Merge branch 'master' of https://git.overleaf.com/67dbe68df0414d63b309f98a

main.tex

@@ -89,12 +89,14 @@ code-for-last-col = \color{blue}
 
 \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 $H \leq SO(4)$ finite
+		\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
-		      $$(\Delta + E_\textcolor{red}{\beta}^\textcolor{blue}{M})\psi_\textcolor{red}{\beta}^{\textcolor{blue}{M}, i} = 0$$
+		\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 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 $H$
 
@@ -133,39 +135,68 @@ code-for-last-col = \color{blue}
 \end{frame}
 
 
-
-
-\begin{frame}[fragile]{Article 2}
-	vsgaegaagjadhvakva
-\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 one inhomogeneous and two homogeneous multi-connected manifolds, up to equivalence.
+		\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}
 
-\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}
+\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}{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}
+\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=0.6\linewidth]{DSE-Test Graph}
+		\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}
 	akbbfbsKJBKJBLJs
 \end{frame}
@@ -180,7 +211,7 @@ code-for-last-col = \color{blue}
 
 \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.