Herschel
The abstract of the paper.
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% Title of the paper, and the short title which is used in the headers.
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\title[Genetic Algorithm on CMB]{Application of Genetic Algorithm to Estimate the Large Angular Scale Features of Cosmic Microwave Background}

% Genetic Algorithm on Internal Linear Combination of Microwave background Maps (GAILCoMM)

% The list of authors, and the short list which is used in the headers.
% If you need two or more lines of authors, add an extra line using \newauthor
\author[Parth Nayak and Rajib Saha]{
Parth Nayak$^{1}$%\thanks{E-mail: mn@ras.org.uk (KTS)}
and Rajib Saha$^{1}$
%Third Author$^{2,3}$
%and Fourth Author$^{3}$
\\
% List of institutions
$^{1}$Department of Physics, Indian Institute of Science Education and Research (IISER) Bhopal, 462066, India\\
%$^{2}$Department, Institution, Street Address, City Postal Code, Country\\
%$^{3}$Another Department, Different Institution, Street Address, City Postal Code, Country
}

% These dates will be filled out by the publisher
\date{Accepted XXX. Received YYY; in original form ZZZ}

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\pubyear{2021}

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% Abstract of the paper
\begin{abstract}
Genetic Algorithm (GA) -- motivated by the natural evolutionary process -- is a robust method to estimate the optimal solutions of problems involving one or more objective functions. In this article, for the first time, we apply GA to reconstruct the cleaned CMB temperature anisotropy map over large angular scales of the sky using (internal) linear combination (ILC) of observations from the final-year WMAP and Planck satellite missions. To avoid getting trapped into a local minimum, we implement the GA with generous diversity in the populations. This is achieved by introducing a small but significant amount of mutation of genes during crossover and selecting pairs with diverse fitness coefficients. We find that the new GA-ILC method produces a cleaned CMB map which agrees very well with the CMB map obtained using the exact and analytical expression of weights in the ILC method. By performing extensive Monte Carlo simulations of the CMB reconstruction using the GA-ILC algorithm we find that residual foregrounds in the cleaned map are minimal and mostly tend to occupy localized regions along the central galactic plane. The CMB angular power spectrum shows no indication of any bias in the entire multipole range $2 \leq \ell \leq 32$ studied in this work. The error in the CMB angular power spectrum is minimal as well and given entirely by the cosmic variance induced error. Our results agree well with those obtained by various other reconstruction methods by different research groups. This problem-independent robust GA-ILC method provides a flexible way towards the complex and challenging task of CMB component reconstruction in cosmology.
\end{abstract}

% Select between one and six entries from the list of approved keywords.
% Don't make up new ones.
\begin{keywords}
cosmic microwave background -- genetic algorithm -- methods: data analysis.
\end{keywords}

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%%%%%%%%%%%%%%%%% BODY OF PAPER %%%%%%%%%%%%%%%%%%

%-------------------------
%-------------------------
\section{Introduction}
\label{sec:intro}
%-------------------------
%-------------------------

\par{The cosmic microwave background (CMB) angular power spectrum is one of the most crucial probes of the cosmological parameters and the dynamical history
of the universe. The thermal radiation is observed by various satellite missions such as WMAP \citep{2013ApJS..208...20B} and Planck \citep{Planck2018}. This primordial signal, however, is largely contaminated by the emissions from outside the Milky Way as well as within. These sources of contamination are collectively called the “foreground” sources since we are concerned with the cosmic microwave \textit{background}. For instance, the hot interstellar dust inside the galaxy emits strongly at frequencies $\gtrsim$ $70$ GHz. This is referred to as the dust emission and is one kind of foreground. Some other kinds of foreground are synchrotron and free-free emissions from galactic and extragalactic sources \citep{10.1093/ptep/ptu065, 2003ApJS..148....1B, 1999NewA....4..443B, 2003ApJS..148..135H}. Hence, an accurate estimation of the angular power spectrum of pure CMB from the foreground-contained observations is crucial for precision cosmology.}

Since the onset of the scientific observations of CMB, extensive research has been done to come up with ways of effective foreground removal and accurate CMB retrieval. As a consequence, several methods exist in literature for getting rid of the foreground. However, most of them make use of the underlying spectral and spatial models of the foreground~\citep{2003ApJS..148..135H, 2003ApJS..148...97B, 2007ApJS..170..288H}. For example, a Gibbs sampling approach \citep{4767596} is proposed and implemented to separate CMB and foreground components by \cite{Eriksen_2004, Eriksen_2007, 2008ApJ...672L..87E, 2008ApJ...676...10E}; and a maximum likelihood approach to reconstruct CMB and foreground components using prior noise covariance information and foreground models is implemented by \cite{Erikson_2006, Gold_2011}. However, they may have uncertainties arising due to incomplete information of the foreground frequency and spatial dependence~\citep{Dodelson_1997,1998ApJ...502....1T}. A completely model-independent method is the internal linear combination (ILC) of multifrequency observations \citep{Tegmark_1996}. \cite{Sudevan_2018_g} explore the CMB posterior using a Gibbs sampling approach within a global ILC framework. \cite{Sudevan_2018} study the method in the wake of the theoretical CMB covariance information. All these works have been carried out considering the Gaussian nature of the primordial anisotropy signal; any non-Gaussianities are small fluctuations~\citep{ALLEN198766,Falk_1992, Gangui_1994,ACQUAVIVA2003119,Maldacena_2003}. \cite{Saha_2011} considers the highly non-Gaussian inheritance of the foreground emissions and proposes an ILC method to get rid of the non-Gaussian contamination based on the kurtosis minimization. Other notable studies of the ILC method include, but not limited to,~\cite{Tegmark_2003,2003ApJS..148...97B, Eriksen_2004a,Saha_2006,Saha_2008}. Some of the previous works also suggest that, depending on the context of the underlying problem, an exact analytical ILC solution may not be possible to find.
%In this work, we have used the ILC method as the basis and applied genetic algorithm to \textit{numerically} estimate the clean CMB map and angular power spectrum.\\

We propose an expression-independent, numerical ILC technique for a robust and effective CMB foreground reconstruction. In this work, we apply the novel numerical method over the global ILC proposed by \cite{Sudevan_2018}, making use of the Planck mission’s theoretical CMB angular power spectrum \citep{Planck2018VI} in our computation of the reduced variance of the output map. We apply the genetic algorithm (GA) for estimating optimal ILC weights to produce a clean map. An evolutionary algorithm such as this involves several inherent parameters that need to be studied extensively for such an implementation. For our study of the method over the large angular scales, we make use of the low-resolution WMAP and Planck observations as input at HEALPix~\footnote{Hierarchical Equal Area isoLatitude Pixelation of sky, e.g., see~\cite{Gorski2005}} $N_{\text{side}} = 16$. We smooth the input maps by a Gaussian beam at 9\textdegree \hspace{1pt} FWHM. Presumably, this also reduces the pixel-uncorrelated noise level (which is dominant on smaller angular scales).

We perform detailed Monte Carlo simulations to ensure the statistical sanity of our method. We find that this novel GA-ILC method produces a clean map with a minimal residual foreground. The residuary contamination tends to occupy small localized regions close to the galactic plane. The angular power spectrum of the clean map contains no apparent bias in the multipole range of $2 \leq \ell \leq 32$. The reconstruction errors in the angular power spectrum are also minimal and conform to the cosmic variance-induced errors.

%%(Paper organization paragraph)
The organization of this paper is as follows. We describe the basic formalism of the method in section \ref{sec:formalism}. We elaborate on the exact implementation of this method in section \ref{sec:meth}. We present our trial GA implementation for validation in section~\ref{sec:valid}. We discuss the GA-ILC implemented with Monte Carlo simulations in section~\ref{sec:MC-simul}. We present the results obtained by using the techniques of this work on WMAP and Planck final-year data and discuss our findings in section \ref{sec:results}. We conclude our paper by outlining some key aspects of this work and its future prospects in section \ref{sec:conclusion}.\\


%-------------------------
%-------------------------
\section{Formalism}
\label{sec:formalism}
%-------------------------
%-------------------------

Suppose there are $N$ channels of different frequencies, each producing a temperature anisotropy map (hereinafter simply, ``map'') upon observation. A map observed from the $i$th frequency channel is denoted by the vector $\Delta \tilde{\mathbf{T}}_i$ in the HEALPix pixelated format. The size of this vector is $p \equiv N_{\text{pix}} = 12N_{\text{side}}^2$.

Since a map is defined over a sphere, it can be expanded in the so-called ``harmonic space'' as
%%%%
\begin{eqnarray}\label{eqn:harm-space}
    \Delta \tilde{T}_i(n) = \sum_{\ell m} a_{\ell m}^i Y_{\ell m} (n),
\end{eqnarray}
where
$n$ denotes the pixel index (corresponding to some direction vector $\mathbf{\hat{n}}$), $Y_{\ell m}$ are the standard spherical harmonics, and $a_{\ell m}^i$ are called the harmonic coefficients of the $i$th map. The multipole index $\ell$ is of special importance because it indicates the underlying angular scale ($\theta_{\ell} \sim 2\pi/\ell$). The cross-correlation of the harmonic coefficients of maps $i$ and $j$ is also called the cross power spectrum, defined by
%%%%%
\begin{eqnarray}\label{eqn:cross-cl-dfn}
    C_{\ell}^{ij} = \sum_{m=-\ell}^{\ell} \frac{a_{\ell m}^{i*} a_{\ell m}^{j}}{2\ell + 1}.
\end{eqnarray}
The auto-correlation of the harmonic coefficients of the $i$th map is called its angular power spectrum, $C_{\ell}^i$. The CMB angular power spectrum is the single, most important quantity we would like to accurately estimate by finding an optimal clean CMB map from multifrequency observations. All observed maps consist primarily of CMB and foreground. Various foreground emissions are not thermalized, hence, they do not have a blackbody spectrum unlike CMB, which follows the thermal blackbody distribution with great accuracy \citep{1994ApJ...420..439M}. Hence, across different frequency bands, the thermodynamic temperature of the foreground patches varies whereas the CMB temperature will be constant. ILC exploits this fact as a benefit of multifrequency observations.

Apart from foreground, detector-noise is also one of the components that contaminates the observed CMB signal. Although small amount of noise is present on all angular scales, it is dominant only on the smaller angular scales of the data which are irrelevant for this work. Furthermore, noise from any two frequency channels is uncorrelated \citep{2003ApJS..148..135H, 2007ApJS..170..288H, 2003ApJS..145..413J, 2007ApJS..170..263J}, more to the advantage of ILC.
% However, in the present work, we focus on the larger angular scales where the noise is not significant.

An observed CMB map in $i$th frequency channel can thus be expressed as
\begin{eqnarray}\label{eqn:cmb_comp}
\Delta \tilde{\mathbf{T}}_i = \Delta \tilde{\mathbf{T}}_{\text{CMB}} 
					   + \Delta \tilde{\mathbf{T}}_{\text{fg}, i}
					    + \Delta \tilde{\mathbf{T}}_{\text{noise}, i}.
\end{eqnarray} 
Notice that the CMB part, $\Delta \tilde{\mathbf{T}}_{\text{CMB}}$, does not contain an $i$ index since the 
CMB temperature is independent of frequency.\\


%-------------------------
\subsection{Internal Linear Combination}
\label{sub:ilc}
%-------------------------


\cite{Tegmark_1996} proposed a model-independent method to estimate the clean CMB signal from a multifrequency observation. ILC attempts to find the optimal estimate of CMB from multifrequency foreground-contaminated maps without using any external templates or models. In this technique, we start by writing a linear combination of $N$ multifrequency input maps to find a clean output map. This is simply
%%%%%
\begin{eqnarray}\label{eqn:ilc}
	\mathbf{P} \equiv \Delta \tilde{\mathbf{T}}_{\text{clean}} (\{w_i\}) = \sum_{i=1}^{N} w_i \Delta \tilde{\mathbf{T}}_i,
\end{eqnarray}
where
\{$w_i$\} are the coefficients, more commonly called the \textit{weights} given to all the input maps before summing them. Using Eqn.~\eqref{eqn:cmb_comp}, this yields
%%%%%%
\begin{eqnarray}\label{eqn:ilc-expn}
	\mathbf{P} = \bigg( \sum_{i=1}^{N} w_i \bigg) \Delta \tilde{\mathbf{T}}_{\text{CMB}} 
	+ \sum_{i=1}^{N} w_i \Delta \tilde{\mathbf{T}}_{\text{trash}, i}, 
\end{eqnarray}
where
$\Delta \tilde{\mathbf{T}}_{\text{trash}, i} = \Delta\tilde{\mathbf{T}}_{\text{fg}, i} + \Delta \tilde{\mathbf{T}}_{\text{noise}, i}$.
In order to preserve the norm of the clean CMB map, a constraint is imposed in the form
%%%%%
\begin{eqnarray}\label{eqn:ilc-constraint}
	\sum_{i=1}^{N} w_i = 1 \text{\hspace{12pt} or \hspace{12pt}}
	\mathbf{e}^{\text{T}}\mathbf{w} = 1,
\end{eqnarray}
where
$\mathbf{e} = (1, 1,\dots,1)^{\text{T}}$ is the column vector of size $N$ with all entries 1. In the usual ILC method, the best estimate of CMB is obtained by minimizing the variance of the clean map, $\mathbf{P}^{\text{T}} \mathbf{P}$, w.r.t. the choice of the weight vector, $\mathbf{w}$. This is a problem of multivariable constrained optimization and the analytical solution can be found by using the Lagrange's multipliers method (e.g., see \cite{Tegmark_1996, 2008PhRvD..78b3003S}) as,
%%%%%
\begin{eqnarray}\label{eqn:w-lm-vector}
	\mathbf{w}_{\text{usual}} = \frac{\mathbf{e}^{\text{T}} 
  \mathbf{C}^{-1}}{\mathbf{e}^{\text{T}} \mathbf{C}^{\text{-1}} \mathbf{e}},
\end{eqnarray}
or following summation notation,
\begin{eqnarray}\label{eqn:w-lm-summ}
	w_{\text{usual}, i} = \frac{\sum_{j=1}^{N}C^{-1}_{ij}}{\sum_{i',j'=1}^{N} \text{ } C^{-1}_{i'j'}},
\end{eqnarray}
where
$\mathbf{C}$ is the $N \times N$ square symmetric covariance matrix of the $N$ input maps. In case the covariance matrix is singular, it is possible to generalize the inverse as the Moore-Penrose pseudoinverse~\citep{Moore_1920,1956PCPS...52...17P} $\mathbf{C}^{-1} \mapsto \mathbf{C}^{\dagger}$. If 
$\mathbf{C}$ is not singular, $\mathbf{C}^{\dagger} = \mathbf{C}^{-1}$. Rewriting Eqn.~\eqref{eqn:w-lm-vector} 
in terms of the pseudoinverse,  
%%%%%%%%
\begin{eqnarray}\label{eqn:w-lm-pi}
	\mathbf{w}_{\text{usual}} = \frac{\mathbf{e}^{\text{T}} \mathbf{C}^{\dagger}}{\mathbf{e}^{\text{T}} \mathbf{C}^{\dagger} \mathbf{e}}.
\end{eqnarray}

%% Here describe the global ILC by Sudevan & Saha, 2018
\cite{Sudevan_2018} (SS, for short) proposed a new, global ILC method making use of the prior knowledge of the theoretical covariance of CMB. The method employs this additional information to assert that the covariance structure of the final clean map is consistent with the expectation of pure CMB. Instead of minimizing the variance of the clean map, this global ILC seeks to minimize the reduced variance, called $\sigma^2$, calculated as
%%%%%%%%%%%
\begin{eqnarray}\label{eqn:sigma-sq-SS}
\sigma^2 =  \mathbf{P}^{\text{T}}
            \mathcal{C}_{\text{th}}^{\dagger} \mathbf{P},    
\end{eqnarray}
where
$\mathcal{C}_{\text{th}}$ is the theoretical CMB \emph{pixel-pixel} covariance matrix of size $p\times p$ and $\dagger$ represents the pseudoinverse. From the definition of $\mathbf{P}$ in Eqn.~\eqref{eqn:ilc}, we can write
%%%%%%%%%%%%%%
\begin{eqnarray}\label{eqn:sigma-sq-A}
    \sigma^2 =  \mathbf{w}^{\text{T}} \mathbf{A w},
\end{eqnarray}
where $\mathbf{A}$ is an $N\times N$ square matrix with elements
%%%%%%%%%%%%%
\begin{eqnarray}\label{eqn:A-elem-dfn}
A_{ij} = (\Delta \tilde{\mathbf{T}}_i)^{\text{T}}
            \mathcal{C}_{\text{th}}^{\dagger} 
            \Delta \tilde{\mathbf{T}}_j.
\end{eqnarray}
Similarly to the usual ILC, using the Lagrange multipliers with the same constraint, the optimal weights under this global ILC can be found as
%%%%%%%%%%
\begin{eqnarray}\label{eqn:w-SS}
    \mathbf{w}_{\text{SS}} =     \frac{\mathbf{e}^{\text{T}} \mathbf{A}^{\dagger}}{\mathbf{e}^{\text{T}} \mathbf{A}^{\dagger} \mathbf{e}}.
\end{eqnarray}

As found by \cite{2018arXiv181008872S}, the matrix $\mathbf{A}$ can be calculated as
\begin{eqnarray}\label{eqn:A-matrix}
	A_{ij} = \sum_{\ell = 2}^{\ell_{\text{max}}} (2\ell + 1) \frac{C_{\ell}^{ij}}{C_{\ell, \text{th}}'},	
\end{eqnarray}
where
$C_{\ell, \text{th}}'$ is the theoretical power spectrum of CMB after accounting for the beam and pixel effects, \textit{i.e.} 
\begin{eqnarray}\label{eqn:beam-pixel}
	C_{\ell, \text{th}}' = C_{\ell, \text{th}} B_{\ell}^2 P_{\ell}^2,
\end{eqnarray}
where
$B_{\ell}$ are the Legendre transforms of the beam function and $P_{\ell}$ are the HEALPix pixel 
window functions. $\ell_{\text{max}}$ can be chosen according to the pixel angular resolution of 
the maps. In this work, we made use of the low resolution maps at $N_{\text{side}} = 16$, for 
which $\ell = 32$ is a sufficiently large multipole. The sum is taken starting from $\ell = 2$ 
because the monopole ($\ell = 0$) and dipole ($\ell = 1$) are uninteresting for cosmological studies, and hence, they are removed from the input maps prior to all the analyses.\\



%-------------------------
\subsection{Genetic Algorithm}
\label{sub:GA_form}
%-------------------------
%%
\begin{figure}
	\begin{center}
		\includegraphics[width=0.9\columnwidth]{figs/GA_flowchart_vert_times.pdf}
	\end{center}
	\caption{The flowchart of a typical GA. The sorting is introduced in order to ensure the better readability and presentability of the solutions in a population.}\label{fig:GA-flow}
\end{figure}
%%

Genetic algorithm (GA) was first proposed by John H. Holland \citep{10.1145/321127.321128} in the mid 1960s and was further extended by David Goldberg in the late 1980s \citep{Goldberg}. GA aims to find global optimal solution(s) to objective functions having a multimodal nature, i.e., it has multiple local optima in the domain of search. As the conventional optima-finding algorithms tend to fail in such cases, GA opens up a new window with its novel concept. The underlying principle of GA is Darwin's theory of ``natural selection''. The biological species evolve by this natural process. In a population, each individual has a different set of genetic characteristics. Thus, the whole population has a diverse gene pool. This community is subject to certain natural circumstances known as the environment. The process of natural selection asserts that the genes which lead to better traits for survival will more likely be passed on to the offspring by reproduction over the generations than those which do not. Occasional mutations may bring in new beneficial genes that were earlier not present. Hence, eventually the whole population will best adapt to its environment after several generations.

