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Observing dark in the galactic spectrum? Lawson, Kyle 2008-11-07

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Observing Dark Matter in theGalactic Spectrum?byKyle LawsonB.Sc., The University of Western Ontario, 2005A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October, 2008c Kyle Lawson 2008AbstractObservations from a broad range of astrophysical scales have forced us tothe realization that the well understood matter comprising the stars andgalaxies we see around us accounts for only a small fraction of the totalmass of the Universe. An amount roughly  ve times larger exist in theform of dark matter about which we have virtually no direct evidence apartfrom its large scale gravitational e ects. It is also known that the largestcontribution to the energy density of the universe is the dark energy, anegative pressure form of energy which will not be dealt with here.I will present a candidate for the dark matter which is based completelyin known physics and which presents several possible observational signa-tures. In this model the dark matter is composed of dense nuggets of bary-onic matter and antimatter in a colour superconducting state. If these objectare su ciently massive their low number density will make them e ectivelydark in the sense that collisions with visible matter become infrequent. Thiswork presents the basics of dark matter as a colour superconductor and thenuses the physical properties of the quark nuggets to extract observationalconsequences.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Bayogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Nugget structure . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Colour superconductivity . . . . . . . . . . . . . . . . . . . . 42.2 The electrosphere . . . . . . . . . . . . . . . . . . . . . . . . 53 Emission mechanisms . . . . . . . . . . . . . . . . . . . . . . . 73.1 Electron annihilation . . . . . . . . . . . . . . . . . . . . . . 73.2 Comparison with observation . . . . . . . . . . . . . . . . . . 93.2.1 The galactic 511keV line . . . . . . . . . . . . . . . . 93.2.2 Di use MeV emissions from the galactic core . . . . . 93.2.3 Other emission mechanisms . . . . . . . . . . . . . . . 104 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16AppendicesA Lepton Distributions . . . . . . . . . . . . . . . . . . . . . . . . 18B Positronium Formation Rates . . . . . . . . . . . . . . . . . . 21iiiTable of ContentsC The direct annihilation spectrum . . . . . . . . . . . . . . . . 23ivList of Figures3.1 Best  t to the high energy di use spectrum of the galacticcentre from Moskalenko et al. based on cosmic ray e ectswith point sources subtracted. Note the observed excess overthe calculated spectrum in the COMPTEL range. . . . . . . . 133.2 Total intensity ( attened by k2 as in 3.2.2) averaged over allelectrosphere heights. The total emission spectrum is shownin solid blue while the cosmic ray background and dark matterspectrum are shown in black dotted and dashed lines respec-tively. The red bars are the COMPTEL data points. . . . . . 14A.1 Chemical potential as a function of height. Here the surfacechemical potential is taken to be  0 = 30MeV. . . . . . . . . 19A.2 Lepton density as a function of height. Here the surface chem-ical potential is taken to be  0 = 30MeV. . . . . . . . . . . . 20C.1 Spectral density as a function of photon energy (with arbi-trary normalization.) Plotted for  = 10MeV (blue),  =30MeV (green) and  = 60MeV (red.) The spectra havebeen rescaled by 1 for to allow multiple curves on the sameaxes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24vChapter 1Introduction1.1 Dark matterA growing list of observations have demonstrated that the majority of themass in the galaxy is in the form of dark matter (DM.) At the same timewe are without an established physical theory to explain it’s fundamen-tal nature. Consistency with observation requires that DM must interactvery minimally with the ordinary matter comprising the visible universe.However, it’s gravitational in uence is critical in shaping the cosmologicalstructures seen today. Any candidate must produce no observational sig-nature and be stable on cosmological time scales. While no de nitive DMcandidate is known the majority of models work within the context of theweakly interacting massive particle (WIMP) paradigm. WIMP models ofdark matter generally introduce a new fundamental  eld which couples onlyweakly to visible matter. As there are no known particles with suitableproperties any such model must invoke new physics. If the mass and cou-pling strength of the  eld comprising the dark matter are known then thepresent day abundance (which is the only de nitively observed parameter)may be calculated.An interesting aspect of dark matter is that its energy density in thepresent universe is of comparable scale to that of the visible matter compo-nent with,  DM  5 vis. If the energy scales at which dark matter formsare dramatically di erent from the QCD scale which determines the mattercomponent of the energy density then  ne tuning of the parameters of thedark sector becomes necessary to explain the similar energy densities.1.2 BayogenesisA second puzzle of modern cosmology which will be addressed here is thenature of baryogenesis. Observations show that the Universe is composedalmost entirely of matter as opposed to antimatter yet physical laws seemto treat the two identically making this dichotomy di cult to explain. The11.3. Motivationnecessary conditions to explain the prevalence of matter are know as theSakharov conditions, any plausible explanation must contain mechanismsto produce Baryon number violations, CP violations and must involve non-equilibrium physics. In practice this generally involves driving a CP violat-ing process at a phase transition when the universe is far from equilibrium.As with the case of dark matter there have been many models put forward toexplain baryogenesis however they all require physics beyond the standardmodel and in many cases involve some  ne tuning of the parameters of thisnew physics.1.3 MotivationThese two, seemingly unrelated, problems of modern cosmology may be ad-dressed through a single theory if the one assumes that both baryogenesisand dark matter formation occur at the QCD phase transition in the earlyuniverse. It has been suggested that there may be mechanisms operating atthe phase transiting which are capable of compacting matter to extremelylarge baryon density and that these processes operate in a manner in whichCP violation happens with a probability of order one. Some evidence forthe charge separation mechanism necessary for this process may in fact beseen in recent QGP experiments at RHIC in Brookhaven [6].In what follows I will accept that these mechanisms do operate at the QCDphase transition and that they are capable of compressing the condensingbaryonic matter to the point where it forms nuggets of a dense colour super-conducting state the properties of which will be discussed in a subsequentsection. In this model there is no baryon number violation involved in theprocess of baryogenesis. The CP violation makes the process of forminganti-nuggets more e cient than forming nuggets of normal quarks. Theremaining matter and antimatter annihilate each other leaving only theheavy quark nuggets and the fraction of excess matter corresponding tothe surplus of antimatter nuggets. Calculating the exact rates of formationis not currently feasible but to match present observations requires that theend result of this process is a universe with a mass distribution of roughly visible :  DM :   DM = 1 : 2 : 3 where DM and  DM are dark matter inquark and antiquark nugget form. This ratio would give the correct visi-ble to dark matter ratio and gives a universe with no net baryonic charge.This model is discussed in greater detail in the original papers [14] and [10].An intensive discussion of the nugget formation mechanisms is beyond thescope of this work. Instead the approach taken here will be to consider the21.3. Motivationpossible observational consequences of this model and test its consistencywith present data and to consider possible future constraints.3Chapter 2Nugget structure2.1 Colour superconductivityAt the relatively low atomic scale densities which are experimentally acces-sible the ground state of baryonic matter is known to be the iron nucleuswhich minimizes the binding energy per nucleon. However, new lowest en-ergy states are allowed as the density is increases. If the baryon density istaken to be asymptotically large the lowest energy state of QCD is know tobe one in which the constituent quarks condense to form Cooper pairs. Thisstate is referred to as colour superconductivity and proceeds through theBCS mechanism familiar from well studied superconductivity in low tem-perature metals. A wide variety of colour superconductors (CS) are knowto be possible depending on the pairing mechanism involved in forming thequark Cooper pairs. It is beyond the scope of the present work to discusthe full range of possible states. A review is available in, for example, [2].Instead I will focus on the generic features of known CS phases in an at-tempt to extract general properties of the proposed dark matter candidate.Much of the physics discussed here depends not on the state of the quarkmatter but simply on the net surface charge. The existence of a net chargeis relatively simple to understand though the exact details rapidly becomecomplicated. At asymptotically large densities the quark chemical poten-tial becomes greater than the strange quark mass, ms  150MeV and itis energetically favorable for the CS to contain equal numbers of up, downand strange quarks with each  avor allowing a new pairing channel and thuslowering the total energy. In a phase with equal numbers of the three lightquark  avors the quark matter is electrically neutral and the presence ofleptons is strongly disfavored energetically. However, near the surface ofthe quark matter the chemical potential drops and with it the fraction ofmuch heavier strange quarks. In this case the quark matter develops a netpositive charge in the case of a matter nugget or a net negative charge in thecase of a nugget composed of antiquarks. An exact calculation of the surfacecharge is a complex calculation and dependent on the