Antimatter as Macroscopic Dark Matter

Antimatter macroscopic dark matter (macros) refers to a generic class of antimatter dark matter candidates that interact with ordinary matter primarily through annihilation with large cross-sections. A combination of terrestrial, astrophysical, and cosmological observations constrain a portion of the anti-macro parameter space. However, a large region of the parameter space remains unconstrained, most notably for nuclear-dense objects.


Introduction
The evidence for dark matter is overwhelming (see, e.g., [1] and references therein), but the nature of dark matter remains one of the great unsolved mysteries of modern cosmology. In recent years the authors have explored the proposition that the dark matter might be macroscopic, in the sense of having a characteristic mass M x and cross-sectional area in the gram and cm 2 range, respectively [2,3,4,5,6,7,8,9]. The dark matter constituents in this model are called "macros." The macro model has two undetermined parameters, the macro mass M x and the interaction cross section σ x . The dominant interaction is assumed to be elastic scattering, with σ x taken to be the geometric crosssection of the macro.
We begin by first briefly reviewing the existing constraints on macros derived in previous works. For macro masses M x ≤ 55 g careful examination of specimens of old mica for tracks made by passing dark matter [2,10,11] has ruled out such objects as the primary dark-matter candidate. For M x ≥ 10 23 g, a variety of microlensing searches have constrained the abundance of macros [12,13,14,15,16] from a lack of magnification of sources by a passing macro along the line of sight of the observer.
A large region of parameter space was constrained by considering thermonuclear runaways triggered by macros incident on white dwarfs [17]. However, it was later shown [8] that the excluded region of macro parameter space for macros providing all of the dark matter was too large, and more accurate constraints were placed. Dark matter-photon elastic interactions were used together with the Planck Cosmic Microwave Background (CMB) data to constrain macros of sufficiently high reduced cross-section σ x /M x [18]. The region of parameter space where macros would have produced obvious devastating injuries was also constrained [6].
In addition to these constraints, limits from possible future observations have also been proposed. Ultra-high-energy cosmic ray detectors that exploit atmospheric fluorescence could potentially be modified to probe parts of macro parameter space [4], including macros of nuclear density. This analysis has led to constraints being placed using networks of cameras that were originally built to study bolides, i.e. extremely bright meteorites [7]. Some of us have also suggested how the approach applied to mica [10,11] could be applied to a larger, widely available sample of appropriate rock [5], and used to search for larger-mass macros. In addition to that, we have identified additional regions of parameter space constrained by the duration between back-to-back superbursts (thermonuclear runaway on the outer surface of a neutron star) [8].
It is unlikely that macro masses beyond ∼ 10 9 g could be probed by any purpose-built terrestrial detector assuming even an observation time of a century and a target area the size of the Earth. Terrestrial probes (eg. ancient rocks [5,10,11]) could have been continuously exposed for up to ∼ 3 × 10 9 years, but we are unlikely to carefully examine the more than 1 km 2 that would be needed to push beyond M x ∼ 10 9 g. It will therefore require innovative thinking about astrophysical probes (eg. [17]) to probe the remaining parameter space at the very highest masses.
In this paper, we consider a related but phenomenologically very different macro model: the possibility that macros are composed of antimatter. We dub these objects "anti-macros". While some macro limits simply carry over to the anti-macro case, we will show that this model yields a rich variety of new phenomena, and we will derive corresponding limits on the anti-macro parameter space.
Anti-macros are most likely composites of more fundamental anti-particles. An intriguing possibility is that the anti-macros could be made of Standard Model quarks or baryons bound by Standard Model forces. This suggestion was originally made by Witten [19], in the context of a first-order QCD phase transition early in the history of the Universe. Others have suggested non-Standard Model versions of such nuclear objects and their formation, for example incorporating the axion [20,21,22,23,24,25,26,27]. The Axion Quark Nugget (AQN) model is the most well-studied model of antimatter macroscopic dark matter in the literature. These nuggets can be made of matter as well as antimatter during the QCD phase transition. A direct consequence of this feature is that the dark matter and baryon densities will automatically assume the same order of magnitude without any fine tuning. However, the nuggets in the AQN model possess a high reflectivity owing to the large potential of the confining layer of axions [21]. In this manuscript, we consider a more generic class of anti-macros in which the anti-macro is not bound by some external layer but by a force sourced by the anti-macro components themselves. Hence, we will assume negligible reflectivity. However, we remain agnostic about the details of this binding force and consider only the phenomenology of such objects.
