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Exploring New Windows Into Fundamental Physics at High

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Exploring New Windows Into Fundamental Physics at High Exploring New Windows into Fundamental Physics at High Energies Published by Proefschriftmaken phone: +31(0)883746777 De Limiet 26 e-mail: [email protected] 4131 NR Vianen http://www.Proefchriftmaken.nl The Netherlands Cover illustration: Planck mapped the strength (color) and polarization (texture) of ra- diation from galactic dust. This is a photo from ESA/PLANCK COLLABORATION. One can find this photo from http://science.sciencemag.org/content/347/6222/595/tab- figures-data. Adrian Cho has used this figure. ISBN: 978-94-6380-312-0 NUR: 925 Copyright c 2019 Leihua Liu. All rights reserved. Exploring New Windows into Fundamental Physics at High Energies Nieuwe Invalshoeken in de Fundamentele Fysica van Hoge Energiee¨n (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof. dr. H.R.B.M. Kummeling, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op woensdag 24 april 2019 des middags te 12.45 uur door Leihua Liu geboren 9 september 1985 te Anhua, China Promotor: Prof. dr. S.J.G. Vandoren Copromoter: Dr. T. Prokopec This thesis is dedicated to my parents and wife, for their endless love and support. Contents List of Tables . ix List of Figures . xi 1 Introduction 1 1.1 A big picture of the Universe . 1 1.2 Why inflation? . 3 1.3 Inflationary model building . 4 1.4 The nature of gravity . 5 1.5 Outline of the thesis . 6 2 Standard cosmological model 7 2.1 A brief history of cosmology . 7 2.1.1 Evolution of the Universe . 7 2.2 FLRW Spacetime . 9 2.2.1 Big Bang Puzzles . 12 2.3 Conditions for inflation . 15 2.4 Quantum fluctuation . 16 3 Inflation induced by f(R) gravity 21 p 3.1 Analysis of R inflationary model as p > 2 . 21 3.1.1 Introduction . 21 3.1.2 Recap of f(R) gravity . 23 3.1.3 Conclusion . 25 3.2 Inflation in an effective gravitational model and asymptotic safety . 27 3.2.1 Introduction . 27 3.2.2 The model . 30 3.2.3 Cosmological perturbations in our model . 31 3.2.4 Implementing constraints . 33 3.2.5 Results . 35 3.2.6 Conclusion . 37 viii 3.3 Appdendix . 39 3.3.1 Appendix A . 39 3.3.2 Appendix B . 41 4 Non-minimally coupled curvaton 45 4.1 Background and introduction . 45 4.2 The model . 46 4.3 Power spectrum . 48 4.3.1 Slow roll analysis . 55 4.3.2 Explicit form for slow roll parameters . 60 4.3.3 The curvaton mechanism . 68 4.4 Curvaton decay . 71 4.5 Conclusion . 72 5 Gravitational microlensing in Verlinde’s emergent gravity 75 5.1 Introduction . 75 5.2 Global monopole metric . 76 5.3 Gravitational lensing . 79 5.4 Discussion . 83 6 Conclusion and Outlook 93 Bibliography . 95 Samenvatting . 109 Acknowledgements . 111 Publications . 113 Curiculum Vitae . 115 List of Tables 2.1 FLRW solutions for a flat universe dominated by radiation, matter or a cosmological constant Λ........................ 11 List of Figures 1.1 The moment of the Universe creation by the creature named Pangu, who is a figure from an ancient Chinese mythology. Pangu uses his super powers to separate the sky and the Earth from a chaotic beginning. 2 2.1 The schematic evolution of universe. We present its main events in dif- ferent stages and how it is related to observations. Acronyms: BBN (Big Bang Nucleosynthesis), LSS (Large-Scale Structure), BAO (Baryon Acoustic Oscillations), QSO (Quasi-Stellar Objects = Quasars), Lyα (Lyman-alpha), CMB (Cosmic Microwave Background), 21cm (hydrogen 21cm-transition). 8 2.2 The conformal diagram of Big Bang cosmology. The CMB occurs at the last scattering surface (recombination) in which consisting of 105 causally disconnected (particle horizons) regions. 13 2.3 The conformal diagram of cosmic inflationary cosmology. Inflation extends the conformal time to very negative values. Due to an expo- nential expansion of space, it can lead to a comparable (or even much larger than) causally connected regions when compared to that of the CMB. 14 3.1 The tensor-to-scalar ratio r as a function of the scalar spectra index ns for p = 2:06; 2:04; 2:02; 2:0 in blue solid lines. We show three curves for N = 50; 60 in red dashed lines. The current upper limit on the tensor-to-scalar ratio (2.49), r < 0:067 is outside the plot’s range. The range of ns from Eq. (2.46) is presented in blue shadow region. 