On the Existence of Low-Mass Dark Matter and its Direct Detection

Dark Matter (DM) is an elusive form of matter which has been postulated to explain astronomical observations through its gravitational effects on stars and galaxies, gravitational lensing of light around these, and through its imprint on the Cosmic Microwave Background (CMB). This indirect evidence implies that DM accounts for as much as 84.5% of all matter in our Universe, yet it has so far evaded all attempts at direct detection, leaving such confirmation and the consequent discovery of its nature as one of the biggest challenges in modern physics. Here we present a novel form of low-mass DM χ that would have been missed by all experiments so far. While its large interaction strength might at first seem unlikely, neither constraints from particle physics nor cosmological/astronomical observations are sufficient to rule out this type of DM, and it motivates our proposal for direct detection by optomechanics technology which should soon be within reach, namely, through the precise position measurement of a levitated mesoscopic particle which will be perturbed by elastic collisions with χ particles. We show that a recently proposed nanoparticle matter-wave interferometer, originally conceived for tests of the quantum superposition principle, is sensitive to these collisions, too.

behind particle production in the early Universe, we at least want to give a snapshot of how things work.
The first point is that at high temperatures, as present in the early Universe, all particles can be regarded as practically massless, i.e. they effectively act as radiation. The other components of the Universe, non-relativistic matter and Dark Energy, are completely negligible at this early stage. All particles χ with a sufficiently large annihilation cross-section, i.e. a sufficiently high rate of producing, or being produced by, two photons (or any other SM particle-antiparticle pair that couples sufficiently strongly to photons), χχ ↔ γγ, are in thermal (and also chemical ) equilibrium [59]. This essentially means that the (thermally averaged) annihilation rate of a particle χ and its antiparticle χ into photons is exactly the same as the inverse rate, within a certain volume. Hence, what matters is the density of the particle species χ. The most important requirement to keep a particle in thermal equilibrium is that its interaction rate with photons (or, more precisely, with SM particles which in turn interact with photons) is large enough. Furthermore, we should note that a massless particle is very easy to produce, since essentially all the photons have a large enough energy to produce the particle χ as long as T m χ , i.e. the temperature of the thermal plasma (and hence the photons) is larger than the mass m χ of the particle χ. Thus, the number density of a species χ in thermal equilibrium will be very large if χ is highly relativistic, i.e. T m χ . As time goes by, the Universe expands and by this it cools down, due to the associated redshift of all the radiation in the Universe. However, this does not only decrease the temperature but it also slows down the effective interaction rates. These are essentially given by the number density times the thermally averaged cross-section, n χ σ ann v , and the number density n χ decreases while the Universe is expanding. Thus the interaction rate further and further decreases until, at some point, it drops below the expansion rate (the Hubble function) H of the Universe. At that point, no particles of the species χ can be annihilated anymore, because their number density is too small and the particles do not find any interaction partners to annihilate with. This is what is called thermal freeze-out. Obviously, the time (or temperature) at which this freeze-out happens depends on the value of the cross-section and can by this be very different, depending on the properties of the species χ. For example, it depends on whether χ is electrically charged or, more generally, which interactions it participates in. Notably, even if the species χ is only weakly interacting this is nevertheless by far enough to keep it in thermal equilibrium in the early Universe. The simple reason for this is that the weakness of the weak interactions purely comes from the relatively large mass of the exchanged W ± and Z 0 bosons, which are also practically massless at high enough temperatures, and the coupling strength itself is comparable to that of electromagnetism. Having explained how a species χ can undergo thermal freeze-out, we also note that a frozen-out species χ will survive in the Universe until today if it is stable (or, at least, if it has a lifetime that is considerably larger than the lifetime of the Universe). This is exactly why, often, DM is thought of as being a massive particle that is only weakly interacting: it is not allowed to be electrically charged, since then it would directly couple to photons and hence not be "dark", but if it is charged under weak interactions only, it can nevertheless enter thermal equilibrium in the early Universe and then undergo thermal freeze-out.
There is still one more subtlety involved which we have to discuss: freeze-out can happen when the species χ is highly relativistic or when it is non-relativistic or anywhere in between. We have already mentioned that the number density n χ of the particle χ is comparatively large when the particle is relativistic, because it is easy to produce such particles. However, in the case of T m χ , only the photons with the highest momenta have enough energy to produce an χχ pair (this is the reason why we need the thermally averaged cross-section, to take into account the thermal momentum distribution of the particles in the Universe). Then, even though the photon density might still be large, it is a rare event that χχ-pairs are produced and its number density decreases considerably. In fact, it even decreases exponentially with the inverse temperature, n χ ∝ e −mχ/T . This will also influence the thermal freeze-out since, as already mentioned, it is actually the product of the number density and the thermally averaged cross-section, n χ σ ann v , which has to be compared to the expansion rate H of the Universe. Now we are ready to understand one basic property of thermal freeze-out: the more relativistic a particle species χ is at freeze-out, the larger its number density will be.
