ddd/bachelor-prep-presentation
1
Fork 0
Code Activity

Clean up formatting

Browse source
This commit is contained in:
prescientmoon 2025-03-21 12:55:41 +01:00
parent 909e0b03db
commit ea3caa20e7
Signed by: prescientmoon
SSH key fingerprint: SHA256:UUF9JT2s8Xfyv76b8ZuVL7XrmimH4o49p4b+iexbVH4
2 changed files with 64 additions and 67 deletions

BIN
main.pdf
View file

Binary file not shown.

131
main.tex
View file

@ -59,10 +59,8 @@ code-for-last-col = \color{blue}
\setbeamercolor{block title}{fg=white, bg=purple!50!black} \setbeamercolor{block title}{fg=white, bg=purple!50!black}
\setbeamercolor{block body}{fg=black, bg=pink!20} \setbeamercolor{block body}{fg=black, bg=pink!20}
\setbeamercolor{titlebox}{fg=black,bg=white} \setbeamercolor{titlebox}{fg=black,bg=white}
\begin{document} \begin{document}
\section{Introduction} \section{Introduction}
@ -70,7 +68,7 @@ code-for-last-col = \color{blue}
{ {
\usebackgroundtemplate{\includegraphics[height=\paperheight]{cmb.png}} \usebackgroundtemplate{\includegraphics[height=\paperheight]{cmb.png}}
\begin{frame} \begin{frame}
\begin{beamercolorbox}[center]{titlebox}% \begin{beamercolorbox}[center]{titlebox}
\titlepage \titlepage
\end{beamercolorbox} \end{beamercolorbox}
\end{frame} \end{frame}
@ -79,12 +77,11 @@ code-for-last-col = \color{blue}
\include{prerequisites} \include{prerequisites}
\include{isospectral} \include{isospectral}
\section{CMB Anisotropy of Spherical Spaces \section{CMB Anisotropy of Spherical Spaces}
}
\begin{frame}[fragile]{Homogeneous Spherical Spaces} \begin{frame}[fragile]{Homogeneous Spherical Spaces}
\begin{itemize} \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 \item The Helmholtz equation on $\textcolor{blue}{M}$ has the spectrum of the underlying manifold as its solutions, and is given by
\begin{align*} \begin{align*}
(\Delta + E_{\textcolor{red}{\beta}}^{\textcolor{blue}{M}})\psi_{\textcolor{red}{\beta}}^{{\textcolor{blue}{M}}, i} = 0. (\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} \section{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}
\begin{frame}{Statistical Isotropy and Hypothesis} \begin{frame}{Statistical Isotropy and Hypothesis}
\small \small
\textbf{Theoretical Expectation:}\\ \textbf{Theoretical Expectation:}\\
From the perspective of an Earth-based observer, we can view the CMB as a function defined on the celestial sphere $\mathbb{S}^2$.\\ 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} \begin{itemize}
\item The CMB temperature fluctuations can be expanded as: \item The CMB temperature fluctuations can be expanded as:
\[ \[
\Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\hat{n}) \Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\hat{n})
\] \]
\item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy: \item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
\[ \[
\langle a_{\ell m} \rangle = 0 \langle a_{\ell m} \rangle = 0
\] \]
\end{itemize} \end{itemize}
\textbf{Key Question:} \textbf{Key Question:}
- Does the observed CMB data deviate from statistical isotropy? - Does the observed CMB data deviate from statistical isotropy?
- If \( \langle a_{\ell m} \rangle \neq 0 \), this suggests a preferred cosmic direction. - If \( \langle a_{\ell m} \rangle \neq 0 \), this suggests a preferred cosmic direction.
\end{frame} \end{frame}
\begin{frame}{Methodology: Sky Masking and the Test Statistic} \begin{frame}{Methodology: Sky Masking and the Test Statistic}
\small \small
\textbf{Challenges in Real Observations:} \textbf{Challenges in Real Observations:}
\begin{itemize} \begin{itemize}
\item We cannot observe the full CMB sky due to foreground contamination. \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 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: \item The test statistic \( S_i \) is used, with a matrix $W$ to correct for these effects:
\[ \[
S_i = \sum_{j} W_{ij} M_j S_i = \sum_{j} W_{ij} M_j
\] \]
\end{itemize} \end{itemize}
\textbf{Monte Carlo Simulations:} \textbf{Monte Carlo Simulations:}
\begin{itemize} \begin{itemize}
\item We generate thousands of \textbf{simulated CMB skies} assuming isotropy. \item We generate thousands of \textbf{simulated CMB skies} assuming isotropy.
\item Each sky has different random \( a_{\ell m} \), drawn from: \item Each sky has different random \( a_{\ell m} \), drawn from:
\[ \[
a_{\ell m} \sim \mathcal{N} (0, C_\ell) a_{\ell m} \sim \mathcal{N} (0, C_\ell)
\] \]
\item The observed WMAP data is compared against these simulations. \item The observed WMAP data is compared against these simulations.
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame}{Results and Interpretation} \begin{frame}{Results and Interpretation}
\small \small
\begin{columns} \begin{columns}
\column{0.5\textwidth} \column{0.5\textwidth}
\begin{figure} \begin{figure}
\centering \centering
\includegraphics[width=6.2cm]{DSE-Test Graph} \includegraphics[width=6.2cm]{DSE-Test Graph}
\caption{The decorrelated band mean test statistic values over multipole ranges [9].} \caption{The decorrelated band mean test statistic values over multipole ranges [9].}
\end{figure} \end{figure}
\column{0.5\textwidth} \column{0.5\textwidth}
\textbf{Key Findings:} \textbf{Key Findings:}
\begin{itemize} \begin{itemize}
\item A statistically significant deviation was found in \( 221 \leq \ell \leq 240 \). \item A statistically significant deviation was found in \( 221 \leq \ell \leq 240 \).
\item This suggests a potential \textcolor{red}{preferred cosmic direction}. \item This suggests a potential \textcolor{red}{preferred cosmic direction}.
\end{itemize} \end{itemize}
\textbf{Possible Explanations:} \textbf{Possible Explanations:}
\begin{itemize} \begin{itemize}
\item A real cosmological signal? → A finite universe or new physics. \item A real cosmological signal? → A finite universe or new physics.
\item A systematic effect? → Foreground contamination or instrumental noise. \item A systematic effect? → Foreground contamination or instrumental noise.
\end{itemize} \end{itemize}
\end{columns} \end{columns}
\end{frame} \end{frame}
\section{Conclusion} \section{Conclusion}
\begin{frame}{Conclusion} \begin{frame}{Conclusion}
\begin{enumerate} \begin{enumerate}
\item We can infer the shape of the universe from its spectrum \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 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 Inhomogeneous spherical spaces exhibit varied behavior of CMB anisotropies
\item Statistical test results suggest possibilities of finite multi-connected topology \item Statistical test results suggest possibilities of finite multi-connected topology
\end{enumerate} \end{enumerate}
\end{frame} \end{frame}