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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}

\section{Introduction}

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\include{prerequisites}
\include{isospectral}

\section{CMB Anisotropy of Spherical Spaces}

\begin{frame}[fragile]{Homogeneous Spherical Spaces}
	\begin{itemize}
		\pause
		\item Finite subgroup $\Gamma \leq \so 4.$
		      \pause
		\item The Helmholtz equation on $\textcolor{blue}{M}$ has the spectrum of the underlying manifold as its solutions, and is given by
		      \begin{align*}
			      (\Delta + E_{\textcolor{red}{\beta}}^{\textcolor{blue}{M}})\psi_{\textcolor{red}{\beta}}^{{\textcolor{blue}{M}}, i} = 0.
		      \end{align*}
		      \pause
		\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.
		      \pause
		\item The set of all possible wave numbers [for which a nonzero solution exists] depends on $\Gamma$ - usually a proper subset of $\mathbb{{N}}$.
		      \pause
		\item Solutions to the Helmholtz equation are calculated for each $\Gamma$ and CMB anisotropy $\frac{\delta T}{T}$ computed as a sum of spherical harmonics.
	\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]{Homogenous Spherical Spaces --- Results}
	\begin{itemize}
		\pause
		\item For the majority of groups $\Gamma$, the anisotropies $\frac{\delta T}{T}$ do not coincide with observations.
		      \pause
		\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.
		      \pause
	\end{itemize}
	\begin{figure}[H]
		\centering
		\includegraphics[width=0.4\linewidth]{binary-octahedron.png}
		\caption{Graphical representation of the binary tetrahedral group
				[5]}
	\end{figure}

\end{frame}

\section{CMB Radiation in an Inhomogeneous Spherical Space}

\begin{frame}{Inhomogeneous spherical space}

	\begin{itemize}
		\item Manifolds of the form $\S^3/\Gamma$ with $\Gamma$ a group acting on $\S^3$.
		\item Multi-connected space: it has non-contractable loops.
		\item Inhomogeneous space: it does not look identical from every point in space.
        \pause
		\item Fixing $|\Gamma|=8$, we have three multi-connected manifolds, up to equivalence: \pause
		      \begin{enumerate}
			      \item homogeneous: $N3$ and $L(8,1)$
			      \item inhomogeneous: $N2 \equiv L(8,3)$
		      \end{enumerate}
		      \pause
		\item Results: inhomogeneous spaces have more variety in the CMB anisotropies than homogeneous spaces, because the strength of anisotropy suppression is observer dependent.
	\end{itemize}


\end{frame}

\section{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}

\begin{frame}{Statistical Isotropy and Hypothesis}
	\small
	\textbf{Theoretical Expectation:}\\
	From the perspective of an Earth-based observer, we can view the CMB as a function defined on the celestial sphere $\mathbb{S}^2$.\\
	\begin{itemize}
		\pause
		\item The CMB temperature fluctuations can be expanded as:
		      \[
			      \Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\hat{n})
		      \]
		      \pause
		\item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
		      \[
			      \langle a_{\ell m} \rangle = 0
		      \]
	\end{itemize}
	\pause
	\textbf{Key Question:}
	- Does the observed CMB data deviate from statistical isotropy?
	- If \( \langle a_{\ell m} \rangle \neq 0 \), this suggests a preferred cosmic direction.
\end{frame}

\begin{frame}{Methodology: Sky Masking and the Test Statistic}
	\small
	\textbf{Challenges in Real Observations:}
	\begin{itemize}
		\item We cannot observe the full CMB sky due to foreground contamination.

		\item A \textbf{sky mask function} \( A(\hat{n}) \) removes contaminated regions, but introduces bias.
		      \pause
		\item The test statistic \( S_i \) is used, with a matrix $W$ to correct for these effects:
		      \[
			      S_i = \sum_{j} W_{ij} M_j
		      \]
	\end{itemize}
	\pause
	\textbf{Monte Carlo Simulations:}
	\begin{itemize}
		\item We generate thousands of \textbf{simulated CMB skies} assuming isotropy.
		\item Each sky has different random \( a_{\ell m} \), drawn from:
		      \[
			      a_{\ell m} \sim \mathcal{N} (0, C_\ell)
		      \]
		\item The observed WMAP data is compared against these simulations.
	\end{itemize}
\end{frame}


\begin{frame}{Results and Interpretation}
	\small
	\begin{columns}
		\column{0.5\textwidth}
		\begin{figure}
			\centering
			\includegraphics[width=6.2cm]{DSE-Test Graph}
			\caption{The decorrelated band mean test statistic values over multipole ranges [9].}
		\end{figure}
		\column{0.5\textwidth}
		\pause
		\textbf{Key Findings:}
		\begin{itemize}
			\item A statistically significant deviation was found in \( 221 \leq \ell \leq 240 \).
			\item This suggests a potential \textcolor{red}{preferred cosmic direction}.
		\end{itemize}
		\pause
		\textbf{Possible Explanations:}
		\begin{itemize}
			\item A real cosmological signal? → A finite universe or new physics.
			\item A systematic effect? → Foreground contamination or instrumental noise.
		\end{itemize}
	\end{columns}
\end{frame}


\section{Conclusion}
\begin{frame}{Conclusion}

	\pause
	\begin{enumerate}
	    \item We can infer the shape of the universe from its spectrum.

\pause
        \item There are two homogeneous spherical manifolds obtained as $\S^3/\Gamma$ which produce CMB similar to observations.

\pause
        \item Inhomogeneous spherical spaces exhibit varied behavior of CMB anisotropies.

\pause
        \item Statistical test results suggest possibilities of finite multi-connected topology.
	\end{enumerate}
\end{frame}


\section{References}
\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}


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		\huge Thank You!
	\end{center}
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