Anti-macros corresponding to the models mentioned in the previous paragraph would most likely have densities that are comparable to nuclear density (which we take to be ρ nuclear ≈ 3.6 × 10 14 g cm −3 ). This is much higher than ordinary "atomic density" (ρ atomic ≈ 1 g cm −3 ), but much lower than the density of black holes of masses in the range we consider. Although anti-macros of approximately nuclear density are of particular interest, other densities are not excluded at this point, so we will consider the full range of possibilities for M X and σ X .
The sensitivity of a detector to anti-macros depends on the energy transferred when the anti-macro transits the detector. Consider an anti-macro with cross section σ X passing through a detector. The energy per unit length deposited by a macro through annihilation in the detector is where ρ is the density of the target and κ is introduced, generically, to account for the fraction of the annihilation energy that is deposited as heat into the surrounding medium. In the case of the AQN, κ 1 in most cases of interest, and the energy deposition is highly suppressed. An order of magnitude estimate is typically κ ∼ 10 −12 (see Appendix C in reference [26]). For such objects, the energy deposition is similar to the case of ordinary macros [2,3,4,5,6,7,8,9] and the constraints presented in those references apply to both macros and anti-macros. The energy transfer expression for ordinary macros is where v x ∼ 250 km s −1 is the speed of the macro. Thus, anti-macros with κ ∼ 1 are expected to deposit ∼ 10 6 times more energy in a target than macros of the same cross-section. They should, in general, be easier to observe. The physical reason for this difference is that the anti-macro collisions with ordinary matter convert rest energy into thermal energy, while macro scattering off of ordinary matter is purely elastic.
To estimate the fraction of energy that thermalizes in the anti-macro, we consider pp annihilation. This predominantly results in multi-pion states, with 43% of the final states including 2 charged pions and 49% including 4 charged pions [30], for an average charged pion multiplicity of 3, and a neutral pion multiplicity of 2 [31]. The charged pion lifetime is 3 × 10 −8 s, while the neutral pion lifetime is 10 −16 s [1]. Thus, all of the neutral pions decay essentially where they were produced into 2 high energy gamma rays [1], while the charged pions or their decay products are likely to escape the macro. We therefore expect that 4 gamma ray photons carrying ∼ 100 MeV are produced per annihilation interaction. Following [27], such photons are expected to thermalize within the anti-macro, resulting in an emission temperature of T surf = 10 7 K from the non-degenerate part of the positron atmosphere, with the emission spectrum expected to be strongly peaked at 1 keV energies. Thus, κ ∼ 0.4.
The preceding arguments are relevant to anti-macros that are able to travel through the overburden of a detector and leave a detectable signal within it. The speed of an anti-macro traveling through a medium can be determined from Newton's second law In the absence of any accelerating forces Equation (3) evolves as where ρx = ρdx is the column density encountered by the anti-macro passing through the medium. Anti-macros with too high a value of σ x /M x would not have been expected to encounter a detector but rather fall vertically reaching some terminal velocity. Indeed for σ x /M x ≥ 10 −12 g cm 2 an anti-macro is not expected to reach far below the surface of the Earth, while for σ x /M x ≥ 10 −9 g cm 2 , the anti-macro is not expected to penetrate to the bottom of the atmosphere with any of its initial kinetic energy. The rest of this paper is outlined as follows. In Section II, we constrain a wide region of parameter space by requiring that the energy deposited by antimacros in the early Universe not alter the CMB significantly. In Section III, we perform a similar calculation for big bang nucleosynthesis (BBN), where the main constraint in this case is the requirement that annihilation with helium-4 not overproduce lighter elements. In Section IV, we place constraints for anti-macros that would have caused unexpected deaths in the well-monitored population of the Western world over the past decade. In section V, we discuss constraints on anti-macros that are of a similar order-of-magnitude to that of regular macros.