exact superconductingphase realized, however, it is quite generically found that the lepton chemical42.2. The electrospherepotential at the quark matter surface should lie in the range from 10MeVup to 100MeV while the quark chemical potential is roughly an order ofmagnitude larger. The binding energy of the surface quarks is at the QCDscale and thus the quark surface is quite sharp, generally on the order offm. By contrast the leptons are bound only through electromagnetic e ectsand the looser binding means that they will extend well beyond the quarksurface. The distribution of these surrounding leptons will be the subject ofthe following section.2.2 The electrosphereRegardless of the phase of superconductivity realized in the quark matterthe nugget will carry a net charge (positive in the case of a matter nuggetand negative in the case of an antinugget.) To compensate for the net chargeof the quark matter the nuggets will build up a layer of leptons, known asthe electrosphere, at its surface. The form of this extended electrospherehas previously been studied in the case of quark stars (see for example [1],[12]). While the quark matter has a very sharp surface at a typical strongforce scale the electrosphere will be an extended object with the leptonsbound only by the relatively weak electric force. At the mean  eld levelwe can model the electrosphere as a fermi gas of leptons. The densitydistribution of electrons may be determined by equating the degeneracypressure of the Fermi gas with the electrical attraction due to the charge ofthe quark matter. This gives a di erential equation of the formr2 = 4  nl = 4  Z 10p2dp1 +e(E  )=T (2.1)Where  is the lepton chemical potential and nl is the lepton density whichis taken to be that of a fermi gas. In the case where the temperature ismuch less than the chemical potential the momentum integration may beapproximated to give,r2 = 4 3   2 m2 3=2(2.2)Combining this equation with the surface chemical potential  0  10 100MeV and the requirement that the chemical potential (and electric  eld)vanishes at spacial in nity allows us to solve for the density at any dis-tance above the quark surface. The general solution to the density equationdoes not have a simple analytic expression however there are two important52.2. The electrosphereregimes in which the solution may be approximated with a simple expres-sion. For reference several numerical solutions are demonstrated in appendixA. The  rst case is the ultrarelativistic limit which occurs close to the quarksurface. In this regime the temperature of the electrosphere and the leptonmass are negligible when compared to the chemical potential. In this casethe the density equation takes the form,d2 dz2 =4 3  3 (2.3)The solution to this second order di erential equation which vanishes asz!1 is, e+(z) =r3 2 1(z +z0); z0 =r3 2 1 0; ne+(z)’ 3e+(z)3 2 (2.4)The other regime in which the solution has a simple form is in whatwill be referred to as the Boltzmann regime. The Boltzmann regime shouldbegin at a density of roughly n = mT2  such that there is less than oneparticle per unit of phase space and degeneracy e ects become weak. Inthis regime the electron mass becomes the largest physically relevant scale.If the Boltzmann regime lies close enough to the quark matter surface thatthe plane parallel approximation holds an analytic solution to equation 2.1may be found. In this case the density takes the simpli ed form,n(z;T) = T2  1(z +  z)2 (2.5)While the chemical potential falls exponentially. Here  z is a constant to bematched to a boundary condition at the start of the Boltzmann regime. Thecritical fact is that whether it is obtained numerically or for a region in whichan approximate solution is known it is possible to express the lepton densityand the fermi momentum as a function of radial distance from the quarkmatter surface. This distribution is essentially independent of the detailsof the colour superconducting core and may be determined completely fromthe central charge.6Chapter 3Emission mechanismsConventional dark matter candidates such as WIMPs are generally madedark by their small interaction cross section with visible matter. Obser-vational consequences of a given model may be determined from the formof these interactions. In this case it is entirely possible that there maybe no observational signature of the dark matter beyond its gravitationalin uence. Currently favoured models tend to give the dark matter par-ticles masses at the TeV scale and thus interaction cross sections at theweak scale. Conversely, if the dark matter is composed of quark nuggets asoutlined above interactions are suppressed by purely geometric factors. Inregions of the galaxy where both the matter and dark matter densities arelarge one would predict collisions to occur with a great enough frequencythat there may be observational consequences. If a quark antinugget doescollide with the matter comprising the visible component of the galaxy weexpect the interactions to proceed at the rapid strong scale. The questionthen becomes whether there will be any clear observational consequences ofthese interactions and if so what form the resulting spectrum will take. Thisquestion