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Jagiellonian U n iv e r s it y D o c t o r a l T h e s is Cosmological models with running cosmological constant Author: Aleksander Stachowski Supervisor: Prof. dr hab. Marek Szydłowski A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Faculty of Physics, Astronomy and Applied Computer Science October 2018 iii Wydział Fizyki, Astronomii i Informatyki Stosowanej Uniwersytet Jagielloński Oświadczenie Ja niżej podpisany Aleksander Stachowski (nr indeksu: 1016516) doktorant Wydziału Fizyki, Astronomii i Informatyki Stosowanej Uniwersytetu Jagiellońskiego oświadczam, że przedłożona przeze mnie rozprawa doktorska pt. „Cosmological models with running cosmological constant" jest oryginalna i przedstawia wyniki badań wykonanych przeze mnie osobiście, pod kierunkiem prof. dr. hab. Marka Szydłowskiego. Pracę napisałem samodzielnie. Oświadczam, że moja rozprawa doktorska została opracowana zgodnie z Ustawą o prawie autorskim i prawach pokrewnych z dnia 4 lutego 1994 r. (Dziennik Ustaw 1994 nr 24 poz. 83 wraz z później­ szymi zmianami). Jestem świadom, że niezgodność niniejszego oświadczenia z prawdą ujawniona w dowolnym czasie, niezależnie od skutków prawnych wynikających z ww. ustawy, może spowodować unie­ ważnienie stopnia nabytego na podstawie tej rozprawy. Kraków, dnia v Abstract The thesis undertakes an attempt to solve the problems of cos­ mological constant as well as of coincidence in the models in which dark energy is described by a running cosmological con­ stant. Three reasons to possibly underlie the constant's change­ ability are considered: dark energy's decay, diffusion between dark energy and dark matter, and modified gravity. It aims also to pro­ vide a parametric form of density of dark energy for the models that involve a running cosmological constant, which would de­ scribe inftation in the early Universe. The principal method used in my investigations was the dy­ namical analysis. The cosmological equations were accordingly recast as a dynamical system, which enabled me to draw up the relevant phase portrait, much useful in considering the possible evolutionary paths of the Universe. The cosmological models were estimated by taking into ac­ count a number of astronomical data, such as observations of type la supernovae, cosmic microwave background, baryon acoustic oscillations, measurements of the Hubble function for galaxies, and the Alcock-Paczynski test. The results of my investigations have been published in eleven papers. Keywords: cosmology, dark energy, dark matter vii Streszczenie Rozprawa podejmuje próbę rozwiązania problemu stałej kos­ mologicznej i problemu koincydencji w modelach kosmologicz­ nych, gdzie ciemna energia jest opisywana zmienną stałą kosmo­ logiczną. Są rozważane trzy możliwe przyczyny zmienności stałej kosmologicznej: rozpadająca się ciemna energia, dyfuzja pomię­ dzy ciemną materią a ciemną energią oraz zmodyfikowana gra­ witacja. Celem jest także wprowadzenie parametryzacji gęstości ciemnej energii w podejściu ze zmienną stałą kosmologiczną, która opisuje inftację we wczesnym Wszechświecie. Najważniejszą metodą użytą w tych badaniach była analiza dy­ namiczna. Równania kosmologiczne były przepisane do postaci układu dynamicznego, który umożliwiał narysowanie odpowied­ niego portretu fazowego przydatnego przy badaniu możliwych ścieżek ewolucji Wszechświata. Modele kosmologiczne były estymowane z uwzględnieniem obserwacji astronomicznych takich jak: obserwacje supernowych typu la, mikrofalowego promieniowania tła, barionowych oscylacji akustycznych, pomiarów wartości funkcji Hubble'a dla galaktyk i testu Alcocka-Paczyńskiego. Wyniki badań zostały zamieszczone wjedenastu opublikowa­ nych pracach. Słowa kluczowe: kosmologia, ciemna energia, ciemna materia ix This dissertation has