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Department of Physics and Astronomy University of Heidelberg Tim Department of Physics and Astronomy University of Heidelberg Master Thesis in Physics submitted by Tim Zimmermann born in Nürtingen (Germany) 2020 An Investigation of the Lower Dimensional Dynamics of Fuzzy Dark Matter This Master Thesis has been carried out by Tim Zimmermann at the Institute for Theoretical Physics under the supervision of Prof. Dr. Luca Amendola and Prof. Dr. Sandro Wimberger. Abstract Due to the lack of experimental evidence for weakly interacting particles (WIMPs) and the unsatisfying numerical predictions of the established cold dark matter (CDM) paradigm on galactic scales, alternative dark matter models remain an intriguing and active field of research. A model of recent interest in cosmology is Fuzzy Dark Matter (FDM) which assumes the nonbaryonic matter component of the universe to consist of scalar bosons with mass m 10−22 eV. FDM possesses a rich phenomenology in (3 + 1) dimensions recovering predictions∼ of CDM on large scales while suppressing structure growth below the de-Broglie wavelength which, due to the miniscule boson mass, attains values of galactic size. As full fledged (3 + 1)-dimensional, cosmological simulations, especially for FDM, are extremely time consuming this thesis investigates to what extend phenomena in three spatial dimensions can be observed with only one geometric degree of freedom. To this end, a first principle derivation of the governing equation of FDM, i.e. (3 + 1)- Schrödinger-Poisson (SP) is presented and dimensionally reduced to two distinct (1 + 1)-FDM representations, namely (i) the standard (1 + 1)-SP equation and (ii) the novel periodic, line adiabatic model (PLAM). After investigating the nature of FDM in linear theory, we present a comprehensive, unified and thoroughly tested numerical method capable of integrating both reduction models into the nonlinear regime. By analyzing an ensemble of high-resolution, cosmological simulation runs, we find standard (1 + 1)-SP to only partially recover the sought after asymptotic dynamics of (3 + 1)-FDM — a result which holds true even under simplified initial conditions or expansion free scenarios. PLAM, on the other hand, shows the most prominent feature of full fledged FDM — a solitonic, dynamical attractor — for both expanding and static background cosmologies. We argue the discrepancy between the models lies in the different nonlocality of the underlying nonlinear interaction and substantiate this with numerical and analytical arguments. Moreover, our results indicate that independent of the reduction model used, the long term dynamics is best understood as a consequence of thermodynamics aiming to maximize the system’s entropy. Zusammenfassung Alternative Dark Matter Modelle stellen aufgrund fehlender experimenteller Evidenz für die Existenz von WIMPs sowie unzureichender numerischer Vorhersagen des etablierten Modells kalter, dunkler Materie (CDM) ein attraktives und aktives Forschungsgebiet dar. Insbesondere Fuzzy Dark Matter (FDM) gilt als vielversprechende Dark Matter Theorie, in der nicht-baryonische Materie als skalare Bosonen mit Masse m 10−22 eV modelliert wird. Zu den wichtigsten Charakteristika FDMs zählen die∼ Übereinstimmung mit dem CDM Paradigma auf großen Längenskalen sowie die Unterdrückung nicht-linearer Materiestrukturen unterhalb der bosonischen de-Broglie Wellenlänge, die aufgrund der winzigen Teilchenmasse kosmisch relevante Größe annimmt. Da großskalige, hochauflösende (FDM) Simulationen dunkler Materie in (3 + 1)- Dimensionen extrem zeitaufwendig sind, stellt sich die Frage ob und wie charakteris- tische Eigenschaften FDMs auch in lediglich einer Raumdimension realisiert werden können. Untersuchung dieses Aspektes ist Gegenstand dieser Arbeit. Ausgangspunkt hierfür bildet die Herleitung der FDM Evolutionsgleichung, die (3 + 1)-Schrödinger- Poisson (SP) Gleichung, sowie eine detaillierte Dimensionsreduktion resultierend in (i) der Standard (1 + 1)-SP Gleichung und (ii) des periodisierten "line adiabatic models"(PLAM). Wir kontrastieren das Verhalten von FDM und CDM in linea- rer Ordnung und präsentieren des Weiteren eine einheitliche, ausführlich getestete, numerische Methode, mit derer beide niederdimensionale FDM Repräsentationen bis ins nichtlineare Regime integriert werden können. Die ausührliche Analyse eines Ensembles hochauflösender, kosmologischer Simulationen zeigt das Versagen (1 + 1)- SPs wesentliche Eigenschaften der (3 + 1)-FDM Phänomenologie in einer Dimension abzubilden — auch nicht unter vereinfachten Anfangsbedigungen oder expansions- freien Szenarien. PLAM, anderseits, realisiert mit der Existenz eines solitonischen, dynamischen Attraktors die wohl wichtigste Eigenschaft FDMs, sowohl für statische wie auch expandierende Hintergrundkosmolgien. Auf Basis analytisch-numerischer Argumente interpretieren wir dieses Verhalten als Konsequenz der verschiedenen Nicht-Lokalitäteten in der Wechselwirkung beider Modelle. Unabhängig vom Re- duktionsmodell indizieren die präsentierten Resultate die Anwendbarkeit thermo- dynamischer Prinzipien für die Beschreibung der asymptotischen Systemzustandes, insbesondere die Gültigkeit des Maximum Entropie Prinzips. Contents 1 Motivation 12 2 Fuzzy Dark Matter from First Principles 16 2.1 General Relativistic Considerations................... 