Axions in cold dark matter and inflation models

The subjects of this thesis are the invisible axion and the more general family of axion-like particles. The invisible axion is a hypothetical elementary particle and a cold dark matter candidate. I present an improved computation of the constraints on the parameter space of the cold dark matter axion in the standard cosmology, that includes the contributions from anharmonicities in the axion potential and from the decay of axionic strings. In this scenario, I update the value of the mass of the cold dark matter axion, fi nding the value (67 ± 17)µeV, approximately one order of magnitude larger than previous computations. The eff ect of nonstandard cosmological scenarios on the parameter space of axion cold dark matter is studied for the first time. In particular, I consider the cases of low-temperature reheating and kination cosmologies, and I show that the mass of the cold dark matter axion can differ from the value in the standard cosmological scenario by orders of magnitude. Finally, I consider the family of axion-like particles, assuming that these particles serve as the inflation in the context; of warm inflation. I find that the axion energy scale f, which in the standard inflation scenario is of the order of the Planck mass, can be lowered to the much safer Grand Uni cation Theory scale f ~ 1016GeV. I also constrain the parameter space and the amount of gravitational waves from this model, using results from the Wilkinson Microwave Anisotropy Probe 7-year data.

AXIONS IN COLD DARK MATTER AND INFLATION MODELS by Luca Visinelli A dissertation submitted to the faculty of The University of Utah in partial ful llment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah December 2011 Copyright c Luca Visinelli 2011 All Rights Reserved THE UNIVERSITY OF UTAH GRADUATE SCHOOL STATEMENT OF DISSERTATION APPROVAL The dissertation of Luca Visinelli has been approved by the following supervisory committee members: Paolo Gondolo , Chair 9/26/2011 Date Approved Yong Shi Wu , Member 9/26/2011 Date Approved Wayne Springer , Member 9/26/2011 Date Approved Oleg Starykh , Member 9/26/2011 Date Approved Aaron Bertram , Member 9/26/2011 Date Approved and by David Kieda , Chair of the Department of Physics and Astronomy, and by Charles A. Wight , Dean of The Graduate School. ABSTRACT The subjects of this thesis are the invisible axion and the more general family of axion- like particles. The invisible axion is a hypothetical elementary particle and a cold dark matter can- didate. I present an improved computation of the constraints on the parameter space of the cold dark matter axion in the standard cosmology, that includes the contributions from anharmonicities in the axion potential and from the decay of axionic strings. In this scenario, I update the value of the mass of the cold dark matter axion, nding the value (67 17) eV, approximately one order of magnitude larger than previous computations. The e ect of nonstandard cosmological scenarios on the parameter space of axion cold dark matter is studied for the rst time. In particular, I consider the cases of low- temperature reheating and kination cosmologies, and I show that the mass of the cold dark matter axion can di er from the value in the standard cosmological scenario by orders of magnitude. Finally, I consider the family of axion-like particles, assuming that these particles serve as the in aton in the context of warm in ation. I nd that the axion energy scale f, which in the standard in ation scenario is of the order of the Planck mass, can be lowered to the much safer Grand Uni cation Theory scale f 1016GeV. I also constrain the parameter space and the amount of gravitational waves from this model, using results from the Wilkinson Microwave Anisotropy Probe 7-year data. A mia moglie Erika CONTENTS ABSTRACT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ix LIST OF TABLES: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x ACRONYMS USED : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xi ACKNOWLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xii CHAPTERS 1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. CONCORDANCE COSMOLOGY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2.1 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 General relativity and cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 The Friedmann-Robertson-Walker metric . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 The Friedmann equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2.1 Derivation of the Friedmann equations . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2.2 The term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2.3 Equivalent forms of the Friedmann equations . . . . . . . . . . . . . . . . 6 2.2.3 Redshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.4 Particular solutions of the Friedmann equation . . . . . . . . . . . . . . . . . . . 8 2.3 Cosmic in ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 The atness problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 The horizon problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.3 Unwanted relics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.4 Small-scale structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.5 In ation building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Fluctuations during in ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.1 The scalar power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.2 The scalar spectral index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.3 The tensor power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.4 Bounds on the Hubble rate at the end of in ation . . . . . . . . . . . . . . . . . 15 2.4.5 Exiting in ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Thermal history of the universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.1 Statistical mechanics at thermal equilibrium . . . . . . . . . . . . . . . . . . . . . 16 2.5.2 Conservation of the number and entropy densities . . . . . . . . . . . . . . . . . 18 2.5.3 Application to the radiation-dominated universe . . . . . . . . . . . . . . . . . . 19 2.5.4 Matter-dominated universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Content of the universe at the present time . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6.1 Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6.2 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6.3 Dark energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3. ELEMENTS OF QUANTUM FIELD THEORY AND QCD : : : : : : : : : 22 3.1 Elements of group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Lagrangian functions for scalar and spinor elds . . . . . . . . . . . . . . . . . . . . . . 24 3.2.1 The Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.2 The Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.3 Equation of motion for a scalar eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.4 Equation of motion for a spinor eld . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.4.1 Representation of dimension two for the Lorentz transformation . . 26 3.2.4.2 Weyl spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.4.3 Equation of motion for the Weyl spinor . . . . . . . . . . . . . . . . . . . . . 28 3.2.5 Equation of motion for the Majorana spinor . . . . . . . . . . . . . . . . . . . . . 28 3.2.6 Equation of motion for the Dirac spinor . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Application of group theory to QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1 The QCD Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.2 Solving the U(1)A problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.3 The strong CP problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4.4 Possible solutions to the strong CP problem . . . . . . . . . . . . . . . . . . . . . 36 3.4.4.1 Calculable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4.4.2 Massless up quark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4. THE AXION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 37 4.1 The axion as a solution to the strong CP problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 On the mass of the axion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.1 Axion mass at zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.2 Finite temperature e ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Coupling of standard model particles with the axion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.1 Coupling of axions to gluons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.2 Coupling of axions to photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3.3 Coupling of axions to fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.4 Axion models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.4.1 The \visible" axion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.4.2 Models for the \invisible" axion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4.2.1 The KSVZ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4.2.2 The DFSZ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4.3 Lifetime of the axion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.5 Astrophysical bounds on axions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.5.1 Constraints from the cooling time of white dwarfs . . . . . . . . . . . . . . . . . 48 4.5.2 Constraints from SN1987A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5.3 Axions from the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5.4 Axions and globular clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 vi 4.6 Direct axion searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.6.1 Axion haloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.6.2 Axion helioscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.6.2.1 Search of solar axions through Bragg scattering . . . . . . . . . . . . . . . 53 4.6.2.2 Solar axions through axion telescopes . . . . . . . . . . . . . . . . . . . . . . 54 4.6.3 Production of axions by laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.6.3.1 Polarization experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.6.3.2 Regeneration experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5. REVISING THE AXION AS THE COLD DARK MATTER : : : : : : : : : 56 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2 Production of axions in the early universe . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.1 Thermal production of axions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.2 Axions from the misalignment mechanism . . . . . . . . . . . . . . . . . . . . . . . 59 5.2.2.1 Equation of motion on the FRW metric . . . . . . . . . . . . . . . . . . . . . 59 5.2.2.2 Computing the oscillation temperature . . . . . . . . . . . . . . . . . . . . . 61 5.2.2.3 Axion energy density from vacuum realignment . . . . . . . . . . . . . . . 61 5.2.2.4 The role of anharmonicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2.3 Axions from string decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2.3.1 The domain wall problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2.3.2 Parameter space of axionic strings . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Axion isocurvature uctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.4 Parameter space of the cosmological axion . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.5 The axion as the CDM particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.5.1 Axion CDM in Scenario I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.5.2 Axion CDM in Scenario II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.6 Constraining the parameter space of the axion . . . . . . . . . . . . . . . . . . . . . . . . 72 5.6.1 Comparison with previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.7 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6. AXION CDM IN NONSTANDARD COSMOLOGIES: : : : : : : : : : : : : : : 78 6.1 Motivations in considering modi ed cosmologies . . . . . . . . . . . . . . . . . . . . . . 78 6.1.1 Cosmological probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2 The misalignment mechanism in nonstandard cosmologies . . . . . . . . . . . . . . . 80 6.2.1 Hubble rate in nonstandard cosmologies . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.2.2 Oscillation temperature in nonstandard cosmologies . . . . . . . . . . . . . . . 81 6.3 Axions from string decays in nonstandard cosmologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.4 Axion CDM in the LTR cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.4.1 Results for the LTR cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.5 Axion CDM in the kination cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.5.1 Results for kination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.6.1 Comparison to previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.6.2 E ects of changing the string decay parameters . . . . . . . . . . . . . . . . . . . 101 6.6.3 Distinguishing nonstandard cosmologies observationally . . . . . . . . . . . . 103 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 vii 7. NATURAL WARM INFLATION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 106 7.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.2 The warm in ation scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.3 Axion-like particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.4 Warm natural in ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.4.1 Slow-roll conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.4.2 Number of E-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4.3 Parameter space of the NWI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.5 Perturbations from in ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.5.1 Scalar power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.5.2 Scalar spectral index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.5.3 Tensor power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.6 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 REFERENCES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 123 viii LIST OF FIGURES 4.1 The function I(T) in Eq.(4.14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 The axion-photon vertex in an axion theory in which ga 6= 0. . . . . . . . . . . . . . 44 4.3 Coupling of an axion with a fermionic eld f. . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.1 Numerical (solid line) and approximated (dashed line) solutions for ( ). . . . . 60 5.2 The parameter space where the axion is 100% CDM in the standard cosmology. 73 5.3 i as a function of fa in the standard cosmology. . . . . . . . . . . . . . . . . . . . . . . . 75 6.1 The PQ scale fLTR a vs. TRH for 100% axion CDM in the LTR cosmology . . . . 