Finally, we have to understand that what is "measured" or, rather, inferred from the observation of the cosmic microwave background (CMB) is in fact an energy density. If we consider a co-moving volume (i.e. a piece of space in a coordinate system that grows together with the expanding Universe), the number density n χ of this volume will remain constant after freeze-out, since no χ particles annihilate or are produced anymore. 1 The χ particles within this co-moving volume will eventually slow down, such that their velocities at late times become negligible in comparison to the speed of light c, and hence the final energy density ρ χ that remains in χ is simply given by the product of their mass and their number density, ρ χ = m χ n χ . The quantity that is derived from observations is called the abundance Ω DM (or, more commonly, Ω DM h 2 , with h being the reduced Hubble constant), and it is essentially the fraction of the total energy density (actually the so-called critical density) of the Universe which resides in DM particles today. The current most up-to-date value comes from the 2013 data release of the Planck satellite [60], and the observed 1σ range derived from Planck data only is given by Whichever DM candidate we consider, if it is to be the only constituent of the DM observed in the Universe, it is absolutely indispensable for it to reproduce the correct abundance. In any case our species χ can clearly not exceed the above measured abundance. On top of that, further constraints arise e.g. from cosmological structure formation, from direct searches, or from indirect bounds, as well as from consistency arguments (the new particle should "fit" with the known ones, loosely speaking). We will investigate in the following whether all these conditions can be fulfilled for the type of DM under consideration.
b. Calculating the Dark Matter abundance Turning now to DM as needed for Ref. [61], we first of all want to get some understanding of the properties of the DM candidate under consideration. Probably the first question a particle physicist would ask is about the spin -is the DM particle a scalar (spin 0), is it a fermion (spin 1/2 or 3/2), or is it a vector (spin 1), as these are the only possibilities which exist in renormalisable theories. However, in fact, the more important question is about the possible suppression of the annihilation cross-section σ ann v: is the leading order an s-wave (such as typical for scalar, Dirac fermion, and vector DM), which is without velocity dependence, or is a p-wave (as typical for Majorana fermions or for cases where certain suppressions apply), which is suppressed by the square of the velocity, v 2 (this being small for nonrelativistic DM). Unfortunately, we cannot easily answer this question, since the exact properties of the DM candidate are not specified in Ref. [61].
On the other hand we want to try to understand what the prospects are for an experiment as suggested in Ref. [61] in general. So, the only way we can proceed is to try a certain generic candidate. For simplicity, we have decided to assume a scalar particle, since then the equations look simplest and since the relation between the DM annihilation cross-section and the scattering on the molecules is also easiest in that case. As we will illustrate later, this choice is in fact probably the best one could make. The reason is that, at several places, the scalar DM case will have a tendency to save itself from strong bounds.
The essential consequence of freeze-out is as follows: the larger the annihilation cross-section (e.g., if it is unsuppressed) the smaller the final abundance will be, due to more of the DM particles annihilating before the freeze-out. Furthermore, the more non-relativistic the particles are at freeze-out the more suppressed their number density and hence their final abundance will be. However, as we had specified earlier, the decisive cross-section is the annihilation cross-section into SM particles σ ann , while what is given in Ref. [61] is the scattering cross-section on nucleons σ, and these two seem to be very different quantities at first sight.
We can try to find an easy estimate for the translation of the scattering cross-section σ on a nucleon to the scattering cross-section σ quark on a quark. Fortunately, this is a standard problem when calculating direct DM detection rates, so that we can make use of the known literature on that subject [62,63]. However note that, although this estimate is fairly generic, it cannot replace the detailed calculation within an explicit model. But, as long as such a concrete model does not exist yet, this estimate is the best one can do and it should be expected to yield at least fair results.
Let us see what we can learn from the known treatments.
In [62], Eq. (9) gives the spin-independent (SI) nuclear cross-section. Since we are only interested in a single nucleon, we have where N = p or n and for the reduced mass we can approximate m r = m χ m N /(m χ + m N ) m χ . Furthermore, their Eq. (10) tells us that where we have kept only q = u-and d-quark DM interactions, which corresponds to the minimal assumption for the treatment of Ref. [61] to be applicable. We have furthermore estimated both of them as equal (hence just the factor 2). Here, the coupling G, which corresponds to the quark-χ effective coupling from Eq. (3) of Ref. [62], is just the coupling that we will use later on, see our Eq. (7) and what follows. We are left to estimate the matrix element f (N ) T q = N |m qq q|N /m N , which has been computed in Ref. [63], see their Eq. (3) and below. One obtains: which leads to f (N ) T q ≈ σ /m N . The quark cross-section is σ quark ∼ m 2 χ G 2 so that, assuming the 4/π has the same kinematical origins for both cases, Eqs. (2) and (3) tell us that: where N eff ≈ 2 σ mq ≈ 20. It is important to note that the u-and d-quarks contained in the nucleons inside the detector are precisely the same particles as present in the early Universe, which leads to the annihilation diagram displayed in FIG. 1 b (main text).
Of course, there is no principal reason that the DM particle χ could not interact with even more types of quarks or even further SM particles, which would require the first equality sign in Eq. (5) to be replaced by ">". That would increase the annihilation cross-section and therefore make the resulting abundance smaller. Indeed, for many popular DM candidates the annihilation cross-section is indeed considerably larger than the direct detection cross-section [12,13,18,[64][65][66][67][68], even though examples for the contrary case exist as well [69], and suppressions as e.g. for Majorana DM can apply. For simplicity, we stick to the minimal assumption that the DM only interacts with the quarks present in the proposed quantum decoherence experiment. Indeed, the estimate in Eq. (5) comprises the minimal assumption on the cross-section.
