Page MenuHomec4science
Files F77477576

doc.tex
No OneTemporary
Actions

File Metadata

Created
Wed, Aug 14, 17:28

doc.tex
View Options

\documentclass{article}
\usepackage{subfigure}
\usepackage{graphicx}
\begin{document}
\section{The sub-halo mass function}
The sub-halo mass function (the mass distribution of dark haloes potentially hosting satellites around a host galaxy)
is computed from the Eq.~17 of Schneider 2015 which provides
the number of dark haloes per mass unit for a given mass of the host galaxy as well for the mass of
the warm dark matter particule considered. The default values for the host galaxy is set to $10^{12}\,\rm{M_\odot}$.
Fig.~\ref{NhCum} shows the comparison between the cumulative number of sub-haloes from a CDM and a WDM model
with particle mass of $2\,\rm{keV}$.
The four different panels show the change in the error bars (shown by the shaded area)
as a function of the number of host galaxy observed, from 100 down to 25.
To estimate the error bars (Poisson noise), we sampled randomly $N$ sub-halo population that matches the sub-halo mass function of Schneider 2015,
$N$ corresponding to the number of host galaxies observed. From those $N$ realisation, a mean distribution is then derived.
This process is then performed hundred times with different random seed, providing us with hundred mean distribution functions.
The edges of the shaded area (estimation of the error) is defined by the mean of the hundred distributions plus or minus
three times the corresponding standard deviation.
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/NhCum100.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/NhCum075.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/NhCum050.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/NhCum025.png}}}
\caption{Comparison of dark halo distribution function between CDM and WDM (with a DM mass of $2\,\rm{keV}$)
as predicted from Schneider 2015.
The shaded area show 3 standard deviations around the mean.}
\label{NhCum}
\end{center}
\end{figure}
%
\newpage
\section{Deriving the luminosity function of satellites}
To estimate the satellite population around a host galaxies we attributed a stellar mass
for each sub-halo obtained in the first step. Contrary to Milky Way-like galaxies, dwarf galaxies
do not follow a well defined stellar mass - halo mass relation ($M_{\star} - M_{\rm h}$).
On the contrary, as seen on Fig.~\ref{Fig2_Sales2022},
a large scatter exists for a given halo mass, showing a variation of up to three dex in luminosity for a given
halo mass. This scatter reflect different buld-up histories of dwarf galaxies.
To convert halo mass to stellar mass, we defined in Fig.~\ref{Fig2_Sales2022} an area that encompass all galaxies,
either including only galaxies formed in the field (right panel) or satellite galaxies (left panel).
The different relations (models) we set-up are shown in Fig.~\ref{MsMh}.
Then for a given $M_{\star} - M_{\rm h}$ relation model, for each sub-halo we randomly determine a luminosity
from a uniform distribution spanning the minimal and maximal luminosity for the given halo mass.
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{1.00\hsize}{!}{\includegraphics[angle=0]{pngs/Fig2_Sales2022.png}}}
\caption{Fig~2. from Sales et al. 2022}
\label{Fig2_Sales2022}
\end{center}
\end{figure}
%
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LvvsMhm1.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LvvsMhm2.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LvvsMhm3.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LvvsMhm4.png}}}
\caption{Comparison of the three $M_{\star} - M_{\rm h}$ relations (\texttt{model1}, \texttt{model2}, \texttt{model3}, \texttt{model4}) used.}
\label{MsMh}
\end{center}
\end{figure}
%
\newpage
\section{Computing the luminosity function of satellites}
Following the previous steps, we can derive the luminosity function of satellites (Fig.~\ref{LCum_m1} to \ref{LCum_m4})
as the cumulative number of satellites observed around a given galaxy.
As for the sub-halo mass function, the errors (shaded area) are estimating by three standard deviation around the mean,
obtained by averaging hundred realisations. In each realisation we estimate the mean distribution obtained after
generating respectively 100, 75, 50 and 25 populations, corresponding to the number of host galaxies observed.
The four figures are obtained for the four $M_{\star} - M_{\rm h}$ models of Fig.~\ref{MsMh}.
The WDM mass is set to $2\,\rm{keV}$.
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum100m1.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m1.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum050m1.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum025m1.png}}}
\caption{Predicted cumulative number of satellites brighter than
a given luminosity (\texttt{model1}), within 300 kpc around a Milky Way-like analogue.
The shaded area show 3 standard deviations around the mean.}
\label{LCum_m1}
