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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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\section{Introduction}
\begin{frame}{Outline}
\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
\begin{align*}
(\Delta + E_{\textcolor{red}{\beta}}^{\textcolor{blue}{M}})\psi_{\textcolor{red}{\beta}}^{{\textcolor{blue}{M}}, i} = 0
\end{align*}
\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]}
\end{figure}
\end{frame}
\begin{frame}{CMB radiation in an inhomogeneous spherical space}
\begin{itemize}
\item Manifolds of the form $\mathbb S/\Gamma$ with $\Gamma$ a group acting on $\mathbb S^3$.
\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 three multi-connected manifolds, up to equivalence:
\begin{enumerate}
\item homogeneous: $N3$ and $L(8,1)$
\item inhomogeneous: $N2 \equiv L(8,3)$
\end{enumerate}
\item Results: inhomogeneous spaces have more variety in CMB anisotropies than homogeneous spaces, because observer dependency changes the suppression of CMB anisotropies.
\end{itemize}
\end{frame}
\section{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}
\begin{frame}{The setup}
From the perspective of an Earth-based observer, we can view CMB as a function defined on the celestial sphere $\mathbb{S}^2$. \\
So, $\Delta T(\hat{n})$ expanded using \textit{spherical harmonics}, with $a_{\ell m}$ its coefficients.\\
We assume the fluctuations:
\begin{itemize}
\item Statistically isotropic and homogeneous in the mean.
\item Gaussian distribution.
\item $\textcolor{blue}{a_{\ell m}}$ are independent Gaussian variables.
\end{itemize}
Coefficients should satisfy:$\langle \textcolor{blue}{a_{\ell m}} \rangle = 0$ \\
\textbf{\textcolor{red}{Remark:}} $\langle \textcolor{blue}{a_{\ell m}} \rangle \neq 0$, \textbf{suggests possible anisotropies or preferred directions in the universe}
\end{frame}
\begin{frame}{The setup}
Real CMB observations are affected by instrumental noise and \textit{sky masking}. So, estimating $C_\ell$ accurately requires simulations.
Given $C^{th}_\ell$, used to generate synthetic realizations of the CMB sky
\[
\textcolor{blue}{a_{\ell m}} \sim \mathcal{N} (0, C_\ell^{\text{th}}).
\]
However, 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.
Giving us the test statistic (tests the assumption of statistical isotropy):
\[
S_i = \sum_{j} W_{ij} M_j,
\]
Which examines $\langle \textcolor{blue}{a_{\ell m}} \rangle$ across multipole ranges. \\
\textbf{\textcolor{red}{Recall:}} $\langle a_{\ell m} \rangle \neq 0$, \textbf{suggests possible anisotropies or preferred directions in the universe}
\end{frame}
\begin{frame}{The Results}
\begin{figure} [h!]
\centering
\includegraphics[width=0.7\linewidth]{DSE-Test Graph}
\caption{The decorrelated band mean spectrum obtained by the seven-year WMAP data [9].}
\end{figure}
\textcolor{red}{Remark}In a positively curved space certain multipoles scales being preferred or suppressed in the CMB power spectrum.
\end{frame}
\section{Conclusion}
\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}
\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}
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\huge Thank You!
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