diff --git a/main.tex b/main.tex
index 63009bb..8f96cfb 100644
--- a/main.tex
+++ b/main.tex
@@ -1,5 +1,6 @@
\documentclass{beamer}
\usetheme{Warsaw}
+\usepackage{amsmath, amssymb}
\usecolortheme{lily}
% boadilla seems to be the only one with enough space for stuff.compile it now it is very toxic but works
@@ -78,18 +79,22 @@ code-for-last-col = \color{blue}
\include{prerequisites}
\include{isospectral}
-\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
+\section{CMB Anisotropy of Spherical Spaces
+}
+
+\begin{frame}[fragile]{Homogeneous Spherical Spaces}
\begin{itemize}
- \item The Helmholtz equation on $\textcolor{blue}{M}$ has the spectrum as its solutions, and is given by
+ \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.
\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.
+ \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.
- \item The set of all possible wave numbers [for which there exists a non-zero solution] depends on $\Gamma$.
+ \item The set of all possible wave numbers [for which a nonzero solution exists] depends on $\Gamma$ - usually a proper subset of $\mathbb{{N}}$.
- \item Solutions to the Helmholtz equation are calculated for each $\Gamma$ and CMB anisotropy $\frac{\delta T}{T}$ computed.
+ \item Solutions to the Helmholtz equation are calculated for each $\Gamma$ and CMB anisotropy $\frac{\delta T}{T}$ computed as a sum of spherical harmonics.
\end{itemize}
\end{frame}
@@ -107,7 +112,7 @@ code-for-last-col = \color{blue}
\end{comment}
-\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces — Conclusion}
+\begin{frame}[fragile]{Homogenous Spherical Spaces --- Results}
\begin{itemize}
\item For the majority of groups $\Gamma$, the anisotropies $\frac{\delta T}{T}$ do not coincide with observations.
\item The only groups for which it does are $\Gamma = O^*$ and $\Gamma = I^*$ — the \textcolor{red}{binary octahedral} and \textcolor{red}{binary icosahedral} groups of order 48 and 120 respectively.
@@ -122,7 +127,7 @@ code-for-last-col = \color{blue}
\end{frame}
-\section{CMB radiation in an inhomogeneous spherical space}
+\section{CMB Radiation in an Inhomogeneous Spherical Space}
\begin{frame}{Inhomogeneous spherical space}
@@ -135,7 +140,7 @@ code-for-last-col = \color{blue}
\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 the strength of anisotropy suppression is observer dependent.
+ \item Results: inhomogeneous spaces have more variety in the CMB anisotropies than homogeneous spaces, because the strength of anisotropy suppression is observer dependent.
\end{itemize}
@@ -143,64 +148,85 @@ 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 $\mathbb{S}^2$. We expand $\Delta T(\hat{n})$ using \emph{spherical harmonics}, yielding coefficients $a_{\ell m}$.
+\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}
- 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{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}{The setup}
- Real CMB observations are affected by instrumental noise and \emph{sky masking}. As a result, estimating $C_\ell$ accurately requires simulations.
-
- Given $C^{th}_\ell$, used to generate synthetic realizations of the CMB sky
- \[
- \textcolor{blue}{a_{\ell m}} \sim \mathcal{N} (0, C_\ell^{\text{th}}).
- \]
-
- Unfortunately, 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. This gives 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}
-
+\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}
\end{frame}
-\begin{frame}{The Results}
- \begin{figure} [h!]
+
+\begin{frame}{Results and Interpretation}
+\small
+ \begin{columns}
+ \column{0.5\textwidth}
+ \begin{figure}
\centering
- \includegraphics[width=0.7\linewidth]{DSE-Test Graph}
- \caption{The decorrelated band mean spectrum obtained by the seven-year WMAP data [9].}
+ \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}
- \textcolor{red}{Remark}: In a positively curved space certain multipoles scales being preferred or suppressed in the CMB power spectrum.
-
+ \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}
- akbbfbsKJBKJBLJs
+ \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
+ \end{enumerate}
\end{frame}
-\begin{frame}{To summerize}
- egqgaaf
- \begin{itemize}
- \item 1
- \item 2
- \item 3
- \end{itemize}
-
-\end{frame}
\section{References}
\begin{frame}{References}
@@ -245,4 +271,5 @@ code-for-last-col = \color{blue}
\end{center}
\end{frame}
+
\end{document}
diff --git a/prerequisites.tex b/prerequisites.tex
index 920cbd1..55e4b5e 100644
--- a/prerequisites.tex
+++ b/prerequisites.tex
@@ -18,7 +18,7 @@
\begin{frame}{Quotients of the $3$-sphere}
\begin{itemize}
- \item $\so 4$ is isomorphic to the isometry group of $\S^3$
+ \item $\so 4$ is isomorphic to the isometry group of $\S^3$.
\item Subgroups $\Gamma \leqslant \so 4$ define equivalence classes of orbits on $\S^3$ by the standard action of $\so 4$ on $\R^3$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$.
\item The obtain space is (\emph{sometimes}) a manifold. In particular, it is well defined and spherical for finite $\Gamma$.
\item This can be easily generalised to the $n$-sphere.
@@ -33,11 +33,11 @@
\end{frame}
\begin{frame}{Lens spaces — the explicit construction}
\begin{definition}[Lens space]
- Given $q \in \mathbb N$ and $s \in \mathbb Z ^n$ each coprime with $q$
+ Given $q \in \mathbb Z$ and $s \in \mathbb Z ^n$ elementwise coprime with $q$
\begin{align*}
\lens q s \coloneq \S^{2n-1}/\langle M_{q,s} \rangle,
\end{align*}
- where $\langle M \rangle$ is the group generated by
+ where $\langle M_{q,s} \rangle$ is the group generated by
\begin{align*}
M_{q,s} \coloneq
\begin{pmatrix}