Clean up formatting
This commit is contained in:
parent
909e0b03db
commit
ea3caa20e7
SSH key fingerprint:
SHA256:UUF9JT2s8Xfyv76b8ZuVL7XrmimH4o49p4b+iexbVH4
131
main.tex
131
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}
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue