Master Thesis Investigation of Cosmic Microwave Background Anomalies with Multipole Vectors ∗ Marvin Pinkwart [email protected] Date of Submission: 14.11.2016 FacultySubmitted of Physics to the Bielefeld University Supervisor and First Referee: Prof. Dr. Dominik Schwarz Second Referee: Prof. Dr. Gernot Akemann ∗ http://www.esa.int/var/esa/storage/images/esa_multimedia/images/2013/03/planck_cmb/12583930- 4-eng-GB/Planck_CMB.jpg ≤ l ≤ Abstract In this thesis we investigate breakingalm of isotropy in the Cosmic Microwave Background on the angular range 2 49. For this purposel we use thousand ensembles of Gaussian and isotropic random in order to receive thousand ensembles of random multipole vectors. Each multipole is represented by multipole vectors which rotate rigidly with the temperature fluctuations. We use four different statistics considering theS multipolel vectors:| l;i · | ;S l | l;i × l;j · | l l l − i ( ) i<j ( ) ( ) 1 1 X 5 2 X S (l) = v |D l;i · l;j(|);S = l (v v ) D| l;i × l;j · l;k |: l l − ( 1) l l − l − i<j ( ) ( ) i<j<k ( ) ( ) ( ) 3 2 X 4 6 X ( ) = v v ( ) = (v v ) v S S ( 1) ( 1)( 2) S S 1 5 3 4 and measure correlation with an outer direction D; and measure intra- multipole correlation. We find< the likelihoods for intra-multipoleS S correlation not to be remarkable, except for the map received by the SEVEM cleaning algorithm, while the 1 5 number of low likelihoods ( 10%) – if considering , and the Cosmic Dipole as well as the Galactic Center as the direction D only – is much higher than one would expect. SEVEM deviates from the other three maps in this regard as well with the Galacticl ; l and Ecliptic Poles impacting it the most. Despite the unusual high number of lower≤ l ≤ likelihoods we do not find a clustering of low likelihoods in some angular patch 1 2 [ ] except for the unusual alignment of multipole vectors with the Cosmic Dipole at 2 4. Kurzfassung≤ ≤ In dieser Arbeit untersuchen wir Isotropiebrechungalm im Kosmischen Mikrowellenhinter- grund in dem Winkelbereich 2 2 49. Zu diesem Zweckel verwenden wir tausend Ensembles Gaußscher und isotroper Zufalls- , um tausend Ensembles von Zufalls- multipolvektoren zu erhalten. Jeder Multipol wird durch Multipolvektoren repräsen- tiert, welche starr mit den Temperaturfluktuationen rotieren. Wir benutzen vier unter- schiedlicheS l Statistiken| l;i · bezüglich| ;S l der Multipolvektoren:| l;i × l;j · | l l l − i ( ) i<j ( ) ( ) 1 1 X 5 2 X S (l) = v |D l;i · l;j(|);S = l (v v ) D| l;i × l;j · l;k |: l l − ( 1) l l − l − i<j ( ) ( ) i<j<k ( ) ( ) ( ) 3 2 X 4 6 X ( ) = v v ( ) = (v v ) v S S ( 1) ( 1)( 2) S S 1 5 3 4 und messen die Korrelation mit einer äußeren Richtung D; und messen Intramultipolkorrelationen. Es stellt sich heraus, dass die Wahrscheinlichkeiten für die gemessenen Intramultipolkorrelationen,< abgesehenS S von der mit dem SEVEM-Algorithmus hergestellten Karte, nicht beachtenswert klein sind, während die Anzahl der geringen 1 5 Wahrscheinlichkeiten ( 10%), wenn man nur , und den kosmischen Dipol, sowie das galaktische Zentrum betrachtet, viel größer