is complicated by the wide range of energy scales involved from thequark chemical potential with  q  500MeV to the lepton chemical po-tential of   10MeV right down to the temperature at the eV scale. Thedominant contribution to the spectrum is expected to be due to annihilationevents involving antinuggets. Based on galactic abundances the majority ofannihilation events are likely to involve a galactic electron or proton. An-nihilation of other forms of galactic matter will also occur but with a lowerfrequency. Events involving larger objects are also less likely to produce acoherent detectable signal. In order to make predictions about the natureof the spectrum we will thus consider the the result of interactions betweena galactic electron or proton and a quark antinugget.3.1 Electron annihilationA galactic electron incident on the electrosphere of an antiquark nugget willproceed through the fermi gas of positrons at a roughly constant velocity73.1. Electron annihilation(at the galactic scale of a few hundred kilometers per second) experiencing arapid increase in both the positron density and the average positron kineticenergy. There are two distinct processes one should consider in treatingannihilation events involving a galactic electron. If the centre of mass mo-mentum is small then the electron positron pair will form an intermediatebound state (positronium) which will subsequently decay in a well knownway producing either a pair of back to back 511keV photons or a three pho-ton continuum. The form of this spectrum has been measured in scatteringexperiments and is well known. This process is a resonance e ect and thus itproceeds at a relatively fast rate which will remain essentially constant overthe regions of the electrosphere at which the densities are not vanishinglysmall. As one moves o resonance, that is to higher momenta, the probabil-ity of forming positronium falls rapidly and direct QED annihilation eventsbegin to dominate. For these events the growth of the density of states withpositron momentum favors annihilations involving positrons near the fermisurface. In this case the form of the spectrum may be determined froma relatively straight forward QED calculation the details of which will beworked through below. The end result is that the annihilation of galacticelectrons will produce a positronium decay line at 511keV as well as a broadcontinuum at the MeV scale with maximum energies near the lepton chem-ical potential at the quark matter surface. The relative strength of thesetwo emission mechanisms may be found from the relative rates at which thetwo processes proceed. While the exact probability of positronium forma-tion is a complicated quantum mechanical problem involving a summationover all positronium states, however a reasonable approximation may be ob-tained by considering the overlap of the incident e+e wavefunctions withthe ground state wavefunction of positronium (for details see appendix B.)From these considerations one predicts a formation rate for positronium of  1Ps  vem where ve is the incident electron velocity which should be ata typical galactic scale of a few hundred kilometers per second. Performinga similar calculation for the annihilation of a galactic electron within theelectrosphere of an antiquark nugget is an exercise in QED. This calculationinvolves computing the rate at which e+e !photons as a function of thelocal chemical potential and the  nal state photon energy. It is performedin detail in appendix C. The essential point of these calculations is that, ifthe dark matter is composed of colour superconducting nuggets as arguedhere, it will necessarily produce a positronium decay line along with it’s as-sociated continuum as well as a broad MeV emission spectrum with a totalenergy  ux below that of the positronium spectrum. The shape of the MeVspectrum will rise quickly to a peak near 511keV and then fall with increas-83.2. Comparison with observationing energy. It should be stressed that these two distinct spectral featuresare required by the dark matter model under consideration and are basedentirely in well known physics with no free parameters. If spectral featuresmatching these results were not to be found in the di use emissions fromthe galactic centre then the model would be invalidated.3.2 Comparison with observation3.2.1 The galactic 511keV lineObservations of the galactic centre do in fact detect a 511keV positron an-nihilation line. SPI/INTEGRAL data shows evidence for both a core andfainter disk component. The spectrum from the core is found to be roughlygaussian with an average  ux of d d ’ 0:025 photons cm 2s 1sr 1 [5, 7].This emission feature implies a large number of low momentum positronsare present in the galactic centre however no astrophysical phenomenon hasbeen proposed to produce them in su cient quantities. For the purposes ofwhat follows it will be assumed that the vast majority of the 511keV linehas as its source the annihilation of galactic electrons in the electrosphereof an antinugget.3.2.2 Di use MeV emissions from the galactic coreThe case of the MeV continuum is more complex than that of the 511keVline. Without a distinct spectral feature