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a/main.tex b/main.tex index 9c549e4..0821640 100644 --- a/main.tex +++ 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}