However, there is one subtlety to discuss. As soon as the temperature of the Universe (or, rather, the temperature of the thermal plasma) falls below the QCD confinement scale between 150 and 450 MeV, u-and d-quarks do not anymore exist as free particles. Instead, the lightest QCD bound states will exist, namely pions with masses of around 150 MeV [70]. However, in the mass range under consideration (where m χ < 0.1 GeV), our DM particles are not able to annihilate into pions at rest for kinematical reasons, since 2m χ < 2m π . Thus, the only annihlation channel which will be open at that stage will be the annihilation into two photons, χχ → γγ. The corresponding Feynman diagram contains a loop, cf. FIG. 1 b in the main text, which means that it is suppressed by a factor of roughly α 2 QED /(16π 2 ) ∼ 3 · 10 −7 , where α QED 1/137 is the fine structure constant, compared to the annihilation cross-section into quarks. Thus, we must correct Eq. (5) by this suppression factor to obtain a reliable estimate, where we have already included the thermal velocity v = 3T m for a particle with mass m and temperature T , as generically obtained from the thermal distributions in the early Universe. 2 This cross-section is not decisive for high temperatures, where quarks can be produced without problems, but late enough in the evolution of the Universe -which is exactly the time that is decisive for our type of DM -it is the only one which is there. This leads to a sudden drop in the crosssection. However, since the cross-sections under consideration are still comparatively high, even this suppressed annihilation rate is enough to keep the DM particles in thermal equilibrium for some time. But at some point the DM particles will nevertheless freeze out due to the suppressed cross-section, which is the reason for their sizable final abundance. This is the first instance where the DM under consideration saves itself from a disaster: typically, a DM particle with such large cross-sections on quarks would stay in equilibrium for a too long time, eventually become very nonrelativistic, and thus have a strongly suppressed final abundance. However, due to the additional kinematical constraint and consequent loop suppression of the cross-section, this does not happen and we are left with a large enough abundance.
The next point to discuss is the variation of the annihilation cross-section with the temperature. As explained before, the decisive quantity is in fact the thermally averaged cross-section times the velocity, σ ann v , which we will approximate by Eq. (6) for low temperatures (this will allow us to estimate the interaction strength). Depending on the temperature T and the mass m χ , the thermal average can be more or less decisive. A particularly delicate region is the one where the particle is neither highly relativistic ("hot") nor fully non-relativistic ("cold") at the time of the freeze-out, but somewhere in between ("warm"). This region looks very different depending on whether the annihilation cross-section is dominated by s-or p-wave contributions. We again have some freedom here and we have decided to assume an s-wave contribution, as generic for a scalar DM particle.
Then we can make use of Ref. [71], where an easy and relatively accurate interpolation formula between the hot, warm, and cold regions had been suggested: where G is an effective coupling constant, 3 cf. Eq. (2), and x ≡ m χ /T is the usual variable which essentially describes the time or, rather, the inverse temperature. Furthermore, its value also distinguishes between the non-relativistic (x 3), semi-relativistic (x ≈ 3), and highly relativistic (x 3) regions. Note that the velocity in terms of the variable x is given by v = 3/x. Note that, in order to express the coupling constant G in terms of the DM-nucleon cross-section as used in Ref. [61], one must take into account that a DM particle detected in an experiment performed "today" (i.e. when the Universe is about 13.8 Gyrs old) is non-relativistic, and a typical value of its velocity today would be v 0 ∼ 10 −3 [72,73]. One can thus estimate: which, using x 3 in (7), leads to the estimate The first step in the actual calculation is to compute the freeze-out temperature. This determines the final abundance in particular in the case of the cold (non-relativistic) freeze-out, due to the exponential suppression of the number density, n χ ∝ e −mχ/T . In order to do this, one needs to equate the interaction rate n χ,eq σ ann v approx of the DM candidate χ in thermal equilibrium with the expansion rate H of the Universe, n χ,eq σ ann v approx = H.
Due to the relatively large cross-section on quarks necessary for a particle to be detected in a set-up as proposed in Ref. [61], it is unavoidable for the DM particle to be in thermal equilibrium with the photons (or, more precisely, the whole thermal plasma) in the early Universe. To give some more technical details, the expansion rate (Hubble function) of the Universe is in this early and radiation-dominated era given by with the time (the age of the Universe) t being related to the temperature T of the Universe (i.e. the temperature of the thermal plasma) by Here, M P = 1.22 · 10 19 GeV is the Planck mass and g * (T ) is the effective number of relativistic degrees of freedom. This number essentially sums up all the spin and colour degrees of freedom of all particles which are relativistic at a given temperature T . For the SM particle content, it is displayed in FIG. 4. If there is unknown physics beyond the SM, i.e. many more particles which have non-negligible interactions strengths and can hence be produced in the early Universe, then this function would need to be modified. However, this modification will not be very significant unless very many new particles are introduced. Although we will use the full g * (T ) for the purposes of producing the plots in FIGs. 1 c, 4-6, it is worth noting that for the freeze-out temperatures T FO < m χ /3 of our eventual choice of DM particle, the effective number of degrees of freedom is to good approximation just g * ≈ 3.36, corresponding to the fact that only (reheated) photons and neutrinos can make a contribution.