%\begin{figure}
%	\begin{center}
%		\includegraphics[width=0.9\columnwidth]{figs/GA_overview_pop_times.pdf}\hrule \vspace{10pt}
%		\includegraphics[width=0.9\columnwidth]{figs/GA_uni-cross_times.pdf}\hrule \vspace{10pt}
%		\includegraphics[width=0.9\columnwidth]{figs/GA_mutation.pdf}
%	\end{center}
%	\caption{A schematic representation of the various parts of GA. The size of this example population is 4 individuals. I-3 and I-4 make a pair on which the uniform crossover is performed and the new individuals I-3$'$ and I-4$'$ are found. These are the offspring which then get mutated into I-3$''$ and I-4$''$.} \label{fig:GA-ovrw}
%\end{figure}

\begin{figure*}
	\includegraphics[width=0.26\linewidth]{figs/GA_pop_times.pdf}
		{\color{gray} \hspace{1pt}\rule{0.4pt}{90pt}\hspace{1pt}} 
	\includegraphics[width=0.2\linewidth]{figs/SUS_wheel.pdf} 
		{\color{gray} \hspace{1pt}\rule{0.4pt}{90pt}\hspace{1pt}}
	\includegraphics[width=0.235\linewidth]{figs/GA_uni-cross_times.pdf}
		{\color{gray} \hspace{1pt}\rule{0.4pt}{90pt}\hspace{1pt}}
	\includegraphics[width=0.255\linewidth]{figs/GA_mutation.pdf}\\
	\hspace*{0.11\linewidth}(A) 
	\hspace*{0.22\linewidth}(B) 
	\hspace*{0.21\linewidth}(C)
	\hspace*{0.23\linewidth}(D)
	\hspace*{0.10\linewidth}
	\caption{A schematic representation of a toy-model of GA. (A) shows a population with 4 individuals and indicates the terminology within a GA framework. (B) shows a Roulette wheel with two fixed-points diametrically opposite pointing to I-3 and I-4. (C) shows a typical uniform crossover event on I-3 and I-4, producing the new individuals I-3$'$ and I-4$'$. These are the offspring which then get mutated into I-3$''$ and I-4$''$ as shown in (D). All the numbers including binary strings and probabilities are only meant for visualization of this toy-model.} \label{fig:GA-ovrw}
\end{figure*}

GA employs this strategy to the optimization problems. The goal at hand is to numerically find the global optimum of an objective function in the given domain of interest. GA works on the principle of convergence with variation. Diversity in a population is a necessary ingredient for the GA to succeed. Over the generations, individual solutions with higher and higher fitness dominate the population. Fig.~\ref{fig:GA-flow} shows the flowchart of a typical GA. In the following we briefly discuss the most common steps of GA. 

\begin{enumerate}
    \item \underline{Population initialization}: GA derives from the basic ideology of population genetics. Hence, GA is based on a ``population'' -- a set of possible solutions to the problem. According to the domain of search for the optimal solution, a population can be randomly created. The size of a population is a heuristic parameter.
    
    \item \underline{Fitness calculation}: A fitness function is chosen such that the fitter individuals will be closer to the optimum (in the functional space) than the less fit ones. The choice of a fitness function depends on the underlying objective and it is not unique. However, in order for a proper implementation of the algorithm, the fitness function should be positive and analytic, at least in the domain of interest.
    
    \item \underline{Parent selection}: This is one of the most crucial steps of the entire algorithm. As Natural selection dictates, the fitter individuals in the population are more likely to pass on their genes by mating than the less fit ones. To simulate this scenario, we must assign some weights to the individuals according to their fitness. One most common method is the fitness-proportionate assignment of weights (probabilities). In this step the pairs are selected for mating and producing the new generation. There is a set of methods for this. We will discuss some of them in section~\ref{sec:meth}.
    
    \item \underline{Crossover and Mutation}: From the parents selected in the previous step, we now make crossovers to generate new individuals, collectively called ``offspring''. The new individuals are occasionally mutated with a small amplitude. In our case, mutation refers to a discrepancy in the passage of genes from one generation to the next. Mutation is a crucial tool to bring in variety in the genetic material. We discuss the importance of mutation in section~\ref{sec:meth}. The offspring thus produced will constitute the next generation of our population.
    
    \item \underline{Termination}: Steps (ii)-(iv) are repeated in a loop for each new generation until the stopping criteria are reached/ GA has ``converged''. In general, a GA implementation is said to have converged if the fitness of the best individual(s) over the generations does not improve any further. The programmer can set a stop-point generation, hitting which will end the generation loop. This stop-point can be found by \textit{hit-and-trail}. Altogether, the fittest individual in the last generation is deemed the solution of the GA run.
\end{enumerate}

Fig.~\ref{fig:GA-ovrw} uses a toy-model population to visually understand the different terms and steps in our GA implementation. (We discuss it further in the due course.) In section~\ref{sec:meth}, we describe the methodology adopted for this implementation of GA-ILC with the details concerning the exact styles and the values of certain parameters. 


%-------------------------
%-------------------------
\section{Methodology}
\label{sec:meth}
%-------------------------
%-------------------------

Python is one of the most commonly used programming languages for scientific computation. It provides many open-source packages for various utilities. One of those packages is the python-port of HEALPix, HEALPy. It comes with many utilities we need for handling and analyzing the data. There are pixel utilities, spherical-harmonic transform utilities, visualization related functions and many miscellaneous tools for an effective manipulation and reduction of a HEALPix data-set. We have implemented the algorithm in Python 3 for numerical computations.

In this work, GA is implemented to find the optimal weights to linearly combine the input maps such that the reduced variance of the clean map is minimized. This is a multivariable constrained optimization problem. In the present work, the problem is converted to an unconstrained optimization problem by a little manipulation of the method. The objective function to be optimized (minimized) is the (reduced) variance function (Eqn.~\eqref{eqn:sigma-sq-A}) 
\begin{eqnarray}\label{eqn:var-func}
	\sigma^2(\mathbf{w}) = \mathbf{w}^{\text{T}}\mathbf{Aw}.
\end{eqnarray}
The constraint is $\sum_{i=1}^{N}w = 1$. Alternatively, for the current work, one of the $N$ weights is found as $w_{i'} = 1 - \sum_{i \neq i'} w_i$. (It is the last weight of the spectrally-ordered system for our work.) Thus, the goal is to estimate $\mathbf{w}_{\text{GA}}$ that minimize $\sigma^2(\mathbf{w})$ using GA.

%%
%\begin{figure*}
%	\begin{center}
%		\includegraphics[width=0.95\linewidth]{figs/weights_diff_GA17.pdf}
%	\end{center}
%	\caption{Differences between analytical `LM' weights and GA weights for Monte Carlo simulations plotted against \texttt{seed}.}
%	\label{fig:weights-seeds}
%\end{figure*}
%%



%-------------------------
\subsection{Implementation of GA-ILC}
\label{sub:implmn}
%-------------------------

In Genetics, an individual can be represented by its genotypic as well as phenotypic characteristics. In our GA, the phenotype of an individual is a decimal number in the case of a single variable problem and a vector (array) of $n$ decimal numbers in the case of a multivariable problem. We chose the binary-string representation as our genotype. The required decimal accuracy is a fixed parameter initially chosen by the programmer. Depending upon the accuracy, each individual fraction can be converted to an integer by multiplying it with 10$^{\text{decimal-accuracy}}$ and rounding off. Then the integer is converted into its binary representation. In the multivariable cases, each element of a vector is converted into a binary string and all of those strings are concatenated in the original order into a single large string. We call this string a ``chromosome''. The binary part of an individual element of this vector chromosome is called a ``segment'', each position of a chromosome is called a ``gene'', and the bit-value of each gene is called the ``allele'' of that gene. Fig.~\ref{fig:GA-ovrw} (A) depicts a schematic of this terminology. It shows an example one-variable population in the genotype representation. By performing the exact inverse operations on a chromosome will return the phenotype (decimal vector) of that individual.

Now that the terminology is clear, we need to initialize a population. The size of the population, \texttt{Popsize}, is fixed by the programmer by \textit{hit-and-trial}. A domain of search must be first decided. For our GA-ILC, we chose $\{ w_{\text{SS},i} \pm 1\}$ as the domain, hereinafter called ``priors''. The initial population is then uniformly randomly generated. The population at any stage of calculation is a 2D array with one dimension containing genes of a single chromosome and the other containing as many as \texttt{Popsize} chromosomes. The allele of each gene of each chromosome is determined by generating a (pseudo-) random number between 0 and 1, and according to whether it is less or greater than 1/2. This is, in some sense, the coin-toss determination and it is frequently used in this work. From the binary population, the decimal population is also found. At this point a check is made on whether all the variables in each individual fall within the priors imposed. The number of variables in our GA-ILC, \texttt{N\_var}, is one less than the number of input maps since one of the weights is dependent on the rest of them.

A suitable fitness function for the GA-ILC is chosen as
\begin{eqnarray}\label{eqn:fitness}
	\bar{f}(\mathbf{w}) = \frac{1}{\sigma^2(\mathbf{w})}.
\end{eqnarray}
Since the $\sigma^2$ is strictly positive, this fitness function satisfies the criteria of being a positive analytic function (in principle, the global minimum of $\sigma^2$ is the reduced variance of a pure CMB map).

As discussed in subsection~\ref{sub:GA_form}, there are various methods for selecting parent pairs from a given fitness-associated population. Some of them are the fitness-proportionate selection methods which assign weights proportional to the fitness of the individuals, whereas some make use of the fitness values indirectly in doing so. One of the most common parent selection methods (which we use in this work) is called ``Stochastic Universal Sampling'' or ``SUS'' in short. It is a Roulette-wheel-based method in which, depending upon the number of individuals to be picked in one turn of the wheel, there is a number of fixed-points. In case of picking pairs, there are two fixed-points, equally separated around the wheel's periphery. A turn of the wheel is simulated by generating a pseudorandom number.

\begin{figure}
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figs/w1_w3_var_times.pdf}
		%\includegraphics[width=0.9\linewidth]{figs/w1_w3_var_rotated.pdf}
	\end{center}
	\caption{The variance function in Eqn.~\eqref{eqn:var-func} plotted against two weights $w_1$ and $w_3$ while keeping the rest of the weights fixed to the SS weights. This is performed on the simulation of \texttt{seed} 100 low-resolution maps. A very broad valley can be seen close to the supposed global minimum.} \label{fig:w-vs-var}
\end{figure}
%%

In a fitness-proportionate (``FP'' for short) selection of parents, the area assigned to each individual in a population on the wheel is directly proportional to their fitness. Thus, depending upon the number of pairs to be picked (a.k.a. size of the ``mating-pool''), the wheel is turned so many times and each turn will return two individuals as a pair. This scenario of fitter individuals having better chance of reproduction is dubbed the ``selection pressure''. However, as the generations progress and the population approaches the desired optimum, the fitness value may get less and less diverse due to a decreasing gradient of the objective function. In such cases, the selection pressure to the fitter individuals wanes since all the individuals will be assigned an \emph{almost} equal area on the wheel. To make sure this does not happen, another selection method such as ``Rank selection'' can be employed. In this technique, the individuals in the population are sorted according to their fitness values and they are assigned ranks. Then the weights are assigned according to their ranks, irrespective of their fitness values. Nonetheless, the individual(s) with the highest fitness value will have the largest weight. The selection is then done by the Roulette-wheel turnings. This method brings the selection pressure back into the picture as the weight-gradient is preserved. When the objective function in the domain of search inherently has a less gradient, rank selection should be preferred from the start of the generation loop. If closer to the optimum point the function is less steep, then after a certain number of generations the selection type can be switched from FP to rank selection. This switching-point generation is another GA parameter called \texttt{switch}. Fig.~\ref{fig:GA-ovrw} (B) shows a Roulette wheel in our toy-model population with some area given to all four individuals under either of the selection settings. Notice that the two fixed-points for picking pairs are diametrically opposite to each other. This way, an individual with low selection pressure may also get picked occasionally, maintaining the genetic diversity in GA.

The variance function in Eqn.~\eqref{eqn:var-func} is plotted in Fig.~\ref{fig:w-vs-var} for a pair of weights while keeping the rest of the weights fixed to the SS-weights. It is done on one of the realizations in the Monte Carlo simulations we performed (we discuss it further in section~\ref{sec:MC-simul}). As can be seen in that figure, the variance function has a very broad valley close to the global minimum. We verified that the feature is present irrespective of the pair of weights chosen. This required us to switch between FP and rank selection types after a \texttt{switch}-point generation. It is yet another parameter that must be fixed by \textit{hit-and-trial} like all the rest of the GA parameters and is variable for a plethora of GA implementations.

The crossover again comes with a variety of types. Some of the most popular types are one-point, two-point, $k$-point, uniform crossovers. We chose the uniform crossover as the universal type for our entire work. An instance of this type of crossover is schematically shown in Fig.~\ref{fig:GA-ovrw} (C) for our toy-model population. Under uniform crossover, the alleles of each gene are exchanged between the two parent chromosomes of a pair using the coin-toss determination technique. Thus, each pair of parents give rise to a pair of children. These children are mutated with a small amplitude, \texttt{P\_mut}. A mutation is introduced as the following. For each gene of each child chromosome, a pseudorandom number \texttt{r} in the range $[0, 1)$ is generated. If \texttt{r<P\_mut}, the allele of that gene is flipped (0$\to$1 and 1$\to$0). The parameter \texttt{P\_mut} is best chosen by the programmer by \textit{hit-and-trial}. Fig.~\ref{fig:GA-ovrw} (D) indicates one of the mutation incidents in the toy-model GA. After these steps, we have a set of offspring chromosomes at hand. Depending upon the fitness of the children and the size of the population, a new generation is recruited by keeping \texttt{Popsize}-many fittest children into the population and discarding the rest of them. The newly deemed population will then undergo the same steps of parent selection and mating. Proper termination criteria in terms of the number of generations, \texttt{N\_gen}, is found by observing the convergence of GA-ILC and subject to the computational resources.


%--------------------
\subsection{Remarks on Genetic Diversity}
\label{sub:diversity}
%--------------------

It is particularly important to ensure a genetically diverse population at \emph{each} step of GA. The initial population is randomly initialized to have the maximum diversity possible. While applying selection pressure towards the fitter individuals, the less fit individuals should not be neglected entirely. A balanced reproduction is only possible if the entire population takes part in mating. We need to make sure that the least fit individuals also get to reproduce at least a couple times. A way to ensure this (in addition to using SUS method) is to make \texttt{N\_pairs} $\sim  \lceil1 /(\text{min}(\{p_i\}))\rceil$ where $\{p_i\}$ is the set of probabilities (weights) assigned to the individuals and $\lceil\rceil$ is the ceiling function. If this is less than half \texttt{Popsize}, \texttt{N\_pairs} can simply be \texttt{Popsize}. Due to the limited resources and increasing requirement of computing power, the bottleneck must be applied to filter-out the fitter individuals after reproduction and preserve the size of the population at each generation. This results in only a \emph{slightly} reduced genetic diversity, which is harmless considering the variety of children produced by mutation. Mutation, thus, is a very crucial ingredient in our GA. Indeed, a mild mutation prevents GA from premature convergence. A GA-ILC implementation without any mutation loses the genetic diversity almost completely as it moves towards the global optimum and causes it to converge to some false local optimum which is not actually present. It is a partly computational -- partly fundamental defect (in that a non-diverse, self-breeding population eventually ceases to survive under natural selection). In Population Genetics, occasional mutation may, by pure chance, introduce a better trait for survival and is preferred through generations. We studied this very interesting ingredient in more detail in our work. We discuss this further in sections \ref{sec:MC-simul} and \ref{sec:results}.




