (1.1) for the force is quantitatively correct.For example, for a spherical target of radius R, we have Next we consider the regime We have kR ≪ 1, so we have s-wave scattering, and we are still in the strong potential regime.For a spherical target of constant density, we have the textbook quantum 'hard sphere scattering' problem.The relevant phase shifts are given by δ 0 = −kR, δ 1 = O (kR) 3 , and the force is given by Once again, this agrees with the estimate Eq. (1.1).Note that the force is 4 times larger than the classical force in this limit; this is the famous factor of 4 enhancement of the s-wave cross section compared to classical geometric scattering.
Next we consider the regime Table 1: Various parametric limits of the drag force on a target of linear size R.
Here F/F classical is the ratio of the dark matter force to the force for classical elastic scattering from the target.The first three entries are for the strong scattering limit (L skin ≪ λdB ) while the last is for the weak scattering limit.All estimates are in the coherent scattering limit where the de Broglie wavelength of the dark matter is large enough that we can approximate the target as a continuous matter distribution.
Because we have R ≪ L skin , the target does not strongly affect the incoming wave, and we can use the Born approximation.In this case, the scattering cross section and the force are proportional to V 2 eff ∝ 1/f 2 .Therefore, the scattering is highly suppressed compared to the strong potential regime, where the cross section is independent of f .For a sphere of radius R with kR ≪ 1, the Born scattering amplitude is given by where q = 2k sin(θ/2) is the momentum transfer.The force is then given by (2.24) Note that the ratio to the maximal force is of order (R/L skin ) 4 ≪ 1.
In the weak potential regime V eff ≪ k 2 /2m ϕ (or L skin ≫ λdB ) we can again use the Born approximation, and the force is again suppressed by 1/f 2 .We summarize these results for a general target of size R in Table 1.To illustrate this, in Fig. 1 we plot the ratio R z between the force on a solid aluminum sphere and the classical particle limit where each dark matter particle collides elastically with the sphere.

Shielding Effects
As we have already stated, ordinary matter can shield test masses from the dark matter wind.We now give some quantitative estimates of this effect.We focus on the strong potential regime L skin ≪ λdB where shielding can be significant.
We first consider a simple geometry where a plane wave of dark matter in vacuum with momentum k = k z ẑ + k y ŷ is incident on an infinite solid plane of thickness ∆R occupying  1: Ratio R z between the force on a solid sphere of aluminum of radius R and the classical particle limit.We fix f = 10 10 GeV, corresponding to a skin depth L skin = 0.56 cm, and we have plotted the ratio for m ϕ = (10 −1 , 10 −2 , 10 −3 , 10 −4 ) eV, which corresponds to the reduced de Broglie wavelength λdB = (0.2, 2.0, 20, 200) cm. the region 0 < z < ∆R.The steady state solution for the wavefunction is where Matching the wavefunction and its first derivative at the boundaries determines the unknown coefficients R, A, B, and T , and gives the for the transmission probability For a plane wave in the z direction we have κ = 1/L skin , and we obtain (2.28) As expected, the transmission coefficient is exponentially suppressed for ∆R > ∼ L skin , and is is close to 1 if ∆R ≪ L skin .This result can be used to approximate the shielding effects if the shield can be approximated as a plane on the scale of the reduced de Broglie wavelength λdB .
We comment briefly on how these results are affected by averaging over the dark matter distribution in our galaxy.We use the standard halo model, where the dark matter has a Maxwellian distribution cut off by the escape velocity: where u is the dark matter velocity in the galactic rest frame.We use σ u = 290 km/s, u esc = 550 km/s.We are interested in the dark matter force on an object at rest in the solar system where the average dark matter wind velocity is v wind ≃ 220 km/s.This is proportional to (2.30) The averaging therefore increases the force by a factor of roughly 1.5.The force exerted by the dark matter on the far side of the barrier is reduced by the ratio This is illustrated in Fig. 2 along with the approximation where |T | 2 is computed for a pure plane wave in the z direction with velocity v wind .We see that the averaging reduces the effect of shielding.These effects are relevant for detailed predictions for experiments, but will not be included in the present exploratory work.
If λdB ≫ R, the shield cannot be approximated by a plane.This limit is relevant only for the very lightest dark matter masses in the phenomenological regime of interest.To investigate this regime, we consider a shield consisting of a spherical shell of of radius R and thickness ∆R ≪ R, surrounding a test mass at the center.To quantify the importance of the shielding, we compute the momentum density of the dark matter at the center of the sphere, ⟨T 0z ⟩ r=0 .This gives the strength of the dark matter 'wind' seen by a target at the center of the sphere.The solution is given in Appendix A (see Eq. (A.27)).In the limit of large λdB , the solution is dominated by the lowest partial wave.The ratio R of the momentum density The ratio R z of the dark matter force on either side of a 1-dimensional aluminum barrier of thickness ∆R that is normal to the average direction of the dark matter wind (see Eq. (2.31)).We take m ϕ = 10 −4 eV and f = 10 12 GeV, which imples L skin = 5.6 cm.The black solid line is the result of averaging over the dark matter velocity distribution in the standard halo model, while the dashed blue line is the result for a pure plane wave with velocity 220 km/s.
. at the center to the unshielded value is given by where the coefficients D ℓ are given in Eq. (A.25c).In the limit we find As we expect, the result is exponentially suppressed for ∆R > ∼ L skin , and near 1 for ∆R ≪ L skin .However, the coefficients differ from the infinite wall limit as in Eq. (2.28).Note that for ∆R ≪ L skin , the deviation from 1 is of order 1/L 2 skin ∼ 1/f rather than 1/f 2 because it results from the interference of a weakly scattered wave and the unscattered wave.For the infinite wall, the the deviation from 1 is of order 1/L 4 skin ∼ 1/f 2 because unitarity dictates that |T | 2 + |R| 2 = 1.
In Fig. 2 we show how the ratio R z depends on the thickness of the shielding sphere.The parameters are chosen to approximately match the satellite experiment proposed in Ref. [9], which will be discussed in §4 below.3: Ratio R pz between the z-component of the momentum density inside a hollow sphere and its classical limit as a function of the thickness of an aluminum shell with outer radius R = 100 cm.We have fixed m ϕ = 10 −5 eV (i.e.λdB = 2×10 3 cm) and plotted the ratio with f = (10 11 , 10 12 , 10 14 , 10 16 ) GeV.