Cosmic Microwave Background Constraints
WIMPs annihilating and dumping energy into the photon-baryon fluid would drastically alter the CMB, leading to changes in both the temperature and polarization power spectra. As such, CMB anisotropies offer an opportunity to constrain the nature of dark matter. Constraints have been placed on the thermally averaged cross section of WIMPS based on the observed spectrum [28]. We will use this result to constrain anti-macros annihilating with protons in the pre-recombination fluid. The way in which dark matter annihilations heat the fluid depends on the nature of the cascade of particles produced following the annihilations. The fraction of the rest mass energy that is injected into the gas can be modelled by an efficiency factor, f (z). Computations for various channels can be found in [29]. For all cases considered in [29], f (1100) ∼ 0.3 and in some cases it is closer to unity. As discussed earlier, in the case of antimacros, the fraction of energy that we expect to contribute to the heating of the surrounding medium is κ ∼ 0.4. Thus, we neglect a detailed calculation and take both values to be equal to each other for simplicity.
Anti-macros would consist of a reasonable fraction of anti-protons that would annihilate with incident protons in the pre-recombination fluid. For WIMPS, which were generically considered in placing the bounds in reference [28], the energy density injection rate iṡ The analogous quantity for macros iṡ By equating the two energy injection expressions and utilizing the bounds from [28], we can determine the constrained region for anti-macros. This bound can be expressed as Macros above this bound, plotted in grey in Figure 1, would have deposited too much energy in the early Universe and altered the observed CMB spectra and are thus ruled out as being all the dark matter.

Big Bang Nucleosynthesis Constraints
The effects of antimatter injection on BBN have long been a topic of study [32,33,34,35,36,37,38]. Anti-macros can affect BBN in several different ways. (See, e.g., Ref. [38] for a detailed discussion). They can annihilate with free protons and neutrons prior to BBN (T > 10 10 K), and they can annihilate with bound nucleons inside nuclei after BBN (T < 10 9 K). Photons or other particles from the annihilation process can themselves interact with (and fission) nuclei, and the light nuclei resulting from these fission and annihilation processes can yield alternative nucleosynthetic pathways. Here we will attempt only a rough order-of-magnitude estimate of these constraints.
We will derive the limit that can be placed on anti-macros from the annihilation of the anti-macro with a proton bound in a 4 He nucleus, along with the requirement that 3 He not be overproduced by this process. We will not consider the additional photofission of 4 He from annihilation-produced photons because the emission spectrum around this epoch will peak in the several hundred keV range, which is far below the scales of ∼ 10 MeV needed to fission 4 He. The tail of the distribution may be important and this will be the subject of a follow up study. As such, we also do not consider the production of 6 Li from this 3 He; both of these processes are discussed in detail in Ref. [38]. Because the 4 He abundance produced by BBN is ∼ 10 5 times the BBN 3 He abundance, even a tiny fraction of destroyed 4 He can be ruled out. The number density of 3 He nuclei, relative to hydrogen, produced by the process in Eq. (8) at time t is given approximately by Because the macros are much more massive (and therefore moving more slowly) than the helium-4 nuclei, we can set v ∼ kT /m4 He . We also have 4 (He/H) ≈ 1/12 and n x = ρ DM /M x , where we will assume in this case that the antimacros make up all of the dark matter (Ω x ≈ 0.25). Then we have n x ≈ 1.0 × 10 −31 cm −3 T 3 /M x g/K 3 . The time t is related to the temperature T during the epoch shortly after BBN by the relation t = 1.8 × 10 20 sec(K/T ) 2 . Combining all of these, we obtain Eq. (10) gives roughly the helium-3 production produced by anti-macro annihilation on helium-4 at the temperature T . To get the best constraint, we set T to be the temperature at which BBN terminates, T ≈ 8 × 10 8 K, giving The primordial abundance of helium-3 is poorly understood; Ref. [36] uses the constraint ( 3 He/H) < 3 × 10 −5 , while [38] takes ( 3 He/H) to be less than the primordial deuterium abundance (D/H) ≈ 2.6 × 10 −5 [39]. Substituting the CMB bound into Eq. (11) gives ( 3 He/H) < 3.0 × 10 −5 , which suggests that the CMB and BBN give similar constraints. However, given the crudeness of the current calculation, the CMB limit is more trustworthy. It is possible that a more detailed calculation utilizing a numerical calculation of the primordial element abundances could yield a tighter limit than our CMB constraint, but we would not expect an order of magnitude difference. This will be checked in a future study.