90 6.2 The parameter space where the axion is 100% CDM in the LTR cosmology. . . 92 6.3 i as a function of fa in the LTR cosmology. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.4 The PQ scale fkin a vs. Tkin for 100% axion CDM in the kination cosmology . . 97 6.5 The parameter space where the axion is 100% CDM in the kination cosmology. 99 6.6 i as a function of fa in the kination cosmology. . . . . . . . . . . . . . . . . . . . . . . . . 100 7.1 i as a function of for di erent values of the number of e-folds Ne. . . . . . . . 113 7.2 i vs. for the axion and the quadratic potentials. . . . . . . . . . . . . . . . . . . . . . 114 7.3 The dissipation term 􀀀 as a function of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.4 The in aton quartic self-coupling as a function of . . . . . . . . . . . . . . . . . . . 117 7.5 The scalar spectral index ns as a function of , for di erent values of Ne. . . . . 118 7.6 Constraints on the NWI model in the r-ns plane, for f = 1016 GeV. . . . . . . . . 121 LIST OF TABLES 4.1 Values of the parameters a, b, c, and d appearing in Eq. (4.21), as a function of the number of relativistic degrees of freedom Nf . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Values of the parameters inst and in Eq. (4.24) for di erent values of QCD. From Ref. [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.1 Expressions for some derived quantities in the NWI theory, valid during slow- roll and Q 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 ACRONYMS USED ADMX Axion dark matter experiment BAO Baryon acoustic oscillations BBN Big Bang nucleosynthesis CDM Cold dark matter CMBR Cosmic microwave background radiation COBE Cosmic background explorer DFSZ Dine-Fischler-Srednicki-Zhitnitsky FRW Friedmann-Robertson-Walker GR General relativity GUT Grand uni cation theory KSVZ Kim-Shifman-Vainshtein-Zakharov LTR Low temperature reheating MSSM Minimal supersymmetric standard model NWI Natural warm in ation QCD Quantum chromodynamics QED Quantum electrodynamics QFT Quantum eld theory SN Supernova WMAP Wilkinson microwave anisotropy probe ACKNOWLEDGEMENTS I would like to thank my advisor, Prof. Paolo Gondolo, for his encouragement and enthusiasm whenever a new project was taking o , and for his support during the developing stages of each work. I am also indebted to my committee members, Yong-Shi Wu, Wayne Springer, Oleg Starykh, and Aaron Bertram, for their valuable time. I would like to thank my family - my mother, Giovanna, and my sister, Laura - for their love and support during my odyssey in Utah. They have always encouraged me towards excellence. Finally, and most importantly, I would like to thank my wife Erika. Her support, patience, encouragement, and unwavering love were undeniably the milestone upon which the past ve years of my life have been built. CHAPTER 1 INTRODUCTION My interest in the invisible axion, a hypothetical pseudo-scalar particle arising in the QCD sector of the Standard Model, addresses the fact that this particle might play the role of the Cold Dark Matter (CDM). This possibility has been intensively studied for more than thirty years; original work on the subject by myself and my advisor prof. Paolo Gondolo is exposed in Chapters 5 and 6, drawn from Refs. [185] and [186], respectively. In Chapter 5, I considered the invisible axion as the CDM particle in the standard cosmological scenario, updating the bounds on the axion parameter space in light of the most recent cosmological data, and improving the treatment of anharmonicities in the axion potential and axionic strings. In Chapter 6, it is shown how the invisible axion can be used as a probe to study the properties of the universe before Big Bang nucleosynthesis (BBN) took place. In fact, we only know that the universe has to be radiation dominated (standard cosmology) at the time of BBN, but a plethora of di erent possibilities could have taken place before this period. We studied the properties of the axion CDM in two di erent nonstandard cosmological scenarios, the low temperature reheating (LTR) cosmology and the kination cosmology. The last chapter of the thesis deals with the role of axion-like particles, a generalization of the invisible axion theory, as the in aton in the speci c scenario of warm in ation, and refers to the paper in Ref. [184]. Axion-like particles have long been successfully considered as the in aton particle in the standard in aton, with the model going under the name of Natural In ation (NI). One problem of the NI model resides in that the energy scale at which the axion symmetry spontaneously breaks is of the order of the Planck mass, which makes it di cult to embed NI into larger Grand Uni cation Theories (GUT). In Chapter 7, I show that this problem can be overcome if we consider the warm in ation scenario in place of standard in ation, thus developing the natural warm in ation model (NWI). CHAPTER 2 CONCORDANCE COSMOLOGY A quantitative approach to cosmology began in the early 20th century, after Einstein's General Relativity (GR) was formulated. In fact, cosmology as we know today is rmly established on the basis of the GR theory, as we will explore in detail in Sec. 2.2. Before relativity, space and time were considered xed, an idea that directly followed from the New- tonian concept of an absolute framework in which physical processes would take place. The birth of GR provided scientists with a new concept, the fact that space-time is a dynamic entity whose evolution is described by Einstein equation. Within the GR theory, gravity can be understood as a geometric property of the space-time. Soon after the introduction of GR in its nal version, the scienti c community began speculating on the origin of the universe, merging observations with the new framework provided by the GR theory. The cosmological solutions of general relativity were rst found, independently, by A. Friedmann, G. Lemaitre, H. P. Robertson, and A. G. Walker and constitute what is known as the Friedmann-Robertson-Walker (FRW) model (see Refs. [108, 192] for exhaustive reviews on the subject). 2.1 General relativity The insight behind GR consists of equating the e ects of the gravitational force on a test particle with the acceleration that such a test particle would experience on a nontrivial manifold. This idea is encoded in the weak equivalence principle, which states that local e ects of motion in a curved space-time are indistinguishable from those of an accelerated observer in a at space-time, and it is depicted by the famous elevator thought-experiment. With this key concept in mind, Einstein derived the famous formula R 􀀀 g R = 8 GT + g : (2.1) The term appearing in Eq. (2.1) is a constant that might play an important role in the description of the universe at present times. We will discuss the e ects of this term in depth in Secs. 2.2.2.2 and 2.2.4. 3 We rst discuss the left-hand side of Eq. (2.1). Here, g is the metric of the manifold considered, from which we can derive the Ricci tensor R via the Riemann tensor R as R = R : (2.2) The Riemann tensor R is de ned through the Christo el symbol (Levi-Civita connection) 􀀀 as R = @ 􀀀 􀀀 @ 􀀀 + 􀀀 􀀀 􀀀 􀀀 􀀀 : (2.3) Here, the Levi-Civita connection is given by 􀀀 = 1 2 g (@ g + @ g 􀀀 @ g ) ; (2.4) and @ means di erentiation with respect to the manifold coordinate x . In Eq. (2.1), the curvature is R = g R , with the inverse of the metric g satisfying g g = : (2.5) Sometimes the combination G = R 􀀀 g R; (2.6) is referred to as the Einstein tensor. On the right-hand-side of Eq. (2.1), the term T describes the energy-momentum content at each point of space-time and is known as the stress-energy tensor. Since T depends implicitly on the metric tensor, in GR a feedback mechanism appears in which the metric structure of space-time and the matter content of the universe mutually in uence each other, a statement that can be summed up with the paraphrase by Wheeler: space-time tells matter how to move; matter tells space-time how to curve. In the following, we will always use the case for an isotropic perfect uid of density eld , pressure p and four-velocity u , whose stress tensor reads T = u u + p (u u 􀀀 g ): (2.7) We remark that the density and the pressure p are time-dependent quantities, although in the text, we do not indicate such dependence explicitly. In Sec. 2.2.2, we will discuss how the density and the pressure for a speci c uid are related by a speci c equation of state. 4 2.2 General relativity and cosmology 2.2.1 The Friedmann-Robertson-Walker metric In the FRW metric, the Copernican principle that demotes Earth's position in the Solar system is extended in a cosmological framework, by stating that no point in the universe is special with respect to the other ones. This concept, called the cosmological principle, can be restated by posing that the universe is isotropic with respect to any point, or equivalently, that it is homogeneous and isotropic with respect to one point. Isotropy and homogeneity can be understood in terms of cosmic in ation, see Sec. 2.3, and lead to the FRW universe, with line element ds2 = dt2 􀀀 a2(t) dr2 1 􀀀 k r2 + r2 d 2 + r2 sin2 d 2 : (2.8) Here a(t) is the scale factor of the universe depending on the cosmic time t, and the term in square brackets indicates the spatial metric, which is specialized to represent a three- manifold of constant curvature k. The only possibilities for a spatial manifold of constant curvature are zero curvature with k = 0 ( at universe), positive curvature with k = +1 (closed universe), and negative curvature with k = 􀀀1 (open universe). Sometimes the conformal time is de ned, d = dt a(t) ; (2.9) so that the FRW metric is written in the conformal form, ds2 = a2(t( )) d 2 􀀀 dr2 1 􀀀 k r2 􀀀 r2 d 2 􀀀 r2 sin2 d 2 : (2.10) 2.2.2 The Friedmann equations The Friedmann equations are a set of two equations that relate the expansion of the universe to its energy density and pressure content. In this section, we expose their derivation from the Einstein equation and some of their uses. In the formulas used for computations, we trade Newton's gravitational constant G, appearing in Eq. (2.1), with the Planck mass MPl, using the relation G = 1 M2 Pl : (2.11) For numerical computations, we will use MPl = 1:221 1019 GeV. 5 2.2.2.1 Derivation of the Friedmann equations It can be shown that the 00 and 11 components of the Einstein Eq. (2.1), expressed on the FRW metric in Eq. (2.8) and with the source in Eq. (2.7), give the relations _a(t) a(t) 2 = 8 3M2 Pl 􀀀 k a2 + 3 ; (2.12) and a(t) a(t) = 􀀀 4 M2 Pl ( + 3p) + 3 : (2.13) A dot over a quantity will always indicate a total derivative with respect to the cosmic time t, so _a(t) = da(t)=dt. Because of isotropy, the 22 and 33 components of Eq. (2.1) give the same expression as Eq. (2.13), while the o -diagonal terms are all zero due to homogeneity. The role of the term , also known as the vacuum energy term, will be discussed in detail in Sec. 2.2.2.2. In a realistic cosmological model, more types of uids are present. Examples include the nonrelativistic matter uid, of density m and pressure pm = 0, and the radiation uid, of density r and pressure pr = r=3. When more than one uid are present in the theory, we de ne the total energy densities tot and the total pressure ptot of the system as tot = X i i; ptot = X i pi; (2.14) where i and pi are the density and the pressure of the i-th uid. To completely specify the system, we also need an equation of state linking the density and the pressure of each uid, pi = wi i; (2.15) with the parameter wi specifying the type of uid. For example, wm = 0 is for a nonrela- tivistic uid and wr = 1=3 is for a relativistic uid. In the multiple elds case, we substitute ! tot and p ! ptot in the Friedmann Eqs. (2.12) and (2.13). This is the case we will discuss from now on. De ning the Hubble rate H(t) = _a(t) a(t) ; (2.16) we obtain the Friedmann Eqs. (2.12) and (2.13) in the form H2(t) = 8 3M2 Pl tot 􀀀 k a2(t) + 3 ; (2.17) and _tot = 􀀀3H(t) (ptot + tot): (2.18) 6 Unless otherwise speci ed, we will always derive our results in the next chapters in the case of at geometry k = 0 and negligible cosmological constant = 0, which are suitable approximations in the cases we consider. For these reasons, we will refer to the Friedmann equation in the form H2(t) = 8 3M2 Pl tot: (2.19) We will keep the parameters k and for the rest of the discussion in the current chapter. 2.2.2.2 The term In cosmology, the constant term is usually identi ed with the vacuum energy, respon- sible for the accelerated expansion we observe at the present time. Using Eq. (2.12), we can associate to the cosmological constant the energy density and the pressure = M2 Pl 8 ; p = 􀀀 : (2.20) From now on, we include, in the de nition of the terms tot and ptot, the contributions from and p , respectively. With these de nitions, the Friedmann Eq. (2.17) is rewritten as H2(t) = 8 3M2 Pl tot(t) 􀀀 k a2(t) : (2.21) From Eq. (2.15), we see that the equation of state for the cosmological constant term has w = 􀀀1. Models in which only a cosmological constant is present have been studied since the early days of relativistic cosmology, and fall under the name of Lemaitre models. To sum up, in order to include the contribution from the vacuum energy in the Friedmann equations, one has to add the energy density to the sum tot, and similarly the pressure p to the sum ptot, keeping in mind that the equation of state for the vacuum energy is = 􀀀p . 2.2.2.3 Equivalent forms of the Friedmann equations The Friedmann Eq. (2.21) is particularly suitable for solving speci c problems in which the content of the universe is speci ed and for explaining the thermal history of the universe. De ning the critical density, crit = 3H2(t)M2 Pl 8 ; (2.22) and the cosmological density ratios, i(t) = i crit ; tot(t) = tot crit = X i i(t); (2.23) 7 we cast the Friedmann Eq. (2.17) in the alternative form k a2(t) = H2(t) ( tot(t) 􀀀 1): (2.24) Eq. (2.24) shows that the universe is open (k = 􀀀1) when tot(t) < 1, at (k = 0) when tot(t) = 1, closed (k = +1) when tot(t) > 1. The second, alternative formulation of the Friedmann equation we show now is partic- ularly suitable for computing the scale factor a(t) given the energy content of the universe at a speci c time, like for example at the present time t0. We indicate the values of the density ratios at the present time with the index 0, and the present value of the critical density is c crit 0 = 3H2 0 M2 Pl 8 = 1:878 10􀀀29 h2 g=cm3; (2.25) where the Hubble expansion rate at present time and the value of h are [110] H0 = 100 hkms􀀀1 Mpc􀀀1 = (70:2 1:4)kms􀀀1 Mpc􀀀1: (2.26) Introducing the density of curvature k(t) = 􀀀 k a2(t)H2(t) ; (2.27) Eq. (2.17) is cast in the form H2(t) H2 0 = r a0 a(t) 4 + m a0 a(t) 3 + k a0 a(t) 2 + + w a0 a(t) 3(1+w) : (2.28) Here and in the following, we have used the notation i = i(t0). We have included the possibility to include in the theory a uid of density ratio w with generic equation of state p = w and w unspeci ed. Notice that setting a(t) = a(t0) = a0 in Eq. (2.28) gives the constraint r + m + k + + w = 1: (2.29) 8 2.2.3 Redshifts The redshift z describes the lowering in the frequency of cosmological photons that have been emitted in the past with some frequency and are received at the present time with frequency 0, and is de ned as 1 + z = i 0 : (2.30) Since it can be shown that the momentum of free particles moving on the FRW metric decreases with 1=a(t), as well as lengths and wavelength stretch proportionally to a(t), the redshift at some time t is related to the scale factor at that time by 1 + z = a(t0) a(t) : (2.31) A precise relation between redshift z and time t is given below in Eq. (2.39). Since the scale factor a(t) ! 