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.
@@ -130,7 +130,7 @@ In what follows we use the concrete definition of the 3-sphere as an embedded su
 \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.
@@ -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.
 
@@ -390,7 +390,7 @@ As a final note, the analysis is limited by the existing data for the CMB temper
 	\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.}
+		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}