ist als man erwarten würde. Auch in diesem Hinblick unterscheidet sich SEVEM von den anderen drei Karten, denn auf SEVEM haben der galaktische und der ekliptische Pol den größten Einfluss.l ; l Trotz der hohen Anzahl an geringen Wahrscheinlichkeiten finden wir keine andere Anhäufung 1 2 von geringen≤ l Wahrscheinlichkeit≤ in einem gewissen Winkelbereich [ ] abgesehen von der ungewöhnlichen Ausrichtung der Multipolvektoren am kosmischen Dipol im Bereich 2 4. Table of Contents Page Table of Contents viii List of Figures x List of Tables xi Conventions and Abbreviations xii I. Introduction 1 II. The Universe 3 II.1. Isotropic Universe / ΛCDM . 4 II.1.1. Isotropy . 4 II.1.2. Friedmann Equations and Solutions . 7 II.1.3. Spatial Part and Full Metric . 9 II.2. Short History of the Universe . 11 II.2.1. Age of the Universe . 11 II.2.2. Contributions of Different Energies . 12 II.2.3. Thermal History . 13 II.3. Problems of ΛCDM . 15 II.3.1. Horizon Problem . 15 II.3.2. Flatness Problem/Entropy Problem . 15 II.3.3. Structure Formation Problem . 16 II.4. Inflation . 16 II.4.1. Idea . 17 II.4.2. Small Field vs. Large Field . 18 II.4.3. Resolution of ΛCDM-Problems . 18 III. The CMB 21 III.1. General Properties . 22 v Table Of Contents III.2. Era of Decoupling . 23 III.2.1. Hydrogen Formation . 24 III.2.2. Transition to Transparency . 25 III.3. Linear Perturbation Theory . 26 III.3.1. Gauges . 27 III.3.2. Helicity Decomposition . 28 III.3.3. Evolution Equations . 29 III.4. Generation of Perturbations/Initial Conditions . 30 III.5. From Initial Conditions to Recombination . 33 III.6. Black Body Spectrum . 34 III.7. CMB Statistics and Angular Spectrum . 35 III.7.1. General Properties and Definitions . 35 III.7.2. Ergodicity and Cosmic Variance . 39 III.7.3. Isotropy vs. Gaussianity . 40 III.7.4. Theoretical Description of The Angular Spectrum 41 IV. Multipole Vectors 45 IV.1. Maxwell’s Multipole Vectors and Sylvester’s Theorem . 45 IV.2. Tensorial Approach . .a .lm . 48 IV.2.1. Mathematical Derivation . 48 IV.2.2. Explicit Calculation of MPVs from ’s . 50 IV.3. Algebraic Approach . 51 IV.3.1. Proof of Sylvester’s Theorem . 51 IV.3.2. ComputationSO ; R of MPVsSL .; C . 53 IV.4. Majorana Sphere Approach . 55 IV.4.1. From (3 ) to (2 ) . 55 IV.4.2. Majorana Vectors . 56 IV.4.3. Application to Sylvester’s Theorem . 60 IV.5. Probability Distribution . 61 IV.5.1. Properties andl Assumptions; ; . 61 IV.5.2. Derivation of the Full Joint PDF . 63 IV.5.3. Special Cases = 1 2 3 . 66 IV.5.4. MPVs, the Gaussian Analytic Function and Ran- dom Matrix Theory . 68 IV.6. Further Remarks and Summary . 70 IV.6.1. Remarks . 70 IV.6.2. Summary . 71 vi Investigation of CMB Anomalies with Multipole Vectors - Marvin Pinkwart Table Of Contents V. Observed Large Scale CMB Anomalies 73 V.1. Types of Anomalies . 73 V.1.1. Lack of Correlation . 74 V.1.2. Hemispherical Asymmetry . 75 V.1.3. Parity Asymmetry . 75 V.1.4. Cold Spot . 75 V.1.5. Low Multipole Alignment Using MPVs . 75 V.2. Possible Explanations . 76 VI. ESA Planck Mission 79 VI.1. General Information . 