Cosmic Reionisation and the Primordial Fluctuations in the Universe Alexander van Engelen Master of Science Department of Physics McGill University Montreal, Quebec August 31, 2007 A thesis submitted to McGill University in partial fulfilment of the requirements of the degree of Master of Science © Alexander van Engelen 2007 Library and Bibliothèque et 1+1 Archives Canada Archives Canada Published Heritage Direction du Bran ch Patrimoine de l'édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A ON4 Ottawa ON K1A ON4 Canada Canada Your file Votre référence ISBN: 978-0-494-51352-1 Our file Notre référence ISBN: 978-0-494-51352-1 NOTICE: AVIS: The author has granted a non­ L'auteur a accordé une licence non exclusive exclusive license allowing Library permettant à la Bibliothèque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l'Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans loan, distribute and sell theses le monde, à des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, électronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriété du droit d'auteur ownership and moral rights in et des droits moraux qui protège cette thèse. this thesis. Neither the thesis Ni la thèse ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent être imprimés ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission. ln compliance with the Canadian Conformément à la loi canadienne Privacy Act some supporting sur la protection de la vie privée, forms may have been removed quelques formulaires secondaires from this thesis. ont été enlevés de cette thèse. While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. ••• Canada ACKNOWLEDGEMENTS 1 thank my supervisor, Gil Holder, for inspiring this work and for helping me immensely during these first two years of graduate school. 1 look forward to more years of working with him as 1 continue my graduate studies. 1 also wish to thank Paul Mercure for technical support on the computer cluster on which sorne of this work was performed; Gaelen Marsden for sharing sorne plotting and other scripts; Sebastien Guillot for the translation of the abstract on page (iv); and my fellow group- and office-mates for much useful discussion. This work has made use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA), support for which is provided by the NASA Office of Space Science; and the publicly-available CosmoMC package, including the CAMB program, by Antony Lewis and Sarah Bridie. Finally 1 would like to thank my mother for her support all these years. ii ABSTRACT We investigate the effect of allowing freedom in the primordial power spec­ trum of curvature perturbations upon the measurement of other cosmological parameters, in particular the Thomson optical depth due to cosmic reionisation which is present in cosmic microwave background (CMB) observations. We find that the constraint on the optical depth from Wilkinson Microwave Anisotropy Probe (WMAP) data broadens by approximately 10% upon allowing spectral freedom on large scales, and by a slightly larger factor when considering data from future experiments with lower noise in measurements of CMB polarisation. We also present a reconstruction of the primordial power spectrum on the largest scales from WMAP, which is jointly obtained from this analysis. iii RÉSUMÉ Nous analysons les effets de l'ajout de degrés de liberté aux perturbations de la courbature originelle sur les mesures des autres paramètres cosmologiques. Nous nous pencherons en particulier sur la profondeur optique de Thomson due a la ré-ionisation que l'on observe dans le fond diffus cosmologique (FDC). Nous re­ marquons que la contrainte pose sur la profondeur optique des donnes du satellite Wilkinson Microwave Anisotropy Probe (WMAP) s'élargie de approximative­ ment 10%, tout en allouant, à grande échelle, de la liberté au spectre. De plus, ce pourcentage augmente légèrement si l'on considère des données d'expériences à venir avec moins de bruit dans les mesures de la polarisation du FDC. Enfin, nous présentons une reconstruction du spectre de puissance originel de WMAP à plus grande chelle, obtenue conjointement a cette analyse. lV TABLE OF CONTENTS ACKNOWLEDGEMENTS 11 ABSTRACT iii RÉSUMÉ .. lV LIST OF FIGURES vii 1 Introduction . 1 2 Background . 7 2.1 The Friedmann Universe and perturbations . 7 2.2 Boltzmann and fluid equations . 11 2.3 General solution . 13 2.4 Super-horizon modes ......... 