249 lines
9.3 KiB
TeX
249 lines
9.3 KiB
TeX
\documentclass{beamer}
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\usetheme{Warsaw}
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\usecolortheme{lily}
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% boadilla seems to be the only one with enough space for stuff.compile it now it is very toxic but works
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\usepackage{graphicx} % Required for inserting images
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\usepackage{caption}
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\usepackage{subcaption}
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\usepackage{tikz}
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\usepackage{pgfplots}
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\usepackage{verbatim}
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\usepackage{mathtools}
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\pgfplotsset{compat = newest}
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\usetikzlibrary{matrix}
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\usepackage[dvipsnames]{xcolor}
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\usetikzlibrary{perspective}
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\DeclareMathOperator{\divergence}{div}
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\DeclareMathOperator{\lensop}{L}
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\DeclareMathOperator{\rotmatop}{R}
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\DeclareMathOperator{\soop}{SO}
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\newcommand*{\lens}[2]{\lensop\left(#1,#2\right)} %
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\newcommand*{\so}[1]{\soop\left(#1\right)}
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\newcommand*{\rotmat}[1]{\rotmatop\left(#1\right)}
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\renewcommand{\S}{\mathbb{S}}
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\newcommand{\R}{\mathbb{R}}
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% cool color
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\usepackage{xcolor}
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\usepackage{nicematrix}
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\NiceMatrixOptions{
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code-for-first-row = \color{red} ,
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code-for-last-row = \color{blue} ,
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code-for-first-col = \color{blue} ,
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code-for-last-col = \color{blue}
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}
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\title{Computing CMB temperature fluctuations for spherical spaces}
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\author{Adriel Matei, Béla Schneider, Javier Vela, Juš Kocutar}
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%\institute{Presenting: Javier, Juš}
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\date{March 24, 2025}
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\DeclareMathOperator{\cl}{cl}
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\DeclareMathOperator{\rank}{r}
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% Define custom headline
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\setbeamertemplate{headline}{%
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\leavevmode%
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\begin{beamercolorbox}[wd=\paperwidth,ht=2.5ex,dp=1.5ex]{section in head/foot}%
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\mbox{}\hspace{.5em}\strut\insertsectionhead\hfill%
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\end{beamercolorbox}%
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}
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\setbeamercolor{block title}{fg=white, bg=purple!50!black}
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\setbeamercolor{block body}{fg=black, bg=pink!20}
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\setbeamercolor{titlebox}{fg=black,bg=white}
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\begin{document}
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\section{Introduction}
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{
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\usebackgroundtemplate{\includegraphics[height=\paperheight]{cmb.png}}
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\begin{frame}
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\begin{beamercolorbox}[center]{titlebox}%
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\titlepage
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\end{beamercolorbox}
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\end{frame}
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}
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\include{prerequisites}
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\include{isospectral}
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\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
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\begin{itemize}
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\item The Helmholtz equation on $\textcolor{blue}{M}$ has the spectrum as its solutions, and is given by
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\begin{align*}
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(\Delta + E_{\textcolor{red}{\beta}}^{\textcolor{blue}{M}})\psi_{\textcolor{red}{\beta}}^{{\textcolor{blue}{M}}, i} = 0.
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\end{align*}
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\item In fact $E_{\textcolor{red}{\beta}}^m = {\textcolor{red}{\beta}}^2-1$ for $\textcolor{red}{\beta} \in \mathbb{N}$ we call $\textcolor{red}{\beta}$ a wave number.
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\item The set of all possible wave numbers [for which there exists a non-zero solution] depends on $\Gamma$.
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\item Solutions to the Helmholtz equation are calculated for each $\Gamma$ and CMB anisotropy $\frac{\delta T}{T}$ computed.
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\end{itemize}
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\end{frame}
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\begin{comment}
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\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
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\begin{itemize}
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\item If $H = \{1\}$ the set of wave numbers is $\mathbb{N}$
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\item If $H = \mathbb Z/m\mathbb Z$ for odd $m$ the set of wave numbers is $$\{1,3,\cdots, m\}\cup \{m+1, m+2, m+3, \cdots \}.$$
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\item If $H = O^*$, the binary tetrahedral group --- a subgroup of $SO(4)$ of size 24, the set of wave numbers is $$\{ 1,7,9 \} \cup \{13,15,17,\cdots \}.$$
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\end{itemize}
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\end{frame}
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\end{comment}
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\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces — Conclusion}
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\begin{itemize}
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\item For the majority of groups $\Gamma$, the anisotropies $\frac{\delta T}{T}$ do not coincide with observations.
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\item The only groups for which it does are $\Gamma = O^*$ and $\Gamma = I^*$ — the \textcolor{red}{binary octahedral} and \textcolor{red}{binary icosahedral} groups of order 48 and 120 respectively.
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\end{itemize}
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.35\linewidth]{binary-octahedron.png}
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\caption{Graphical representation of the binary tetrahedral group
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[5]}
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\end{figure}
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\end{frame}
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\section{CMB radiation in an inhomogeneous spherical space}
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\begin{frame}{Inhomogeneous spherical space}
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\begin{itemize}
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\item Manifolds of the form $\S^3/\Gamma$ with $\Gamma$ a group acting on $\S^3$.
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\item Multi-connected space: it has non-contractable loops
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\item Inhomogeneous space: it does not look identical from every point in space
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\item Fixing $|\Gamma|=8$, we have three multi-connected manifolds, up to equivalence:
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\begin{enumerate}
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\item homogeneous: $N3$ and $L(8,1)$
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\item inhomogeneous: $N2 \equiv L(8,3)$
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\end{enumerate}
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\item Results: inhomogeneous spaces have more variety in CMB anisotropies than homogeneous spaces, because the strength of anisotropy suppression is observer dependent.