Existing Constraints
In this section, we summarize constraints on the dark matter model from existing observations.Much of this section is a summary of previous work, but we also consider additional effects related to the dark Meissner effect that have not been previously considered in the literature; we find that these effects do not affect the existing constraints.
For definiteness, we consider the constraints a benchmark models with effective couplings to nucleons and electrons given by For most purposes, we can assume that there is an approximately equal coupling to both protons and neutrons with coefficient

Supernova Cooling
The first constraint we consider is the cooling of stars due to ϕ emission.Our model contains an irrelevant interaction, so the strongest astrophysical cooling constraint comes from supernova SN1987A, since this has the highest relevant temperature scale (T SN ∼ 30 MeV).The constraint can be approximated using the 'Raffelt criterion' [10], which states that the instantaneous luminosity for new light particles with effective masses smaller than T SN cannot exceed the neutrino luminosity observed by the SN1987A.
For the nucleon coupling, this constraint was estimated in [11], and gives The bound for electron couplings does not appear in the literature, so we derived it using the approximations described in [11].We constrain the production rate by Because of the high density of the supernova core, the mass of the ϕ particles inside the core is much larger than the mass in vacuum.However, this does not affect the bounds in this model because the mass is still small compared to the temperature T SN . 4For example, for couplings to nucleons, we have where we assumed n N ∼ 2 × 10 38 /cm 3 .The effects for the electron coupling are even weaker, since n e /n N ∼ 10 −2 .

Big Bang Nucleosynthesis
Next we consider the constraints from big bang nucleosynthesis.During nucleosynthesis ⟨ϕ 2 ⟩ was larger than it is today, and this affects the proton-neutron mass difference and the electron mass via the couplings Eq. (3.1).The relic abundances of nuclei are very sensitive to these quantities, so this puts a bound on the parameters of the model.The nucleon and electron mass modification is given by where the dark matter density ρ ϕ is fixed by the cosmological evolution.Therefore, nucleosynthesis primarily puts a bound on the parameter combination 1/m 2 ϕ f ψ .These bounds were first obtained in [13], and have been refined in [14,15].For electron couplings, the constraint in the parameter region of interest to us can be summarized as For the nucleon couplings, the bounds are model-dependent: they depend on the form of the couplings of ϕ to quarks and gluons above the QCD confinement scale.The reason is that the nucleosynthesis bounds are primarily sensitive to the neutron-proton mass difference, while the matter effects we are considering in this paper are primarily sensitive to the sum of the proton and nucleon couplings.To illustrate the range of possibilities, we consider two benchmark models, one where the ϕ field couples only to the down quark, and the second where it couples only to gluons: Each of these models can be approximately realized by specific UV completions of the model, as discussed in Appendix B. The nucleosynthesis bounds are weaker for the second model because the gluon coupling contributes to the neutron-proton mass difference only through small isospin-breaking effects.In both models, we have f p ≃ f n ≃ f N , and the respective bounds are (3.9) These constraints are illustrated in Fig. 3 along with the constraints from supernova cooling.

