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+++ b/main.tex
@@ -59,10 +59,8 @@ code-for-last-col = \color{blue}
\setbeamercolor{block title}{fg=white, bg=purple!50!black}
\setbeamercolor{block body}{fg=black, bg=pink!20}
-
-
-
\setbeamercolor{titlebox}{fg=black,bg=white}
+
\begin{document}
\section{Introduction}
@@ -70,7 +68,7 @@ code-for-last-col = \color{blue}
{
\usebackgroundtemplate{\includegraphics[height=\paperheight]{cmb.png}}
\begin{frame}
- \begin{beamercolorbox}[center]{titlebox}%
+ \begin{beamercolorbox}[center]{titlebox}
\titlepage
\end{beamercolorbox}
\end{frame}
@@ -79,12 +77,11 @@ code-for-last-col = \color{blue}
\include{prerequisites}
\include{isospectral}
-\section{CMB Anisotropy of Spherical Spaces
-}
+\section{CMB Anisotropy of Spherical Spaces}
\begin{frame}[fragile]{Homogeneous Spherical Spaces}
\begin{itemize}
- \item Finite subgroup $\Gamma \leq \so 4.$
+ \item Finite subgroup $\Gamma \leq \so 4.$
\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.
@@ -149,81 +146,81 @@ code-for-last-col = \color{blue}
\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}
- \item The CMB temperature fluctuations can be expanded as:
- \[
- \Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\hat{n})
- \]
- \item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
- \[
- \langle a_{\ell m} \rangle = 0
- \]
- \end{itemize}
+ \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}
+ \item The CMB temperature fluctuations can be expanded as:
+ \[
+ \Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\hat{n})
+ \]
+ \item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
+ \[
+ \langle a_{\ell m} \rangle = 0
+ \]
+ \end{itemize}
- \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.
+ \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.
- \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}
-
- \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}
+ \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.
+ \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}
+
+ \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}
- \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}
+ \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}
+ \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}
- \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}
+ \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}
\begin{enumerate}
- \item We can infer the shape of the universe from its spectrum
- \item There are two homogeneous spherical manifolds obtained as $\S^3/\Gamma$ which produce CMB similar to observations
- \item Inhomogeneous spherical spaces exhibit varied behavior of CMB anisotropies
- \item Statistical test results suggest possibilities of finite multi-connected topology
+ \item We can infer the shape of the universe from its spectrum
+ \item There are two homogeneous spherical manifolds obtained as $\S^3/\Gamma$ which produce CMB similar to observations
+ \item Inhomogeneous spherical spaces exhibit varied behavior of CMB anisotropies
+ \item Statistical test results suggest possibilities of finite multi-connected topology
\end{enumerate}
\end{frame}