From a0d9612e04006bdfa7e5048dcc7cff360d5c7272 Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Fri, 21 Mar 2025 12:21:43 +0000
Subject: [PATCH] Update on Overleaf.
---
main.tex | 46 +++++++++++++++++++++++-----------------------
1 file changed, 23 insertions(+), 23 deletions(-)
diff --git a/main.tex b/main.tex
index d0a2196..b5a8f5f 100644
--- a/main.tex
+++ b/main.tex
@@ -96,7 +96,7 @@ code-for-last-col = \color{blue}
\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.
+ \item Solutions to the Helmholtz equation are calculated for each $\Gamma$ and CMB anisotropy $\textcolor{purple}{\frac{\delta T}{T}}$ computed as a sum of spherical harmonics.
\end{itemize}
\end{frame}
@@ -137,16 +137,16 @@ code-for-last-col = \color{blue}
\begin{itemize}
\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 \textcolor{red}{Multi-connected} space: it has non-contractable loops.
+ \item \textcolor{red}{Inhomogeneous} space: it does not look identical from every point in space.
\pause
- \item Fixing $|\Gamma|=8$, we have three multi-connected manifolds, up to equivalence: \pause
+ \item Fixing $|\Gamma|=8$, we have \textcolor{red}{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.
+ \item Results: inhomogeneous spaces have \textcolor{red}{more variety} in the CMB anisotropies than homogeneous spaces, because the strength of anisotropy suppression is \textcolor{red}{observer dependent}.
\end{itemize}
@@ -162,18 +162,18 @@ code-for-last-col = \color{blue}
\pause
\item The CMB temperature fluctuations can be expanded as:
\[
- \Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell m}(\hat{n})
+ \Delta T(\hat{n}) = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} \textcolor{blue}{a_{\ell m}} Y_{\ell m}(\hat{n})
\]
\pause
\item If the universe is isotropic, the \textbf{mean of the harmonic coefficients} should satisfy:
\[
- \langle a_{\ell m} \rangle = 0
+ \langle \textcolor{blue}{a_{\ell m}} \rangle = 0
\]
\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.
+ - If \( \langle \textcolor{blue}{a_{\ell m}} \rangle \neq 0 \), this suggests a preferred cosmic direction.
\end{frame}
\begin{frame}{Methodology: Sky Masking and the Test Statistic}
@@ -195,7 +195,7 @@ code-for-last-col = \color{blue}
\item We generate thousands of \textbf{simulated CMB skies} assuming isotropy.
\item Each sky has different random \( a_{\ell m} \), drawn from:
\[
- a_{\ell m} \sim \mathcal{N} (0, C_\ell)
+ \textcolor{blue}{a_{\ell m}} \sim \mathcal{N} (0, C_\ell)
\]
\item The observed WMAP data is compared against these simulations.
\end{itemize}
@@ -250,11 +250,11 @@ code-for-last-col = \color{blue}
\section{References}
\begin{frame}{References}
\begin{itemize}
- \item R. Aurich, P. Kramer, and S. Lustig. \textit{Cosmic microwave background radiation in an inhomogeneous spherical space}. Physica Scripta, 2011.
- \item R. Aurich, P. Kramer, and S. Lustig. \textit{A survey of lens spaces and large-scale CMB anisotropy}. Monthly Notices of the Royal Astronomical Society, 2012.
- \item R. Aurich, S. Lustig, and F. Steiner. \textit{CMB anisotropy of spherical spaces}. Classical and Quantum Gravity, 2005.
- \item P. Bielewicz, K. M. G´orski, and A. J. Banday. \textit{Low-order multipole maps of cosmic microwave background anisotropy derived from WMAP}. Oxford University Press, 2004.
- \item G. Egan. \textit{Symmetries and the 24-cell}. 2021. Accessed at https://www.gregegan.net/SCIENCE/24-cell/24-cell.html.