15 2.5 Inflation and the origin of fluctuations 17 2.6 The Observed CMB .......... 19 2.7 Reionisation and the peak in the EE spectrum 20 2.8 Deviations from powerlaw spectra, and constraining them from data ....... 25 3 Application to current data 30 3.1 Choice of parameterisation 30 3.2 CMB power spectra . 32 3.3 Using Monte Carlo Markov chains to constrain cosmology . 33 3.4 Results ....... 42 4 Forecasting for future data . 49 4.1 The Fisher information matrix for CMB experiments 49 4.2 Application 56 4.3 Results ......................... 57 v 5 Conclusions . 63 vi LIST OF FIGURES Figure page 2-1 Sorne sample results of cosmological perturbation theory: the tem­ perature Cz's for a wide range of l's. The three regions indicated in the figure are SW, the super-horizon Sachs-Wolfe plateau; AP, the acoustic peaks; and SD, the region where there is noticeable Silk damping. The vertical lines are not meant to denote hard bound- aries between the regions, since there is sorne overlap. 16 2-2 The effect of changing the Thompson optical depth to reionisation. The CMB EE power spectrum is plotted for an instant reionisa­ tion model with seven values of T between 0 and 0.32; this corre­ sponds to redshifts of reionisation between 0 (i.e. no reionisation) and 28. These plots are obtained using the cosmological Boltz­ 2 2 mann code CAMB with Ddmh = 0.12, Dbh = 0.022, Ho= 70km s -1 M pc-1 ............................... 23 3-1 Primordial (top panel), CMB TT (centre panel), and CMB EE (bot- tom panel) power spectra for several 11-parameter "broken" spec- trum models, showing how the freedom allowed in the primordial spectrum propagates into the CMB. In addition to the seven pa- rameters describing the primordial power spectrum, the four other 2 2 parameters allowed to vary are {Dbh , Dch , h, Zre}, which for these models are {0.028, 0.092, 0.89, 13.3} (solid line), {0.050, 0.212, O. 72, 10.1} (dotted line), {0.039, 0.185, 0.75, 7.4} (dashed line) and {0.038, 0.165, 0.77, 4.9} (dot-dashed line). Note that for l < 20 the values of CfE increase with increasing Zre· The vertical grey bars are the binned uncer- tainty from the WMAP 3-year data from Hinshaw et al. (2007) (TT data) and from Page et al. (2007) (EE data). These particu- lar models are not good fits to the data but are shown for demon- strative purposes; they have values of -2ln L between 3586 and 3774. 34 vii 3-2 The effect on the err and crE spectra of doubling or nulling each of the first 5 power spectrum parameters. Note that for the TE case the y-axis plots only one power of l. 35 3-3 Samples from a Markov chain for the 11-parameter model, showing (top) the angle subtended by the sound horizon, a slow parame­ ter, and (bottom) the amplitude of the primordial power spectrum 3 1 at k4 = 1.5 x 10- Mpc- . Note that the fast parameter changes position much more often than the slow parameter. 39 3-4 The constraints on the six power spectrum amplitude parameters, Bi = ln(1010 Ai), in the 11-parameter model. The 68% and 95% confidence limits for the amplitude in each bin, marginalised over all other cosmological parameters, shown in black and grey, re­ spectively. The crosses indicate the values of the parameters corre­ sponding to the maximum-likelihood point in the Markov chains. The dashed line is the best fit powerlaw to the WMAP data. The first 2a error bar reaches to low values and is not effectively con- strained away from zero. 43 3-5 The constraints from the MCMC on the six power spectrum ampli­ tude parameters that are allowed to vary in the model. The thick and thin lines indicate 68% and 95% confidence limits, respectively; the crosses indicate the values of the parameters corresponding to the maximum-likelihood point in the chains. The constraint on 1 the seventh parameter, the spectral index above 0.05 Mpc- , is not shown. As in Fig. 3-4 the first power spectrum parameter is not constrained to be above zero for the 2a contour. 45 3-6 The matrix indicating the degree of correlation between pairs of am­ plitude parameters (Bi, Bj), as defined in eq. 3.7. The rows and columns correspond to the amplitude parameters B 1 through B6 , and the pairings in the bottom triangle of this matrix correspond to the panels in Fig. 3-5. 46 viii 3-7 The constraints on the four non-power spectrum parameters which are allowed to vary in the model, with (grey) the ACDM model and (black) the 11-parameter model described in the text.