# 10.3. Mass-concentration relations¶

Possible keyword choices for gXXX_FLAG_CDELTAMDELTA variables

Warning

Be aware that different $$c_\Delta-M_\Delta$$ parametrisations are only valid for restricted mass scales, which are given in the table below.

Also note that the distant-dependent descriptions kPIERI11_AQUARIUS, kPIERI11_VIALACTEA, kB01_VIR_RAD, and kMOLINE17_200 are at the moment only usable for Galactic subhalos in the -g module. Their usage for subhalos in the -h and -e modules is still under development.

Note

Various $$c_\Delta-M_\Delta$$ parametrisations use the same fitting formula from Lavalle et al. (2008),

$\ln (c_{\Delta} ) = \sum_{i=0}^{n} C_i \cdot \left[ \ln\left(\frac{M_{\Delta}}{M_\odot}\right) \right]^{i} + \ln\left(\frac{1}{1+z}\right)\,.$

In these cases, we provide the $$C_i$$ coefficients in the table below.

 Keyword Description kB01_VIR extrapolation down to any mass range. $$C_{i} = \{ - 4.34,\, 0.0384,\, -3.91\times 10^{-4},\, - 2.2\times 10^{-6},\, - 5.5\times 10^{-7}\}$$ kB01_VIR_RAD adaption of Bullock et al. (2001) with radial dependence. kCORREA15_PLANCK_200 Only for extragalactic haloes and Planck cosmology. kENS01_VIR extrapolation down any mass range. $$C_{i} = \{3.14,\;-0.018,\; -4.1\times 10^{-4}\}$$ kNETO07_200 valid for $$M>10^8 M_\odot$$. $$c_{200} = 4.67\times\left(\frac{M_{200}}{10^{14}M_\odot}\right)^{-0.11}\times\;(1+z)^{-1}\;\;$$ [their Eq. 5]. kDUFFY08F_VIR Duffy et al. (2008), $$\Delta=\Delta_{\rm vir}(z)$$: valid for $$M>10^8 M_\odot$$. $$c_{\rm vir}=A \left( \frac{M_{\rm vir}}{2\times10^{12}M_{\odot}}\right)^B(1+z)^C\;\;$$ [see their Table 1 for A, B, C]. kDUFFY08F_200 Duffy et al. (2008), $$\Delta=200$$: valid for $$M>10^8 M_\odot$$. kDUFFY08F_MEAN Duffy et al. (2008), $$\Delta=200\times\Omega_{\rm mean}(z)$$: valid for $$M>10^8 M_\odot$$. kETTORI10_200 valid for $$M>10^8 M_\odot$$ $$c_{200} = 10^{0.62} \times \left(\frac{M_{200}}{10^{15}M_\odot}\right)^{-0.1}\times\;(1+z)^{-1}\;\;$$ [see their Eq. 5]. kGIOCOLI12_VIR extrapolation down to any mass range. kLUDLOW16_200 Only for extragalactic haloes and Planck cosmology. kMOLINE17_200 Moliné et al. (2017), Eq. (6): improvement on Sánchez-Conde & Prada (2014) for Galactic subhalos accounting for spatial dependence of subhalos in field halos. kPIERI11_VIALACTEA Pieri et al. (2011), Eq.(11): extrapolation down to any mass range, inspired by Via Lactea II simulation results (also has an explicit spatial dependence). kPIERI11_AQUARIUS extrapolation down to any mass range, inspired by Aquarius simulation results (also has an explicit spatial dependence). kPRADA12_200 Prada et al. (2012), Eqs.(12-17,19-23): valid for $$M>10^8 M_\odot$$. kROCHA13_SIDM_VIR Only applicable to a Burkert-profile, resulting in core densities according to Eq. (20) of Rocha et al. (2013). kSANCHEZ14_200 extrapolation down to any mass range. $$C_{i} = \{ 37.5153,\, -1.5093,\, 1.636\times 10^{-2},\, 3.66\times 10^{-4},\, -2.\times 10^{-5},$$ $$\;5.32\times 10^{-7}\}$$