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Quark nugget dark matter : cosmic evidence and detection potential Lawson, Kyle 2015

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Quark Nugget Dark Matter : CosmicEvidence and Detection PotentialbyKyle LawsonB.Sc., The University of Western Ontario, 2005M.Sc., The University of British Columbia, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)February, 2015c© Kyle Lawson 2015AbstractI present a dark matter model in which the dark matter is composed of veryheavy “nuggets” of Standard Model quarks and antiquarks. This modelwas originally motivated by the fact that the matter and dark matter massdensities are observed to have similar scales. If these two forms of matteroriginate through completely distinct physical processes then their densitiescould easily have existed at vastly different scales. However, if the darkand the visible matter are co-produced, this similarity in scales is a naturaloutcome. In the model considered here dark matter and the baryonic mattershare an origin in Standard Model strong force physics.The main goal of this work is to establish the testable predictions ofthis model. The physical properties of the nuggets are set by well under-stood nuclear physics and quantum electrodynamics, allowing many definiteobservable consequences to be predicted. To this end, I devote special at-tention to the structure of the surface layer of the nuggets from which themajority of observable consequences arise.With this basic picture of nugget structure in place, I will discuss theconsequences of their interactions with a number of different environments.Particular attention is given to the galactic centre and to the early universe,as both are sufficiently dense to allow for significant levels of matter-darkmatter interaction. The emitted radiation, in both cases, is shown to beconsistent with observations.Finally, I discuss the consequences of a nugget striking the earth. In thiscontext, I will demonstrate that the nuggets produce effects observable incosmic ray detectors. Based on these considerations, I discuss the nuggetdetection potential for experiments primarily devoted to the study of highenergy cosmic rays.iiPrefaceMuch of the original research contained in this dissertation has previouslybeen published in several versions.Many of the details of Chapter 2 are adapted from work previously pub-lished in, The Electrosphere of Macroscopic ’Quark Nuclei’: A Source forDiffuse MeV Emissions from Dark Matter, (Forbes, Lawson and Zhitnitsky,2010 [40]) as is the material of Chapter 4. Many of the detailed nuclearphysics calculations of that work were due to Michael Forbes, my primaryefforts came in translating those results in observational consequences. Theresults of Chapter 5 are taken from, Isotropic Radio Background from QuarkNugget Dark Matter, (Lawson and Zhitnitsky, 2013 [81]). The underlyingresearch was originally proposed by Ariel Zhitnitsky and conducted collab-oratively. Many of the details of Chapter 6 are taken from independentresearch presented in the solo author papers, Quark Matter Induced Exten-sive Air Showers [73] and, Atmospheric Radio Signals From Galactic DarkMatter [77]. This research arose out of, and was strongly influenced by,extensive discussions with Ariel Zhitnitsky, but was pursued largely inde-pendently.I have previously presented aspects of this work at; the Winter Nuclearand Particle Physics Conference, Banff, AB (2009); the Lake Louise WinterInstitute, Lake Louise, AB (2009) [75]; UWO Physics and Astronomy De-partment Colloquium, London, ON (2009); the Winter Nuclear and ParticlePhysics Conference, Banff, AB (2010); the Lake Louise Winter Institute,Lake Louise, AB (2010) [76]; APS Northwestern Meeting, Vancouver, BC(2012); the International Symposium on Very High Energy Cosmic Ray In-teractions, Berlin, Germany (2012) [78] and CERN, Geneva, Switzerland(2014) [79]; as well as the Snowmass meeting, Stanford, California (2013)[82].iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi1 Dark Matter and Baryogenesis . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Dark matter as compact composite objects . . . . . . . . . . 41.3 Baryogenesis through charge separation . . . . . . . . . . . . 62 Nugget Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Quark matter . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 The electrosphere . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 The Boltzmann limit . . . . . . . . . . . . . . . . . . 172.2.2 The ultrarelativistic limit . . . . . . . . . . . . . . . . 172.3 Charge equilibrium . . . . . . . . . . . . . . . . . . . . . . . 183 Motivation from Galactic Observations . . . . . . . . . . . . 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Thermal emission : the WMAP “haze” . . . . . . . . . . . . 253.3 Surface proton annihilations : the Chandra x-ray background 303.4 Low momentum electron annihilations : the galactic 511keVline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Near surface electron annihilations : the COMPTEL MeVexcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 Relative Emission Strengths . . . . . . . . . . . . . . . . . . . 454.1 Electron-positron annihilation . . . . . . . . . . . . . . . . . 45ivTable of Contents4.2 Nuclear annihilations . . . . . . . . . . . . . . . . . . . . . . 485 Cosmological Consequences . . . . . . . . . . . . . . . . . . . 535.1 Temperature evolution . . . . . . . . . . . . . . . . . . . . . 565.2 The isotropic radio background . . . . . . . . . . . . . . . . . 585.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 636.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2 Air shower production and scale . . . . . . . . . . . . . . . . 666.3 Atmospheric fluorescence . . . . . . . . . . . . . . . . . . . . 716.4 Lateral surface profile . . . . . . . . . . . . . . . . . . . . . . 766.5 Geosynchrotron emission . . . . . . . . . . . . . . . . . . . . 797 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93AppendicesA Cosmology Review . . . . . . . . . . . . . . . . . . . . . . . . . 104A.1 The expanding universe . . . . . . . . . . . . . . . . . . . . . 104A.2 The cosmic microwave background . . . . . . . . . . . . . . . 106A.3 Radiation from distant objects . . . . . . . . . . . . . . . . . 107B QCD Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 109B.2 The QCD vacuum . . . . . . . . . . . . . . . . . . . . . . . . 110B.3 The strong CP problem and axions . . . . . . . . . . . . . . 111B.4 Domain walls in QCD . . . . . . . . . . . . . . . . . . . . . . 112B.5 Stability and lifetimes . . . . . . . . . . . . . . . . . . . . . . 114C Nugget Thermodynamics . . . . . . . . . . . . . . . . . . . . . 117D Muon Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 120vList of Figures2.1 Schematic picture of the structure of a quark nugget showingthe sources of associated emission processes . . . . . . . . . . 132.2 Radial density profile of the electrosphere of an antiquarknugget as well as the associated annihilation rate of incidentgalactic electrons. . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 The diffuse γ-ray spectrum of the inner galaxy as observedby COMPTEL and EGRET. . . . . . . . . . . . . . . . . . . 434.1 Survival fraction of incident galactic electrons as a functionof penetration depth within the electrosphere. . . . . . . . . 474.2 Spectral density emitted by electron annihilations within theelectrosphere of an antiquark nugget compared to observa-tions by comptel. . . . . . . . . . . . . . . . . . . . . . . . 495.1 The rise in extragalactic sky temperature as observed by thearcade2 experiment overlaid with the predicted contributionfrom quark nugget dark matter. . . . . . . . . . . . . . . . . . 626.1 Total rate of nuclear annihilations within an antiquark nuggetwithin the earth’s atmosphere . . . . . . . . . . . . . . . . . . 706.2 Laterally integrated particle flux as a function of height foran antiquark nugget at a height of 1km. . . . . . . . . . . . . 746.3 Integrated particle flux generated by a quark nugget as a func-tion of height above the earth’s surface. . . . . . . . . . . . . 756.4 Integrated particle flux as a function of height above theearth’s surface for various particle production saturation tem-peratures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.5 Total particle flux received over the entire air shower as afunction of radial distance from the shower core. . . . . . . . 796.6 Laterally integrated flux of particles recorded at the earth’ssurface as a function of time. . . . . . . . . . . . . . . . . . . 80viList of Figures6.7 Total electric field strength generated by geosynchrotron emis-sion as received 250m from the shower core. . . . . . . . . . . 846.8 Total electric field strength generated by geosynchrotron emis-sion as a function of radial distance from the shower core. . 856.9 Total radio band electromagnetic intensity received on theshower axis for various heights of the antiquark nugget. . . . 866.10 Total radio band electromagnetic intensity received on showeraxis as a function of time . . . . . . . . . . . . . . . . . . . . 876.11 Projected ANITA constraints on the flux of antiquark nuggets. 89viiChapter 1Dark Matter andBaryogenesis1.1 IntroductionThe matter content of the universe is predominantly in the form of darkmatter, which carries an average energy density five times larger than thatof the visible matter1. The large scale behavior of the dark matter is wellunderstood in terms of its gravitational interaction with the surroundingvisible matter. However, at present we have no microscopic understanding ofits properties or origin. While many theories have been put forward none hasreceived any observational verification2 and most require the introductionof new physics beyond the standard model for which we have no evidence.As such, the physical nature of the dark matter remains one of the mostimportant open questions in cosmology.The physical properties of visible matter, as represented by the StandardModel, are better understood than those of the dark matter. However, thereremains an important outstanding question related to the origin of the visiblematter content of the universe. While the basic physical laws apparentlytreat matter and antimatter identically, we observe a large global asymmetrybetween the two - namely the visible universe is almost entirely composed ofmatter with only trace amounts of antimatter. The source of this asymmetryhas not been established, nor does it have any mechanism by which toarise within the context of Standard Model physics. The process by whichthe present day matter dominated universe emerges from a (presumably)1Current best measurements suggest a universe dominated by the cosmological con-stant (Λ) and cold dark matter, the so called ΛCDM cosmology, with the relative energydensities divided as ΩΛ = 0.7181,ΩDM = 0.236,ΩB = 0.0461 and negligible contributionsfrom photons and neutrinos [56]. Here Ω is the fraction of the critical density representedby each of the components of the universe as defined in appendix A.2There have been some tantalizing recent results in ground based detectors, but thedata remains confusing and, at times, seemingly contradictory. The current state of theseinvestigations will be briefly reviewed at the beginning of chapter 6.11.1. Introductionmatter/antimatter symmetric initial state is known as baryogenesis, and isthe second outstanding question in cosmology to be addressed here.It is intriguing that, while the dark matter dominates over visible, it doesso by what is seemingly only a geometric factor: ΩDM ≈ 5Ωvis. Had thesetwo forms of matter originated at different epochs and through radicallydifferent physical processes their densities could easily be separated by manyorders of magnitude. This seemingly coincidental common scale in energydensity may, in fact, hint at a deeper relation between baryogenesis andthe origin of the dark matter. Motivated by this possible connection thiswork will consider a model in which the dark matter emerges as a necessarybyproduct of baryogenesis. This connection is made possible if the baryonasymmetry develops at the QCD3 phase transition in the early universe. Thedetails of this model will be outlined in the following section. In this context,it is important to note where the dominant component of the visible mass ofthe universe resides. The majority of visible matter is baryonic and, as such,its mass is determined primarily by the QCD binding energy associated withprotons and neutrons, rather than the masses of the individual quarks4. Assuch, the similar mass densities of the visible and dark matter may imply alink between the dark matter and the QCD scale.Before turning to the specific model to be considered here, it will beuseful to review some general considerations relating to baryogenesis. Animportant measure of the matter/antimatter asymmetry is the baryon-to-photon ratio [56]η ≡nB − nB¯nγ= 6× 10−10. (1.1)Here nB and nB¯ are the observed baryon and antibaryon number densitiesand nγ is the photon density. This ratio will remain constant with theuniverse’s expansion once whatever process creates the baryon asymmetryceases to act and the CMB photons5 decouple from the matter content. Hadno form of baryogenesis occurred, the universe would contain equal numbersof protons and antiprotons, with a total density much lower than presentlyobserved. In fact, were the asymmetry not present baryon annihilation3Quantum chromodynamics (QCD) is the theory governing the strong force interactionsof the Standard Model. Some of its basic details are outlined in appendix B.4In the Standard Model the light quarks within the proton and neutron acquire massesat the MeV scale through the Higgs mechanism. The observed mass of the nucleons ishowever, much larger than that of three quarks due to the contribution from the strongforce binding energy.5The cosmic microwave background (CMB), along with some other relevant cosmology,is reviewed in appendix A.21.1. Introductionwould continue until expansion dilutes the universe to the point where themean free path of a proton is longer than the Hubble scale. In this freeze outscenario the matter-to-photon ratio would be some ten orders of magnitudesmaller than presently observed with, nB = nB¯ ≈ 10−20nγ . It would seemthat there must be some process which is capable of generating the matterdominance observed today. It should be noted that the asymmetry as afraction of the total baryonic matter content, ∆n/N = (nB − nB¯)/(nB +nB¯), remains small until the QCD phase transition even if it originatesat an earlier epoch. Before the phase transition the early universe plasmacontained deconfined light quarks which are thermally abundant in numbersessentially equal to the photon abundance. After the phase transition, attemperatures below T ∼ 100MeV, the quarks are confined in nucleons whichare too heavy to be created in thermal collisions. At this point, the onlypossible interactions are inelastic scattering or the annihilation of matterwith antimatter and the ratio ∆n/N rapidly drops to its present day valueof ∆n/N ∼ 1. Thus, even if the baryon asymmetry is present before theQCD phase transition, it does not become an order one effect until this time.Any dynamical process which generates a baryon asymmetry must meetthe three Sakharov Conditions [105]. These are:• The violation of baryon number ,• Charge parity (CP) symmetry violation,• Non-equilibrium processes.The first condition allows for the creation or removal of baryons or an-tibaryons independently. CP violation is necessary as the relevant processmust preferentially remove antibaryons over baryons6. Finally these pro-cesses must be out of equilibrium in order to assure that antibaryon de-struction occurs at a different rate than its reverse process of antibaryoncreation.In the model of interest here the baryon asymmetry is only an apparentone. Consequently, it does not require the introduction of a baryon num-ber violating process and leaves the global baryonic charge of the universeunchanged. The required CP violation and non-equilibrium conditions areprovided by the physics of the QCD phase transition. To demonstrate howthis is possible, I will now turn from general considerations to the specificsof the model.6Charge Parity (CP) symmetry is the combination of the charge reversal and mirrorreflection symmetries.31.2. Dark matter as compact composite objects1.2 Dark matter as compact composite objectsSince the existence of dark matter was firmly established a wide variety ofmodels for its physical nature have been proposed. The majority of thesemodels assume that the dark matter is comprised of a new fundamentalparticle whose properties should be chosen to match the observational con-straints coming from dark matter searches. There is presently no evidencefor the existence of suitable beyond Standard Model particles (except possi-bly the existence of the dark matter itself) and their physical properties arenot well constrained. Rather than treating the dark matter as a new, yet tobe discovered, particle this work will consider the possibility that the darkmatter may be composite in nature and involve large numbers of knownparticles.The first proposal that dark matter may be not a new fundamentalparticle but conventional Standard Model particles in a novel phase wasthat described by Witten [122]. This model suggested that, at sufficientlylarge densities, the presence of strange quarks could make a quark matterstate energetically favorable to nuclear matter. This is possible becausePauli exclusion requires that each additional particle be placed in a higherenergy state. Thus, at some large density, it becomes favorable to beginadding heavier strange quarks rather than ultrarelativistic u and d quarks.Droplets of matter in this phase are referred to as strangelets and have beensuggested as a dark matter candidate. There is, at present, no evidence forthe existence of strangelets but their possible stability is not ruled out (seefor example [85, 86].)The original suggestion for the formation of strangelets [122], in sufficientnumbers to explain the dark matter, required that the QCD phase transitionbe first-order (though it is now believed to be a second-order crossover.) Inthis picture, bubbles of the nucleating low temperature phase grow and it isthe pressure of the bubble walls that is responsible for compressing regionsof the high temperature phase to sufficient density that they form strangequark matter. In addition to the now disfavored requirement of a first-order phase transition, this model also requires more efficient cooling of theshrinking bubbles than is theoretically predicted7.The model under consideration here is a modification of this original pro-posal in that it invokes axion domain walls, which may form at the phase7The strangelet model also requires that baryogenesis occur at an early epoch of theuniverse’s history and provides no mechanism for generating the observed asymmetry.While not an explicit failure of the model, which was intended only to explain the darkmatter, it does leave unanswered one of the fundamental problems to be addressed here.41.2. Dark matter as compact composite objectstransition, as a means by which to compress the high temperature quarkgluon plasma to sufficient densities to form quark matter (the propertiesand dynamics of these walls are reviewed in appendix B.4.) The resultingobjects, known as quark nuggets or antiquark nuggets depending on theircomposition, will end up in a high density quark matter state which is nowthought more likely to be a colour superconductor than a form of strangequark matter. If the QCD phase transition results in the production ofnuggets of quarks and antiquarks, the question becomes whether these ob-jects could possible serve as the dark matter.Large inherent uncertainties remain in the formation process and in thehigh density structure of the QCD phase diagram (particularly in the case ofnonzero θ which, as discussed below, will be of relevance here8.) However, itis not necessary for our purposes to present an in depth discussion of quarkmatter. Instead, I will use generic considerations and energy scales to extractthe basic properties any such objects must display. A brief review of theproperties of quark matter and colour superconductivity, with a particularemphasis on the properties relevant to the present analysis, is given in section2.1.This idea may at first seem counterintuitive as these objects are macro-scopically large and interact strongly with visible matter. However, there isa range of allowed parameter space in which models of this type of feasible.The basic idea is that gravitational probes of the dark matter are sensitive tothe mass density of the dark matter, while all possible non-gravitational ob-servations depend on the product of the number density and the interactioncross section. As such, it is not the interaction strength which is obser-vationally constrained but rather the cross section to mass ratio (σ/M).Thus, even strongly interacting, macroscopically large objects may serve asthe dark matter, provided they are sufficiently dense.The local dark matter mass density is estimated at roughly 1 GeV/c2cm−3 (i.e. ∼ 2×10−27 kg/cm3). In order to remain bound to the galaxy, thedark matter should carry an average velocity near 200km/s as determinedfrom virial equilibrium. This translates to a dark matter flux ofΦDM = nDMvgal ≈(1GeV/c2MDM)1011m−2s−1. (1.2)For WIMP dark matter with a mass near the 100GeV/c2 scale this impliesa relatively large flux and, in order to avoid direct detection constraints,8The vacuum angle θ is a free parameter appearing in the fundamental theory of QCDwhich parameterizes the degree of CP violation present in the theory. For further detailssee appendix B.51.3. Baryogenesis through charge separationrequires that the dark matter is coupled to visible matter at a level belowthe electroweak scale. For comparative value, a state of the art dark matterexperiment such as CDMSII publishes exclusion limits up to, at most, adark matter mass at the TeV scale [14].Alternatively, if the dark matter is sufficiently massive, the flux maybecome small enough to evade detection even without the requirement of astrongly suppressed interaction strength. In the case of quark nuggets witha baryonic charge BN the flux expression 1.2 is more usefully formulated asΦNuggets ≈(1024BN)km−2yr−1. (1.3)Given that the nuggets must carry a baryonic charge larger (and possiblyseveral orders of magnitude larger) than 1024, these events are infrequentenough to avoid detection by conventional dark matter searches.Similar considerations apply in the case of constraints on the dark mattercoming from astrophysical observations. The frequency of direct scatteringevents is determined by the ratio of the nugget’s physical cross section tomass ratio, and the strength of their electromagnetic coupling to variousastrophysical plasmas scales with the charge to mass ratio. Both of theseare increasingly suppressed for increasing nugget mass. As such, the nuggetsbehave almost identically to any other type of collisionless cold dark matter.Their coupling to the lighter baryonic matter is primarily gravitational sothat they form extended dark matter halos rather than clumping as thevisible matter does.To place meaningful constraints on this class of dark matter modelsrequires either the indirect analysis of galactic and cosmological data ableto sample over large volumes, or the use of much larger direct detectionexperiments. These two complementary search techniques will be discussedin the following chapters.1.3 Baryogenesis through charge separationAs suggested above, the similar energy densities of the baryonic and darkmatter components of the universe may argue for some deeper connection intheir origins. This work draws on the possibility that both the baryon asym-metry and the dark matter may be generated at the time of the QCD phasetransition [95, 126, 127]. Some background on the QCD physics relevant tothe following discussion may be found in appendix B.61.3. Baryogenesis through charge separationAt temperatures well above the phase transition the universe containsa quark-gluon plasma rather than the discrete baryons observed today. Asthe temperature drops, the effective QCD coupling strength increases, andthe quarks are forced into colour singlet protons and neutrons which are thelow energy particles of the low temperature phase.The phase transition provides the requisite non-equilibrium physics and,as discussed further in appendix B, may also allow for sufficient CP violationto satisfy the second Sakharov condition. In the high temperature phase, theCP violating θ term is expected to have been non-zero, so that CP violatinginteractions were generally as common as those respecting CP symmetry.During the phase transition QCD physics contains no small parameters and,as such, all processes must occur at essentially the same rate9. After thephase transition the dynamics of the axion allow the value of θ to relax fromθ ∼ 1 to its present near zero value. At this point QCD becomes a CPpreserving theory as it is observed to be today. Once this relaxation hasoccurred the Standard Model (and many of its proposed extensions) allowsfor too little CP violation to explain the observed degree of baryogenesis.In this way, the QCD phase transition may satisfy the requirements of non-equilibrium physics and CP violation without contradicting present limitson the scale of strong CP violation. However, there is no mechanism at thisscale for explicitly violating baryon number conservation.Rather than relying on the introduction of a new baryon number vi-olating process, the baryogenesis model considered here preserves globalbaryonic charge, instead using CP violating processes to separate the mat-ter from the antimatter. In this sense it is not a “baryogenesis” processbut rather one of charge separation. A more detailed description of thisprocess is given in [126]. Possible evidence of a related process in heavy ioncollisions is discussed in [68]. In this picture axion domain walls, related tothe 2pi periodicity of the θ parameter, form at the QCD phase transitionand carry sufficient energy to compress the quarks and antiquarks of thequark-gluon plasma down to densities at or above the nuclear scale. As CPsymmetry is strongly violated along these walls, the reflection coefficientsof quarks and antiquarks may be quite different. The differential escapeprobabilities will result in an excess of either quarks or antiquarks insidethe contracting wall. The excess quarks remaining within the nugget, donot have antimatter particles with which to annihilate, and are compressedby the collapsing wall until the internal Fermi pressure becomes sufficient9For comparison the theory of QED includes the fine structure constant, the small valueof which favours processes involving the fewest possible photon-fermion interactions.71.3. Baryogenesis through charge separationto halt collapse. On formation these objects essentially become dark in themanner discussed above.As a natural consequence of the order one CP violation present whenθ '= 0 the rate of nugget formation is likely to differ from that of antinuggetformation by a geometric factor. If the efficiency of forming nuggets ofantiquarks is higher than for quarks the result, at the end of the nuggetformation process, will be a universe with an excess of antimatter bound inthe nuggets and a corresponding excess of matter in the plasma of hadronicmatter. Annihilation of the free matter and antimatter not confined tothe nuggets continues within the early universe plasma until the antimatterhas completely annihilated away and only the excess of matter remains. Itis this component which makes up the visible universe as observed today.It should be noted that the small value of the baryon to photon ratio inexpression 1.1 implies that the process of nugget formation need not behighly efficient, the vast majority of the original baryonic content of theuniverse does annihilate to photons. Observationally, the antiquark nuggetsmust be favoured over quark nuggets by a factor of ∼ 3/2. This would resultin a baryon distribution between visible matter, nuggets and antinuggets of,Bvis : Bn : Bn¯ ≈ 1 : 2 : 3 (1.4)consistent with the observed matter to dark matter ratio (ΩDM ≈ 5Ωvis)and a universe with zero net baryon number. While this ratio cannot beestimated with any level of precision, it can be argued that this order oneproportionality is to be expected. This is because the nuggets continue tointeract with the surrounding baryons as long as the temperatures are atthe QCD scale. At these energies all processes, including those violatingCP symmetry, are expected to occur at the same scale. As the value of thistemperature is critical in establishing the degree of baryogenesis it will befurther discussed below, a more extensive estimation may be found in [95].This general picture may be made more specific if we assume that therelaxation of the θ term occurs through the axion mechanism (a brief re-view of axion physics is given in appendix B or, for more details, see, forexample, the recent review article [109] and references therein.) In this casethe domain walls are associated with transitions in the axion field, and wemay estimate the basic properties of the nuggets from the assumed prop-erties of the (as yet unobserved) axion. The following discussion will givea qualitative outline of the formation of the nuggets in this scenario, goingonly as far as needed to motivate some of the basic assumptions required toestablish the phenomenological consequences of quark nugget dark matter.It is these consequences that are to be the primary focus of this work.81.3. Baryogenesis through charge separationAs discussed in appendix B the axion domain wall has a sandwich struc-ture, with a hard core capable of reflecting some fraction of the quarksincident on it. As these walls form, some will collapse into closed surfaceswhich then further collapse down to smaller sizes, condensing the nuggetsout of the quark gluon plasma. From basic considerations of this process weare able to estimate the size of the nuggets that will be produced.Across the axion domain wall the energy density will be at the typicalquark condensate scale:ρ ∼ mq < ψ¯ψ >∼ mqΛ3QCD. (1.5)Here mq is the quark mass, and < ψ¯ψ > is the vacuum expectation value(VEV) of the quark field. The quark VEV is generally expected to occur atthe characteristic energy scale of QCD interactions, ΛQCD ∼ 100MeV. Thethickness of the wall is set by the axion length scale, la ∼ m−1a , where mais the axion mass. It is the introduction of this new length scale into thedynamics of the phase transition which allows the development of macro-scopically large objects carrying a very large baryonic charge. Without sucha scale all structures would evolve at the much smaller femtometer scaleassociated with QCD energies.Suppose that, as argued above, there is a high density phase of hadronicmatter which is energetically favourable at low energies to free baryons byan amount ∆. In this case, the change in potential energy associated witha quark nugget of radius R and baryon number B is,U = 4piσaR2 −∆B. (1.6)Where σa is the surface tension of the axion domain wall. From this expres-sion we can see that there will be some minimum baryonic charge for