Add gitignore

.gitignore

@@ -0,0 +1,11 @@
+*.aux
+*.bbl
+*.blg
+*.fdb_latexmk
+*.fls
+*.log
+*.out
+main.pdf
+paper.pdf
+technology.pdf
+*.synctex.gz

main.tex

@@ -61,7 +61,7 @@ spaces} \\\ Preparation Bachelor's Project}
 
 \section{Abstract}
 
-We present a study of cosmic microwave background (CMB) temperature fluctuations in spherical spaces, which are the models of the universe where the space is a 3-dimensional sphere or its quotient. We give an overview of the preliminary machinery used in the study of CMB temperature fluctuations for spherical space. We examine the behavior of isospectral but not isometric spherical forms. We explore the problem of testing for anisotropies in the mean of CMB temperature fluctuations for spherical spaces.
+We present a study of cosmic microwave background (CMB) temperature fluctuations in spherical spaces, which are the models of the universe where the space is a 3-dimensional sphere or its quotient. We give an overview of the preliminary machinery used in the study of CMB temperature fluctuations for spherical space. Next, we examine the behavior of isospectral but not isometric spherical forms. Finally, we explore the problem of testing for anisotropies in the mean of CMB temperature fluctuations for spherical spaces.
 
 \section{Introduction}
 The cosmic microwave background (CMB) is the electromagnetic radiation left from the Big Bang.  Temperature fluctuations provide an idea of the early state of the universe, as well as possible hints on its current shape. Observations of space missions such as COBE, WMAP and PLANK have revealed an unexpectedly low variance in CMB anisotropies, which are small variations in the radiation, at really large angular scales \cite{aurich_2012}. This goes against the expected results of the infinite and flat universe from the standard $\Lambda$CDM model.
@@ -78,25 +78,25 @@ Metric and topological spaces allow us to generalize concepts from analysis like
 The definition of a topological space is too general to meaningfully describe physical reality. The concept of topological manifolds tries to limit the potential pathological examples of topological spaces by requiring the spaces to look Euclidean at a local level. Formally, this means that any point on the set has a neighborhood around it equivalent (homeomorphic) to an open set of Euclidean space.
 
 \begin{figure}[h]
-    \centering
-    \includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
-    \caption{The prototypical example of a manifold -- a mug. Image source: \cite{serri}}
-    \label{fig:mug-neighbourhoods}
+	\centering
+	\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
+	\caption{The prototypical example of a manifold -- a mug. Image source: \cite{serri}}
+	\label{fig:mug-neighbourhoods}
 \end{figure}
 
 \begin{definition}
-    A topological space $M$ is a \textbf{topological manifold} of dimension $n$, or topological $n$-manifold, if it has the following properties: 
-    \begin{itemize}
-        \item $M$ is a Hausdorff space,
-        \item $M$ is second countable,
-        \item $M$ is locally euclidean of dimension $n$, that is, for any point $p \in M$ there exist an open subset $U \subset M$ with $p \in U$, and open subset $V \subset \mathbb{R}^n$ and a homeomorphism $\varphi: U \rightarrow V$.
-    \end{itemize}
+	A topological space $M$ is a \textbf{topological manifold} of dimension $n$, or topological $n$-manifold, if it has the following properties:
+	\begin{itemize}
+		\item $M$ is a Hausdorff space,
+		\item $M$ is second countable,
+		\item $M$ is locally euclidean of dimension $n$, that is, for any point $p \in M$ there exist an open subset $U \subset M$ with $p \in U$, and open subset $V \subset \mathbb{R}^n$ and a homeomorphism $\varphi: U \rightarrow V$.
+	\end{itemize}
 \end{definition}
 
 Where second-countability is defined as follows.
 
