     I. What is CLUMPY?
     II. Main ingredients
     III. $\gamma$ -ray flux and the J-factor: definitions and conversions

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I. What is CLUMPY?
This code is dedicated to the calculation of the J-factor from any DM distribution, i.e.:

smooth DM halo in our Galaxy;
sub-clump DM distribution in the Galaxy;
any DM halo object (e.g. dwarf spheroidal galaxies in our Galaxy);
sub-clumps DM distribution in any host halo.

and hopefully, more to come in the future (extragalactic, Sommerfeld enhancement, charged particles...)

II. Main ingredients

The focus of the code is on the astrophysical term $J$ (see below). For the various pieces of information required to do such calculations, we refer to:

geometry.h - defines the geometry and coordinate transformations of the problem;
inlines.h and misc.h - definitions and useful low-level functions;
integr.h and integr_los.h - integration functions and algorithms;
janalysis.h - J-term analyses (astrophysics);
params.h - contains the user-defined input parameters;
profiles.h and clumps.h - DM profiles and relationships for the clump distributions respectively;
spectra.h - $d\Phi_{\gamma}^{PP}/dE_{\gamma}$ term (particle physics);
stat.h - statistical operations (mean, variance) on several clump properties.

Remarks:: The .h contain the function definitions and the doxygen documentation, whereas the .cc contains the code, written in C/C++.

III. $\gamma$ -ray flux and the J-factor: definitions and conversions

Studies of DM annihilations or decay involves both particle physics and astrophysics. The obvious difference of scales between the two fields and habits among the two communities have given rise to a plethora of notations and unit choices throughout the literature. This makes comparisons between studies particularly difficult. Below, we provide some explanatory elements and conversion factors to ease comparison between the different works published on the subject.

In CLUMPY, we define the differential $\gamma$ -ray flux as integrated over the solid angle $\Delta\Omega$ (in the direction $(\psi,\theta)$ , see here) as

$\frac{\mathrm{d}\Phi_{\gamma}}{\mathrm{d}E_{\gamma}}(E_{\gamma},\Delta\Omega) = \Phi^{\rm pp}(E_{\gamma}) \times J(\Delta\Omega)\,.$

Annihilating DM

The particle physics term (spectra.h) is

$\Phi^{\rm pp}(E_{\gamma})\equiv \frac{\mathrm{d}\Phi_{\gamma}}{\mathrm{d}E_{\gamma}} = \frac{1}{4\pi}\frac{\langle\sigma_{\rm ann}v\rangle}{2m_{\rm WIMP}^{2}} \cdot \frac{dN_{\gamma}}{dE_{\gamma}}\nonumber\;,$

with $<\sigma_{\rm ann}v>$ is the velocity average annihilation cross-section, $m_{\rm WIMP}$ is the mass of the WIMP candidate, and $dN_{\gamma}/dE_{\gamma}$ is the $\gamma$ -ray spectrum produced per annihilation. The astrophysics term is

$J(\Delta\Omega) = \int_{\Delta\Omega}\int \rho_{\rm DM}^2 (l,\Omega) \,dld\Omega,$

with $\rho_{\rm DM}$ the dark matter density. The solid angle is simply related to the integration angle $\alpha_{\rm int}$ by

$\Delta\Omega = 2\pi\cdot(1-\cos(\alpha_{\rm int})) \,.$

In our work, the units of these quantities are as follows:

$\left[\mathrm{d}\Phi_{\gamma}/\mathrm{d}E_{\gamma}\right] = {\rm cm}^{-2}{\rm ~s}^{-1} {\rm ~GeV}^{-1}$ ;
$\left[ \Phi^{\rm pp}(E_{\gamma})\right] = {\rm cm}^{3}{\rm ~s}^{-1} {\rm ~GeV}^{-3} ({\rm sr}^{-1})$ ;
$[J] = M_\odot^2{\rm ~kpc}^{-5} ({\rm sr})$ .

First of all, note that the location of the $1/4\pi$ factor appearing in $\Phi^{\rm pp}$ is arbitrary. We included it in the particle physics factor, but in other works, it can appear in the astrophysical factor J. Therefore, to compare the astrophysical factors between several studies, one must first ensure to correct the value of J by $4\pi$ if needed. The conversion factor (once the $4\pi$ issue is resolved) from our J units to that traditionally found in the literature are:

$1\; M_\odot^2{\rm ~kpc}^{-5}=10^{-15}~M_\odot^2{\rm ~pc}^{-5}$
$1\; M_\odot^2{\rm ~kpc}^{-5}=4.45\times10^{6}{\rm ~GeV}^2{\rm ~cm}^{-5}$
$1\; M_\odot^2{\rm ~kpc}^{-5} ~(\rm sr) = 1.44\times10^{-15}{\rm ~GeV}^2{\rm ~cm}^{-6} {\rm ~kpc} \;(\rm sr)$

Before comparing any number, one must also ensure that the solid angle $\Delta\Omega$ over which the integration is performed is the same. In most works, a $\alpha_{\rm int}=0.1^\circ$ angular resolution is chosen, corresponding to $\Delta\Omega=10^{-5}$ sr. However this is not always the case, as in the present study where we explore several angular resolutions.

Decaying DM

In that case, e.g., Bertone et al. (2007), the particle physics term is given by

$\Phi^{\rm pp}(E_{\gamma})\equiv \frac{\mathrm{d}\Phi_{\gamma}}{\mathrm{d}E_{\gamma}} = \frac{1}{4\pi}\frac{\Gamma_{\rm DM}}{m_{\rm DM}} \cdot \frac{2}{m_{\rm DM}} \delta\left( 1-\frac{2E_\gamma}{m_{\rm DM}}\right)\nonumber\;,$

with $\Gamma_{\rm DM}$ is the dark matter decay rate, and $m_{\rm DM}$ is the mass of the DM candidate. The astrophysics term is

$J(\Delta\Omega) = \int_{\Delta\Omega}\int \rho_{\rm DM} (l,\Omega) \,dld\Omega.$

In CLUMPY, the unit of $J$

is $M_\odot{\rm ~kpc}^{-2} ({\rm sr})$ .

Instrument response: Note that in the above integrations, it is assumed that the instrument is 'perfect'. To take into account the PSF (instrumental response) in a skymap, the best route is to calculate the value at a resolution much better than the PSF, and post-process the result by convolving it with the appropriate instrument response. This is not included in CLUMPY.