whichthe energy cost of the domain wall is overcome by the lower binding energyof the quarks10. The radial size at which this occurs is,R = 2σanB∆= 2 mqma∆< ψ¯ψ >nB, (1.7)where nB is the baryon density in the nugget such that B = 43piR3nB. Thequark condensate, the nugget baryon density and the binding gap must beat the QCD scale up to geometric factors so that R ∼ mqmaΛQCD .10This analysis is conducted in a more rigorous way in [95] which also incorporatespressure terms and reflection coefficients at the domain wall. That analysis gives resultssimilar to those obtained in this simplified static case as all processes, with the exceptionof the axion wall, must occur at or near the scale set by ΛQCD.91.3. Baryogenesis through charge separationIf we assume that the axion mass falls in the allowed range of 10−6eV <ma < 10−3eV, then the nugget radius may vary by,10−6cm < RN < 10−2cm. (1.8)Assuming that the resulting quark nugget is of roughly nuclear density thismay be converted into a total baryon number for the nuggets of1023 < BN < 1033. (1.9)These values will be taken as an estimate of the basic scale at which thenuggets are likely to exist.Finally, a brief discussion of the baryon to photon ratio, as given inequation 1.1, is in order. At the phase transition all the quarks not confinedto nuggets form into baryons. Baryon-antibaryon pairs are too heavy tobe formed in thermal collisions, and annihilations begin to rapidly decreasetheir density. At this time the baryons remain energetic enough that theyare able to penetrate into the still forming nuggets. As the baryons cool, andthe nuggets settle into an ordered high density phase, the probability that abaryon will be reflected from a nugget grows. Once the baryon temperaturefalls below the superconducting gap of the quark matter the probability thata baryon or antibaryon can penetrate the nugget becomes very small, andthe nuggets effectively freeze out. At this time the baryon number is fallingrapidly, nB ≈ nB¯ ∼ exp(−mN/T ), where mn is the nucleon mass and T isthe temperature of the plasma. If the matter decouples from the nuggetsat a temperature Tf , then the scale of baryogenesis will be the same as thebaryon density at this temperature. In this case the baryon to photon ratiois given byη ∼(mNTf)3/2e−mN/Tf . (1.10)As it is exponentially dependent on temperature, this fraction can vary byseveral orders of magnitude across the possible physical values of Tf . Whilethis means that the value of η cannot be predicted with any precision inthis model, one can work backwards from the observed value and make aconsistency check on the model. The value of η, as given in equation 1.1,implies a nugget freeze out temperature of Tf ≈ 40MeV [95]. This value isfully consistent with the structure of quark matter as presently understood.The quark matter, and the domain wall which binds it, exist at the QCDscale set by the temperature at which chiral symmetry breaking and quarkconfinement occur. This transition happens at T ∼ 100MeV, and nugget101.3. Baryogenesis through charge separationformation must occur entirely below this energy scale. Once the nuggetshave started to form the next relevant energy scale is the binding gap ofthe colour superconducting phase. The exact value of the gap is not wellestablished, and depends on the form of quark matter realized in the nuggets.Across a wide range of possible quark matter phases the binding gap isfound to be in the few tens of MeV range [16, 17, 102]. Below this scalethe transmission coefficient at the quark matter surface falls rapidly, so thisshould represent a lower limit on the possible freeze out temperature for thenuggets. It should also be noted that the absolute lower limit on the freezeout temperature is set by the point where baryon-antibaryon annihilationwould cease due to the universe’s expansion, even if no form of baryogenesishad occured. This will happen when the baryon collision rate falls belowthe Hubble time, which coincides with a temperature of T ∼ 22MeV andwould result in a baryon to photon ratio ten orders of magnitude lowerthan observed. The limits set by the phase transition and the freeze outof nuclear annihilations between, 100MeV< T <20MeV naturally cover theenergy scale of nuclear physics, and the formation temperature of the quarknuggets should be expected to fall in this range. It should also be noticedthat, as the nuggets effectively decouple from the remaining nucleons atan energy in the tens of MeV range, they have no impact on big bangnucleosynthesis which occurs at energies an order of magnitude lower.While this discussion of nugget formation remains qualitative becauseof the inherent complexity of any quantitative details and the accompany-ing uncertainties, I will take the basic properties of the nuggets and theorder of magnitude estimates made here as indicative of the range of possi-ble physical properties for the nuggets and as an argument that they mayrepresent a viable dark matter candidate. These preliminary arguments willallow for the formulation of some basic properties of the nuggets, and allowestimations of their phenomenological consequences for present and futureobservations.11Chapter 2Nugget StructureThe structure of a quark nugget may be divided into two basic components;the central quark matter, composed of light standard model quarks; and thesurrounding electromagnetically bound leptons. This surface layer, knownas the electrosphere, consists of electrons in the case of a quark nugget andpositrons in the case of an antiquark nugget. These two structures, andtheir relevant properties, will be discussed below. Much of this discussionis influenced by previous work, my own participation was primarily to thedetailed calculations related to the structure of the electrosphere as discussedin section 2.2.For reference a schematic picture of the nuggets’ structure, and the waythat it maps onto various sources of electromagnetic emission discussed be-low, is given in figure Quark matterThe nuggets of this model are composed of quark matter of densities withina few orders of magnitude of nuclear density, ρQM ∼ 1−100 MeV fm−3. Thisdensity range is not sufficiently large that asymptotic freedom11 may be ex-ploited to study the ground state structure, which must instead be studiedusing more complicated non-perturbative means. A full determination ofthe structure of the QCD phase diagram remains an outstanding problem,and the exact form of quark matter realized in the nuggets must, therefore,remain uncertain for the present. Fortunately, the surrounding electrosphere(to be discussed in the following section) prevents direct observation of thequark matter surface in all relevant contexts. For this reason, the inter-nal structure of the nugget is mostly important for establishing the lowerboundary conditions of the electrosphere (such as surface density, electricfield strength and the total radius of the nugget) and for estimating the scale11The asymptotic freedom of QCD implies that, while strongly coupled at low ener-gies, the high energy limit of the theory is weakly interacting and may be treated usingperturbation theory [53, 99]. For a limited review of this phenomenon see appendix B.122.1. Quark matterAntiquark nugget structure.    Source of emission  e-e+e+e+e+511 keV~ 100 MeV - 1 GeVe+Antimattercolor superconductore+e+ee+e+p,n electrospheree1-20 MeVX rays ~10 keV thermalization(10−4 − 1)eVbremsstrahlung  radiation( largest fraction)q¯q → 2 GeVenergyγ(rare events)  (finite fraction)axion domain wall pressureFermi PressureR ∼ 10−5cm, B ∼ 1025Figure 2.1: Schematic picture of the structure of a quark nugget showingthe sources of various emission processes to be discussed in chapters 3 and5. Figure adapted from [82].at which nuclear annihilations, occurring within the quark matter, transferenergy to the electrosphere from which it is emitted in a modified form.Many of the details considered here were originally discussed in the con-text of strangelets and strange stars or quark stars [15, 67, 84]. These arehypothetical objects similar to neutron stars composed, either wholly orpartially, of quark matter rather than nuclear matter. Stars of this formare possible if, at sufficiently large densities, quark matter is energeticallyfavourable to nuclear matter. The quark matter of which these objectswould be composed is essentially identical to that which is found in thequark nuggets considered here and, as such, many of the results may becarried over.The form of quark matter most likely to be realized in the nuggets, isa colour superconductor. As in the case of a conventional superconduc-tor, this state occurs when a weak attraction between charge carriers nearthe Fermi surface causes the elementary charges to form pairs. In such132.1. Quark mattera state the fundamental modes are no longer individual charges but thesebound Cooper pairs12. In a sense, the pairing mechanism involved in acolour superconductor is easier to understand than that in a conventionalsuperconductor. Within the context of QCD there are several attractivequark-quark interactions (the most obvious being the mechanism by whichquarks are bound in colourless mesons and baryons) and these naturallybecome weak at large momenta. The paired quarks are now bosonic, andform a condensate which is the quantum ground state of the superconduc-tor. As individual quark pairs cannot form colour singlets, this groundstate breaks local colour SU(3) symmetry. At asymptotically large densi-ties, the Fermi surface quarks are sufficiently energetic that they interactin the weak coupling limit of asymptotic freedom, and the problem may betreated perturbatively. In the case where Fermi surface quarks are in theasymptotically free limit, and the difference in mass between the u,d ands quarks is negligible, it is found that the ground state is a colour-flavourlocked (CFL) superconductor [16]. In this state, the pairing mechanismconnects the colour and flavour indices of the quarks forming Cooper pairs.All fermionic excitations and all eight gluons are gapped at the 10-100MeVscale and, because the quarks carry electric charge, the photon also picks upa gap at this scale.Moving to lower densities the momentum carried by quarks at the Fermisurface drops, and the effects of the larger s quark mass begin to becomerelevant. At these densities the CFL phase is stressed by the increasedenergetic cost of adding an s quark, rather than a u or d, in a particularmomentum state. This stress favours a depletion of strange quarks andgives a net electric charge to the quark matter. Lowering the density alsocauses the coupling between quarks to run towards its vacuum value inthe strong coupling regime. As the interaction strength grows, ever higherorder interactions must be considered until any suitable description becomesfully non-perturbative. This makes it impossible to predict exactly how thephase structure of quark matter extrapolates between the asymptoticallydense CFL phase and the low density nuclear phase.For the purposes of the present work, which is to take a mainly phe-nomenological standpoint, the exact nature of the quark matter realized inthe core of the nugget is not a primary concern. It will simply be noted herethat, in order for this proposal to explain the presence of dark matter it is12At large densities fermions can only be scattered to states which are not alreadyoccupied. As such, the low energy interactions of a given system involve those states nearthe Fermi surface which can be scattered to an unoccupied higher momentum space statewith only a small additional energy.142.2. The electrospherenecessary that the nuggets must be stable on timescales longer than the ageof the universe. At present there are several proposed quark matter phaseswhich may meet this condition.One final component of the quark nugget which is unique to this modelis the axion domain wall, mentioned in section 1.3, and appendix B andanalyzed in detail in [41]. This structure is important in the initial formationof the quark nuggets, introducing the macroscopic length scale necessary inthis process, and may be important in maintaining their absolute stabilityon cosmic timescales. A more detailed description of the axion domain walland its properties and evolution is given in appendix B.Having established some basic properties of the core of the quark nuggets,I now turn to the surface layer of leptons, which will be more important inextracting observable properties of the nuggets.2.2 The electrosphereIndependent of the form of quark matter realized in the core of the quarknugget, the decreasing pressure near the quark surface will necessarily resultin the accumulation of a net electric charge at the quark surface, positive inthe case of quark matter and negative in the case of antiquark matter. Thisis evident in the case where the nugget is composed of nuclear matter but,even if the core of the nugget is in a charge neutral phase (for example theCFL phase) the falling chemical potential near the quark surface results ina depletion of s quarks relative to the much lighter u and d quarks. Thiswill, as in the purely nuclear case, result in a net electric charge.The structure of this outer layer of leptons has been considered previ-ously in the context of strangelets and quark stars [15, 67]. As the electricfields established by the quark matter in these situations will be similar tothose at the surface of a quark nugget many of these earlier results alsoapply in the model considered here. However, these earlier studies weregenerally limited to treating the leptons as massless and considering planeparallel geometries. As will be seen below, the low energy emission fromthe nuggets originates primarily from the lower density outer regions of theelectrosphere where the lepton density scales as ne ∼ (meT )3/2. A detailedtreatment of these emission mechanisms will therefore require a full descrip-tion of the electrosphere’s structure accounting for the mass of the electronand for thermal effects. To this end the remainder of this section will estab-lish the density profile of the electrosphere from the quark matter surfaceout to the low density limit far from the nugget. In this outer limit the full152.2. The electrospherespherical geometry of the nuggets will also need to be taken into account.The strong electric potential near a quark matter surface will support asurrounding distribution of leptons, much as typical nuclei are surroundedby a distribution of electrons. To model the electrosphere structure we beginwith the Poisson equation, which relates the electrostatic potential to thecharge distribution:∇2φ()r) = −4pien()r). (2.1)Here φ is the electric potential and n is the number density of charges inthe electrosphere. This may be re-expressed in terms of the lepton chemicalpotential µ()r) = −eφ()r). In the two examples to be considered below, I willdiscuss regions of the electrosphere close enough to the quark nugget that thespherical geometry of the nugget may be neglected and the density will varyonly with the height (z) above the quark surface. Under this simplificationI may reformulate the Poisson equation, in terms of chemical potential anddensity, asd2dz2µ(z) = 4pie2ne(z). (2.2)Outside the quark matter the only relevant charges are electrons andpositrons and their number density is determined by the local chemical po-tential:n(µ) = 2∫d3)p(2pi)3[ 11 + e(E−µ)/T −11 + e(E+µ)/T]. (2.3)Here E =√p2 +m2e is the particle energy. The first term in the integrandof equation 2.3 represents the contribution from electrons while the secondis due to positrons. In [40] the equations 2.1 and 2.3 were self consistentlysolved in the spherically symmetric case to obtain a solution to the leptondensity valid across all radial distances and dependent only on the chemicalpotential at the surface of the nugget. The results of this computation areshown in figure 2.2. The details of the extrapolation between the lowerdensity outer layers and the ultrarelativistic regime near the nugget surfacewill allow for the relative strength of the 511keV line and the MeV continuumto be exactly computed in the following chapter.Rather than an in depth discussion of the details of figure 2.2 and thecalculations behind it, I will simply demonstrate the basics of the calculationin two limiting cases for which analytic results may be obtained. In whatfollows these results are sufficient to discuss the 511keV and MeV bandemission profiles separately, while the full numerical treatment is necessaryfor an analysis of their relative scales.162.2. The electrosphere2.2.1 The Boltzmann limitFirst consider the low density regime far from the quark matter surface. Inthis regime, which will be denoted as the “Boltzmann” regime, the electricfield strength is screened by the high density inner regions of the electro-sphere and thermal effects come to dominate the lepton distribution. In thislimit we can take the number density in equation 2.3 to be,n(µ) ≈ 2∫d3)p(2pi)3 e(µ−me−p2/2m)/T ≈√2(meTpi)3/2e(µ−me)/T (2.4)Here me is the mass of the electron. The Boltzmann regime may extend overa substantial fraction of the electrosphere as seen in figure 2.2. To furthersimplify matters I will assume that we may neglect the curvature of thequark matter surface so that the density is dependent only on the heightabove the quark surface and we may use the plane parallel form of thePoisson equation. In terms of the number density this gives the expression,1nd2ndz2−1n2(dndz)2= 4piαTn (2.5)which is solved bynB(z) =T2piα1(z + z0)2. (2.6)Here z0 is the height within the electrosphere at which these approximationsbecome valid and above which the fall off in density is fixed by this expres-sion. This regime will persist so long as the height (z + z0) remains smallwith respect to the radial size of the quark nugget. Once the height becomescomparable to the nugget size the spherical terms in the Poisson equationbecome relevant and the fall off in the number density becomes exponentialwith distance from the nugget.2.2.2 The ultrarelativistic limitNear the quark matter surface the chemical potential is in the 10-100MeVrange, the average positron energy is also be near this scale and the restenergy in expression 2.3 may safely be neglected. Under astrophysical con-ditions the electrosphere temperature will also be well below the chemicalpotential so that the full density expression given in equation 2.3 reducesto,ne[µ] ≈ 2∫ µ0d3)p(2pi)3 ≈µ33pi2 . (2.7)172.3. Charge equilibriumThe extent of the ultrarelativistic regime is considerably smaller than thephysical size of the quark nugget itself so that we may also simplify thePoisson equation to the plane parallel limit as in the case of the Boltzmannregime. In this limit the number density falls as,nUR ≈µ303pi(1 + z/z0)3, z0 ≡√3pi2α1µ0. (2.8)This density expression may also be formulated in terms of the chemicalpotential,µUR =√3pi2α1z + z0(2.9)which may in turn be converted to an electric field strength near the quarksurface,E(z) = −1edµdz=√23piµ20(1 + z/z0). (2.10)Note that for typical values of µ0 this implies that the surface electric fieldscarry nuclear scale energy densities, a fact that will become important ina discussion of positrons ejected from the quark surface that will come insection 3.3.The number densities given in equations 2.6 and 2.8 may be used toestablish the production rates of various types of emission from the nuggetas discussed in the main body of this work as well as the thermal propertiesof the nuggets discussed in the following appendix C.2.3 Charge equilibriumAs discussed in section 2.2.1 the distribution of leptons, far from the quarkmatter surface, is influenced by the temperature of the nugget. As theoutermost leptons are only weakly bound this temperature also influencesthe total ionization levels of the nuggets. As the temperature increasespositrons further evaporate from the electrosphere. In the case of matternuggets the nugget will fall into a static equilibrium with the surroundinggalactic matter. The temperature will be determined by the rate of energyabsorption through collisions with surrounding matter and photons whichis balanced by the rate of thermal emission. The temperature will be low aswill the net ionization of the nuggets.In the case of nuggets of antimatter the situation is more complicated.The annihilation of galactic matter within the quark nuggets will increasethe temperature of the nuggets (thus affecting the ionization levels) while182.3. Charge equilibriumany difference in the flux of galactic ions and electrons onto the nuggetmay also change the overall charge dynamics. This net charge, in turn, isimportant in determining the rate at which charged particles annihilate withthe nugget.As the positron density increases gradually across the electrosphere, agalactic electron incident on an antiquark nugget annihilates with a prob-ability very near one. Conversely, the quark matter surface is very sharp,resulting in a relatively large probability of galactic ions being reflected.The preferential annihilation of positrons will begin to generate a negativecharge on the nuggets. This charge will increase until the electric field ofthe nugget is sufficient to prevent the escape of charged ion with a velocitytypical of the interstellar medium. Once this field strength is reached theion will become bound to the nugget and may reflect off the surface as manytimes as is necessary for annihilation to occur.This situation is complicated by the possibility of charge exchange inter-actions between the nugget and the incident ion. If such a charge exchangeprocess occurs it may neutralize the ion, for example converting a proton toa neutron. In this case the incident particle may escape without annihilat-ing. The ratio of charge exchange interactions to annihilation interactionswill set the relative rate at which electrons and baryons annihilate with thenugget. As the interstellar medium is primarily composed of hydrogen, thisratio would be very near one in the absence of charge exchange processes.However, the possibility of charge deposition without annihilation meansthat this ratio may be less than one. This process will be discussed furtherwhen we estimate the relative strengths of various forms of emission fromthe nugget is section 4.2.With the physical properties of the electrosphere established from con-ventional physical properties we are now in a position to discuss the ob-servational consequences of this dark matter model. These observationalconsequences will primarily arise through the annihilation of galactic mat-ter within an antiquark nugget.192.3. Charge equilibriumFigure 2.2: Radial density profile of the electrosphere of a quark nugget.The positron density in Bohr units are shown for nuggets with baryon num-ber 1020 (red), 1024 (black) and 1033 (blue). The solid curves assume anuclear density core while the dashed curves assume a density 100 timeslarger than nuclear. The thick black band is the density profile neglectingnugget curvature. The cyan curves show the relativistic (dotted) and Boltz-mann (dot-dash) approximations discussed in the text. The yellow bandindicates the region from which the microwave emission discussed in section3.2 originates. The upper two curves give the annihilation rate of incidentelectrons relative to the maximum positronium formation rate. These ratesare used to establish the relative emission strengths as discussed in chapter4. Figure taken from [40].20Chapter 3Motivation from GalacticObservations3.1 IntroductionIn the search for dark matter, indirect detection techniques rely on astro-physical observations to reveal the presence of dark matter through itspotential non-gravitational interactions. Thus far, no such detection hasyielded an unambiguous dark matter signal. There have, however, been sev-eral suggestive observations warranting further consideration. This chapterwill highlight several galactic observations, spanning many orders of magni-tude in energy, which have been suggested as possibly containing signaturesof the dark matter. Based on these observations it will be argued that thecontribution to the galactic spectrum of quark nugget dark matter, in themass range considered here, is fully consistent with present observationsand may offer a source for several observed emission features. The under-lying uncertainty in the diffuse galactic backgrounds means that none ofthese observations may be attributed to the dark matter with any certainty.However, the observations discussed below are generally taken as being in-dicative of the presence of emission sources which have not, at present, beendirectly identified. These sources may, with further investigation, prove to beconventional astrophysical populations but, at present, they have attractedinterest as possible indications of non-gravitational dark matter interactions.If the dark matter does consist of nuggets of quark matter then they couldprovide emission much like that observed. If, however, the apparent excessemission is found to be attributed to conventional astrophysical sources thenthese observations will serve to impose strong constraints on the existenceof quark nuggets.The material of this chapter serves primarily as background and moti-vation for the material to follow, and is primarily based on work done bymyself and others predating my thesis research. It is presented here strictlyfor completeness, readers interested in further details should consult the213.1. Introductioncited original works.The scale of the possible observational consequences of quark nuggetsis strongly suppressed by their small cross section to mass ratio. However,there is nothing fundamentally weak about the interactions of these objectswith the surrounding visible matter. As they are entirely governed by wellknown QED and nuclear physics, it is possible to calculate the emissionspectrum expected when a quark nugget or antinugget interacts with visiblematter in a particular environment. Once this emission spectrum is estab-lished, we are in a position to observationally constrain the allowed rangeof nugget mass scales. This is done by comparing the predicted spectrumto observations of regions where both the visible and dark matter densitiesare high. Following the standard terminology I will refer to this processfor constraining dark matter properties as indirect detection, that is, tech-niques in which the astrophysical consequences of a dark matter candidateare searched for, generally in the form of an additional component in diffuseemission.In this analysis, I will emphasize diffuse emission sources which mayarise from either self interaction of the dark matter or from the interactionof the dark matter with the visible matter of the interstellar medium. In thecase of self interacting dark matter (for example the annihilation of a darkmatter particle with its antimatter partner) the interaction rate is scaled bythe line of sight integral∫dr n2DM v σDM−DM (3.1)in which the integral runs over the thickness of the dark matter distribution,v is the relative velocity, nDM is the dark matter density and σDM−DM isthe dark matter self interaction cross section. Alternatively, for interactionsbetween dark and visible matter we have to include both the dark matterand visible matter distributions,∫dr nDM v σDM−visnvis (3.2)with the integral again running across the thickness of the interaction region.In this case the relevant cross section is that for interactions between visibleand dark matter. For most dark matter models, the contribution from darkmatter self interaction and interaction with visible matter can be of similarmagnitude unless the interaction strengths are tuned to suppress one or theother. However, in the case of quark nugget dark matter the self interactionrate is suppressed, with respect to interactions with visible matter, by the223.1. Introductionextra factor of the nugget baryonic charge appearing in the dark matternumber density. As such, the following section will focus on determiningthe spectrum generated by matter striking a quark nugget, rather than theinteraction between nuggets13.The dark matter number density, as it appears in equation 3.2, is notknown directly. The mass density of the dark matter is inferred from thekinematics of the visible matter and from simulations of large scale structureformation. It is generally assumed that the dark matter has a sphericallysymmetric distribution rather than tracing the disk and bulge structuresobserved in the the visible matter. The mass distribution is frequently takento have a Navarro-Frenk-White type profile [94] with the density scaling as,ρDM (r) ∼1r(1 + r/rs)2(3.3)where rs is a characteristic scale length of a given dark matter profile. Thisprofile give a good description of the dark matter distribution on large scales,but seems to predict a stronger than expected cusp in the galactic centre.Within numerical simulations the central divergence is regulated by the res-olution of the simulation, however the discrepancy with observation seemsto extend to scales beyond this resolution limit. Weak lensing measurementsstrongly favour a central, constant density, core to the dark matter distribu-tion. This distribution is expected independent of the actual form taken bythe dark matter. While they do carry a baryonic charge the nuggets will notbehave like conventional baryonic matter. Their small cross section to massratio prevents any significant level of clumping as discussed in section 1.2.This uncertainty in the structure of the dark matter should be kept in mindfor the following discussion, as it directly affects the scale and morphologyof any dark matter contribution to the direct spectrum.As the observational consequences highlighted below are strongly asso-ciated with the galactic centre, within the presumed core of the dark matterdistribution, and as the emission traces both visible and dark matter (asin equation 3.2) they will be strongly correlated with the visible matterdistribution with a slightly stronger spherical morphology favoured by thecontribution from the dark matter distribution.In addition to the underlying uncertainty in the dark matter distributionthe total scale of any quark nugget contribution to the spectrum will depend13This is further justified by the fact that the nuggets exist as complex many bodyobjects, represented by macroscopically large multiparticle wave functions. In any givencollision there is unlikely to be a large wave function overlap and thus the most likelyoutcome is simply elastic scattering.233.1. Introductionon the average baryon number of the nuggets. Once the dark matter massdistribution, ρ(r), has been estimated in a particular region the quark nuggetnumber density (which is the factor actually appearing in the line of sightintegral of expression 3.2) is given by,nN (r) =ρ(r)MN, MN = mBB, (3.4)where MN is the nugget mass and mB is the mean mass per baryon withinthe nuggets. The estimates to be made below are generally confined to thegalactic centre. In this region it will be assumed that the dark matter formsa relatively constant density, spherically symmetric, core with a mean massdensity comparable to that of the visible matter.Detailed estimations of the consequences of this model for the galacticspectrum have been worked out in a series of previous papers [96], [128],[80], [42], [40], [43] and [74]. In each of these works a particular source ofdiffuse galactic