0 for t ! 0 (Big Bang) and a(t) ! a0 at present time, we also have z ! +1 when t ! 0 and z = 0 at the present time. Using Eq. (2.31), one can trade the time or scale factor dependence appearing in some equation with a z-dependence. In this view, a useful form of Eq. (2.28) that provides the Hubble expansion rate in terms of the redshift z is H(z) = H0 h r (1 + z)4 + m (1 + z)3 + k (1 + z)2 + + w (1 + z)3(1+w) i1=2 : (2.32) 2.2.4 Particular solutions of the Friedmann equation We show the explicit solution of Eq. (2.19) in various cosmologically interesting scenarios. In general, we expect the energy density i that dominates the universe at some particular time to depend on the scale factor as i = i0 a0 a 3(1+wi) : (2.33) The Friedmann Eq. (2.19) with = i is written in the form _a a 2 = 8 3M2 Pl i0 a0 a 3(1+wi) = H2 0 i a0 a 3(1+wi) ; (2.34) with solution a(t) = a0 ( 3(1+wi) 2 p i (H0 t) 2 3(1+wi) ; with wi 6= 􀀀1; Exp H0 tp i ; with wi = 􀀀1: (2.35) In particular, the solutions when the universe is dominated by matter (wm = 0), radiation (wr = 1=3), and vacuum (w = 􀀀1) are, respectively, a(t) / t2=3 for matter domination; (2.36) 9 a(t) / t1=2 for radiation domination; (2.37) a(t) / e tp =3 for vacuum domination: (2.38) Notice that when wi = 􀀀1, the scale factor expands exponentially. A family of cosmological models that use such accelerated expansion in the very early stages of the universe has been used in order to solve serious problems linked with the standard Big Bang model and fall under the name of in ation. We will discuss these problems and the solution posed by the in ationary scenarios in Sec. 2.3. When more than one uid is present, a general expression for the cosmological time t as a function of the redshift z can be found by integrating Eq. (2.32), t = Z +1 z dz (1 + z)H(z) ; (2.39) we derived Eq. (2.39) setting our initial time t = 0 at redshift z = +1. The age of the universe t0 is given by Eq. (2.39) by setting z = 0, so t0 = Z +1 0 dz (1 + z)H(z) : (2.40) For a universe where only matter is present we have H0 t0 = 8>>< >>: 1 1􀀀 m 􀀀 m 2(1􀀀 m)3=2 ArcCosh 2 m 􀀀 1 ; for m < 1; 2 3 ; for m = 1; 􀀀1 m􀀀1 + m 2( m􀀀1)3=2 ArcCos 2 m 􀀀 1 ; for m > 1: (2.41) This equation, for m = 1 and with H0 = 70 km=s=Mpc yields t0 = 9:3 Gyr, which underestimates the e ective age of the universe by almost 35%. To obtain the correct result for t0, we need to include the dark energy contribution in Eq. (2.39); in fact, considering a at (k = 0) universe where matter m and vacuum energy 1 􀀀 m are present (with negligible radiation), we obtain H0 t0 = 1 3p1 􀀀 m ln 2(1 + p1 􀀀 m) m 􀀀 1 : (2.42) Eq.(2.42) with H0 = 70 km/s/Mpc and m = 0:25 gives the correct result for the age of the universe, t0 = 13:7 Gyr. In Sec. 2.6.3, we will discuss how current observations motivate a nonzero value of . 10 2.3 Cosmic in ation The standard Big Bang theory yields striking successes in explaining a large number of cosmological observations. However, in order for this model to actually be consistent, the universe has to emerge from the Big Bang with very speci c initial conditions, in order to match measurements of the quantities we observe at the present time. The most challenging problems that the standard Big Bang theory faces are the horizon, atness, and unwanted relics problems. The in ationary scenario was introduced in order to solve these speci c problems [102, 165, 78, 151, 7, 118], and it also proved to be a valid model in which the seed of local inhomogeneities form [137, 79, 87, 166, 18]. For reviews of the in ationary mechanism, see Refs. [120, 108]. 2.3.1 The atness problem If the initial value of the total energy density of the universe slightly di ers from the critical density, or j tot(t) 􀀀 1j 6= 0 in Eq. (2.24), any deviation from unity will eventually be ampli ed by the present time; alternatively, it is required that the universe be extremely at at early times, in order to explain the close-to- at geometry we observe today. To illustrate the atness problem in cosmology, we assume that the universe had an initial deviation from unity tot;i􀀀1. Indicating with tot = tot(t0) the value of tot(t) at the present time t0, we compute the deviation of tot from unity by considering Eqs. (2.24) in the form k = _a2 ( tot;i 􀀀 1): assuming radiation domination at all times a(t) t1=2, we obtain j tot;i 􀀀 1j = j tot 􀀀 1j tPl t0 ; (2.43) where tPl = 1=MPl = 5:3 10􀀀44 s is the Planck time and t0 = 13:7 Gyr is the age of the universe. Numerically it is j tot;i 􀀀 1j 10􀀀60 j tot 􀀀 1j: (2.44) Measurements of the anisotropies in the Cosmic Microwave Background Radiation (CMBR) by the WMAP7+BAO+SN data constraint j tot 􀀀 1j at 95% C.L. as [110] 􀀀0:0178 < tot 􀀀 1 < 0:0063; (2.45) we state that j tot􀀀1j . 0:01, implying that tot;i initially di ered from one by one part over 1062, any initial deviation of tot;i from unity being ampli ed at the present time according to Eq. (2.43) above. Summing up, we have that the primordial universe has to exit from 11 the Big Bang being extremely at in order to explain the tiny anisotropy in the density observed. In ation solves this problem because, during the in ationary period, the value on the right hand side of Eq. (2.24) drops exponentially due to the exponential increase of the scale factor. In ation has to last su ciently long in order to solve the atness problem: a parameter used to describe such a requirement is the number of e-folds Ne, de ned as the logarithm of the ratio of the scale factors at the time when in ation ends tend and at the beginning of in ation ti, Ne ln a(tend) a(tPl) = Z tend tPl H(t) dt: (2.46) Su cient in ation requires Ne & 60. 2.3.2 The horizon problem The horizon problem deals with the fact that a large degree of homogeneity is observed in the sky, although most of the patches have never been in causal contact before. For example, inhomogeneities in the CMBR are observed only at a T=T 10􀀀5 scale, even though the sky contains about a hundred patches that never interact causally in the standard picture. A very simpli ed view of this problem can be sketched by computing the age of the universe using Eq.(2.42) with H0 = 70 km/s/Mpc and m = 0:25, which yields t0 = 20 Gyr. This computation overestimates the e ective age of the universe t0 = 14:3 Gyr and implies that in the past, there was not enough time for the universe to be in causal contact. An early in ationary period would solve such a problem because in this scenario, an initial region that was in causal contact before in ation is then expanded and appears at the present time as distinct patches not causally connected. 2.3.3 Unwanted relics Generic Grand Uni cation Theories (GUT) predict heavy particles and topological defects to be copiously produced in the early universe (see Ref. [182] for a review), to the extent that these relics and the particles emitted by them would eventually dominate the expansion of the universe [22]. Since such relics are not observed in the present universe, it is believed that some other mechanisms have diluted their number to the present day. One viable explanation for this fact is that defects were produced before or during in ation, with monopoles being separated from each other (or \washed out") as the accelerated expansion progressed and their number density being consistently reduced to a safe cosmological value. 12 2.3.4 Small-scale structures Since the standard FRW model describes a homogeneous and isotropic universe, it does not account for the structures we observe today at various scales, like stars, galaxies, and clusters of galaxies. For this reason, perturbations in the FRW metric have been long studied, see Refs. [17, 106], and following the evolution of these perturbations, it is possible to explain the spectrum of inhomogeneities we observe in the CMBR. The initial power spectrum of perturbations that describes such variations in the densities will be reviewed in Sec. 2.4. 2.3.5 In ation building A period of in ation is de ned as a period of accelerated expansion: using Eq. (2.13) with p = w , it follows that it must be 1 + 3w < 0: (2.47) If the condition in Eq. (2.47) is satis ed, a solution to the Friedmann Eq. (2.12) when k = 0 and is constant is (see Eq. (2.38) above) a(t) = a(ti) Exp t s 8 3M2 Pl ! : (2.48) An exponential growth like in Eq. (2.48) can be obtained from the Friedmann equation whenever the density, and thus the Hubble, are constant. This situation can be achieved in a -dominated universe, the so-called Lemaitre models, or when we have some form of energy that is dominating the expansion rate of the universe whose energy density is constant. It is generally believed that a constant term might account for the accelerated expansion at the present time, see Sec. 2.6.3, whereas in order to give a microscopic explanation of primordial in ation, the early domination of an exotic form of energy has been invoked. It is usually postulated that the expansion rate of the universe during the in ation epoch is dominated by a hypothetical scalar eld called the in aton . In most in ation models, the in aton is initially stuck in a high energy state and it is the slow release of energy that governs the in ationary period. The dynamics of the in ation is governed by some at potential U( ), so that the energy density and pressure associated with this eld are = 1 2 _ 2 + U( ); (2.49) p = 1 2 _ 2 􀀀 U( ): (2.50) 13 If the eld does not develop enough kinetic energy, p = 􀀀 and w = 􀀀1, meeting the condition in Eq. (2.47) for in ation to occur. If the value of the potential U( ) remains approximately unchanged during the eld evolution, then the density and the pressure do not change as well during the in ationary period and Eq. (2.48) applies with = U( ). A precise mechanism for in ation is yet to be found, the major problems facing in aton models being new ne-tuning troubles concerning the self-interaction of the in aton eld itself. Moreover, the in aton eld is, in the simplest models, a scalar eld, and motivations from string theories justify the use of such elds, although fundamental scalar elds are yet to be discover experimentally. 2.4 Fluctuations during in ation As mentioned in Sec. 2.3.4, one attractive feature of in ation is that scalar and tensor perturbations emerge during this epoch: these features later evolve into uctuations in the primordial density and gravitational waves that might lead an imprint in the CMBR anisotropy and on the large scale structures [137, 79, 87, 166, 18]. Each uctuation is characterized by a power spectrum and a spectral index, respectively 2 R(k), ns for density perturbations and 2 T (k), nT for tensor perturbations. 2.4.1 The scalar power spectrum The spectrum of the adiabatic density perturbations generated by in ation is speci ed by the power spectrum 2 R(k) that depends mildly on the comoving wavenumber k accordingly to a spectral index ns and its tilt dns=d ln k as [111] 2 R(k) k3 PR(k) 2 2 = 2 R(k0) k k0 ns(k)􀀀1 : (2.51) The function 2 R(k) describes the contribution to the total variance of the primordial density perturbation due to perturbations at a given scale per logarithmic interval in k [109]. The WMAP collaboration reports the combined measurement from WMAP7+BAO+SN of 2 R(k0) at the reference wavenumber k = k0 = 0:002 Mpc􀀀1 [110], 2 R(k0) = (2:430 0:091) 10􀀀9; (2.52) where the uncertainty refers to a 68% likelihood interval. The RHS of Eq. (2.51) is evaluated when a given comoving wavelength crosses outside the Hubble radius during in ation, and the LHS when the same wavelength re-enters the horizon. In Eq. (2.51), we have used the 14 notation in Refs. [109, 110] for the density perturbations. Other authors use the symbol PR(k) for our 2 R(k) and R2 k for our PR(k), and might di er by factors of 2 2. From a theoretical computation that describes the uctuations in the in aton eld , the scalar power spectrum has the form 2 R(k) = H _ 2 hj j2i; (2.53) where _ is the time derivative of the in aton eld and hj j2i describes the variance of the uctuations in the in aton scalar eld, related to the spectrum of uctuations. The LHS of Eq. (2.53) is computed at the time at which the largest density perturbations on observable scales are produced, corresponding Ne e-foldings before the end of in ation. For any massless and nearly-massless elds, the theory of quantum uctuations predicts [79] hj j2iquantum = H 2 2 : (2.54) Since the Hubble rate H is approximately constant during in ation, see Sec. 2.3.5, we can substitute H in Eq. (2.54) with its value when in ation ends HI , with the index I standing for \In ation": hj j2iquantum = HI 2 2 : (2.55) The Hubble expansion rate at the end of in ation HI is bound by the WMAP measurements, as we will discuss in Sec. 2.4.4, and parametrizes the e ectiveness of in ation. Using Eqs. (2.53) and (2.55), the scalar power spectrum when quantum uctuations in the in aton eld dominate is 2 R(k0) = H2 I 2 _ 2 : (2.56) 2.4.2 The scalar spectral index The scalar spectral index ns describes the mild dependence of the scalar power spectrum on the wavenumber k, as in Eq. (2.51). We expand the spectral index around the reference scale k0 as ns(k) = ns + 1 2 ln k k0 ; (2.57) where ns ns(k0), and the spectral tilt is = dns(k) d ln k=k0 k=k0 : (2.58) Using Eq. (2.51), the scalar spectral index is [111] ns 􀀀 1 = @ @ ln k=k0 ln 2 R(k) 2 R(k0) : (2.59) 15 2.4.3 The tensor power spectrum Fluctuations in the gravitational wave eld are statistically described by a power spec- trum for tensor perturbations 2 h(k). Writing this spectrum in a similar fashion to 2 R(k0), we have 2 T (k) k3 PT (k) 2 2 = 2 T (k0) k k0 nT ; (2.60) where the tensor spectral index nT is assumed to be independent of k, because current measurement cannot constraint its scale dependence. WMAP does not constraint 2 T (k0) directly, but rather the tensor-to-scalar ratio r 2 T (k0) 2 R(k0) : (2.61) which qualitatively measures the amplitude of gravitational waves per density uctuations. The WMAP5+BAO+SN measurement constrains the tensor-to-scalar ratio as [110] r < 0:20 at 95% C.L. (2.62) 2.4.4 Bounds on the Hubble rate at the end of in ation We combine the results from the WMAP-7 plus BAO and SN in Eqs. (2.52) and (2.62) to obtain an upper bound on the spectrum of primordial gravitational waves, 2 h(k0) . 