 \begin{definition}[]
-    A topological space $(X,T)$ is said to be \textbf{second countable} if there exists a countable set $B \subset T$ such that any open set can be written as a union of sets from $B$. In such case, $B$ is called a (countable) basis for the topology $T$ .
+	A topological space $(X,T)$ is said to be \textbf{second countable} if there exists a countable set $B \subset T$ such that any open set can be written as a union of sets from $B$. In such case, $B$ is called a (countable) basis for the topology $T$ .
 \end{definition}
 
 %This is very useful, as it allows us to `pull back' many familiar concepts from Euclidean space (for instance, derivatives). Still, this turns out not to be enough. Analysis is built upon limits and sequences, yet those building blocks can behave in unexpected ways when taken out of the Euclidean setting.
@@ -116,21 +116,21 @@ A natural application of the above is detecting the curvature of the universe it
 Topologically, holes can be detected using homology groups (the abelian group of loops in the space, under the identification of loops that can be smoothly transformed into each other as equivalent). In particular, a space is contractible if any loop can be smoothly transformed into the trivial loop (a path that never leaves the initial point). That is, the (first) homology group must be trivial. We say a space is simply connected when it is both contractible and path connected (i.e. every two points are connected by a path).
 
 \begin{figure}[H]
-    \centering
-    \includegraphics[width=0.25\linewidth]{amogus.png}
-    \caption{The yellow path is non contractible on the space highlighted in red (there's an additional hole in the back of the figure, thus the path cannot simply contract through there)}
-    \label{fig:amogus}
+	\centering
+	\includegraphics[width=0.25\linewidth]{amogus.png}
+	\caption{The yellow path is non contractible on the space highlighted in red (there's an additional hole in the back of the figure, thus the path cannot simply contract through there)}
+	\label{fig:amogus}
 \end{figure}
 
 
 In what follows we use the concrete definition of the 3-sphere as an embedded submanifold of $\mathbb{R}^4$ lying in Euclidean space meaning $\mathbb{S}^3 := \{x \in \mathbb{R}^4 \; |\; \|x\| = 1\}.$ Next we define a certain special set of matrices which induce well-defined maps on the sphere.
 
 \begin{definition}
-    We define $$SO(4) = \{A \in \mathbb{R}^{4 \times 4}:\; A^TA = I_4, \; \det(A) = 1\}.$$
+	We define $$SO(4) = \{A \in \mathbb{R}^{4 \times 4}:\; A^TA = I_4, \; \det(A) = 1\}.$$
 \end{definition}
 
 By the standard action of $\mathbb{R}^{4 \times 4} $ on $\mathbb{R}^4$ we have that $SO(4)$ is naturally a subgroup of the group of isometries of $\mathbb{S}^3$. This is because for any $x \in \mathbb{S}^3$ we get $$\|Ax\| = (Ax)^T(Ax) =
-x^T(A^TA)x = x^Tx = \|x\| =1$$
+	x^T(A^TA)x = x^Tx = \|x\| =1$$
 hence $Ax \in \mathbb{S}^3$ implying $A|_{\mathbb{S}^3}$ is a well-defined map and a bijection. By analogous reasoning, we find that:
 
 $$\|Ax - Ay\| = \|A(x-y)\| = \|x-y\|,$$ hence $A$ is an isometry.
@@ -145,14 +145,14 @@ The set $\mathbb{S}^3/_\sim$ can be given a manifold structure if certain condit
 There is an alternative way of viewing the quotient $\mathbb{S}^3/\Gamma$ using the concept of a covering space.  In particular for our universe, the spaces are interested in are quotients $\mathbb S^3 /_\sim$ for various groups $\Gamma$ of covering transformations (i.e. transformations that shuffle around the different copies of the underlying manifold which tile the covering space).
 