\end{center}
\end{figure}
%
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum100m2.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m2.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum050m2.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum025m2.png}}}
\caption{Predicted cumulative number of satellites brighter than
a given luminosity (\texttt{model2}), within 300 kpc around a Milky Way-like analogue.
The shaded area show 3 standard deviations around the mean.}
\label{LCum_m2}
\end{center}
\end{figure}
%
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum100m3.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum050m3.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum025m3.png}}}
\caption{Predicted cumulative number of satellites brighter than
a given luminosity (\texttt{model3}), within 300 kpc around a Milky Way-like analogue.
The shaded area show 3 standard deviations around the mean.}
\label{LCum_m3}
\end{center}
\end{figure}
%
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum100m4.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m4.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum050m4.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum025m4.png}}}
\caption{Predicted cumulative number of satellites brighter than
a given luminosity (\texttt{model4}), within 300 kpc around a Milky Way-like analogue.
The shaded area show 3 standard deviations around the mean.}
\label{LCum_m4}
\end{center}
\end{figure}
%
\newpage
\section{Dependency of the satellite luminosity function on the WDM mass}
Fig.~\ref{LCum_MDM} is similar to Fig.~\ref{LCum_m1} to \ref{LCum_m4} but shows
the influence of the mass of WDM particles, increasing from $1.1$ to $4\,\rm{keV}$.
The $M_{\star} - M_{\rm h}$ is set to \texttt{model3} and we assume that 75 host galaxies are
observed.
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM1.1.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM2.0.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM3.0.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM4.0.png}}}
\caption{Dependency of the predictions as a function of the dark matter mass.}
\label{LCum_MDM}
\end{center}
\end{figure}
%
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM5.0.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM6.0.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM7.0.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM8.0.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM9.0.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM10.0.png}}}
\caption{Dependency of the predictions as a function of the dark matter mass (suite).}
\label{LCum_MDMb}
\end{center}
\end{figure}
%
\newpage
\section{Uncertainties related to the mass of the host galaxies}
Fig.~\ref{LCum_Mgal} shows the influence of uncertainties on the mass determination of the host
galaxies (and thus on the sub-halo mass function) on the uncertainties related to the satellite population.
In addition to the previous plots, for each realisation, we randomly attribute the mass of the host galaxy.
To this end,
we used a uniform distribution with minimal and maximal values corresponding to respectively 0, 0.1, 0.3 and 1 dex
around our fiducial host halo mass of $10^{12}\,\rm{M_\odot}$. The correspondence between dex and masses is provided
in the following table.
%
\begin{table}[h]
\centering
\begin{tabular}{ c c c }
dex & $M_{\rm{h,min}}$ & $M_{\rm{h,max}}$ \\
& $[10^{12}\,\rm{M_\odot}]$ & $[10^{12}\,\rm{M_\odot}]$ \\
0 & $1$ & $1$\\
0.1 & $0.79$ & $1.25$\\
0.3 & $0.5$ & $2$\\
1.0 & $0.1$ & $10$\\
\end{tabular}
\end{table}
%
For those models, the $M_{\star} - M_{\rm h}$ is set to \texttt{model3},
the WDM mass is set to $2\,\rm{keV}$,
and we assume that 75 host galaxies are
observed.
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM2.0.d0.0.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM2.0.d0.1.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM2.0.d0.5.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum075m3DM2.0.d1.0.png}}}
\caption{Influence of uncertainties on the mass determination of the host
galaxies on the uncertainties related to the satellite population.
From left to right,
the total mass of the main galaxy may vary between plus or minus 0, 0.1, 0.5 and 1 dex around $10^{12}\,\rm{M_\odot}$.}
\label{LCum_Mgal}
\end{center}
\end{figure}
%
\newpage
\section{Dependency of the satellite luminosity function on the WDM mass in magnitude}
%
\begin{figure}
%\vspace{-20pt}
\begin{center}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum_VIS_075m3DM1.1.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum_VIS_075m3DM2.0.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum_VIS_075m3DM3.0.png}}}
\subfigure{\resizebox{0.49\hsize}{!}{\includegraphics[angle=0]{pngs/LCum_VIS_075m3DM4.0.png}}}
\caption{Dependency of the predictions as a function of the dark matter mass.}
\label{LCum_VIS_MDM}
\end{center}
\end{figure}
%
\end{document}

Event Timeline

c4science · Help