Human Detectors
For a range of regular macro masses and cross sections, collisions with the human population would have caused a detectable number of serious injuries and deaths with obvious and unusual features, while there have been no reports of such injuries and deaths in regions of the world in which the human population is well-monitored. The region of parameter space where macros would have produced a devastating injury similar to a gunshot wound on the carefully monitored population of the Western world was thus constrained [6]. We use this same null result to constrain the same range of anti-macro masses, which deposit significantly more energy in human tissue. At 1 keV energies, the photons possess a scattering length in human tissue of roughly 10 4 cm [40]. Thus, 1 − exp(−0.001) ≈ 10 −3 of the energy from the emission at the surface of the anti-macro is deposited in a cylinder of radius 10 cm. To determine the total energy deposited, we multiply dE/dx in Eq. (1) by the path length of the macro inside the human body, which we assume to be ∼ 10 cm. The energy deposited in this radius of length 10 cm is where α = 0.001 is the fraction of energy that remains in the human tissue through the 1 keV photons that interact within 10 cm of the anti-macro surface and deposit their energy in the human tissue and κ = 0.4 is the fraction of energy that thermalizes and is eventually deposited into the human tissue. Requiring ∆E ≥ 100 J = 10 9 ergs, a bound used previously [6] to constrain regular macros, we can rule out σ x ≥ 10 −12 cm 2 To constrain M x , we consider the number of encounters between an antimacro and the total number of humans in our sample where f = Ω x /Ω DM is the maximum allowed abundance of anti-macros that can contribute to the dark matter energy density in the Universe, ρ DM = 5 × 10 −25 gcm −3 [41] and we have considered a monochromatic distribution of antimacros with n x = ρ DM /M x , N = 8 × 10 8 is the total number of humans in our sample A ∼ 1 m 2 is the cross-sectional area of a human, t = 10 years is the exposure time of our detector and v x is the (relative) speed between the anti-macro and a human. (For more details, see the corresponding discussion in Ref. [6]). Since the impact of an anti-macro is a Poisson process, the probability P (n) of n impacts over the exposure time t follows the Poisson distribution where N events is the expected number of events per interval and was given in Equation (13). If no events are observed, then the value N events ≥ 3 may be ruled out at 95% confidence, i.e. by requiring that the probability of no detected signals be less than 5%. This allows us to constrain the abundance of anti-macros as a function of the mass M

Other constraints
For many physical processes, the limits that can be placed on anti-macros are identical to the corresponding limits on macros. We discuss those limits briefly here.

Microlensing
For very large masses (M x ≥ 10 23 g), a variety of microlensing searches have constrained heavy composite object candidates [12,13,14,15,16] to make up at most a sub-leading component of the dark matter, regardless the nature of their non-gravitational interactions.