 \begin{definition}[Covering spaces]
-    A covering space $\tilde X$ for a space $X$ is a (continuous) map $\pi : \tilde X \to X$, such that every point $x \in X$ has a neighborhood $U_x$ such that $\pi ^{-1} (U_x) = \coprod_{d \in D_x} V_d$ for some indexing set $D_x$ and sets $V_d$ such that $\pi |_{V_x} : V_x \to U_x$ is a homeomorphism.
+	A covering space $\tilde X$ for a space $X$ is a (continuous) map $\pi : \tilde X \to X$, such that every point $x \in X$ has a neighborhood $U_x$ such that $\pi ^{-1} (U_x) = \coprod_{d \in D_x} V_d$ for some indexing set $D_x$ and sets $V_d$ such that $\pi |_{V_x} : V_x \to U_x$ is a homeomorphism.
 \end{definition}
 
 \begin{figure}[h]
-    \centering
-    \includegraphics[width=0.25\linewidth]{covering-space.png}
-    \caption{The space $Y \coloneq [0, 1] \times \mathbb R$ is a covering space for $X \coloneq [0,1] \times \S^1$. The disjoint open sets $S_i$ are mapped homeomorphically onto $U$. The fiber of $x$ consists of the points $y_i$. Source: \cite{wiki:covering} }
-    \label{fig:enter-label}
+	\centering
+	\includegraphics[width=0.25\linewidth]{covering-space.png}
+	\caption{The space $Y \coloneq [0, 1] \times \mathbb R$ is a covering space for $X \coloneq [0,1] \times \S^1$. The disjoint open sets $S_i$ are mapped homeomorphically onto $U$. The fiber of $x$ consists of the points $y_i$. Source: \cite{wiki:covering} }
+	\label{fig:enter-label}
 \end{figure}
 \end{comment}
 
@@ -168,16 +168,16 @@ In \cite{emilio}, the authors go over the theoretical grounding of the literatur
 An important class of manifolds are the so-called `lens spaces', which correspond to the case where $\Gamma$ is cyclic and the dimension of the underlying manifold is odd. As we are interested in studying the shape of our (spatially) three-dimensional universe, all the spaces we are interested in will be lens spaces as long as $\Gamma$ is cyclic.
 
 \begin{definition}[Lens spaces]
-    Formally, we denote by $\lens q s$ (where $q \in \mathbb N$, $s \in \mathbb Z ^n $ and $\gcd(q, s_i) = 1$ for all $i$), the space $\mathbb S^{2n - 1} / \Gamma _{q;s}$, where $\Gamma_{q;s}$ is the (cyclic) group generated by the diagonal matrix with diagonal entries $R(2 \pi s_i/ q)$ (where $R(-)$ is a two-dimensional rotation matrix by the given angle).
+	Formally, we denote by $\lens q s$ (where $q \in \mathbb N$, $s \in \mathbb Z ^n $ and $\gcd(q, s_i) = 1$ for all $i$), the space $\mathbb S^{2n - 1} / \Gamma _{q;s}$, where $\Gamma_{q;s}$ is the (cyclic) group generated by the diagonal matrix with diagonal entries $R(2 \pi s_i/ q)$ (where $R(-)$ is a two-dimensional rotation matrix by the given angle).
 \end{definition}
 The condition $\gcd(q, s_i) = 1$ is necessary to ensure that $\Gamma$ acts freely on $\mathbb S^{2n-1}$.
 
 The Laplace-Beltrami operator generalizes the Laplace operator on smooth functions defined on $\mathbb{R}^n$ to general manifolds.
 \begin{definition}[Laplace-Beltrami operator]
-  The Laplace-Beltrami operator is defined as the divergence of the gradient:
-  \begin{align*}
-      \Delta f \coloneqq \divergence (\nabla f)
-  \end{align*}
+	The Laplace-Beltrami operator is defined as the divergence of the gradient:
+	\begin{align*}
+		\Delta f \coloneqq \divergence (\nabla f)
+	\end{align*}
 \end{definition}
 