%-------------------------
%-------------------------
\section{Validation of the GA implementation}
\label{sec:valid}
%-------------------------
%-------------------------

%
\begin{figure}
	\begin{center}
		\includegraphics[width=0.9\linewidth]{figs/trial_fn_2d_times.pdf}
	\end{center}
	\caption{The two-variable case of the trial function in Eqn.~\eqref{eqn:trial-fn} within an appropriate domain of interest. Observe the highly multimodal nature with multiple local maxima located closely together.}\label{fig:trial-fn}
\end{figure}
% 

Implementation of GA is a complicated task. In particular, the programmer needs to find the optimal values of various GA parameters mostly by \textit{hit-and-trial} as mentioned quite frequently throughout subsection~\ref{sub:implmn}. The most common GA parameters are the population size \texttt{Popsize}, number of total generations \texttt{N\_gen}, decimal accuracy required \texttt{Dec}, probability of mutation \texttt{P\_mut}, etc. Hence, it is wise to start with a very simple implementation and then move towards more complicated problems. We have employed this strategy for this work. We initially implemented GA for single variable test-problems. Then we moved on to multivariable test-problems and we obtained satisfactory results in both of them. The test function we chose is
\begin{eqnarray}\label{eqn:trial-fn}
	f(\mathbf{x}) = \frac{1}{1 + \sum_{i=1}^{N_{\text{var}}}[ x_i^2 + 10(\cos(2\pi x_i) - 1)]},
\end{eqnarray}
with the same function as the fitness function. This function is well-defined for any number of variables. The two-variable case of this function is shown in fig.~\ref{fig:trial-fn} for visualization. It is observed from the figure that this trial function is a highly multimodal function \textit{i.e.} it has multiple local maxima with a single global maximum at $\mathbf{x = 0}$ with $f(\mathbf{0}) = 1$, within the domain of interest as plotted therein. Also, it is a steep function with large gradient. Hence, for such implementations, no mutation was necessary (in other words, \texttt{P\_mut} = 0) and after $\sim$60\% of total generations, the selection type was switched from FP to rank selection to compensate for having no mutation. The best of those results for different $N_{\text{var}}$ are summarized in table~\ref{tab:primary-GA}. Table \ref{tab:primary-GA-11} shows the results obtained for the fixed number of variables in the trial function -- which is the same as that in our GA-ILC implementation -- with different population sizes. As expected, the best function value improves as we increase the size of the population. The results of the trials as discussed here are sufficient to conclude that our GA implementation is robust and accurate in the multivariable scenario of our GA-ILC method, and that it is safe and desirable to incorporate ILC with it in the manner described in section~\ref{sec:meth}.
\begin{table}
	%%\caption{The results of the primary implementations of GA on the trial function 
	%%in Eqn.~\eqref{eqn:trial-fn}.} 
	%\begin{center}
	%\begin{tabular}{ll}
	%%\begin{subtable}%{0.47\linewidth}
	\caption{Results of the trial implementation for different number of variables. Notice that, for larger and larger number of variables, to get the best results, we need to have a larger and larger population.}\label{tab:primary-GA}
	\centering 
	\begin{tabular}{|c|c|c|c|}
		\hline
		$N_{\text{var}}$ & \texttt{Popsize} & \texttt{N\_gen} & Best $f(\mathbf{x^*})$\\
		\hline
		1 	& 200 	& 60	&1.0 \\
		2	& 200	& 60 	&1.0\\
		5	& 400	& 200 	&1.0\\
		10	& 600	& 250	&1.0\\
		11	& 600	& 250	&1.0\\
		12	& 600 	& 250	&1.0\\
		\hline
	\end{tabular}
	%\end{table}
	
	%\begin{table}%{0.47\linewidth}
	\caption{Variation of the final function value (in the trial implementation) \emph{w.r.t} \texttt{Popsize} for fixed number of generations \texttt{N\_gen} = 250 and $N_{\text{var}}$ = 11.
		Notice the very bad function value for small \texttt{Popsize}.}\label{tab:primary-GA-11}
	\centering
	\begin{tabular}{|c|c|}
		\hline
		\texttt{Popsize}  	& Best $f(\mathbf{x^*})$\\
		\hline 
		400		& 0.5011925167207855\\
		450		& 1.0000000000000000\\
		500		& 0.9999999999980158\\
		550		& 0.9999999999682601\\
		600		& 1.0000000000000000\\
		650		& 1.0000000000000000\\
		\hline
	\end{tabular}
	%\begin{tabular}{|r|r|r|}
	%\end{tabular}
	%\end{center}
\end{table}
%





%%
\begin{figure}%[!ht]
	\begin{center}
		\includegraphics[width=0.95\columnwidth]{figs/mean_diff_std_maps_GA17_times.pdf}
		%\includegraphics[width=0.48\linewidth]{figs/cl_simul_err.pdf}
	\end{center}
	\caption{The statistical difference maps from the Monte Carlo simulations. Only a very small residual contamination seems to be present, only close to the plane of Milky Way. The unit of the values is $\mu$K.}
	\label{fig:simul-diff-maps}
\end{figure}
%%
%%
\begin{figure}%[!ht]
	%\includegraphics[width=0.98\columnwidth]{figs/cl_GA17_diff_err.pdf}
	\includegraphics[width=\columnwidth]{figs/cl_GA_th_err.pdf}
	%\includegraphics[width=0.48\linewidth]{figs/cl_simul_err.pdf}
	\caption{The angular power spectrum of the GA clean simulated mean difference map overplotted with the Planck theoretical power spectrum. Note that the beam and pixel effects are removed. Middle: The difference of the power spectra of GA simulations and Planck theory with errors in mean-estimation. It can be seen that the deviation is within 3$\sigma_{\mu}$ level which indicates that there is no bias. Bottom: The standard deviation errors in the angular power spectrum of the final mean difference map. Notice that these errors conform with the minimal errors induced by the cosmic variance, $\Delta C_{\ell, \text{cv}} = \sqrt{\frac{2}{2\ell+1}}C_{\ell}$.}
	\label{fig:simul-diff-cls}
\end{figure}
%%

%-------------------------
%-------------------------
\section{Monte Carlo Simulations}
\label{sec:MC-simul}
%-------------------------
%-------------------------

As a statistical test for our GA-ILC method, we performed Monte Carlo simulations using WMAP and Planck simulated maps. From the satellite missions' observed angular power spectra, using HEALPix IDL \footnote{IDL is the interactive data language for Unix and Linux based systems, see, e.g., \url{https://caligari.dartmouth.edu/doc/idl/}} routine \texttt{isynfast}, anisotropy maps were randomly generated by passing an argument called \texttt{seed} which seeds the pseudorandom number generator. The sample size of our simulations is 200, with \texttt{seed} varying from 1 to 200 in steps of 1. A total of 12 low-resolution maps at $N_{\text{side}} = 16$ were generated for each \texttt{seed} as input of GA-ILC. Of the 12 input maps, the 5 WMAP frequency bands are K (23 GHz), K$_{\text{a}}$ (33 GHz), Q (41 GHz), V (61 GHz), W (94 GHz); and the seven Planck frequency bands are 30, 44, 70, 100, 143, 217 and 353 GHz. These simulated maps are 9\textdegree  beam-smoothed. Corresponding to each \texttt{seed}, a pure CMB map was also generated using the same routine from the Planck theoretical angular power spectrum. These CMB maps were used as references to compare with the clean maps that GA-ILC produced. Some of the GA parameters were kept fixed for the simulations, namely, \texttt{Popsize} = 400, \texttt{N\_gen} = 400, decimal accuracy \texttt{Dec} = 7, and switching between FP and rank selection types after \texttt{switch} generation \# 20, all of them optimally chosen. To explore the effect of mutation on the results, GA-ILC was run for 128 different values of \texttt{P\_mut} starting from 0.05\% in steps of 0.05\%, for each \texttt{seed} value. The best results (of all \texttt{P\_mut}) for every \texttt{seed} were compiled and then we found the statistical measures. In particular, we found the best map of all \texttt{P\_mut} for every \texttt{seed} and subtracted the pure CMB simulated map for that \texttt{seed} from it. This map is called a simulated difference map here. 200 such simulated difference maps were produced and the mean and standard deviation difference maps were calculated from it. Fig.~\ref{fig:simul-diff-maps} shows these statistical maps. The mean difference map contains very small amount of residual contamination and that is limited to small regions around the galactic equator, as can also be inferred from the standard deviation difference map.

The angular power spectrum corresponding to the simulated mean clean map along with the standard deviation errors is overplotted with the Planck theoretical CMB power spectrum in the top panel of fig.~\ref{fig:simul-diff-cls}. In the middle panel of the figure, the mean difference power spectrum is plotted with the errors in the estimation of mean ($\sigma_{\mu, \ell}$). We observe that the estimated power spectrum is within a 3$\sigma_{\mu}$ deviation to the theoretical CMB power spectrum. This confirms that there is no bias in the entire multipole range $2<\ell<32$ under study. In the bottom panel of the same figure, the standard deviation errors ($\sigma_{\ell}$) are overplotted with the cosmic-variance induced errors. The observed errors also agree well with the expected minimal errors purely of the cosmic origin.





%-------------------------
%-------------------------
\section{Application on WMAP and Planck data}
\label{sec:results}
%-------------------------
%-------------------------

%%
\begin{figure}
	\includegraphics[width=0.99\columnwidth]{figs/GA18.2_mutation-plot.pdf}
	%\includegraphics[width=0.48\linewidth]{figs/cl_simul_err.pdf}
	\caption{The minimal variance obtained by GA-ILC on data against the coefficient of mutation. The minimum of these minimal values -- that occurs at a low-to-moderate mutation by 1.15\% -- is also shown distinctively. Bottom: a zoomed-in version for a closer look at the variation.} \label{fig:GA18.2_mut-plot}
\end{figure}
%%\hspace{10pt}
\begin{figure}
	\centering
	\includegraphics[width=\linewidth]{figs/trace_plots_1.15_times.pdf}
	\caption{Top: The overplot of the traces of mutated and unmutated cases. The mutated case is the best among all the 144 mutation values with \texttt{P\_mut} = 1.15\% implemented on data. Middle and Bottom: closer look at the convergence in the mutated case.} \label{fig:GA18.2_trace-plot}	
	%\caption{\small Effects of mutation on the GA-ILC implementation within the simulations.}
	%\label{fig:mut-effects}
\end{figure}
%%

We applied our GA-ILC on 12 multifrequency low-resolution input maps observed by WMAP and Planck to find an optimal clean map as output (we refer to this as the implementation on ``data'' as shorthand; the 12 input frequencies are those mentioned in section~\ref{sec:MC-simul}). GA parameters like \texttt{Popsize}, \texttt{N\_gen}, \texttt{Dec}, and \texttt{switch} were kept at the fixed values, same as the Monte Carlo simulations. Similarly to the simulations, we applied GA-ILC with 144 values of \texttt{P\_mut}, from 0\% to 7.15\% in steps of 0.05\%, to analyze the final reduced variance in each case. The best case of all these 144 was deemed to represent the best version of our GA-ILC. The plot of final reduced variance against the mutation probability is shown in fig.~\ref{fig:GA18.2_mut-plot}. As we can see from there, a very low (or negligible) mutation does not help in our pursuit of optimization, and a large amount of mutation is also not useful. A moderately small value of 1.15\% is the best mutation value in this case. There is no exact analytical problem-based expression for the optimal amount of mutation as is demonstrated by this figure. We plotted the variance values against the cumulative individuals through generations (the population is sorted in the ascending order of fitness). This type of plots are called ``trace-plots'' here. These plots help us understand the convergence of GA and the effects of various parameters on it. Fig.~\ref{fig:GA18.2_trace-plot} shows an example of trace-plots. Therein we can see that the population without any mutation converges prematurely due to the loss of genetic diversity as there exists a broad valley close to the global optimum. On the other hand, the optimally mutated case is able to find better solutions since it is able to explore the variable space. It is also evident that even when the unmutated case has long since converged, the mutated case is slowly able to find better and better solutions after generations.

\begin{figure}
	\includegraphics[width=0.99\columnwidth]{figs/ilcm_GA18.2_1.150_400_400_times.pdf}
	\includegraphics[width=0.99\columnwidth]{figs/ilcm_GA18.2_400_400_diff_GA-SS_times.pdf}
	\caption{The clean map produced by the GA-ILC on WMAP and Planck data with 1.15\% mutation. Bottom: Difference map of GA-ILC and the SS-ILC. All the values are in $\mu$K. Notice that the residual contamination is very small and exists only close to the galactic plane. In other parts of the sky, a very good agreement can be seen.} \label{fig:GA18.2_clean-maps}
\end{figure}
%%
\begin{figure}
	\includegraphics[width=\linewidth]{figs/cl_GA_SS_err.pdf}
	\caption{The clean angular power spectrum produced by GA-ILC implemented on data with 1.15\% mutation. The reconstruction errors in GA-ILC here are found using 200 Monte Carlo simulations. The two power spectra seem to agree quite well with each other.} \label{fig:GA18.2_clean-cls}
\end{figure}


Fig.~\ref{fig:GA18.2_clean-maps} shows the clean CMB map produced by GA-ILC on data. Therein the bottom panel shows the difference map of GA-ILC and SS-ILC \citep{Sudevan_2018}. It confirms that the minimal residual contamination is present in the GA-ILC clean map and that it tends to occupy small areas close to the galactic plane. The clean angular power spectrum calculated from the GA-ILC clean map (data implementation) is overplotted with SS-ILC clean power spectrum in fig.~\ref{fig:GA18.2_clean-cls}. We observe that the two match very well with each other. This also indicates that the purely numerical GA-ILC technique gives as good results as SS-ILC with analytical expression of weights.

%%
\begin{figure*}
	\includegraphics[width=0.49\linewidth]{figs/diffmap_COMM_GA_pm30_times.pdf}
	\includegraphics[width=0.49\linewidth]{figs/diffmap_NILC_GA_pm30_times.pdf}
	%\includegraphics[width=0.49\linewidth]{figs/diffmap_SEVEM_GA_pm30.pdf}
	\includegraphics[width=0.49\linewidth]{figs/diffmap_SMICA_GA_pm30_times.pdf}
	\includegraphics[width=0.49\linewidth]{figs/diffmap_WMAP_GA_pm30_times.pdf}
	%\includegraphics[width=0.48\linewidth]{figs/cl_simul_err.pdf}
	\caption{Difference maps of several other foreground reconstruction techniques and GA-ILC implemented on data. Notice that except the small residual foreground close to the galactic equator -- which itself varies from one method to the other, our results agree very nicely with those of other methods. The values are in $\mu$K.
	}
	\label{fig:comp_diff_maps}
\end{figure*}
%%

%%
\begin{figure*}
	%\includegraphics[width = 0.8\paperwidth]{figs/cl_comparison_new2_GA18.2.pdf}
	%\includegraphics[width=0.48\linewidth]{figs/cl_simul_err.pdf}
	\includegraphics[width = \linewidth]{figs/cl_GA_comp_new2.pdf}
	\caption{Comparison of angular power spectrum of the GA-ILC clean map with those of several different foreground reconstruction techniques. Bottom panel: differences of those various clean angular power spectra and GA-ILC clean power spectrum (data implementation). A good agreement between GA-ILC and other methods is seen here as well.}
	\label{fig:cl_comp}
\end{figure*}
%%

We compare our GA-ILC results with those obtained by various component reconstruction methods of other science groups, namely, COMMANDER, NILC, SMICA \citep{Planck2018IV}, and WMAP-ILC \citep{WMAP-9yr-maps}. Shown in fig.~\ref{fig:comp_diff_maps} are the difference maps of each of those clean maps and the GA-ILC clean map (data implementation, of course). It is evident from here that our method produces a clean map that agrees well with some other completely different methods. We also observe that the residual contamination near the Milky Way plane is different in different methods. This indicates that those other clean maps also contain some (minimal) amount of residual foreground, same as ours. Finally we show, in fig.~\ref{fig:cl_comp}, each of the angular power spectra of GA-ILC and that of COMMANDER, NILC, SMICA, and WMAP-ILC for comparison. As expected, the spectra agree well with each other, with the small difference between all the different methods. Again, the other methods' results differ from each other same as our method's.

%%%%-------
% GA17 corresponds to the Monte-Carlo simulations of GA-ILC
% GA18.2 corresponds to the GA-ILC implementation on WMAP-Planck data
% The sources of WMAP-Planck data need to be mentioned/ discussed/ referred/ cited.
%%%%-------

%%

\section{Conclusion}\label{sec:conclusion}

In this paper, for the first time in literature, we develop and implement the biological-selection-rule-motivated genetic algorithm to reconstruct the CMB component over large angular scales by removing foregrounds using linear combination of multifrequency observations of WMAP and Planck satellite missions. We validate our methodology by performing 200 Monte Carlo simulations using realistic observations from the final-year WMAP and Planck missions. The results of the simulations show that the CMB map and the angular power spectrum can be accurately recovered by using our method. The outcome of our method is in close agreement with the results obtained by using weights following the exact analytical method which demonstrates usefulness of the new method of this paper. We compare the cleaned CMB maps and recovered angular power spectrum obtained by the new method with those obtained by WMAP and Planck science groups. These results agree well with each other, which shows that the CMB results obtained by the satellite missions are robust with respect to the data analysis algorithms applied. Enriched in inherent parametric characteristics, GA-ILC is a very flexible and volatile method to work with. Nonetheless, since GA is expected to produce robust results with respect to fluctuations in the optimizing system, our method opens up a new avenue to apply the method when the weight factors for linear combination in a modified ILC method can no longer be obtained in a closed analytical form, e.g., when one is interested in an objective function which deviates from quadratic nature. The larger domain of application of the GA-ILC is a very promising feature and will be explored in the future communications by the authors.
	



\section*{Acknowledgements}
We thank the HPC cluster facility,  ``Kanad'' of IISER Bhopal for providing access and resources for the crucial part of this research.