17 2.1.1 Derivation of the Klein-Gordon-Einstein Equation....... 17 2.1.2 Recovering the Homogeneous Universe............. 19 2.2 The Non-Relativistic Limit of KGE — (3 + 1)-SP........... 20 2.2.1 Overview of Competing Interpretations............. 23 2.2.2 SP as Nonlinear Schrödinger Equation............. 25 2.3 Dimension Reduction........................... 28 2.3.1 Homogeneous Matter Sheets — (1 + 1)-SP........... 29 2.3.2 Strong Confinement — PLAM.................. 32 2.4 Symmetries and Conserved Quantities................. 38 3 Fuzzy Dark Matter in the Linear Growth Regime 41 3.1 Statistical Description of Fluctuation.................. 42 3.1.1 Statistics in Real Space - The Correlation Function...... 42 3.1.2 Statistics in Reciprocal Space - The Power Spectrum..... 43 3.2 Time Evolution of the Matter Power Spectrum............. 44 3.2.1 The Quantum-Euler-Poisson Equation............. 45 3.2.2 Stability Analysis and Growth Factors............. 47 3.2.3 Transfer Functions........................ 51 3.3 Initial Conditions for (1 + 1) SP..................... 53 3.3.1 Initial Density Contrast..................... 53 3.3.2 Initial Phase............................ 55 4 Numerical Considerations 57 4.1 Numerical Challenges and Existing Methods.............. 58 4.2 Spatial Discretization........................... 59 9 4.3 Time Integration............................. 62 4.3.1 Integration in a Static Space Time............... 62 4.3.2 Integration in an Expanding Space Time............ 64 4.4 Convergence and Stability........................ 65 5 Fuzzy Dark Matter in the Nonlinear Growth Regime 67 5.1 Parameter Choices and Experimental Setup.............. 68 5.2 Evolution in Reciprocal Space...................... 71 5.2.1 Small Scale Suppression — The Heisenberg Scale....... 72 5.2.2 Large Scale Growth........................ 74 5.2.3 Nonlinear Mode Coupling.................... 77 5.3 Evolution in Real Space......................... 79 5.3.1 Correlation Function....................... 80 5.3.2 Failure of (1 + 1)-SP in Reaching the Soliton State...... 81 6 Asymptotic Dynamics 87 6.1 Relaxation Processes and Equilibrium States.............. 88 6.1.1 Quantum Virial Equilibrium................... 88 6.1.2 Thermal Equilibirum....................... 90 6.2 The FDM Groundstate.......................... 91 6.2.1 Construction of the (1 + 1)-FDM Ground State........ 91 6.2.2 Properties of the (1 + 1)-FDM Ground State.......... 94 6.3 Static Space-Time Dynamics....................... 98 6.3.1 (1 + 1)-Schrödinger-Poisson................... 98 6.3.2 PLAM............................... 103 6.3.3 Self-Organization Processes in Nonlinear Dynamics...... 107 6.4 Expanding Space-Time Dynamics.................... 109 6.4.1 (1 + 1)-Schrödinger-Poisson................... 110 6.4.2 PLAM............................... 112 7 Conclusion and Perspectives 115 7.1 Summary of Results........................... 115 7.2 Future Extensions............................. 117 7.2.1 Next Generation Numerics.................... 117 7.2.2 Nonlocal NLSE as Distinct Physical Problem.......... 118 7.2.3 Towards (3 + 1)-FDM — Spherical Symmetry......... 119 CONTENTS A Fuzzy Dark Matter from First Principles 123 A.1 Derivation of Periodic Greens Functions................ 123 A.2 Momentum Conservation of (1 + 1) Fuzzy Dark Matter........ 126 A.3 Scaling Symmetry............................. 128 B Numerical Considerations 130 B.1 An Augmented Fourth Order Scheme.................. 130 B.2 Convergence Analysis........................... 132 B.2.1 Dominance of the Temporal Error................ 133 B.2.2 Behavior of the Temporal Error................. 133 B.2.3 [2] vs. [4] ........................ 137 SM BM C Asymptotic Dynamics 146 C.1 Quantum Virial Theorem......................... 146 C.2 Discrete Normalized Gradient Flow................... 147 C.3 Mass-Size Relation from Dimensional Analysis............. 148 11 Chapter 1 Motivation While ordinary, baryonic matter contributes only 5% to the total energy budget of the observable universe, origin and physical nature∼ of the remaining 95% are still an open question. In fact, gaining a fundamental understanding of the∼ "dark sector", which consists of dark matter ( 25%) and dark energy ( 70%), is arguably the largest challenge
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