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\title{Computing CMB temperature fluctuations for spherical spaces}
\author{Adriel Matei, Béla Schneider, Javier Vela, Juš Kocutar}
%\institute{Presenting: Javier, Juš}


\date{March 24, 2025}

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\begin{document}


\maketitle

\section{Introduction}

\begin{frame}{Outline}
	%Should we make an outline?
	\begin{enumerate}%[&lt;+-&gt;]
		\item Motivation
		\item Part 1
		\item Part 2
		\item Part 3
		\item etc
	\end{enumerate}
\end{frame}

% should we make is so that these bullet points appear one after another? Yes
\begin{frame}{Abstracting Independence - Motivation}
	\begin{itemize}
		\item 1
		\item 2
	\end{itemize}

\end{frame}

\include{prerequisites}
\include{isospectral}

\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
	\begin{itemize}
		\item Manifolds $\textcolor{blue}{M} := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$

		\item Helmholtz equation on $\textcolor{blue}{M}$ given by
		      $$(\Delta + E_\textcolor{red} {\beta}^\textcolor{blue}{M})\psi_\textcolor{red}{\beta}^{\textcolor{blue}{M}, i} = 0$$

		\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

		\item The set of all possible wave numbers [for which there exists a non-zero solution] depends on $H$

		\item Solutions to the Helmholtz equation are calculated for each $H$ and CMB anisotropies $\frac{\delta T}{T}$ computed
	\end{itemize}
\end{frame}


\begin{comment}

\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
	\begin{itemize}
		\item If $H = \{1\}$ the set of wave numbers is $\mathbb{N}$
		\item If $H = \mathbb Z/m\mathbb Z$ for odd $m$ the set of wave numbers is $$\{1,3,\cdots, m\}\cup \{m+1, m+2, m+3, \cdots \}.$$
		\item  If $H = O^*$, the binary tetrahedral group --- a subgroup of $SO(4)$ of size 24, the set of wave numbers is $$\{ 1,7,9 \} \cup \{13,15,17,\cdots \}.$$
	\end{itemize}
\end{frame}

\end{comment}


\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces - Conclusion}
	\begin{itemize}
		\item For the majority of groups $H$ the anisotropies $\frac{\delta T}{T}$ do not coincide with observations
		\item The only for which it does are $H = O^*$ and $H = I^*.$ the \textcolor{red}{binary octahedral} and \textcolor{red}{binary icosahedral} groups of order 48 and 120


	\end{itemize}
	\begin{figure}[H]
		\centering
		\includegraphics[width=0.4\linewidth]{binary-octahedron.png}
		\caption{Graphical representation of the binary tetrahedral group
				[5]}
		\label{fig:binary-octahedron}
	\end{figure}

\end{frame}




\begin{frame}[fragile]{Article 2}
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\end{frame}


\begin{frame}{CMB radiation in an inhomogeneous spherical space}

	\begin{itemize}
		\item Multi-connected space: it has non-contractable loops
		\item Inhomogeneous space: it does not look identical from every point in space
		\item fixing $|\Gamma|=8$, we have one inhomogeneous and two homogeneous multi-connected manifolds, up to equivalence.
	\end{itemize}

\end{frame}

\begin{frame}{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}
	\begin{itemize}
		\item this section goes through an overview of a mathematical method for computing and interpreting CMB temperature fluctuations in spherical spaces, incorporating statistical isotropy, covariance structures, and Monte Carlo simulations, which are later compared with the Wilkinson Microwave Anisotropy Probe (WMAP) seven-year observation data.
		      \textit{statistical isotropy}, meaning that the statistical properties of the CMB should be the same in all directions. Testing this assumption is essential, as deviations from isotropy could indicate alternative topologies for the universe.
	\end{itemize}
\end{frame}

\begin{frame}{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}
	\begin{figure} [h!]
		\centering
		\includegraphics[width=0.6\linewidth]{DSE-Test Graph}
		\caption{The decorrelated band mean spectrum obtained by the seven-year WMAP data [9].}
		\label{fig:Decorrelated Statistical Spectrum}
	\end{figure}
\end{frame}


\begin{frame}{Conclusion}
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\end{frame}

\begin{frame}{To summerize}
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	\begin{itemize}
		\item 1
		\item 2
		\item 3
	\end{itemize}

\end{frame}


\begin{frame}{References}
	\begin{itemize}
		\item R. Aurich, P. Kramer, and S. Lustig. \textit{Cosmic microwave background radiation in an inhomogeneous spherical space}. Physica Scripta, 2011.
		\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.
		\item  R. Aurich, S. Lustig, and F. Steiner. \textit{CMB anisotropy of spherical spaces}. Classical and Quantum Gravity, 2005.
		\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.
		\item G. Egan. \textit{Symmetries and the 24-cell}. 2021. Accessed at https://www.gregegan.net/SCIENCE/24-cell/24-cell.html.
	\end{itemize}
\end{frame}


\begin{frame}{References}
	\begin{itemize}

		\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.
		\item M. Herlihy and N. Shavit. \textit{The Topological Structure of Asynchronous Computability}. Journal of the ACM, 1996.
		\item A. Ikeda. \textit{On the spectrum of a Riemannian manifold of positive constant curvature}. Osaka Journal of Mathematics, 1980.
		\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.
		\item P. Labrana. \textit{Emergent Universe Scenario and the Low CMB Multipoles}. Physical Review, 2013.
	\end{itemize}
\end{frame}

\begin{frame}{References}
	\begin{itemize}
		\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.
		\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.
		\item M. Serri. \textit{Analysis on Manifolds}. AMS Open Math Notes, 2025.
		\item B. Tai-Dana, T. Bryson, and J. Terilla. \textit{Topology, a categorical approach}. The MIT Press, 2020.
	\end{itemize}
\end{frame}


\begin{frame}{}
	\begin{center}
		\huge Thank You!
	\end{center}
\end{frame}

\end{document}
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