25 3.2 The running of the scalar spectral index α as a function of the scalar spectral index ns for p = 2:06; 2:04; 2:02; 2:0 in blue solid lines. We show three curves for N = 50; 60 in red dashed lines. The shadow shows the allowed values of ns according to (2.46). The current Planck Collaboration limits on α 2 [−0:0112; 0:0022]. 26 xii 3.3 The running of the scalar spectral index α as a function of the tensor- to-scalar ratio r for p = 2:06; 2:04; 2:02; 2:0 in blue dashed lines. We show three curves for N = 50; 60 in red solid lines. The lines N = 50 and N = 60 are shown as dashed lines. The current upper limit on the tensor-to-scalar ratio (2.49) r < 0:067 . 26 −2 3.4 The Einstein frame potential VE from Eq. (3.17) is shown for b = 10 (top blue curve), b = 10−3 (middle green curve) and b = 10−4 (lower red dashed curve). In 8 −5 this figure a = 7 × 10 , µ = 10 HE . The upper (lower) black dashed curve shows −10 4 −10 4 VE = 6×10 MP (VE = 1:2×10 MP) and they represent the upper and lower limits of the effective potential corresponding to the COBE constraint (3.21) evaluated for r = 10−2 (r = 2 × 10−3). ........................ 34 3.5 The number of e-foldings N as a function of a~ = 10−10a and b (brown plane). The upper blue and lower green planes represent N = 50 and N = 65, respectively. 34 −10 3.6 The scalar spectral index ns as a function of parameters a~ = 10 a and b (brown plane). The allowed range of ns is between the upper blue and lower green planes. 34 3.7 The tensor-to-scalar ratio r as a function of the scalar spectral index −4 −2 ns for b in the range from 10 to 10 . We show two curves: N = 50 (upper red dotted curve) and N = 65 (lower red dotted curve). The shadowed region shows the allowed values of ns (2.48). The current upper limit on the tensor-to-scalar ratio, r < 0:067 (2.49) is outside the plot’s range. The COBE constraint (3.21) is imposed and µ = −5 10 HE. ................................ 35 3.8 The scalar spectrum index running parameter α as a function of the −4 −2 scalar spectral index ns for b in the range from 10 to 10 . We show two curves: N = 50 (upper dotted curve) and N = 65 (lower dotted curve). The shadowed region shows the allowed values of ns in Eq. (2.48). The current Planck Collaboration limits on the running of scalar spectral index, α 2 [−0:010; 0:004] lie outside the plot’s range. −5 The COBE constraint (3.21) is imposed and µ = 10 HE. 36 3.9 The the running of scalar spectrum index α as a function of the tensor- to-scalar ratio r for b in the range from 10−4 to 10−2. We show two curves: N = 50 (upper dotted curve) and N = 65 (lower dot- ted curve). The shadowed region shows the allowed values of ns in Eq. (2.48). The current upper limit on the tensor-to-scalar ratio, r < 0:067 (2.49) is outside the plot’s range. The COBE constraint (3.21) −5 is imposed and µ = 10 HE. ..................... 37 3.10 Potential for RP inflationary model with p=2(red thick line), p=1.9(blue dashed line) and p=2.06(green dashed line). 39 xiii 2 4.1 The configuration space scalar Ricci curvature (4.13) MPR as a func- 2 2 tion of f = 1−ξχ =MP for negative non-minimal couplings: ξ = −1 (bright yelow curve asymptoting ≈ −0:28), ξ = −0:1 (light blue curve asymptoting ≈ −0:12), ξ = −0:01 (solid red curve asymptot- ing ≈ −0:02) and ξ = −0:001 (solid black curve asymptoting almost 2 zero). When the coupling is very large and negative, MPR asymptotes −1=3, which is a hyperbolic space H2 of constant curvature. 49 4.2 The spectral index nR from Eq. (4.81) is shown as a function of the curvaton field f (horizontal axis) and the inflaton condensate φ/MP for negative non-minimal couplings: ξ = −0:001 (left panel) and ξ = −7 −0:01 (right panel). The values of the parameters are: mχ = 10 , −1 −12 −9 λ = 10 , λχ = 10 and V0 = 10 . 60 2 2 2 4.3 The dimensionless turning rate η! = ! =H calculated in slow roll approximation (4.86) as a function of the curvaton field f and for neg- ative non-minimal couplings: ξ = −0:01 (blue dashed curve on the left panel), ξ = −0:001 (black solid curve on the left panel), ξ = −0:1 (solid black curve on the right panel), ξ = −1 (blue dashed curve on the right panel).
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