to search for one is forced to performa complicated background subtraction across a relatively large energy range.In the relevant regime there are relatively few contributing point sources,instead the majority of the di use background may be attributed to thescattering of cosmic rays o the galactic medium. Fortunately the spectrumattributed to cosmic ray processes has been studied for purposes apart fromthose discussed here [11]. While current models produce accurate energyand spacial distributions across a wide range in energy there is a distinctexcess of emissions detected in the MeV energy range where the COMPTELsatellite has made several observations. This excess seems to be stronglycorrelated with the galactic centre. Details of this spectrum along withthe cosmic ray spectrum are depicted in  gure 3.2.2. While the precedingsection gave a relative scale for the 511keV line and the MeV continuum noabsolute energy scale was given. The di culty in arriving at an absolutemagnitude for the emission mechanisms discussed is due to uncertainties inthe number density of both visible and dark matter in the galactic core.93.2. Comparison with observationEven if the mass density was precisely known there would remain a largeuncertainty in the number density of the nuggets due to their wide rangein possible baryon number. To circumvent this di culty the strength ofthe 511keV line from the galactic center will be used as normalization fromwhich the scale of the MeV emission may be determined. Comparison ofthe predicted MeV spectrum with observations of the galactic centre willthen provide a nontrivial test of the quark nugget dark matter model. Thecomplete annihilation spectrum integrated over all depths was obtained inappendix C C.7. However the overall scale was at that point arbitrary as theentire spectrum was scaled by an undetermined constant n1. To predict theresulting MeV spectrum from the galactic centre we will use the strengthof the 511keV line as normalization. The total strength of the positroniumdecay line is also dependent on n1. Similar to the calculations for thedirect annihilation spectrum the total number of positronium events maybe written as,NPstot =Z dzvednPsdz =Zdz ne(z) Psve: (3.1)Where ne is the surviving number of galactic electrons as a function of depthwithin the electrosphere as evaluated in C.6 and the rate of positroniumformation is  Ps = vem as discussed in appendix B. The strength of thegalactic 511keV line is then,NPstot = n1m Zdz e  =ve (3.2)The value of n1 will then be chosen so that the total measured  ux matchesthe SPI/INTEGRAL data. Numerically this gives a result ofn1 NUMBER.This result may then be used to evaluate the  nal spectrum C.7. As notedin appendix C once the surface chemical potential  0 > 20MeV only a verysmall fraction of incident electrons will survive to the inner region of theelectrosphere and thus the spectrum is almost independent of the surfaceconditions. A plot of this spectrum is given in  gure 3.2.2 for referencethe COMPTEL data points have been overlaid, a second curve is plottedin which the established cosmic ray background shown in 3.2.2 has beenadded to the calculated nugget spectrum. Note that both the shape of thespectrum as well as the magnitude are a good match to the COMPTELdata.3.2.3 Other emission mechanismsWhile it is not the focus of this work nearly identical considerations to thosemade above about the annihilation of galactic electrons can be made about103.2. Comparison with observationgalactic protons. The brief discussion presented here is based on the detailedcalculations presented in [3] [4]. A proton incident on an antiquark nuggetwill pass though the electrosphere and strike the colour superconductingcore. The CS, being a complicated many body quantum state with a verysharp edge, has a large probability of Andreev re ecting the incident protonrather than annihilating it. Those protons which do annihilate directly willrelease 2GeV of energy in a pair of back to back jets with one jet directedtowards the surface while the other thermalizes deep within the nugget. Asthe outward moving jet reaches the surface it will transfer most of its energyto the positrons which are the lightest available degrees of freedom. Theseejected positrons with relativistic energies then pass through the strong elec-tric  elds at the quark matter surface. It can be shown that the result ofthis process is a burst of bremsstrahlung radiation from with a characteris-tic energy of roughly 10keV. As mentioned, the second jet will expend itsenergy deep inside the nugget. This e ect proves to be the dominant factorin determining the average temperature of the nuggets. The temperaturescale is set by the energy balance between proton annihilation and thermalemission from the nuggets. This thermal emission will occur from an ef-fective surface in the electrosphere near the Boltzmann regime where thethermal photons, with energy E  T  eV, are  rst able to escape thenugget without scattering. This result makes the calculation of the thermalemission mechanism quite general regardless of the exact form of colour su-perconductivity realized at the core. By calculating the emissivity of