emission, centred on the galactic centre, was considered andfound to be consistent with emission generated by a galactic population ofquark nugget dark matter, provided that the nuggets carry a baryonic chargegreater than ∼ 1023. It is not the purpose of the present work to fully repeatthese estimates, but rather to reproduce only the basic arguments necessaryto motivate the further analysis which follows. For this discussion I will beginwith the lowest frequency contributions to the galactic spectrum and movefrom there to the highest. Several independent observations of apparentexcesses in diffuse emission are relevant here:1. WMAP has observed a possible excess in microwave radiation asso-ciated with the galactic centre [36]. This so called “WMAP haze”seems to require an additional diffuse microwave source or a harderthan predicted galactic synchrotron spectrum.2. The Chandra X-ray Observatory has measured diffuse keV emissionfrom the galactic centre [93]. This emission has been modeled as origi-nating from a hot diffuse plasma, but the temperature of such a plasmawould exceed the energy thought to be available.3. SPI/INTEGRAL show a strong 511 keV line associated with the galac-tic centre [60, 70]. This line indicates the rate of low momentumelectron-positron annihilations is higher than had been previously es-timated and more strongly spherical than anticipated based on knownpositron sources and propagation models.243.2. Thermal emission : the WMAP “haze”4. COMPTEL detects diffuse emission across the 1-20 MeV range thatexceeds the previous estimates based on cosmic ray interactions andthe decay of radionuclides [112].Each of the following four sections will describe the features of a particularband of observed diffuse emission. These features are then mapped onto theproperties of the nuggets discussed above. The resulting spectra, and theirrelative strengths, are uniquely predicted within the model with the onlyfree parameter being the size of the nuggets. The spectral properties of theemission are largely independent of this parameter which is responsible onlyfor the overall normalization of the emission spectrum14. As such the limitson any possible excess in each of the four emission bands discussed abovemay be translated into constraints on the minimum size of the nuggets.3.2 Thermal emission : the WMAP “haze”The Wilkinson Microwave Anisotropy Probe (WMAP) has produced fullsky temperature maps across the microwave band [30]. Of particular impor-tance to this work are observations made by WMAP of the galactic planein the tens of GHz range15[49]. These observations show an excess of diffuseemission across the galactic centre above what would be expected from ex-trapolating the synchrotron emission measured at lower energies [36]. Thisapparent excess has been dubbed the “WMAP haze.” Subsequent observa-tions by the Planck satellite, across a similar frequency band, have supportedthe existence of this diffuse hard spectral component. Analysis of the Planckdata has found that the haze follows an approximate power law spectrumwith a spectral index of βh = −2.55 ± 0.05, such that T ∝ νβh [9]. This isa significantly harder spectrum than was expected for galactic synchrotronemission based on extrapolation from earlier measurements at 408MHz [54]which predict an index of βs = −3.1 from all synchrotron sources in this en-ergy range 16. This implies the existence of a non-thermal component to thegalactic spectrum with a spectral index considerably higher (βh − βs ≈ 0.5)than is observed from typical galactic synchrotron emission. The strength14As the total baryon number of the nuggets may range over several orders of magnitudewe are not able to directly estimate the scale of emission in any particular band. However,it is possible to calculate the relative scales of the different bands, this will be demonstratedin chapter 4.15The CMB measurements, which were the primary purpose of WMAP, will be impor-tant in a radically different context to be discussed in chapter 5.16The relative strength and distinct spectral index may clearly be seen in figure 7 of thePlanck Collaboration’s investigation of the haze [9]253.2. Thermal emission : the WMAP “haze”of this feature, its spectral index, and possibly its very existence are, how-ever, highly dependent on the galactic model used. The galactic spectrumalso contains contributions from other diffuse interstellar medium compo-nents, such as dust and the free-free emission of hot plasma, in addition tothe galactic synchrotron emission, and the degree to which these may con-tribute to the haze remains an open question. Several physical mechanismswhich may be responsible for producing the haze have been proposed. It wasoriginally modeled as either a hard synchrotron component [29] or free-freeemission from a hot (104K < T < 106K) gas [36]. However, the first ofthese explanations requires a much harder synchrotron component than isobserved at lower energies, while the second should necessarily produce anassociated Hα line which is not observed. The haze has also been interpretedas evidence for dark matter annihilations [58], a larger than expected pulsarcontribution [63, 66] or the result of a modified dust spectrum [18]. It hasalso been argued that the galactic synchrotron emission may evolve withenergy, more than is allowed by a power-law extrapolation from lower en-ergy measurements, and that the uncertainty inherent in this extrapolationis on the same level as the observed haze [89]. However, at this time, noneof these explanations is strongly preferred over the others, and the natureof the haze remains an open research question. I will argue that, if the darkmatter is in the form of quark nuggets, it will necessarily produce microwaveemission which could produce all, or a significant fraction, of this apparenthaze effect. Before discussing the means by which this emission arises in thequark nugget model I will offer a brief review of the association of the hazewith potential dark matter sources. This discussion is, of course, contingenton the uncertainties listed above and is necessarily somewhat speculative.It has been argued that the haze has a roughly spherical morphologywith an approximate 1/r fall off in intensity with distance from the galacticcentre [58]. The haze may also be described as a hard synchrotron compo-nent on top of the expected, relatively soft, galactic synchrotron emission.This combination of morphology and spectrum lead to speculation that thehaze could be produced by a distinct population of electron-positron pairs,injected into the galactic centre at high energies by the annihilation or decayof galactic dark matter [37]. In this model the spectral index of the hazeis determined by the decay or annihilation spectrum of the dark matterparticles, which may be chosen to match the spectrum of the haze. Thisinterpretation is challenged by the apparent lack of polarization in the hazesignal, as would be expected from synchrotron radiation [49]. However, anypolarization of the signal could be masked if the galactic magnetic fields gen-erating the synchrotron radiation heavily tangled [114]. It has also been ar-263.2. Thermal emission : the WMAP “haze”gued that the electron injection spectra expected from standard dark mattercandidates will generate significant emission at higher energies that shouldalso be detectable. Dark matter which undergoes hadronic decays will pro-duce a γ-ray signal through subsequent pion decays, such a signal wouldvery closely trace the dark matter distribution. The same is true for de-cays or annihilations which directly produce photon pairs. Even if the darkmatter decays or annihilates exclusively to electrons and positrons these willbe injected into the interstellar medium at high energies (E > 1GeV) andwill inverse Compton scatter off interstellar photons. This higher energyspectrum should be present along with the microwave range synchrotronemission at a magnitude which should be visible to the Fermi Large AreaTelescope, but which is not observed17[7]. It has also been argued thatthe production of high energy electron-positron pairs in dark matter decaysor annihilations could explain the growth in the cosmic ray positron frac-tion with energy, as observed by the PAMELA satellite [10] and the AlphaMagnetic Spectrometer (AMS) experiment on board the International SpaceStation [98]. While possible, there seems to be some tension between thedark matter interaction cross section required by the different experimentalresults [7].With this background in place I will now illustrate how a diffuse mi-crowave component of the galactic spectrum is required within the darkmatter model under consideration. Much of this discussion will rely on thethermodynamic properties of the nuggets as discussed in appendix C. Theannihilation of visible matter within an antiquark nugget causes its temper-ature to increase. This thermal energy must then be radiated from the layerof the nugget where the electrosphere becomes transparent to low energyphotons. The temperature of the nuggets in a given environment is thusdetermined by the flux of visible matter onto the nugget, and the fractionof the energy released in the subsequent annihilations which is thermalizedwithin the quark matter. This fraction, which is important in determiningthe temperature of the nuggets and in estimating the relative strength ofthermal and non-thermal components of their emission spectrum, will becalled fT in the following discussions. While the fraction of annihilationenergy thermalized is determined purely by the physical properties of thenuggets, the flux of matter is set by the visible matter distribution through17It has been argued that the Fermi constraints may be avoided if the large galacticlatitude component of the diffuse γ-ray sources know as the “Fermi Bubbles” [114] isgenerated by the required inverse Compton scatterings [58]. However, this argument isdisfavored by the morphology of the bubbles [114].273.2. Thermal emission : the WMAP “haze”which the nugget moves18. The nugget temperature will be largest in envi-ronments where the visible matter density is relatively high, and where theaverage velocity is large.If the nuggets are in thermodynamical equilibrium with the surroundingmatter, then the total thermal emission, as given by equation C.5, mustbe balanced by the rate at which annihilations deposit energy within thenugget. This will result in a temperature as given by expression C.7. In thegalactic centre, where the matter density is estimated as ρvis ≈ 300GeV/cm3and the velocity as v ≈ 10−3c, the antiquark nuggets19 carry a temperatureof T ∼ 1eV.The thermal spectrum of the nuggets, as given in equation C.4, runsup to energies near or slightly above the temperature of the nugget (theeV band in the case considered here) but at lower energies, displays onlya weak, logarithmic, dependence on emission frequency. By contrast, atypical blackbody spectrum falls off as the second power of frequency belowthe thermal peak. For this reason, thermal radiation emitted by the nuggetsis distributed over a much wider range of frequencies below the eV scale.Radiation emitted in the eV band will easily be lost in the background ofradiation from visible matter which is many orders of magnitude brighter inthis range. But, at much lower energies, the relatively slow fall off in emissionfrom the nuggets means that their contribution to the diffuse backgroundmay become competitive with that of the visible matter20.In [43] the thermal emission of the nuggets was applied to the distributionof dark matter across the galactic centre. At that time the more detailedobservational data from Planck was not yet available and only the basicscale of the haze emission was known:dEdt dA dω dΩ ≈ (3− 6)× 10−20 ergs cm2 Hz sr . (3.5)In the case where we assume that quark nuggets near the galactic centrecarry an average temperature of 1eV it is possible to estimate the totalnugget contribution to the haze21. This is done by integrating the individual18The rate of annihilations is also partially dependent on the charge of the nuggets, asnoted in section 2.3.19Nuggets composed of quarks rather than antiquarks do not annihilate incident mat-ter and have a much lower radiating temperature, thus they do not make a significantcontribution to the total diffuse thermal emission.20A similar argument will be made in chapter 5 where the same considerations areapplied in a very different context.21A calculation in which the nugget temperature is allowed to vary with the properties ofthe surrounding visible matter is also possible, however the resulting temperature variationis relatively small and such a calculation adds more complexity than is currently warented.283.2. Thermal emission : the WMAP “haze”nugget contributions along a given line of sight,dEdt dA dω dΩ ≈∫dr4pinN¯dEdt dω, (3.6)where nN¯ is the number density of antiquark nuggets, and the integral sumsthe spectral contribution from each nugget (as given by equation C.4) alongthe line of sight through the galactic centre. Uncertainty in the distributionof nuggets makes this integral impossible to evaluate exactly. However, ifwe assume an approximately uniform dark matter distribution across thegalactic centre and that emission is dominated by the inner few kpc of thegalaxy we arrive at the approximation,dEdt dA dω dΩ ∼(104B)ergs cm2 Hz sr . (3.7)Comparing this estimation with the observed intensity suggests that thehaze emission could be entirely produced by a population of quark nuggetswith a mean baryonic charge of B ∼ 1024. While this crude estimation issubject to large uncertainties the suggested baryonic charge falls within theallowed parameter space. This implies that the nuggets are at least capableof providing some or all of the required haze. In chapter 4 a scaling argumentwill be made that the intensity of the haze is also fully consistent with itscoproduction with the other diffuse emission mechanisms discussed below.The spectral index of the haze as measured by Planck was not availableat the time of the publication of [43]. As such, that work simply pointedout that the spectrum of the haze component would be relatively hard.However, a simple estimate of of the index may be made from basic physicalconsiderations. When one attempts to determine a background temperatureby fitting a non-thermal spectrum to a blackbody radiation curve it leadsto an frequency dependent temperature. In the case of a constant intensityacross a range of wavelengths one has,8pihν3c21ehν/T − 1 = Φ0. (3.8)In the case where hν < T it is easy to see that the temperature must scale asT ∼ ω−2. Similarly, a spectrum that is inversely proportional to frequencywill have a temperature scaling T ∼ ω−3. The emission spectrum of thenuggets has a logarithmic decrease in intensity with wave length (as seen inequation A.6) and must fall between these two cases. As such, one expectsa spectral index of,−2 > βN > −3. (3.9)293.3. Surface proton annihilations : the Chandra x-ray backgroundWhich is the range in which the spectral index of the haze falls. The spectralindex of the low energy emission from the nuggets will be discussed in moredetail in chapter 5, where the same considerations are applied in a verydifferent physical scenario.With further study of the spectrum and distribution of the haze, obser-vations may come to favor any of the other proposed sources for this diffuseemission. For example, if the haze is found to be more strongly polarizedthan is presently believed thermal emission from the quark nuggets would bestrongly disfavored. However, it is important to point out that the emissionstrength from the nuggets could easily have been much larger than observedif either the matter distribution, or the thermal emission spectrum from thenuggets had been dramatically different than their estimated values. Thereis no way to avoid the basic physics which goes into the flux approximatedin [43], making it difficult to avoid these constraints. It is thus non-trivialthat this model, which was proposed to explain very different phenomena,not only avoids the constraints imposed by WMAP and Planck, but mayoffer an explanation for an observed spectral feature.3.3 Surface proton annihilations : the Chandrax-ray backgroundImaging of the galactic centre by the Chandra X-ray Observatory has shownevidence of diffuse emission in the x-ray band, even following the subtrac-tion of the contribution of known point sources [93]. This radiation has beenfit by assuming the presence of a two component thermal plasma, with thecooler component having a temperature Tcool ≈ 0.8keV and the hot compo-nent an order of magnitude warmer with Thot ≈ 8keV. The analysis leadingto this model was performed by a best fit to specific regions of the galacticcentre and, consequently, provides little information on the spatial distribu-tion of these two components apart from their general association with thegalactic centre. The cool component is consistent with being produced bysupernova occurring in this region. An energy budget analysis suggests thatthe supernova rate is sufficient to provide the required energy input, andobservations have confirmed that supernova do, in fact, heat the interstel-lar medium up to the 1keV scale [93]. The spatial distribution of the coolplasma is also relatively patchy, consistent with supernova heating. Thehot component is, however, more difficult to understand. Its temperatureis greater than the gas typically observed surrounding supernova or clustersof young stars, making its origin uncertain. While its morphology is not303.3. Surface proton annihilations : the Chandra x-ray backgroundwell constrained the emission from the hot component appears to be morehomogeneous than the cool component with an observed surface brightnessofΦhot = (1.5− 2.6)× 1013ergcm2 s arcmin2 . (3.10)The higher temperature also implies that the plasma should expand outwardand cool more rapidly than the low temperature plasma. Consequently, thepower required to sustain the hot plasma is several orders of magnitudelarger than that required by the cool component. This power requirementseems beyond the level provided by mechanisms, such as supernova andstellar winds, known to heat the interstellar medium.It has been suggested that some of the problems of gas expansion maybe solved if the hot plasma has a higher than expected helium content asthe heavier helium ions would be more strongly gravitationally bound to thegalaxy than hydrogen [27]. This scenario could be realized by the preferen-tial evaporation of the hydrogen component. This would lower the overallpower requirements of the plasma, but does not provide an actual heatingmechanism.In addition to the continuum emission the observed hot plasma containsa number of emission lines, indicating that the plasma must be opticallythin. These lines include contributions from Hydrogen like and Helium likeions of Mg, Si, S, and Fe. An analysis of the relative strength of these linesis consistent with production in a two component plasma and also findsthat the spectrum is similar to that associated with point sources in theregion of the galactic centre [93]. This analysis also suggests that, evenafter point source subtraction, there may remain some contribution to thediffuse emission spectrum from point sources below the detection thresholdof Chandra [92, 93]. The contribution of these objects does not however,seem sufficient to explain the total emission from the galactic centre region,particularly in the case of the hot emission component. The most promisingcandidates considered were cataclysmic variables, but even these provide anx-ray contribution an order of magnitude below what is observed. If thisis the case then a new source of diffuse emission in the x-ray band may berequired to produce this apparent hot plasma component. I will argue thatquark nugget dark matter may provide just such a source. In this picture theemission lines arise from conventional diffuse astrophysical processes and thepoint source contribution, while the nuggets contribute a significant portionof the observed x-ray continuum.The previous section dealt with the fraction of energy, produced in nu-clear annihilations, which is thermalized within the nuggets. However, these313.3. Surface proton annihilations : the Chandra x-ray backgroundannihilations occur relatively near the surface, and we must also consideremission resulting from annihilation products that escape the nugget beforethermalizing. This requires a more detailed description of the dynamics ofthe annihilation process. Within the quark matter, individual quarks arenot bound in colour singlet hadronic states, but exist as Cooper pairs. Thewave functions of these spatially extended pair states have only a small over-lap with the proton wave function. Thus, in order for a galactic proton toannihilate within the quark matter, these two wave functions must first be-come aligned. The alignment process requires a longer time than in the caseof proton-antiproton annihilation and the galactic protons, therefore, havetime to penetrate deeply into the nugget, with respect to the QCD scale.If annihilation rates maintained their vacuum value the incoming protonwould survive for roughly 2fm/c. However, it has been estimated that evenin ordinary nuclear matter this lifetime could easily be an order of magni-tude larger [90], resulting in a correspondingly longer penetration depth forthe incident proton.The annihilation of a galactic proton within the nugget typically pro-duces a pair of back to back jets (as observed in standard proton-antiprotonannihilations.) These jets then rapidly cascade down to the lightest modesof the colour superconductor. The exact decay chain of these hadronic pro-cesses will be very complicated, and will depend on the quark matter phaserealized within the nugget. However, near the quark surface the lightestmodes are quite generically the positrons which are only electromagneticallycoupled to the quark matter. The result of a near surface proton annihilationwill, therefore, be a stream of energetic positrons crossing the quark surfaceinto the electrosphere. As with most electromagnetic processes, the transferof energy from the initial hadronic jets to the positrons will be dominatedby exchanges of the lowest possible energy photons. Within the Fermi gasof positrons near the quark surface, electromagnetic effects are screened bythe presence of background charges. The plasma frequency in a Fermi gaswith chemical potential µ is given by,ωp ≈√4α3piµ. (3.11)with the chemical potential at typical QCD scales, µ ∼ 100 MeV, in thiscase. This gives a plasma frequency ωp ∼ 5MeV and photon exchange atlower energies is then strongly suppressed. Consequently, the majority ofexcited positrons will carry energies at this scale.These excited positrons will be the primary source of non-thermal emis-sion generated by nuclear annihilations. As such they carry a fraction 1−fT323.3. Surface proton annihilations : the Chandra x-ray backgroundof the total energy released in a typical nuclear annihilation22. As only anni-hilation products directed towards the quark surface have any possibility ofescaping the nugget we require 1− fT < 1/2. Taking each positron to carrya momentum near the plasma frequency in expression 3.11, a single protonannihilation will produce, at most, roughly a hundred relativistic positronscrossing the quark matter surface.As the positrons move through the strong electric field at the quarksurface (as described by equation 2.10), they will be decelerated and emitbremsstrahlung radiation. The momentum of these positrons is sufficientthat they penetrate upwards in the electrosphere to a regime where thepositron chemical potential has dropped to the keV scale. As the excitedpositrons cross this region of the electrosphere, the bremsstrahlung photonsmust be emitted above the local plasma frequency. Across the electrospherethe plasma frequency falls from the MeV scale down to below a keV. This,combined with the growth of the electron-photon scattering cross section atlow energies produces emission primarily at 1-10keV [42]. As with thermalemission, the observational consequences of near surface annihilation, x-rayemission will be strongest towards the galactic centre, where both the visibleand dark matter densities are largest.As the emission from a hot dilute plasma is primarily through thermalbremsstrahlung, it should not be surprising that the spectrum of photonsemitted by an ejected positron is similar to that of a hot plasma. Thisspectrum tends to be relatively flat up to a sudden exponential cutoff. Inthe thermal case this cutoff is determined by the temperature of the plasmaas the production of photons with energies above the average interactionenergy of the plasma components is strongly disfavored. Near the cutoff theemission spectrum will scale as,dEdt dω∼ e−ω/T . (3.12)Estimating the high energy cutoff in the case of emission from the nuggetsis more complicated as it involves many body interactions rather than thecollective effect of many interactions between independent plasma compo-nents. At any given depth in the electrosphere the emission of a photon withan energy below the plasma frequency, as given in equation 3.11 is stronglysuppressed. Once the positron reaches a regime of the electrosphere wherethe plasma frequency is well below the positron’s momentum the nugget’s22Recall that fT was introduced in section 3.2 and represents the fraction of energythermalized within the nugget, 1 − fT is then the fraction emitted through non-thermalprocesses.333.3. Surface proton annihilations : the Chandra x-ray backgroundelectric field will be unscreened and the emission of bremsstrahlung radiationwill quickly slow the positron.An analysis of the classical path of a positron in the near surface electricfield, as given by expression 2.10, was performed in [42]. That analysissuggested that the response time of the system could be approximated as,τ ≈µ0 + /0eE≈√3pi2α(z + z0z0)2 µ0 + /0µ20(3.13)where z should be taken as the height above the quark surface from whichthe majority of bremsstrahlung radiation is emitted and /0 ≈ ωp is thepositron injection energy. As this timescale limits the emission of radiationwe should expect that the bremsstrahlung spectrum should be cut off atfrequencies of ωc ∼ τ−1. Numerically, if we take µ0 = 10MeV, /0 = 5MeVand z ∼ z0 we obtain ωc ≈ 30keV. This estimation (which is accurate onlyat the order of magnitude level) turns out to be quite close to the spectralcutoff ωc ≈ 10keV suggested by the Chandra data. This ten keV scale arisesfrom the basic physical properties of the nuggets and is in no way tunedto fit the observed emission. Near the point of maximum positron energyloss the emission spectrum may be given in terms of the modified Besselfunction, K5/3:dEdt dω∼ωωc∫ ∞ω/ωcK5/3(x)dx, (3.14)dEdt dω∼ e−ω/ωc ,ωωc> 1. (3.15)The similar scaling of equations 3.12 and 3.14, with T ∼ ωc ∼ 10keV, impliesthat the basic continuum shape of the bremsstrahlung spectrum generatedin these two very different physical scenarios may be indistinguishable23.In this framework, the heated plasma observed by Chandra is actuallythe localized hot spots , generated by the annihilation of galactic protons,on the surface of many individual antiquark nuggets. In this case, thereis no need to explain how the plasma remains bound to the galaxy, as thepositrons involved in each annihilation event remain bound to the nuggetby the strong electric fields at the surface of the quark matter.While the basic form of the spectrum generated by x-ray emission fromthe quark nuggets is consistent with the spectral shape observed by Chandra23The actual spectra observed by Chandra, as well at the two component plasma fitsto the data, may be seen in figure 6 of [93]. From that figure it is clear that a hot plasmacomponent is required to fit the spectrum, as significant x-ray emission is observed acrossthe entire 1-8keV range observable with Chandra.343.4. Low momentum electron annihilations : the galactic 511keV linethe overall scale of the nugget contribution is not yet established. Thisemission strength is dependent on the details of the matter and dark matterdistributions as well as the distribution of nugget masses. As with any otherobservational quantities the contribution of the nuggets to the galactic x-ray background will be lower if the nuggets have a larger average mass. Inchapter 4 the scaling of the various emission features discussed here will bediscussed allowing the total intensity to be estimated.3.4 Low momentum electron annihilations : thegalactic 511keV lineA strong 511 keV line, along with the associated three photon continuum,has been observed by the SPI spectrometer on board the INTEGRAL ob-servatory [60, 70]. While this gamma ray source in the galactic centre hasbeen known for four decades [62], the source of the roughly 1043s−1 annihi-lating positrons remains an active research question (for a recent review ofthe situation see, for example, [100].)The observed e+e− annihilation spectrum at 511keV is modeled as a twocomponent spectral line, combined with the associated ortho-positroniumcontinuum. The spectral line displays a broad component with a width of5.4 ± 1.2 keV FWHM and a narrow component with width 1.3 ± 1.2 keVFWHM [61]. The narrow line is consistent with the low momentum anni-hilation of a positron with a free electron while the broad component hasa width typical of annihilation through charge exchange with neutral hy-drogen. This width is also entirely consistent with formation within theelectrosphere of a quark nugget. In the latter case, the additional broaden-ing comes from annihilations involving slightly higher momentum positronscloser to the quark surface as well as the capture of electrons from incidentneutral hydrogen.Morphologically, the 511 keV emission is strongly associated with thegalactic centre and the galactic bulge [118], displaying only a much fainterdisk component [70], [119]. However, the exact spatial distribution of theemission strength remains model dependent in that one may always intro-duction additional components (beyond disk and bulge) to obtain a betterfit to the data. This is particularly true of components with low surfacebrightness and a large spatial extent [117]. A contribution of this form canrepresent a large total number of positrons but make only a small contribu-tion to the observed spectrum. While the exact ratio is strongly dependenton the model employed, the bulge to disk ratio of the 511keV emission is353.4. Low momentum electron annihilations : the galactic 511keV linegenerally found to be ≥ 1. This is a stronger spherical component than isseen in most astrophysical sources, and seems to require more complicatedpropagation of cosmic rays through the galaxy than had previously beenassumed. This in turn generated interest in dark matter models capable ofproducing the requisite number of positrons through either decays or anni-hilations. The reasoning being that, as the dark matter is generally taken tohave a spherically symmetric distribution, any associated emission shouldbe dominantly in the spheroid component of the galactic centre, rather thanbeing associated with the disk. These models, however, tend to producepositrons with sufficient energy that they will produce a significant inflightannihilation spectral component rather than just a clean 511 keV