The Dark Drag Force
Because of the dark matter Meissner effect, moving ordinary baryonic matter object will experience a force in the dark matter rest frame that tends to make it come to rest relative to the dark matter.For obvious reasons, we call this the dark drag force.In this subsection, we consider the effect of this force on ordinary matter objects in a galaxy such as the Milky way.
Our galaxy consists of a dark matter halo, with ordinary matter orbiting inside the halo with speed ∼ v ϕ .The dark drag force tends to make baryonic matter come to rest relative to the halo, possibly modifying galactic dynamics in an observable way.For sufficiently large objects, the collective effects become important for the drag force.To estimate this maximum size of this effect, we assume a maximal acceleration given by Eq. (1.2).For a given density ρ of the object, the force is proportional to the area, while the mass is proportional to the Fig. 4: Summary of constraints for couplings of ϕ to nucleons (left panel) and electrons (right panel).The blue regions are excluded by supernova cooling, while the red and green regions are excluded by nucleosynthesis.In the left figure, the red region corresponds to a coupling to gluons, while the green region corresponds to the coupling to the down quark.Above the black dotted line, the dark matter wind is shielded by the earth's atmosphere.The red solid lines give the acceleration of a solid aluminum target with radius 1 cm in the absence of shielding.For comparison, the horizontal blue dashed lines give the skin depth L skin for aluminum.
volume, so the acceleration is proportional to 1/R, where R is the size of the object.The dark drag force is then large enough to slow the object over the lifetime of the galaxy for where we have normalized ρ to the density of ordinary matter5 .(Note that the average density of the sun and Jupiter are both ∼ 1 g/cm 3 , which is not so different.)Such small objects do not play an important role in the dynamics of the galaxy.It is intriguing that small chunks of ice (for example) cannot freely orbit our galaxy, but we know of no observational constraint arising from this effect.

Detecting the Dark Matter Wind
In this section we discuss experimental sensitivity to the dark matter force.We first discuss the magnitude of the force neglecting shielding effects.We then consider a number of existing Fig. 5: Sensitivity on our parameter space from the proposed space mission test of modification of gravity at distances ∼ 10 AU [9], for the nucleon coupling (left panel) and the electron coupling (right panel).We have assumed a ranging accuracy of 10 cm, over the 7 year lifetime of the mission, corresponding to an accuracy on the measurement of the acceleration of δa ∼ 4 × 10 −18 m/s 2 .
force experiments and explain why they are not sensitive due to shielding effects.We then show that the proposed space mission described in ?? is sensitive.Finally, we give a general discussion of some aspects of the signal that may be relevant for new experiments.

Force without Shielding
To illustrate the magnitude of the force neglecting shielding effects, we plot the acceleration of a 10 g spherical aluminum target with radius of 1 cm in Fig. 4, along with the constraints on the model discussed in §3.For comparison, the sensitivity of the Eöt-Wash torsion balance experiment for a similar target (m ≃ 5 g) is δa ≃ 9 × 10 −15 m/s 2 [1,2].The Microscope satellite experiment, which currently puts the strongest limits on violations of the equivalence principle, has a sensitivity of δa ≃ 5 × 10 −14 m/s 2 , but for a larger target (m ∼ 300 g) [16].
The upgraded version of this experiment (Galileo) is expected to increase the sensitivity by another 2 orders of magnitude [17] .We see that without any shielding, the accelerations that can be detected in existing or planned weak force experiments are nominally sensitive to the dark matter wind a wide range of parameter space of the model.

