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MODIFIED GRAVITY in COSMOLOGY and FUNDAMENTAL PARTICLE PHYSICS by DE-CHANG DAI Submitted in Partial Fulfillment of the Requireme

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MODIFIED GRAVITY in COSMOLOGY and FUNDAMENTAL PARTICLE PHYSICS by DE-CHANG DAI Submitted in Partial Fulfillment of the Requireme MODIFIED GRAVITY IN COSMOLOGY AND FUNDAMENTAL PARTICLE PHYSICS by DE-CHANG DAI Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Thesis Adviser: Dr. Glenn Starkman Department of Physics CASE WESTERN RESERVE UNUVERSITY May, 2008 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of De-Chang Dai candidate for the Doctor of Philosophy degree * (signed)Dr. Glenn D. Starkman (chair of the committee) Dr. Lawrence M. Krauss Dr. Harsh Mathur Dr. Colin McLarty (date) Mar 2008 *We also certify that written approval has been obtained for any proprietary material contained therein. Table of Contents 1 Introduction 1 1.1TheStandardModelofParticlePhysics................ 1 1.1.1 ParticlesandInteractionoftheStandardModel........ 2 1.2TheoryBeyondtheStandardModel................... 3 1.2.1 HierarchyProblem........................ 3 1.2.2 Supersymmetry(SUSY)..................... 4 1.2.3 Extra-Dimensions......................... 5 1.3ModifiedGravityinCosmology..................... 7 2 BlackMax: A Black-Hole Generator for the LHC 9 2.1Introduction................................ 9 2.2Black-holeproduction.......................... 13 2.2.1 Black-HoleFormationinBlackMax............... 24 2.3Gray-bodyFactors............................ 26 2.4Black-HoleEvolution .......................... 28 2.4.1 Electric and Color Charge Suppression ............. 31 2.4.2 MovementoftheBlack-HoleduringEvaporation........ 32 2.4.3 Rotation.............................. 34 2.5FinalBurst................................ 38 2.6 Input and Output . ............................ 39 2.7Results................................... 43 2.7.1 MassoftheInitialBlack-Holes................. 45 2.7.2 MovementofBlack-HolesintheBulk.............. 45 2.7.3 InitialBlack-HoleChargeDistribution............. 46 2.7.4 InitialBlack-HoleColorDistribution.............. 46 2.7.5 Evolution of Black-Hole Color and Charge during the Hawking RadiationPhase......................... 47 2.7.6 Numberofemittedparticles................... 57 2.7.7 EnergyDistributionsoftheEmittedParticles......... 58 2.7.8 PseudorapidityDistributionsoftheEmittedParticles..... 65 2.7.9 EmittedParticleTypes...................... 66 2.8Conclusion................................. 67 3 Birkhoff’s Theorem (BT) 72 3.1Introduction................................ 72 3.2Spacetimegeometry........................... 74 iii 3.2.1 Vacuumsolution......................... 76 3.2.2 Weaksourceonthebrane.................... 77 3.3Gravitationaleffectsofmassshells................... 80 3.3.1 Modeling............................. 80 3.3.2 Multipleshells........................... 81 3.4Conclusions................................ 84 4Conclusion 86 A Particle Emission Spectra 88 iv List of Tables A.1Literaturesourcesforparticleemissionspectra............. 93 A.2 Degrees of freedom of Standard-Model particles which are emitted from a black hole. For gravitons, the table shows 1, because the appropriate growth in the number of degrees of freedom is included explicitly in the graviton emission spectrum. ns is the number of extra dimensions in which vector and scalar fields can propagate. ............ 93 A.3 The fraction of emitted particles of different types (including final burst particles) in a variety of extra dimension scenarios. ∗Note that the ab- sence of gravitons in the case of a rotating black hole is due exclusively toourcurrentignoranceofthecorrectgray-bodyfactor........ 94 v List of Figures 1.1SomequadraticallydivergentHiggsself-energies............. 4 1.2 Split fermion model: Different fermions are constrained on different branes and gauge bosons are moving between different brances. 6 2.1 Horizon radius (in GeV−1) of a non-rotating black-hole as a function ofmassfor4-10spatialdimensions.................... 16 2.2 Hawking temperature(in GeV) of a non-rotating black-hole as a func- tionofmassfor4-10spatialdimensions................. 16 2.3 Horizon radius (in GeV−1) of a black-hole as a function of mass for differentBind=5spatialdimensions. ................. 17 2.4 Hawking temperature(in GeV) of a black-hole as a function of mass fordifferentBind=5spatialdimensions................. 18 2.5 Horizon radius (in GeV−1) of a rotating black-hole as a function of mass for different angular momentum in d=5 spatial dimensions. Angular momentum J is in units of ........................ 18 2.6 Hawking temperature(in GeV) of a rotating black-hole as a function of mass for different angular momentum in d=5 spatial dimensions. Angular momentum J is in units of ................... 