\section{Data Availability}
The data pertaining to this article will be shared on reasonable request to the corresponding author.
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\usepackage{bm}		% Bold maths symbols, including upright Greek
\usepackage{pdflscape}	% Landscape pages

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%%%%%% AUTHORS - PLACE YOUR OWN MACROS HERE %%%%%%

% Please keep new commands to a minimum, and use \newcommand not \def to avoid
% overwriting existing commands. Example:
%\newcommand{\pcm}{\,cm$^{-2}$}	% per cm-squared
\newcommand{\kms}{\,km\,s$^{-1}$} % kilometres per second
\newcommand{\bibtex}{\textsc{Bib}\!\TeX} % bibtex. Not quite the correct typesetting, but close enough

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% Use vector fonts, so it zooms properly in on-screen viewing software
% Don't change these lines unless you know what you are doing
\usepackage[T1]{fontenc}
\usepackage{ae,aecompl}

% MNRAS is set in Times font. If you don't have this installed (most LaTeX
% installations will be fine) or prefer the old Computer Modern fonts, comment
% out the following line
\usepackage{newtxtext,newtxmath}
% Depending on your LaTeX fonts installation, you might get better results with one of these:
%\usepackage{mathptmx}
%\usepackage{txfonts}

%%%%%%%%%%%%%%%%%%% TITLE PAGE %%%%%%%%%%%%%%%%%%%

% Title of the paper, and the short title which is used in the headers.
% Keep the title short and informative.
\title[MNRAS \LaTeX\ guide for authors]{\textit{Monthly Notices of the Royal Astronomical
  Society}:  \LaTeX\ guide for authors}

% The list of authors, and the short list which is used in the headers.
% If you need two or more lines of authors, add an extra line using \newauthor
\author[K. T. Smith]{Keith T. Smith$^{1}$%
\thanks{Contact e-mail: \href{mailto:mn@ras.ac.uk}{mn@ras.ac.uk}}%
\thanks{Present address: Science magazine, AAAS Science International, \mbox{82-88}~Hills Road, Cambridge CB2~1LQ, UK}%
\\
% List of institutions
$^{1}$Royal Astronomical Society, Burlington House, Piccadilly, London W1J 0BQ, UK}

% These dates will be filled out by the publisher
\date{Last updated 2020 June 10; in original form 2013 September 5}

% Enter the current year, for the copyright statements etc.
\pubyear{2020}

% Don't change these lines
\begin{document}
\label{firstpage}
\pagerange{\pageref{firstpage}--\pageref{lastpage}}
\maketitle

% Abstract of the paper
\begin{abstract}
This is a guide for preparing papers for \textit{Monthly Notices of the Royal Astronomical Society} using the \verb'mnras' \LaTeX\ package.
It provides instructions for using the additional features in the document class.
This is not a general guide on how to use \LaTeX, and nor does it replace the journal's instructions to authors.
See \texttt{mnras\_template.tex} for a simple template.
\end{abstract}

% Select between one and six entries from the list of approved keywords.
% Don't make up new ones.
\begin{keywords}
editorials, notices -- miscellaneous
\end{keywords}

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%%%%%%%%%%%%%%%%% BODY OF PAPER %%%%%%%%%%%%%%%%%%

% The MNRAS class isn't designed to include a table of contents, but for this document one is useful.
% I therefore have to do some kludging to make it work without masses of blank space.
\begingroup
\let\clearpage\relax
\tableofcontents
\endgroup
\newpage

\section{Introduction}

The journal \textit{Monthly Notices of the Royal Astronomical Society} (MNRAS) encourages authors to prepare their papers using \LaTeX.
The style file \verb'mnras.cls' can be used to approximate the final appearance of the journal, and provides numerous features to simplify the preparation of papers.
This document, \verb'mnras_guide.tex', provides guidance on using that style file and the features it enables.

This is not a general guide on how to use \LaTeX, of which many excellent examples already exist.
We particularly recommend \textit{Wikibooks \LaTeX}\footnote{\url{https://en.wikibooks.org/wiki/LaTeX}}, a collaborative online textbook which is of use to both beginners and experts.
Alternatively there are several other online resources, and most academic libraries also hold suitable beginner's guides.

For guidance on the contents of papers, journal style, and how to submit a paper, see the MNRAS Instructions to Authors\footnote{\label{foot:itas}\url{http://www.oxfordjournals.org/our_journals/mnras/for_authors/}}.
Only technical issues with the \LaTeX\ class are considered here.


\section{Obtaining and installing the MNRAS package}
Some \LaTeX\ distributions come with the MNRAS package by default.
If yours does not, you can either install it using your distribution's package manager, or download it from the Comprehensive \TeX\ Archive Network\footnote{\url{http://www.ctan.org/tex-archive/macros/latex/contrib/mnras}} (CTAN).

The files can either be installed permanently by placing them in the appropriate directory (consult the documentation for your \LaTeX\ distribution), or used temporarily by placing them in the working directory for your paper.

To use the MNRAS package, simply specify \verb'mnras' as the document class at the start of a \verb'.tex' file:

\begin{verbatim}
\documentclass{mnras}
\end{verbatim}
Then compile \LaTeX\ (and if necessary \bibtex) in the usual way.

\section{Preparing and submitting a paper}
We recommend that you start with a copy of the \texttt{mnras\_template.tex} file.
Rename the file, update the information on the title page, and then work on the text of your paper.
Guidelines for content, style etc. are given in the instructions to authors on the journal's website$^{\ref{foot:itas}}$.
Note that this document does not follow all the aspects of MNRAS journal style (e.g. it has a table of contents).

If a paper is accepted, it is professionally typeset and copyedited by the publishers.
It is therefore likely that minor changes to presentation will occur.
For this reason, we ask authors to ignore minor details such as slightly long lines, extra blank spaces, or misplaced figures, because these details will be dealt with during the production process.

Papers must be submitted electronically via the online submission system; paper submissions are not permitted.
For full guidance on how to submit a paper, see the instructions to authors.

\section{Class options}
\label{sec:options}
There are several options which can be added to the document class line like this:

\begin{verbatim}
\documentclass[option1,option2]{mnras}
\end{verbatim}
The available options are:
\begin{itemize}
\item \verb'letters' -- used for papers in the journal's Letters section.
\item \verb'onecolumn' -- single column, instead of the default two columns. This should be used {\it only} if necessary for the display of numerous very long equations.
\item \verb'doublespacing' -- text has double line spacing. Please don't submit papers in this format.
\item \verb'referee' -- \textit{(deprecated)} single column, double spaced, larger text, bigger margins. Please don't submit papers in this format.
\item \verb'galley' -- \textit{(deprecated)} no running headers, no attempt to align the bottom of columns.
\item \verb'landscape' -- \textit{(deprecated)} sets the whole document on landscape paper.
\item \verb"usenatbib" -- \textit{(all papers should use this)} this uses Patrick Daly's \verb"natbib.sty" package for citations.
\item \verb"usegraphicx" -- \textit{(most papers will need this)} includes the \verb'graphicx' package, for inclusion of figures and images.
\item \verb'useAMS' -- adds support for upright Greek characters \verb'\upi', \verb'\umu' and \verb'\upartial' ($\upi$, $\umu$ and $\upartial$). Only these three are included, if you require other symbols you will need to include the \verb'amsmath' or \verb'amsymb' packages (see section~\ref{sec:packages}).
\item \verb"usedcolumn" -- includes the package \verb"dcolumn", which includes two new types of column alignment for use in tables.
\end{itemize}

Some of these options are deprecated and retained for backwards compatibility only.
Others are used in almost all papers, but again are retained as options to ensure that papers written decades ago will continue to compile without problems.
If you want to include any other packages, see section~\ref{sec:packages}.

\section{Title page}

If you are using \texttt{mnras\_template.tex} the necessary code for generating the title page, headers and footers is already present.
Simply edit the title, author list, institutions, abstract and keywords as described below.

\subsection{Title}
There are two forms of the title: the full version used on the first page, and a short version which is used in the header of other odd-numbered pages (the `running head').
Enter them with \verb'\title[]{}' like this:
\begin{verbatim}
\title[Running head]{Full title of the paper}
\end{verbatim}
The full title can be multiple lines (use \verb'\\' to start a new line) and may be as long as necessary, although we encourage authors to use concise titles. The running head must be $\le~45$ characters on a single line.

See appendix~\ref{sec:advanced} for more complicated examples.

\subsection{Authors and institutions}

Like the title, there are two forms of author list: the full version which appears on the title page, and a short form which appears in the header of the even-numbered pages. Enter them using the \verb'\author[]{}' command.

If the author list is more than one line long, start a new line using \verb'\newauthor'. Use \verb'\\' to start the institution list. Affiliations for each author should be indicated with a superscript number, and correspond to the list of institutions below the author list.

For example, if I were to write a paper with two coauthors at another institution, one of whom also works at a third location:
\begin{verbatim}
\author[K. T. Smith et al.]{
Keith T. Smith,$^{1}$
A. N. Other,$^{2}$
and Third Author$^{2,3}$
\\
$^{1}$Affiliation 1\\
$^{2}$Affiliation 2\\
$^{3}$Affiliation 3}
\end{verbatim}
Affiliations should be in the format `Department, Institution, Street Address, City and Postal Code, Country'.

Email addresses can be inserted with the \verb'\thanks{}' command which adds a title page footnote.
If you want to list more than one email, put them all in the same \verb'\thanks' and use \verb'\footnotemark[]' to refer to the same footnote multiple times.
Present addresses (if different to those where the work was performed) can also be added with a \verb'\thanks' command.

\subsection{Abstract and keywords}

The abstract is entered in an \verb'abstract' environment:
\begin{verbatim}
\begin{abstract}
The abstract of the paper.
\end{abstract}
\end{verbatim}
\noindent Note that there is a word limit on the length of abstracts.
For the current word limit, see the journal instructions to authors$^{\ref{foot:itas}}$.

Immediately following the abstract, a set of keywords is entered in a \verb'keywords' environment:
\begin{verbatim}
\begin{keywords}
keyword 1 -- keyword 2 -- keyword 3
\end{keywords}
\end{verbatim}
\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.
Do \emph{not} make up new keywords!
For the current list of allowed keywords, see the journal's instructions to authors$^{\ref{foot:itas}}$.

\section{Sections and lists}

Sections and lists are generally the same as in the standard \LaTeX\ classes.

\subsection{Sections}
\label{sec:sections}
Sections are entered in the usual way, using \verb'\section{}' and its variants. It is possible to nest up to four section levels:
\begin{verbatim}
\section{Main section}
 \subsection{Subsection}
  \subsubsection{Subsubsection}
   \paragraph{Lowest level section}
\end{verbatim}
\noindent The other \LaTeX\ sectioning commands \verb'\part', \verb'\chapter' and \verb'\subparagraph{}' are deprecated and should not be used.

Some sections are not numbered as part of journal style (e.g. the Acknowledgements).
To insert an unnumbered section use the `starred' version of the command: \verb'\section*{}'.

See appendix~\ref{sec:advanced} for more complicated examples.

\subsection{Lists}

Two forms of lists can be used in MNRAS -- numbered and unnumbered.

For a numbered list, use the \verb'enumerate' environment:
\begin{verbatim}
\begin{enumerate}
 \item First item
 \item Second item
 \item etc.
\end{enumerate}
\end{verbatim}
\noindent which produces
\begin{enumerate}
 \item First item
 \item Second item
 \item etc.
\end{enumerate}
Note that the list uses lowercase Roman numerals, rather than the \LaTeX\ default Arabic numerals.

For an unnumbered list, use the \verb'description' environment without the optional argument:
\begin{verbatim}
\begin{description}
 \item First item
 \item Second item
 \item etc.
\end{description}
\end{verbatim}
\noindent which produces
\begin{description}
 \item First item
 \item Second item
 \item etc.
\end{description}

Bulleted lists using the \verb'itemize' environment should not be used in MNRAS; it is retained for backwards compatibility only.

\section{Mathematics and symbols}

The MNRAS class mostly adopts standard \LaTeX\ handling of mathematics, which is briefly summarised here.
See also section~\ref{sec:packages} for packages that support more advanced mathematics.

Mathematics can be inserted into the running text using the syntax \verb'$1+1=2$', which produces $1+1=2$.
Use this only for short expressions or when referring to mathematical quantities; equations should be entered as described below.

\subsection{Equations}
Equations should be entered using the \verb'equation' environment, which automatically numbers them:

\begin{verbatim}
\begin{equation}
 a^2=b^2+c^2
\end{equation}
\end{verbatim}
\noindent which produces
\begin{equation}
 a^2=b^2+c^2
\end{equation}

By default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \verb'\numberwithin{equation}{section}' to the preamble.

It is also possible to produce un-numbered equations by using the \LaTeX\ built-in \verb'\['\textellipsis\verb'\]' and \verb'$$'\textellipsis\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.

\subsection{Special symbols}


\begin{table}
 \caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}
 \label{tab:anysymbols}
 \begin{tabular*}{\columnwidth}{@{}l@{\hspace*{50pt}}l@{\hspace*{50pt}}l@{}}
  \hline
  Command & Output & Meaning\\
  \hline
  \verb'\sun' & \sun & Sun, solar\\[2pt] % additional height spacing for enhanced symbol legibility
  \verb'\earth' & \earth & Earth, terrestrial\\[2pt]
  \verb'\micron' & \micron & microns\\[2pt]
  \verb'\degr' & \degr & degrees\\[2pt]
  \verb'\arcmin' & \arcmin & arcminutes\\[2pt]
  \verb'\arcsec' & \arcsec & arcseconds\\[2pt]
  \verb'\fdg' & \fdg & fraction of a degree\\[2pt]
  \verb'\farcm' & \farcm & fraction of an arcminute\\[2pt]
  \verb'\farcs' & \farcs & fraction of an arcsecond\\[2pt]
  \verb'\fd' & \fd & fraction of a day\\[2pt]
  \verb'\fh' & \fh & fraction of an hour\\[2pt]
  \verb'\fm' & \fm & fraction of a minute\\[2pt]
  \verb'\fs' & \fs & fraction of a second\\[2pt]
  \verb'\fp' & \fp & fraction of a period\\[2pt]
  \verb'\diameter' & \diameter & diameter\\[2pt]
  \verb'\sq' & \sq & square, Q.E.D.\\[2pt]
  \hline
 \end{tabular*}
\end{table}

\begin{table}
 \caption{Additional commands for mathematical symbols. These can only be used in maths mode.}
 \label{tab:mathssymbols}
 \begin{tabular*}{\columnwidth}{l@{\hspace*{40pt}}l@{\hspace*{40pt}}l}
  \hline
  Command & Output & Meaning\\
  \hline
  \verb'\upi' & $\upi$ & upright pi\\[2pt] % additional height spacing for enhanced symbol legibility
  \verb'\umu' & $\umu$ & upright mu\\[2pt]
  \verb'\upartial' & $\upartial$ & upright partial derivative\\[2pt]
  \verb'\lid' & $\lid$ & less than or equal to\\[2pt]
  \verb'\gid' & $\gid$ & greater than or equal to\\[2pt]
  \verb'\la' & $\la$ & less than of order\\[2pt]
  \verb'\ga' & $\ga$ & greater than of order\\[2pt]
  \verb'\loa' & $\loa$ & less than approximately\\[2pt]
  \verb'\goa' & $\goa$ & greater than approximately\\[2pt]
  \verb'\cor' & $\cor$ & corresponds to\\[2pt]
  \verb'\sol' & $\sol$ & similar to or less than\\[2pt]
  \verb'\sog' & $\sog$ & similar to or greater than\\[2pt]
  \verb'\lse' & $\lse$ & less than or homotopic to \\[2pt]
  \verb'\gse' & $\gse$ & greater than or homotopic to\\[2pt]
  \verb'\getsto' & $\getsto$ & from over to\\[2pt]
  \verb'\grole' & $\grole$ & greater over less\\[2pt]
  \verb'\leogr' & $\leogr$ & less over greater\\
  \hline
 \end{tabular*}
\end{table}

Some additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\ref{tab:anysymbols}--\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.

Many other mathematical symbols are also available, either built into \LaTeX\ or via additional packages. If you want to insert a specific symbol but don't know the \LaTeX\ command, we recommend using the Detexify website\footnote{\url{http://detexify.kirelabs.org}}.

Sometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.

To produce bold symbols in mathematics, use \verb'\bmath' for simple variables, and the \verb'bm' package for more complex symbols (see section~\ref{sec:packages}). Vectors are set in bold italic, using \verb'\mathbfit{}'.

For matrices, use \verb'\mathbfss{}' to produce a bold sans-serif font e.g. \mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\nabla$ (del, used in gradients, divergence etc.) use \verb'$\nabla$'.

\subsection{Ions}

A new \verb'\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.
For example, to typeset singly ionised calcium use \verb'\ion{Ca}{ii}', which produces \ion{Ca}{ii}.

\section{Figures and tables}
\label{sec:fig_table}
Figures and tables (collectively called `floats') are mostly the same as built into \LaTeX.

\subsection{Basic examples}
\begin{figure}
 \includegraphics[width=\columnwidth]{example}
 \caption{An example figure.}
 \label{fig:example}
\end{figure}
Figures are inserted in the usual way using a \verb'figure' environment and \verb'\includegraphics'. The example Figure~\ref{fig:example} was generated using the code:
\begin{verbatim}
\begin{figure}
 \includegraphics[width=\columnwidth]{example}
 \caption{An example figure.}
 \label{fig:example}
\end{figure}
\end{verbatim}

\begin{table}
 \caption{An example table.}
 \label{tab:example}
 \begin{tabular}{lcc}
  \hline
  Star & Mass & Luminosity\\
   & $M_{\sun}$ & $L_{\sun}$\\
  \hline
  Sun & 1.00 & 1.00\\
  $\alpha$~Cen~A & 1.10 & 1.52\\
  $\epsilon$~Eri & 0.82 & 0.34\\
  \hline
 \end{tabular}
\end{table}
The example Table~\ref{tab:example} was generated using the code:
\begin{verbatim}
\begin{table}
 \caption{An example table.}
 \label{tab:example}
 \begin{tabular}{lcc}
  \hline
  Star & Mass & Luminosity\\
   & $M_{\sun}$ & $L_{\sun}$\\
  \hline
  Sun & 1.00 & 1.00\\
  $\alpha$~Cen~A & 1.10 & 1.52\\
  $\epsilon$~Eri & 0.82 & 0.34\\
  \hline
 \end{tabular}
\end{table}
\end{verbatim}

\subsection{Captions and placement}
Captions go \emph{above} tables but \emph{below} figures, as in the examples above.

The \LaTeX\ float placement commands \verb'[htbp]' are intentionally disabled.
Layout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.
Simply place the \LaTeX\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.