line [26].As further observations revealed greater detail in the spatial distributionof the 511 keV emission it became clear that dark matter models involv-ing the production of relativistic positrons would generally overproduce thehigh energy γ-ray background. As such, rather than naturally explainingthe morphology, these models require modifications of the cosmic ray prop-agation models similar to those that they sought to avoid [83].The difficulty in explaining the 511 keV line lies not in producing therequired number of positrons, but in concentrating them so strongly in thegalactic centre and at low momentum. The observed morphology does notseem to trace the spatial distribution of known positron sources, and thenarrow width of the spectral line suggests that the positrons must slow dra-matically between production and annihilation. As such, an understandingof the contribution of known positron sources to the observed 511 keV linerequires extensive modeling of positron propagation. This remains an openresearch question and I will give only a brief overview here.The main astrophysical source of positrons is thought to be supernovae.These produce a large number of radionuclei which may then β-decay andemit a positron, generally with an MeV scale energy. The supernova ratein the galaxy is believed to be sufficient to provide the required number ofpositrons. However, many of the relevant radionuclei have a short half life,which may lead to a large number of the positron produced being annihi-lated before they escape from the supernova remnant itself. This sourcealso requires rather complicated transport of the resulting positrons as su-pernovae occur primarily in the disk while the majority of annihilations areobserved to occur in the galactic bulge [101].The disk component of the 511 keV emission may, however, be under-estimated as the positron annihilation is likely to be more spatially dilute.This could result in sections of he disk having a surface brightness belowthe sensitivity of SPI/INTEGRAL. If this is the case the number of disk363.4. Low momentum electron annihilations : the galactic 511keV linepositrons could be much larger than is observed. In fact, the AMS [98]has measured the electron to positron ratio in the cosmic ray spectrum atenergies up to 500 GeV. This spectrum shows a rise in the positron frac-tion above ∼ 10 GeV, indicating the presence of more positrons in the localdisk than would be extrapolated from lower energy measurements [33] andsupports the possibility of an extended disk component. However, thesepositrons are at energies well above where they could contribute to the 511keV line and there have not been convincing models put forward as to howthey may be slowed and transported to the galactic bulge.Several other potential astrophysical sources have been put forward suchas pulsars or stellar winds These may increase the number of positronsavailable but do not solve the problems of distribution and energy discussedabove. Keeping in mind this underlying uncertainty in the nature of the511keV emission I will now discuss how such emission will arise in the quarknugget dark matter model.In addition to protons, we may also consider the contribution made to thediffuse galactic spectrum by electrons annihilating on an antiquark nugget.The properties of these annihilations, and the related emission, will be gov-erned by the properties of the nugget’s electrosphere which were outlinedin section 2.2. When a galactic electron strikes a nugget, it first movesthrough the low density “Boltzmann” region of the electrosphere, where thepositron densities and momenta are at typical atomic scales. Under theseconditions, the dominant annihilation channel is through the resonance for-mation of a positronium bound state (an “atom” consisting of a bound e+e−pair.) The positronium state subsequently decays, to either a pair of backto back 511 keV photons, or a three photon continuum, depending on itsinitial spin configuration24. Unlike the emission mechanisms, related to pro-ton annihilation, considered in the previous two sections, the decay of a 1S0positronium state results in a narrow spectral line, rather than continuumemission extending over a wide frequency range. This relatively clean signal,independent of the nuclear physics of the quark matter, is a particularly at-tractive observational target. The line width is determined by the fact thatpositronium formation is strongest in the region of the electrosphere justabove the atomic scale. This may be seen in figure 2.2 which shows therapid increase in positronium annihilations where the electrosphere density24One quarter of the time the electron-positron pair will form a positronium atom inthe spin zero 1S0 state which can decay to a pair of back to back photons with oppositehelicities. The remainder of the time the positronium will decay from the 3S1 state. Asthis state carries a half integer spin the state must decay to three photons in order toconserve angular momentum.373.4. Low momentum electron annihilations : the galactic 511keV lineapproaches a−3B . In this background the positronium is likely to form withan atomic scale momentum, well below its rest energy. The resulting linewill then be Doppler broadened at the keV scale [128]. This diffuse emissionsource should track the matter and dark matter distributions as describedby equation 3.2 and, as such, will be strongly peaked towards the galacticcentre with a fainter disk component. This is not necessarily true of modelsin which the positrons are produced by astrophysical processes which arestrongly associated with the disk or of models involving exclusively darkmatter interactions which are purely spherical.Finally, we may make a rough estimate of the number of positroniumannihilations expected from the galactic core. The rate at which positroniumis formed in the galactic bulge may be expressed as,ΓPs = fPsnenN¯σnveVB, (3.16)where fPs ≈ 0.9 is the fraction of annihilating electrons expected to producea 511 keV photon, ne and nN¯ are the electron and antiquark nugget densities,ve is the average electron speed and VB is the volume of the galactic bulge.If we take the same matter distribution as was used in the estimation of thetotal haze contribution and take the observed bulge radius to be a few kpcthen we arrive at the approximation,ΓPs ≈ 1051s−1B−1/3. (3.17)Comparing this to the observed rate of 1043 s−1 implies that nuggets witha mean baryonic charge of 1024 could easily produce the required numberof positronium decays. This approximation is very approximate, and maybe affected by the charge of the nuggets in various environments, by gives abasic feel for the scale involved.At present the source of the positrons responsible for the galactic 511keV line remains ambiguous without a definite preferred origin. As such, it ispossible that a considerable fraction of them could be associated with quarknugget dark matter without any contradictions with observation. In this casethe positrons are naturally present at low energies and do not need to beproduced in high energy astrophysical processes and then decelerated beforeundergoing low energy annihilation through the positronium channel. Themorphology is also consistent with the mixed bulge/disk distribution as theemission will track the product of the matter and dark matter distributions.As the 511 keV line is the cleanest, and best known, of the diffuse emis-sion excesses discussed here it will be used to estimate the total rate ofannihilation events involving quark nuggets in chapter 4. Once this basic383.5. Near surface electron annihilations : the COMPTEL MeV excessrate has been established the scale of all other forms of diffuse emission maybe estimated in a more detailed way than has been done here.3.5 Near surface electron annihilations : theCOMPTEL MeV excessThe galactic MeV background is dominated by nuclear decays and cosmicray interactions. These processes require extensive modeling to compareastrophysical predictions to observational data. As in the case of the galac-tic 511 keV line there are many complications, relating to production anddiffusion models which may dramatically change the scale of the predictedastrophysical background. Several attempts have been made to fit the galac-tic γ-ray spectrum based on known astrophysical emission mechanisms [112].In each case, these calculations have produced a good fit to both the spec-trum and the spatial distribution of the observed diffuse γ-ray background,except in the MeV band in the galactic centre region. In this region themodels predict MeV band emission significantly below what is observed.The observational data for this analysis was taken from the COMPTELtelescope for energies from 1-30 MeV [113], while the data from 30MeVup to 50GeV comes from the Energetic Gamma Ray Experiment Telescope(EGRET) using the data set described in [111]. As with the other diffusebackgrounds discussed above it is difficult to extract an exact morphologyfrom the COMPTEL excess. However, of the regions considered in [112]only the inner galaxy (l = 330o−30o, |b| = 0o−5o) show signs of this excess,indicating that whatever mechanism is responsible is strongly concentratedtowards the galactic centre. Unfortunately the MeV scale excess cannotbe verified with follow up with higher resolution observations by the Fermigamma-ray telescope which is not sensitive to photons below 30MeV.Several proposals have been made to fit this excess in diffuse emissionby modifying the emission spectrum of known astrophysical objects, intro-ducing a new class of unresolved point sources, or through dark matterinteractions. In each of these cases, the modification to known physical pro-cesses must be very carefully chosen, so as not to ruin the good fit to thegalactic spectrum at other energies or outside of the higher density galacticcentre.In the case of quark nugget dark matter this MeV excess must necessarilybe coproduced with the 511keV line. It was demonstrated in [80] that thishigh energy component of the nugget spectrum could provide the additionalsource, necessary to explain the observed MeV galactic background. This is393.5. Near surface electron annihilations : the COMPTEL MeV excesspossible as the spectrum expected from the annihilation of galactic electrons,passing through the high density regime of the electrosphere, maps neatlyonto the MeV background observed by COMPTEL [113].A fraction of the galactic electrons incident on the quark nugget willavoid annihilation in the low density outer layers, these will penetrate throughto the high density region of the electrosphere near the quark surface. Oncethere the annihilation energy scale becomes significantly larger, and emis-sion cannot be treated as in the nonrelativistic positronium formation caseconsidered above. The positronium resonance favours annihilation of theincident electron with the lowest momentum positrons available. This isbecause the probability of forming a positronium bound state scales as,PPs = | < ψPs|ψe+e− > |2 ≈(11 + a2Bq2)2(3.18)where aB is the Bohr radius and q is the centre of mass momentum. Thus,once the positron carries a momentum above the atomic scale, the formationrate falls with the fourth power of momentum. However, the density ofstates of a Fermi gas scales as dndk ∼ k2, so that the integrated number ofhigh energy positrons will grow as Nk>>m ∼ k3. This effect will dominateover the decrease in scattering cross section which falls only as σ ∼ k−2.The density of low momentum positron states remains fixed with increasingchemical potential and thus the positronium formation rate saturates quitehigh in the electrosphere. Nearer to the quark matter surface the growthof the density of states at large momenta favors the off resonance, directe+e− → 2γ channel despite its smaller cross-section. The relative scaleof resonant and direct annihilation processes, as a function of height in theelectrosphere, are shown at the top of figure 2.2. The exact relation betweenthese two processes will be the primary focus of chapter 4.Near the quark surface the positrons carry momenta on the order ofpF ∼ eΛQCD ∼ 10MeV. The photons produced in these annihilations willcarry energies up to this scale. Some forms of quark matter are predictedto support a larger lepton density at their surface, and thus allow a largermaximum possible annihilation energy. However, electrons passing througha medium of this density have only a very low probability of reaching a depthat which the chemical potential is much greater than ∼ 10 MeV. Conse-quently, these annihilations represent the highest energy emission from thenuggets produced in significant quantities, and no signature will be producedabove this scale. As such, this model is not subject to the strong constraintsimposed by the Fermi gamma-ray telescope [7, 8].403.5. Near surface electron annihilations : the COMPTEL MeV excessThe details of this spectrum can be made more explicit through argu-ments based on some simple QED based calculations [80]. Consider anelectron in a Fermi gas of positrons with chemical potential µ >> me. Thedifferential cross section for a positron of momentum p to annihilate witha stationary electron and produce a final state containing a photon of mo-mentum k may be found, at tree level, by a standard QED calculation:dσdk= piα2mp2[−(3m+ E)(E +m)(E +m− k)2 − 2]+ piα2mp2[(3m+ E)(E +m)2/k − (m/k)2(E +m)2(E +m− k)2]. (3.19)Here E =√p2 +m2 is the positron energy. From this we may calculate therate at which photons of momentum k will be produced by integrating thecross section over the distribution of positron momenta in the Fermi gas,this givesdNγ(k, µ)dk dt=∫p2pi2dσdkdE, (3.20)with the differential cross section as given in expression 3.19. The integrationin this expression should run from the threshold positron energy required toproduce a photon of momentum k up to the chemical potential of the Fermigas. This integral may be evaluated explicitly and the result is given in [80],however, the exact form of this solution is rather cumbersome and is beyondthe level required here. Near the peak of the spectrum, when both k and µare much larger than the electron mass it has the approximate form,dNγdk dt≈α2kpim(12 + ln[2(µ− k)m])(3.21)which is valid up to photon energies where µ−k ∼ m. Above these energiesthe spectrum quickly falls to zero as kinematic constraints require that k <µ+2m. In the limit where the chemical potential is well above the electronmass, µ >> m, the total annihilation rate has a relatively simple form:Γan(µ) ≡dNγdt=∫dNγdk dtdk ≈α2m2pi(µm)2ln(µm). (3.22)This rate may then be used to estimate the fraction of electrons annihilatingat a specific height through direct annihilation. This fraction is given by,F(z) = exp(−∫ zz0Γan(µ(z))dzve). (3.23)413.5. Near surface electron annihilations : the COMPTEL MeV excessIn this expression ve is the velocity of the electron incident on the nugget andis assumed to be at the galactic scale. The lower limit for the integration maybe taken as the beginning of the ultrarelativistic regime in the electrospherewhere the evolution of the chemical potential is as given in expression 2.9.We may get a feel for the rate of this exponential fall off by noting that,from equation 2.9 we find, µ2dz =√3pi/2α dµ. Using this substitution andthe form of the total rate given in equation 3.22 gives a survival fractionF ≈ exp[−µµ¯(1 + ln(µm))], µ¯ ≈2pimveα2√2α3pi . (3.24)Here I have defined µ¯ as an energy scale that sets the rate of the exponentialdecay. If I take the electron velocity to be at a typical galactic scale ofve = 10−3c, then the decay energy scale is µ¯ ≈ 3MeV. This decay ratebecomes relevant at chemical potentials above roughly an MeV, as discussedabove.The spectrum given in equation 3.21, combined with the weighting fac-tor in expression 3.24, determines the spectral contribution of the quarknuggets from the ultrarelativistic limit regime of the electrosphere. Theuncertainties in dark matter distribution, and the distribution of nuggetmasses, prevents an explicit calculation of the overall scale we should expectfor this diffuse emission source. However, following the estimations madein [80], we may assume that high energy emission from the ultrarelativis-tic regime is co-produced with the galactic 511keV line, and that nuggetemission dominates the 511keV signal. This procedure packages all the un-certainties in the estimated emission strength in a single parameter, herecalled fur, which sets the fraction of annihilating electrons able to penetrateinto the ultrarelativistic regime (and thus contributing to the galactic dif-fuse emission above 1MeV.) Using this scaling procedure it is possible toestimate the nugget contribution to the galactic spectrum. The result for atypical value of fur = 0.1, along with observational data and the predictedgalactic diffuse emission, may be seen in figure 3.1. The diffuse spectrum,taken from [91], provides a good fit to the observed spectrum over muchof the γ-ray range, but is predicted to fall too rapidly below 100MeV to beable to fit the COMPTEL data. It is precisely in this range that the nuggetsprovide an additional contribution to the galactic γ-ray spectrum.The introduction of the scaling parameter fur allows us to treat the over-all normalization of the spectrum in a relatively simple way. However, it ispossible to go beyond this very approximate treatment by using the detailedmodeling of the electrosphere presented in section 2.2. This will be done in423.5. Near surface electron annihilations : the COMPTEL MeV excessDiffuse cosmic gamma-rays at 1-20 MeV: A trace of the dark matter? 4Figure 1. γ ray spectrum of inner galaxy for optimized model[24]. Green verticalbars: COMPTEL data. Heavy solid line: total calculated flux for optimized model.Heavy black dots: Combination of calculated emission spectrum from electron-nuggetannihilation processes with the optimized model of [24].2. Dark Matter as Compact Composite Objects (CCOs).Unlike conventional dark matter candidates, dark matter/antimatter nuggets arestrongly interacting, macroscopically large objects. Such a seemingly counterintuitiveproposal does not contradict any of the many known observational constraints on darkmatter or antimatter in our universe for three main reasons: 1) the nuggets carry ahuge (anti)baryon charge |B| ≈ 1020 – 1033, so they have a macroscopic size and a tinynumber density. 2) They have nuclear densities in the bulk, so their interaction cross-section per unit mass is small σ/M ≈ 10−13 – 10−9 cm2/g. This small factor effectivelyreplaces a condition on weakness of interaction of conventional dark matter candidatessuch as WIMPs. 3) They have a large binding energy (gap ∆ ≈ 100 MeV) such thatbaryons in the nuggets are not available to participate in big bang nucleosynthesis(BBN) at T ≈ 1 MeV. On large scales, the CCOs are sufficiently dilute that theybehave as standard collisionless cold dark matter (CCDM). However, when the numberdensities of both dark and visible matter become sufficiently high, dark-antimatter–Figure 3.1: The diffuse γ-ray spectrum of the inner galaxy as observed byCOMPTEL (Green bars) and EGRET (re bars). Also shown are the contri-butions from expected backgrounds including; inverse Compton scattering,bremsstrahlung and pion decay as well as the extragalactic background. Thesolid blue line gives the predicted total γ-ray intensity and the black dotsshow the effect of d ing a quark nugget contribution as described in thetext. Figure taken from [80] where it was adapted from [91].chapter 4, where the exact rates of positronium formation and direct anni-hilation will be calculated and a precise electron annihilation spectrum willbe determined without the necessity of introducing the phenomenologicalparameter fur. A more version of figure 3.1 based on this calculation ispresented in figure 4.2.Some important aspects of the spectral properties outlined here shouldbe highlighted. The emission falls in the band between approximately 1MeV,the chemical potential at which direct annihilation first becomes importantand the chemical potential at the nugget’s surface. As mentioned above,the lepton chemical potential should be expected to be a few tens of MeV.However, emission from increasing depths are suppressed by the exponential433.6. Conclusionsdecay of the number of galactic electrons penetrating to a given depth. Itmay be estimated that this effect will limit the emission spectrum to a fewtimes µ¯. These features are consistent with generating a diffuse excess in theMeV range while not contributing strongly to the well fit diffuse backgroundat higher energies.It is also important to note that, had the MeV spectrum from the galacticcentre been more tightly constrained, the quark nugget model could notserve as a source for the galactic 511 keV line. The two components mustboth be present if the nuggets are to be invoked to explain the galacticdiffuse γ-ray background. The fact that these two spectral components areproduced through exactly the same physical process will be exploited in thefollowing chapter to predict the exact ratio between the 511keV line and theMeV diffuse emission. That analysis will demonstrate that this ratio mustbe close to that which is observed.3.6 ConclusionsThis chapter has argued that the presence of quark nugget dark matter isnot ruled out by observations of the galactic centre, where the contributionof the nuggets to the diffuse background should be largest. In fact, theircontribution to the galactic spectrum may help to explain some apparentanomalies in galactic observations from the microwave band up to γ-rays.Taken together these observations span more than ten orders of magnitudein energy, but may all be understood in terms of a single emission model.Improved constraints on any one of the spectral bands discussed above,whether through further observations or improved modeling of astrophysicalbackgrounds, must necessarily lower any potential nugget contribution to allthe others. This correlation will be the subject of the following chapter wheremore precise estimations of the relative strengths of different components ofthe nugget spectrum will be calculated.44Chapter 4Relative Emission StrengthsIn this chapter I will estimate the relative emission strength of the vari-ous components of the nugget emission spectrum outlined in the previouschapter. As the baryonic charge of the nuggets is unknown the line of sightintegral, from expression 3.2, cannot be directly evaluated. Thus, we can-not fix the absolute emission strength of the various spectral components.However, once this overall scaling is fixed it must be identical for all theforms of emission. As such, if we assume that the galactic 511keV line isdominated by annihilations involving positrons in the electrosphere of anantiquark nugget, then we may estimate the relative strengths of all theother spectral components. The discussion of the relative strength of thetwo electron-positron annihilation processes (those which we associate withthe 511keV line and the diffuse MeV continuum observed by COMPTEL)is derived from original research conducted as part of this thesis [40]. Thefollowing discussion of the relation of the nuclear annihilation processes isbased on previous work [42, 43] and, as such, only briefly reviewed here.4.1 Electron-positron annihilationBoth the 511keV line and the MeV continuum discussed above arise fromthe same physical process; the impact of a galactic electron on an antiquarknugget and its subsequent annihilation with a positron. The relative scale ofemission expected from these two processes is, therefore, directly calculable.As discussed in section 3.5 the ratio between the two spectral componentsis essentially determined by the fraction of low momentum galactic elec-trons penetrating deeply into the electrosphere to a point where the Fermimomentum of the positron gas is large.The survival fraction of electrons depends on the annihilation rate ateach depth (through both the positronium and direct annihilation channels)and this is, in turn, depends on how the positron density grows with depth.The structure of the electrosphere was discussed in section 2.2, and manyof its specifics have been established in detail in [40].454.1. Electron-positron annihilationThe rate at which a galactic electron of velocity ve binds to a positronto form a positronium may be estimated asΓPs(z) =∫p<mαve σPsd3)p(2pi)3 ∼ 4pivea2bne+, pF < mα (4.1)∼4ve3piab, pF > mα. (4.2)Here the integral runs over the local density of states with momenta smallenough to sit within the positronium resonance. Far from the quark surface,where the density is below the atomic scale, the positrons carry only smallmomenta and they all contribute to positronium formation. This gives thefactor of the positron density, ne+, appearing in the expression for ΓPs at lowdensities. Above the atomic scale, the low energy positron states are fullyoccupied, and any additional positrons must be added in momentum stateslarge enough to be off of the positronium resonance. At these densities, onlythe low lying states contribute to positronium formation, and the formationrate saturates. This saturation effect, and the radial distance at which itoccurs are plotted at the top of figure 2.2.The cross section for an electron-positron pair to scatter to a pair ofphotons is easily calculated within the framework of QED, an approximateversion of this rate was given in expression 3.22 while the exact form maybe obtained by performing the integration in expression 3.20 and then in-tegrating the result over the final state photon momenta, as was done in[80]. Within the dense region of the electrosphere scattering is limited tocases where the emitted photons are above the local plasma frequency sothat the phase space for annihilation is partially restricted by the kinematicconstraints on the final state photons. The resulting annihilation rate wascalculated analytically in the zero temperature limit in [80] and numericallyfor non-zero temperatures in [40]. Within the context of this full numericalsolution, it is possible to calculate directly the fraction of incident electronsthat survive to a given depth in the electrosphere. This survival fractionis plotted in figure 4.1. Two important features of the annihilation rateare immediately obvious from figure 4.1. First, the rapid drop in the sur-vival fraction at Fermi momenta in the keV range is almost entirely due topositronium formation, and accounts for the majority of annihilation events.The steepening of the decay curve beginning at pF ∼ 1MeV is due to thegrowth of the direct annihilation rate, and accounts for roughly a tenthof the total annihilation events. Second, at very large positron densities,annihilations occur at an ever increasing rate, to the point where there is464.1. Electron-positron annihilationFigure 4.1: The fraction of incident electrons which survive to a givendepth within the electrosphere of an antiquark nugget. The three differentcurves represent different initial electron velocities: v=0.01c (red), v=0.005c(green) and v=0.001c (blue). The thickness of the curves represents a 10%variation in the positronium formation rate. The yellow band indicates theregion of the electrosphere well modeled by the Boltzmann approximationsas discussed in section 2.2. Figure taken from [40].virtually no chance of an electron penetrating to a depth where its annihila-tion produces a photon with an energy of more than a few tens of MeV. In[80] these features were built into the emission model, in a phenomenologi-cal way, to demonstrate that the presence of quark nuggets in the galacticcentre was entirely compatible with the COMPTEL measurements. In thefull treatment of the annihilation rates presented in [40], these values werederived from a precise microscopic modeling of the properties of the electro-sphere. In that analysis, it was found that the MeV excess and the 511keVline must, not only both be present, but must be present in the ratio inwhich they seem to be observed.To further establish the importance of this correlation between the 511keVline and the MeV continuum, I will now repeat the estimations of [40] usedin determining the ratio between the two spectral features. From the annihi-lation rates presented in figure 4.1 it is possible to determine the fraction ofgalactic electrons which annihilate through the positronium channel. These474.2. Nuclear annihilationsrepresent roughly 90% of all annihilations, and must account for the observed511keV line and its associated three photon continuum. The strength of the511keV line from the galactic centre, as observed by SPI/INTEGRAL, is∼ 0.025 photons cm−2 s−1sr−1, coming from a circle of half angle 6o centredon the galactic centre [60]. The majority of this emission does not have awell established origin and, for present purposes, will be assumed to be pro-duced in annihilations within a quark nugget. Under this assumption theflux of 511keV photons essentially fixes the value of the line of sight integralin equation 3.2. Fixing this emission rate allows us to scale all of the otheremission mechanisms, without adopting a particular matter density profileor nugget size distribution.An exact comparison between the SPI/INTEGRAL data at 511keV andthe COMPTEL data in the 1-30 MeV range is made difficult by the com-plicated background subtraction required to extract the MeV continuum.The diffuse background contribution, due to known astrophysical processesin the interstellar medium, was modeled in [112] which provides a detailedspectrum across the relevant energy range, but averages emission over asomewhat larger angular extent than the SPI/INTEGRAL data, coveringgalactic longitudes in the range l = 330o − 30o and latitudes |b| = 0o − 5o.Across this region the average strength in the MeV band, as measured byCOMPTEL, is approximately k2 dΦdk ∼ 10−2MeV s−1cm−2sr−1, where I havegiven the spectral density flattened by the energy squared following the orig-inal work in [112]. Directly scaling the MeV continuum, such that the totalnumber of photons involved (i.e.