The equilibrium number density of the DM candidate χ is (for scalars) given by the ordinary Bose-Einstein distribution, where E = m 2 χ + p 2 is the total energy of a particle with mass m χ and momentum p. The inverse freeze-out temperature x FO is obtained by numerically solving Eq. (10). The result is displayed, for different masses m χ in FIG. 5, as a function of σ. We have indicated the region where the DM would be hot (i.e. highly relativistic at freeze-out), which is excluded because this scenario would not lead to a successful formation of structures in the Universe [74,75]. Furthermore, in the region right of the purple point (where m χ = 10 −8 ), the freeze-out would happen too late, i.e. after the time where the energy densities of matter and radiation must have been equal according to observations (corresponding to the temperature T eq ≈ 0.8 eV), significantly affecting the measured fluctuations in the CMB. 4 However, as we will see later on, that part of the curve is in any case excluded. This problem does not appear for larger masses m χ , which is why there are no corresponding markings in the plot.
Writing p = m χ y in (13), where y is a dimensionless integration variable, and using the fact that we want x = m χ /T > 3 in order for DM not to be too hot, the integral may be approximated to better than 6% by Using g * ≈ 3.36 and combining the expressions above we thus have: Although it is slightly counter-intuitive to instead express σ as a function of m χ and x FO in this way, we thus get a closed expression which agrees with the plots of FIG. 5; for example it accounts for the σ ∝ 1/m χ -dependence clearly visible in the plots. Using the standard techniques, one can then expand the cross-section as where G 2 was reported in Eq. (9). The standard formula for the final DM abundance is given by [71]: where the cross-section is measured in square metres. Note that, as pointed out in Ref. [71], using in Eq. (17) the value of x FO obtained by numerically solving Eq. (10) may lead to errors of something like 10% in the case of non-relativistic freeze-out, so that in the worst case, there could FIG. 6: Final DM abundances for different masses of the assumed DM particle, for the whole parameter space considered in Ref. [61] (left panel) and for the most interesting region drawn to a larger scale (right panel). The parts drawn in light colours correspond to hot DM (HDM), which is excluded. Notice the intersection of several of the lines which is visible on the right. This originates from the fact that, depending on the exact value of the freeze-out temperature, the degrees of freedom need to be evaluated before or after e + e − annihilation, which leads to a jump in the abundance due to entropy dilution. be a further O(1) factor involved in our result. However, this uncertainty, while present, is less than the uncertainty introduced by using already generic estimates for the cross-sections.
The final result for the abundance is presented in FIG. 6, where the whole parameter region (masses and cross-sections on nucleons as taken in Fig. 5a of Ref. [61]) is displayed on the left panel, and a blow-up of the most interesting region is shown on the right. As can be seen, the resulting DM abundance can hit the observed value for a ballpark of masses, from m χ = 10 −1 GeV to 10 −7 GeV. Smaller masses are excluded even though the correct abundance could be in principle be obtained, because that part of the parameter space would correspond to HDM which is ruled out (or, rather, bound to make up at most about 1% of the total DM in the Universe [74,75]).
Recalling the cold DM (CDM) constraint x FO > 3 and the DM abundance (1), and substituting g * ≈ 3.36 and the typical value v 0 ∼ 10 −3 into (17), we can write compactly that the fraction of DM made up of χ species is f = Ω χ /Ω DM ≈ 9.1 · 10 −30 x FO /σ[m 2 ]. For a fixed fraction f , for example f = 1 corresponding to the case where DM is entirely made up of χ, the mass m χ is then related to the freeze-out temperature x FO > 3 as: .
This may be combined with Eq. (15) to get directly the relevant points in FIG. 6. These results may look somewhat surprising at first sight, since for scattering cross-sections around σ = 10 −28 m 2 = 10 −24 cm 2 , where an abundance in the correct ballpark is generated, one might naively expect a much larger annihilation cross-section which would keep the DM in equilibrium until it is very cold, thereby suppressing its abundance by a huge number. However, because of the DM particles being so light, the strong suppression of the annihilation cross-section (due to it necessarily being a loop process in this case) saves our DM candidate from that fate. c. Particle physics constraints While the calculation up to this point looks in fact quite good, we should mention that there could be potentially dangerous bounds, because after all our DM candidate does couple quite strongly to quarks.
As we had mentioned, one could expect a potentially strong bound from hadron colliders, such as the LHC. While the most natural choice to search for a χ particle may seem to be a protonantiproton collider such as the Tevatron, the most natural reaction pp → χχ would probably be invisible since the χ would just remain on the beam direction due to their small masses and the associated large boost factors. However, if one of the quarks inside the baryons radiated off a photon before the annihilation process, this could lead to a classic signature of one single photon plus missing energy. 5 Thus, the most stringent bounds come from experiments performed at hadron colliders which have searched for such a signature, for example CDF [76], D0 [77], or CMS [78]. 6 However, depending on the true ultraviolet completion of the effective vertex in FIGs. 1 a,b (main text), this may not be a problem. Imagine the existence of another scalar particle ξ with mass m ξ which couples with strength g q to quark/antiquark pairs qq, and with the dimensionful 3-point coupling f χ to DM/anti-DM pairs χχ. If it holds that m χ m ξ E collider , i.e. the new particle is much heavier than χ but has a mass much smaller than typical collider energies, the interaction between quarks and χ's would be an effective four-point coupling for all low energy purposes as already discussed, but fundamental at collider level. If furthermore g q is very small, the ξ would not show up in any electroweak precision data.