theBoltzmann regime of the electrosphere it is possible to determine the result-ing spectrum which is found to extend well into the microwave ( 10 4eV.)As in the case of electron annihilation it is found that proton annihilationwill produce a de nite observational signature of dark matter as colour su-perconducting nuggets. As with electron annihilations, observations in therelevant energy range show an excess of  ux from the galactic centre beyondthat predicted by standard astrophysical considerations. In particular theChandra satellite has detected what has been described as a plasma with atemperature of roughly 8keV closely associated with the galactic centre [9].However, such a plasma would not remain bound to the galactic core and nopossible heating mechanism is known. Within our model this \plasma" isactually the combined e ect of the numerous proton annihilations occuringin the galactic centre. One consequence of this model is that, as it is basedentirely on events within the nugget, the form of the resulting spectrumshould be independent of the surrounding galactic environment. In the mi-crowave range it has been suggested that the WMAP data would be better t if the galactic spectrum includes a di use emission source distributed in113.2. Comparison with observationa roughly gaussian way around the galactic centre. Exactly this type ofdi use emissions are required by the dark matter model presented here. Asin the case of the MeV emissions one can extract the matter/dark matterinteraction rates from the SPI/INTEGRAL 511keV line strength and use itpredict the energy scale at which the proton annihilation spectra should fall.In both the 10keV and microwave cases the total energy scales are found tobe in good agreement with the Chandra and WMAP data.123.2. Comparison with observationFigure 3.1: Best  t to the high energy di use spectrum of the galacticcentre from Moskalenko et al. based on cosmic ray e ects with point sourcessubtracted. Note the observed excess over the calculated spectrum in theCOMPTEL range.133.2. Comparison with observationFigure 3.2: Total intensity ( attened by k2 as in 3.2.2) averaged over allelectrosphere heights. The total emission spectrum is shown in solid bluewhile the cosmic ray background and dark matter spectrum are shown inblack dotted and dashed lines respectively. The red bars are the COMPTELdata points.14Chapter 4ConclusionThis work is intended to convince the reader that the dark matter in theuniverse may in fact be ordinary matter and antimatter condensed into anon-baryonic phase and that such a model presents no contradictions withpresent observations. The interactions between the quark nuggets of thismodel and the ordinary visible matter of the galaxy may help to explaincomponents of the galactic spectrum over a broad range on energies. Itshould be noted just how broad this range is, from the di use MeV emissioncontinuum, to the positron annihilation line and di use x-ray spectrum downto the microwave contribution the emission mechanisms discussed here spansome thirteen orders of magnitude. In each case the total energy budgetand the form of the resulting spectrum are consistent with observation. Itshould also be noted that this is achieved entirely within the context ofwell known QED and QCD physics. In principle all quantities discussedhere may be directly calculated from  rst principle arguments. Only theuncertainty in the dark matter distribution of the galactic core requires theuse 511keV line for normalization. As this model requires no new physicsand contains no tunable parameters it is much more tightly constrained thanthe majority of dark matter candidates. This o ers the prospect of severalstrong tests of the model using data either presently available or which islikely to be available in the near future. In particular there should exist astrong spacial correlation between all the spectral features discussed here.This correlation must also mirror the product of the matter and dark matterdensities. Also, the proton annihilation spectrum must be independent ofthe region of the galaxy from which it is observed. Future modeling ofgalactic astrophysics is also likely to tighten the constraints on this modelas the relevant background contributions to the observed spectra becomebetter understood.As has been demonstrated here it is possible to explain both the nature ofdark matter and the mechanism of baryogenesis without appealing to newphysics for which we have no evidence and upon which we can place veryfew constraints.15Bibliography[1] Charles Alcock, Edward Farhi, and Angela Olinto. Strange stars. As-trophys. J., 310:261{272, 1986.[2] Mark G. Alford, Andreas Schmitt, Krishna Rajagopal, and ThomasSchafer. Color superconductivity in dense quark matter. 2007.[3] Michael McNeil Forbes and Ariel R. Zhitnitsky. Di use x-rays: Directlyobserving dark matter? JCAP, 0801:023, 2008.[4] Michael McNeil Forbes and Ariel R. Zhitnitsky. WMAP Haze: DirectlyObserving Dark Matter? Phys. Rev., D, to appear, 2008.[5] Pierre Jean et al. Early SPI/INTEGRAL measurements of galactic511 keV line emission from positron annihilation. Astron. Astrophys.,407:L55, 2003.