 In particular, the spectrum of Laphlace-Beltrami operator (collection of eigenvalues) in $L^2(M, g)$ is a discrete subset of the non-negative reals, in which every value occurs with a finite multiplicity. Two such manifolds are said to be isospectral if they share a spectrum.
@@ -318,13 +318,13 @@ Thus, if the mean is nonzero $\langle a_{\ell m} \rangle \neq 0$, it \textbf{sug
 
 Since real CMB observations are affected by instrumental noise and sky masking, estimating $C_\ell$ accurately requires simulations. Given a theoretical power spectrum $C^{th}_\ell$, the corresponding harmonic coefficients are drawn from a Gaussian distribution:
 \[
-a_{\ell m} \sim \mathcal{N} (0, C_\ell^{\text{th}})
+	a_{\ell m} \sim \mathcal{N} (0, C_\ell^{\text{th}})
 \]
 
 Applying a sky mask function $M(\hat{n})$ modifies the observed coefficients:
 
 \[
-a^{\text{mask}}_{\ell m} = \sum_{\ell' m'} M_{\ell m, \ell' m'} a^{\text{all sky}}_{\ell' m'}
+	a^{\text{mask}}_{\ell m} = \sum_{\ell' m'} M_{\ell m, \ell' m'} a^{\text{all sky}}_{\ell' m'}
 \]
 
 where $a^{\text{all sky}}$ is taken from the all sky CMB map (which we can
@@ -333,7 +333,7 @@ the mask. This introduces mode coupling, which must be corrected using a decorre
 
 To test the assumption of statistical isotropy, we examine the mean of $a_{\ell m}$ across multipole bins. The test statistic is defined as:
 \[
-S_i = \sum_{j} W_{ij} M_j
+	S_i = \sum_{j} W_{ij} M_j
 \]
 where $W$ is a decorrelation matrix derived from the covariance structure. If significant anomalies appear in the mean values of $a_{\ell m}$, this suggests deviations from isotropy that may indicate the need for new theories.
 
@@ -346,16 +346,16 @@ $\theta \approx \frac{180^\circ}{\ell}$.\
 
 Plotting the results for the frequency bands Q,V and W, we get the graph for the decorrelated Statistical Spectrum in Figure \ref{fig:Decorrelated Statistical Spectrum} (This is the main graph from \cite{Kashino_2012})
 \begin{figure} [h!]
-    \centering
-    \includegraphics[width=0.8\linewidth]{DSE-Test Graph}
-    \caption{The decorrelated band mean spectrum obtained by the seven-year WMAP data. The boxes and solid line are the result of the overall combined map (Q+V+W) and color data points are the results of the individual frequency maps (Q, V, W). The horizontal width of boxes indicates the bin size $\Delta \ell = 20$ for calculating the binned mean spectrum. Image source: \cite{Kashino_2012}}
-    \label{fig:Decorrelated Statistical Spectrum}
+	\centering
+	\includegraphics[width=0.8\linewidth]{DSE-Test Graph}
+	\caption{The decorrelated band mean spectrum obtained by the seven-year WMAP data. The boxes and solid line are the result of the overall combined map (Q+V+W) and color data points are the results of the individual frequency maps (Q, V, W). The horizontal width of boxes indicates the bin size $\Delta \ell = 20$ for calculating the binned mean spectrum. Image source: \cite{Kashino_2012}}
+	\label{fig:Decorrelated Statistical Spectrum}
 \end{figure}
 \\
 We should note that Figure \ref{fig:Decorrelated Statistical Spectrum} contains a bin of size 20. The bin size in this context refers to the range of multipole moments $\ell$ that are grouped together when calculating the test statistic $S_i$. Or, in other words, instead of analyzing each individual $\ell$ separately, we sum over a range of values (a bin).
 Moreover, Figure \ref{fig:Decorrelated Statistical Spectrum} presents the plots for: \begin{itemize}
-    \item $Q+V+W$: A combined dataset that merges all three frequency bands to improve the signal-to-noise ratio.
-    \item $Q1-Q2, V1-V2, W1-W2$: These are difference maps, created by subtracting different detector readings of the same frequency band. Due to this, they primarily contain instrumental noise, since the real CMB signal cancels out when differencing measurements from the same frequency.
+	\item $Q+V+W$: A combined dataset that merges all three frequency bands to improve the signal-to-noise ratio.
+	\item $Q1-Q2, V1-V2, W1-W2$: These are difference maps, created by subtracting different detector readings of the same frequency band. Due to this, they primarily contain instrumental noise, since the real CMB signal cancels out when differencing measurements from the same frequency.
 \end{itemize}
 These maps help verify whether anomalies seen in the main dataset are truly cosmological or just instrumental artifacts.
 