The question remains, however, if this could possibly lead to a large enough scattering crosssection for χ. This point can be easily answered. Taking the collider limits to amount to roughly σ ann.+γ < 1 fb for a centre-of-mass energy of √ s ∼ 100 GeV (which is already conservative), then the cross-section can be estimated on dimensional grounds as σ ann.+γ ∼ g 2 q f 2 χ s 2 α QED , where the additional suppression factor comes from the photon being radiated off. Taking α QED ∼ 10 −2 , one can thus estimate g 2 q (f χ /GeV) 2 < 0.1. At low energies, in turn, the scattering cross-section would be σ scatt. ∼  6, which has to be divided by another factor of roughly 400 to translate 5 As already mentioned, χ would not necessarily show up as missing energy. For low energies, it would be shielded by the detector material, but for high energies it could deposit energy in a calorimeter. However such a signal would be generated primarily from a χ particle embedded in a jet, through multiple t channel exchanges, and thus easily confused with other particles. (This follows from the high energy s-channel suppression we are about to discuss.) 6 Note that bounds from lepton colliders such as LEP do not play any role as long as our DM candidate does not couple directly to leptons. Thus the classic limits from detectors like DELPHI [79,80] may not apply in this case.
it from nucleons into quarks. Using the collider bound, one can estimate m ξ 0.3 GeV, which is still much larger than the required masses for χ. Thus, a sufficiently light ξ can compensate for the small coupling g q such that the scattering cross-section can be large at low energies but suppressed at collider level. This argumentation is quite generic and would make the strong limits from colliders much less problematic (if not completely harmless).
Strong bounds may also originate from the invisible decay width of the SM Z-boson, which could in principle decay as Z 0 → χχ by a 1-loop diagram with u-and d-quarks intermediate states ( cf. left panel of FIG. 7), and from the invisible decay of a neutral pion, π 0 → χχ, again involving a u-and/or d-loop (cf. right panel of FIG. 7). This might lead to a problem, because the invisible decay width of the Z-boson is well measured, Γ inv Z = 499.0 ± 1.5 MeV [70], and it agrees with the SM prediction so that any additional contribution could only modify it by an amount of the order of the uncertainty 1.5 MeV of the measurement. There is no actual measurement on the invisible pion decay width available, but a bound can be estimated from the branching ratio of 2.7 · 10 −7 (at 90% C.L.) of π 0 [70] decaying into neutrinos which results into a tiny invisible decay width of Γ inv π 0 < 2 · 10 −12 MeV. Indeed, such very model-independent bounds can exist for light DM candidates (see, e.g., Refs. [39,81] for concrete examples).
However, for the scalar DM case, both these processes in fact have vanishing amplitudes. This is relatively easy to understand. In the case of the Z-boson, since the χ particle has no charge or hypercharge, there is no gauge-invariant effective coupling to mediate Z 0 → χχ. Although gauge symmetry is spontaneously broken, the one-loop diagram illustrated above is insensitive to this. In fact by this argumentation one sees that the simplest coupling involving Z and χ is via the dimension six operator ∼ B 2 µν χχ, where B µν is the U (1) Y field strength. This allows for processes such as Z 0 → χχγ, but they will be suppressed by α QED . For the pion decay, in turn, one has to take into account that the pion is in fact a pseudo-scalar (i.e. the corresponding field changes its sign under parity transformations), but the s-wave final state containing two scalars will always be of positive parity. Thus, the amplitude is zero as long as no parity breaking interactions are assumed, since the kinematics of the situation always enforce back-to-back emission of the final states and their spinlessness makes it impossible to compensate for that by orbital angular momentum. Again, the simplest process that allows the pion to decay to χ particles must involve also radiating a photon and thus is suppressed by α QED . In spite of these arguments, we have calculated both processes explicitly (using a scalar vertex L eff =ξ|χ| 2 qq for the Z-boson diagram and the chiral Lagrangian in combination with L eff = ξ|χ| 2 pp, which results into a Feynman rule (−i)ξ, for the pion decay diagram), and we can confirm that the amplitudes are indeed zero. While these bounds would certainly be strong for a number of possible DM candidates, a scalar χ evades them completely. Again, this type of particle seems to save itself from the most dangerous bounds. However, this situation could be very different for other possible DM candidates, such as fermionic particles. Furthermore, fermionic DM would necessarily obey the so-called Tremaine-Gunn bound [82], which is essentially based on the Fermi pressure of fermions. Recent analyses [83] of dwarf satellite galaxies show that, using the limit of a degenerate Fermi gas, a lower bound of m fermionic DM > 0.41 keV is derived on the mass of a fermionic DM particle. This would indeed cut significantly into our parameter space. Note that, in our case, it is not unthinkable that only a certain fraction of the DM, say p = 1%, is made up of the χ particles studied in this work. In that case, the mass bound does become marginally lower by a factor of p 1/12 [83], e.g., m fermionic DM > 0.28 keV for p = 1%.
d. The annihilation signal The annihilation of two DM particles into two photons, cf. FIG. 1 b (main text), is the only possible annihilation channel whenever the two initial state particles are non-relativistic. Thus in regions where a lot of DM particles accumulate, such as the Galactic centre, the same process will be active and could possibly lead to an observable signal.