∫ dΦdk dk) matches the annihilation rate setby the strength of the 511keV line results in the spectrum shown in figure4.2. Even neglecting the difference in the angular distributions in [60] and[112], and ignoring the very complicated distribution of matter densities andvelocities across the galactic centre, it can be seen that the spectrum gen-erated by quark nugget dark matter will fall in precisely the range whereCOMPTEL observes an excess above the predicted galactic background bothin terms of energy and intensity.4.2 Nuclear annihilationsA similar analysis may be applied to the WMAP and Chandra data both ofwhich are, in this model, associated with the annihilation of a galactic pro-ton within an antiquark nugget. There is, however, a complication in thatthe division of energy, between thermal emission and surface bremsstrahlungfrom accelerated positrons, is highly dependent on the depth in the quark484.2. Nuclear annihilationsFigure 4.2: Spectral density scaled by ω2 emitted by electrons annihilatingon an antiquark nugget. The three curves assume incident electron temper-atures v=0.01c (red), v=0.005c (green) and v=0.001c blue). The thicknessof the bands assumes a 10% variation in the positronium formation rateas in figure 4.1. The overall normalization of the curves is obtained fromthe galactic 511keV line as discussed in the text. The vertical bars are thecomptel data points [113] and the dashed line is the predicted astrophysicalbackground calculated in [112]. Figure taken from [40].matter at which the annihilation occurs, and on the efficiency with whichenergy is distributed between the various light modes of the colour super-conductor. These are difficult problems to address theoretically and havelittle experimental input. The structure of interactions, and even the na-ture of the light modes of the superconductor, are also highly dependenton the particular phase of quark matter realized in the nuggets. As such,the relative scale of the various emission mechanisms cannot be extractedas cleanly as was the case with electron-positron annihilations. Instead Iwill introduce a pair of phenomenological parameters which parameterizethe uncertainty inherent in this process. The values of these parameters willthen be extracted from observation and it will be argued that their valuesare reasonable based on the generic properties of quark matter. Much ofthis analysis follows previous work done in [42] and [43].The first parameter to be introduced is the fraction of energy, produced494.2. Nuclear annihilationsin a nuclear annihilation, that is thermalized within the nugget. This pa-rameter, which will be called fT , was previously referred to in estimatingthe nuggets’ temperature in the discussion of the WMAP haze in section3.2. As all thermalized energy is emitted in the thermal spectrum while thedominant non-thermal emission process is x-ray bremsstrahlung from ex-cited positrons this fraction sets the relative scaling of the microwave band“haze” component and the diffuse x-ray background. As the physics of thisratio is entirely determined by the internal properties of the nuggets it shouldbe independent of the environment in which the nuggets are found.The second parameter is related to the relative rate of electron and pro-ton annihilations, it will be labeled fep. This will set the relative scaleof electron annihilation processes (positronium decay and the direct anni-hilation MeV continuum) and those associated with nuclear annihilation(microwave band thermal emission and x-ray bremsstrahlung.) The valueof fep will be dependent on the relative rates of charge exchange processesand inelastic collisions with the nugget as discussed in section 2.3. As theseproperties may depend on the nugget temperature and ionization, as well asthe ionization of the surrounding interstellar medium, this parameter mayshow more spatial variation than is expected for fT .As was demonstrated in the discussion of electron-positron annihilationin section 4.1 the photon flux from all these events is,Φe+e− ≈ 0.1photonscm2 s sr . (4.3)This is related to the total energy flux from proton annihilations by theparameter fep, such that,Ipp¯ = Itherm + Ibrem = 2mpfepΦe+e− . (4.4)Here Itherm is the intensity generated by thermal emission from the nuggets(which I will associate with the WMAP haze) and Ibrem is the non-thermalbremsstrahlung emission from the nugget surface (which I associate with thediffuse x-ray background.) The fraction of energy thermalized in the nuggetmay be expressed in terms of the same parameters,fT =IthermIpp¯= Itherm2mpfepΦe+e−, (4.5)so that, by extracting the total intensities associated with the two spectralcomponents, it is possible to determine observational values for fT and fep.The Planck analysis gives the haze contribution in the 23GHz channelas ∆T = 0.1mK, which translates to an intensity of dIdν ≈ 10−9 eVcm2 s sr Hz .504.2. Nuclear annihilationsIf I want to associate this with the thermal emission from a distribution ofquark nuggets then the frequency dependence must be as given in equationC.4 so that I may generally writedI(ν)dν= I0(1 + hνTN)e−hν/TNF(hνTN)(4.6)with the function F as defined in expression C.2. Evaluating this expres-sion at 23GHz and equating it with the flux, measured by Planck at thisfrequency, gives I0 = 7 × 10−11 eVcm2 s sr Hz . This normalization must holdacross the entire thermal emission spectrum so that,Itherm =∫I0(1 + hνTN)e−hν/TNF(hνTN)dν (4.7)= I0Th∫(1 + x) exF (x)≈ 106 eVcm2 s sr .This gives the total energy flux to be expected from thermal emission fromthe nuggets.The case of the diffuse x-ray background is somewhat simpler as themajority of emission falls in the 2-8 keV range which is observed. As suchthe observed surface brightness may be taken to be approximately equal tothe total intensity and Ibrem ≈ 4 × 105 eVcm2 s sr . These intensities allow foran estimation of the parameters fT and fep so that,fT ≈ 0.7, fep ≈ 10−2 (4.8)The remainder of this chapter will be devoted to arguing that these, obser-vationally derived, values are well motivated by the physical properties ofthe nuggets.As argued above any annihilation products directed downward into thenugget will necessarily be thermalized. This means that we require fT > 1/2.Any inefficiency in the energy transfer from the initial hadronic annihilationwill act to further lower the value of fT . Some energy loss must be expectedin the cascade of secondary particles down to the positrons which actuallyproduce the x-ray bremsstrahlung emission. However, these energy lossesshould be expected to remove only a fraction of the total cascade energyas the annihilation occurs at a depth to which the incident proton was ableto penetrate unimpeded. These considerations make the value of fT ≈ 0.7perfectly reasonable within the context of the quark nugget model. It should514.2. Nuclear annihilationsalso be noticed that the actual value must be larger than stated here as atleast some of the diffuse x-ray background must come from sources capableof producing the observed spectral lines.An estimation of the factor fep is more complicated. As suggested abovethe probability that a proton will scatter inelastically off the quark surfaceis much greater than for it to penetrate into the quark matter. However,if the nuggets are sufficiently charged, a reflected proton will remain boundto the nugget. As discussed in section 2.3, the bound proton will then re-main bound to the nugget until it either annihilates or undergoes a chargeexchange reaction, either through induced beta decay or direct charge ex-change with the quark matter. As the net rate of charge deposition on thenugget must be near zero the relative rate of electron and proton annihila-tions will be set by the probability of these charge exchange interactions.In all the cases discussed above, there remains much uncertainty relatedto the distribution of matter across the galactic centre, and to the natureof the astrophysical backgrounds contributing to the galactic spectrum atthe same frequencies as the quark nuggets. It is compelling that estimationsof the energy scales and spectra expected from the nuggets, which are allbased on well known physics, predict an excess of diffuse emission at pre-cisely the frequencies where the galactic spectrum seems to require a largerthan anticipated set of sources. These excesses are also found to occur ina ratio very much like that anticipated by the quark nugget dark mattermodel. In order to further test this model, it would be beneficial to exploreits consequences in a regime in which the physical processes, backgrounds,and overall scales involved are dramatically different from the galactic datadiscussed above. To this end, the following chapter will be devoted to ap-plying many of the considerations we have just dealt with to much largerscale cosmological observations.52Chapter 5Cosmological ConsequencesAs argued in the previous chapter, antimatter nuggets, in an environmentwith a matter density comparable to that of the galactic centre, will pro-duce thermal radiation across the microwave band up to the eV scale. Theother emission mechanisms discussed above will also occur, but here we areconcerned only with the thermal component which represents the majorityof the energy involved. At present, these densities are reached only in re-gions where gravitational collapse has produced an over-density of matter,well above the cosmological average25. However, the expansion of the uni-verse implies that, in the distant past, the matter density must have beenat this scale even in the highly isotropic early universe. We may then ask ifthe collective thermal contribution of these quark nuggets at large distancesmake a significant contribution to the isotropic radio background. The fol-lowing section, which addresses this question, is based on original researchfirst published in collaboration with Ariel Zhitnitsky [81].At microwave wavelengths the isotropic background is saturated by theCMB radiation, a blackbody spectrum with a temperature of 2.7K (up towell known anisotropies below the mK scale.) As should be expected fromthermal radiation, the spectral intensity of this background falls off as essen-tially the second power of frequency below its peak. Conversely, as discussedin appendix C and section 3.2, the thermal spectrum of emission from theelectrosphere displays only a relatively weak, logarithmic, dependence onfrequency. As a result of this relatively flat spectrum, the nugget contri-bution to the radio background may come to exceed the CMB contributionat frequencies well below the thermal peak of the CMB. If this is the casethere may be a window in the radio band, below the CMB peak but abovethe low frequency cut off in the nugget spectrum, in which the nuggets haveobservational consequences.Recent observations by the Arcade 2 experiment show just such a lowenergy excess over the isotropic background of the CMB [39]. In the contextof these observations, it has been suggested that several earlier observations25The galactic centre has a density of ρ ∼ 100GeV cm−3, roughly eight orders of mag-nitude above the average matter density in the universe today.53Chapter 5. Cosmological Consequences[54, 87, 103, 104] may show evidence for a similar excess at even lowerfrequencies [39, 46]. The Arcade 2 analysis suggests that their data isconsistent with a power law rise with a spectral index of ∼ 2.6. Whencombined with the earlier radio data the best fit power law obtained is,T (ν) = T0 + TR(νν0)β(5.1)with the CMB temperature T0 = 2.729 ± 0.004K, and with the fit param-eters TR = 1.19 ± 0.14K and β = −2.62 ± 0.04 for the reference frequencyν0 = 1GHz26. According to the Arcade analysis, the total magnitude andthe spectral index observed seem to require an additional spectral compo-nent beyond that expected based on a range of complementary observations.At these wavelengths and angular scales the primary astrophysical contri-butions (apart from the CMB) are galactic foreground emission and thecontribution of distant point sources below the resolution of Arcade 2.The treatment of these galactic and extragalactic contributions is discussedextensively in the original Arcade papers [39, 72, 106] and will only bebriefly reviewed here.The galactic contribution to the Arcade 2 data in the 3-10GHz rangewas analyzed in [72]. In this frequency band galactic emission is domi-nated by synchrotron emission but also includes smaller contributions frombremsstrahlung emission from ionized particles and discrete radio sources.Determination of the extragalactic background requires the removal of theforeground galactic component. This process is rather complicated and in-volves introducing various models for galactic emission determined by boththe Arcade 2 data and earlier radio sky maps27. However, the subtrac-tion process performed in [72] is not believed to contribute to uncertaintiesabove 5mK in the 3GHz channel, well below the ∼ 60mK excess observed.The Arcade 2 data plotted in figure 5.1 has had the galactic contribu-tion subtracted and the error bars include the cited uncertainty level in thissubtraction. To extract an extragalactic radio background, data from theearlier radio surveys [54, 87, 103, 104] incorporated into the analysis of the26It should be noted that the spectral index of the Arcade 2 excess is suggestivelysimilar to that which the Planck results suggest for the WMAP haze as discussed insection 3.2. This must be the case if both are to arise from the same thermal emissionprocess.27In particular the Arcade 2 analysis uses a galactic emission map derived either froma simple plane parallel galactic model or a more complicated map constructed from asurvey at 408 MHz [54] and the CII line map made by COBE/FIRAS [38]. Both modelsproduce a similar result in terms of the extragalactic background extracted.54Chapter 5. Cosmological ConsequencesArcade 2 results in [106] were subjected to the same galactic foregroundsubtraction procedure as the Arcade 2 data.The contribution of various background sources was analyzed in [106].That analysis used deep surveys of radio sources in the 1-10GHz range andfar-IR surveys, which may be correlated with radio band emission, to esti-mate the total discrete source contribution to the observed isotropic back-ground. It was concluded that the 58mK excess observed in the Arcade 2data at 3GHz would require an implausibly large source density, and thateven optimistic estimates are likely to provide a temperature increase of only5-10mK above the CMB. A separate assessment of the radio background wasmade in [48] based on previous radio surveys. That work extracted the con-tribution from potential unresolved extragalactic sources from source countsin the 150MHz to 10GHz range, modeled by a two component power lawbackground allowing for populations of both hard and soft radio emitters.In this case the observed contribution to sky temperature from unresolvedsources take the form,∆T = T0(νν0)γ0+ T1(νν0)γ1(5.2)with the reference frequency taken as ν0 = 610MHz. The best fit was ob-tained for T0 = 876± 22 mK, T1 = 18.9± 0.2 mK and γ0 = −2.707± 0.027,γ1 = −2.0. This implies a relatively weak flat component and a steep com-ponent that exceeds the CMB emission only below the Arcade 2 data butwith steeper slope than observed. This implies the existence of an addi-tional, highly isotropic radio source which provides the Arcade 2 excess.Subsequent to the publication of the Arcade 2 results several reanalyses ofthe astrophysical contribution to the radio background (through large lowsurface brightness regions, radio supernova, quasars and distant star forminggalaxies) have found that these sources seem able to account for, at most, asmall fraction of the observed excess [110, 123].Since its detection the Arcade 2 radio excess has motivated a numberof dark matter models in which processes involving dark matter contributeto the radio background [44, 45, 57, 125]28. In the dark matter modelconsidered here, the radio excess is explained in exactly the same way as thegalactic backgrounds discussed in the previous chapter. As such, the scaleof the cosmological background in this model is directly predicted, with no28Direct annihilation of the χχ¯ → 2γ occurring at a rate sufficient to produce theobserved excess are excluded by other constraints, so these models generally assume thatthe dark matter must annihilate to pairs of leptons. These then emit radio band radiationas they are deflected by the magnetic fields of early galaxies.555.1. Temperature evolutiontunable parameters, based on the strength of the diffuse galactic emission.The following sections will argue that, if the galactic backgrounds discussedabove are due to a quark nugget dark matter contribution, then the isotropicradio background generated by the nuggets must occur at exactly the scaleand energies which are observed by Arcade 2.5.1 Temperature evolutionThe temperature of an antiquark nugget in a given environment is deter-mined by the flux of matter onto the nugget. When matter annihilateswithin the nugget much of the released energy is thermalized, raising thenugget’s temperature. In a gas of hydrogen atoms, with temperature T , thisflux may be estimated as,dEdt dA= ρc2v = ρc√2TmH. (5.3)Here ρ is the background density and v is the mean velocity of the hydrogenatoms (which scales with the square root of the temperature) and mH is themass of a hydrogen atom. As the universe expands both the matter densityand the temperature of the matter fall, resulting in a lower flux onto thenuggets. In a matter dominated universe the temperature drops with thesquare of scale factor while density falls as the third power. The matter fluxat redshift z may then be expressed asdEdt dA= ρ0c2v0(1 + z)4. (5.4)The transport of thermal energy across the nuggets occurs rapidly in com-parison to the Hubble time, so it may safely be assumed that the nuggets willremain in thermal equilibrium as the universe expands. The total thermalemissivity of the nuggets, as a function of temperature, is given in appendixC, equation C.5. Under the assumption of thermal equilibrium, this emis-sion rate must be directly proportional to the flux of energy into the nuggets,as given in equation 5.4. The exact proportionality will depend on the ef-ficiency with which the background matter is converted to thermal energy.This, in turn, depends on the thermal properties of the nuggets as well as thephysics of the early universe plasma. Rather than introducing this constantof proportionality as a phenomenological parameter, it is more intuitive tonote that, once its value is fixed at any point in the cosmological history,the subsequent evolution of temperature, as a function of redshift, is given565.1. Temperature evolutionby expression 5.4. As such, we may equivalently take the temperature ofthe nuggets at the time of last scattering as a phenomenological parameterto be estimated from observed data. In this case, the temperature evolutionis given by,T (z) = TLS( 1 + z1 + zLS)17/4(5.5)where the subscript LS denotes values at the surface of last scattering. Theeffective radiating temperature of the nuggets at the time of last scatter-ing is, in principle, calculable from first principles within the context of aspecific quark matter model. However, the inherent uncertainties in sucha calculation make it unjustified at the present level of analysis. Instead,I will use a comparison with microwave emission from the galactic centre(which was argued in section 3.2 to be attributable to the same thermalemission process) to make a consistency argument that, if the microwaveexcess observed from the galactic centre arises through the interaction ofthe interstellar plasma with nuggets of quark antimatter, then the sameprocess will necessarily produce a cosmological background at the observedscale.The similarity in temperature in the galactic centre and at the time oflast scattering may be seen by estimating the energy flux onto the nuggetsin these two different environments. The estimation of the nugget contribu-tion to the galactic spectrum made in [43] adopted a typical visible matternumber density, in the galactic centre, of nvm ∼ 150cm−3 and assumed aninterstellar medium dominated by hydrogen, and a velocity at typical galac-tic scales of v ∼ 200km/s. In this environment the energy flux onto thenuggets will scale asρv ≈(150GeVcm3)(2× 107cm/s)= 3× 109 GeVcm2s . (5.6)At the time of last scattering the photons decouple from the baryonic matter.Up until this moment the photons are thermally abundant, afterwards thephoton number density dilutes with the expansion in the same way as thebaryon density, so that the baryon to photon ratio, η given in equation1.1, remains fixed. At last scattering the motion of the baryonic matter isprimarily thermal. Under these conditions the matter flux onto the nuggetmust scale as,ρv ≈ ηnγ√2Tmp≈ 3× 108 GeVcm2s . (5.7)575.2. The isotropic radio backgroundNoting the similar scales of equations 5.6 and 5.7, as well as the fact thatthe temperature scales as only a weak (∼ 4/17)29 power of the energy flux,we should anticipate that the temperature reached by the nuggets in theearly universe should be similar to, if slightly lower than, that reached inthe galactic centre where they were estimated to have a temperature ofT ∼ 1eV [43].5.2 The isotropic radio backgroundCombining the emission spectrum of a quark nugget at a given temperature,from expression C.4, with the redshift evolution of nugget temperature, fromexpression 5.5, allows us to determine the contribution of quark nugget darkmatter to the isotropic radio background. This may be done by integratingeach individual nugget’s contribution over the entire dark matter distribu-tion back to the time of CMB formation. A full treatment of this processwould include the reheating of the nuggets, as a result of structure formationat late times. While technically possible, such a calculation would be com-putationally expensive, and would detract from the simplicity of the basicideas presented here. Instead, I will neglect the effects of structure forma-tion and consider an isotropic universe, in which the nuggets’ temperaturefalls with the universe’s expansion precisely as given in expression 5.530.In the limit in which the universe remains homogeneous, the dark mattercontribution to the radio background is given by the line of sight integral,through the increasingly dense dark matter distribution, of the spectral con-tribution from each distance. This intensity must then be redshifted to thewavelength at which it will appear today. The line of sight integral throughthe dark matter background is discussed in appendix A.3, and results in anisotropic background intensity given by,I(ν) =∫c dz(1 + z)H(z)ρDMMNdEdν dt[ν(1 + z), T (z)] . (5.8)29This scaling differs from the typical fourth root as the nuggets do not emit a standardblackbody spectrum. See appendix C for details.30While this may seem a dramatic simplification it is, to some extent, justified by thefact that the total radio background is strongly dominated by early time emission. Evenif the calculations are modified to allow a tenth of the dark matter to reheat to 1eV (thepresent temperature in the galactic centre) it is found to result in variations in the radiobackground at only the few percent level. As such, reheating will make little contributionto the scale of the radio background though it may result in slight anisotropies, an analysisof which is beyond the scope of this work and is not strongly motivated by the resolutionof present data.585.2. The isotropic radio backgroundHere H is the Hubble constant and MN is the average mass of a nugget sothat nDM = ρDM/MN is the dark matter number density. The emissionspectrum, dEdν dt , is as given in equation C.4 and is to be evaluated at thetemperature of the nuggets at redshift z, and at the frequency ν(1+ z) thatis redshifted to the present day observed value ν. The low energy cutoff, asgiven in expression C.3, will also be redshifted to lower energies so that weshould expect little or no spectral contribution below ∼ 6MHz, the observedfrequency today of a photon emitted at the minimum allowed frequency inthe early universe.Rather than dealing directly with the measured intensity this discussionwill follow the analysis of Arcade 2 [39] and convert the measured intensityinto an apparent sky temperature. This is done by assuming that the skyacts as a blackbody emitter (as it does in the case of the CMB) and invertingthe Planck spectrum to find the temperature which would account for theintensity at a given wavelength. In this representation a universe with noradio background apart from the CMB contribution would have an observedsky temperature of 2.7K across all frequencies. The presence of any typeof cosmological radio sources will then produce sky temperatures above thisvalue across the frequencies over which they emit.Figure 5.1 shows the sky temperature values reported by Arcade 2 aswell as those extracted from earlier radio maps by the Arcade group. Thebest fit power spectrum determined in [39] is then plotted over this data. Thestrong rise in the isotropic radio background above the CMB contribution,after subtraction of the galactic foreground, can clearly be seen31.Having established the basic properties of the observed isotropic radiobackground it is now possible to assess the contribution which quark nuggetdark matter could make to this spectrum. Based on the arguments of sec-tion 5.1 the initial temperature of the nuggets should be expected to fallin the range 0.1eV < TLS < 1eV. As the baryon number of the nuggetsincreases the spectral contribution falls, due to the decreasing dark mat-ter number density, as can be seen in equation 5.8. Demanding that thenugget contribution to the isotropic radio background does not exceed theArcade 2 data, while also having an initial temperature in the physicallyacceptable range, requires that the nuggets have a baryon number largerthan B ∼ 1024 but does not produce an upper limit on the baryonic charge.However, to associate the Arcade 2 excess with emission from the nuggets,31As this plot shows measured temperature as a function of frequency a CMB dominatedspectrum would remain flat at T=2.7K across all frequencies as opposed to the power lawrise in temperature at low frequencies.595.3. Conclusionswhile constraining the initial temperature to be less than 1eV, requires abaryon number less than B ∼ 1028. Obviously, a larger nugget size cannotbe ruled out, but these objects would have no observational consequences atthe frequencies covered by Arcade 232. Within the range of mean baryonnumbers from 1024 to 1028 the nuggets may produce some or all of the excessradio band emission observed by Arcade 2. A representative spectrum fornuggets (with B = 1025 and initial temperature T0 = 0.2eV) is includedin figure 5.1 and, as can be seen, is able to match the observed rise in skytemperature across the Arcade 2 range.For the background sky temperature extracted from the pre-existing skymaps of [54, 87, 103, 104]. The resulting temperature estimates are likelyto contain at least some contamination from the galactic foregrounds, butcertainly set an upper limit on any extragalactic contribution. The trendof increasing effective sky temperature at lower frequencies, observed in theArcade 2 data, is seen to continue down to the ∼ 10MHz scale.It should be noted that the low energy cutoff in the spectrum, discussedin appendix C, is likely to be considerably more complicated than the hardcutoff imposed here. As such, the spectrum should be considered only arough estimate near its low energy peak. A more complete treatment ofthe finite size effects that lead to this cutoff could alter the details of spec-trum at low energies. Even given these uncertainties, it may be seen thatthe contribution of quark nugget dark matter to the isotropic backgroundfall below the constraints imposed by low frequency observations, and maycontribute, in part, to the radio excess argued for in [39] and [46].5.3 ConclusionsThe consequences of quark nugget dark matter have now been investigatedin a range of environments, both galactic and cosmological, and across awide range of energies, from MHz radio emission up to γ-rays in the tens ofMeV range. In each case, the predicted diffuse emission associated with thenuggets has been found to be entirely consistent with observational data.In fact, this dark matter model may offer a single mechanism capable ofexplaining the origin of several observed emission features. These were pre-viously thought to be unrelated, and each requires substantial modificationsto predicted astrophysical spectra to explain.32For nuggets with B > 1030 the nugget contribution to the radio background remainswell below the CMB contribution down to the cut off frequency of the spectrum. As suchthese objects would remain unobservable for the foreseeable future.605.3. ConclusionsIt is important to note that this model was not introduced in order toexplain any of these observed sources of diffuse emission, but was insteadproposed as a solution to cosmologically motivated questions about the na-ture of dark matter and baryogenesis. The only unknown parameter withinthis model is the mean baryonic charge of the nuggets, and thus their num-ber density. This unknown factor will scale the total strength of each of theemission mechanisms discussed above, but will not alter the shape of theactual spectrum. Once the scale of one emission source, for example thestrength of the galactic 511keV line, is established the relative scales of allother spectral contributions are also fixed.Possible future experiments may more tightly constrain the strength andmorphology of the various diffuse emission sources discussed here. This ex-perimental progress will, necessarily, have to be accompanied by a deeperunderstanding of the contribution of known astrophysical sources to thevarious diffuse backgrounds. However, it is difficult at present to take thesesignals as more than suggestive of particular models or ideas. The uncer-tainty remaining in both the measurements themselves, and particularly inthe astrophysical backgrounds, means that attributing any particular ob-servation to dark matter is necessarily speculative. Even in well studiedexamples of a known diffuse excess, such as the galactic 511keV line, nu-merous explanations have been offered. Most of these are subject to largeenough uncertainties that they would be difficult to distinguish, even withsignificantly improved observational data.Given the inherent difficulties of this sort of analysis, it makes sensethat it be complemented by direct searches for dark matter. The prospectsfor applying this type of detection to quark nugget dark matter will be thesubject of the remainder of this work.615.3. ConclusionsFigure 5.1: Antenna temperature as extracted from Rogers and Bowman[104] (red), Maeda et al. [87] (green), Haslam et al. [54] (blue), Reichand Reich [103] (yellow) and the Arcade2 data [39] (black). The data isoverlaid with spectra calculated from equation 5.8 for nuggets of baryoniccharge B = 1025 and an initial temperature T = 0.2eV chosen for a bestfit to the high frequency Arcade2 data. The dotted line is the best fitobtained in the Arcade analysis, as given in equation 5.1.Insert: The background temperature, as a function of frequency, measuredby Arcade 2 [39] showing the reported GHz scale excess. This data isoverlaid with spectra calculated from equation 5.8 for nuggets with baryonnumber B = 1025 (blue) and the best fit curve of equation 5.1 (red). Avariation of this plot was originally published in [81].62Chapter 6Direct Detection6.1 IntroductionThe indirect detection techniques discussed in previous chapters may of-fer hints as to the nature of the dark matter, and the observations behindthem have certainly inspired a wide range of dark matter models apart fromthe one discussed here. However, while these unexpected excesses in dif-fuse emission are difficult to explain through known astrophysical processes,particularly when considered collectively, it remains possible to fit this datathrough significant modification of the background astrophysics. As the ex-act nature of the relevant astrophysical processes remains an open researchquestion, it is difficult to know to what degree, if any, dark matter mustbe invoked to explain the galactic and cosmic diffuse backgrounds. Evenif the contributions of conventional astrophysical sources were known pre-cisely, it is possible that any remaining dark matter signature could be fitby a variety of alternative models. This is particularly true of models withan extended dark sector involving a large number of tunable parameters33.As such, indirect dark matter searches are