Paleo-detectors
Macros that would have penetrated a few km into the Earths crust would have left tracks in ancient muscovite mica. Searches for grand-unified theory magnetic monopoles [10,11] sought to detect lattice defects left in ancient mica through chemical etching techniques. These limits were used to place limits on regular macros, over a wide range of cross-sections, up to M x = 55 g [2] (this idea of using paleo-detectors to detect dark matter is not new; see [42,43] for an approach to detect WIMPs with paleo-detectors). We expect the elastic scattering cross-section of such objects to be of the same order as the annihilation cross-section for anti-macros [44] 1 and so the same results apply as in the case of regular macros.
We have also suggested that, for appropriate M x and σ x , the passage of a macro through granite would form long tracks of melted and re-solidified rock that would be distinguishable from the surrounding unmelted granite [5]. A search for such tracks in commercially available granite slabs is planned. Using the same reasoning that allowed us to use the null result from the etching of ancient muscovite mica, anti-macros of the same minimum cross-section would have left visible tracks in the granite and so we expect the same results of hold. Thus, the results from the search of slabs of mica will apply to anti-macros as well.

Thermonuclear runaway
As discussed in [17] (and references therein), for thermonuclear runaway to be ignited, there is a minimum sized region, λ trig that must be raised above a threshold temperature T crit ∼ 3 × 10 9 K, where λ trig is strongly dependent on density. Constraints were placed on elastically scattering macros using the continued existence of white dwarfs [17]. However, it was later determined that these constraints were too stringent; more accurate bounds were placed [8] although these bounds are subject to additional uncertainties. It has not been confirmed through numerical simulations that the conditions identified in [8,17] are indeed sufficient to initiate thermonuclear runaway, i.e., there remains some uncertainty whether in fact heating a region of size at least λ trig to T ∼ few ×10 9 K necessarily causes type 1A supernovae in white dwarfs and superbursts in neutron stars. In the case of anti-macros, the energy deposition is not expected to be much higher than the case of macros. This is because the emission temperature of the anti-macro is expected to be high enough that most of the positrons in the non-degenerate regime will be ionized. 2 Thus, λ trig is still determined primarily by the cross-section of the anti-macro similar to the case of regular macros [8]. Emission from denser regions near the core is highly suppressed (see Appendix 4 in [21]).

Discussion and Conclusions
We have considered a phenomenological approach and constrained the abundance of anti-macros over the relevant mass range based on several terrestrial, astrophysical and cosmological probes. Atomic density anti-macros are entirely ruled out by a combination of the CMB and microlensing constraints. Nuclear density macros are ruled out below 5 × 10 4 g, and possibly at some higher range mass windows through thermonuclear runaway. However those results are subject to additional uncertainties as discussed in Section IV.
We also note that such anti-matter objects could in principle alter the reionization history, as discussed in [45]. However, the most stringent constraints come from the early Universe due to the large number density of both the antimacros and the protons. For objects below the grey bound in Figure 1, which are significantly denser than atomic density objects, the number of encounters of an anti-macro with hydrogen atoms was found to be low enough that it will not significantly alter the recombination history nor produce a higher extragalactic photon background around the 100 MeV range. Atomic density Nuclear density Black holes Figure 1: Constraints on the anti-macro cross section and mass (assuming the anti-macros have a single mass). Constraints in brown come from various microlensing experiments, in yellow from a lack of tracks in an ancient slab of mica [10,11], in cyan from a lack of a signal in the Icecube experiment [27]; in grey from the Planck Cosmic Microwave Background, in red from a lack of human impacts, in light blue from thermonuclear runaway in white dwarfs and in light purple from a lack of superbursts over a period of a decade on the superburster 4U 1820-30. The hatched light purple region shows the region of parameter space that may be constrained from an analysis of all known local X-ray binaries and a better understanding of the mean background superburst rate [8]. The black and green lines correspond to objects of constant density 1 g cm −3 and 3.6 × 10 14 g cm −3 respectively. Black hole candidates lie on the magenta line.