300 lines
11 KiB
TeX
300 lines
11 KiB
TeX
\documentclass{beamer}
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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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\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]{\textcolor{red}{\rotmatop\left(\textcolor{black}{#1}\right)}}
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\renewcommand{\S}{\mathbb{S}}
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\newcommand{\R}{\mathbb{R}}
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\newcommand{\sparkles}{\includegraphics[height=0.9em]{sparkles.png}}
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\newcommand{\ghostzero}{\textcolor{lightgray}{0}}
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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{University of Groningen}
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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\mbox{}\hfill\strut {\insertframenumber/\inserttotalframenumber}\mbox{}\strut\hspace{.5em}\mbox{}%
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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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\definecolor{silver}{RGB}{192, 192, 192}
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\definecolor{darkyellow}{RGB}{186, 142, 35}
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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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\section{CMB Anisotropy of Spherical Spaces}
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\begin{frame}[fragile]{Homogeneous Spherical Spaces}
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\begin{itemize}
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\pause
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\item Finite subgroup $\Gamma \leq \so 4.$
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\pause
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\item The Helmholtz equation on $\textcolor{blue}{M}$ has the spectrum of the underlying manifold 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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\pause
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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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\pause
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\item The set of all possible wave numbers [for which a nonzero solution exists] depends on $\Gamma$ - usually a proper subset of $\mathbb{{N}}$.
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\pause
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\item Solutions to the Helmholtz equation are calculated for each $\Gamma$ and CMB anisotropy $\textcolor{purple}{\frac{\delta T}{T}}$ computed as a sum of spherical harmonics.
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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]{Homogenous Spherical Spaces --- Results}
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\begin{itemize}
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\pause
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\item For the majority of groups $\Gamma$, the anisotropies $\textcolor{purple}{\frac{\delta T}{T}}$ do not coincide with observations.
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\pause
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\item The only groups for which they do 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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\pause
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\end{itemize}
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\begin{figure}[H]
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\centering
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\includegraphics[width=0.4\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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\pause
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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 \textcolor{red}{Multi-connected} space: it has non-contractable loops.
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\item \textcolor{red}{Inhomogeneous} space: it does not look identical from every point in space.
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\pause
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\item Fixing $|\Gamma|=8$, we have \textcolor{red}{three} multi-connected manifolds, up to equivalence: \pause
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\begin{enumerate}
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\item homogeneous: $N3$ and $\lens 8 1 $.
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\item inhomogeneous: $N2 \equiv \lens 8 3 $.
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\end{enumerate}
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\pause
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\item Results: inhomogeneous spaces have \textcolor{red}{more variety} in the CMB anisotropies than homogeneous spaces, because the strength of anisotropy suppression is \textcolor{red}{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}{Statistical Isotropy and Hypothesis}
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\small
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\textbf{Theoretical Expectation:}\\
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From the perspective of an Earth-based observer, we can view the CMB as a function defined on the celestial sphere $\mathbb{S}^2.$\\
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\begin{itemize}
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\pause
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\item The CMB temperature fluctuations can be expanded as:
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\[
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\Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \textcolor{blue}{a_{\ell m}} Y_{\ell m}(\hat{n})
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\]
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\pause
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\item If the universe is \textcolor{red}{isotropic}, the \textbf{mean of the harmonic coefficients} should satisfy:
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\[
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\langle \textcolor{blue}{a_{\ell m}} \rangle = 0
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\]
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\end{itemize}
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\pause
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\textbf{Key Question:}
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- Does the observed CMB data deviate from statistical isotropy?
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- If \( \langle \textcolor{blue}{a_{\ell m}} \rangle \neq 0 \), this suggests a preferred cosmic direction.
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\end{frame}
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\begin{frame}{Methodology: Sky Masking and the Test Statistic}
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\small
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\textbf{Challenges in Real Observations:}
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\begin{itemize}
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\item We cannot observe the full CMB sky due to foreground contamination.
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\item A \textbf{sky mask function} \( A(\hat{n}) \) removes contaminated regions, but introduces bias.
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\pause
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\item The test statistic \( S_i \) is used, with a matrix $W$ to correct for these effects:
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\[
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S_i = \sum_{j} W_{ij} M_j
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\]
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\end{itemize}
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\pause
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\textbf{Monte Carlo Simulations:}
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\begin{itemize}
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\item We generate thousands of \textbf{simulated CMB skies} assuming isotropy.
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\item Each sky has different random \( a_{\ell m} \), drawn from:
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\[
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\textcolor{blue}{a_{\ell m}} \sim \mathcal{N} (0, C_\ell)
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\]
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\item The observed WMAP data is compared against these simulations.
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\end{itemize}
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\end{frame}
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\begin{frame}{Results and Interpretation}
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\small
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\begin{columns}
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\column{0.5\textwidth}
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\begin{figure}
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\centering
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\includegraphics[width=6.2cm]{DSE-Test Graph}
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\caption{The decorrelated band mean test statistic values over multipole ranges [9].}
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\end{figure}
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\column{0.5\textwidth}
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\pause
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\textbf{Key Findings:}
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\begin{itemize}
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\item A statistically significant deviation was found in \( 221 \leq \ell \leq 240 \).
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\item This suggests a potential \textcolor{red}{preferred cosmic direction}.
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\end{itemize}
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\pause
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\textbf{Possible Explanations:}
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\begin{itemize}
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\item A real cosmological signal? → A finite universe or new physics.
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\item A systematic effect? → Technological or theory related limitations.
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\end{itemize}
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\end{columns}
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\end{frame}
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\section{Conclusion}
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\begin{frame}{Conclusion}
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\pause
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\begin{enumerate}
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\item We can infer the \textcolor{blue}{shape} of the universe from its \textcolor{blue}{spectrum}.
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\pause
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\item There are two homogeneous spherical manifolds obtained as $\S^3/\Gamma$ which produce CMB similar to observations.
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\pause
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\item Inhomogeneous spherical spaces exhibit varied behavior of CMB anisotropies.
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\pause
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\item Statistical test results suggest possibilities of \textcolor{red}{finite multi-connected} topology.
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\end{enumerate}
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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 [1] 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 [2] 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 [3] R. Aurich, S. Lustig, and F. Steiner. \textit{CMB anisotropy of spherical spaces}. Classical and Quantum Gravity, 2005.
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\item [4] 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 [5]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 2 --- Electric Boogaloo}
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\begin{itemize}
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\item [6] 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 [7] M. Herlihy and N. Shavit. \textit{The Topological Structure of Asynchronous Computability}. Journal of the ACM, 1996.
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\item [8] A. Ikeda. \textit{On the spectrum of a Riemannian manifold of positive constant curvature}. Osaka Journal of Mathematics, 1980.
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\item [9] 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 [10] 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 3 --- the References Strike Again}
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\begin{itemize}
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\item [11] 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 [12] 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 [13] M. Serri. \textit{Analysis on Manifolds}. AMS Open Math Notes, 2025.
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\item [14] 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 \sparkles\: Thank You! \sparkles
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\end{center}
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\end{frame}
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\end{document}
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