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\end{itemize}
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\end{frame}
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\section{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}
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\begin{frame}{The setup}
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From the perspective of an Earth-based observer, we can view CMB as a function defined on the celestial sphere $\mathbb{S}^2$. We expand $\Delta T(\hat{n})$ using \emph{spherical harmonics}, yielding coefficients $a_{\ell m}$.
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We assume the fluctuations:
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\begin{itemize}
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\item Statistically isotropic and homogeneous in the mean.
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\item Gaussian distribution.
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\item $\textcolor{blue}{a_{\ell m}}$ are independent Gaussian variables.
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\end{itemize}
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Coefficients should satisfy:$\langle \textcolor{blue}{a_{\ell m}} \rangle = 0$.
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\textbf{\textcolor{red}{Remark:}} $\langle \textcolor{blue}{a_{\ell m}} \rangle \neq 0$, \textbf{suggests possible anisotropies or preferred directions in the universe}
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\end{frame}
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\begin{frame}{The setup}
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Real CMB observations are affected by instrumental noise and \emph{sky masking}. As a result, estimating $C_\ell$ accurately requires simulations.
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Given $C^{th}_\ell$, used to generate synthetic realizations of the CMB sky
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\[
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\textcolor{blue}{a_{\ell m}} \sim \mathcal{N} (0, C_\ell^{\text{th}}).
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\]
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Unfortunately, this introduces bias, which must be corrected using a decorrelation matrix $W$, and a \textit{sky mask} matrix $M$ which accounts for issues caused by masking effects. This gives us the test statistic (tests the assumption of statistical isotropy):
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\[
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S_i = \sum_{j} W_{ij} M_j,
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\]
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which examines $\langle \textcolor{blue}{a_{\ell m}} \rangle$ across multipole ranges.
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\textbf{\textcolor{red}{Recall:}} $\langle a_{\ell m} \rangle \neq 0$, \textbf{suggests possible anisotropies or preferred directions in the universe}
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\end{frame}
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\begin{frame}{The Results}
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\begin{figure} [h!]
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\centering
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\includegraphics[width=0.7\linewidth]{DSE-Test Graph}
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\caption{The decorrelated band mean spectrum obtained by the seven-year WMAP data [9].}
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\end{figure}
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\textcolor{red}{Remark}: In a positively curved space certain multipoles scales being preferred or suppressed in the CMB power spectrum.
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\end{frame}
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\section{Conclusion}
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\begin{frame}{Conclusion}
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akbbfbsKJBKJBLJs
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\end{frame}
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\begin{frame}{To summerize}
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\begin{itemize}
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\item 1
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\item 2
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\item 3
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\end{itemize}
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\end{frame}
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\section{References}
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\begin{frame}{References}
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\begin{itemize}
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\item R. Aurich, P. Kramer, and S. Lustig. \textit{Cosmic microwave background radiation in an inhomogeneous spherical space}. Physica Scripta, 2011.
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\item R. Aurich, P. Kramer, and S. Lustig. \textit{A survey of lens spaces and large-scale CMB anisotropy}. Monthly Notices of the Royal Astronomical Society, 2012.
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\item R. Aurich, S. Lustig, and F. Steiner. \textit{CMB anisotropy of spherical spaces}. Classical and Quantum Gravity, 2005.
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\item P. Bielewicz, K. M. G´orski, and A. J. Banday. \textit{Low-order multipole maps of cosmic microwave background anisotropy derived from WMAP}. Oxford University Press, 2004.
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\item G. Egan. \textit{Symmetries and the 24-cell}. 2021. Accessed at https://www.gregegan.net/SCIENCE/24-cell/24-cell.html.
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\end{itemize}
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\end{frame}
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\begin{frame}{References}
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\begin{itemize}
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\item N. Jarosik et. al. \textit{Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results}. The American Astronomical Society, 2011.
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\item M. Herlihy and N. Shavit. \textit{The Topological Structure of Asynchronous Computability}. Journal of the ACM, 1996.
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\item A. Ikeda. \textit{On the spectrum of a Riemannian manifold of positive constant curvature}. Osaka Journal of Mathematics, 1980.
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\item D. Kashino, K. Ichiki, and T. Takeuchi. \textit{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}. Physics Review, 2012.
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\item P. Labrana. \textit{Emergent Universe Scenario and the Low CMB Multipoles}. Physical Review, 2013.
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\end{itemize}
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\end{frame}
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\begin{frame}{References}
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\begin{itemize}
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\item E. A. Lauret and B. Linowitz. \textit{The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems}. New York Journal of Mathematics, 2025.
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\item R. Lehoucq, J. Weeks, J. P. Uzan, E. Gausmann, and J.P. Luminet. \textit{Eigenmodes of three-dimensional spherical spaces and their application to cosmology}. Classical and Quantum Gravity, 2002.
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\item M. Serri. \textit{Analysis on Manifolds}. AMS Open Math Notes, 2025.
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\item B. Tai-Dana, T. Bryson, and J. Terilla. \textit{Topology, a categorical approach}. The MIT Press, 2020.
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\end{itemize}
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\end{frame}
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\begin{frame}{}
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\begin{figure}[H]
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\centering
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\includegraphics[width=.4\paperwidth]{qrcode.png}
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\end{figure}
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\begin{center}
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\huge Thank You!
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\end{center}
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\end{frame}
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\end{document}
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