4:86 10􀀀10: (2.63) Expressing 2 h(k0) in terms of HI , 2 h(k0) = 2H2 I 2M2 Pl ; (2.64) leads to an upper bound on HI , HI < 6:0 1014 GeV: (2.65) A lower limit on HI comes from requiring the Universe to be radiation-dominated at T ' 4 MeV, so that primordial nucleosynthesis can take place [98, 99, 85]. Equating the highest temperature of the radiation TMAX (T2R HHIMPl)1=4; (2.66) to the smallest allowed reheating temperature TRH = 4MeV gives HI > H(TRH) = 7:2 10􀀀24 GeV: (2.67) 16 2.4.5 Exiting in ation After a certain period in which the universe experienced an exponential growth, a transition towards the standard cosmology occurred. In the literature, this transition period goes under the general name of \reheating", the designation referring to the idea that it was at this time that most of the particles and radiation that formed the primordial soup were created. Since the detailed mechanism behind in ation is still obscure, the theory of the reheating process is yet to be speci ed in its entirety. Generally speaking, in ation ends when the potential of the in aton eld is no longer at enough for the exponential solution to occur: the in aton eld starts rolling down and oscillates around a minimum of its potential U( ), with damped oscillations. The energy stored in the in aton eld is transferred through these damped oscillations into Standard Model particles and possibly other exotic particles, which make up the primordial soup [56, 1]. An alternative mechanism is the decay of the in aton through a broad parametric resonance into intermediary particles, which then decay into Standard Model particles [55, 176, 107]. In the standard cosmological scenario, the universe quickly becomes radiation-dominated, with Eq. (2.37) describing the growth of the scale factor with time, a(t) t1=2. 2.5 Thermal history of the universe Right after the end of in ation and the subsequent re-ionization, the early universe was lled with a hot plasma of Standard Model particles and possibly dark matter and other exotic particles and forms of energy. At such high temperature, most interaction rates were capable of keeping these constituents in thermal equilibrium; a speci c particle i would go out of the thermal equilibrium when its annihilation rate into other particles 􀀀i falls below the Hubble expansion rate H(T) at some temperature Tdec;i de ned via H(Tdec;i) 􀀀i: (2.68) From that moment on, the number density of such a relic is xed to its value at temperature Tdec;i. 2.5.1 Statistical mechanics at thermal equilibrium Because of the thermal equilibrium existing among particles participating in this primor- dial soup, we can use statistical tools for describing the properties of each species. Here we review these methods, following the treatments in Refs. [108, 29]. In statistical mechanics, 17 the properties of a species i of mass mi in a thermal bath at temperature T are described by a distribution function over the momentum p, fi(p) = Exp Ei 􀀀 i T 1 􀀀1 ; (2.69) where Ei = q p2 + m2i i is the total energy, i is the chemical potential, and the minus sign (plus sign) speci es bosons (fermions). An additional number that characterizes the species is the number gi of degrees of freedom, describing the possible number of polarization of the species. In terms of these quantities, the number density ni and energy density i are, respectively, ni = gi (2 )3 Z d3p fi(p); (2.70) and i = gi (2 )3 Z d3pEi fi(p): (2.71) The pressure of the i uid is obtained from the equation of state pi = wi i; (2.72) where wi is speci ed by the type of uid itself, see Sec. 2.2.2.1. These quantities can be readily computed in the nonrelativistic T mi and ultrarela- tivistic T mi cases. In the former case we obtain, for both boson and fermion particles ni = gi mi T 2 3 e􀀀mi=T ; for T mi; (2.73) and i = mi ni = gimi mi T 2 3 e􀀀mi=T ; for T mi: (2.74) In the ultra-relativistic case, di erent results are obtained for the two statistics, ni = ( (3) 2 gi T3; for T mi and Bose-Einstein statistics; 3 4 (3) 2 gi T3; for T mi and Fermi-Dirac statistics; (2.75) where (z) is the Riemann zeta function of argument z, and i = ( 2 30 gi T4; for T mi and Bose-Einstein statistics; 7 8 2 30 gi T3; for T mi and Fermi-Dirac statistics: (2.76) Notice that the number density in the Fermi-Dirac statistics di ers from the one in the Bose-Einstein statistics for a factor 3=4, and similarly the result for the energy density in the two cases di ers by a factor 7=8. 18 2.5.2 Conservation of the number and entropy densities We now quote some important results concerning the evolution of a system of particles in thermal equilibrium, remanding to Ref. [29] for further details. The second Friedmann Eq. (2.18) can be cast in the form d dt a3 p + T = 0; (2.77) where the expression between square brackets is identi ed with the total entropy of the universe within a volume a3 and = P i i. Introducing the entropy density s(T) = p + T ; (2.78) we have that the entropy density scales with the Hubble volume a3. Using the fact that for a relativistic bath of particles p = =3 and using the expressions for the various i in Eq. (2.76), we obtain s(T) = 2 2 45 g S(T) T3; (2.79) with the entropy degrees of freedom g S(T) at temperature T being de ned in a similar way as g (T) in Eq. (2.85), g S(T) X i2bosons gi Tdec;i T 4 + 3 4 X j2fermions gj Tdec;j T 4 : (2.80) For T & 1 MeV, we essentially have g S(T) g (T). From the conservation of the entropy in a comoving volume a3(T) and using Eq. (2.79), we derive the relation between the scale factor and temperature as g (T) T3 a3(T) = constant; (2.81) valid only when no release of entropy occurs. If there is no release of entropy and g S(T) is constant, the total number density is also conserved, dn dt = 0: (2.82) The two conditions for the conservation of the number and the entropy densities can be cast in a single expression [108], n s = 0: (2.83) 19 2.5.3 Application to the radiation-dominated universe We now turn our attention to a thermal bath of particles in cosmology. Considering an ensemble of relativistic particle species in thermal equilibrium, we evaluate the Hubble rate Hrad(T) from Eq. (2.19) using the expression for the energy density in Eq. (2.76) as H2 rad = 8 3MPl X i i = g (T) 8 3 90MPl T4; (2.84) where we de ned the number of relativistic degrees of freedom at temperature T for an ensemble of species i that decouple at temperature Tdec;i in Eq. (2.68) as g (T) X i2bosons gi Tdec;i T 4 + 7 8 X j2fermions gj Tdec;j T 4 : (2.85) Numerically, the function that gives the number relativistic degrees of freedom as a function of temperature for values of T around the QCD scale QCD can be approximated by a step function, g (T) = 8>< >: 61:75; for T & QCD; 10:75; for QCD & T & 4MeV; 3:36; for T . 4MeV: (2.86) When all standard model particles can be treated as relativistic, we have g (T) = 106:75, while the number of relativistic degrees of freedom in the Minimal Supersymmetric Standard Model (MSSM) is g (T) = 228:75. We simplify Eq. (2.84) by writing Hrad = r g (T) 8 3 90 T2 MPl = 1:66 p g (T) T2 MPl : (2.87) This expression describes the Hubble rate when the universe is dominated by radiation. 2.5.4 Matter-dominated universe As the universe cools down, relativistic particles lose momentum due to the redshift e ect and eventually become nonrelativistic. For a stable nonrelativistic particle, the number density scales with n a􀀀3 T3, with the latter relation coming from Eq. (2.81). Knowing the matter energy density at the present time M, the energy density at a time t is matter = M a0 a 3 ; (2.88) from which the Hubble rate for a matter-dominated universe scales as H(T) T3=2. 20 2.6 Content of the universe at the present time Even though we are still lacking a fundamental theory for in ation, the in ationary scenario has been embedded in the history of the cosmos due to its capability in solving all of the problems posed in Sec. 2.3 and make predictions on the amplitude of the seeds for inhomogeneities. This paradigm on the history of the universe explains the exceptionally at universe we live in, tot;0 1, as evidenced from distinct measurements on the CMBR, the baryon acoustic oscillations and the redshift of supernovae. These same measurements also point out that the expansion of the universe is accelerating at the present time, a fact that is in sharp contrast with the naive expectation that the universe be matter-dominated at the present time. Clearly, the speci c content of the universe is yet to be determined, although in the last decade we have been able to determine some general features of the di erent uids that make it up. Here, we review the most important components of the present total energy density. 2.6.1 Baryons In cosmology, the term \baryon" indicates the totality of the Standard Model species, and not only the color-neutral bound system made of three quarks in the acceptation of particle physics. The WMAP-7 data constrain the density of baryons today as b = 0:0458 0:0016: (2.89) 2.6.2 Dark matter It has long been known that baryons only account for a very small fraction of the present energy density. This conclusion was rst obtained from considering the rotation curves of outer objects in galaxies, which reveal that the average galactic mass not only consists of dust and gases, but also and for its most part of a nonluminous halo of unknown composition, hence dubbed dark matter. The majority of the dark matter observed has to be in the form of CDM, which means that this exotic component has to be nonrelativistic at the time of galaxy formation. The fact that dark matter is a nonrelativistic eld has been established with the rst results on the CMBR anisotropy from COBE. In the CDM theory, small structures clump and grow hierarchically \from the bottom up", forming larger structures. This scenario is opposite to the Hot Dark Matter (HDM) paradigm, in which larger structure form earlier and subsequently fragment, following a \top - down" evolution. The predictions of the 21 CDM model are in general agreement with the observations, whereas the HDM paradigm disagrees with large-scale structure observations. The WMAP 7-years data, once combined with the BAO and SNe data, yield the value CDM = 0:229 0:015: (2.90) for the totality of CDM observed in the present universe. Although the CDM paradigm explains current data and the evolution of large-scale structures, there are some major discrepancies with observation of other features within galaxies and clusters of galaxies. In particular, CDM models predict that the density distribution of dark matter halos be much more peaked than what is inferred by the rotation curves of galaxies, the so-called cuspy halo problem. Moreover, CDM models predict a large amount of low angular momentum dust, in contrast with observations. 2.6.3 Dark energy The dark matter and baryon components are not su cient to explain the atness observed, since the total abundance of nonrelativistic matter only accounts for about 26% of the total content of the Universe. Instead, what comes out of the measurements on the content of the universe reveals that a large fraction of the present energy density is due to the so-called dark energy, responsible for the current period of accelerated expansion. In some models, dark energy is identi ed with the constant appearing in the Friedmann Eq. (2.17) and which would lead to an accelerated expansion as discussed in Sec. 2.2.4. For anthropic reasons for this choice, see Ref. [191]. Another popular explanation for the dark energy introduces a new light scalar eld whose equation of state, similarly to the in aton eld , resembles that of a cosmological constant. Mechanisms of this latter type include quintessence [148, 193], or the landscape of string theory (see Ref. [170]). Evidence that the universe is experiencing a period of accelerated expansion comes from measuring the distances of type Ia supernovae (SNe). In fact, the lifespan of a SN is directly correlated with its luminosity, so that SNe can be used as standard candles to measure distances of neighboring stars to us via the \cosmic distance ladder" technique. Experiments reporting distances of SNe have shown that a nonzero cosmological term better ts data than a vanishing . For a at universe, the favored region of the parameter space has = 0:725 0:016: (2.91) CHAPTER 3 ELEMENTS OF QUANTUM FIELD THEORY AND QCD 3.1 Elements of group theory Group theory is a branch of mathematics devoted to the study of groups. A group G is a mathematical structure consisting of a set V together with an operation that acts on a pair of elements in V . For this reason, the particular group structure is often explicitly indicated by writing G = G(V; ). In order for G to qualify as a group, the operation must satisfy the following conditions: 1. Closure: 8 x; y 2 V; x y 2 V ; 2. Associativity: 8 x; y; z 2 V; (x y) z = x (y z); 3. Identity: 9! e x e = e x = e 8x 2 V ; 4. Invertibility: 8x 2 V 9! x􀀀1 2 V x x􀀀1 = x􀀀1 x = e. If further the commutativity relation holds, the group G(V; ) is called Abelian. Of central importance in the context of Quantum Field Theory (QFT) is the concept of the compact Lie group. The adjective compact refers to the compactness of the topology of the group, namely the topological space associated to the group G is compact; the notion of the Lie group refers to the smoothness of the di erentiable manifold associated to G. Due to the importance of the subject, the theory of compact Lie groups is particularly well-developed. The properties of a compact Lie group assure that the group G contains elements x( ) that are arbitrarily close to the identity e for small values of the group parameter . For continuous groups such as Lie groups, we can write x( ) = e + i a Ta + O( 2); (3.1) 23 where the index a runs over the dimension of the group and the Ta's are called the generators of the group. These generators satisfy the commutation relation [Ta; Tb] = i fabc Tc; (3.2) and the coe cients fabc are called the structure constants of the group. If the set of generators cannot be further divided into two subsets of mutually commuting generators, the generated group is called simple. For compact Lie groups that are simple, a complete classi cation of these groups is known [5], and it is indicated as the Cartan classi cation system. Given two N-dimensional complex vectors u and v, a general transformation is, respectively, a 1. unitary transformations if it preserves the inner product u a va. A unitary transfor- mation belongs to the so-called special unitary group in N dimensions, SU(N). The generators ta of this group are N N traceless Hermitian matrices, and there are N2 􀀀 1 of these generators. 