79 VI.2. Full Sky Maps . 82 VI.2.1. HEALPix . 82 VI.2.2. General . 85 VI.2.3. Foreground Cleaned CMB Temperature Maps . 85 VI.3. 2015 Results . .l . 90 VII. Planck 2015 MPV CMB Anomalies Up to = 49 93 VII.1. Statistics Used . 93 VII.2. Numerical Method . .a .lm . 97 VII.2.1. Receiving MPVs from Planck Maps . 98 VII.2.2. Calculating MPVs for Random . 100 VII.2.3. p-Values and Likelihoods . 102 VII.2.4. Remark . 104 VII.3. Results . 105 VII.3.1. Distribution of MPVs . 105 VII.3.2. Qualitative Behaviour of Statistics and Appar- ent Similarity of Maps . 106 VII.3.3. SEVEM . 109 VII.3.4. Absence of Intra-Multipole Anomalies . 110 VII.3.5. Outer Correlations . 112 VII.3.6. Cumulative Behaviour . 115 VIII. Conclusion and Outlook 117 Appendices 121 A. Spherical Harmonics 122 A.1. Definition and Properties . 122 Investigation of CMB Anomalies with Multipole Vectors - Marvin Pinkwart vii SO L S Table Of Contents 2 2 A.2. Basis Functions of (3) on ( ) (Multipole Decom- position) . 123 B. Differential Geometry and General Relativity 125 C. Statistics 128 C.1. Bayesian Analysis . 128 C.2. Gaussian Random Variables . 129 C.3. Change of Variables . 130 D. Data 131 E. Plots 132 References 135 viii Investigation of CMB Anomalies with Multipole Vectors - Marvin Pinkwart List of Figures II.1. Distribution of galaxies measured by the 2dF Galaxy Redshift Survey . 5 II.2. Hubble length for different energy contributions . 12 II.3. Thermal History of the Universe. 14 III.1. CMB Intensity spectrum . 22 III.2. CMB Temperature Map . 23 III.3. CMB Angular Band Power Spectrum . 38 IV.1. Visualisation of relationship between distributions . 62 IV.2. 2-point density:l MPV vs. uniform . 68 IV.3. Multipole Vectors and spherical harmonics for different angular scales . ....................... 72 V.1. Lack of Correlation . 74 V.2. Angular Scales Compared to Hubble Scale . 77 VI.1. Planck Mission Timeline . 79 VI.2. Planck Scan Strategy . 81 VI.3. HEALPix pixelation scheme . 83 VI.4. HEALPix NESTED and RING ordering in Mollweide projection . 83 VI.5. Comparison of Full Sky Maps . 86 l; b VII.1. MPVs in Mollweide View . 105 VII.2. MPVs in ( )-plane . 106 VII.3. Outer statistics - All maps, Dipole . 107 VII.4. Inner statistics - All maps . 108 VII.5. Full likelihood plotted for all maps and the Galactic Pole.109 VII.6. Abundance of likelihoods - SEVEM . 110 VII.7. Abundance of inner likelihoods - all . 111 VII.8. Abundance of outer likelihoods - all . 112 VII.10. Odd modes vs. even modes . 113 ix List of Figuresχ VII.9. Outer Likelihood, NILC, Dipole2 and Center . 114 VII.11. Cumulative p-values and . 115 E.1. Full likelihood plotted for all maps and the Galactic Pole.132 E.2. Abundance of likelihoods - SEVEM - Zoom . 133 E.3. Inner likelihood of COMMANDER. 133 E.4. NILC - full and inner likelihood . 134 x Investigation of CMB Anomalies with Multipole Vectors - Marvin Pinkwart List of Tables l D.1. List of NILC MPVs up to = 5 . 131 xi c h kB − ; ; ; (1)Conventions We set = = = 1. and Abbreviations t η (2) We use the mostly plus convention for the metric signature ( 1 1 1 1).