Existing Experiments
We consider force experiments that search for violations of the equivalence principle [1,16,17], since these typically involve very precise measurements of forces on large (cm scale) targets.The gravitational acceleration is independent of the target, while the acceleration produced by the dark matter wind is proportional to the area of the target and inversely proportional to its mass.In this sense, the dark matter wind induces a violation of the equivalence principle.
However, to estimate the signal, we must take into account shielding effects.For example, terrestrial experiments will be sensitive only if the dark matter can penetrate the earth's atmosphere.The atmosphere will block dark matter wind if ∆m This is the region above the dashed black line in Fig. 4. The effects of shielding in an experiment cannot be determined quantitatively without detailed modeling the experiment and its environment.Rough estimates for existing experiments indicate that the shielding is too large for them to be sensitive to the dark matter wind.For example, torsion balance experiments such as the Eöt-Wash experiment [1,2] are performed in a vibration-isolated laboratory surrounded by meters of dense matter.The probe masses in the Microscope experiment [16] are surrounded by instrumentation with thickness ∼ 5 cm, in addition to the shielding due to the surrounding satellite.Simple estimates of the shielding based on simplified geometries such as the ones discussed in §2.3 indicate that these experiments are not sensitive to the dark matter wind.The lesson is that future experiments will need to be carefully designed to ensure that shielding effects are small.

Spacecraft Ranging
One proposed experiment that is sensitive to the dark matter wind is the space mission proposed in Ref. [9].This experiment was designed to test the inverse square law of the gravitational force on a distance scale of 1-100 AU.The spacecraft is sent out of the orbital plane of the solar system, where the gravitational force is dominated by the sun.The force measurement is made by a 'drag free' spacecraft that steers around a proof mass floating in its center.A second spacecraft ∼ 10 km away sends ranging information back to earth to measure the distance from the proof mass to an earth station with an accuracy in the range 10-100 cm.The expected sensitivity to a deviation in the radial acceleration assuming an optimistic ranging accuracy of 10 cm is δa ∼ 4 × 10 −18 m/s 2 .The sensitivity benefits from the fact that the deviation from the expected geodesic orbit builds up over the 7 year run time of the experiment.The drag-free spacecraft has a simple spherical geometry to control various background effects.Using the parameters in Ref. [9] we approximate the satellite as a spherical aluminum shell of radius 1 m and thickness 1 mm.The proof mass in the design is a solid platinum sphere of radius 5 cm, with a mass of ∼ 10 kg.In principle the directionality of the dark matter force could be used to distinguish it from a violation of the inverse-square law for the gravitational force, but this experiment is sensitive only to the distance to the earth, and hence the component of the acceleration along this direction.In Fig. 5 we give the sensitivity to the dark matter wind for the proposed experimental parameters.The force on the proof mass is estimated as follows.For m ϕ < ∼ 10 −4 eV, the de Broglie wavelength is large compared to the size of the spacecraft, and we use the result of Eq. (2.32) for the momentum density inside a spherical shell.For m ϕ > ∼ 10 −4 eV we approximate the shielding effect by approximating the shielding by a planar wall.In the intermediate region, we simply interpolate. 6We see that the experiment is sensitive to a large part of the parameter space due to a combination of minimal shielding and high acceleration sensitivity.

General Signal Characteristics
There are a number of characteristics of the signal that could be used to design other experiments sensitive to the dark matter wind.First, we note that the dark matter wind is known to be coming from the direction of the constellation Cygnus, which is visible in the northern hemisphere.Terrestrial experiments therefore have a daily modulation in the direction of the force, including the disappearance of the force when Cygnus is below the horizon.The reflection of the dark matter from the surface of the earth means that the net direction of the wind is horizontal in the approximation where the only matter near the experiment is the (nearly) flat surface of the earth.However, the dark matter wind will be sensitive to other dense objects nearby, and modeling of the interaction of the dark matter with the experimental environment will be required to determine if there is a signal whose characteristics can be understood.For satellite-based experiments, there is a modulation once per orbit due to the orbit of the satellite.The earth will have a dark matter shadow where the force is suppressed, and reflection from the earth's surface must be taken into account.
In addition, both terrestrial and satellite experiments will experience an annual modulation of the signal due to the earth's motion around the sun.The earth orbits the sun with a speed of approximately 30 km/s, approximately 10% of the average speed of the dark matter wind.The dark matter force is proportional to the square of the wind velocity (see Eq. (1.1)), so this will give rise to a significant modulation in the magnitude of the force.The phase and magnitude of this modulation is known, and provides an additional handle on the signal.