2. orthogonal transformations if it preserves the inner product ua ab vb. An orthogonal transformation belongs to the so-called orthogonal group in N dimensions, SO(N). The elements of this group are N N orthogonal matrices of determinant one, and there are N(N 􀀀 1)=2 generators. 3. symplectic transformations if it preserves the inner product ua Eab vb, with Eab being the symplectic matrix. A symplectic transformation belongs to the so-called symplec- tic group in N dimensions (N even), Sp(N). There exist N(N + 1)=2 generators. There also exist ve exceptional Lie groups that are simple but do not belong to the above classi cation. These exceptional groups are of interest in uni ed theories, but they will not be treated in this thesis. Summing up, the dimension of the simple Lie groups, not including the exceptional groups, can be written as d(G) = 8>< >: N2 􀀀 1 for SU(N); N(N 􀀀 1)=2 for SO(N); N(N + 1)=2 for Sp(N): (3.3) In the following, we will specify the relations in Eq. (3.2) for some groups of interest in particle physics, namely SU(2), SO(3), SU(3), and the homogeneous Lorentz group SO(1,3). 24 3.2 Lagrangian functions for scalar and spinor elds 3.2.1 The Lorentz group As discussed in Sec. 3.1, a group can be de ned through the algebra of its generators. For the case of the Lorentz group SO(1,3), we have that the six generators for the angular momentum Ji and for the boost Ki, with i 2 f1; 2; 3g, satisfy the commutation relations: [Ji; Jj ] = i ijk Jk; (3.4) [Ki;Kj ] = 􀀀i ijk Jk; (3.5) [Ji;Kj ] = i ijk Kk: (3.6) We see from Eq. (3.4) that the three Ji form a closed subalgebra of the Lorentz group, more precisely the algebra that de nes the SU(2) or SO(3) Lie groups. Representations for both these groups are three hermitian matrices of dimension 2n + 1, with n assuming integer values when referring to a representation of the SO(3) group, or n taking half-integer values when representing the SU(2) group. Nevertheless, each speci c value of n in physics refers to a di erent particle that is associated with this group. For each di erent choice of n, the wave function for such a particle will transform di erently when a Lorentz transformation is applied. We will focus only on the cases n = 0 (boson) and n = 1=2 (Weyl spinor), with other values of n not being treated here. 3.2.2 The Lorentz transformation In the following, we denote with the matrix describing the homogeneous linear transformation of a four-vector x , x ! x0 = x : (3.7) Imposing the invariance of the pseudo-length of the four-vector, x2 = x x = x x = x20 􀀀 x21 􀀀 x22 􀀀 x23 ; (3.8) we nd the relation = : (3.9) A Lorentz transformation, represented by the matrix , belongs to the Lorentz group SO(1,3), and thus the product of two Lorentz matrices is a Lorentz matrix; the identity belongs to the group and a speci c transformation has an inverse 􀀀1. 25 Taking the determinant of Eq. (3.9), we see that Det = 1; (3.10) so that Lorentz transformations are called proper if Det = +1 and improper if Det = 􀀀1. A further, independent subdivision can be made by noticing that Eq. (3.9) implies that 􀀀 0 0 2 = 1 + 􀀀 i 0 2 ; (3.11) so that a transformation is orthochronous when 0 0 1, and nonorthochronous when 0 0 􀀀1. Transformations that are both proper and orthochronous are connected to the identity; in particular, we can write an in nitesimal transformation as = + ; (3.12) where the matrix of in nitesimal quantities is forced by Eq. (3.9) to be antisymmetric: = 􀀀 : (3.13) These are six parameters, corresponding to the three spatial rotations and the three boosts. Explicitly, a Lorentz transformation that rotates the system by an in nitesimal angle about the direction ^n and gives an in nitesimal boost along ^n with rapidity to the system has parameters ij = 􀀀 ijk ^nj ; i0 = ^ni : (3.14) 3.2.3 Equation of motion for a scalar eld We consider a generic complex scalar eld = (x). The term \scalar" indicates that, when a Lorentz transformation is applied to the coordinate four-vector x as in Eq. (3.7), the scalar eld transforms as (x) ! 0(x) = ( 􀀀1 x0): (3.15) The Lagrangian density for a classical scalar eld moving in a potential U( ) is L = @ @ 􀀀 U( ); (3.16) and the equation of motion is @ @ 􀀀 @U( ) @ = 0: (3.17) 26 In particular, when the potential is quadratic in the eld, U( ) = m2 j j2; (3.18) Eq. (3.17) is cast as the Klein-Gordon equation 􀀀 @ @ 􀀀 m2 (x) = 0: (3.19) A similar equation holds for the complex conjugate eld (x). 3.2.4 Equation of motion for a spinor eld 3.2.4.1 Representation of dimension two for the Lorentz transformation We focus again on Eqs. (3.4)-(3.6) that give the algebra of the Lorentz group in terms of the generators. We de ne two sets of linear combinations of these generators, Ni = 1 2 (Ji 􀀀 iKi) ; (3.20) and Ny i = 1 2 (Ji + iKi) ; (3.21) so that the Lorentz algebra is rewritten as [Ni;Nj ] = i ijk Nk; (3.22) [Ny i ;Ny j ] = 􀀀i ijk Ny k; (3.23) [Ni;Ny j ] = 0: (3.24) This means that the algebra of each set Ni and Ny i is separately closed, and the Lorentz algebra consists of two sets of SU(2) algebras, whose generators are hermitian conjugate. For this reason, a Lorentz transformation is speci ed by two indices n1, n2, and the representation has dimension (2n1+1) (2n2+1). The convention for Weyl spinors is that the choice (n1 = 1=2; n2 = 0) leads to a left-handed transformation, while (n1 = 1=2; n2 = 0) yields a right-handed transformation. These are two distinct representations of the Lorentz group with dimension two. 27 3.2.4.2 Weyl spinors Because of the two di erent representations of the Lorentz group with the same dimen- sion of two, two distinct type of spinors are possible, called the left-handed and right-handed Weyl spinors. These two representations are distinguished by the action of the boost transformations Ki, which in this representation are K i = i 2 i; (3.25) with the plus (minus) sign acting on the left (right) handed Weyl spinor. For both spinors, the generators of the angular momentum are given by Ji = 1 2 i: (3.26) The nonzero entries of the left-handed generator T L are de ned as Tij L = ijk Jk = 1 2 ijk k; and Tk0 L = Kk+ = i 2 i: (3.27) We now discuss the action of a generic Lorentz transformation on these spinors, considering rst a left-handed Weyl spinor. In the following, we write the left-handed spinor as , with the spinor index raised and lowered by the spinor metric tensor . Under the transformation in Eq. (3.7), the left-handed Weyl spinor transforms as (x) ! 0 (x) = L ( ) ( 􀀀1x): (3.28) Here, L( ) is a matrix in the (1/2, 0) representation of the Lorentz group: for the in nites- imal Lorentz transformations in Eq. (3.12), it is L (1 + ) = + i 2 􀀀 T L ; (3.29) where we have indicated explicitly the spinor structure of the generator TL, that is, the entries of the speci c matrix with given and . The right-handed Weyl spinor is de ned as y _ = [ (x)]y: (3.30) A dotted index _ is used for the right-handed spinor, in order to distinguish it from the undotted index for the left-handed spinor. The metric tensor used for raising and 28 lowering dotted indices is _ _ . Under a Lorentz transformation, the right-handed Weyl spinor transforms as _ (x) ! 0 _ (x) = R _ _ ( ) _ ( 􀀀1x); (3.31) where R( ) is a matrix in the (0,1/2) representation of the Lorentz group. For an in nites- imal Lorentz transformation, we have R _ _ (1 + ) = _ _ + i 2 􀀀 T R _ _ ; (3.32) where T R is a set of matrices, linked to the set of T L matrices by 􀀀 T R _ _ = 􀀀 h􀀀 T L i : (3.33) 3.2.4.3 Equation of motion for the Weyl spinor Under speci c conditions, it is possible to study the equation of motion for just one spinor, say the left-handed spinor (x). The Lagrangian is [164] LL = i y _ ( ) _ @ + m 2 + y _ y _ ; (3.34) with the set of matrices = (I2; ); = (I2;􀀀 ): (3.35) From Eq. (3.34), we derive two equations of motion, one for and one for y _ as i ( ) _ @ y _ 􀀀 m = 0; (3.36) and i ( ) _ @ 􀀀 m y _ = 0: (3.37) These last two equations can be combined in the matrix form 􀀀m i ( ) _ @ i ( ) _ @ 􀀀m _ _ ! y _ = 0: (3.38) 3.2.5 Equation of motion for the Majorana spinor We de ne the four-spinor M = y _ ; (3.39) which is known in the literature as the Majorana spinor. The key element characterizing the Majorana spinor is that it is invariant under the charge conjugation operation, that is, 29 when we invert the electric charge of the particle, we obtain the same spinor. In order for this to be consistent, the electric charge of a Majorana eld must be zero. To see this in more detail, we introduce two operations that act on generic four-spinors . A bar over a spinor indicates the operation = y ; (3.40) with the matrix = 0 _ _ 0 : (3.41) For the Majorana eld, M = ; y _ : (3.42) We also introduce the charge conjugation operator ^ C, whose matrix representation in four dimensions reads C = 0 0 _ _ : (3.43) Given a generic four-spinor , the charge-conjugated spinor is C = C T : (3.44) For a Majorana eld, we obtain indeed C M = M. Examples of Majorana particles are the chargeless supersymmetric partners of the Higgs, Z0 and photon. A question mark is still posed on the nature of the neutrino, which may also be described by a Majorana eld. De ning the Dirac matrices in the chiral representation = 0 0 ; (3.45) we have the Dirac equation for a Majorana eld (i @ 􀀀 m) M = 0: (3.46) This equation of motion can be derived from the Lagrangian LM = i 2 T M C @ M 􀀀 1 2 m T M C M; (3.47) 30 3.2.6 Equation of motion for the Dirac spinor In order to describe spin one-half particles that possess charge, such as electrons, we consider a four-spinor in which two di erent and unrelated Weyl spinors , y _ appear. We construct the Dirac four-spinor as D = y _ : (3.48) Under charge conjugation, the Dirac eld does not transform into itself; in fact we have C D = y _ 6= D; (3.49) so that nonzero values of the electric charge are allowed. The Dirac eld satis es the equation 􀀀m i ( ) _ @ i ( ) _ @ 􀀀m _ _ ! y _ = 0; (3.50) which is put in the compact form (i @ 􀀀 m) D = 0: (3.51) The fact that the Dirac eld allows us to describe particles with a nonzero charge can be seen from the Dirac Lagrangian, LD = i D @ D 􀀀 m D D; (3.52) which is invariant under a U(1) transformation of parameter !, D ! ei! D; D ! e􀀀i! D: (3.53) 3.3 Application of group theory to QFT We now turn our attention to gauge theories in Quantum Field Theory (QFT), focusing on Dirac elds. For this reason, we drop the index D and simply denote with a collection of one or more Dirac elds. The Dirac Lagrangian in Eq. (3.52), describing a free eld theory for a massless complex-valued collection of Dirac elds, reads L = i (x) @ (x): (3.54) We assume that the collection of elds (x) is invariant under a global symmetry that generalizes Eq. (3.53), (x) ! U( a) (x); (3.55) 31 where the transformation of the eld can be written in terms of a set of constant parameter a as U( ) = exp (i a ta) : (3.56) For example, if (x) represents a single Dirac eld, then the Lagrangian in Eq. (3.54) is invariant under the U(1) transformation in Eq. (3.53), while if (x) represents a doublet of Dirac elds then the Lagrangian in Eq. (3.54) is invariant under a SU(2) transformation, (x) ! exp i 2 (x); (3.57) where = f 1; 2; 3g are the usual 2 2 -matrices. In QFT, a general recipe is adopted in order to include the interaction of the Dirac eld (x) with a gauge eld Aa . We rst promote the symmetry from global to local by making the parameters depending on the space-time coordinate x, ! (x): (3.58) The transformation of the Dirac eld (x) under this local symmetry reads (x) ! U( a(x)) (x) = exp (i a(x) ta) (x): (3.59) This local transformation resembles that in Eq. (3.55), the main di erence being that the parameters a(x) now depend on the speci c space-time point x at which the transfor- mation takes place. This justi es the use of the adjective \local" to describe such kinds of transformation. In order for this transformation to be imposed on the Lagrangian in Eq. (3.54), we de ne a covariant derivative D = @ 􀀀 ig Aa ta; (3.60) where g is the coupling of the Dirac eld to the gauge eld. The Lagrangian in Eq. (3.54) then reads L = (x) i D (x) = (x) i (@ 􀀀 ig Aa ta) (x): (3.61) We see that the introduction of the covariant derivative introduces in the Lagrangian above the interaction term Lint = g (x) Aa ta (x): (3.62) The requirement that the Lagrangian in Eq. (3.61) be invariant with respect to the local transformation in Eq. (3.59) imposes the gauge eld Aa to transform as Aa ! Aa + 1 g @ a ta + fabc Ab a c; (3.63) 32 and fabc are the structure constants de ned in Eq. (3.2) for the group G with generators ta. Finally, we can write the complete Lagrangian for a Dirac multiplet (x) of mass matrix M and belonging to an irreducible representation of a gauge group G, known as the Yang- Mills Lagrangian, as LYM = (x) (i D 􀀀M) (x) 􀀀 1 4 F a Fa : (3.64) In the Yang-Mills Lagrangian, the index a is intended summed over the generators of the group G, while we have introduced the eld strength Fa = @ Aa 􀀀 @ Aa + g fabc Ab Ac : (3.65) In Eq. (3.64), we have included the mass term M which is manifestly invariant under the symmetry in Eq. (3.59), and the term LGauge = 􀀀 1 4 F a Fa ; (3.66) that describes the self-interaction and the dynamics of the gauge eld. 3.4 Quantum Chromodynamics We apply the tools developed in the last sections to describe the theory of Quantum Chromodynamics (QCD), and provide a theory to describes the mechanism of the strong interactions. The QCD theory is nonabelian since gluons carry color charge causing them to interact with each other in a more complicated structure: in fact, the underlying gauge group for QCD is the nonabelian group G = SU(3). As we will see, a yet-to-be-solved problem arises in QCD, the so-called strong Charge-Parity (CP) problem. In fact, as in the theory of weak interactions, the CP symmetry is expected to be violated in strong interactions, whereas experiments show that the CP symmetry is preserved by strong interactions to a high degree of precision. The lack of a CP-violating term then requires a ne tuning of QCD parameters in its mathematical description. 3.4.1 The QCD Lagrangian We denote the QCD gauge eld (the gluon tensor eld) living in the adjoint represen- tation of the group as Ga , where a 2 f1; :::; 8g. The Lagrangian obtained using the SU(3) gauge group with Nf quark (the QCD Lagrangian) is [143] LQCD = q (i D 􀀀M) q 􀀀 1 4 G a Ga ; (3.67) 33 where q is the collection of quark spinor elds, M is the quark mass matrix [138] and Ga is the gluon tensor eld, which can be written in terms of the QCD eld strength Aa and of the structure constant fabc SU(3) of the SU(3) group as Ga = @ Aa 􀀀 @ Aa + g fabc SU(3) Ab Ac : (3.68) The covariant derivative D appearing in Eq. (3.67) is de ned in terms of the Gell-Mann matrices ta SU(3) by D = @ 􀀀 i g Aa ta SU(3): (3.69) In the limit of vanishing quark masses M ! 