By default a figure or table will occupy one column of the page.
To produce a wider version which covers both columns, use the \verb'figure*' or \verb'table*'  environment.

If a figure or table is too long to fit on a single page it can be split it into several parts.
Create an additional figure or table which uses \verb'\contcaption{}' instead of \verb'\caption{}'.
This will automatically correct the numbering and add `\emph{continued}' at the start of the caption.
\begin{table}
 \contcaption{A table continued from the previous one.}
 \label{tab:continued}
 \begin{tabular}{lcc}
  \hline
  Star & Mass & Luminosity\\
   & $M_{\sun}$ & $L_{\sun}$\\
  \hline
  $\tau$~Cet & 0.78 & 0.52\\
  $\delta$~Pav & 0.99 & 1.22\\
  $\sigma$~Dra & 0.87 & 0.43\\
  \hline
 \end{tabular}
\end{table}
Table~\ref{tab:continued} was generated using the code:

\begin{verbatim}
\begin{table}
 \contcaption{A table continued from the previous one.}
 \label{tab:continued}
 \begin{tabular}{lcc}
  \hline
  Star & Mass & Luminosity\\
   & $M_{\sun}$ & $L_{\sun}$\\
  \hline
  $\tau$~Cet & 0.78 & 0.52\\
  $\delta$~Pav & 0.99 & 1.22\\
  $\sigma$~Dra & 0.87 & 0.43\\
  \hline
 \end{tabular}
\end{table}
\end{verbatim}

To produce a landscape figure or table, use the \verb'pdflscape' package and the \verb'landscape' environment.
The landscape Table~\ref{tab:landscape} was produced using the code:
\begin{verbatim}
\begin{landscape}
 \begin{table}
  \caption{An example landscape table.}
  \label{tab:landscape}
  \begin{tabular}{cccccccccc}
    \hline
    Header & Header & ...\\
    Unit & Unit & ...\\
    \hline
    Data & Data & ...\\
    Data & Data & ...\\
    ...\\
    \hline
  \end{tabular}
 \end{table}
\end{landscape}
\end{verbatim}
Unfortunately this method will force a page break before the table appears.
More complicated solutions are possible, but authors shouldn't worry about this.

\begin{landscape}
 \begin{table}
  \caption{An example landscape table.}
  \label{tab:landscape}
  \begin{tabular}{cccccccccc}
    \hline
    Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\
    Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\
    \hline
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
    \hline
  \end{tabular}
 \end{table}
\end{landscape}

\section{References and citations}

\subsection{Cross-referencing}

The usual \LaTeX\ commands \verb'\label{}' and \verb'\ref{}' can be used for cross-referencing within the same paper.
We recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.
This ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).

It is best to give each section, figure and table a logical label.
For example, Table~\ref{tab:mathssymbols} has the label \verb'tab:mathssymbols', whilst section~\ref{sec:packages} has the label \verb'sec:packages'.
Add the label \emph{after} the section or caption command, as in the examples in sections~\ref{sec:sections} and \ref{sec:fig_table}.
Enter the cross-reference with a non-breaking space between the type of object and the number, like this: \verb'see Figure~\ref{fig:example}'.

The \verb'\autoref{}' command can be used to automatically fill out the type of object, saving on typing.
It also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.
For example, \verb'\autoref{tab:journal_abbr}' produces \autoref{tab:journal_abbr}.

\subsection{Citations}
\label{sec:cite}

MNRAS uses the Harvard -- author (year) -- citation style, e.g. \citet{author2013}.
This is implemented in \LaTeX\ via the \verb'natbib' package, which in turn is included via the \verb'usenatbib' package option (see section~\ref{sec:options}), which should be used in all papers.

Each entry in the reference list has a `key' (see section~\ref{sec:ref_list}) which is used to generate citations.
There are two basic \verb'natbib' commands:
\begin{description}
 \item \verb'\citet{key}' produces an in-text citation: \citet{author2013}
 \item \verb'\citep{key}' produces a bracketed (parenthetical) citation: \citep{author2013}
\end{description}
Citations will include clickable links to the relevant entry in the reference list, if supported by your \LaTeX\ compiler.

\defcitealias{smith2014}{Paper~I}
\begin{table*}
 \caption{Common citation commands, provided by the \texttt{natbib} package.}
 \label{tab:natbib}
 \begin{tabular}{lll}
  \hline
  Command & Ouput & Note\\
  \hline
  \verb'\citet{key}' & \citet{smith2014} & \\
  \verb'\citep{key}' & \citep{smith2014} & \\
  \verb'\citep{key,key2}' & \citep{smith2014,jones2015} & Multiple papers\\
  \verb'\citet[table 4]{key}' & \citet[table 4]{smith2014} & \\
  \verb'\citep[see][figure 7]{key}' & \citep[see][figure 7]{smith2014} & \\
  \verb'\citealt{key}' & \citealt{smith2014} & For use with manual brackets\\
  \verb'\citeauthor{key}' & \citeauthor{smith2014} & If already cited in close proximity\\
  \verb'\defcitealias{key}{Paper~I}' &  & Define an alias (doesn't work in floats)\\
  \verb'\citetalias{key}' & \citetalias{smith2014} & \\
  \verb'\citepalias{key}' & \citepalias{smith2014} & \\
  \hline
 \end{tabular}
\end{table*}

There are a number of other \verb'natbib' commands which can be used for more complicated citations.
The most commonly used ones are listed in Table~\ref{tab:natbib}.
For full guidance on their use, consult the \verb'natbib' documentation\footnote{\url{http://www.ctan.org/pkg/natbib}}.

If a reference has several authors, \verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \bibtex\ (see section~\ref{sec:ref_list}) then this is handled automatically. If not, the \verb'\citet*{}' and \verb'\citep*{}' commands can be used at the first citation to include all of the authors.

\subsection{The list of references}
\label{sec:ref_list}

It is possible to enter references manually using the usual \LaTeX\ commands, but we strongly encourage authors to use \bibtex\ instead.
\bibtex\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.
An MNRAS \bibtex\ style file, \verb'mnras.bst', is distributed as part of this package.
The rest of this section will assume you are using \bibtex.

References are entered into a separate \verb'.bib' file in standard \bibtex\ formatting.
This can be done manually, or there are several software packages which make editing the \verb'.bib' file much easier.
We particularly recommend \textsc{JabRef}\footnote{\url{http://jabref.sourceforge.net/}}, which works on all major operating systems.
\bibtex\ entries can be obtained from the NASA Astrophysics Data System\footnote{\label{foot:ads}\url{http://adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.
Simply copy this into your \verb'.bib' file or into the `BibTeX source' tab in \textsc{JabRef}.

Each entry in the \verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.
Simply cite it in the usual way, as described in section~\ref{sec:cite}, using the specified key.
Compile the paper as usual, but add an extra step to run the \texttt{bibtex} command.
Consult the documentation for your compiler or latex distribution.

Correct formatting of the reference list will be handled by \bibtex\ in almost all cases, provided that the correct information was entered into the \verb'.bib' file.
Note that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.
If in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\ref{foot:itas}}$ for the current guidelines on how to format the list of references.

\section{Appendices and online material}

To start an appendix, simply place the \verb'\appendix' command before the next \verb'\section{}'.
This will automatically adjust the section headings, figures, tables, and equations to reflect the fact that they are part of an appendix.
It is only necessary to enter the \verb'\appendix' command once -- everything after that command is in an appendix.
Remember that appendices should be placed \textit{after} the list of references.

Unlike other astronomy class files, there are no special commands for online material.
If your paper has any online material, it should be placed in a separate file.
See our instructions to authors$^{\ref{foot:itas}}$ for guidance.

\section{Packages and custom commands}
\label{sec:packages}
\subsection{Additional packages}

Sometimes authors need to include additional \LaTeX\ packages, which provide extra features.
For example, the \verb'bm' package provides extra bold maths symbols, whilst the \verb'pdflscape' package adds support for landscape pages.
Packages can be included by adding the \verb'\usepackage{}' command to the preamble of the document (not the main body).

Please \emph{only include packages which are actually used in the paper}, and include a comment to explain what each one does.
This will assist the typesetters.
If you are using \texttt{mnras\_template.tex}, it includes a specific section for this purpose, near the start of the file with the header 'authors - place your own packages here'.

For example, to include \verb'pdflscape', use:
\begin{verbatim}
\usepackage{pdflscape}	% Landscape pages
\end{verbatim}
Consult the documentation for that package for instructions on how to use the additional features.


\subsection{Custom commands}

Authors should avoid duplicating or redefining commands which are already available in \LaTeX\ or \verb'mnras.cls'.
However it may sometimes be necessary to introduce a custom command e.g. as a shortcut while writing the paper.

Please \emph{only include commands which are actually used in the paper}, and include a comment to explain what each one does.
This will assist the typesetters.
Use \verb'\newcommand', \emph{not} \verb'\def', as this will avoid accidentally overwriting existing commands.
Place custom commands in the preamble of the document (not the main body).
If you are using \texttt{mnras\_template.tex}, it includes a specific section for this purpose, near the start of the file with the header 'authors - place your own commands here'.

As an example, a shortcut for the unit \kms can be defined like this:
\begin{verbatim}
\newcommand{\kms}{\,km\,s$^{-1}$}	% kilometres per second
\end{verbatim}
Velocities can then be written as e.g. \verb'2.3\kms' which produces 2.3\kms.
Similar shortcuts can be used for frequently quoted object designations.


\section*{Acknowledgements}
% Entry for the table of contents, for this guide only
\addcontentsline{toc}{section}{Acknowledgements}

This guide replaces an earlier one originally prepared by Cambridge University Press (CUP) in 1994, and last updated in 2002 by Blackwell Publishing.
Some code segments are reproduced from, and some examples are based upon, that guide.
The authors were: A.~Woollatt, M.~Reed, R.~Mulvey, K.~Matthews, D.~Starling, Y.~Yu, A.~Richardson (all CUP), and Penny~Smith, N.~Thompson and Gregor~Hutton (all Blackwell), whose work is gratefully acknowledged.

The accompanying \bibtex\ style file was written by John Sleath, Tim Jenness and Norman Gray, without whom \bibtex\ support would not have been possible.

Some special symbols in tables~\ref{tab:anysymbols}--\ref{tab:mathssymbols} were taken from the Springer Verlag \textit{Astronomy \& Astrophysics} \LaTeX\ class, with their permission.

KTS thanks Nelson Beebe (University of Utah) for helpful advice regarding CTAN.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Data Availability}

 
The inclusion of a Data Availability Statement is a requirement for articles published in MNRAS. Data Availability Statements provide a standardised format for readers to understand the availability of data underlying the research results described in the article. The statement may refer to original data generated in the course of the study or to third-party data analysed in the article. The statement should describe and provide means of access, where possible, by linking to the data or providing the required accession numbers for the relevant databases or DOIs.


%%%%%%%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%%

% The best way to enter references is to use BibTeX:

%\bibliographystyle{mnras}
%\bibliography{example} % if your bibtex file is called example.bib


% Alternatively you could enter them by hand, like this:
\begin{thebibliography}{99}
\bibitem[\protect\citeauthoryear{Author}{2013}]{author2013}
Author A.~N., 2013, Journal of Improbable Astronomy, 1, 1
\bibitem[\protect\citeauthoryear{Jones}{2015}]{jones2015}
Jones C.~D., 2015, Journal of Interesting Stuff, 17, 198
\bibitem[\protect\citeauthoryear{Smith}{2014}]{smith2014}
Smith A.~B., 2014, The Example Journal, 12, 345 (Paper I)
\end{thebibliography}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%% APPENDICES %%%%%%%%%%%%%%%%%%%%%

\appendix
\section{Journal abbreviations}
\label{sec:abbreviations}
Abbreviations for cited journals can be accessed using the commands listed in table~\ref{tab:journal_abbr}.
Although some of these may appear to be outdated or rarely cited, they have been selected to be compatible with the \bibtex\ output by the NASA Astrophysics Data System$^{\ref{foot:ads}}$, commands used by other astronomy journals, and with additional entries for journals with non-standard abbreviations in MNRAS.
For journals which are not on this list, see our instructions to authors$^{\ref{foot:itas}}$ for guidance on how to abbreviate titles.

\begin{table*}
\caption{Commands for abbreviated journal names, see appendix~\ref{sec:abbreviations}.}
\label{tab:journal_abbr}
\begin{tabular}{@{}l@{\:}l@{\:}l@{}} % manual @ spacing to prevent this being too wide for a page
\hline
Command & Output & Journal name\\
\hline
\verb'\aap' or \verb'\astap' & \aap & Astronomy and Astrophysics$^a$\\
\verb'\aapr' & \aapr & The Astronomy and Astrophysics Review\\
\verb'\aaps' & \aaps  & Astronomy and Astrophysics Supplement Series\\
\verb'\actaa' & \actaa & Acta Astronomica\\
\verb'\afz' & \afz & Astrofizika\\
\verb'\aj' & \aj & The Astronomical Journal\\
\verb'\ao' or \verb'\applopt' & \ao & Applied Optics\\
\verb'\aplett' & \aplett & Astrophysics Letters\\
\verb'\apj' & \apj & The Astrophysical Journal\\
\verb'\apjl' or \verb'\apjlett' & \apjl & The Astrophysical Journal Letters$^a$\\
\verb'\apjs' or \verb'\apjsupp' & \apjs & The Astrophysical Journal Supplement Series\\
\verb'\apss' & \apss & Astrophysics and Space Science\\
\verb'\araa' & \araa & Annual Review of Astronomy and Astrophysics\\
\verb'\arep' & \arep & Astronomy Reports$^b$\\
\verb'\aspc' & \aspc & Astronomical Society of the Pacific Conference Series\\
\verb'\azh' & \azh & Astronomicheskii Zhurnal$^c$\\
\verb'\baas' & \baas & Bulletin of the American Astronomical Society\\
\verb'\bac' & \bac & Bulletin of the Astronomical Institutes of Czechoslovakia\\
\verb'\bain' & \bain & Bull. Astron. Inst. Netherlands\\
\verb'\caa' & \caa & Chinese Astronomy and Astrophysics\\
\verb'\cjaa' & \cjaa & Chinese Journal of Astronomy and Astrophysics\\
\verb'\fcp' & \fcp & Fundamentals of Cosmic Physics\\
\verb'\gca' & \gca & Geochimica Cosmochimica Acta\\
\verb'\grl' & \grl & Geophysics Research Letters\\
\verb'\iaucirc' & \iaucirc & International Astronomical Union Circulars\\
\verb'\icarus' & \icarus & Icarus\\
\verb'\japa' & \japa & Journal of Astrophysics and Astronomy\\
\verb'\jcap' & \jcap & Journal of Cosmology and Astroparticle Physics\\
\verb'\jcp' & \jcp & Journal of Chemical Physics\\
\verb'\jgr' & \jgr & Journal of Geophysics Research\\
\verb'\jqsrt' & \jqsrt & Journal of Quantitiative Spectroscopy and Radiative Transfer\\
\verb'\jrasc' & \jrasc & Journal of the Royal Astronomical Society of Canada\\
\verb'\memras' & \memras & Memoirs of the Royal Astronomical Society\\
\verb'\memsai' & \memsai & Memoire della Societa Astronomica Italiana\\
\verb'\mnassa' & \mnassa & Monthly Notes of the Astronomical Society of Southern Africa\\
\verb'\mnras' & \mnras & Monthly Notices of the Royal Astronomical Society$^a$\\
\verb'\na' & \na & New Astronomy\\
\verb'\nar' & \nar & New Astronomy Review\\
\verb'\nat' & \nat & Nature\\
\verb'\nphysa' & \nphysa & Nuclear Physics A\\
\verb'\pra' & \pra & Physical Review A: Atomic, molecular, and optical physics\\
\verb'\prb' & \prb & Physical Review B: Condensed matter and materials physics\\
\verb'\prc' & \prc & Physical Review C: Nuclear physics\\
\verb'\prd' & \prd & Physical Review D: Particles, fields, gravitation, and cosmology\\
\verb'\pre' & \pre & Physical Review E: Statistical, nonlinear, and soft matter physics\\
\verb'\prl' & \prl & Physical Review Letters\\
\verb'\pasa' & \pasa & Publications of the Astronomical Society of Australia\\
\verb'\pasp' & \pasp & Publications of the Astronomical Society of the Pacific\\
\verb'\pasj' & \pasj & Publications of the Astronomical Society of Japan\\
\verb'\physrep' & \physrep & Physics Reports\\
\verb'\physscr' & \physscr & Physica Scripta\\
\verb'\planss' & \planss & Planetary and Space Science\\
\verb'\procspie' & \procspie & Proceedings of the Society of Photo-Optical Instrumentation Engineers\\
\verb'\rmxaa' & \rmxaa & Revista Mexicana de Astronomia y Astrofisica\\
\verb'\qjras' & \qjras & Quarterly Journal of the Royal Astronomical Society\\
\verb'\sci' & \sci & Science\\
\verb'\skytel' & \skytel & Sky and Telescope\\
\verb'\solphys' & \solphys & Solar Physics\\
\verb'\sovast' & \sovast & Soviet Astronomy$^b$\\
\verb'\ssr' & \ssr & Space Science Reviews\\
\verb'\zap' & \zap & Zeitschrift fuer Astrophysik\\
\hline
\multicolumn{3}{l}{$^a$ Letters are designated by an L at the start of the page number, not in the journal name}\\
\multicolumn{3}{l}{\footnotesize$^b$ In 1992 the English translation of this journal changed its name from Soviet Astronomy to Astronomy Reports}\\
\multicolumn{3}{l}{\footnotesize$^c$ Including the English translation Astronomy Letters}\\
\end{tabular}
\end{table*}

\clearpage % to avoid the long table breaking up the formatting examples
\section{Advanced formatting examples}
\label{sec:advanced}

Sometimes formatting doesn't behave exactly as expected when used in titles or section headings, and must be modified to obtain the correct appearance.
Generally the publishers can fix these problems during the typesetting process after a paper is accepted, but authors may wish to adjust these themselves to minimise the possibility of errors and/or for the benefit of the refereeing process.
Below are some examples of output, followed by the \LaTeX\ code which produces them.