@@ -365,11 +365,11 @@ Therefore, $S_i < 0$ corresponds to fewer temperature fluctuations than expected
 Moreover, to understand the multipole moment $\ell$ and what it represents in the graph, we can break it down into three main scenarios:
 
 \begin{itemize}
-  \item Deviations only at high $\ell$ (small scales, $\ell > 300$): This would correspond to deviations caused by instrumental noise or foreground contamination \cite{Kashino_2012}.
+	\item Deviations only at high $\ell$ (small scales, $\ell > 300$): This would correspond to deviations caused by instrumental noise or foreground contamination \cite{Kashino_2012}.
 
-  \item Deviations only at low $\ell$ (large scales, $\ell < 50$): This would correspond to deviations caused by large-scale anisotropies, potentially challenging the assumption of a statistically isotropic universe \cite{Bielewicz_2004}\cite{Pedro_2013}.
+	\item Deviations only at low $\ell$ (large scales, $\ell < 50$): This would correspond to deviations caused by large-scale anisotropies, potentially challenging the assumption of a statistically isotropic universe \cite{Bielewicz_2004}\cite{Pedro_2013}.
 
-  \item Deviations on both large and small scales: This could indicate a more fundamental cosmological effect affecting multiple scales.
+	\item Deviations on both large and small scales: This could indicate a more fundamental cosmological effect affecting multiple scales.
 \end{itemize}
 
 In Figure \ref{fig:Decorrelated Statistical Spectrum} we can see that the most significant anomaly appears in the mid-range multipoles ($221 \leq \ell \leq 240$), where $S_i$ is strongly negative, suggesting that the mean spherical harmonic coefficients at this scale are lower than expected compared to a statistically isotropic model. In other words, this means that temperature fluctuations at these angular scales are weaker than predicted by standard cosmological models.
@@ -387,11 +387,11 @@ As a final note, the analysis is limited by the existing data for the CMB temper
 \section{Conclusion}
 
 \begin{figure}[H]
-    \centering
-    \includegraphics[width=0.25\linewidth]{binary-octahedron.png}
-    \caption{
-In conclusion, the universe has a strange shape, and everything you knew is a lie. Fear the binary octahedral, for the end is near.}
-    \label{fig:binary-octahedron}
+	\centering
+	\includegraphics[width=0.25\linewidth]{binary-octahedron.png}
+	\caption{
+		In conclusion, the universe has a strange shape, and everything you knew is a lie. Fear the binary octahedral, for the end is near.}
+	\label{fig:binary-octahedron}
 \end{figure}
 
 We have designed and outlined a study of CMB temperature fluctuations in spherical spaces. We introduced spherical 3-manifolds as a plausible class of cosmological models and developed the mathematical tools (topology and eigenmode analysis) to handle them. Using these tools, we describe how one can compute CMB anisotropy patterns for a given spherical topology. We then connected these predictions to real data, discussing how one would recognize the fingerprint of a closed spherical universe in CMB observations ( such as through statistical anisotropies). Our synthesis of results from the articles by Aurich et al., Kashino et al., Pranav et al., and others shows that while no specific spherical topology is confirmed, the concept remains consistent with current data and even offers potential explanations for certain anomalies. It is relevant to mention that ongoing refinements in both theory and measurement could yet reveal a topological signature.