0, the QCD Lagrangian is invariant under the global transformation UL(Nf ) UR(Nf ). However, such symmetry implies that hadrons come in doublets [144], which is not seen experimentally. In fact, it turns out that the axial part of the chiral symmetry is spontaneously broken into the subgroup SU(Nf )A U(1)A. Because of the symmetry breaking, nearly massless Goldstone bosons arise: these bosons are experimentally identi ed with the pseudo-scalar octet formed by the u, d, and s quarks and explain the SUA(Nf ) breaking. However, there exists no candidate in the particle spectrum that would play the role of the Goldstone boson associated with the spontaneous breakdown of U(1)A. In fact, such a particle should be a light pseudo-scalar boson of mass ma < p3m : the latter requirement excludes the 0 meson which would otherwise be a suitable candidate having the proper quantum numbers. The absence of this light pseudo-scalar particle is known as the U(1)A problem [189]. 3.4.2 Solving the U(1)A problem As proposed by t'Hooft [172, 173], the U(1)A problem might be solved by adding a term to the QCD Lagrangian in Eq. (3.67) that explicitly breaks the U(1)A symmetry, LQCD ! LQCD + L~ ; (3.70) with L~ = g2 32 2 ~ G a ~G a : (3.71) Here, we have de ned the dual of the eld strength tensor ~G a = 1 2 G a; (3.72) with the Levi-Civita completely antisymmetric tensor having 0000 = 1, g is a coupling constant, and ~ is an observable [91, 33] that might take values form zero to 2 . The origin 34 of the variable ~ in the QCD Lagrangian can be explained by the following argument. The classical gluon eld equations admit an anti-instanton solution, which satis es the antiduality condition G = ~G ; (3.73) and which has an integer index n = 1 32 2 Z d4xGa ~G a: (3.74) The angular variable parametrizes the linear combination of di erent jni-vacua in the the- ory corresponding to di erent values of the integer n, with the particular linear combination known as the -vacuum, j~ i = X+1 n=􀀀1 e􀀀in~ jni: (3.75) Here, every value of n characterizes the winding number of the U(1)A symmetry in Eq. (3.74). Although the term in Eq. (3.71) is a surface term, it is possible to show that if the U(1)A problem is solved and none of the quark is massless, a nonzero value of ~ implies that the CP symmetry is broken [157]. Thus, QCD physics depends on the value of ~ , or better, when electroweak interactions are included, on the combination = ~ + weak; (3.76) with the extra term arising in the QCD Lagrangian of the form L = g2 32 2 weak G a ~G a : (3.77) The parameter weak is expressed in terms of the quark mass matrix M as weak = arg (DetM) : (3.78) Eventually, the QCD Lagrangian including the 'tHooft and electroweak interaction terms reads LQCD+ = LQCD + L = LQCD + g2 32 2 G a ~G a : (3.79) 35 3.4.3 The strong CP problem As we discussed in Sec. 3.4.2, it is expected that the CP symmetry is violated by strong interactions due to the CP-breaking term L . However, experimentally no violation of the CP symmetry is observed, and the strong interactions preserve this symmetry to a high degree of accuracy. This problem in merging theoretical motivations and experimental observations is known in the literature as the strong CP problem, which can be restated by asking why strong interactions do not violate the CP symmetry when CP violation is not forbidden in the theory. The most stringent constrain on the violation of the CP symmetry by the strong interaction comes from the measurement of the electric dipole moment of the neutron, de ned as dN emud m2 N ; (3.80) where e is the absolute value of the electron charge, mN is the neutron mass and mud is given in terms of the masses of the up and down quarks mu, md as mud = mumd mu + md : (3.81) The latest experimental bound on the neutron electric dipole moment is [15] jdNj < 2:9 10􀀀26 e cm at 90% C.L.; (3.82) while we take the value of dN from the review by Kim and Carosi [104], dN = 4:5 10􀀀15 e cm: (3.83) Eqs. (3.82) and (3.83) imply the constraint j j < 0:7 10􀀀11: (3.84) From the de nition of in Eq. (3.76), we see that this quantity is the sum of the two terms ~ and weak, whose physical origin is completely unrelated; it is thus a mystery why two quantities which are naturally of order one have to cancel out in a sum with such a high degree of accuracy as in Eq. (3.84). 36 3.4.4 Possible solutions to the strong CP problem As of today, three main solutions to the strong CP problem have been proposed. These are, respectively: calculable , massless up quark, the axion particle. As the axion is the most favorable solution and the subject of this thesis, we will brie y review here only the rst two cases and concentrate on the axion throughout the subsequent chapters. 3.4.4.1 Calculable The underlying idea of the proposed calculable is to impose CP invariance in the QCD Lagrangian, setting ~ = 0. Taking into account that the only source of CP violation comes from weak interactions, as explicit in phenomena like the neutral K-meson oscillations and the B-meson decay, the term weak is still nonzero and can be calculable in an underlying theory, with a small value constrained by observations as in Eq. (3.84). An example of such a model is the Barr-Nelson CP violating model, in which heavy singlet quarks are introduced and whose vacuum expectation value lies much above the electroweak scale. These quarks mix with the ordinary light quarks after symmetry breaking, so that at low energy they can be integrated out and the light quarks acquire the required amount of CP symmetry through the Cabibbo-Kobayashi-Maskawa mixing. In this theory, the condition arg(DetM) = 0 is imposed at tree level. A nonzero value of weak arises through higher-loop corrections, weak loop, with the ne structure constant loop bound by the stringent bound in Eq. (3.84). 3.4.4.2 Massless up quark If the lightest quark (the up quark) is massless, it is possible to eliminate by a rotation of the quark elds q, q ! Exp i 2 5 q; (3.85) so that is no longer an observable. In fact, the QCD Lagrangian in Eq. (3.79) transforms under the rotation above as LQCD+ ! LQCD; (3.86) making the term L disappear. The massless up quark possibility has been ruled out by measurements, as quoted in Eq. (4.10) below that states mu=md zd = 0:568 0:042. CHAPTER 4 THE AXION 4.1 The axion as a solution to the strong CP problem The third and the most compelling solution to the strong CP problem discussed in Secs. 3.4.3 and 3.4.4 introduce an additional chiral symmetry that promotes the parameter to a dynamical eld: the dynamics of this new eld naturally relaxes its expectation values towards arbitrarily small values, thus eliminating the CP problem. This mechanism is achieved by introducing a new global, chiral symmetry U(1)PQ which is spontaneously broken at an unknown energy scale fa, known as the PQ energy scale. The axion is the Goldstone boson resulted from such symmetry breaking. The Lagrangian term resulting after the spontaneous symmetry breaking of the U(1)PQ symmetry reads La = 􀀀 1 2 (@ a) (@ a) + g2 32 2 a fa=N G a ~G a ; (4.1) where a a(x) is the axion eld and N = P f Xf is a new parameter called the PQ color anomaly, de ned as the sum of the PQ charges Xf over the fermions in the theory. In Eq. (4.1), the rst term represents the kinetic term, while the second term describes the interaction of the axion with the gluon eld. When considering the Lagrangian LQCD+ +a = LQCD + L + La; (4.2) an e ective potential Ve (a) for the axion appears, whose minimum is reached when ha(x) + fa N i = 0: (4.3) The expectation value of the axion eld in La at its minimum cancels out the term in L , thus eliminating the CP-violating term. Since we can shift the axion eld by a constant amount without changing the physics, we de ne the (axion) misalignment angle = + a fa=N : (4.4) 38 Summing up, with the PQ solution to the strong CP problem, the free parameter has been traded for a dynamical eld a(x) that evolves to its CP-conserving minimum = 0 through the spontaneous breaking of a U(1) symmetry. Essentially, can be seen as the phase of a new complex scalar eld a = j aj ei ; (4.5) that takes a nonzero expectation value at the energy scale fa=N. The axion potential thus has the usual Mexican hat form VPQ( ) = 4 j aj2 􀀀 fa N 2 !2 + m2 a(T) f2 a (1 􀀀 cos ); (4.6) where is a coupling constant. Here, the second term in Eq. (4.6) comes from nonpertur- bative QCD e ects associated with instantons [77], that break the U(1)PQ symmetry down to a Z(N) discrete subgroup [159]. As we will see later in Sec. 4.2.2, the presence of this second term justi es the fact that the axion mass ma(T) depends on the temperature of the plasma considered [181]. After the spontaneous breaking of the PQ symmetry, we are thus left with the residual potential (but see Refs. [62, 64, 63] for di erent potentials) V ( ) = m2 a(T) f2 a (1 􀀀 cos ): (4.7) This potential is approximately quadratic for small values of , but quickly shows deviations from a pure quadratic potential for 1. 4.2 On the mass of the axion 4.2.1 Axion mass at zero temperature We can expand the term L + La around the minimum in Eq. (4.3) to obtain the value of the axion mass as m2 a = 􀀀 g2 32 2 N fa @ @a D G a ~G a E ha(x)i=􀀀fa =N : (4.8) The axion mass has been computed in Ref. [190] including the axion mixing with the other neutral pions through the axion-gluon-gluon coupling, and is written in terms of the quark ratios zd = mu=md, zs = mu=ms and of the 0 mass and decay constant m 0 and f 0 as ma = f 0 m 0 fa=N r zd (1 + zd)(1 + zd + zs) ; (4.9) 39 Using the experimental values [116] zd = 0:568 0:042; zs = 0:029 0:003; (4.10) and the values [9] m 0 = (134:9766 0:0006) MeV; f 0 = 1 p2 (130 5) MeV; (4.11) we obtain the numerical value ma = (5:91 0:53) eV 1 fa;12=N : (4.12) Here and in the following, a number y indexing some quantity with units of energy indicates that such quantity has been divided by 10y GeV: for example, in the equation above, it is fa;12 = fa=1012 GeV. The axion mass in Eq. (4.12) is exact in the context of the simplest hadronic axion model [103, 157] (see Sec. 4.4.2.1), where Standard Model fermions do not possess a PQ charge, and axions only interact with an exotic heavy quark. Above the QCD phase transition nonperturbative e ect arise, giving the axion a temperature-dependent mass as we now discuss. 4.2.2 Finite temperature e ects In a quark-gluon plasma of nite temperature T, nonperturbative e ects modify the properties of the components of the plasma itself. Nonperturbative e ects in QCD have been studied via lattice simulations, see ex. Ref. [38], or phenomenology, see ex. Refs. [50, 158]. For axions, the mass acquires a temperature dependence at su ciently high T due to the instanton e ects: In the high temperature limit, T QCD, the axion mass is given by an integral over noninteracting instantons of all sizes [77] as (see also Refs. [178, 60]) m2 a(T) ^c 33 􀀀 2Nf 6 6 ^ Nf 4􀀀Nf QCD (fa=N)2 QCD T 7+ Nf 3 Det (M) I(T); (4.13) with I(T) = Z +1 0 dx x6+ Nf 3 ln T QCD x 6􀀀^a e􀀀f(x); (4.14) f(x) = 2 3 (6 + Nf ) x2 + 9 􀀀 Nf 6 " 12 ^ 􀀀 1 + ^ ( x)􀀀3=2 8 􀀀 ln 1 + 2 3 x2 # ; (4.15) and the numerical values ^c = 0:130078, ^ = 1:33876, ^ = 0:01289764, ^ = 0:15858 and ^a = (153 􀀀 19Nf )=(33 􀀀 2Nf ). The parameter Nf gives the number of quarks that are 40 relativistic at temperature T, that is, whose mass is . T. It is worth stressing here that, using di erent computation techniques, various authors have obtained di erent results for ma(T). For later convenience, we parametrize the general expression for the temperature- dependent mass with a general broken-law function, ma(T) = ma ( a 􀀀 QCD T c (1 􀀀 ln QCD T )d; T & QCD; 1; T . QCD; (4.16) with the parameter c = 7=2 + Nf =6 from Eq. (4.13). Before showing our computation for the parameters a and d, we will brie y review the literature on the subject for the nontrivial case T & QCD. Earlier work on the temperature-dependent axion mass are in Refs. [145, 2, 53], where the authors use the method by Gross, Pisarski, and Ya e in Ref. [77] in the dilute gas approximation. Preskill et al. [145] obtain, for the case Nf = 3, the expression ma(T) = 2 10􀀀2 1=2 QCD fa p Det (M) QCD T 4 9 ln T QCD 3 ; (4.17) which we put in the form of the rst line in Eq. (4.16) with QCD = 200 MeV and Det (M) = 1000 MeV3 around T 4 QCD [145] as ma(T) = apreskillma QCD T 4 ; (4.18) with apreskill = 0:035. Sikivie [161] quotes the general result from Refs. [145, 2, 53] as ma(T) = 4 10􀀀9 eV 1012 GeV fa GeV T 4 : (4.19) We notice that this expression can be written as ma(T) = asikiviema QCD T 4 ; (4.20) with asikivie = 0:417= 4 0:2 and 0:2 = QCD=200MeV. Turner [178] uses the tools developed in Ref. [77] to compute ma(T) in the case T QCD and gives the general expression ma(T) = ma a b 0:2 QCD T c 1 􀀀 ln QCD T d ; (4.21) where ma ma(T = 0) is given in Eq. (4.12). Table. 4.1 shows the dependence of the parameters a, b, c, and d on the number Nf of relativistic quarks at the temperature T1 at which the axion eld starts oscillating, see Eq. (5.15) 41 Table 4.1. Values of the parameters a, b, c, and d appearing in Eq. (4.21), as a function of the number of relativistic degrees of freedom Nf . Nf a b c d 1 0.277 3/2 3.67 0.84 2 0.0349 1 3.83 1.02 3 0.0256 1/2 4.0 1.22 4 0.0421 0 4.17 1.46 5 0.118 -1/2 4.33 1.74 6 0.974 -1 4.5 2.07 The exact value of Nf is not well determined, because the value of T1 is of the order of the current mass of the charm quark, mc 1 GeV, so it is possible for it to be Nf = 3 or Nf = 4. Because of this uncertainty, Ref. [178] considers the function ma(T) = 7:7 10􀀀2 0:5ma QCD T 3:7 0:1 ; (4.22) where the uncertainties interpolate between the various values of the parameters in Table 4.1 for di erent plausible values of Nf . DeGrand et al. [49], using the formula in Ref. [77], quote an expression for the axion mass valid for T QCD, as ma(T) = 15 2 QCD fa s Det (M) QCD QCD T 4 ln T QCD 3 : (4.23) This is basically the same expression found by Preskill in Eq. (4.18). Bae, Huh, and Kim [14] improved the results by Turner by considering updated values of current quark masses and the e ects of the QCD phase transition. They parametrized the axion mass with m2 a(T) = inst GeV4 f2 a (T=GeV) : (4.24) For the parameters inst and , the authors nd a QCD dependence like in Table 4.2 Wantz and Shellard [187, 188], within the interacting instanton liquid model [153], obtained a numerical expression for the temperature dependence of ma(T) valid at all temperatures; in particular, they were able to follow the ma(T) function around the QCD scale QCD, where analytic methods break. The analytic t in Refs. [187, 188] reports, for T & QCD, m2 a(T) = a 4 QCD f2 a (T= QCD) ; (4.25) with a = 1:68 10􀀀7 and = 6:68. 