Most mathematics and text formatting works as expected, but some commands might not be the correct size, bold or italic.
If so they can be finessed by hand, as in the bold mathematics here:
\boxit{\huge\bf \textit{Herschel} observations of galaxies at $\bm{\delta > 60\degr}$}
\begin{verbatim}
\title{\textit{Herschel} observations of galaxies at
 $\bm{\delta > 60\degr}$}
\end{verbatim}

Most fonts do not provide bold and italic versions of small capitals, so the \verb'\ion{}{}' command doesn't produce the expected output in headings.
The effect has to be `faked' using font size commands, remembering that the running head is a different style:
\boxit{\huge\bf Abundances in H\,{\Large \textbf{II}} regions}
\begin{verbatim}
\title
[Abundances in H\,{\normalsize \textit{II}} regions]
{Abundances in H\,{\Large \textbf{II}} regions}
\end{verbatim}

Complex mathematics can cause problems with links, so might require adding a less formatted short version of the heading:
\boxit{\bf 2\quad FINDING Mg\,{\sevensize II} ABSORBERS AT $\bm{z > 2}$}
\begin{verbatim}
\section
[Finding Mg II absorbers at z > 2]
{Finding M\lowercase{g}\,{\sevensize II} absorbers
 at $\lowercase{\bm{z > 2}}$}
\end{verbatim}


Using square brackets in headings can cause additional linking problems, which are solved by wrapping them in \{\textellipsis\}:
\boxit{\bf 2.1\quad [C\,{\sevensize II}] 158$\bmath{\umu}$m emission}
\begin{verbatim}
\subsection
[{[C II] 158$\umu$m emission}]
{[C\,{\sevensize II}] 158$\bmath{\umu}$m
 emission}
\end{verbatim}

Use \verb'\text{}' (not \verb'\rm') for non-variables in mathematics, which preserves the formatting of the surrounding text.
For the same reasons, use \verb'\textit{}' for italics (not \verb'\it').
\boxit{\bf 3.1\quad Measuring $\bm{T}_\text{eff}$ from \textit{Gaia} photometry}
\begin{verbatim}
\subsection{Measuring $\bm{T}_\text{eff}$ from
 \textit{Gaia} photometry}
\end{verbatim}

\section{Additional commands for editors only}
The following commands are available for the use of editors and production staff only.
They should not be used (or modified in the template) by authors.

\begin{description}
 \item \verb'\maketitle' inserts the title, authors and institution list in the correct formatting.
 \item \verb'\nokeywords' tidies up the spacing if there are no keywords, but authors should always enter at least one.
 \item \verb'\volume{}' sets the volume number (default is 000)
 \item \verb'\pagerange{}' sets the page range. The standard template generates this automatically, starting from 1.
 \item \verb'\bsp' adds the `This paper has been typeset\textellipsis' comment at the end of the paper.
The command name refers to Blackwell Science Publishing, who were the publishers at the time when MNRAS began accepting \LaTeX\ submissions in 1993.
 \item \verb'\mniiiauth{}' used by the \bibtex\ style to handle MNRAS style for citing papers with three authors. It should not be used manually.
 \item \verb'\eprint{}' used by the \bibtex\ style for citing arXiv eprints.
 \item \verb'\doi{}' used by the \bibtex\ style for citing Digital Object Identifiers.
\end{description}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% Don't change these lines
\bsp	% typesetting comment
\label{lastpage}
\end{document}

% End of mnras_guide.tex

main.bbl
\begin{thebibliography}{}
\makeatletter
\relax
\def\mn@urlcharsother{\let\do\@makeother \do\$\do\&\do\#\do\^\do\_\do\%\do\~}
\def\mn@doi{\begingroup\mn@urlcharsother \@ifnextchar [ {\mn@doi@}
  {\mn@doi@[]}}
\def\mn@doi@[#1]#2{\def\@tempa{#1}\ifx\@tempa\@empty \href
  {http://dx.doi.org/#2} {doi:#2}\else \href {http://dx.doi.org/#2} {#1}\fi
  \endgroup}
\def\mn@eprint#1#2{\mn@eprint@#1:#2::\@nil}
\def\mn@eprint@arXiv#1{\href {http://arxiv.org/abs/#1} {{\tt arXiv:#1}}}
\def\mn@eprint@dblp#1{\href {http://dblp.uni-trier.de/rec/bibtex/#1.xml}
  {dblp:#1}}
\def\mn@eprint@#1:#2:#3:#4\@nil{\def\@tempa {#1}\def\@tempb {#2}\def\@tempc
  {#3}\ifx \@tempc \@empty \let \@tempc \@tempb \let \@tempb \@tempa \fi \ifx
  \@tempb \@empty \def\@tempb {arXiv}\fi \@ifundefined
  {mn@eprint@\@tempb}{\@tempb:\@tempc}{\expandafter \expandafter \csname
  mn@eprint@\@tempb\endcsname \expandafter{\@tempc}}}

\bibitem[\protect\citeauthoryear{Acquaviva, Bartolo, Matarrese  \&
  Riotto}{Acquaviva et~al.}{2003}]{ACQUAVIVA2003119}
Acquaviva V.,  Bartolo N.,  Matarrese S.,   Riotto A.,  2003, \mn@doi [Nuclear
  Physics B] {https://doi.org/10.1016/S0550-3213(03)00550-9}, 667, 119

\bibitem[\protect\citeauthoryear{Allen, Grinstein  \& Wise}{Allen
  et~al.}{1987}]{ALLEN198766}
Allen T.,  Grinstein B.,   Wise M.~B.,  1987, \mn@doi [Physics Letters B]
  {https://doi.org/10.1016/0370-2693(87)90343-1}, 197, 66

\bibitem[\protect\citeauthoryear{{Bennett} et~al.}{{Bennett}
  et~al.}{2003a}]{2003ApJS..148....1B}
{Bennett} C.~L.,  et~al., 2003a, \mn@doi [\apjs] {10.1086/377253}, \href
  {https://ui.adsabs.harvard.edu/abs/2003ApJS..148....1B} {148, 1}

\bibitem[\protect\citeauthoryear{{Bennett} et~al.}{{Bennett}
  et~al.}{2003b}]{2003ApJS..148...97B}
{Bennett} C.~L.,  et~al., 2003b, \mn@doi [\apjs] {10.1086/377252}, \href
  {https://ui.adsabs.harvard.edu/abs/2003ApJS..148...97B} {148, 97}

\bibitem[\protect\citeauthoryear{{Bennett} et~al.}{{Bennett}
  et~al.}{2013a}]{2013ApJS..208...20B}
{Bennett} C.~L.,  et~al., 2013a, \mn@doi [\apjs] {10.1088/0067-0049/208/2/20},
  \href {https://ui.adsabs.harvard.edu/abs/2013ApJS..208...20B} {208, 20}

\bibitem[\protect\citeauthoryear{{Bennett} et~al.}{{Bennett}
  et~al.}{2013b}]{WMAP-9yr-maps}
{Bennett} C.~L.,  et~al., 2013b, \mn@doi [\apjs] {10.1088/0067-0049/208/2/20},
  \href {https://ui.adsabs.harvard.edu/abs/2013ApJS..208...20B} {208, 20}

\bibitem[\protect\citeauthoryear{{Bouchet} \& {Gispert}}{{Bouchet} \&
  {Gispert}}{1999}]{1999NewA....4..443B}
{Bouchet} F.~R.,  {Gispert} R.,  1999, \mn@doi [\na]
  {10.1016/S1384-1076(99)00027-5}, \href
  {https://ui.adsabs.harvard.edu/abs/1999NewA....4..443B} {4, 443}

\bibitem[\protect\citeauthoryear{Dodelson}{Dodelson}{1997}]{Dodelson_1997}
Dodelson S.,  1997, \mn@doi [The Astrophysical Journal] {10.1086/304157}, 482,
  577

\bibitem[\protect\citeauthoryear{Eriksen et~al.}{Eriksen
  et~al.}{2004a}]{Eriksen_2004}
Eriksen H.~K.,  et~al., 2004a, \mn@doi [The Astrophysical Journal Supplement
  Series] {10.1086/425219}, 155, 227–241

\bibitem[\protect\citeauthoryear{Eriksen, Banday, Gorski  \& Lilje}{Eriksen
  et~al.}{2004b}]{Eriksen_2004a}
Eriksen H.~K.,  Banday A.~J.,  Gorski K.~M.,   Lilje P.~B.,  2004b, \mn@doi
  [The Astrophysical Journal] {10.1086/422807}, 612, 633

\bibitem[\protect\citeauthoryear{{Eriksen} et~al.}{{Eriksen}
  et~al.}{2006}]{Erikson_2006}
{Eriksen} H.~K.,  et~al., 2006, \mn@doi [\apj] {10.1086/500499}, \href
  {https://ui.adsabs.harvard.edu/abs/2006ApJ...641..665E} {641, 665}

\bibitem[\protect\citeauthoryear{Eriksen, Huey, Saha  et~al.}{Eriksen
  et~al.}{2007}]{Eriksen_2007}
Eriksen H.~K.,  Huey G.,  Saha R.,   et~al., 2007, \mn@doi [The Astrophysical
  Journal] {10.1086/509911}, 656, 641

\bibitem[\protect\citeauthoryear{{Eriksen}, {Dickinson}, {Jewell}, {Banday},
  {G{\'o}rski}  \& {Lawrence}}{{Eriksen} et~al.}{2008a}]{2008ApJ...672L..87E}
{Eriksen} H.~K.,  {Dickinson} C.,  {Jewell} J.~B.,  {Banday} A.~J.,
  {G{\'o}rski} K.~M.,   {Lawrence} C.~R.,  2008a, \mn@doi [\apjl]
  {10.1086/526545}, \href
  {https://ui.adsabs.harvard.edu/abs/2008ApJ...672L..87E} {672, L87}

\bibitem[\protect\citeauthoryear{{Eriksen}, {Jewell}, {Dickinson}, {Banday},
  {G{\'o}rski}  \& {Lawrence}}{{Eriksen} et~al.}{2008b}]{2008ApJ...676...10E}
{Eriksen} H.~K.,  {Jewell} J.~B.,  {Dickinson} C.,  {Banday} A.~J.,
  {G{\'o}rski} K.~M.,   {Lawrence} C.~R.,  2008b, \mn@doi [\apj]
  {10.1086/525277}, \href
  {https://ui.adsabs.harvard.edu/abs/2008ApJ...676...10E} {676, 10}

\bibitem[\protect\citeauthoryear{Falk, Rangarajan  \& Srednicki}{Falk
  et~al.}{1992}]{Falk_1992}
Falk T.,  Rangarajan R.,   Srednicki M.,  1992, \mn@doi [Phys. Rev. D]
  {10.1103/PhysRevD.46.4232}, 46, 4232

\bibitem[\protect\citeauthoryear{{Gangui}, {Lucchin}, {Matarrese}  \&
  {Mollerach}}{{Gangui} et~al.}{1994}]{Gangui_1994}
{Gangui} A.,  {Lucchin} F.,  {Matarrese} S.,   {Mollerach} S.,  1994, \mn@doi
  [\apj] {10.1086/174421}, \href
  {https://ui.adsabs.harvard.edu/abs/1994ApJ...430..447G} {430, 447}

\bibitem[\protect\citeauthoryear{{Geman} \& {Geman}}{{Geman} \&
  {Geman}}{1984}]{4767596}
{Geman} S.,  {Geman} D.,  1984, IEEE Transactions on Pattern Analysis and
  Machine Intelligence, PAMI-6, 721

\bibitem[\protect\citeauthoryear{{Gold} et~al.}{{Gold}
  et~al.}{2011}]{Gold_2011}
{Gold} B.,  et~al., 2011, \mn@doi [\apjs] {10.1088/0067-0049/192/2/15}, \href
  {https://ui.adsabs.harvard.edu/abs/2011ApJS..192...15G} {192, 15}

\bibitem[\protect\citeauthoryear{Goldberg}{Goldberg}{1989}]{Goldberg}
Goldberg D.~E.,  1989, Genetic algorithms in search, optimization, and machine
  learning.
Addison-Wesley Publishing Company, Inc.

\bibitem[\protect\citeauthoryear{{G{\'o}rski}, {Hivon}, {Banday}, {Wandelt},
  {Hansen}, {Reinecke}  \& {Bartelmann}}{{G{\'o}rski}
  et~al.}{2005}]{Gorski2005}
{G{\'o}rski} K.~M.,  {Hivon} E.,  {Banday} A.~J.,  {Wandelt} B.~D.,  {Hansen}
  F.~K.,  {Reinecke} M.,   {Bartelmann} M.,  2005, \mn@doi [\apj]
  {10.1086/427976}, \href
  {https://ui.adsabs.harvard.edu/abs/2005ApJ...622..759G} {622, 759}

\bibitem[\protect\citeauthoryear{{Hinshaw} et~al.}{{Hinshaw}
  et~al.}{2003}]{2003ApJS..148..135H}
{Hinshaw} G.,  et~al., 2003, \mn@doi [\apjs] {10.1086/377225}, \href
  {https://ui.adsabs.harvard.edu/abs/2003ApJS..148..135H} {148, 135}

\bibitem[\protect\citeauthoryear{{Hinshaw} et~al.}{{Hinshaw}
  et~al.}{2007}]{2007ApJS..170..288H}
{Hinshaw} G.,  et~al., 2007, \mn@doi [\apjs] {10.1086/513698}, \href
  {https://ui.adsabs.harvard.edu/abs/2007ApJS..170..288H} {170, 288}

\bibitem[\protect\citeauthoryear{Holland}{Holland}{1962}]{10.1145/321127.321128}
Holland J.~H.,  1962, \mn@doi [J. ACM] {10.1145/321127.321128}, 9, 297–314

\bibitem[\protect\citeauthoryear{Ichiki}{Ichiki}{2014}]{10.1093/ptep/ptu065}
Ichiki K.,  2014, \mn@doi [Progress of Theoretical and Experimental Physics]
  {10.1093/ptep/ptu065}, 2014

\bibitem[\protect\citeauthoryear{{Jarosik}, {Bennett}  et~al.}{{Jarosik}
  et~al.}{2003}]{2003ApJS..145..413J}
{Jarosik} N.,  {Bennett} C.~L.,   et~al., 2003, \mn@doi [\apjs]
  {10.1086/346080}, \href
  {https://ui.adsabs.harvard.edu/abs/2003ApJS..145..413J} {145, 413}

\bibitem[\protect\citeauthoryear{{Jarosik} et~al.}{{Jarosik}
  et~al.}{2007}]{2007ApJS..170..263J}
{Jarosik} N.,  et~al., 2007, \mn@doi [\apjs] {10.1086/513697}, \href
  {https://ui.adsabs.harvard.edu/abs/2007ApJS..170..263J} {170, 263}

\bibitem[\protect\citeauthoryear{Maldacena}{Maldacena}{2003}]{Maldacena_2003}
Maldacena J.,  2003, \mn@doi [Journal of High Energy Physics]
  {10.1088/1126-6708/2003/05/013}, 2003, 013

\bibitem[\protect\citeauthoryear{{Mather} et~al.}{{Mather}
  et~al.}{1994}]{1994ApJ...420..439M}
{Mather} J.~C.,  et~al., 1994, \mn@doi [\apj] {10.1086/173574}, \href
  {https://ui.adsabs.harvard.edu/abs/1994ApJ...420..439M} {420, 439}

\bibitem[\protect\citeauthoryear{{Moore}}{{Moore}}{1920}]{Moore_1920}
{Moore} E.~H.,  1920, Proceedings of the American Mathematical Society

\bibitem[\protect\citeauthoryear{{Penrose}}{{Penrose}}{1956}]{1956PCPS...52...17P}
{Penrose} R.,  1956, \mn@doi [Proceedings of the Cambridge Philosophical
  Society] {10.1017/S0305004100030929}, \href
  {https://ui.adsabs.harvard.edu/abs/1956PCPS...52...17P} {52, 17}

\bibitem[\protect\citeauthoryear{{Planck Collaboration} et~al.}{{Planck
  Collaboration} et~al.}{2020a}]{Planck2018}
{Planck Collaboration} et~al., 2020a, \mn@doi [A\&A]
  {10.1051/0004-6361/201833880}, 641, A1

\bibitem[\protect\citeauthoryear{{Planck Collaboration} et~al.}{{Planck
  Collaboration} et~al.}{2020b}]{Planck2018IV}
{Planck Collaboration} et~al., 2020b, \mn@doi [{A\&A}]
  {10.1051/0004-6361/201833881}, \href
  {https://ui.adsabs.harvard.edu/abs/2020A&A...641A...4P} {641, A4}

\bibitem[\protect\citeauthoryear{{Planck Collaboration} et~al.}{{Planck
  Collaboration} et~al.}{2020c}]{Planck2018VI}
{Planck Collaboration} et~al., 2020c, \mn@doi [A\&A]
  {10.1051/0004-6361/201833910}, 641, A6

\bibitem[\protect\citeauthoryear{Saha}{Saha}{2011}]{Saha_2011}
Saha R.,  2011, \mn@doi [The Astrophysical Journal]
  {10.1088/2041-8205/739/2/l56}, 739, L56

\bibitem[\protect\citeauthoryear{Saha, Jain  \& Souradeep}{Saha
  et~al.}{2006}]{Saha_2006}
Saha R.,  Jain P.,   Souradeep T.,  2006, \mn@doi [The Astrophysical Journal]
  {10.1086/506321}, 645, L89–L92

\bibitem[\protect\citeauthoryear{Saha, Prunet, Jain  \& Souradeep}{Saha
  et~al.}{2008a}]{Saha_2008}
Saha R.,  Prunet S.,  Jain P.,   Souradeep T.,  2008a, \mn@doi [Phys. Rev. D]
  {10.1103/PhysRevD.78.023003}, 78, 023003

