From 325d12ba7e2b9cecd50b1d58e06b8de66b8b3a20 Mon Sep 17 00:00:00 2001
From: prescientmoon <git@moonythm.dev>
Date: Thu, 20 Mar 2025 17:06:00 +0100
Subject: [PATCH] Remove matroid thingy
---
main.tex | 73 +++++++++++++++++++++++++-------------------------------
1 file changed, 32 insertions(+), 41 deletions(-)
diff --git a/main.tex b/main.tex
index aa16097..ffadfd8 100644
--- a/main.tex
+++ b/main.tex
@@ -75,16 +75,7 @@ code-for-last-col = \color{blue}
\end{enumerate}
\end{frame}
-% should we make is so that these bullet points appear one after another? Yes
-\begin{frame}{Abstracting Independence - Motivation}
- \begin{itemize}
- \item 1
- \item 2
- \end{itemize}
-\end{frame}
-
\include{prerequisites}
-
\include{isospectral}
\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
@@ -92,9 +83,9 @@ code-for-last-col = \color{blue}
\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
- \begin{align*}
- (\Delta + E_{\textcolor{red}{\beta}}^{\textcolor{blue}{M}})\psi_{\textcolor{red}{\beta}}^{{\textcolor{blue}{M}}, i} = 0
- \end{align*}
+ \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
@@ -138,15 +129,15 @@ code-for-last-col = \color{blue}
\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 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 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.
+ \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}
@@ -155,34 +146,34 @@ code-for-last-col = \color{blue}
\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:$\langle \textcolor{blue}{a_{\ell m}} \rangle = 0$ \\
+ 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}
+ \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}{The setup}
-Real CMB observations are affected by instrumental noise and \textit{sky masking}. So, estimating $C_\ell$ accurately requires simulations.
+ 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.
+ 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}
+ 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}
@@ -192,7 +183,7 @@ Which examines $\langle \textcolor{blue}{a_{\ell m}} \rangle$ across multipole r
\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.
+ \textcolor{red}{Remark}In a positively curved space certain multipoles scales being preferred or suppressed in the CMB power spectrum.
\end{frame}