strongly complemented by paral-lel direct searches. These involve contact between a detector and the darkmatter itself, rather than the light produced by its interactions, and mayprovide a cleaner detection signal and a greater ability to distinguish be-tween alternate dark matter models. The following discussion is based onoriginal research conducted as a component of this thesis and closely followsthe results first published in [73, 77].There are, at present, many dark matter searches underway around theworld [11, 22, 31]. There have been several intriguing results from this33It is generally assumed that the dark matter consists of a single particle type, left asa relic from the early universe. In this case its self interactions would be limited to theannihilation of particle antiparticle pairs. However, in attempts to explain observationssuch as the diffuse emission discussed here, the DAMA annual modulation [31] and theapparent positron excess observed by PAMELA [10] and AMS [12], dark matter modelsinvolving an extended dark sector involving many particle types have been introducedto allow for a richer phenomenology than is possible with a single dark matter particlespecies.636.1. Introductionsearch program, for example the DAMA experiment [31] at the Gran Sassolaboratory has observed an unexplained seasonal variation in particles hits.This type of seasonal variation is expected as the earth’s velocity relative tothe dark matter distribution varies with motion around the sun. This wasassumed to be a signal clearly attributable to dark matter interactions andits detection over many years and with the correct phase for a dark mattersignal, would seem strong evidence for dark matter detection. This strongdetection signal is, however, problematic. The DAMA result favors a light(M ∼ 10GeV) dark matter particle mass and a nucleon scattering cross sec-tion at the σ ∼ 10−40cm2 level. However, much of the allowed parameterspace is excluded by non-detection in collider searches and by higher sen-sitivity direct dark matter searches, such as CDMS [13] and XENON [21].Conversely, the CoGeNT experiment reports an excess of low energy, single-hit detector events which may be consistent with a light dark matter particlewith similar properties to those required to explain the DAMA oscillation[2]. The CoGeNT signal also seems to show a seasonal variation with aphase similar, though not identical, to that of DAMA [1]. In light of theirconfusing, and partially contradictory, observations the results of these vari-ous underground cryogenic experiments will be taken as interesting, thoughfar from conclusive.The main focus of the underground direct detection search program isthe rare interactions of WIMP scale dark matter passing through a detec-tor. Given the relatively large flux, as expressed in equation 1.2, and thesmall interaction cross section of WIMP dark matter, these experiments at-tempt to push the sensitivity limit in as large a detection mass as possible.However, even the largest present or proposed dark matter searches do nothave the multiple square kilometer detector dimensions capable of placingmeaningful limits on the presence of very high mass dark matter such as themodel considered here. Instead, this work will focus on the largest area par-ticle physics detectors available and their prospects for detecting this classof heavy dark matter.The most important experiments in this context are those designed tostudy ultrahigh energy cosmic rays34. In recent years there has been signifi-cant interest in the study of the origin and propagation of cosmic rays at thehighest energies. Experiments targeting these ultrahigh energy cosmic raysare particularly interested in gathering statistically significant numbers of34There have previously been several searches for a flux of high mass neutral objectssuch as strangelets or monopoles but none impose serious constraints in the mass rangeconsidered here. A review of several of these experiments and the resulting mass and fluxbounds are given in [23].646.1. Introductionevents at or above the Greisen-Zatsepin-Kuzmin (GZK) cutoff in the cosmicray spectrum35. The flux of cosmic rays falls with energy as a simple powerlaw over many orders of magnitude up to the GZK cutoff which occurs neara flux of roughly one per square kilometer per year and steepens the spec-trum significantly at higher energies. Consequently, the detectors intendedto study these events must have areas at or above the square kilometer scaleif they hope to provide statistically significant data on cosmic rays at rel-evant energies. This scale is of interest not only for cosmic rays physicsbut also, I will argue, may prove useful in the search for quark nugget darkmatter.Ultrahigh energy cosmic rays are detected via the extensive air showerthey initiate through the particle cascade extending downward from thefirst collision between the ultrahigh energy primary and the molecules ofthe upper atmosphere. The developing air shower may be detected by par-ticle detectors on the ground, which directly observe any secondary particlesreaching the earth’s surface, through atmospheric fluorescence generated asthe passage of charged particles excites UV band transitions in the surround-ing nitrogen molecules or, through the radio emission generated by the largenumber of secondary charged particles moving through the earth’s magneticfield.In order to extract meaningful limits on the flux of quark nugget darkmatter from cosmic ray observatories it is necessary that their passagethrough the atmosphere initiate a large scale air shower, which may bedetected, rather than a highly concentrated energy release which, while pos-sibly quite intense, would not lend itself to this type of detection. The re-mainder of this work will be devoted to the phenomenology of the air showerassociated with the passage of a quark nugget through the atmosphere andwill argue that the rate of these events may be strongly constrained withdata from the current generation of cosmic ray experiments.A nugget of quark matter passing through the atmosphere at galactic ve-locities will interact primarily through the elastic scattering of atmosphericmolecules, possibly generating some amount of ionization. As the scatteredparticles remain at relatively low energies, the trail of accelerated moleculeswill be limited to roughly the cross section of the nugget rather than trig-gering a much larger air shower. If we assume that every molecule along thenugget’s path is accelerated to a typical galactic velocity of vN ∼ 200km/s35The GZK cutoff is a feature in the cosmic ray spectrum above 1019eV originallypredicted in 1966 [52, 124] and only recently observed [3, 4]. It is due to the limiting ofthe cosmic ray horizon by the scattering of cosmic rays off the CMB above the thresholdfor photo-pion production.656.2. Air shower production and scalethen the total deposited energy will be on the order of a few joules36. Theseevents will release most of their energy low in the atmosphere, and at dif-ficult to observe thermal levels. Consequently, they will have little or nodetectable signature for a detector whose primary target is high energy cos-mic rays. Conversely, a nugget composed of antiquarks will annihilate muchof the atmospheric material in its path and the products of these annihi-lations, many of which will be produced at nuclear scale energies, may becapable of initiating a much larger shower of high energy secondary particles.The maximum total energy generated by a quark nugget can be estimatedby assuming that all the matter lying in its path is annihilated and that theenergy released is twice the energy equivalent of this atmospheric mass:∆E ≈ 2piR2NXatc2 ≈ 107J(RN10−5cm)2. (6.1)Here Xat ≈ 1kg/cm2 is the total atmospheric depth and RN is the radial sizeof the nugget. This is obviously a substantial amount of energy and, thoughmuch of it will be thermalized within the nugget or emitted in difficult todetect channels, it is easily capable of producing a signal which may beobserved with a range of present experiments. The following sections willdiscuss the physics of the passage of a quark nugget through the earth’satmosphere and extract the basic observable properties of such an event.6.2 Air shower production and scaleThe primary interaction of a nugget of quark matter will be nuclear anni-hilations involving atmospheric molecules. Any initial imbalance betweenthe rate at which molecular electrons and nuclei annihilate will quickly becompensated by a net charge developed by the nugget. As such, it is safeto assume that electron and proton annihilations proceed at essentially thesame rate.Electron annihilations will proceed in much the same way they do in themuch lower density environment of the interstellar medium, so much of theanalysis of the previous chapter is directly applicable. The majority of these36The nuggets carry substantial momentum relative to the much lower density atmo-sphere so that their velocity is essentially unaltered as they travel downwards. Assumingthat the entire atmospheric column swept out is accelerated to the velocity of the incom-ing nugget then the kinetic energy loss is simply given by the ratio of the mass of this aircolumn to that of the nugget. Using typical nugget parameters the fractional change inkinetic energy is ∆T/T ∼ 1013/B which is negligible across the allowed range of nuggetbaryon number.666.2. Air shower production and scaleannihilations will result in the production of photons in the energy range0.1-10MeV. These photons contribute to the electromagnetic component ofthe shower.Tracing the energy generated by nuclear annihilations is a more com-plicated problem. As previously discussed, the majority of the energy gen-erated will be thermalized within the nugget, to eventually be emitted asthermal photons with a spectrum as given in equation C.4. Annihilationshappening near the nugget’s quark surface produce hadronic jets, which cas-cade down to lighter modes of the quark matter. Much of this energy will bedissipated as the charged components pass through the electrosphere, result-ing in x-ray emission as discussed in section 3.3. Hadronic components of theannihilation jets are likely to remain bound to the nugget, while electronsand positrons will either annihilate or be captured within the high densitylayers of the electrosphere. The only components to escape the nugget arelikely to be secondary muons generated in annihilations very near the quarksurface. All other particles are likely to eventually transfer their energy tosome form of photon emission from the nugget electrosphere.First, consider the thermal evolution of the nugget as it passes throughthe atmosphere. Before entering the atmosphere the nugget will have atemperature of ∼ 1eV as discussed in section 3.2. As it annihilates atmo-spheric material some fraction (which I label fT as in section 3.2) of thereleased energy will thermalize within the quark matter. The thermaliza-tion process will occur on QCD timescales, much shorter than the evolutionof the atmospheric density along the nugget’s path, so we may safely assumethat the nugget will remain in radiative equilibrium as it crosses the atmo-sphere. That is, the rate at which energy is deposited in the nugget shouldbe balanced by its net thermal radiation as given by expression C.5. Thiscan be translated into a temperature evolution in terms of the surroundingatmospheric density:(T10keV)17/4=(ρat(h)5× 10−2kg/m3)(vN200km/s)fT . (6.2)Given that the atmospheric density at ground level is ρat ≈ 1.2 kg/m3, thisimplies a maximum nugget temperature of roughly 20keV provided that allmatter swept up by the nugget fully annihilates.There is, however, an upper limit to the rate at which atomic scale mattercan be forced onto the quark surface. As the flux of matter onto the surfaceincreases, so must the flow of energy away from the surface. Independentof the exact mechanism of outward energy transfer it must, at some level,676.2. Air shower production and scaleimpede the inward flow of matter, setting up a feedback mechanism by whicha maximum annihilation rate is established. The temperature at which theannihilation rate saturates was estimated in [73] based on the rate of inelasticscattering of positrons off of an incoming molecule. This analysis suggestedTmax ∼ 10keV. However, as pointed out in [50] this analysis did not considerthe photoionization of surrounding matter, an effect which may increase themaximum temperature by an order of magnitude. A more precise estimationof the maximum temperature reached by the quark nugget while it crossesthe atmosphere requires a detailed treatment of the plasma surrounding thenugget by the time it reaches the lower atmosphere. As the evolution of thisplasma will influence the emission spectrum of the nuggets its developmentremains an important open question. Despite this remaining uncertaintyit is possible to extract a rough phenomenological description of the airshower’s development based on a simplified set of energetic constraints.One crucial feature of the development of a quark nugget induced airshower is the timescale over which it will develop. In a conventional airshower, initiated by a single ultrahigh energy proton or nucleus, all of theshower components are ultrarelativistic and move at essentially the speed oflight. The resulting shower thus crosses the entire atmosphere on a ∼ 10µstimescale. All observables related to the shower should occur over timescalesshorter than this. In the case of a quark nugget initiated shower, the sec-ondary particles emitted by the nugget will move at the speed of light butthe nugget itself, which sources the surrounding shower, moves at galacticscale velocities, some three orders of magnitude slower. Consequently, theair shower initiated by a quark nugget will develop on a much slower ∼ mstimescale.Beginning with this qualitative picture of air shower development, it ispossible to estimate some of the most important observable properties of aquark nugget induced air shower relevant for cosmic ray observatories.The scale of an air shower, generated by an antiquark nugget passingthrough the atmosphere, will depend on the rate of nuclear annihilationswithin the nugget. As this rate increases with atmospheric density, ratherthan depth, the shower develops deep in the atmosphere where the densityis well modeled as an exponential decay with heightnat(h) = n0e−h/H (6.3)with a typical scale height H ≈ 7.5km and a surface nucleon density of n0 =ρ0/mN ≈ 7×1020 cm−3. If we neglect the thermal saturation effect discussedin the previous section the rate of nuclear annihilations is simply given by686.2. Air shower production and scalethe product of the nugget cross section and velocity with the atmosphericdensity:Γan = piR2NvNnat(h). (6.4)This would result in an annihilation rate ∼ 1018s−1 near the surface. If,however, the annihilation rate saturates at a temperature Tmax then themaximum annihilation rate may be obtained by solving expression C.5 forthe rate at which annihilating material must deposit energy :Γan =1mNc2dEdT= 643 α5/2R2NT4maxmN4√Tmax10keV (6.5)≈ 2× 1017s−1(Rn10−5cm)2 ( Tmax10keV)17/4. (6.6)Figure 6.1 shows the annihilation rate as a function of height in the atmo-sphere.This analysis of the annihilation rate sets the energy scale available todrive air shower development at a given height. Having established this scaleit is now necessary to determine how efficiently this energy can propagateout from the nugget and create the type of large scale events to which cosmicray detectors are sensitive.As stated above, the annihilation of electrons results in the productionof high energy photons across the x-ray and γ-ray bands. The majorityof the energy released in nuclear annihilations is thermalized within thenugget and subsequently radiated according to the thermal spectrum givenin equation C.4. Of the energy emitted directly from the annihilation pointwithout thermalizing the majority is transferred to surface positrons whichthen radiate in the x-ray band as discussed in section 3.3. Finally, somesmall fraction arising from annihilations very near the surface will releasehigh energy particles with near the GeV scale.Hadronic particles are strongly coupled to the quark matter, and areunlikely to escape the nugget. Electrons and positrons created in nuclearannihilations are unlikely to be able to penetrate through the high densityelectrosphere, with the electrons rapidly annihilating as was discussed insection 3.5 and the positrons being rapidly decelerated as discussed in section3.3. As such, the only charged particles likely to be able to escape from thenugget and propagate through the atmosphere in significant numbers arehigh energy muons produced in nuclear annihilations very near the quarksurface. Thermal photons will be readily emitted from the outer regions ofthe electrosphere and higher energy photons may escape from deeper withinthe nugget where they are generated by either nuclear of electron-positron696.2. Air shower production and scaleFigure 6.1: The total rate of nuclear annihilation as a function of heightin the earth’s atmosphere. The red curve assumes no saturation of thisrate and exactly follows expression 6.4. The blue curve assumes a thermalsaturation effect that switches on at a nugget temperature of T = 15eV.annihilations. As the temperature of the nuggets in the lower atmospheremay be as high as a few tens of keV, the majority of the radiation emitted bythe nuggets will be in the form of x-rays. There will also be a much smallercomponent of higher energy photons with energies up to about 1GeV. Atlower energies the very flat thermal spectrum means that there will also benon-negligible emission right down to radio frequencies.The atmosphere is relatively opaque to high energy photons, so thatthe majority of emitted radiation will be absorbed quite close to the nugget.This will produce a localized ionization trail, but will not lead to emission ona large enough scale to present a clear target for large cosmic ray detectors.For present purposes, I will focus on the high energy muons and the longwavelength radiation which is capable of propagating over long distances inthe atmosphere.The production of low energy photons is governed by the thermal spec-706.3. Atmospheric fluorescencetrum in expression C.4, and the nugget’s temperature evolution is as given inequation 6.2. The production of muons is more difficult to estimate. A rel-ativistic muon may be produced directly in a nuclear annihilation, but thisprocess is disfavored relative to the production of strongly coupled mesonsor other light superconductor modes. As a rough estimate of the scale ofthis process we may assume that the production of electromagnetically cou-pled particles is suppressed by a factor of (α/αs)2 where α = 1/137 is thefine structure constant and αs is its order one strong force equivalent. Thissuppression factor would imply that the probability of producing a muon issmaller than annihilation to more strongly coupled modes by a factor of atleast 104. Assuming that high energy muons are produced at this level wewould expect roughly one muon to be produced in every 104 events involvingnon-thermal emission which in turn represent roughly a tenth of all anni-hilation events. Assuming that this is the only mode of muon production,the rate of muon production will be given in terms of the total annihilationrate by approximately 10−5Γan. There are, however, other processes whichare capable of producing high energy muons such as the decay of a meson-like excitation very near the quark surface. The range of mechanisms whichmay lead to emission of a relativistic muon is large and rather complicated,and will depend on the form of quark matter realized near the nugget’s sur-face. For the purposes of this analysis I will comress all of this informationinto a single muon production factor fµ defined so that the rate of muonproduction is given byΓµ = fµΓan, (6.7)where I will assume fµ ≥ 10−5 though it is not likely to greatly exceed thisvalue. Taking fµ at this scale would give a production rate of 1012s−1 deepin the atmosphere. While muon production may be strongly suppressed thesheer number of annihilations leads to the production of a large number ofrelativistic charged particles surrounding the nugget.6.3 Atmospheric fluorescenceOne of the primary detection techniques for high energy cosmic rays isthrough the observation of atmospheric fluorescence generated as chargedparticles move through the atmosphere. A relativistic particle dissipates en-ergy along its path through the excitation and ionization of stationary back-ground molecules. For nitrogen molecules, which constitute the majority ofthe atmosphere, excitation by a charged particle is followed by radiative de-excitation emitting light in the UV band with wavelengths λ ∼ 300 − 430716.3. Atmospheric fluorescencenm. As the atmosphere is relatively transparent to UV radiation this fluo-rescence light may be used for detection of cosmic rays of sufficient energyto produce a large number of secondary particles. This radiation is emittedisotropically from the track of excited nitrogen molecules left in the shower’swake so that it may be observed from positions well off the shower axis (asopposed to Cherenkov radiation, for example, which is also produced byrelativistic particles moving through a medium, but which is emitted in aforward directed cone.) The total fluorescence yield is proportional to thetotal number of charged particles in the shower so that its intensity directlytraces shower development. The main drawback of this detection techniqueis that the fluorescence light is relatively faint and can only be observed onclear moonless nights. These restrictions give fluorescence detection a dutycycle of only about 10%.There are currently several large experiments which use this techniqueto detect high energy cosmic rays including the Pierre Auger Observatory[5] and the Telescope Array [115].As discussed above, the air shower initiated by an antiquark nugget willbe dominated by ultrarelativistic muons emitted in annihilations near thequark matter surface. These are generated at a rate given by expression 6.7and, from this estimate of muons production at a given height, it is possibleto predict the fluorescence yield of these muons. The motion of charged par-ticles through the atmosphere is rather complicated and, as such, the detailsof their treatment have been relegated to appendix D to better concentratethis discussion on the most fundamental properties of the shower.The rate at which muons are produced will scale with the atmosphericdensity, becoming quite large near the surface. However higher atmosphericdensity also reduces the average distance that each muon travels. As arguedin appendix D, the drop in scattering cross-section with energy implies thathigh energy muons lose very little of their total energy before they decay.As such, this length scale is independent of the height at which the muonis initially produced. However, muons at lower energies may lose enoughenergy to be stopped by collisions with surrounding matter in the lower at-mosphere. This means that the average path length of a muon is dependenton the energy spectrum with which the muons are emitted.If we consider a model in which all muons are injected at typical QCDscales with about a GeV of energy, the energy loss to the atmosphere slowsthe muons only negligibly and the total number simply decays exponentiallyover the muon decay length (ld ≈ 7km in this case.) In this case the numberdensity of particles as a function of radial distance from the nugget and the726.3. Atmospheric fluorescenceangle between the observer and the direction of nugget motion is,nµ(r,φ) =Γµ2pir2c cosφ e−r/ld . (6.8)This expression uses the radial dependence of muon emission given in equa-tion D.7. For demonstrative purposes, consider the case of a nugget whosemotion is vertically downwards as this will allow us to discuss the showerproperties in a relatively simply geometric case. More general cases maybe dealt with using an identical procedure, a more complicated geometrywould simple obscures the basic arguments on which I want to focus. Inthe vertical geometry we may parameterize the muon distribution purely interms of height (h) and distance from the shower core (b):nµ(∆h, b) =Γµ2pic∆h(b2 +∆h2)3/2exp(−√b2 +∆h2ld). (6.9)Here I have defined ∆h = hN −h as the difference between the height of thenugget and the height at which muon density is being evaluated. We maythen integrate this expression over all b values to get the area integratedflux of particles at a given height :dNµdt= 2pic∫ ∞0nµ(∆h, b) cosφ b db =Γµ∆h2l2d∫ ∞∆h/lde−xdxx4. (6.10)The exponential integral is easily evaluated numerically and the resultingintegrated flux, as given by expression 6.10 assuming that the annihilationrate does not saturate at large atmospheric densities, is shown in figure 6.2.We may also use this expression to find the total number of particleswhich will move past a given height over the course of the entire shower. Todo this one simply integrates the muon flux at a given height over the entiretime that the nugget spends above that height in the atmosphere:Nµ =∫ ∞h(dNµdt)dhNvN(6.11)where the integrated flux dNµdt is as given in equation 6.10. This expressiongives a scale for the total number of charged particles contributing to atmo-spheric fluorescence over the entire duration of the shower. The results ofthis integration are shown in figure 6.3.This gives a basic idea of the scale of the shower, but we may alsoconsider geometries more complex than in this simple estimate. Consider, for736.3. Atmospheric fluorescenceFigure 6.2: Laterally integrated particle flux as a function of height whenthe antiquark nugget is at a height of 1km. This example uses a constantmuon injection energy Eµ = 1GeV and takes the thermal maximum to occurbelow the earth’s surface.example, a case in which the annihilation rate saturates at some height abovethe earth’s surface. If all the shower components are highly energetic thentheir path length is set by the decay time and the profile simply levels outat a constant particle number after saturation. Less energetic muons mayhowever have their path length limited by energy loses to the atmosphere. Ifthe shower involves a sufficient number of marginally relativistic muons theirdecreasing path length near the surface will cause the total shower size toshrink near the surface much as it does in a conventional cosmic ray shower.For example [73] considered a model in which the muons rapidly lose energyto the surrounding positrons as they pass through the electrosphere. In thiscase the energy spectrum is peaked near the plasma frequency of the highdensity regime of the electrosphere but includes a high energy tail :dnµdk= 1ωpe(ωp−k)/ωp , mp > k > ωp. (6.12)746.3. Atmospheric fluorescenceFigure 6.3: Total integrated particle flux as a function of height. Thisexample uses a constant muon injection energy Eµ = 1GeV and takes thethermal maximum to occur below the earth’s surface and is the total showerintegrated counterpart to figure 6.2.In this case many of the muons are rapidly slowed in the dense lower atmo-sphere and the shower size begins to decline as shown in figure 6.4.For comparison a large cosmic ray shower may have a maximum particlecontent on the order of 1010 or more. As can be seen in figures 6.3 and 6.4the air shower initiated by an antiquark nugget can easily have a particlecontent at or above this level. Consequently, the total fluorescence yieldshould be at levels observable to cosmic ray fluorescence detectors.The complication in this basic analysis comes in the cuts made to thedata based on timing. As the fluorescence light generated by a quark nuggetpersists over a timescale much longer than that associated with a cosmic ray,it is possible that data cuts made to avoid backgrounds such as meteors ordistant lightning may limit the ability of some experiments to detect thistype of long duration air shower. The chances of making a detection willbe increased if the fluorescence signal is accompanied by a signal in an756.4. Lateral surface profileFigure 6.4: Total integrated particle flux as a function of height. Thisexample uses the muon energy spectrum 6.12 and results in a reduction ofthe total particle count near the surface as low energy muons are lost fromthe shower. The curves shown are for saturation temperatures of 10keV(solid), 15keV(dashed) and 20keV (dotted). This model also uses a muonproduction factor of fµ ∼ 10−3 thus the larger overall shower scale. Figuretaken from [73].associated surface detector array. To this end, I will now turn to the surfaceparticle flux associated with an antiquark nugget.6.4 Lateral surface profileA complementary cosmic ray detection technique to the fluorescence de-tection discussed in the previous section involves the use of ground basedparticle detectors, and is generally referred to as surface detection. An airshower initiated by a cosmic ray of sufficiently high energy will produce alarge number of secondary particles as it cascades down towards the earth’ssurface. Pions produced in the cascade decay before they are able to reach766.4. Lateral surface profilethe surface and electrons are quickly slowed by energy loss to the surround-ing atmosphere. However, if they are produced at sufficiently large energies,muons produced in the air shower are able to reach the surface in significantnumbers 37. On reaching the surface these particles may be detected usinga grid of particle detectors, the properties of the initial cosmic ray may thenbe reconstructed from the number of particle counts at each station andthe arrival time of the particles at different stations. While this proceduregives less information about shower evolution, and requires more extensivemodeling to reconstruct the properties of the initial cosmic ray, it has theadvantage over fluorescence detection of operating day and night and undermost weather conditions and thus has a duty cycle of nearly 100%. For thisreason surface detectors gather statistics at roughly ten times the rate ofthe fluorescence detectors.On reaching the earth’s surface the secondary air shower componentsare spread into a flat disk with a thickness of 1-10 m depending on theamount of atmosphere through which the shower has passed. This impliesthat all of the relativistic components of the shower will strike the surfacewithin approximately 10−7s with some electromagnetic shower componentstrickling in at later times.Both the Pierre Auger Observatory [19] and the Telescope Array Project[6] have large scale surface detector grids. Auger employs Cherenkov detec-tors consisting of photomultiplier tubes submersed in tanks of water andspread across 3000 km2 while Telescope Array uses an array of plastic scin-tillation panels spread over roughly 700 km2. The array spacing for Auger is1.5km while the Telescope Array has a grid spacing of 1.2 km. In order foreither of these observatories to trigger on a surface detection event adjacentsurface detectors must record particle hits at the same time, thus the surfacearrays are only sensitive to events spread over multiple square kilometers38.As surface particle detectors are primarily sensitive to relativistic parti-cles distributed across multiple square kilometers they will only detect themuonic component of a quark matter induced air shower 39. In these show-37The muon has a lifetime of 2.2 × 10−6s. Most are produced high in the atmospherewhere the shower components are energetic enough for pair production to occur and theywill be able to travel the tens of kilometers to the surface provided that they are producedwith a boost factor of γ ∼ 10− 2038Both projects have smaller sections of the array where the spacing between surfacedetectors is smaller, thus allowing for the detection of lower energy showers. These howeverrepresent only a small fraction of the total array and, for the purpose of a search for quarknugget dark matter, I am interested mainly in the detection capability of the entire areacovered by surface detectors.39Obviously if an antiquark nugget were to directly strike a surface detector it would776.4. Lateral surface profileers the nugget emits muons with energies of up to about 1GeV which arecapable of traveling a few kilometers before they decay. The nugget pro-ducing these relativistic muons has a speed of only about 200km/s and thepresence of this slower moving primary extends the timescale over whichparticles will arrive at the surface. The first particle counts in a surfacedetector will occur when the nugget is still several kilometers above thesurface when the highest energy muons it emits are first able to reach theground. Surface detection will end when the nugget strikes the surface ofthe earth. Therefore, the difference between the arrival time of the firstand last shower components may be as large as a few tens of milliseconds.This timescale is considerably longer than that of a cosmic ray induced airshower and may present difficulties in the triggering of detectors built withthe specific parameters of a cosmic ray air shower in mind.The surface detection of particles emitted from a quark nugget may bestudied in much the same way as particle motion through the atmospherewas treated in section 6.3. In particular the approximations leading to theexpressions for the charge density in equation 6.9 and integrated flux 6.10may be directly applied to the case where h = 0. It is also useful to know thetotal number of particle hits received at a detector a given distance from theshower core. I can define a local muon flux as nµ)vµ and then take the dotproduct of this with the surface to get the local flux along the surface as afunction of the nugget’s height. The total flux over the course of the showeris then given by integrating this expression back along the nugget’s paththrough the atmosphere. If we again work in the simplified geometry wherethe nugget moves vertically downwards this gives a total particle count ofdNµdA(b) =∫ ∞0dhNvNc nµ(hN , b)∆h√b2 + h2N. (6.13)This expression gives the surface flux as a function of radial distance fromthe shower core, the lateral profile produced is shown in figure 6.5 and theintegrated flux arriving at the earth’s surface as a function of time duringthe shower is shown in figure 6.6.The most important feature of figure 6.5 is its lateral extent. The parti-cles of the air shower arrive at the earth’s surface distributed over a multiplekilometer radius. This is of particular importance for the relatively sparseobserve a very large deposit of energy. However, the individual surface detectors havecross sections of only a few square meters so the probability of a direct hit is very small.It is only when taken collectively that the surface array of a cosmic ray detector has asufficient collection area to impose useful constraints on the flux of quark nuggets.786.5. Geosynchrotron emissionFigure 6.5: Total particle flux received over the entire air shower as a func-tion of radial distance from the shower core.surface detector grids employed at large scale cosmic ray detectors. If thelateral scale of the shower had been less than a kilometer across it wouldbe unlikely to cause coincidental particle hits in adjacent detectors and,consequently, would not be able to trigger a shower detection. While thetimescale involved in the shower is longer than that of a typical ultra highenergy cosmic ray initiated shower the particle flux is steeply peaked at latetimes, as may be seen from figure Geosynchrotron emissionThe same relativistic particles responsible for generating atmospheric flu-orescence and triggering surface detectors will also produce a contributionto the radio band emission of the nugget. This occurs as charged particlesare deflected by the earth’s magnetic field resulting in the emission of syn-chrotron radiation. A similar effect occurs with the secondary particles ofan air shower induced by an ultrahigh energy cosmic ray. Radio detection796.5. Geosynchrotron emissionFigure 6.6: Laterally integrated flux of particles recorded at the earth’ssurface as a function of time. Here the nugget is taken to strike the surfaceat t = a promising addition to hybrid detectors as it can operate with a muchhigher duty cycle than fluorescence detection, which requires clear moonlessnights, and is sensitive to many of the same air shower properties. As such,there are several experiments currently operating which use radio detectionto study cosmic ray showers [24, 59, 65]. As with the case of fluorescenceand surface detection, large scale radio detection arrays also have the abilityto set strong constraints on the flux of quark nuggets. This section will out-line the basic mechanisms by which the radio band signal is generated, andthen use those properties to extract the basic observable quantities associ-ated with such a shower. The results presented here are based on originalresearch published in [77].The earth’s magnetic field near the surface has a strength in the rangeof a few times 10µT. For this field strength a muon will undergo circularmotion with a frequency ωB = eB0/mµ ∼ 104 s−1. The muon will onlyfollow this path until it decays, and since the product of this frequency withthe muon life time is small ωBτµ ≈ 5 × 10−2 we can simplify the problem806.5. Geosynchrotron emissionconsiderably by taking the limit in which moving charges separate in a linearway within the magnetic field,)v(t) ≈ )v0 + )˙v0t ≈ )v0 +(qmγ))v0 × )B0, (6.14)where )v0 is the initial velocity of the charged particle emitted from thenugget. The velocity distribution will generally be rather complicated as themuons are produced in complex many body annihilations and subsequentlylose energy as they propagate through the quark matter and surroundingelectrosphere. Rather than attempting to estimate the initial energy spec-trum with which muons are produced and then tracking the energy loss asthey escape from the nugget I will simply consider all muons capable of es-caping the nugget to carry nuclear scale energies, so that they have a boostfactor of β ∼ 0.9. While this may be a serious simplification of a rather com-plex physical process it allows much of the shower evolution to be treatedin a relatively transparent way and elucidates some of its basic properties.The acceleration term in 6.14 leads to the emission of synchrotron radi-ation. The electric field produced by an accelerating particle is given by)E()r, t) = q4pi/0R()R · )u)3[(c2 − v2))u+ )R× ()u× )a)](6.15)with)u(t) ≡ cRˆ− )v(tR), (6.16)where )R points from the charged particle to the observation point and tR =t−R/c is the retarded time. While the emitted muons do not arise throughpair production the emission process is essentially charge independent, so wecan assume that µ+ and µ− production proceeds at the same rate. Applying6.15 to a net neutral µ+µ− pair and keeping only leading order terms wearrive at the electric field strength| )E()r, t)| = q2pi/0ωBcR(t)β0 sin θvBγ(1− β0)2, (6.17)where sin θvB is the angle between the initial muon velocity and the earth’smagnetic field and R(t) is the distance between the muon and the observa-tion point. Integrating this expression over the muons’ entire path we obtainthe frequency space representation|)E()r,ω)| = q(2pi)3/2/0ωB sin θvBc2γ(1− β0)2|I(R,ω)|, (6.18)816.5. Geosynchrotron emissionwhere, for notational convenience, I have defined the integral|I(R,ω)|2 =(∫ 1∆R/R0dxxcos(R0ωcβ0x))2+(∫ 1∆R/R0dxxsin(R0ωcβ0x))2(6.19)with R0 being the nugget to observer distance and ∆R the smallest separa-tion between the emitting muon and the observer. Note that in cases where∆R → 0 (i.e. cases where the muon pairs reach the observer) the integral6.19 diverges. In order to regulate this divergence the integral will be cut offat distance scales where ∆R becomes comparable to the separation distancebetween the µ+µ− pair. All of the geometry of the shower is carried by thesine function and the unitless integral. Writing 6.19 in a form that makesthe scale of the emission clear we have,|)E()r,ω)| ≈ (5× 10−10µV m−1 MHz−1)( B0.5G) sin θvB|I|γ(1− β)2 . (6.20)As should be expected, the contribution of each muon pair is relatively small.The total field strength is then obtained by scaling this expression up bythe total number of particles involved in the shower at a given time.The total number of relativistic particles was estimated in section 6.2 soit remains to estimate which fraction of these contribute to the synchrotronemission along a given line of sight. I will assume that the angular depen-dence of particle emission is scaled by the forward directed surface area asin equation D.7 and once again focus on the geometrically simple case of aquark nugget with a purely vertical trajectory.Following the analysis in [77] I will begin by noting that the radia-tion from a relativistic particle is sharply peaked along the direction ofparticle motion with the intensity showing an angular dependence of I ∼(1 − β cos θ)4. For the order of magnitude estimates here it will serve toconsider contributions to the radio flux across the angular distribution overwhich the intensity falls to half its maximum value. This scale may beestimated as(1− β1− β cos θ1/2)= 12 (6.21)θ21/2 ≈ 0.38(1− ββ). (6.22)826.5. Geosynchrotron emissionThe second equation here uses the small angle approximation for the cosine.The solid angle of the nugget’s surface from which emitted particles con-tribute to the radio flux will be estimated as dΩ ≈ θ21/2. This gives a totalrate of muons contributing to the geosynchrotron flux ofdNµdt= Γµθ21/22pihN√h2N + b2(6.23)with Γµ as given in 6.7. Finally I need to evaluate the timescale over whichthe particles are contributing to the emission. This, as with many showerproperties, is dependent on the energy loss rate of the particles as theymove through the atmosphere and a full treatment of the problem wouldinvolve large scale numerical simulations. However, for the demonstrativepurposes of this work I will assume that muon propagation may be treatedin the simple model presented in appendix D. Furthermore I will assumethat the muons are injected at sufficiently large energies that their totalpath length in the atmosphere is limited by their decay rate so that thetimescale for emission is roughly ∆t ∼ γτµ. Combining this with the emis-sion rate, expression 6.23 gives the total number of particles contributing tothe synchrotron flux,Nµ = Γµγτµθ21/22pihN√h2N + b2. (6.24)This factor allows us to scale the field strength for a single muon pair givenin equation 6.20 to give an estimate of the radio contribution from the entireshower. This will depend on the height of the nugget in the atmosphere andthe observer’s distance from the shower core (or, in a more general non-vertical shower it will depend on the observer’s orientation with respect tothe plane of the nugget’s motion. The electric field strength as a functionof frequency across the radio MHz band is shown in figure 6.7. Figure 6.8gives the field intensity as a function of distance from the shower core acrossmultiple radio frequency bins.It should be noted that the oscillations appearing in figure 6.7 are un-physical and arise from the assumption that all muons are emitted withidentical energies. Even a small spread in emission energy will smooth outthis effect which is related to the cutoff imposed on the integral 6.19.Finally, I will work out the total intensity produced by geosynchrotronemission. This is given by the Poynting vector:|)S| = dEdt dA= |)E|2µ0c(6.25)836.5. Geosynchrotron emissionFigure 6.7: Total electric field strength generated by geosynchrotron emis-sion in units of (µV m−1 MHz−1) as received 250m from the shower corewhen the nugget is at heights of h=500m (blue), h=1000m (green) andh=1500m (red). Figure taken from [77].| )S| = dEdω dA= |)E(ω)|2µ0c(6.26)where the second expression is formulated in frequency space. As arguedabove this energy will be deposited over the lifetime of the boosted muon sothat the total flux along a given line of sight is given by,dEdω dt dA= 1γτµ|NµE|2µ0c(6.27)=(10−16J m−2s−1MHz−1)( Γanfµ1016s−1)2 ( B0.5G)2×γ |I|2β2(1− β)2(h2Nb2 + h2N)sin2 θvB.This expression gives the total geosynchrotron intensity as a function ofposition on the earth’s surface. I want to add to this the contribution from846.5. Geosynchrotron emissionFigure 6.8: Total electric field strength generated by geosynchrotron emis-sion in units of (µV m−1 MHz−1) as a function of radial distance from theshower core. The individual curves are the field strength as measured atω = 5MHz (blue), ω = 20MHz (green) and ω = 20MHz (red). Figure takenfrom [77].thermal emission in the radio band. This has the relatively simple form,dEdω dt dA= 14pi(b2 + h2N )dEdω dt. (6.28)To give a feel for the scales involved the thermal and geosynchrotron flux asa function of frequency are plotted in figure 6.9. It is also possible to useexpressions 6.27 and 6.28 to extract the intensity received as a function oftime. This time dependence is shown in figure 6.10.As may be seen from figure 6.8 the radio signal produced by an anti-quark nugget crossing the atmosphere will extend over a few square kilo-meters. This makes the radio signal a potentially valuable search channel,particularly at radio facilities designed to look for the radio signal from highenergy cosmic rays. As in the case of fluorescence detection, the simulta-neous arrival of a millisecond duration radio pulse with a particle shower856.5. Geosynchrotron emissionFigure 6.9: Total radio band electromagnetic intensity (J s−1m−2MHz−1)received on the shower axis when the nugget is at heights of h=1km (blue),h=1.5km (green) and h=2km (red). The solid curves are the geosynchrotroncontribution while the dashed lines are the thermal contribution. Figuretaken from [77].detected by a surface array would be a strong signal of a quark nugget initi-ated shower as conventional mechanisms producing a large number of highenergy secondary particles evolve over much shorter timescales.Of particular interest in this context are experiments such as the AugerEngineering Radio Array (AERA) [65], LOPES [59] at the KASCADE-Grande array and CODALEMA [24]. Each of these radio detection ex-periments have sufficient spatial extents to observe square kilometer scaleevents and a sufficient sensitivity in the MHz range to detect radio signalsat the scale relevant for a quark nugget search.This work was originally intended to allow constraints to be made onthe flux of antiquark nuggets based on ground based radio detection pro-grams. It has since been pointed out in [50] that suborbital observations bythe Antarctic Impulsive Transient Antenna (ANITA) ballon borne experi-866.5. Geosynchrotron emissionFigure 6.10: Total radio band electromagnetic intensity (J s−1m−2MHz−1)as a function of time received on the shower axis from both thermal andgeosynchrotron emission. Here the nugget is taken to hit the surface at t=0after which the radio intensity will drop to zero. Intensity profiles are shownat 5MHz (blue), 20MHz (green) and 60MHz (red). Figure taken from [77].ment [51] will also be sensitive to this radio signal. The ANITA detector isintended to detect radio signals produced by ultrahigh energy cosmic raysscattering in the radio transparent antarctic ice. The detector views theice out to the horizon at a radius of approximately 600km, and has the di-rectional sensitivity to use the nugget velocity to discriminate an antiquarknugget from many of the backgrounds. The ANITA project involves threeflights ANITA-1 (2006-2007), ANITA-2 (2009-2010) and ANITA-3 which isplanned for future flight. ANITA-2 does not have the sampling rate to trackthe entire evolution of a nugget induced radio pulse, however, it recordsdata in multiple independent channels, allowing the nugget signal to bedetected through a coincidence signal in all channels. The ANITA-3 exper-iment will increase the integration time of the detectors and thus have aneven higher sensitivity. The ANITA-2 data is currently available and is be-876.5. Geosynchrotron emissioning analyzed to look for events characteristic of an antiquark produced radiosignal. The anticipated sensitivity range of ANITA is shown in figure 6.11(taken from[50]) and can been seen to cover several decades of the allowedmass range. Figure 6.11 also shows the limits which may be derived fromearlier searches conducted by the Gyrlanda array at lake Baikal [28], theanalysis of lunar seismology [55], and the IceCube detector in its 22 and 80string configurations [64].886.5. Geosynchrotron emissionFigure 6.11: Projected ANITA constraints on the flux of antiquark nuggets.If the quark nuggets do comprise all of the dark matter then they must sitsomewhere on the solid black line. The geothermal exclusion line is derivedbased on a limit to the amount of thermal energy that annihilating nuggetscan add to the earth’s temperature. All limits other than those from ANITA(shown here in blue) are based on completed analysis. The ANITA-2 datais currently under analysis while the ANITA-3 experiment has not yet beenconducted. Figure taken from [50].89Chapter 7ConclusionsThe fundamental nature of the dark matter is one of the most importantopen questions in physics today. We have had strong evidence of its exis-tence, through gravitational effects, for a long time but, currently, we haveno clear picture of its origins or physical properties. In this absence of directobservational evidence many dark matter models have been proposed. Mostof these models introduce a new fundamental particle (or particles) withtheir mass and interaction strength chosen to match the known propertiesof the dark matter.This work has attempted to take a different approach, asking whether thedark matter can be composed of known Standard Model particles in a novelconfiguration. While the possibility that heavy nuggets of quark matter formin the early universe remains conjectural, the physical properties of theseobjects are strongly constrained by the well tested theories of QCD andQED. Beginning with the established properties of any form of high densityquark matter from which the nuggets may be composed, I have attemptedto extract their basic, unavoidable observational consequences.If the galactic dark matter does consist of heavy quark nuggets, thenit will necessarily have an observable signature in the high density galacticcentre. In this environment the nuggets will emit thermal radiation, x-raysfrom hot spots near the site of proton annihilations and high energy photonsfrom the annihilation of galactic electrons. All of these forms of radiation,spread across vastly different scales, must necessarily be generated by thenuggets. Furthermore, once we fix the emission scale from a single spec-tral feature (for example the galactic 511keV line) the scale is fixed acrossthe rest of the emission spectrum. Following this procedure it has beenfound that emission from nuggets in the preferred mass range of this modelis entirely compatible with observations. These possible diffuse excess fea-tures include the WMAP haze, the Chandra x-ray background, the galactic511keV line and its associated three photon positronium decay continuumand the MeV band galactic excess observed by COMPTEL. Individuallynone of these features provides a smoking gun signature for quark nuggetdark matter but, if any of these apparent excesses had been dramatically90Chapter 7. Conclusionssmaller than measured, the model could have been strongly constrained.Future observations may well reduce the “excess” of emission in any one ofthese channels and pose a serious challenge to quark nugget dark matter.But, at present, it is non-trivial that all of these correlated emissions, acrossten orders of magnitude in energy, are simultaneously allowed, and that thedark matter model considered here may explain them all with only a singleparameter.The consequences of this model may be extrapolated from the galaxyup to a vastly larger scale. The matter density at the time the universebecame transparent is only slightly below that in the galactic centre today.As the nugget temperature is determined by the background matter density,thermal emission from the nuggets at the time of CMB formation must havebeen similar to that from the galactic centre today. The redshift of thesethermal photons implies that they will now fall primarily in the MHz radioband. This is an unavoidable consequence of the quark nugget dark mattermodel once the emission scale from the galactic centre is fixed. As such, it ishighly non-trivial that the isotropic radio background shows an increase insky temperature at low frequencies, with the temperature growing inverselywith the third power of frequency, as predicted by the thermal spectrum ofthe nuggets which was originally computed to compare to a very differentset of data.Given the intriguing, if inconclusive, evidence offered by these galacticand cosmological observations, it is worthwhile to ask whether any current orplanned experiments may be directly sensitive to the flux of quark nuggets.In this context, I have examined the observable consequences of the passageof a nugget through the earth’s atmosphere. In particular, I have focusedon the extensive air shower that will surround an antiquark nugget. Theseair showers have been demonstrated to extend over a multiple kilometerscale and, consequently, may prove observable by large scale cosmic raydetectors. I have given a qualitative description of the basic properties ofthese air showers and highlighted their distinguishing features. The primarydistinguishing feature in this context is the production of an air shower witha millisecond or longer duration. There are very few mechanisms, outsideof this model, able to deposit enough energy to trigger a kilometer scale airshower without carrying a velocity well above the typical galactic scale. It isprecisely this low velocity primary which allows for the production of a longduration air shower. With this main distinguishing feature in mind I havereviewed the prospect for nugget detection at cosmic ray detectors, whetherthrough atmospheric fluorescence, surface detection or radio measurements.I have also pointed out the constraints which may be provided by data from91Chapter 7. Conclusionsthe ANITA experiment which is currently under analysis.It has been the basic purpose of this work to highlight that the quarknugget dark matter model has essentially one free parameter, the averagemass of an individual nugget. While this quantity is, in principle, calculablein the theory of QCD, it is dependent on dynamics at the QCD phase tran-sition, at θ '= 0, far from equilibrium and at strong coupling. As such, it isunlikely that significant theoretical progress will be made on this front in thenear future. However, once the mass scale is set by a single observation itautomatically fixes the scale of a highly diverse range of other consequences.These range from galactic scale astrophysics up to Hubble scale contribu-tions to the isotropic background. Once this mass scale is established, italso determines the flux of nuggets through the earth. It is also possibleto directly constrain this flux. 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Sov.J.Nucl.Phys., 31:260, 1980.103Appendix ACosmology ReviewThis appendix is intended to provide a brief review of some basic concepts incosmology relevant, but not directly related, to the main focus of this work.In addition to providing general background this material is particularlyrelevant to the discussion of the nugget contribution to the cosmic radiobackground presented in chapter 5.A.1 The expanding universeWe live in an expanding universe that emerged from a hot big bang approx-imately 13.7 billion years ago. The expansion of the universe is governed bygeneral relativity and may, in the isotropic and flat limit, be described bythe Friedmann equations:H2 ≡(a˙a)2= 8piGρ + Λc23 (A.1)a¨a= −4piG3(ρ + 3pc2)+ Λc23 . (A.2)Here a is the scale factor, a measure of the scale of the universe, and the rateof change of this scale (the Hubble parameter H) describes the expansionof the universe. This expansion rate is related to the energy density (ρ)and pressure (p) of the contents of the universe and is also sourced by thecosmological constant Λ. In terms of their impact on the dynamics of theexpansion, the contents of the universe may be expressed by their equationof state, that is, the ratio between density and pressure (w = p/ρ.) Non-relativistic matter carries the majority of its energy as mass and exerts nopressure, thus wm = 0. Radiation or ultrarelativistic matter has wr = 1/3and the cosmological constant has wΛ = −1. From the Friedmann equationswe can determine the rate at which different types of energy densities dilute:ρ˙ = −3 a˙a(p+ ρ) . (A.3)104A.1. The expanding universeThis expression can be solved for the various types of energy density tofind their dependence on the scale factor. The energy of non-relativisticmatter is mass dominated, thus, the energy content remains constant whilethe volume it occupies expands and ρm ∼ a−3. The expansion of spacetimecauses a redshifting of radiation so that its energy density dilutes fasterthan that of matter, and ρr ∼ a−4. Finally, the cosmological constant is aform of vacuum energy, it scales in proportion to the size of the spacetimeit occupies. That is to say, it does not dilute with the universe’s expansion.A useful parameterization of the contents of the universe is in terms ofthe fraction of the critical density that a given form of energy represents.The critical density is defined as the energy density for which, in the absenceof a cosmological constant, the universe just avoids collapsing back on itself(analogous to the escape speed at which an object is no longer gravitationallybound to the earth.) This density may be found by inverting expression A.1when Λ = 0ρc =3H28piG. (A.4)From this we may define the density parameter of a substance, of energydensity ρ, as Ω = ρ/ρc. This provides a simple way of characterizing thecontents of the universe and, if we assume that the total energy density isequal to the critical density, allows us to write the first Friedmann equationas, (a˙a)2= H20(Ωra−4 + ΩDMa−3 + Ωvisa−3 + ΩΛ)(A.5)where H0 is the Hubble constant as measured today and the present valueof a is normalized to one. In this expression, I have separated the visi-ble matter (Ωvis) and the dark matter components, despite the fact thatthey display identical dynamics in terms of the expansion. In is only whenwe consider their microscopic properties that these two components displayunique behaviours. From expression A.5, it is clear that at different timesin the history of the universe the form of energy density which drives thedynamics of the expansion will be different. In the early universe thereare many ultrarelativistic species, and their energy density is large, so thatthe universe is radiation dominated. As the universe expands the energydensity of radiation drops relatively quickly and eventually the dynamicsbecome matter driven. Finally, at relatively late times in the cosmic his-tory, the other forms of energy have diluted away and the dynamics becomedominated by the cosmological constant.At present the universe is dominated by dark energy which, thus far,seems consistent with a cosmological constant. The dark matter and visible105A.2. The cosmic microwave backgroundmatter make secondary contributions with ΩΛ = 0.7181,ΩDM = 0.236 andΩB = 0.0461 [56].When we observe distant objects, we see the light they emit as havingbeen redshifted by the expansion according to νemitaemit = νobsaobs. Herethe subscripts emit and obs denote the values at the time of emission and ob-servation respectively. Using this relation, and normalizing the present dayscale factor to one, the scale factor at any given distance may be expressedin terms of the redshift of objects observed at that distance a = 1/(1 + z).A.2 The cosmic microwave backgroundThe cosmic microwave background (CMB) is one of the most importantsources of our knowledge of the early universe, and it has also providedsignificant insight into the development of large scale structure. This reviewwill give only a very brief discussion of the details of CMB formation andevolution as required for an understanding of the main discussion of thiswork.The expansion of the universe has diluted and cooled the contents which,at earlier times, were much hotter and more dense. In the early universe thetemperature was high enough that the matter existed as a highly ionizedplasma which was fully coupled to the radiation. At this time the meanfree path of a photon was short, and the number density of photons wasset by the temperature of the plasma. As the plasma cooled its ionizationfraction fell and the photons’ mean free path increased until they couldtravel unimpeded across the universe. This transition represents the furthestdistance in the universe from which electromagnetic radiation can reach us.In the time since it was emitted, this radiation has diluted and cooled to thepoint where it now represents a thermal bath of photons with a temperatureof TCMB = 2.7K. These photons carry too little energy to influence thedynamics of the expansion, but there are enough of them that they stilldominate the thermodynamics of the universe. As a thermal collection ofphotons, the spectrum of the CMB is given by,dEdt dA dν= 8pihν3c21ehν/T − 1 . (A.6)Once the CMB has decoupled from the matter its temperature falls with thescale factor so that this expression holds at all later times with a radiationtemperature given by T = T0(1 + z), where T0 = 2.7K is the present dayphoton temperature. The form of this spectrum will be important in the106A.3. Radiation from distant objectsanalysis of the isotropic radio background in chapter 5. The scattering ofultrahigh energy cosmic rays off this photon background determines theenergy of the GZK cutoff as discussed in the introduction to chapter 6.A.3 Radiation from distant objectsA photon emitted at a scale factor a will experience a redshift of z = a0/a−1,where a0 is the present day scale factor, generally normalized to one. Inpractice, it is easier to speak about a distant source as being at a particularredshift which, unlike the scale factor, is a directly observable quantity. Inthis case, the frequency of a photon observed today will be redshifted fromits original value down toν = νemit1 + z (A.7)with the energy carried by the photon falling accordingly.When observing a distant object the energy flux received is reduced bythe redshift of the photons. We can write the intensity asI(ν) ≡ dEdt dν dA= (1 + z)4pid2LdEdt dν(ν[1 + z]) (A.8)where dL is the luminosity distance of the object. A more complex case isthat of emission from a distribution of sources