\bibitem[\protect\citeauthoryear{{Saha}, {Prunet}, {Jain}  \&
  {Souradeep}}{{Saha} et~al.}{2008b}]{2008PhRvD..78b3003S}
{Saha} R.,  {Prunet} S.,  {Jain} P.,   {Souradeep} T.,  2008b, \mn@doi [\prd]
  {10.1103/PhysRevD.78.023003}, \href
  {https://ui.adsabs.harvard.edu/abs/2008PhRvD..78b3003S} {78, 023003}

\bibitem[\protect\citeauthoryear{Sudevan \& Saha}{Sudevan \&
  Saha}{2018a}]{Sudevan_2018_g}
Sudevan V.,  Saha R.,  2018a, An Application of Global ILC Algorithm over Large
  Angular Scales to Estimate CMB Posterior Using Gibbs Sampling (\mn@eprint
  {arXiv} {1810.08872})

\bibitem[\protect\citeauthoryear{{Sudevan} \& {Saha}}{{Sudevan} \&
  {Saha}}{2018b}]{2018arXiv181008872S}
{Sudevan} V.,  {Saha} R.,  2018b, arXiv e-prints, \href
  {https://ui.adsabs.harvard.edu/abs/2018arXiv181008872S} {p. arXiv:1810.08872}

\bibitem[\protect\citeauthoryear{Sudevan \& Saha}{Sudevan \&
  Saha}{2018c}]{Sudevan_2018}
Sudevan V.,  Saha R.,  2018c, \mn@doi [The Astrophysical Journal]
  {10.3847/1538-4357/aae439}, 867, 74

\bibitem[\protect\citeauthoryear{{Tegmark}}{{Tegmark}}{1998}]{1998ApJ...502....1T}
{Tegmark} M.,  1998, \mn@doi [\apj] {10.1086/305905}, \href
  {https://ui.adsabs.harvard.edu/abs/1998ApJ...502....1T} {502, 1}

\bibitem[\protect\citeauthoryear{Tegmark \& Efstathiou}{Tegmark \&
  Efstathiou}{1996}]{Tegmark_1996}
Tegmark M.,  Efstathiou G.,  1996, \mn@doi [Monthly Notices of the Royal
  Astronomical Society] {10.1093/mnras/281.4.1297}, 281, 1297–1314

\bibitem[\protect\citeauthoryear{Tegmark, de Oliveira-Costa  \&
  Hamilton}{Tegmark et~al.}{2003}]{Tegmark_2003}
Tegmark M.,  de Oliveira-Costa A.,   Hamilton A. J.~S.,  2003, \mn@doi
  [Physical Review D] {10.1103/physrevd.68.123523}, 68

\makeatother
\end{thebibliography}

main.bib
@article{Planck2018,
	author = {{Planck Collaboration} and others},
	title = {Planck 2018 results - I. Overview and the cosmological legacy of Planck},
	DOI= "10.1051/0004-6361/201833880",
	url= "https://doi.org/10.1051/0004-6361/201833880",
	journal = {A\&A},
	year = 2020,
	volume = 641,
	pages = "A1",
}


@ARTICLE{WMAP-9yr-maps,
	author = {{Bennett}, C.~L. and others},
	title = "{Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results}",
	journal = {\apjs},
	keywords = {cosmic background radiation, cosmology: observations, dark matter, early universe, instrumentation: detectors, space vehicles, space vehicles: instruments, telescopes, Astrophysics - Cosmology and Nongalactic Astrophysics},
	year = 2013,
	month = oct,
	volume = {208},
	number = {2},
	eid = {20},
	pages = {20},
	doi = {10.1088/0067-0049/208/2/20},
	archivePrefix = {arXiv},
	eprint = {1212.5225},
	primaryClass = {astro-ph.CO},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2013ApJS..208...20B},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}


@ARTICLE{Planck2018IV,
	author = {{Planck Collaboration} and others},
	title = "{Planck 2018 results. IV. Diffuse component separation}",
	journal = "{A\&A}",
	keywords = {ISM: general, cosmology: observations, cosmic background radiation, diffuse radiation, Galaxy: general, Astrophysics - Cosmology and Nongalactic Astrophysics},
	year = 2020,
	month = sep,
	volume = {641},
	eid = {A4},
	pages = {A4},
	doi = {10.1051/0004-6361/201833881},
	archivePrefix = {arXiv},
	eprint = {1807.06208},
	primaryClass = {astro-ph.CO},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2020A&A...641A...4P},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}


@article{Planck2018VI,
	author = {{Planck Collaboration} and others},
	title = {Planck 2018 results - VI. Cosmological parameters},
	DOI= "10.1051/0004-6361/201833910",
	url= "https://doi.org/10.1051/0004-6361/201833910",
	journal = {A\&A},
	year = 2020,
	volume = 641,
	pages = "A6",
}


@ARTICLE{1965ApJ...142..419P,
	author = {{Penzias}, A.~A. and {Wilson}, R.~W.},
	title = "{A Measurement of Excess Antenna Temperature at 4080 Mc/s.}",
	journal = {\apj},
	keywords = {Microwave Background, Cosmic Background Radiation},
	year = 1965,
	month = jul,
	volume = {142},
	pages = {419-421},
	doi = {10.1086/148307},
	adsurl = {https://ui.adsabs.harvard.edu/abs/1965ApJ...142..419P},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{1965ApJ...142..414D,
	author = {{Dicke}, R.~H. and {Peebles}, P.~J.~E. and {Roll}, P.~G. and
	{Wilkinson}, D.~T.},
	title = "{Cosmic Black-Body Radiation.}",
	journal = {\apj},
	keywords = {Big Bang},
	year = 1965,
	month = jul,
	volume = {142},
	pages = {414-419},
	doi = {10.1086/148306},
	adsurl = {https://ui.adsabs.harvard.edu/abs/1965ApJ...142..414D},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{1990ApJ...354L..37M,
	author = {{Mather}, J.~C. and others},
	title = "{A Preliminary Measurement of the Cosmic Microwave Background Spectrum by the Cosmic Background Explorer (COBE) Satellite}",
	journal = {\apjl},
	keywords = {Astronomical Spectroscopy, Cosmic Background Explorer Satellite, Relic Radiation, Black Body Radiation, Cosmic Dust, Infrared Spectrophotometers, Red Shift, Space Radiation, COSMIC BACKGROUND RADIATION},
	year = 1990,
	month = may,
	volume = {354},
	pages = {L37},
	doi = {10.1086/185717},
	adsurl = {https://ui.adsabs.harvard.edu/abs/1990ApJ...354L..37M},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{1992ApJ...396L...1S,
	author = {{Smoot}, G.~F. and others},
	title = "{Structure in the COBE Differential Microwave Radiometer First-Year Maps}",
	journal = {\apjl},
	keywords = {Cosmic Background Explorer Satellite, Microwave Radiometers, Normal Density Functions, Relic Radiation, Root-Mean-Square Errors, Astronomical Maps, Milky Way Galaxy, Space Radiation, COSMOLOGY: COSMIC MICROWAVE BACKGROUND, COSMOLOGY: OBSERVATIONS},
	year = 1992,
	month = sep,
	volume = {396},
	pages = {L1},
	doi = {10.1086/186504},
	adsurl = {https://ui.adsabs.harvard.edu/abs/1992ApJ...396L...1S},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2003ApJ...583....1B,
	author = {{Bennett}, C.~L. and others},
	title = "{The Microwave Anisotropy Probe Mission}",
	journal = {\apj},
	keywords = {Cosmology: Cosmic Microwave Background, Cosmology: Observations, Cosmology: Dark Matter, Cosmology: Early Universe, Space Vehicles: Instruments, Astrophysics},
	year = 2003,
	month = jan,
	volume = {583},
	number = {1},
	pages = {1-23},
	doi = {10.1086/345346},
	archivePrefix = {arXiv},
	eprint = {astro-ph/0301158},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2003ApJ...583....1B},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2010A&A...520A...1T,
	author = {{Tauber}, J.~A. and others},
	title = "{Planck pre-launch status: The Planck mission}",
	journal = {A\&A},
	keywords = {cosmic microwave background, space vehicles: instruments, instrumentation: detectors, instrumentation: polarimeters, submillimeter: general, radio continuum: general},
	year = 2010,
	month = sep,
	volume = {520},
	eid = {A1},
	pages = {A1},
	doi = {10.1051/0004-6361/200912983},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2010A&A...520A...1T},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@article{10.1093/ptep/ptu065,
	author = {Ichiki, Kiyotomo},
	title = "{CMB foreground: A concise review}",
	journal = {Progress of Theoretical and Experimental Physics},
	volume = {2014},
	number = {6},
	year = {2014},
	month = {06},
	abstract = "{Observations of Cosmic Microwave Background (CMB) anisotropies have become the most powerful probe of the early universe and cosmology. Recent satellite CMB experiments have revealed the all-sky view at microwave frequencies, and provided us with plenty of information about not only the cosmological CMB but also about various astrophysical processes in the Galaxy through foreground emissions. Accurate understanding about the foreground emissions is essential in order to precisely determine the cosmological CMB signal, and especially so if one tries to estimate weaker polarization signals from the CMB anisotropies. Here we briefly review the recent progress on the foreground subtraction methods and current knowledge about the foregrounds mainly focusing on the large-scale, diffuse, Galactic components.}",
	issn = {2050-3911},
	doi = {10.1093/ptep/ptu065},
	url = {https://doi.org/10.1093/ptep/ptu065},
	note = {06B109},
	eprint = {https://academic.oup.com/ptep/article-pdf/2014/6/06B109/4447714/ptu065.pdf},
}

@ARTICLE{2013ApJS..208...20B,
	author = {{Bennett}, C.~L. and others},
	title = "{Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results}",
	journal = {\apjs},
	keywords = {cosmic background radiation, cosmology: observations, dark matter, early universe, instrumentation: detectors, space vehicles, space vehicles: instruments, telescopes, Astrophysics - Cosmology and Nongalactic Astrophysics},
	year = 2013,
	month = oct,
	volume = {208},
	number = {2},
	eid = {20},
	pages = {20},
	doi = {10.1088/0067-0049/208/2/20},
	archivePrefix = {arXiv},
	eprint = {1212.5225},
	primaryClass = {astro-ph.CO},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2013ApJS..208...20B},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2013ApJS..208...19H,
       	author = {{Hinshaw}, G. and others},
       	title = "{Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results}",
      	journal = {\apjs},
     	keywords = {cosmic background radiation, cosmology: observations, dark 						matter, early universe, instrumentation: detectors, space vehicles, space vehicles: instruments, telescopes, Astrophysics - Cosmology and Nongalactic Astrophysics},
      	year = 2013,
        month = oct,
       	volume = {208},
       	number = {2},
       	eid = {19},
        pages = {19},
       	doi = {10.1088/0067-0049/208/2/19},
		archivePrefix = {arXiv},
       	eprint = {1212.5226},
 		primaryClass = {astro-ph.CO},
       	adsurl = {https://ui.adsabs.harvard.edu/abs/2013ApJS..208...19H},
      	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2005ApJ...622..759G,
	author = {{G{\'o}rski}, K.~M. and {Hivon}, E. and {Banday}, A.~J. and {Wandelt}, B.~D. and {Hansen}, F.~K. and {Reinecke}, M. and {Bartelmann}, M.},
	title = "{HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere}",
	journal = {\apj},
	keywords = {Cosmology: Cosmic Microwave Background, Cosmology: Observations, Methods: Statistical, Astrophysics},
	year = 2005,
	month = apr,
	volume = {622},
	number = {2},
	pages = {759-771},
	doi = {10.1086/427976},
	archivePrefix = {arXiv},
	eprint = {astro-ph/0409513},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2005ApJ...622..759G},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@article{Tegmark_1996,
	title={A method for subtracting foregrounds from multifrequency CMB sky maps},
	volume={281},
	ISSN={1365-2966},
	url={http://dx.doi.org/10.1093/mnras/281.4.1297},
	DOI={10.1093/mnras/281.4.1297},
	number={4},
	journal={Monthly Notices of the Royal Astronomical Society},
	publisher={Oxford University Press (OUP)},
	author={Tegmark, M. and Efstathiou, G.},
	year={1996},
	month={Aug},
	pages={1297–1314}
}

@ARTICLE{2008PhRvD..78b3003S,
	author = {{Saha}, Rajib and {Prunet}, Simon and {Jain}, Pankaj and
	{Souradeep}, Tarun},
	title = "{CMB anisotropy power spectrum using linear combinations of WMAP maps}",
	journal = {\prd},
	keywords = {98.70.Vc, 07.05.Kf, 98.80.Es, Background radiations, Data analysis: algorithms and implementation, data management, Observational cosmology, Astrophysics},
	year = "2008",
	month = "Jul",
	volume = {78},
	number = {2},
	eid = {023003},
	pages = {023003},
	doi = {10.1103/PhysRevD.78.023003},
	archivePrefix = {arXiv},
	eprint = {0706.3567},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2008PhRvD..78b3003S},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@article{Moore_1920,
	author = {{Moore}, E. H.},
	title = {On the reciprocal of the general algebraic matrix},
	journal = {Proceedings of the American Mathematical Society},
	year = "1920",
	OPTvolume = {26},
	OPTpages = {394-395}
}


@ARTICLE{1956PCPS...52...17P,
	author = {{Penrose}, R.},
	title = "{On best approximate solutions of linear matrix equations}",
	journal = {Proceedings of the Cambridge Philosophical Society},
	year = "1956",
	month = "Jan",
	volume = {52},
	number = {1},
	pages = {17-19},
	doi = {10.1017/S0305004100030929},
	adsurl = {https://ui.adsabs.harvard.edu/abs/1956PCPS...52...17P},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2018arXiv181008872S,
	author = {{Sudevan}, Vipin and {Saha}, Rajib},
	title = "{An Application of Global ILC Algorithm over Large Angular Scales to Estimate CMB Posterior Using Gibbs Sampling}",
	journal = {arXiv e-prints},
	keywords = {Astrophysics - Cosmology and Nongalactic Astrophysics},
	year = 2018,
	month = oct,
	eid = {arXiv:1810.08872},
	pages = {arXiv:1810.08872},
	archivePrefix = {arXiv},
	eprint = {1810.08872},
	primaryClass = {astro-ph.CO},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2018arXiv181008872S},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@article{10.1145/321127.321128,
	author = {Holland, John H.},
	title = {Outline for a Logical Theory of Adaptive Systems},
	year = {1962},
	issue_date = {July 1962},
	publisher = {Association for Computing Machinery},
	address = {New York, NY, USA},
	volume = {9},
	number = {3},
	issn = {0004-5411},
	url = {https://doi.org/10.1145/321127.321128},
	doi = {10.1145/321127.321128},
	journal = {J. ACM},
	month = jul,
	pages = {297–314},
	numpages = {18}
}

@Book{Goldberg,
	author = {Goldberg, D. E.},
	title = {Genetic algorithms in search, optimization, and machine learning},
	publisher = {Addison-Wesley	Publishing Company, Inc.},
	year = {1989}
}

@ARTICLE{2016arXiv161000976C,
	author = {{Chehouri}, Adam and {Younes}, Rafic and {Perron}, Jean and
	{Ilinca}, Adrian},
	title = "{A Constraint-Handling Technique for Genetic Algorithms using a Violation Factor}",
	journal = {arXiv e-prints},
	keywords = {Computer Science - Artificial Intelligence, Mathematics - Optimization and Control},
	year = 2016,
	month = oct,
	eid = {arXiv:1610.00976},
	pages = {arXiv:1610.00976},
	archivePrefix = {arXiv},
	eprint = {1610.00976},
	primaryClass = {cs.AI},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2016arXiv161000976C},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2003ApJS..148....1B,
	author = {{Bennett}, C.~L. and others},
	title = "{First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results}",
	journal = {\apjs},
	keywords = {Cosmology: Cosmic Microwave Background, Cosmology: Observations, Cosmology: Dark Matter, Cosmology: Early Universe, Instrumentation: Detectors, Space Vehicles: Instruments, Astrophysics},
	year = "2003",
	month = "Sep",
	volume = {148},
	number = {1},
	pages = {1-27},
	doi = {10.1086/377253},
	archivePrefix = {arXiv},
	eprint = {astro-ph/0302207},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2003ApJS..148....1B},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{1999NewA....4..443B,
	author = {{Bouchet}, Fran{\c{c}}ois R. and {Gispert}, Richard},
	title = "{Foregrounds and CMB experiments. I. Semi-analytical estimates of contamination}",
	journal = {\na},
	keywords = {Astrophysics},
	year = "1999",
	month = "Sep",
	volume = {4},
	number = {6},
	pages = {443-479},
	doi = {10.1016/S1384-1076(99)00027-5},
	archivePrefix = {arXiv},
	eprint = {astro-ph/9903176},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/1999NewA....4..443B},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2003ApJS..148..135H,
	author = {{Hinshaw}, G. and others},
	title = "{First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: The Angular Power Spectrum}",
	journal = {\apjs},
	keywords = {Cosmology: Cosmic Microwave Background, Cosmology: Cosmological Parameters- Cosmology: Observations, Cosmology: Early Universe, Cosmology: Large-Scale Structure of Universe, Space Vehicles: Instruments, Astrophysics},
	year = "2003",
	month = "Sep",
	volume = {148},
	number = {1},
	pages = {135-159},
	doi = {10.1086/377225},
	archivePrefix = {arXiv},
	eprint = {astro-ph/0302217},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2003ApJS..148..135H},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2003ApJS..148...97B,
	author = {{Bennett}, C.~L. and others},
	title = "{First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Foreground Emission}",
	journal = {\apjs},
	keywords = {Cosmology: Cosmic Microwave Background, Cosmology: Observations, Cosmology: Diffuse Radiation, Galaxy: Halo, Galaxy: Structure, ISM: Structure, Astrophysics},
	year = "2003",
	month = "Sep",
	volume = {148},
	number = {1},
	pages = {97-117},
	doi = {10.1086/377252},
	archivePrefix = {arXiv},
	eprint = {astro-ph/0302208},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2003ApJS..148...97B},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2007ApJS..170..288H,
	author = {{Hinshaw}, G. and others},
	title = "{Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Temperature Analysis}",
	journal = {\apjs},
	keywords = {Cosmology: Cosmic Microwave Background, Cosmology: Observations, Cosmology: Dark Matter, Cosmology: Early Universe, Instrumentation: Detectors, Space Vehicles: Instruments, Telescopes, Astrophysics},
	year = "2007",
	month = "Jun",
	volume = {170},
	number = {2},
	pages = {288-334},
	doi = {10.1086/513698},
	archivePrefix = {arXiv},
	eprint = {astro-ph/0603451},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2007ApJS..170..288H},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@article{Dodelson_1997,
	doi = {10.1086/304157},
	url = {https://doi.org/10.1086%2F304157},
	year = 1997,
	month = {jun},
	publisher = {{IOP} Publishing},
	volume = {482},
	number = {2},
	pages = {577--587},
	author = {Scott Dodelson},
	title = {Determining Cosmic Microwave Background Anisotropies in the Presence of Foregrounds},
	journal = {The Astrophysical Journal},
	abstract = {Separating foregrounds from the signal is one of the major challenges in cosmic microwave background (CMB) experiments. A simple way to estimate the CMB temperature in a given pixel is to fit for the amplitudes of the CMB and the various foreground components. The variance squared of this estimator is shown to be equal to (FDF)2 σ+σ, where σ is the variance in the absence of foregrounds, σshape is the variance due to the uncertainty in the shapes of the foreground components, and FDF is the foreground degradation factor. This one number, the FDF, gives a good indication of the ability of a given experiment to disentangle the CMB from foreground sources. A variety of applications relating to the planning and analyzing of experiments is presented. This approach, viz., analyzing the variance, can be used to assess the usefulness of any estimator.}
}