19 2.7 Cross section for production of a black-hole (rotating or non-rotating) as a function of the number of spatial dimensions, for a tensionless brane,withnofermionbrane-splitting.................. 22 2.8 Cross section for production of a non-rotating black-hole as a function of the deficit angle parameter for d =5andns =2........... 22 2.9 Cross section for production of a non-rotating black-hole as a function of the number of fermion brane-splitting dimensions for d = 10. 23 2.10 Cross section for formation of a black-hole (rotating or non-rotating) as function of the minimum mass of black-hole, for a zero-tension brane, with no fermion brane-splitting. The vertical lines are the error bars. 23 2.11 Cross section for the two-body final-state scenario as a function of number of spatial dimensions where Mmin = M∗ =1TeV,Mmin = M∗ =3TeVandMmin =5TeV...................... 24 2.12Angularmomentumtiltgeometry..................... 27 vi 2.13 The emitted fermion intensity (normalized to one) as a function of the distance between the black-hole and the center of the gaussian distribu- tion of a fermionic brane. As a black-hole increases the distance from a fermionic brane due to recoil the intensity of the emitted fermions of that type falls down quickly. The radius of black-hole is set to be −1 −1 2M∗ . The width of the fermionic brane is M∗ . The plot is for the caseofoneextradimension........................ 29 2.14 Parameter.txt is the input file containing the parameters that one can change. The words in parentheses are the parameters that are used in thepaper.................................. 49 2.15 Output.txt: There are three parts to this file. The first part is a copy of parameter.txt. The second part includes information about the black-hole and the emitted particles. The first column identifies the what type of information each row is supplying – parent is information about the two incoming partons; Pbh is information on the energy and momenta of the produced black-holes; trace describes the location of the black-holes; Pem characterizes the emitted particles in the lab frame; Pemc characterizes the emitted particles in the center-of-mass frame; Elast describes the final burst. The third part of the file is the black-hole production cross-section as inferred from the events in this generatorrun. .............................. 50 2.16 Lines in the output file headed by the ID = Parent contain information abouttheinitialpartonswhichformedtheblack-hole.......... 51 2.17 Lines in the output file headed by the ID = Pbh contain the energies and momenta of the black-holes for each emission step. In case of rotating black-holes, the last column in the line is the angular momentum. 51 2.18 Lines in the output file headed by the ID = trace contain the location oftheblack-holeforeachemissionstep.................. 51 2.19 Lines in the output file headed by the ID = Pem contain the types of the emitted particles, their energies and momenta in the lab frame and thetimesoftheiremission......................... 51 2.20 Lines in the output file headed by the ID = Elast contain the types, energiesandmomentaofparticlesofthefinalburst........... 52 2.21 Mass distribution of initial black-holes (rotating and non-rotating) on atensionlessbraneforvariousnumbersofextradimension. ..... 52 2.22 Mass distribution of initial (non-rotating) black-holes on a non-zero tension brane for B =1.0, B =0.8andB =0.6............. 52 2.23 Mass distribution of initial (non-rotating) black-holes on a tensionless brane for d =10anddifferentnumbersofsplit-fermionbranes..... 53 2.24 black-hole movement in the mini-bulk due to recoil. X1, Y1 are coordi- nates in two extra dimensions. The red circle indicates the width of the quark brane. The blue circle indicates the width of the lepton brane. The black lines are black-holes traces. The size of the mini-bulk is 10 TeV−1 ×10 TeV−1. black-holes with non-zero standard model gauge charges bounce back from the wall of the mini-bulk. black-holes with zeroStandardModelgaugechargescanleavethemini-bulk...... 53 vii 2.25Electromagneticchargedistributionoftheinitialblack-holes...... 54 2.26 Initial color distribution of the created black-holes. The vertical lines areerrorbar................................ 54 2.27 Cumulative color distribution for non-rotating black-holes on a ten- sionless brane with d = 4 and no fermion brane splitting. Histogram with the black squares (open circles) is with (without) color suppression. 54 2.28 Number of particles emitted by a non-rotating black-hole on a tension- less brane prior to the “final burst” as function of number of extra dimension. Here ns = 0. The error bars denote one standard deviation range.................................... 55 2.29 Number of particles emitted by a non-rotating black-hole in the split- fermion model prior to the “final burst” as a function of the number of brane-splitting dimensions, with d = 10. Error bars denote one standarddeviationrange........................
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