The differential flux of photons per area and time, stemming from the annihilation of two DM particles, can easily be calculated [84,85]: Since the broadening of the differential spectrum is small and not of any relevance if we are only interested in the total photon flux, we can take dN γ /dE γ = 2δ(E γ − m χ ), where the factor of 2 originates from the fact that two photons are produced per annihilation process. We calculate the expected flux from the Galactic centre, for the sake of an example, for now assuming that the signal could reach us without being perturbed (as we will see later on, our Galaxy is in fact opaque for part of the photon spectrum). For a Navarro-Frenk-White (NFW) profile [86] with the parameters (α, β, γ) = (1.0, 3.0, 1.0) 7 and a scale radius of r S = 20 kpc, one expects J GC ∆Ω ∆Ω = 0.13 sr for the line-of-sight integral, where ∆Ω = 10 −5 sr. 8 Thus, the integrated total flux at an energy of E γ m χ is given by: As can be seen already from this formula, the smallness of m χ leads to a huge photon flux. This can be understood intuitively: since the mass of the DM candidate under consideration is comparatively small, its number density must be quite high to compensate for the small mass so that the correct DM abundance can be met. Since there are no kinematical restrictions associated with the annihilation into two photons (the only restrictions could come from angular momentum related issues, but these are not present here), this rate cannot depend strongly on the initial state masses, apart from them being the only dimensionful quantities involved. Thus, the large number density translates into a high photon rate, since the fact that two DM particles have to meet to annihilate yields to a proportionality of the signal rate to a square of the DM number density. The resulting fluxes are plotted in FIG. 1 c (main text). Indeed, the fluxes turn out to be very large. So large, in fact, that they would exceed the known bounds by orders of magnitude, if taken at face value (i.e. if no further subtleties such as strong atomic lines are considered). This would also be true for annihilation in regions other than the Galactic centre. As we will show, we have as example computed the expected rate for a DM mass of around 3.56 keV in order to see whether we could reproduce the recently reported X-ray signal from galaxy clusters [56,57], and indeed our DM candidate would, due to the large annihilation cross-sections related to the large direct detection cross-sections, exceed the observed signal strength by several orders of magnitude. Similar results would be obtained when computing the signal for some other regions in the parameter space.
Accordingly, the natural reaction one could (and probably should) have is to discard the DM candidate particle discussed in this article, because its large annihilation signal would already have been seen for sure. 9 But would this conclusion be correct? As it turns out, it would not! The simple reason is that, first of all, not all energy ranges relevant here have been thoroughly investigated by observations, as some of them were considered to be "uninteresting" from an astrophysical point of view, but furthermore a galaxy can also be quite opaque to certain wavelengths, so that the signal, even if present, would not necessarily reach us. As we will see, this leaves us with an unconstrained window in the parameter space. Furthermore, even for the regions where we exceed the bounds, there are many subtleties involved with detecting a line signal, ranging from potentially strong backgrounds by neighbouring atomic lines to the modelling of the continuum background. We are aware of these subtleties but we cannot discuss them here, since in particular many of them are very specific to certain energy ranges, while our global analysis spans over many orders of magnitude in energy. However, to make clear that one would need to investigate certain regions in greater detail to be absolutely sure that our DM candidate could not hide in there, we marked the regions threatened by astrophysics as "disfavoured" rather than "excluded" in FIG. 1 c (main text), and we indicate that by using only a light gray background colour.
Before discussing the tentative bounds, we need to be clear which photon energy range we are talking about. Since E γ m χ , any strong constraint on m χ will directly translate into a constraint on E γ . As we have already seen, cf. FIG. 6 (in particular for m χ = 10 −8 GeV= 10 eV), for too small masses of χ we are in fact hitting the HDM region, which would conflict with cosmological structure formation. In addition recall that the freeze-out temperature T FO has to be greater than the equality temperature T eq . This sets a lower limit m χ > 3T FO > 3T eq = 2.4 eV for all crosssections under consideration, which results in the upper dark gray exclusion region in FIG. 1 c  (main text). If m χ > m π , in turn, the suppression of the annihilation cross-section, cf. FIGs. 1 a,b (main text), would not work anymore, thereby completely destroying any abundance of χ, which translates into the lower dark gray region in the plot. The region in between these boundaries can be constrained by observations.
The first question to answer is where to look for signals. Dwarf spheroidal galaxies are generally considered to be very good places to search for DM decay or annihilation lines, as they have a very high mass to light (M/L) ratio and are therefore thought to contain a particularly high fraction of DM. Clusters of galaxies are also good as they have a high M/L ratio, too, except that they are usually strong sources of bremstrahlung X-ray emission which forms a high background against which to search for the X-ray signal lines. Some other searches have looked at various angles in our own Galaxy. DM decay or annihilation lines would be expected to be stronger nearer to the Galactic centre, due to the accumulation of DM, and so any lines which did not vary appreciably around the sky are probably not decay or annihilation lines.
