# 6.6. (Sub-)haloes (clumps.h & stat.h)¶

CLUMPY needs to manipulate and calculate many quantities related to generic DM (sub-)haloes. Their most general description takes into account that the total DM distribution is the sum of a smooth contribution and a distribution of sub-halos. The sub-halos can then be seen themselves as scale-down versions of their parent (“host”) halo. Hence, multi-level of substructures should be accounted for. For the specific case of DM decay, if only the mean contribution is calculated, all these details are irrelevant (the signal is basically proportional to the mass, which remains the same whether we put some of it in sub-halos or not). For DM annihilation, they are important as sub-halos can boost the signal (note however that after the second level, the contributions are generally negligible).

The libraries clumps.h and stat.h contain functions to calculate,

• mass of DM halo $$M_\Delta$$, $$[M_\Delta]={\rm M}_\odot$$;
• the properties of subhaloes (distributions, mass-concentration, etc.);
• the intrinsic luminosity (see below), and J-factor for any host halo, spherical and triaxial, with or without sub-structures;
• the average and variance of galactic subhaloes or field haloes (not yet implemented for the extragalactic calculation).

## 6.6.1. Overdensity $$\Delta$$ ($$R_\Delta$$ and $$M_\Delta$$)¶

All integrations for (sub-)haloes must be carried out to their maximum radius. However, the radius of a halo is an ill-defined quantity. In general, the mass $$M_{\Delta}$$ of a halo is connected to its size, $$R_{\Delta}$$, via the relation

(6.8)$R_{\Delta}( M_{\Delta}, z) =\left( \frac{3\, M_{\Delta}}{ 4\pi \times \Delta(z) \times \varrho_{\rm c}(z)}\right)^{1/ 3} \times (1 + z)\,,$

where the subscript $$\Delta$$ denotes a characteristic collapse overdensity above the critical density of the Universe, $$\varrho_{\rm c}= 3H^2(z)/8\pi G$$. All calculations can be chosen to be performed with respect to any of the following overdensity definitions:

$\begin{split}\Delta(z) & = {\rm const.},\\ \Delta(z) &= {{\rm const.} \times \Omega_{\rm m}(z)} =: {\Delta_{\rm m} \times \Omega_{\rm m}(z)},\\ \Delta(z) &= 18\pi^2 + 82\,[\Omega_{\rm m}(z) - 1)]- 39\,[\Omega_{\rm m}(z) - 1]^2 \,.\end{split}$

If a mass-concentration $$c_{\Delta}(M_{\Delta},z)$$ or halo mass function $${\mathrm{d}n}/{\mathrm{d}M}(M,\,z)$$ are natively provided for a $$\Delta$$ different from the user’s choice, $$c_{\Delta}(M_{\Delta},z)$$ or $${\mathrm{d}n}/{\mathrm{d}M}(M,\,z)$$ are internally rescaled to the user-chosen Delta using the algorithm described in appendix A of Hütten et al. (2018). Note that this rescaling presumes a given halo profile (see Section 6.5).

## 6.6.2. Concentration: $$c_\Delta-M_\Delta$$¶

Definition

The concentration $$c_\Delta$$, at a given characteristic overdensity $$\Delta$$, is defined to be

$c_\Delta\equiv \frac{R_\Delta}{r_{-2}},$

where $$R_\Delta$$ is the radius of the DM halo for which the density equals this overdensity (see above), and $$r_{-2}$$ is the position where the slope of the DM halo density profile reaches $$-2$$ (see Section 6.5). A list of all implemented $$c_\Delta-M_\Delta$$ descriptions is given in Mass-concentration relations.

Implementation in CLUMPY

We chose to code the generic $$c_\Delta-M_\Delta$$ relationship (with $$\Delta=200,\, {\rm vir},\,...)$$ and then rely on a conversion function to translate between different choices of $$\Delta$$. The redshift dependence of the concentration is taken into account; haloes formed at earlier times are less concentrated. There is generally a dependence on the environment and an intrinsic scatter of the $$c_\Delta-M_\Delta$$ relationship (see below). However, effects of halo selection (relaxed or not…) are not implemented. They can lead to $$\sim 10\%$$ differences in the $$c_\Delta-M_\Delta$$ relationship (see, e.g. Duffy et al. 2008 and App. of

Note

Most of the parametrisations deal with galaxy cluster or down to galaxy size objects, $$M_{\Delta}\gtrsim [10^{10}-10^{15}]\,M_\odot$$. Only kB01_VIR, kENS01_VIR, kGIOCOLI12_VIR, and kSANCHEZ14_200 can be extrapolated down to the lowest halo masses, $$M_{\Delta}\ll 10^{10}\,M_\odot$$.