+ \item [1] R. Aurich, P. Kramer, and S. Lustig. \textit{Cosmic microwave background radiation in an inhomogeneous spherical space}. Physica Scripta, 2011.
+ \item [2] R. Aurich, P. Kramer, and S. Lustig. \textit{A survey of lens spaces and large-scale CMB anisotropy}. Monthly Notices of the Royal Astronomical Society, 2012.
+ \item [3] R. Aurich, S. Lustig, and F. Steiner. \textit{CMB anisotropy of spherical spaces}. Classical and Quantum Gravity, 2005.
+ \item [4] P. Bielewicz, K. M. G´orski, and A. J. Banday. \textit{Low-order multipole maps of cosmic microwave background anisotropy derived from WMAP}. Oxford University Press, 2004.
+ \item [5]G. Egan. \textit{Symmetries and the 24-cell}. 2021. Accessed at https://www.gregegan.net/SCIENCE/24-cell/24-cell.html.
\end{itemize}
\end{frame}
@@ -262,20 +262,20 @@ code-for-last-col = \color{blue}
\begin{frame}{References}
\begin{itemize}
- \item N. Jarosik et. al. \textit{Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results}. The American Astronomical Society, 2011.
- \item M. Herlihy and N. Shavit. \textit{The Topological Structure of Asynchronous Computability}. Journal of the ACM, 1996.
- \item A. Ikeda. \textit{On the spectrum of a Riemannian manifold of positive constant curvature}. Osaka Journal of Mathematics, 1980.
- \item D. Kashino, K. Ichiki, and T. Takeuchi. \textit{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}. Physics Review, 2012.
- \item P. Labrana. \textit{Emergent Universe Scenario and the Low CMB Multipoles}. Physical Review, 2013.
+ \item [6] N. Jarosik et. al. \textit{Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results}. The American Astronomical Society, 2011.
+ \item [7] M. Herlihy and N. Shavit. \textit{The Topological Structure of Asynchronous Computability}. Journal of the ACM, 1996.
+ \item [8] A. Ikeda. \textit{On the spectrum of a Riemannian manifold of positive constant curvature}. Osaka Journal of Mathematics, 1980.
+ \item [9] D. Kashino, K. Ichiki, and T. Takeuchi. \textit{Test for anisotropy in the mean of the CMB temperature fluctuation in spherical harmonic space}. Physics Review, 2012.
+ \item [10] P. Labrana. \textit{Emergent Universe Scenario and the Low CMB Multipoles}. Physical Review, 2013.
\end{itemize}
\end{frame}
\begin{frame}{References}
\begin{itemize}
- \item E. A. Lauret and B. Linowitz. \textit{The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems}. New York Journal of Mathematics, 2025.
- \item R. Lehoucq, J. Weeks, J. P. Uzan, E. Gausmann, and J.P. Luminet. \textit{Eigenmodes of three-dimensional spherical spaces and their application to cosmology}. Classical and Quantum Gravity, 2002.
- \item M. Serri. \textit{Analysis on Manifolds}. AMS Open Math Notes, 2025.
- \item B. Tai-Dana, T. Bryson, and J. Terilla. \textit{Topology, a categorical approach}. The MIT Press, 2020.
+ \item [11] E. A. Lauret and B. Linowitz. \textit{The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems}. New York Journal of Mathematics, 2025.
+ \item [12] R. Lehoucq, J. Weeks, J. P. Uzan, E. Gausmann, and J.P. Luminet. \textit{Eigenmodes of three-dimensional spherical spaces and their application to cosmology}. Classical and Quantum Gravity, 2002.
+ \item [13] M. Serri. \textit{Analysis on Manifolds}. AMS Open Math Notes, 2025.
+ \item [14] B. Tai-Dana, T. Bryson, and J. Terilla. \textit{Topology, a categorical approach}. The MIT Press, 2020.
\end{itemize}
\end{frame}