@ARTICLE{1998ApJ...502....1T,
	author = {{Tegmark}, Max},
	title = "{Removing Real-World Foregrounds from Cosmic Microwave Background Maps}",
	journal = {\apj},
	keywords = {COSMOLOGY: COSMIC MICROWAVE BACKGROUND, METHODS: DATA ANALYSIS, Cosmology: Cosmic Microwave Background, Methods: Data Analysis, Astrophysics},
	year = "1998",
	month = "Jul",
	volume = {502},
	number = {1},
	pages = {1-6},
	doi = {10.1086/305905},
	archivePrefix = {arXiv},
	eprint = {astro-ph/9712038},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/1998ApJ...502....1T},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2003ApJS..145..413J,
	author = {{Jarosik}, N. and {Bennett}, C.~L. and others},
	title = "{Design, Implementation, and Testing of the Microwave Anisotropy Probe Radiometers}",
	journal = {\apjs},
	keywords = {Cosmology: Cosmic Microwave Background, Instrumentation: Detectors, Space Vehicles: Instruments, Astrophysics},
	year = "2003",
	month = "Apr",
	volume = {145},
	number = {2},
	pages = {413-436},
	doi = {10.1086/346080},
	archivePrefix = {arXiv},
	eprint = {astro-ph/0301164},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2003ApJS..145..413J},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2007ApJS..170..263J,
	author = {{Jarosik}, N. and others},
	title = "{Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Beam Profiles, Data Processing, Radiometer Characterization, and Systematic Error Limits}",
	journal = {\apjs},
	keywords = {Cosmology: Cosmic Microwave Background, Instrumentation: Detectors, Space Vehicles: Instruments, Astrophysics},
	year = "2007",
	month = "Jun",
	volume = {170},
	number = {2},
	pages = {263-287},
	doi = {10.1086/513697},
	archivePrefix = {arXiv},
	eprint = {astro-ph/0603452},
	primaryClass = {astro-ph},
	adsurl = {https://ui.adsabs.harvard.edu/abs/2007ApJS..170..263J},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@article{Eriksen_2004,
   title={Power Spectrum Estimation from High‐Resolution Maps by Gibbs Sampling},
   volume={155},
   ISSN={1538-4365},
   url={http://dx.doi.org/10.1086/425219},
   DOI={10.1086/425219},
   number={2},
   journal={The Astrophysical Journal Supplement Series},
   publisher={IOP Publishing},
   author={Eriksen, H. K. and others},
   year={2004},
   month={Dec},
   pages={227–241}
}

@article{Eriksen_2007,
	doi = {10.1086/509911},
	url = {https://doi.org/10.1086\%2F509911},
	year = 2007,
	month = {feb},
	publisher = {{IOP} Publishing},
	volume = {656},
	number = {2},
	pages = {641--652},
	author = {H. K. Eriksen and Greg Huey and R. Saha and others},
	title = {A Reanalysis of the 3 {YearWilkinson} Microwave Anisotropy {ProbeTemperature} Power Spectrum and Likelihood},
	journal = {The Astrophysical Journal},
}

@ARTICLE{2008ApJ...672L..87E,
       author = {{Eriksen}, H.~K. and {Dickinson}, C. and {Jewell}, J.~B. and {Banday}, A.~J. and {G{\'o}rski}, K.~M. and {Lawrence}, C.~R.},
        title = "{The Joint Large-Scale Foreground-CMB Posteriors of the 3 Year WMAP Data}",
      journal = {\apjl},
     keywords = {cosmic microwave background, cosmology: observations, methods: numerical, Astrophysics},
         year = 2008,
        month = jan,
       volume = {672},
       number = {2},
        pages = {L87},
          doi = {10.1086/526545},
    archivePrefix = {arXiv},
       eprint = {0709.1037},
     primaryClass = {astro-ph},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2008ApJ...672L..87E},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{2008ApJ...676...10E,
       author = {{Eriksen}, H.~K. and {Jewell}, J.~B. and {Dickinson}, C. and {Banday}, A.~J. and {G{\'o}rski}, K.~M. and {Lawrence}, C.~R.},
        title = "{Joint Bayesian Component Separation and CMB Power Spectrum Estimation}",
      journal = {\apj},
     keywords = {cosmic microwave background, cosmology: observations, methods: numerical, Astrophysics},
         year = 2008,
        month = mar,
       volume = {676},
       number = {1},
        pages = {10-32},
          doi = {10.1086/525277},
    archivePrefix = {arXiv},
       eprint = {0709.1058},
    primaryClass = {astro-ph},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2008ApJ...676...10E},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{4767596,  
    author={S. {Geman} and D. {Geman}},  
    journal={IEEE Transactions on Pattern Analysis and Machine Intelligence},             
    title={Stochastic Relaxation, Gibbs Distributions, and the Bayesian     Restoration of Images},   
    year={1984},  
    volume={PAMI-6},  
    number={6},  
    pages={721-741},}
    
@ARTICLE{Erikson_2006,
       author = {{Eriksen}, H.~K. and others},
        title = "{Cosmic Microwave Background Component Separation by Parameter Estimation}",
      journal = {\apj},
     keywords = {Cosmology: Cosmic Microwave Background, Cosmology: Observations, Methods: Numerical, Astrophysics},
         year = 2006,
        month = apr,
       volume = {641},
       number = {2},
        pages = {665-682},
          doi = {10.1086/500499},
    archivePrefix = {arXiv},
       eprint = {astro-ph/0508268},
    primaryClass = {astro-ph},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2006ApJ...641..665E},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@ARTICLE{Gold_2011,
       author = {{Gold}, B. and others},
        title = "{Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Galactic Foreground Emission}",
      journal = {\apjs},
     keywords = {cosmic background radiation, cosmology: observations, diffuse radiation, Galaxy: halo, Galaxy: structure, ISM: structure, Astrophysics - Astrophysics of Galaxies, Astrophysics - Cosmology and Nongalactic Astrophysics},
         year = 2011,
        month = feb,
       volume = {192},
       number = {2},
          eid = {15},
        pages = {15},
          doi = {10.1088/0067-0049/192/2/15},
archivePrefix = {arXiv},
       eprint = {1001.4555},
 primaryClass = {astro-ph.GA},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2011ApJS..192...15G},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@article{Tegmark_2003,
   title={High resolution foreground cleaned CMB map from WMAP},
   volume={68},
   ISSN={1089-4918},
   url={http://dx.doi.org/10.1103/PhysRevD.68.123523},
   DOI={10.1103/physrevd.68.123523},
   number={12},
   journal={Physical Review D},
   publisher={American Physical Society (APS)},
   author={Tegmark, Max and de Oliveira-Costa, Angélica and Hamilton, Andrew J. S.},
   year={2003},
   month={Dec}
}

@article{Eriksen_2004a,
	doi = {10.1086/422807},
	url = {https://doi.org/10.1086\%2F422807},
	year = 2004,
	month = {sep},
	publisher = {{IOP} Publishing},
	volume = {612},
	number = {2},
	pages = {633--646},
	author = {H. K. Eriksen and A. J. Banday and K. M. Gorski and P. B. Lilje},
	title = {On Foreground Removal from {theWilkinson} Microwave Anisotropy {ProbeData} by an Internal Linear Combination Method: Limitations and Implications},
	journal = {The Astrophysical Journal},
}

@article{Saha_2006,
   title={A Blind Estimation of the Angular Power Spectrum of CMB Anisotropy from WMAP},
   volume={645},
   ISSN={1538-4357},
   url={http://dx.doi.org/10.1086/506321},
   DOI={10.1086/506321},
   number={2},
   journal={The Astrophysical Journal},
   publisher={IOP Publishing},
   author={Saha, Rajib and Jain, Pankaj and Souradeep, Tarun},
   year={2006},
   month={Jul},
   pages={L89–L92}
}

@article{Saha_2008,
  title = {CMB anisotropy power spectrum using linear combinations of WMAP maps},
  author = {Saha, Rajib and Prunet, Simon and Jain, Pankaj and Souradeep, Tarun},
  journal = {Phys. Rev. D},
  volume = {78},
  issue = {2},
  pages = {023003},
  numpages = {23},
  year = {2008},
  month = {Jul},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevD.78.023003},
  url = {https://link.aps.org/doi/10.1103/PhysRevD.78.023003}
}

@article{Sudevan_2018,
   title={A Global ILC Approach in Pixel Space over Large Angular Scales of the Sky Using CMB Covariance Matrix},
   volume={867},
   ISSN={1538-4357},
   url={http://dx.doi.org/10.3847/1538-4357/aae439},
   DOI={10.3847/1538-4357/aae439},
   number={1},
   journal={The Astrophysical Journal},
   publisher={American Astronomical Society},
   author={Sudevan, Vipin and Saha, Rajib},
   year={2018},
   month={Nov},
   pages={74}
}

@article{Saha_2011,
   title={A FOREGROUND-CLEANED COSMIC MICROWAVE BACKGROUND MAP FROM NON-GAUSSIANITY MEASUREMENT},
   volume={739},
   ISSN={2041-8213},
   url={http://dx.doi.org/10.1088/2041-8205/739/2/L56},
   DOI={10.1088/2041-8205/739/2/l56},
   number={2},
   journal={The Astrophysical Journal},
   publisher={IOP Publishing},
   author={Saha, Rajib},
   year={2011},
   month={Sep},
   pages={L56}
}

@ARTICLE{Ujjal_2017,
       author = {{Purkayastha}, Ujjal and {Saha}, Rajib},
        title = "{Separating CMB Stokes Q and U polarization signals from Non-Gaussian Emissions}",
      journal = {arXiv e-prints},
     keywords = {Astrophysics - Cosmology and Nongalactic Astrophysics},
         year = 2017,
        month = jul,
          eid = {arXiv:1707.02008},
        pages = {arXiv:1707.02008},
    archivePrefix = {arXiv},
       eprint = {1707.02008},
     primaryClass = {astro-ph.CO},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2017arXiv170702008P},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@misc{Sudevan_2018_g,
    title={An Application of Global ILC Algorithm over Large Angular Scales to Estimate CMB Posterior Using Gibbs Sampling},
    author={Vipin Sudevan and Rajib Saha},
    year={2018},
    eprint={1810.08872},
    archivePrefix={arXiv},
    primaryClass={astro-ph.CO}
}

@article{ALLEN198766,
title = "Non-gaussian density perturbations in inflationary cosmologies",
journal = "Physics Letters B",
volume = "197",
number = "1",
pages = "66 - 70",
year = "1987",
issn = "0370-2693",
doi = "https://doi.org/10.1016/0370-2693(87)90343-1",
url = "http://www.sciencedirect.com/science/article/pii/0370269387903431",
author = "T.J. Allen and B. Grinstein and Mark B. Wise",
abstract = "The primordial mass density fluctuations may have arisen from quantum fluctuations in a (massless) scalar field that occurred during an inflationary era. We show that it is possible for primordial mass density fluctuations, which arose in this way, to be highly non-gaussian. We also show that the “bad” infrared properties of the propagator for a massless scalar field in de Sitter space can translate itself into a power spectrum, for the two-point spatial correlation of objects that do not trace the mass, which behaves like k−3, at small wave numbers k."
}

@article{Falk_1992,
  title = {Dependence of density perturbations on the coupling constant in a simple model of inflation},
  author = {Falk, Toby and Rangarajan, Raghavan and Srednicki, Mark},
  journal = {Phys. Rev. D},
  volume = {46},
  issue = {10},
  pages = {4232--4234},
  numpages = {0},
  year = {1992},
  month = {Nov},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevD.46.4232},
  url = {https://link.aps.org/doi/10.1103/PhysRevD.46.4232}
}

@ARTICLE{Gangui_1994,
       author = {{Gangui}, Alejandro and {Lucchin}, Francesco and {Matarrese}, Sabino and
         {Mollerach}, Silvia},
        title = "{The Three-Point Correlation Function of the Cosmic Microwave Background in Inflationary Models}",
      journal = {\apj},
     keywords = {Astronomical Models, Correlation Coefficients, Cosmic Background Explorer Satellite, Gravitational Fields, Perturbation Theory, Temperature Distribution, Angular Resolution, Astrophysics, Cosmology, Data Correlation, Microwave Radiometers, Skewness, Stochastic Processes, Universe, Astrophysics, COSMOLOGY: COSMIC MICROWAVE BACKGROUND, COSMOLOGY: THEORY, COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE, Astrophysics},
         year = 1994,
        month = aug,
       volume = {430},
        pages = {447},
          doi = {10.1086/174421},
archivePrefix = {arXiv},
       eprint = {astro-ph/9312033},
 primaryClass = {astro-ph},
       adsurl = {https://ui.adsabs.harvard.edu/abs/1994ApJ...430..447G},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@article{ACQUAVIVA2003119,
title = "Gauge-invariant second-order perturbations and non-Gaussianity from inflation",
journal = "Nuclear Physics B",
volume = "667",
number = "1",
pages = "119 - 148",
year = "2003",
issn = "0550-3213",
doi = "https://doi.org/10.1016/S0550-3213(03)00550-9",
url = "http://www.sciencedirect.com/science/article/pii/S0550321303005509",
author = "Viviana Acquaviva and Nicola Bartolo and Sabino Matarrese and Antonio Riotto",
abstract = "We present the first computation of the cosmological perturbations generated during inflation up to second order in deviations from the homogeneous background solution. Our results, which fully account for the inflaton self-interactions as well as for the second-order fluctuations of the background metric, provide the exact expression for the gauge-invariant curvature perturbation bispectrum produced during inflation in terms of the slow-roll parameters. The bispectrum represents a specific non-Gaussian signature of fluctuations generated by quantum oscillations during slow-roll inflation. However, our findings indicate that detecting the non-Gaussianity in the cosmic microwave background anisotropies emerging from the second-order calculation will be a challenge for the forthcoming satellite experiments."
}

@article{Maldacena_2003,
	doi = {10.1088/1126-6708/2003/05/013},
	url = {https://doi.org/10.1088%2F1126-6708%2F2003%2F05%2F013},
	year = 2003,
	month = {may},
	publisher = {Springer Science and Business Media {LLC}},
	volume = {2003},
	number = {05},
	pages = {013--013},
	author = {Juan Maldacena},
	title = {Non-gaussian features of primordial fluctuations in single field inflationary models},
	journal = {Journal of High Energy Physics},
	abstract = {We compute the three point correlation functions for
primordial scalar and tensor fluctuations in single field inflationary
models. We obtain explicit expressions in the slow roll limit where
the answer is given terms of the two usual slow roll parameters.  In a
particular limit the three point functions are determined completely
by the tilt of the spectrum of the two point functions. We also make
some remarks on the relation of this computation to dS/CFT and
AdS/CFT. We emphasize that (A)dS/CFT can be viewed as a statement
about the wavefunction of the universe.}
}

@ARTICLE{Gorski2005,
       author = {{G{\'o}rski}, K.~M. and {Hivon}, E. and {Banday}, A.~J. and {Wandelt}, B.~D. and {Hansen}, F.~K. and {Reinecke}, M. and {Bartelmann}, M.},
        title = "{HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere}",
      journal = {\apj},
     keywords = {Cosmology: Cosmic Microwave Background, Cosmology: Observations, Methods: Statistical, Astrophysics},
         year = 2005,
        month = apr,
       volume = {622},
       number = {2},
        pages = {759-771},
          doi = {10.1086/427976},
archivePrefix = {arXiv},
       eprint = {astro-ph/0409513},
 primaryClass = {astro-ph},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2005ApJ...622..759G},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}


@ARTICLE{1994ApJ...420..439M,
       author = {{Mather}, J.~C. and others},
        title = "{Measurement of the Cosmic Microwave Background Spectrum by the COBE FIRAS Instrument}",
      journal = {\apj},
     keywords = {Background Radiation, Cosmic Rays, Cosmology, Far Infrared Radiation, Infrared Spectrophotometers, Microwave Spectra, Universe, Black Body Radiation, Brightness Distribution, Calibrating, Cosmic Background Explorer Satellite, Data Processing, Error Analysis, Red Shift, Astrophysics, COSMOLOGY: COSMIC MICROWAVE BACKGROUND, COSMOLOGY: EARLY UNIVERSE, COSMOLOGY: OBSERVATIONS},
         year = 1994,
        month = jan,
       volume = {420},
        pages = {439},
          doi = {10.1086/173574},
       adsurl = {https://ui.adsabs.harvard.edu/abs/1994ApJ...420..439M},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