been based on the scientific results previ­ ously reported in the following articles: • Marek Szydłowski, Aleksander Stachowski, Cosmology with decaying cosmological constant - exact solutions and model testing, JCAP 1510 (2015) no. 10, 066, doi:10.1088/1475-7516/2015/10/0 66. • Marek Szydłowski, Aleksander Stachowski, Cosmological models with running cosmological term and decaying dark mat­ ter, Phys. Dark Univ. 15 (2017) 96-104, doi:10.1016/j.dark.2017.01.002. • Aleksander Stachowski, Marek Szydłowski, Krzysztof Urba­ nowski, Cosmological implications o f the transition from the false vacuum to the true vacuum state, Eur. Phys. J. C77 (2017) no. 6, 357, doi:10.1140/epjc/s10052-017-4 934-2. • Marek Szydłowski, Aleksander Stachowski, Krzysztof Urba­ nowski, Quantum mechanical look at the radioactive-like decay of metastable dark energy, Eur. Phys. J. C77 (2017) no. 12, 902, doi:10.1140/epjc/s10052-017-5471-8. • Zbigniew Haba, Aleksander Stachowski, Marek Szydłowski, Dynamics o f the difusive DM-DE interaction - Dynamical system approach, JCAP 1607 (2016) no. 07, 024, doi:10.1088/1475-7516/2016/07/024. • Marek Szydłowski, Aleksander Stachowski, Does the difusion dark matter-dark energy interaction model solve cosmological puzzles?, Phys. Rev. D94 (2016) no. 4, 043521, doi:10.1103/PhysRevD.94.043521. • Aleksander Stachowski, Marek Szydłowski, Dynamical system approach to running A cosmological models, Eur. Phys. J. C76 (2016) no. 11, 606, doi:10.1140/epjc/s10052-016-4439-4. x • Aleksander Stachowski, Marek Szydłowski, Andrzej Borowiec, Starobinsky cosmological model in Palatini formalism, Eur. Phys. J. C77 (2017) no. 6, 406, doi:10.1140/epjc/s10052-017-4981-8. • Marek Szydłowski, Aleksander Stachowski, Andrzej Borowiec, Emergence o f running dark energy from polynomial f(R) theory in Palatini formalism, Eur. Phys. J. C77 (2017) no. 9, 603, doi:10.1140/epjc/s10052-017-5181-2. • Marek Szydłowski, Aleksander Stachowski, Simple cosmologi­ cal model with inûation and late times acceleration, Eur. Phys. J. C78 (2018) no. 3, 249, doi:10.1140/epjc/s10052-018-5722-3. • Marek Szydłowski, Aleksander Stachowski, Polynomial f (R) Palatini cosmology: Dynamical system approach, Phys. Rev. D97 (2018) 103524, doi:10.1103/PhysRevD.97.103524. xi Acknowledgements I am most grateful to my Supervisor, Prof. Marek Szydłowski, for his help and support during preparation of this thesis as well as careful supervision over the past years. I would like to thank Adam Krawiec, Marek Krośniak, and Aleksander Kurek for remarks, comments and stimulating discussion. 1 Contents 1 Introduction 3 2 Statistical analysis of cosmological models 7 3 Cosmology with decay of metastable dark energy 11 3.1 Decay of metastable dark energy from quantum vac­ uum ................................................................................. 11 3.2 Late-time approximation of decaying metastable dark energy .................................................................... 13 3.3 Model te s tin g ................................................................. 16 3.4 Modified scaling law of matter density....................... 17 3.5 Cosmological implications of transition from false to true vacuum state ......................................................... 18 3.6 Radioactive-like decay of metastable dark energy . 20 3.7 Main results ................................................................... 21 4 Diffusion dark matter-dark energy interaction model 23 4.1 Relativistic diffusion interacting of dark matter with dark energy .................................................................... 23 4.2 Diffusive DM-DE interaction: coincidence problem . 25 4.3 Diffusive DM-DE interaction: non-relativistic case and statistical analysis ......................................................... 