From ec4b25adbd25072719e833c24d4a63f815203f46 Mon Sep 17 00:00:00 2001
From: Javier Gustavo Vela Castro <j.g.vela.castro@student.rug.nl>
Date: Fri, 21 Mar 2025 12:05:25 +0000
Subject: [PATCH] Update on Overleaf.
---
isospectral.tex | 1 +
main.tex | 35 +++++++++++++++++++++++++++--------
2 files changed, 28 insertions(+), 8 deletions(-)
diff --git a/isospectral.tex b/isospectral.tex
index 3492287..be72010 100644
--- a/isospectral.tex
+++ b/isospectral.tex
@@ -33,6 +33,7 @@
\begin{frame}{The silver lining}
\begin{itemize}
+
\item Isospectral pairs are cool and all, but they cannot occur in dimension $3$.
\item We can thus attempt to infer the shape of our universe based on its spectrum.
\end{itemize}
diff --git a/main.tex b/main.tex
index 9c549e4..87dadc5 100644
--- a/main.tex
+++ b/main.tex
@@ -84,16 +84,18 @@ code-for-last-col = \color{blue}
\begin{frame}[fragile]{Homogeneous Spherical Spaces}
\begin{itemize}
+ \pause
\item Finite subgroup $\Gamma \leq \so 4.$
+ \pause
\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*}
-
+\pause
\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.
-
+\pause
\item The set of all possible wave numbers [for which a nonzero solution exists] depends on $\Gamma$ - usually a proper subset of $\mathbb{{N}}$.
-
+\pause
\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}
@@ -114,9 +116,11 @@ code-for-last-col = \color{blue}
\begin{frame}[fragile]{Homogenous Spherical Spaces --- Results}
\begin{itemize}
+ \pause
\item For the majority of groups $\Gamma$, the anisotropies $\frac{\delta T}{T}$ do not coincide with observations.
+ \pause
\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.
-
+\pause
\end{itemize}
\begin{figure}[H]
\centering
@@ -135,11 +139,13 @@ code-for-last-col = \color{blue}
\item Manifolds of the form $\S^3/\Gamma$ with $\Gamma$ a group acting on $\S^3$.
\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:
+ \pause
+ \item Fixing $|\Gamma|=8$, we have three multi-connected manifolds, up to equivalence: \pause
\begin{enumerate}
\item homogeneous: $N3$ and $L(8,1)$
\item inhomogeneous: $N2 \equiv L(8,3)$
\end{enumerate}
+ \pause
\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}
@@ -153,16 +159,18 @@ code-for-last-col = \color{blue}
\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}
+ \pause
\item The CMB temperature fluctuations can be expanded as:
\[
\Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\hat{n})
\]
+ \pause
\item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
\[
\langle a_{\ell m} \rangle = 0
\]
\end{itemize}
-
+ \pause
\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.
@@ -173,13 +181,15 @@ code-for-last-col = \color{blue}
\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.
+ \pause
\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}
-
+ \pause
\textbf{Monte Carlo Simulations:}
\begin{itemize}
\item We generate thousands of \textbf{simulated CMB skies} assuming isotropy.
@@ -202,12 +212,13 @@ code-for-last-col = \color{blue}
\caption{The decorrelated band mean test statistic values over multipole ranges [9].}
\end{figure}
\column{0.5\textwidth}
+ \pause
\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}
-
+ \pause
\textbf{Possible Explanations:}
\begin{itemize}
\item A real cosmological signal? → A finite universe or new physics.
@@ -219,10 +230,18 @@ code-for-last-col = \color{blue}
\section{Conclusion}
\begin{frame}{Conclusion}
+
+\pause
\begin{enumerate}
\item We can infer the shape of the universe from its spectrum
+
+\pause
\item There are two homogeneous spherical manifolds obtained as $\S^3/\Gamma$ which produce CMB similar to observations
+
+\pause
\item Inhomogeneous spherical spaces exhibit varied behavior of CMB anisotropies
+
+\pause
\item Statistical test results suggest possibilities of finite multi-connected topology
\end{enumerate}
\end{frame}