Conclusions
In this paper, we have explored a model of dark matter where the interaction between dark matter and ordinary matter can be maximally strong, in the sense that dark matter scatters elastically from sufficiently dense and large matter targets.The dark matter in this model consists of a scalar ϕ with mass m ϕ < ∼ eV and an effective coupling to nucleons and/or electrons given by Eq. (1.3).This coupling increases the mass of the ϕ particle inside ordinary matter, which suppresses the propagation of dark matter inside the target and can lead to elastic scattering.This is a collective effect due to the coherent scattering of the dark matter from many nucleons/electrons.It is similar to the Meissner effect that gives photons a mass inside a superconductor, so we call it the 'dark Meissner effect.'Because of the rotation of the Milky Way inside the dark matter halo, the dark matter around the earth is moving with an average velocity v ϕ ∼ 300 km/s from the direction of Cygnus.The maximal force from this dark matter 'wind' that arises for elastic scattering is very small (see Eq. (1.2)), but is large enough to be detected in sensitive force measurements.However, the strong interaction between dark matter and ordinary matter also means that the dark matter is shielded from existing experiments.
We have shown how this force can be computed quantitatively, using both the classical and quantum pictures for the dark matter.We present a number of explicit calculations to illustrate how the force depends on the dark matter de Broglie wavelength, target size and density, amount of shielding, etc.These calculations confirm that the dark matter force can indeed be maximal in a wide range of parameters.Based on these estimates, we believe that existing fifth force experiments (both terrestrial and satellite based) are not sensitive to the dark matter force because of shielding effects.We also show that this model is consistent with astrophysical and cosmological constraints in a region where the induced acceleration on the test object with radius of cm is within the sensitivity of current technology as long as the shielding effect can be reduced (see Fig. 4).This leads us to consider a novel experimental signal of dark matter: multiple elastic scatterings of dark matter from ordinary matter that are accumulated during a long period of time, leading to a collective force that can be detected using sensitive force experiments.This is a unique signal with many characteristics that can be used to distinguish it from backgrounds: the force has a known direction, annual modulation due to the earth's orbit, and time dependence due to the earth's shadow.Perhaps most uniquely, the force is proportional to the area of the target, and dark matter can be shielded and/or controlled by ordinary matter.Detecting such a signal would not only give direct evidence of dark matter, but also information about its local velocity distribution.
To detect this signal, one needs a sufficiently sensitive force probe that is not too shielded from the dark matter wind.The size of the force to be measured is larger than forces already probed in these experiments although special design should be explored to reduce the shielding effect, for example thinner shielding made with low density materials.As an example, we have presented the prospective sensitivity on our parameter space (see Fig. 5) from proposed space mission test of long-distance modification of the gravity inverse-square law by Ref. [9], which consists of an aluminum shell of thickness 1 mm with radius 1m.We hope that our results will stimulate work in this direction.
Note: As we were completing this work, we became aware of Ref. [18], which demonstrates that the force mediated by the ϕ field has a range of order the de Broglie wavelength (rather than the Compton wavelength) due to the presence of a background ϕ field.That work does not give results for the case where the collective effects discussed in this paper are important.Ref. [18] also gives a stronger bound on the ϕ 4 coupling from the bullet cluster, modifying the tuning estimates in our Appendix B. and we obtain With the normalization Eq. (2.17), this gives the result Eq. (2.16) for the force.
We now compare this to the momentum transfer computed in the quantum mechanics picture.In this picture, the momentum is transferred to the target by the scattering of individual ϕ particles.The scattering cross section is determined by the r → ∞ behavior of the scattered wave [19] The differential scattering cross section is given by so the momentum transferred to the target is given by where we used k = m ϕ v ϕ .This agrees with the classical result Eq. (2.16).

A.1 Solid sphere
We first consider scattering from a sphere of radius R and number density n 0 .This is a standard problem in quantum mechanics.The wavefunction ψ(r) outside the sphere is given by the expansion Eq. (A.2), while the wavefunction inside is given by where Note that the wavefunction for r < R does not have a y ℓ term because this is singular at the origin.The coefficients A ℓ (which are equivalent to S ℓ , see Eq. (A.3)) and B ℓ that define the solution are obtained by requiring that the wavefunction and its first derivative are continuous at r = R.The result is ℓ (kR) where f ℓ (r) is the r-dependent component of the wavefunction in the region where the force is applied.In this case, we have for the region outside the target sphere and inside the shell.Although the result appears to be r-dependent, it is independent of r in the region where f ℓ (r) is defined.Eq. (A.33) holds for any spherical geometry, and we have checked that it gives the correct result for the force on a solid sphere with f ℓ (r) = j ℓ (kr) + A ℓ h ℓ (kr) (see Eq. (A.22)).

Appendix B: UV Completions
In this appendix, we consider two possible UV completions of the effective interactions Eq. (1.3) and show how they connect with the different cases for the nucleosynthesis bound discussed in §3.We also briefly discuss the tuning in these models.
As one would expect for a model of a light scalar with non-derivative couplings, the mass m ϕ is very fine-tuned.The dark matter relic abundance depends sensitively on m ϕ (for example through the misalignment mechanism [20][21][22]), so this tuning may have an anthropic origin [23].See also Ref. [5] for possible mechanisms to obtain light scalar particles without fine-tuning.On the other hand, a coupling of the form is allowed by all symmetries and is also UV sensitive. 8This coupling is tightly constrained by constraints from structure formation [24] λ ϕ < ∼ 3 × 10 −7 m ϕ eV 4 .

(B.2)
A value of λ ϕ that violates this bound modifies the spectrum of density fluctuations, but structure formation still takes place, so there is no obvious anthropic constraint on λ ϕ .
The simplest UV completion of the model comes from adding the scalar ϕ to the Standard Model with the most general renormalizable couplings compatible with the ϕ → −ϕ symmetry: This gives a ϕ dependent shift in the Higgs VEV: where m h = 125 GeV is the physical Higgs mass.Below the electroweak symmetry breaking scale, this induces various couplings of ϕ 2 to standard model fields.For example, the coupling to SM fermions ψ is given by For light fermions such as the electron and the up and down quarks, this is suppressed by the fermion mass.The low-energy couplings that are not suppressed in this way are couplings to the SM gauge field strength operators, for example ϕ 2 G a G µνa , where G µνa is the gluon field strength.Integrating out the heavy quarks gives a common contribution to the neutron and proton masses At low energies, this model dominantly couples to the nucleons, and this coupling is approximately isospin preserving.This corresponds to the weaker nucleosynthesis constraint in the left panel of Fig. 4.9 In this model, the ϕ 4 coupling has a UV divergent 1-loop contribution of order where we have assumed that log(Λ UV /m h ) ∼ 1.As long as this is ratio is smaller than 1, the model is not fine-tuned.
Next, we consider another UV completion which can dominantly couple to electrons at low energies.This comes from adding the following interaction to the SM: In this UV completion, the dark matter couples dominantly to the electron at low energies.
We now discuss the fine-tuning of the quartic coupling in this model.The dominant UV divergent contribution is given by a 2-loop diagram, and is of order (B.12) A variation of this model is to couple dominantly to a light quark, for example the down quark: Fig.1: Ratio R z between the force on a solid sphere of aluminum of radius R and the classical particle limit.We fix f = 10 10 GeV, corresponding to a skin depth L skin = 0.56 cm, and we have plotted the ratio for m ϕ = (10 −1 , 10 −2 , 10 −3 , 10 −4 ) eV, which corresponds to the reduced de Broglie wavelength λdB = (0.2, 2.0, 20, 200) cm.
Fig. 2:The ratio R z of the dark matter force on either side of a 1-dimensional aluminum barrier of thickness ∆R that is normal to the average direction of the dark matter wind (see Eq. (2.31)).We take m ϕ = 10 −4 eV and f = 10 12 GeV, which imples L skin = 5.6 cm.The black solid line is the result of averaging over the dark matter velocity distribution in the standard halo model, while the dashed blue line is the result for a pure plane wave with velocity 220 km/s.

L
Fig.3: Ratio R pz between the z-component of the momentum density inside a hollow sphere and its classical limit as a function of the thickness of an aluminum shell with outer radius R = 100 cm.We have fixed m ϕ = 10 −5 eV (i.e.λdB = 2×10 3 cm) and plotted the ratio with f = (10 11 , 10 12 , 10 14 , 10 16 ) GeV.
UV is the UV cutoff, which we identify with the scale of new physics.Comparing this to the bound Eq. (B.2) from structure formation we have ∆λ ϕ λ ϕbound ∼ 10 −6 m ϕ eV

2 e ϕ 2 (
LHe c + h.c.).(B.9)This requires UV completion at a scale of order M e , so we require M e > ∼ TeV.Below the electroweak symmetry breaking scale, this generates a coupling of ϕ 2 to the electron given

2 d ϕ 2 (
QHd c + h.c.).(B.13)Below the electroweak symmetry breaking scale, the dominant coupling is to the down quark, maximally, corresponding to the stronger nucleosynthesis bound in the left panel of Fig.4.