27 5 Dynamical system approach to running A cosmological models 29 5.1 A(H)CDM cosmologies.............................................. 29 5.2 A(R)CDM cosmologies.............................................. 31 5.3 Non-canonical scalar field cosmology....................... 32 5.4 Cosmology with emergent A(a) relation .................. 33 6 Starobinsky cosmological model in Palatini formalism 37 6.1 Palatini formalism in Jordan frame ............................ 37 6.2 Palatini formalism in Einstein frame ............................ 39 6.3 Starobinsky cosmological model in Palatini formal­ ism: dynamical system approach ............................... 41 6.4 Extended Starobinsky cosmological model in Palatini formalism ...................................................................... 43 2 6.5 Inftation in Starobinsky cosmological model in Pala­ tini fo rm a lism ................................................................... 44 6.6 Einstein frame vs Jordan frame .................................... 45 7 Conclusions 47 Bibliography 51 Appendix 59 A.1 Cosmology with decaying cosmological constant - exact solutions and model testing ............................... 61 A.2 Cosmological models with running cosmologicalterm and decaying dark matter .............................................. 85 A.3 Cosmological implications of the transition from the false vacuum to the true vacuum sta te ........................ 95 A.4 Quantum mechanical look at the radioactive-like de­ cay of metastable dark energy.......................................109 A.5 Dynamics of the diffusive DM-DE interaction - Dy­ namical system approach...............................................123 A.6 Does the diffusion dark matter-dark energy interac­ tion model solve cosmological puzzles? .....................147 A.7 Dynamical system approach to running A cosmolog­ ical models ....................................................................
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  • Supergravity Domain Walls in Basic Theory 64 8.1 Connection to Topological Defects in Superstring Theory

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    IASSNS-HEP-96/25 CERN-TH/96-97 Supergravity Domain Walls Mirjam Cvetiˇc Institute for Advanced Study, School of Natural Sciences Olden Lane, Princeton, NJ 80540. U.S.A. Phone: (+1) 609 734 8176 FAX: (+1) 609 924 8399 e-mail: [email protected] and Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, U.S.A. Phone: (+1) 215 898 8153 FAX: (+1) 215 898 8512 e-mail: [email protected] Harald H. Soleng CERN, Theory Division, CH-1211 Geneva 23, Switzerland Phone: (+41) 22–767 2139 FAX: (+41) 22–768 3914 arXiv:hep-th/9604090v1 16 Apr 1996 e-mail: [email protected] April 16, 1996 1 Contents 1 Introduction 5 1.1 Classesofdomainwalls ....................... 5 1.2 Walls in N =1supergravity..................... 6 2 Supergravity theory 9 2.1 Field content of N =1supergravity ................ 9 2.2 BosonicpartoftheLagrangian . 10 2.3 Bosonic Lagrangianand topological defects . .. 12 2.4 Supersymmetrytransformations. 13 3 Topological defects and tunneling bubbles 14 3.1 Overview ............................... 14 3.1.1 Topologicaldefectsinphysics . 14 3.1.2 Cosmological implications of topological defects . .... 15 3.2 Thekink................................ 16 3.2.1 Topologicalcharge . 17 3.2.2 Higher-dimensionaldefects . 19 3.3 Homotopy groupsand defect classification . 19 3.4 Formationoftopologicaldefects. 20 3.4.1 TheKibblemechanism. 20 3.4.2 Quantumcreation . 21 3.4.3 Falsevacuumdecay . 21 3.5 Domain walls as a particular type of topological defect . ..... 21 4 Isotropic domain walls 24 4.1 Domainwallsymmetries. 24 4.2 Inducedspace–timesymmetries . 24 4.2.1 Metricansatz......................... 24 4.3 Thethinwallformalism . 27 4.3.1 Lagrangianandfieldequations .