42 Table 4.2. Values of the parameters inst and in Eq. (4.24) for di erent values of QCD. From Ref. [14] QCD inst=10􀀀12 320 0.9967 6.967 380 3.964 6.878 440 12.74 6.789 Here, we consider the case Nf = 3, because, as we will show later, the axion eld starts oscillating at a temperature T1 that lies below the GeV; with only three quarks being relativistic at T1 we have Nf = 3 and thus, from Eq. (4.13) and from Table 4.1, the axion mass decreases with T as T􀀀4 to the leading order. For this reason, we set c = 4 in Eq. (4.16). To compute the parameters a and d, we proceed as follows. Based on the latest available data for the masses of the quarks u, d, s, which are the relativistic quarks at temperature T1, we write the determinant of the quark mass matrix as Det (M) = mumdms = m3 u zd zs = (1:64 0:29) 103 MeV3 mu 3MeV 3 ; (4.26) where zd and zs are given in Eq. (4.10) and mu lies in the range (1.7 - 3.3) MeV [138]: here we take the value mu = 3 MeV. To evaluate ma(T) we use Eq. (4.13) with Nf = 3. We are particularly interested in evaluating the function I(T), whose plot is shown in Fig. 4.1. Since the integral in the function I(T) in Eq. (4.13) depends on the temperature, we t the numerical solution of such function and obtain I(T) 5:84 10􀀀6 1 + ln T QCD 2:41 : (4.27) Using this t for I(T) with the values of the parameters as above and Nf = 3, we nd for the temperature-dependent axion mass in Eq. (4.13) the expression ma(T) = (0:017 0:003)ma 1=2 0:2 QCD T 4 1 􀀀 ln QCD T 1:2 ; for T & QCD; (4.28) while ma(T) = ma in Eq. (4.12) for T . QCD. The uncertainty in the prefactor a = 0:017 0:003 is due to the uncertainties in zd and zs only, since we neglected the uncertainty in the mass of the up quark. Comparing our Eq. (4.28) above with the result by Turner in Table 4.1, we see that we con rm the exponent d = 1:2, but we di er from Turner's result because of the prefactor a = 0:017, which is about 50% lower than a = 0:0256 in Table 4.1. We attribute this discrepancy to the di erent value of the up quark mass used in Ref. [178]. 43 0 1000 2000 3000 4000 5000 0.0000 0.0002 0.0004 0.0006 0.0008 T HMeVL IHTL Figure 4.1. The function I(T) in Eq.(4.14) as a function of temperature T for QCD = 200 MeV. Notice that the value of our prefactor a is compatible with the value a = 0:018 used in Refs. [24, 60, 88, 185]. For T of the order of QCD, we neglect the logarithmic term; in the following, we will always assume this is the case, adopting the following expression for the axion mass ma(T) = ma ( a 􀀀 QCD T 4; T & QCD; 1; T . QCD; (4.29) with a = 0:017 0:003. 4.3 Coupling of standard model particles with the axion Thanks to the Lagrangian term La, axions couple at tree level to gluons and photons. However, depending on the speci c particle model, axions may couple di erently to the Standard Model fermions and other gauge bosons, and the coupling to photons might be suppressed. 4.3.1 Coupling of axions to gluons In all axions models, axions couple to gluons through the second term in the Lagrangian in Eq. (4.1), 44 Lag = s 8 a fa=N G a ~G a ; (4.30) where we included the expression for the ne structure constant of strong interactions s = g2=4 . Thanks to this coupling to gluons, the axion mixes with pions and acquires the zero-temperature axion mass in Eq. (4.9). 4.3.2 Coupling of axions to photons The axion-photon coupling is described by the term La = 􀀀 ga 4 a F ~ F = ga aE B: (4.31) where ga is the axion-photon coupling, given by ga = 2 fa E N 􀀀 2(4 + z + zs) 3(1 + z 􀀀 zs) = 2 fa E N 􀀀 1:92 0:08 ; (4.32) where 1=137 is the ne structure constant. The ratio E=N can be written in terms of the PQ charges Xf , the electric charges Qf and the number of colors in a multiplet Df as E N = P f Xf Df Q2f P f Xf : (4.33) Depending on the model, ga can be enhanced when the number of quarks Df is large, or suppressed when E=N 2. In the following, we indicate the Feynman diagram associated to the Lagrangian term in Eq. (4.32) with the graph shown in Fig. 4.2. Models in which there exists an axion-photon coupling at tree level predict that the Primako e ect for axions might have an important role in a number of astrophysical processes like the cooling of stars, and enable us to attempt the detection of axions through a \shining through a wall" experiment. γ γ a gaγ Figure 4.2. The axion-photon vertex in an axion theory in which ga 6= 0. 45 4.3.3 Coupling of axions to fermions In the Lagrangian term describing the interaction of axions with the fermionic eld f , the axion eld always appears in the derivative term @ a(x), so that the invariance a ! a + const: proper to Nambu-Goldstone bosons is imposed: Laff = Cf 2fa f 5 f @ a: (4.34) In the expression above, f is the fermionic eld of mass mf and Cf is a model-dependent parameter that gives the PQ charge of the fermion f. Writing the interaction Lagrangian as Laff = 􀀀i Cf mf 2fa f 5 f a; (4.35) it appears that the combination gaf = Cf mf fa ; (4.36) plays the role of a Yukawa interaction, with af g2 af =4 being analogous to the ne structure constant in this model. Fig. 4.3 shows the Feynman diagram associated with Laff for the fermion f. 4.4 Axion models 4.4.1 The \visible" axion model The rst viable axion model is known as the PQWW model from the initials of the authors Peccei and Quinn [141], Weinberg [190], and Wilczek [194], or the \visible" axion because this theory would lead to sizable e ects in various physical phenomena due to the strong axion interaction with light and light mesons. In the PQWW model, the PQ energy scale fa is related to the weak interaction scale weak 250GeV. A pair of Higgs elds 1 f f a gaf Figure 4.3. The Feynman diagram showing the axion-fermion vertex for a fermion eld f, in an axion theory in which gaf 6= 0. 46 and 2 is introduced, the rst giving mass to the up quark and the second to the down quark, with each Higgs eld having a vacuum expectation value h 1;2i = 1;2=p2, with p 21 + 22 = weak. From these premise it is possible to bound the PQ scale as fa . 42 GeV and so the axion mass ma & 200 keV. The PQWW axion model has long been abandoned because its very premise fa weak has been ruled out by various astrophysical considerations (see Sec. 4.5) and because this model predicts a lifetime of various mesons like K+ or J/ that would be too short compared to observations. 4.4.2 Models for the \invisible" axion After experiments showed that the PQ energy scale cannot be related to the weak interaction scale, authors have either considered fa as a free parameter or they have considered more complicated models in which fa relates to the Grand Uni cation Theory (GUT) scale GUT 1016 GeV or to the Planck scale Planck 1019 GeV. In the rst case, the axion is studied in a model-free theory and its generic properties are constrained by astrophysical considerations, direct and indirect searches, and accelerator experiments. In the second case, the axion is embedded in a more fundamental theory such as a GUT or a string theory. All these models share a PQ scale much higher than the weak energy scale, so that the coupling of axions to Standard Model particles is considerably weaker, thus making the experimental detection of the axion even more challenging. 4.4.2.1 The KSVZ model The rst invisible axion model has been proposed by Kim [103] and by Shifman, Vain- shtein, and Zakharov [157], and it is thus known as the KSVZ model. Since, in this model, the axion does not couple with SM fermions at tree level, being the Cf = 0 in Eq. (4.34) for SM fermions, in the literature this model is also referred to as the hadronic axion model. The KSVZ axion couples at tree level to gluons and to an exotic heavy quark Q; interaction with the other SM particles occur at one loop through the intermediation of gluons and of the new Q quark. Depending on the charge of Q, the ratio E=N in Eq. (4.33) may take a value in the range [0; 6]. In particular, the KSVZ model allows the value E=N = 2 at which the axion-photon coupling ga in Eq. (4.32) is suppressed. 47 4.4.2.2 The DFSZ model In the DFSZ model, introduced by Dine, Fischler, Srednicki [54], and by Zhitnitskii [202], the axion couples at tree level to SM photons and charged leptons, besides nucleons. One of the main purposes for the introduction of this model, and an advantage for it, is that it can be easily embedded in GUT models. Any GUT model predicts E=N = 8=3, and so a precise value of the axion-photon coupling, ga = (8:67 0:93) 10􀀀4 fa : (4.37) The model also predicts the value of the axion-electron coupling Ce appearing in Eq. (4.34) with f = e, as Ce = cos2 H 3 ; (4.38) with cot H the ratio of two Higgs vacuum expectation values of this model. The coupling of axions to nucleons are also calculable in the model, and they are related by generalized Goldberger-Treiman relations to nucleon axial-vector current matrix elements. Finally, the axion coupling to the up and down quarks are Cu = sin2 H 3 ; and Cd = cos2 H 3 : (4.39) 4.4.3 Lifetime of the axion Similarly to the neutral pion, the coupling of the axion to two photons arises through the electromagnetic anomaly of the PQ symmetry, and the decay of an axion into two photons is allowed with lifetime a!2 = 64 g2 a m3 a = (6 1) 1024 s (eV=ma)5 [(E=N 􀀀 1:92 0:08)=0:75]2 : (4.40) In the last expression, we used the values of fa and ga in Eqs. (4.12) and (4.32). In analogy with the theory of pion decay, the lifetime of the axion depends on the fth power of the axion mass. When considering light axions of mass ma . 1 eV, the lifetime a!2 1025 s implies that these particles are cosmologically stable. 4.5 Astrophysical bounds on axions If axions existed, they would be produced in the hot plasma that constitutes stars and other astrophysical objects. The presence of axions in this dense environment would open up additional channels for the occurrence of well-studied astrophysics processes, and thus it would alter star evolution. 48 4.5.1 Constraints from the cooling time of white dwarfs When helium-burning stars reach their latest stages of helium consumption, they ascend in the Hertzsprung-Russell (HR) diagram through the red giant branch and evolve to the asymptotic giant branch (AGB). It is then possible that a AGB star, consisting of a degenerate carbon-oxygen core and an outer helium-burning shell, to evolve into a white dwarf star by rst cooling down due to neutrino emission and later by surface photon emission. If axions existed, an additional channel for the cooling of AGB into a white dwarf star exists given by the axion bremsstrahlung process e + Ze ! e + Ze + a: (4.41) Computing the theoretical luminosity function by taking into account the processes de- scribed above and comparing it to the observed cooling rate derived from the measured decrease of the rotational period _P =P, it is possible to constrain any new physics contribu- tion to the cooling process and in particular the contribution from axion bremsstrahlung. The constraint on the axion-electron coupling thus obtained is gae < 1:3 10􀀀13; (4.42) which set the most stringent bound on the axion-electron coupling constant. We derive a constraint on the PQ scale fa by inserting the expression for gae in Eq. (4.36) and (4.38) for Ce in the bound in Eq. (4.42), to obtain fa > 1:3 108 GeV cos2 H; (4.43) where cot H is the ratio of the two Higgs vacuum expectation values. Recently, a method for constraining the axion-photon coupling ga using the amount of linear polarization in the radiation emerging from magnetic white dwarfs has been proposed [71]. This method sets an upper limit on ga that depends on the axion mass ma. For ma . 10􀀀4 eV, the authors in Ref. [71] derive the bound ga < 10􀀀10 GeV􀀀1: this bound is much more stringent than the one obtained using measurements on the lifetime of horizontal branch stars in globular clusters, see Sec. 4.5.4. 49 4.5.2 Constraints from SN1987A A supernova (SN) event of type II consists of a core collapse of a massive star which subsequently leads to a proto-neutron star. Axions may be produced in SN events via axion-nucleon bremsstrahlung N + N ! N + N + a; (4.44) and this additional process can a ect the cooling time of the SN and shorten the duration of the burst. The pattern of the cooling time as a function of the axion-neutron coupling gaN is described in Ref. [147]. For very low values of the coupling, the axion emission does not a ect the burst duration. As gaN increases, the burst duration shortens because the emission of bremsstrahlung axions increases, reaching its minimum value when the mean free path of axions in the medium is of the order of the size of the SN. For even larger values the axions are trapped in the medium and the axion emission decreases, until the burst duration becomes una ected by the presence of these new particles. In 1987 it was possible to test models of supernovae explosions, due to the observation of the emission from a distant supernova named 1987A. The ux of antineutrinos e com- ing from SN1987A was detected at both Kamiokande II and Irvine-Michigan-Brookhaven experiments, allowing us to compare the data with theoretical expectations. Due to this analysis, it is possible to exclude axion models with an axion-nucleon coupling in the range 3 10􀀀10 . gaN . 3 10􀀀7: (4.45) This range corresponds to the high yield emission of axions excluded by the statistical analysis on the SN1987A data. 4.5.3 Axions from the Sun In the Sun, axions might be produced through the Primako process: an incoming photon interacts with the electromagnetic eld of a nearby electron or nucleus, converting into an axion, see the Feynman diagram in Fig. 4.2. Since this new channel can in principle shorten the lifetime of the Sun, it is possible to constrain the coupling of axions to photons, electrons, and nucleus by imposing that the presence of axions does not spoil the standard solar model (SSM). To see this in mode details, we consider solar limits on the axion-photon coupling ga . in the approximation in which the electron or nucleus is nonrelativistic, the rate 50 of conversion of photons of energy E into axions of the same energy in a nondegenerate plasma of temperature T is 􀀀 !a = g2 a T k2 s 32 1 + k2 s 4E2 log 1 + 4E2 k2 s 􀀀 1 : (4.46) Here, the screening factor ks is given by the Debye-Huckel formula, k2 s = 4 nB T 0 @Ye + X j Z2 j Yj 1 A; (4.47) where nB is the nondegenerate baryon density and Ye, Yj are the fractions of electrons and nuclear species j per baryon. The energy loss rate in axions per unit volume is then Q !a = g Z d3 k (2 )3 E e􀀀E=T 􀀀 1 􀀀 !a; (4.48) where g = 2 and k is the three-momentum of the incoming photon of energy E. Eventually, using the solar temperature distribution obtained from the SSM, the ux of solar axions on Earth is evaluated as La = 1:7 10􀀀3 ga 10􀀀10 GeV􀀀1 2 L ; (4.49) with L indicating the Sun luminosity. Here, we use the conservative limit on the solar axion luminosity La . 