Illustration

In Fig. 6.12, we plot the $$c_\Delta-M_\Delta$$ relationships for selected prescriptions available in CLUMPY, and in Fig. 6.13 the intrinsic luminosity $${\cal L}(M_{vir})$$ for all profiles: Fig. 6.12 Concentration-mass relations. Fig. 6.13 Intrinsic halo luminosities.

Fig. 6.12 and Fig. 6.13 can be reproduced via

$clumpy -e2 -D  which creates a default parameter file. Then the gEXTRAGAL_FLAG_CDELTAMDELTA_LIST can be expanded within the parameter file or simply overwritten in the command line: $ clumpy -e2 -i clumpy_params_e2.txt --gEXTRAGAL_FLAG_CDELTAMDELTA_LIST=kB01_VIR,kENS01_VIR,kNET007_200,kDUFFY08F_VIR,kDUFFY08F_200,kDUFFY08F_MEAN,kETTORI10_200,kPRADA12_200,kGIOCOLI12_VIR,kSANCHEZ14_200


As calculating ten times the substructure contribution for each description may be rather time-consuming, the signal boost from substructures can be disregarded by overwriting the default value in the parameter file via the command line:



## 6.6.5. Mean and variance¶

A typical DM halo of $$10^{12}M_\odot$$ (Milky-Way like) contains up to $$10^{14}$$ substructures, which renders the explicit calculation of the signal summed over all haloes prohibitive. This huge number allows the use of the continuum limit as the subhalo positions and masses are random variables, drawn from distribution functions (described above) obtained by N-body numerical simulations and/or semi-analytical calculations

At first order, a random variable (e.g. the mass and position of substructures) is described by its average value and variance. Departure from the average can arise if a small number of objects contribute significantly to the total J-factor, which happens if a massive subhalo dominates, or if one of the smallest subhaloes (the smaller, the more numerous they are) is sitting almost at the observer location. The latter configuration only happens for subhaloes in the Galaxy, since substructures in dSphs or extragalactic objects are far away.

As presented in Section 6.1 for the galactic halo, J-factor skymaps rely on a combination of the calculation of the average signal and the calculation of individual drawn clumps above and below a critical distance $$l_{\rm crit}$$: the critical distance is obtained by requiring the relative error of the signal integrated from $$l_{\rm crit}$$ to remain lower than a user-defined precision requirement. This strategy ensures a controlled and extremely quick calculation of skymaps: the number of subhaloes to draw in the Galaxy is reduced from a few tens of thousands to a few hundreds depending on the configuration.

Average mass and the mean of (some power of) the distance:

$\begin{split}\langle M\rangle \!&=\!\!\! \int_{M_{\rm min}}^{M_{\rm max}}\!\!\!\!\! M\frac{{\rm d} {\cal P}_M}{{\rm d} M} {{\rm d} M},\\ \langle l^n \rangle \!&=\!\!\! \int_0^{\Delta\Omega}\!\!\!\! \int_{l_{\rm min}}^{l_{\rm max}}\!\!\!\! l^{\,n} \frac{{\rm d} {\cal P}_V}{{\rm d} V} l^{\,2} {\rm d} l \, {\rm d}\Omega.\end{split}$

Mean luminosity over $$M$$ and $$c$$:

$\langle {\cal L} \rangle \!=\!\!\! \int_{M_{\rm min}}^{M_{\rm max}} \!\!\frac{{\rm d} {\cal P}_M}{{\rm d} M}(M) \!\!\! \int_{c_{\rm min}(M)}^{c_{\rm max}(M)} \!\!\frac{{\rm d} {\cal P}_c}{d c}(M,c) \,{\cal L}(M,c)\, {\rm d} c \,{\rm d} M,$

Average and variance on J:

\begin{align}\begin{aligned}\langle J_{\rm subs}\rangle &= N_{\rm tot}\!\!\int_0^{\Delta\Omega} \!\! \int_{l_{\rm min}}^{l_{\rm max}} \frac{{\rm d}{\cal P}_V}{{\rm d} V}(l,\Omega) \,{\rm d} l \,{\rm d}\Omega \langle {\cal L} \rangle\\\sigma^2_{\rm subs} &= \langle {\cal L}^2\rangle \left\langle\frac{1}{l^4}\right\rangle - \langle{\cal L}\rangle^2 \left<\frac{1}{ l^2}\right>^2.\end{aligned}\end{align}