[6] D. Kharzeev and A. Zhitnitsky. Charge separation induced by P-oddbubbles in QCD matter. Nucl. Phys., A797:67{79, 2007.[7] Jurgen Knodlseder et al. Early SPI/INTEGRAL contraints on themorphology of the 511 keV line emission in the 4th galactic quadrant.Astron. Astrophys., 411:L457{L460, 2003.[8] Kyle Lawson and Ariel R. Zhitnitsky. Di use cosmic gamma-rays at1-20 MeV: A trace of the dark matter? JCAP, 0801:022, 2008.[9] Michael P. Muno et al. Di use X-ray Emission in a Deep ChandraImage of the Galactic Center. Astrophys. J., 613:326{342, 2004.[10] David H. Oaknin and Ariel Zhitnitsky. Baryon asymmetry, dark matterand quantum chromodynamics. Phys. Rev., D71:023519, 2005.[11] Andrew W. Strong, Igor V. Moskalenko, and Olaf Reimer. Di useGalactic continuum gamma rays. A model compatible with EGRETdata and cosmic-ray measurements. Astrophys. J., 613:962{976, 2004.16Chapter 4. Bibliography[12] V. V. Usov, Tiberiu Harko, and K. S. Cheng. Structure of the electro-spheres of bare strange stars. Astrophys. J., 620:915{921, 2005.[13] Ariel Zhitnitsky. The Width of the 511 KeV Line from the Bulge of theGalaxy. Phys. Rev., D76:103518, 2007.[14] Ariel R. Zhitnitsky. ‘Nonbaryonic’ dark matter as baryonic color su-perdonductor. JCAP, 0310:010, 2003.17Appendix ALepton DistributionsThe distribution of leptons in the electrosphere surrounding a CS nuggetmay be determined based on the surface charge alone. However this is avery complex many body problem (equivalent to calculating the electronstructure of an atom with Z > 1020.) As a primary step to be used through-out what follows the electrosphere will be modeled as a fermi gas for whichthe density is given by,n =Z 102d3~p(2 )311 +e(E  )=T: (A.1)This approximation treats the leptons of the electrosphere as non-interactingand should be quite good in the highly degenerate regime near the quarksurface where all interactions are gapped. Obviously this will break downin the long distance tail of the electrosphere where degeneracy e ects be-come progressively less important. However the densities in this regimeare su ciently small that they have little impact on the  nal interactioncalculations.The simplest result is obtained by taking the temperature to be muchsmaller than any other scales in the problem at which point the fermi func-tion behaves as a step function and the electrosphere is governed by theexpression 2.2. First we consider the plane parallel case. Here the secondorder di erential equation may be reduced to  rst order as, d dz 2= 8 3 Zd   2 m2 3=2;d dz = 3 8  3=2  4  2 m2  38 m2( 2 m2)1=2+ 38m4ln  + ( 2 m2)1=2  ( 2 m2)1=2! : (A.2)This di erential equation does not have a simple closed form however itis quite simple to evaluate numerically. This may be done by convertingthe di erential equation to an integral equation and performing a numerical18Appendix A. Lepton DistributionsFigure A.1: Chemical potential as a function of height. Here the surfacechemical potential is taken to be  0 = 30MeV.integration starting from the quark surface taking the chemical potential atthis point as a boundary condition. The result of this integration is depictedin  gure A. Once the chemical potential at all radial distances is known thedensity may be established using the T = 0 expression for the density of afermi gas, n = ( 2 m2)3=23 2 . A plot of positron density is given in  gure A19Appendix A. Lepton DistributionsFigure A.2: Lepton density as a function of height. Here the surface chemicalpotential is taken to be  0 = 30MeV.20Appendix BPositronium FormationRatesA complete calculation of the rate at which positronium forms from lowmomentum electron positron collisions is beyond the scope of this work.Such a calculation would entail calculating the overlap of the incident par-ticle wavefunctions with all states of positroium and a summating over allresulting terms. Instead, following [13], consider the overlap between a lowmomentum electron positron pair represented by incoming plain waves withthe positronium ground state only. First we note that the ground state ofpositronium is j 1;0;0i= 1( a3)1=2e r=a. where a = (m ) 1  10 8cm is theBohr radius. The overlap of this wave function with the center of mass planewaves corresponding to the e+e pair is,h 1;0;0j e+e (q)i Zd3re r=aeirq 11 +a2q2 (B.1)Where q is the centre of mass momentum and the overall constant, whichis not relevant to our purposes, has been neglected. The main result ofthis calculation is that the probability of forming positronium (which is ofcoursejh 1;0;0j e+e (q)ij2) falls rapidly when the centre of mass momentumbecomes larger than m , when the momentum is smaller than this the prob-ability of order one. For simplicity the branching fraction to positroniumwill be taken asn(e+e !Ps)n(total) = 1 if <m (B.2)0 if >m The exponential behavior of the positronium wave function also introducesthe Bohr radius as a natural length scale. If the colliding electron andpositron are within one Bohr radius then the probability of positroniumformation is high, consequently the cross section for positronium formationshould be  Ps  4 (m ) 2. Under these assumptions the total rate at21Appendix B. Positronium Formation Rateswhich positronium is formed will be, Ps =Zdn(p)v  nve (m ) 2  ve