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main.tex
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\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}
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\end{center}
\end{frame}
\end{document}

6
prerequisites.tex
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@ -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}