Which observations are relevant for us? Let us start by the upper energy boundary and work (roughly) down in energy. There have been a number of searches for decay lines in the Fermi data. In particular Ref. [87] presented observations of dwarf spheroidal galaxies. Above 100 MeV they find no detections and derive upper limits on any line of approximately a few times 10 −9 cm −2 s −1 (per 10 −5 sr), which is below the signal expected from our region of interest. In Ref. [51], a search for lines in data taken with the high resolution spectrometer, SPI, on the INTEGRAL γ-ray observatory has been presented. The search is performed in the energy range from 40 keV to 14 MeV, in blank sky data at various distances from our own Galactic centre. They find a number of lines, many of which are identified as instrumental lines, but some of which are unidentified. However, none of these lines varies enough to be considered a likely DM line. They place upper limits on the DM origin of each line, and these limits typically lie in the range 10 −7 to 10 −8 cm −2 s −1 . Again, a signal from our DM candidate in that mass range would completely overshoot these bounds, if taken at face value.
In the keV region, a vast variety of bounds exist [47][48][49][50][51][52][53][54][55][88][89][90][91][92]. However, since these bounds are typically interpreted in terms of keV sterile neutrino DM (see, e.g., Refs. [45,46] for reviews), the corresponding fluxes, which on top of that come from many different observations of various (dwarf) satellite galaxies, are usually given in terms of the so-called active-sterile mixing angle θ. For our purpose, it is most reasonable to translate these bounds into event rates, in order to compare them to our cross-sections. This job is made easy by recently proposed fit formulas [93] to the combination of the bounds reported above.
To perform the comparison, it is easiest to look at event rates instead of fluxes. For sterile neutrino decays, the photon rate per second in a given volume V can be estimated as V n DM Γ γ , where is the decay rate of a sterile neutrino with mass m s [56]. For the corresponding annihilation rate of our DM candidate, the equivalent event rate is V n 2 DM σ ann , where σ ann is given in Eq. (6) and the square on the number density arises from the fact that two DM particles have to meet in order for the annihilation to take place. Since the event rate from our candidate has to be smaller than the bound, we can easily derive where we have used the fact that θ is small, and we have expressed the DM number density by the local DM energy density, n DM = ρ χ /m χ with ρ χ = 0.4 GeV/cm 3 . Note that, for the comparison, we need to set m s = 2m χ , due to the sterile neutrino signal arising from a 2-body decay ν s → νγ, whereas our signal would arise from an annihilation process.
Using the conservative bounds from Ref. [93], the upper bound on the cross-section σ turns out to be around 10 −38 m 2 , for the whole mass range from m χ = 0.25-25 keV. Thus, a signal from our DM candidate would probably not have been missed in this mass range, so that it is strongly disfavoured for our purpose as well. Alternatively, we could try to reproduce the recently reported 3.56 keV X-ray line signal [56,57]. The derived mixing angle of sin 2 (2θ) = 7 · 10 −11 would again require a cross-section of σ ≈ 10 −38 m 2 , which is off our plot.
We should compare the above non-detections with the typical value of the X-ray background, which provides a lower ball-park sensitivity limit. At 1 keV, Ref. [94] gives the background as 1.1 · 10 −4 photons per cm 2 s keV (per 10 −5 sr). Taking a conservative energy resolution of 10%, a 3σ limit on the flux would be approximately 3.3 · 10 −5 cm −2 s −1 . Using typical broad band diffuse X-ray background measurements (e.g. [95]), we obtain 3σ limits which are factors of about 10 below those listed in Fig. 2 of [53]. That is the direction that one would expect so the observations of the background are merely a weak consistency check, and again, a signal of a DM candidate like ours would have been highly visible.
However, in the range of approximately 10 eV (13.6 eV, to be precise) to about 100-200 eV, observations are severely limited due to the absorption by neutral hydrogen in our own Galaxy. Thus, in that energy range our Galaxy is in fact opaque and a photon signal from the Galactic centre cannot be expected to reach us. So, indeed, this mass/energy range at the moment comprises an astrophysical window, in which our DM candidate could live, cf. the white band in FIG. 1 c  (main text). As indicated by the purple band in the plot, there is a surviving and distinctive region in which our DM candidate could live, even when putting all known constraints together. This narrows us down so far that we can characterise the properties of the DM particle presented here to be a scalar particle χ with m χ ≈ 100 eV and σ ≈ 5 · 10 −29 m 2 .
This is the parameter region which should be scrutinised and where, ultimately, experimentalists should search if they plan to probe our proposal.
A cautionary note at the end: The predictions given in the current paper relate to the surface brightness of the expected emission. Surface brightness, at least in the local universe, is independent of the distance. Thus, to first order, the surface brightness from DM decay radiation around a line of sight through our Galactic Centre (GC) should be similar to that through the centre of a similar nearby galaxy. However, many of the nearby dwarf spheroidal galaxies, which are promising targets for DM detection, have angular sizes less than a degree, making them, at best, marginally resolved in Fermi observations (> 1 GeV) and thus Fermi search papers give total integrated fluxes (e.g. [87]). A proper comparison with theory then requires that the emission from an assumed DM density profile is integrated for comparison with observation. The resulting integrated flux will thus be less than if the central surface brightness flux prevailed over the whole of the resolution element. However, for the nearby dwarf spheroidals, where the angular scale size of the galaxy is not much smaller than the resolution size of the instrument, the difference in integrated fluxes is unlikely to be more than a factor of 10, which is pretty marginal for the regions we can rule out, since a potential signal would overshoot the bounds by much more than that.
B. Acceleration of a spherical test particle As described in the main text, a χ particle at a typical velocity has a de Broglie wavelength large compared with the inter-nuclear separation of normal matter, and so the overall effect of multiple scattering events is well described as an interaction with an effective potential; cold neutrons interact with normal matter in a similar way [96]. We use partial waves to treat a sphericallysymmetric test particle which we describe as a finite potential well with radius given by the size of the particle and a depth chosen to match the scattering cross-section (23) at low energy.
The wavefunction for a χ particle incident from a distant source is well approximated by a plane wave, and the total wavefunction after scattering by a localised particle is where k is the wavenumber of the incident particle, and r, θ are the radial coordinates relative to the scatterer and the z axis, respectively. From scattering theory, we identify the differential cross-section which, using partial waves, we can express as where c l ≡ (2l + 1)e iδ l sin δ l , δ l is the phase-shift for angular momentum l, and P l are the Legendre polynomials. For a pressure P , the force on the particle is the flux multiplied by the cross-section, less the recoil at angle θ: Expanding by using Eq. (26), we find sin θ cos θP l (cos θ)P m (cos θ)dθ where we have used Here, δ a,b is the Kronecker delta. Relabelling with l = l − 1, the second summation becomes the complex conjugate of the first, and Using the definitions of c l to expand c * l c l+1 , we obtain (2l + 1) sin 2 δ l − (2l + 1)(2l + 2)(2l + 3) (2l + 2) 2 − 1 cos (δ l − δ l+1 ) sin δ l sin δ l+1 where we have used (2l + 1) sin 2 δ l .
This expression is numerically efficient to evaluate. To accurately describe a χ particle of wavenumber k scattering from a test particle of radius r, we must include at least kr terms. For a finite spherically-symmetric potential-well of radius r and depth parameterised by a wavenumber κ [97], the phase-shifts are δ l = j l (kr) − Dj l (kr) n l (kr) − Dn l (kr) , where D ≡ (K/k)j l (Kr)/j l (Kr) and K = √ k 2 + κ 2 . The functions j l and n l are related to the Bessel functions of first-and second-kind: j l (x) = π 2x J l+ 1 2 (x) and n l (x) = π 2x Y l+ 1 2 (x), with j l (x) = dj l (x)/dx and n l (x) = dn l (x)/dx. In the low-energy limit, where k → 0 and only s-wave (l = 0) scattering is significant, we have valid for small rκ, where V is the particle volume. By equating this to the cross-section under the Born approximation σ eff = (nV ) 2 σ, where n is the number density of nucleons, we obtain Equivalently, and as in the field of neutron optics, we may parameterise by a 'critical wavelength' λ c = 2π/κ = π/na s where σ = 4πa 2 s .

C. Decoherence in a matter-wave interferometer
A full phase-space treatment is detailed elsewhere [98]. We use the result for decoherence induced by isotropic elastic scattering including recoil; while this was originally derived for the case of Rayleigh scattering of black-body radiation, it is parameterised only in terms of a momentum and is therefore valid for the isotropic elastic scattering considered here: where sinc(x) ≡ sin(x)/x and Si(x) ≡ x 0 sinc(x )dx is the Sine integral. It is possible to treat the full spectrum γ(k) of incident χ particles; however, for our purposes it is sufficient to take the spectrum of incident wavenumbers to be narrow: γ(k) = Γ δ(k − k 0 ), where k 0 = m χv0 / is the typical wavenumber andv 0 ≡ v 0 c ∼ 10 −3 c. While the exact value of the argument to this resolution function f (x) in the expression for decoherence depends on the time scales and on the geometry of the interferometer experiment, it is very close to the grating spacing Λ.
The number of expected events is the flux of χ particles multiplied by the effective cross-section and the interaction time: where we have used σ eff = σN 2 which is valid for small particles. The characteristic time-scale for interference is given by the 'Talbot time' τ T = M Λ 2 /(2π ), where M is the mass of the target particle; for the proposed interference experiment, τ = 3.6 τ T .

D. Penetration of Earth's atmosphere
The refractive index model implies that, for a sharp boundary between vacuum and dense matter, χ DM particles will be strongly reflected. The critical wavelength for air at standard temperature and pressure (n ≈ 7.3 · 10 26 m −3 ) is λ c ≈ 1.5 µm, which falls within the expected range for the χ particle's de Broglie wavelength λ: the expected refractive index η could be close to unity or as large as 20 (20i) for an underlying attractive (repulsive) interaction.
In the attractive case, the very slow increase in atmospheric mass density on approach to the Earth's surface would strongly suppress Fresnel reflections; for the repulsive case, χ particles would penetrate only to a finite depth. Density fluctuations in the atmosphere, and the relatively large associated refractive index contrast, could lead to multiple reflections and hence to an effective finite penetration depth even in the attractive case. Such uncorrelated multiple events would reduce the kinetic energy of the incident χ particle.
In the case that χ particles do reach the Earth's surface, it may be possible (although challenging) to create a wavelength-scale structure of varying material densities which uses multiple reflections to engineer penetration of the particles into a dense solid. With a such a device, χ particles could be focused and injected into an experimental vacuum chamber in which detection could be performed much as in the space-based experiment proposed in the main text.