extended over a range of dif-ferent redshifts. This flux calculation is needed in determining cosmologicalbackgrounds such as the isotropic cosmological background associated withdark matter (analyzed, for example in [125].) This may be formulated as anintegral of the comoving source density over all redshifts back to the surfaceof last scattering:I(ν) ≡ dEdt dν dA=∫c dz(1 + z)H(z)ns(z)L(ν(1 + z), z) (A.9)where ns is the number density of sources and L(ν(1+ z), z) is the luminos-ity per frequency interval, evaluated at the frequency from which it will beredshifted down to ν in the present day universe. In evaluating this expres-sion it is useful to invert expression A.5 and note that, over the cosmologicalhistory we are interested in, the radiation term is negligible so that,H(z) = H0√ΩM (1 + z)3 + ΩΛ. (A.10)Thus, if we know how sources evolve with redshift, we can integrate theircombined intensity across the observable universe and determine their con-tribution to the cosmological background. This procedure is followed in107A.3. Radiation from distant objectssection 5.2 to determine the isotropic radio background arising from a pop-ulation of antiquark nuggets in the early universe.108Appendix BQCD ReviewThis appendix will offer a brief review of some of the properties of quantumchromodynamics (QCD) as necessary background to the main body of thiswork. After a brief introduction, it will focus primarily on the strong CPproblem, and its potential resolution through the introduction of the axion.Finally, these concepts will be applied, in a qualitative way, to the problemof nugget formation in the early universe.B.1 IntroductionQCD is the theory governing the strong force interactions of hadronic mat-ter. It is an SU(3) gauge symmetry in which the colour charge carryingquarks interact through the exchange of eight gauge bosons known as glu-ons. As the gluons also carry colour charge they couple to each other directlyas well as to the quarks. In addition to the more complicated set of gaugeinteractions, QCD is differentiated from the U(1) gauge theory of quantumelectrodynamics (QED) by the strength of the quark-gluon coupling. Thestrength of coupling between a photon and an electrically charged particleis indicated by the scale of the fine structure constant, α = 1/137. Thesmall value of this parameter ensures that processes involving additionalinteractions occur with significantly lower probability. This allows QED tobe treated perturbatively, so that transition probabilities may be expandedin powers of α, corresponding to an expansion in the number of charge-photon interaction vertices. In QCD the strong force coupling constant, αs,is of order one and, consequently, the theory is not suited to the type ofperturbative treatment which has proved so useful in QED.There is, however, a regime in which the above considerations do not ap-ply. The values of the coupling constants, as given above, are measured inthe zero momentum exchange limit, but these values run with increasing mo-mentum. In the case of QED the coupling strength grows with momentum,as a result of vacuum polarization. In QCD the vacuum polarization, in-volving the non-abelian gauge fields, produces an antiscreening effect, ratherthan screening. As a result, the coupling strength in QCD falls with growing109B.2. The QCD vacuummomentum exchange, an effect know as asymptotic freedom [53, 99] . Thiseffect has been critical in the understanding of deep inelastic scattering ex-periments and, more germane to this work, has allowed for detailed studiesof the QCD phase diagram at asymptotically large densities where the fermisurface quarks are in the asymptotically free regime, as discussed in section2.1.B.2 The QCD vacuumThe ground state of QCD is fully non-perturbative and is found to exhibita complex phenomenology. This discussion will, however, limit itself to theintroduction of some basic concepts, and the properties necessary to moti-vate the formulation of the strong CP problem. The basic QCD Lagrangianis,LQCD = ψ¯j [iγµDµ −m]ψj −14GµνGµν , (B.1)which is the standard formulation for a gauge field (here the gluons, withfield strength tensor Gµν) coupled to charged fermions (the quarks ψj , withthe index j running over all the quark flavours.) This formulation of theLagrangian uses a summation convention whereby repeated indices are tobe summed over. In the limit where the quarks are massless the Lagrangianhas a conformal symmetry and a symmetry under chiral rotations, bothof these are spontaneously broken by the QCD vacuum. The breaking ofconformal symmetry introduces the fundamental scale of the theory ΛQCD,and establishes the mass scale of the baryons. Chiral symmetry is sponta-neously broken by the non-vanishing vacuum expectation values of the quarkcondensates. The nine broken generators of the chiral-flavour symmetry cor-respond to the pseudoscalar meson nonet. The nonet may be broken downinto an octet (the pions, kaons and the η) and a singlet (the η′.) An exampleof the impact of the vacuum may be seen in the mass splitting of the η andthe η′ meson, with the η′ having a mass comparable to the proton, while theη has a mass roughly half this value despite having the same quark content.This mass splitting is traceable to the fact that the η′ is associated with theaxial U(1)A symmetry which is explicitly broken by the chiral anomaly. Assuch, the η′ mass is not protected against quantum corrections and acquiresa mass at the QCD scale. Conversely, the mesons of the octet are genuinepseudo-Goldstone bosons which would be exactly massless in the limit ofvanishing quark masses. The properties of the η′ will be further discussedin section B.4.110B.3. The strong CP problem and axionsWhile the U(1)A symmetry is explicitly broken by the anomaly, theremaining chiral-flavour symmetry (associated with the meson octet) is bro-ken spontaneously by the formation of chiral condensates < q¯RqL > in theQCD ground state. Because they couple quark fields of opposite chiral-ity these vacuum states break the chiral symmetry of the Lagrangian B.1.The energy density scale for a quark condensate is approximately given by,mq < q¯RqL >∼ mqΛ3QCD.In the high temperature phase of QCD the condensates break down andchiral symmetry is restored. It is also believed that the U(1)A symmetry isat least partially restored in the high temperature phase as the instantoneffects responsible for breaking it are screened at large temperatures [108].B.3 The strong CP problem and axionsCombined charge conjugation and parity symmetry (CP) is known to be vi-olated in many physical processes. For example CP violating phases appearin the CKM matrix [71] and PMNS matrix [88] describing the electroweakmixing of quarks and neutrinos, respectively. The resulting CP violationmay then be observed in the decay rates of heavy mesons40. However, nosuch violation has been observed within the strong force interactions, wherethe gluons couple to vector currents rather than to chiral currents as do thegauge bosons of the weak interactions. The best constraints on the degree ofallowed strong CP violation come from measurements of the electric dipolemoment of the neutron, which will be exactly zero only if QCD respects CPsymmetry. As CP symmetry is known to be broken in both the quark andneutrino sectors, we may well ask why it seems to be respected by QCD.That QCD apparently preserves CP symmetry is particularly interestinggiven that there exists a mechanism within QCD which would allow for CPviolation. In addition to the standard terms in the QCD Lagrangian, givenin equation B.1, we are free to introduce an additional term,∆L ∼ iθGµνG˜µν (B.2)G˜µν ≡ /µνστGστ . (B.3)At this point θ may be thought of as a free parameter of the Standard Modelwhose value is to be determined experimentally, and which sets the degree of40For example the kaons, mentioned in the previous section, have been observed todecay into pion states with opposite CP quantum numbers. This is possible as kaons,while strongly interacting, decay through the weak interactions where CP violation isknown to occur.111B.4. Domain walls in QCDCP violation present in the theory41. At present observational constraintsdemand |θ| < 10−10. As there is no explicit reason for this value to besmall, this raises a possible fine-tuning problem. This problem is made evenmore important by the fact that, even if the value of θ were to be set tozero by hand, quantum fluctuations would generate non-zero corrections tothis term, just as they do to the η′ mass. The fact that QCD remains aCP preserving theory, despite this mechanism through which CP violationshould be expected to occur, is known as the strong CP problem.The strong CP problem has not yet been resolved, but one of the pre-ferred solutions is that of the Peccei-Quinn mechanism [97]. In this modelan additional symmetry is introduced which is then broken by the QCD vac-uum. The pseudo-Goldstone boson resulting from this symmetry is knownas the axion42, and it enters the Lagrangian in the same way as the θ param-eter. In this case θ may be absorbed by a redefinition of the axion field. Theaxion field then relaxes down to θ → 0, resulting in the CP preserving theorywe observe today. The axion has not been observed, but there are severalexperiments currently searching for it [25, 116] and, more than three decadesafter it was first proposed, it remains the only viable proposed solution tothe strong CP problem. The properties of the axion, as relevant to this work,are its mass, which is believed to fall in the 10−3eV> ma > 10−6eV range,and the fact that it is a singlet of the colour gauge group. These propertiesbecome important in the mechanism for compressing quarks of the earlyuniverse plasma into the high density phase which forms the quark nuggetsin the dark matter model presented here. This will be briefly outlined inthe following section.B.4 Domain walls in QCDFinally, I will present a brief overview of the domain walls involved in theformation of the quark nuggets. Domain walls are topological defects, knowto exist in a wide range of field theories. These objects are generally the clas-sical solutions of a field theory, which extrapolate between distinct ground41This term is, in fact, a total derivative term representing a choice of vacuum state. Assuch, θ is a periodic variable such that the values θ and θ+2pin describe identical physics.Where n is any integer.42The axion model originally formulated by Wilczek and Weinberg [120, 121] producedan axion that was relatively heavy and strongly coupled to the particles of the StandardModel and has been ruled out by experiment. However, two subsequent models, the KSVZaxion [69, 107] and the DFSZ axion [34, 129], remain possible solutions to the strong CPproblem.112B.4. Domain walls in QCDstates of the theory. In each domain the light degrees of freedom are excita-tions around the particular ground state configuration of the fields. Domainwalls are the two dimensional surfaces at which two such distinct domainsmeet, and have energies at the scale of the barrier separating the differentvacuum states. Associated with these objects is a conserved topologicalcharge which makes them classically stable.The domain walls considered here are a slight variation on this basicidea. As the θ parameter is 2pi periodic, the states with θ and θ + 2pirepresent exactly the same physical ground state. Across the domain wallsof this model the axion field (which has absorbed the bare value of θ into adynamical field variable) varies by 2pi, arriving back at a physically identicalvacuum on the other side of the wall. These objects, involving a singlewinding through 2pi are known as N = 1 domain walls. The physical lengthscale over which this transition occurs is set by the Compton wavelength ofthe axion, λ ∼ m−1a , and is much larger than the typical QCD scale. Despiteextrapolating between identical vacua, these field configurations still carryan associated topological charge which protects them from decay at theclassical level [41].The distinct vacuum states of QCD may be parameterized in the phasesof the various condensates which they form. One may also consider transi-tions in the phase of the chiral condensates of the QCD ground state. Aswith the axion wall these involve the transition between two identical vacua,and possess a distinct topological charge. The transitions in these phasesmay be classified as belonging to rotations of the singlet or triplet config-urations of the chiral-flavour symmetry group. The triplet field will havea relatively large length scale (somewhat larger than m−1pi .) However, thesinglet field (associated with the η′) receives a correction from the gluoncondensate, so that it exists at the QCD scale and has a length ∼ Λ−1QCD.This range in length scales, from a few Fermi up to the macroscopic, willprove important in the considerations that follow.It is also found that these transitions in the axion and condensate fieldsmay overlap with each other forming a “sandwich” structure [41, 47]. Theη′ wall carries an energy density similar to that of the axion wall, but itsnarrower width implies that it is extended over a much smaller space, andcontributes relatively little to the total energy associated with the wall.Partial mixing of the axion field with the η′, which carries the same quantumnumbers, also implies that a transition in one field may be accompanied bya transition in the other and, as such, some fraction of any axion walls whichform will contain a hard η′ core. The relatively small energetic cost impliesthat this fraction could be quite large.113B.5. Stability and lifetimesWith these preliminaries it is possible to outline the basic picture ofnugget formation. As the early universe cools from the initial quark-gluonplasma to the baryonic phase, the axion fields condense out, on scales oforder m−1a , and relax the CP violating θ term down to zero. These fieldconfigurations will include domain walls of the type described above. Ifthe axion wall does not include an η′ core43 it is essentially transparent tofermions and the wall collapses down to R=0. If, however, the wall is ofthe sandwich type, with both a large axion scale component and a QCDscale core, it will be capable of trapping fermions within the collapsing wall.As the reflection process occurs at the centre of the wall where θ '= 0 thisprocess will be CP violating, and the wall will preferentially trap eitherbaryons or antibaryons. This process will continue until the Fermi pressureexerted by the trapped matter becomes comparable to the surface tensionof the domain wall. As the trapped matter subsequently cools, the quarkswill settle into a high density, superconducting phase forming nuggets withthe properties discussed in the main body of this work.B.5 Stability and lifetimesThe purpose of this section is to offer some brief arguments about the lifetimeand stability of the quark nuggets and the axion domain walls which formand stabilize them. This discussion will necessarily be qualitative in natureas many exact calculations in high density QCD at θ '= 0 are not presentlytractable. An analysis of the stability of the nuggets over a range of physicalparameters was originally performed in [126] and the basic arguments willbe presented here. An analysis of the stability of axion domain walls similarto those considered here was presented in [41] and in a slightly differentgeometry in [32], some qualitative results of this analysis will be sketchedbelow.The primary mode for the nuggets to decay into normal baryonic matteris through the emission of a nucleon or light nuclei. This process will beenergetically favorable so long as the mass of the emitted nucleon is lessthan the decrease in the mass of the nugget, mN < MB − MB−1. Thestability of quark matter at zero external pressure is found to be highlymodel dependent [35, 122]. If the form of quark matter realized in thenuggets is absolutely stable with respect to nuclear matter then no external43We could also consider η′ walls without a surrounding axion wall, however thesestructures will form only over typical QCD length scales and be too small to be relevanthere.114B.5. Stability and lifetimespressure from the axion wall is required and the wall’s lifetime need onlybe long enough to provide the initial compression necessary to form thenuggets. This requires that the walls survive from the time of the QCDphase transition until the nuggets settle into the stable high density phase.A wider range of high density phases may become either stable or metastablein the presence of a confining domain wall. If this is the case the nuggetswill remain stable over the lifetime of the wall and then begin to decay intofree nucleons. For quark nuggets to serve as the dark matter this scenariowould require domain wall stability over cosmological timescales.In general the energy associated with a nugget of radius R and totalbaryonic charge B may be approximated as,E = 4piR2σ + 43piR3[gµ48pi2 + /B]. (B.4)Here σ is the surface tension of the axion domain wall, µ is the quark chem-ical potential in the bulk of the nugget and /B is the binding energy ofthe quarks. As the chemical potential fixes the baryon density it is notindependent from the baryon number and radius:B = 43piR3nB(µ) =∫ µ0gd)p(2pi)3 =2gµ39pi R3 (B.5)where g is the number of particle types present in the Fermi gas so that g ≈(2 spins)×(3 light flavours)×(3 colours) ≈ 18. The stability of these objectsmay then be determined by minimizing the energy per baryon, E/B for agiven value of σ and /B. If the energy at the minimization point is below theproton binding energy then nuggets in such a configuration will be stable. Itis not necessary here to explore the full range of nugget parameters, allowedby expression B.4, that will result in long term stability. I will simply notethat, as stable quark matter solutions have been found in the absence ofa domain wall, adding this new component, which further compresses thequark matter, will necessarily increase the range of allowed phases. It shouldalso be noted that at large radial sizes the pressure of the domain wall perquark in the nugget decreases. Thus, in the large R limit, the first term inequation B.4 becomes negligible and the situation becomes identical to thatof quark matter not supported by the external pressure of the domain wall.The domain walls themselves are topological defects which represent asolution to the classical field equations. This class of topological config-urations generally arises when the theory under consideration contains atopological charge (for example a winding number) for which a conserved115B.5. Stability and lifetimescurrent may be derived. The conservation of this topological charge preventsthe decay of the defect even in the case where such a decay would lower thetotal energy of the system.The situation is more complicated when the walls are treated quantummechanically. In that case the walls may decay through tunneling eventseven at temperatures well below the wall energy. A domain wall containingno hadronic matter will collapse down to an arbitrarily small size, and candecay relatively easily. If, however, the wall is supported against collapseby the Fermi pressure of matter trapped within it, the decay must occurthrough the nucleation of a hole in the wall. Holes of a sufficient size willexpand rapidly and the wall will decay. The exact dynamics of this processare more complicated than will be considered here, but some basic scalearguments may be made. As with any tunneling process the rate of holeformation will be suppressed by a factor ∼ e−Sc/h¯, where Sc is the classicalaction for the formation of the hole. In order for the hole to begin expandingit should be large with respect to the thickness of the wall t ∼ m−1a . As thisrepresents a macroscopically large area (comparable to the size of the quarknuggets themselves) and the energy density across the wall is large, theaction for the formation of a hole will be large Sc >> h¯. This leads toa strong exponential suppression of the wall decays and may allow theseobjects to be stable over cosmological timescales.116Appendix CNugget ThermodynamicsIt is the purpose of this appendix to establish some of the basic thermalproperties of the nuggets as required to determine their observational prop-erties. Within the quark matter itself, the photon is screened at the QCDscale, so low energy thermal photons within the quark matter have no chanceof escaping. Screening is also very efficient deep in the electrosphere wherethe positrons are at high densities. The first layer of the electrosphere fromwhich low energy photons are able to escape occurs when the mean kineticenergy of the positrons becomes comparable to the temperature. This isprecisely the Boltzmann regime discussed in section 2.2, and it will be theproperties of this region which determine the thermal emission spectrum ofthe nuggets. In [43] the power emitted per unit volume at frequency ω of aBoltzmann gas of positrons of mass me was estimated as,dEdt dω dV≈8α15(αme)2n2e(z, T )√2Tmpi(1 + ωT)e−ω/TF(ωT)(C.1)where the function F (x) is defined asF (x) = 17 − 12ln(x2)x < 117 + 12ln(2) x > 1. (C.2)Here we may take the number density ne to be given by 2.6 for the Boltzmannregime of the electrosphere. Note that the function F (ω/T ) diverges in theω → 0 limit. This divergence is unphysical and is a result of not imposing along wavelength cutoff on the resulting photons. This cutoff will be imposedby the limited size of the Boltzmann regime of the electrosphere from whichthis radiation is emitted. As argued in section 2.2, the Boltzmann regimewill persist from very near the nugget surface out to scales at which the planeparallel approximation fails. As this occurs at heights in the electrospherefor which the nugget’s spherical geometry becomes important we shouldexpect the emitting region to have a length on the order of the nugget’sradius. The arguments leading to expression C.1 assumed a set of plane117Appendix C. Nugget Thermodynamicswave positron states of arbitrary long wavelength. If, however, we limit theallowed positron states44 to a region with a scale of the size of the nugget wewould expect a low momentum cutoff near pmin ∼ R−1N . For photons emittedthrough elastic positron scattering this corresponds to a low frequency cutoffon the order of,ωmin ≈p2min2me≈h¯2R2Nme≈ 6× 109s−1 (C.3)while lower frequency radiation will be strongly suppressed. Thus, the emis-sivity given in C.1 and the emission spectrum to be derived from it shouldbe considered valid only for frequencies above this limit. The exact way inwhich the spectrum falls from the value determined in C.1 near the cutoffwill depend on the microscopic details of the electrosphere. For present pur-poses it will be sufficient to impose a hard cutoff at ∼ 6GHz, as impliedby expression C.3, and limit ourselves to details of the spectrum above thisfrequency.The expression C.1 may be integrated over the height of the Boltzmannregime to obtain a total surface emissivity as a function of temperature forthe nuggets. In the plane parallel approximation roughly half of the emittedphotons move upward and escape, the other half move downward and arereabsorbed. On performing this integration, one arrives at a final expressionfor the surface emissivity of a quark nugget of a given temperature:dEdt dA dω= 445T 3α5/2pi4√Tme(1 + ωT)e−ω/T F(ωT). (C.4)This will be the primary mechanism by which thermal energy is releasedfrom the nugget and, in addition to determining the spectral properties ofthe thermal emission, it is important for estimating the temperature evolu-tion of the nuggets. As a general rule, the nuggets will be heated by nuclearannihilations and then radiate off this energy through low energy thermalradiation. The nuggets’ temperature may then be estimated as the temper-ature for which the total thermal emission balances the energy input. Tomake this comparison we must integrate the surface emissivity across all44Such a constraint is not placed on the photon states as they do not have to remainbound to the system. As such, the long wavelength radiation is not in equilibrium with theelectrosphere and emission at these energies may exceed that expected from a blackbody.118Appendix C. Nugget Thermodynamicsfrequencies 45 to obtain the total emitted energy:dEdt dA=∫dEdt dA dωdω ≈163T 4α5/2pi4√Tme. (C.5)This expression gives the total power output per surface area so that, if weassume the nugget to have a uniform surface temperature, the total poweremitted through thermal radiation is simplyP (T ) = 4piR2N(dEdt dA)(C.6)where RN is the radius of the emitting quark nugget. In cases where thenugget is heated by the annihilation of matter falling onto the antiquarksurface this total emitted power will be balanced by the flux of matter ontothe nugget. This balance implies a temperature relation,(T1eV)17/4≈ 10−9fTΦvis1GeVcm−2s−1 ≈ 30fTρvisβ1GeVcm−3 (C.7)where Φvis is the visible matter energy flux onto the nugget, fT is the fractionof this energy which is thermalized, ρvis is the visible matter density andβ = v/c is the boost factor of the matter. Note that for typical galacticvalues ρvis ∼ 100 GeV cm−3 and β ∼ 10−3 so that this expression is oforder one.45As only a small fraction of the total energy is radiated at low frequencies this integralis not highly sensitive to the value of the low frequency cutoff given in expression C.3119Appendix DMuon PropagationMany of the observable properties of the air shower associated with a quarknugget passing through the atmosphere are dependent on its charged particlecontent. As argued in section 6.2, the charged particle content of the showerwill primarily consist of muons. This appendix will discuss the treatmentof muon propagation through the atmosphere to be used in estimating theobservable properties of a quark nugget induced air shower.In the study of cosmic ray induced air showers, the development of theshower as the secondary particles are produced and lose energy to the sur-rounding atmosphere is generally handled with large scale Monte Carlo sim-ulations [20]. This level of detail is necessary in order to accurately calibratethe energy and composition of the primary cosmic ray, but goes well beyondthe level of sophistication required for the simplified analysis presented here.Instead, I will use a highly simplified model of muon propagation which willallow a rough estimate of such important shower properties as the fluo-rescence profile, the surface distribution and the production of radio bandgeosynchrotron emission.Here I will focus on the importance of a number of basic scales involvedin muon propagation through the atmosphere. The lifetime of a muon atrest is τµ = 2.2×10−6s. For a relativistic muon this timescale is made longerby time dilation. The distance a muon can travel before it decays will begiven by,ld = γvτµ =pmµcτµ (D.1)where v is the muon’s velocity and p is its momentum. This is the largestpossible scale involved in muon propagation, and neglects energy losses tothe atmosphere. Muons at the highest energies have very small interactioncross-sections with the surrounding matter and will generally travel aboutthis distance. Muons decay to an electron and a neutrino pair. The neutrinosdisappear from the shower while the electron is stopped much more rapidlythan a muon and quickly becomes non-relativistic.Muons at lower energies will loose a significant fraction of their energy tothe atmosphere and may be stopped before they decay. The rate of energy120Appendix D. Muon Propagationloss will be dependent on the density of the background through whichmuons propagate. As the atmospheric molecules are neutral on scales largerthan a few Bohr radii, scattering is only possible through the exchange ofa photon with an energy of at least ∆E ∼ meα. The cross section for thistype of low energy scattering is approximately given by,σµe ≈2piαm2e1v2≈ 7× 10−23cm2( cv). (D.2)The scattering length for a given muon is then given by finding the pathlength over which it is likely to encounter a single atmospheric molecule.This may be found by solving the expressionσµe∫ l0ds nat(s) = 1 (D.3)where l is the scattering length, s parameterizes the path along which themuon travels and nat is the density of particles off of which the muon mayscatter. The atmospheric depth is defined asX =∫ds ρ(s) (D.4)and gives a measure of the amount of atmospheric material through which aparticle moves 46. Using this definition it is easier to express the scatteringlength in terms of the depth interval through which a muon can move:∆Xs =mpσµe. (D.5)Here mp is the mass per scattering site which is taken to be the mass of oneproton.The scattering amplitude is strongly peaked at small momentum transferso that most scattering events will involve the muon losing roughly meα inenergy. In this case the total number of scatterings required to stop a muonwith total energy E is (E −mµ)/meα. Thus the total depth interval acrosswhich a muon can travel is,∆Xtot =E −mµmeαmpσµe≈ 6 kg cm−2( E1GeV)(D.6)46This is a useful measure in studying a cosmic ray shower because the shower devel-ops with depth rather than with the surrounding density as in the case of an antiquarknugget. This means, for example, that steeply inclined showers will leave a much longerfluorescence track than vertical ones. The atmospheric depth at which fluorescence peaksis referred to as Xmax and is strongly correlated with the total energy initially carried bythe primary cosmic ray.121Appendix D. Muon Propagationwhere the final expression, intended only to give the scale involved, assumesthat the muon energy is much larger than its rest mass. Note that the totalatmospheric depth is on the order of 1kg cm−2. So high energy particleswill have no problem reaching the surface from most relevant heights. It isonly relatively low momentum muons for which this energy loss scale willbe important.The basic muon propagation model I will use is simply to have the muonpath length determined by either its decay length in equation D.1 or themaximum depth interval it can cross as given by expression D.6, dependingon which is shorter. Until it reaches this distance the muon will be assumedto remain relativistic so that its velocity may be taken to be roughly thespeed of light.The muons originate at the nugget’s location and propagate outward ata speed much larger than that of the nugget. As muons must be producednear the surface it will be assumed that the rate of muon emission is directlyproportional to the matter flux onto the surface at any given time. This leadsto preferential emission from the side of the nugget facing along the directionof motion. It will also be assumed that muon emission happens essentiallyperpendicular to the nugget surface as this implies the smallest distancewithin the quark matter that the muon must cross. Under these conditionsthe muon emission geometry may be expressed as the rate at which muonsare emitted into a given solid angle:dNµdΩ dt =Γµ2pi cosφ. (D.7)Here Γµ is the total muon production rate as given in 6.7 and φ is the anglebetween the direction that the muon is emitted and the direction of thenugget’s motion. All emission will be taken to originate from the forwarddirected surface of the nugget so that −pi/2 < φ < pi/2. In this geometry themuons are directed into a cone along the forward direction of the nuggets’motion.122


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