m (B.3)Where we have taken the relative velocity to be at the typical galactic veloc-ity of ve 0:001 and have noted that the density corresponding to a fermimomentum of pF  m is simply n (m )3 up to geometric factors. It isthis value of the positronium formation rate that will be used in subsequentcalculations.22Appendix CThe direct annihilationspectrumThis appendix details the process by which the spectrum arising from galac-tic electrons annihilating in the electrosphere of an antiquark nugget may beobtained. An essentially identical calculation was performed in [8]. Thosegalactic electrons which do not annihilate through the positronium channelwill instead annihilate through a direct e+e !photons QED process. Thedi erential cross section for this process is well known (it seems to have been rst calculated by Dirac) and is found to be,d dk =  2mp2  (3m+E)(m+E)(m+E k)2  2 +   2mp2" 1k(3m+E)(m+E)2 (mk )2(m+E)2(m+E k)2#(C.1)Here E is the energy of the positron, m is the electron mass and k is the nal state photon momentum. The rate at which photons of a given energyare produced for a given chemical potential then follows from a scatteringcalculation.dN(k; )dkdt =Zdn(p)v(p)d (p;k)dk =Z 2d3(p)(2 )3pEd (p;k)dk : (C.2)Where dN(k; )dk dt is the number of photons produced per energy per unit timeand the T ! 0 limit has been taken for simplicity. The integration overpositron momenta may be performed to give the exact form of the resultingspectrum.dI(k; )dkdt = 2 mk2"k(k2 + 2mk 2m2) ln (2k m)( +m k)mk (C.3) 32k3 ( + 5m)k2 + (12 2 + 3 m+ 92m2)k m2( +m)+ k2m2 +m k + (8k4 8mk3 52m2k2 + 4m3k m4) k(2k m)2#:23Appendix C. The direct annihilation spectrumFigure C.1: Spectral density as a function of photon energy (with arbitrarynormalization.) Plotted for  = 10MeV (blue),  = 30MeV (green) and = 60MeV (red.) The spectra have been rescaled by 1 for to allow multiplecurves on the same axes.This function is plotted for several values of the positron chemical potentialin  gure C. While the function C.3 is rather complicated its general formis easily understood. The di erential cross section falls o due to phasespace constraints as the photon momentum is increased while the densityof positrons in the fermi gas increases rapidly as you approach the fermisurface. As such the resultant spectrum spans energies from slightly belowthe electron mass to just above the chemical potential and is roughly  atacross much of this range. The peak at the lower end of the spectrum is dueto the pole at k!0.As discussed in appendix A there are several increasingly complicatedmethods for modeling the positron distribution in the electrosphere of anantiquark nugget. Here it will be su cient to consider the T !0 limit. Thisgives a reasonably accurate density pro le over a large fraction of the radialextent of the electrosphere while remaining simple enough for demonstrativepurposes. Thus the density distribution is as shown in  gure A and thechemical potential as in  gure A. In order to obtain the spectrum whichresults from a galactic electron moving through the entire electrosphere anintegration over all heights weighted by the probability of reaching eachheight must be performed.To begin this calculation one must calculate the rate at which galactic24Appendix C. The direct annihilation spectrumelectrons will be annihilated for a given chemical potential. As noted abovepositronium events occur at a constant rate through the electrosphere of Ps  ve m . The direct annihilation rate is obtained by integrating thespectral density C.3 over all outgoing photon energies  dir = R dk dIdkdt. Thenat each height the number of incident electrons will be extinguished at a ratedetermined by the di erential equationdnedt = ne(z) ( dir(z) +  Ps) (C.4)dnedz = ne(z)ve ( dir(z) +  Ps) (C.5)where ne is the number of galactic electrons surviving to reach a givenheight and ve is the average velocity of incident electrons 1. The di erentialequation has been transformed into the second form presented here for futureconvenience. The dependence of  dir on the chemical potential (and thusheight) makes this equation di cult to evaluate explicitly. The solution willtake the form,ne(z) = n1e Rz1dz( dir(z)+ Ps)=ve (C.6)where n1 is the number of galactic electrons incident on the nugget. Inorder to extract the resultant spectrum the annihilation rate as a functionof chemical potential must be combined with the spectrum produced at aparticular chemical potential as given in C.3. This is done by an integrationover all heights weighted by the annihilation rate at each height.dItotdkdt =Zdzdndir(z)dz dI(z)dkdt (C.7)dItotdkdt =Zdzne dirdI(z)dkdtdItotdkdt = n1Zdze Rdz tot=ve dirdI(z)dkdtWhile this integration is di cult due to the z dependence in the exponentialfactor it may be evaluated numerically. The results are depicted in  gure3.2.2 along with COMPTEL data points as discussed above.1There remains a large uncertainty in the distribution of electron velocities in thegalactic centre as the  nal spectrum will be found do have an exponential factor withdependence on ve this value will dominate the uncertainty in the calculations which follow.25


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