0:2L [155] which yields ga . 1:1 10􀀀9 GeV􀀀1: (4.50) Using the fact that axions a ect the sound speed-pro le of the Sun, the authors in Ref. [155] obtained a similar bound, ga . 1:0 10􀀀9 GeV􀀀1: (4.51) Ra elt and Gondolo [75] derived a restrictive bound on ga using the sensitive dependence of the production rate of neutrinos from 8B. Using the measurements of the solar neutrino ux by the Sudbury Neutrino Observatory, the authors in Ref. [75] obtain ga . 7 10􀀀10 GeV􀀀1: (4.52) To conclude, we remark here that the best constraint on ga from astrophysical measure- ments comes from considering the production of axions in globular clusters. 51 4.5.4 Axions and globular clusters A globular cluster (GC) is a gravitationally-bound ensemble of stars which formed at about the same epoch. Since these stars share the same age, but not other physical parameters like the mass or the surface temperature, a GC is particularly suitable for testing models of stellar evolution. A typical method used for comparing stars within the same GC is to plot the color luminosity vs. the total brightness of each star in a color-magnitude diagram. The typical quantities used are the color B 􀀀 V and the brightness V . When axions are included in the model, helium-burning stars may consume helium faster than their expected rate because of the extra axion-production channel. The lifetime of stars which are in the horizontal branch is reduced by a factor 1 + 3 8 ga 10􀀀10 GeV􀀀1 2 􀀀1 ; due to the production of axions. When a statistically signi cant ensemble of helium-burning stars is considered, it is found that the lifetime of these stars agrees with theoretical expectations within 10%, which in turns leads to the bound ga . 1 10􀀀10 GeV􀀀1: (4.53) This bound on the axion-photon coupling is a much more stringent bound than those obtained from considerations on the solar activity. 4.6 Direct axion searches As rst proposed by Sikivie [160], it is possible to directly search for axions in labo- ratories by using the conversion of an axion into a photon in the presence of an external electromagnetic eld. This process, which is the inverse process of the Primako e ect, has been extensively used in a number of experiments. Direct searches of axions fall into three primary categories: search of axions of galactic origin (axion haloscope), produced in the Sun (axion helioscope), and pure laboratory experiments, in which virtual axions are produced by shining lasers in strong magnetic elds. 4.6.1 Axion haloscope Axion haloscope searches rely on the possibility that relic axions from the Big Bang would be gravitationally bound to the Milky way, having a nonrelativistic velocity v with dispersion v 10􀀀3 c, corresponding to the rotational velocity of a virialized object in the 52 vicinity of the solar system. This would correspond to an axion mean energy and energy dispersion E ma c2 1 + v2 2c2 ; and E = ma v v: (4.54) In order to detect these particles, a sensitive technique known as the microwave cavity has been developed. In such a cavity, a strong electromagnetic eld is produced, with a frequency related to the size of the cavity. For a given frequency, there exists a narrow range of the axion mass ma for which the axion would interact with the electromagnetic eld and convert into a light pulse which would be eventually detected by a receiver. In order to search for di erent values of the axion mass, the size of the microwave cavity is adjustable. The search of axions through a microwave cavity assumes that the totality of the CDM is in the form of axions. As we will explore in depth in this thesis, this is possible if the mass of the axion is of the order of the eV, corresponding to the energy of a microwave with a wavelength of approximately 10 cm. The probability that an axion of energy E converts into a detectable photon while traveling in the homogeneous and transverse magnetic eld B of a microwave cavity is Pa! = (ga B L=2)2 L2 (q2 + 􀀀2=4) 1 + e􀀀􀀀L 􀀀 2E􀀀2􀀀L cos(q L) ; (4.55) where L is the length of the path, 􀀀 is the inverse absorption length for photons in the medium and q is the momentum transfer between the axion and photon. In the vacuum, the momentum transfer q reads q = m2 a 􀀀 m2 2E ; (4.56) where the result is expressed in terms of the e ective photon mass in a plasma with electrons density ne, m = r 4 ne me : (4.57) Notice that the result in Eq. (4.56) is similar to that obtained in the neutrino oscillations theory. When the absorption 􀀀 is neglected, Eq. (4.55) reduces to Pa! = ga B q 2 sin2 q L 2 ; (4.58) The rst experiments of this type were performed at Brookhaven Laboratories [195] and at the University of Florida [82]: the axion mass is excluded from the former and the latter experiments in the ranges 5:4 5:9 eV and 4:5 16:3 eV, respectively. As of today, the most sensitive axion haloscope is the Axion Dark Matter eXperiment (ADMX) at Laurence 53 Livermore National Laboratory (LLNL). In its rst direct scan of the parameter space, ADMX directly excluded axions with mass in the range [12] 1:9 eV < ma < 3:3 eV: (4.59) A later upgrade of the experiment [13] exploited a Superconducting QUantum Interference Device (SQUID), which replaced the microwave receiver and allowed us to extend the scan to higher values of the axion mass. The improved ADMX excluded the region [13] 3:3 eV < ma < 3:53 eV; (4.60) so that the overall region excluded by the ADMX experiment so far is 1:9 eV < ma < 3:53 eV: (4.61) 4.6.2 Axion helioscope The Sun produces axions through nuclear interactions within its core; the expected axion ux at Earth due to the solar activity is expressed in Eq. (4.49). The search for solar axions is mainly conducted via two types of experiments, which respectively exploit the intense Coulomb eld in a crystal to convert axions into photons (Bragg scattering experiments), or use intense magnetic elds that point at the Sun in which solar axions can convert into photons (magnet helioscope experiments). 4.6.2.1 Search of solar axions through Bragg scattering This type of experiment takes advantage of the fact that the mean energy of solar axions is around 4keV, so that the axion wavelength is of the same order as the lattice spacing in a typical crystal. The expected Bragg scattering of the resulting photon would increase the signal of the axion interaction with the lattice to around 104 with respect to the interaction of an impurity in the same crystal. Moreover, the signal would be distinctive because of the relative movement of the Sun within 24 hours (daily modulation of the signal). Various experiments undergoing such a search use a di erent crystal such as Sodium Iodide at DAMA [30], Germanium at SOLAX [65] and COSME [132], or Germanium and Silicon at CDMS [6]. All of these experiments constrain the axion-to-photon coupling to about the same level, ga . 2 10􀀀9 GeV􀀀1, which however is more than one order of magnitude above the bound from haloscope searches. 54 4.6.2.2 Solar axions through axion telescopes The main component of an axion telescope consists of a magnet in which the strong magnetic eld can be pointed at the Sun. An axion in the strong magnetic eld may convert to a low-energy X-ray, through a mechanism similar to the conversion of CDM axions in a microwave cavity. In fact, the probability of conversion is given by Eq. (4.55) also in this situation, provided that the same conditions in which the latter equation has been derived are met. Searches of solar axions via this method were conducted by Lazarus and collaborators [115] and by the Tokyo Axion Helioscope collaboration [133]: the latter group constrained ga < 6 10􀀀10 GeV􀀀1 for axion masses ma < 0:03 eV. A parallel search has been conducted by CERN Axion Solar Telescope (CAST) exper- iment, which to date is the most sensitive axion helioscope in use. In fact, CAST in its two phases has been able to constrain ga more stringently than astrophysical bounds, setting ga < 8:8 10􀀀11 GeV􀀀1 at 95% C.L. and for ma < 0:02 eV in Phase I [203], and ga . 2:2 10􀀀10 GeV􀀀1 at 95% C.L. and for the mass range 0:02 eV < ma < 0:4 eV during Phase II [11]. 4.6.3 Production of axions by laser Axion searches conducted in laboratories make use of an intense laser beam that might partially convert into axions or other pseudo-scalar particles in the presence of a strong magnetic eld, via the Primako e ect. We brie y discuss two di erent types of experiments that exploit this technique. 4.6.3.1 Polarization experiments When axions are produced by the interaction of a strong laser beam with an external magnetic eld, the polarization of the laser beam is a ected because of such a nonzero coupling with the pseudo-scalar particles. Moreover, when a photon-axion-photons re- conversion takes place, the emerging photon has a component of the E and B elds retarded with respected to a freely-propagating photon. This signal had been claimed by the Polarizzazione del Vuoto LASer (PVLAS) collaboration [35], but the event was later labeled as arti cial. 55 4.6.3.2 Regeneration experiments When a polarized laser beam propagating in a constant transverse magnetic eld is blocked by a thick absorber (a wall), it is possible to detect photons of the same wavelength as the laser beam on the other side of the blocking medium itself. This process is called photon regeneration, and the experiment is colloquially referred to as the shining light through a wall experiment. This unusual phenomenon is possible if some photons from the laser beam convert into axions in the magnetic eld, before being absorbed by the wall. In this case, the produced axions are able to travel through the wall with little absorption. Applying a second magnetic eld on the other side of the wall makes it possible for some axions to reconvert into photons through the inverse Primako process. This type of experiments has been conducted by the Brookhaven-Fermilab-Rutherford- Trieste collaboration [34] and at CERN with the Optical Search for QED vacuum birefrin- gence, Axions and photon Regeneration (OSQAR) experiment. In addition, the shining light through a wall experiment is able to put limits on more general models in which photons couple to axion-like particles (ALPs), theoretical particles that share similar properties with the axion but for which the mass-energy scale relation in Eq. (4.12) does not apply. We will talk in detail about ALPs in Sec. 7. The bound on the ALP-photon coupling constant Ga from this experiment is Ga . 1 10􀀀10 GeV􀀀1: (4.62) CHAPTER 5 REVISING THE AXION AS THE COLD DARK MATTER 5.1 Introduction The recent measurements by the WMAP mission [109] have established the relative abundance of dark and baryonic matter in our universe with great precision. About 84% of the nonrelativistic content in the universe is in the form of CDM, whose composition is yet unknown. One of the most promising hypothetical particles proposed for solving the enigma of the dark matter nature is the axion [190, 194]. This particle was rst considered in 1977 by R. Peccei and H. Quinn [142] in their proposal to solve the strong-CP problem of the QCD theory. Although the original PQ axion is by now excluded, other axion models are still viable [103, 157, 53, 202]. The hypothesis that the axion can be the dark matter particle has been studied in various papers (see e.g. [145, 2, 53, 167, 180, 123, 24, 88] and the reviews in [60, 161]). Here we examine the possibility that the invisible axion may account for the totality of the observed CDM, in light of the WMAP5 mission, baryon acoustic oscillations (BAO), and supernovae (SN) data. We also upgrade the treatment of anharmonicities in the axion potential, which we nd important in certain cases. We consider invisible axion models, in which the breaking scale of the PQ symmetry fa is well above the electroweak scale. The axion parameter space is described by three parameters, the PQ energy scale fa, the Hubble parameter at the end of in ation HI , and the axion initial misalignment angle i. 5.2 Production of axions in the early universe 5.2.1 Thermal production of axions In the early universe, a population of thermal axions originates together with standard model particles and possibly other exotic components. Scattering and annihilation processes keep this population of axions in thermal equilibrium with the remaining of the hot plasma at temperature T, with the axion number density nth a following the Boltzmann equation 57 d nth a dt + 3H(t) nth a = 􀀀ann neq;th a 􀀀 nth a : (5.1) Here, the number density of axions at thermal equilibrium is, see Eq. (2.75), neq;th a = (3) 2 T3: (5.2) In Eq. (5.1), the rate at which axions annihilate and are created in the plasma is 􀀀ann = X i ni h i vi; (5.3) where ni is the number density of the particle species i. The index i runs over processes of the type a + i other particles, with cross section i, and with v the relative velocity between i and the axion. Finally, the angular brackets in Eq. (5.3) indicate an average over the momentum distributions of the particles involved. Using the conservation of the number density at equilibrium, d neq;th a dt + 3H(t) neq;th a = 0; (5.4) we can rewrite Eq. (5.1) as d dt h a3(t) neq;th a 􀀀 nth a i = 􀀀􀀀ann a3(t) neq;th a 􀀀 nth a : (5.5) The solution of Eq. (5.5) implies that a thermal population of axions reaches its equilibrium value neq;th a exponentially fast whenever 􀀀ann > H(t): (5.6) The time tchem at which axions chemically decouple from the plasma is then de ned with 􀀀ann = H(tchem). Depending on the axion model, axions couple di erently to standard model fermions, photons, and exotic particles, whereas the coupling to gluons is model-independent and is given in Eq. (4.1). Thermal axions were rst studied by Turner [179], who considered the Primako and photoproduction processes. Given a heavy quark Q of mass mQ, whenever 58 the temperature of the plasma T . mQ the Primako process dominates and axions are created via q + q + a (q is a light quark), with cross section Primako = 3 f2 a : (5.7) At higher temperatures T & mQ, photoproduction of axions via Q + Q + a dominates with cross section 􀀀prod = mQ fa T 2 : (5.8) Ref. [179] concluded that a thermal population of axions exists today whenever fa . 109 GeV; moreover, for fa . 4 108 GeV, the thermal population is greater than the nonthermal. These considerations were revised in Ref. [125], where the authors extended the work in Ref. [179] by including other processes than Primako and photoproduction, using the model-independent axion-gluon vertex derived from Eq. (4.1). In detail, the authors in Ref. [125] include the following reactions: a + g q + q a + q g + q a + g g + g These processes have a cross section of the order of agg = 3 s=f2 a . With these additional reactions included, axions thermalize if fa . 1012 G