Model definitions

5.5 Mass modelling technique: LENSTOOL

5.5.2 Model definitions

The lens modelling method of J07 requires a first attempt at a model for the mass distribution of the lens. The model for the lensing potential given in equation 5.7 is initially defined using two sets of components: cluster-scale halos, which trace the overall mass distribution, and galaxy scale halos which trace the mass distribution on a these smaller scales and thus allow the model to take into account galaxy-galaxy lensing effects. N-body simulations have shown that the distribution of subhalo masses inside a cluster halo follows the Schechter function (Shaw et al., 2006) and thus the 2D cluster gravitational potential can be separated as follows (Natarajan and Kneib, 1997):

φ_tot =X

i

φ_cluster,i+X

j

φ_subhalo,j. (5.36)

Here the smooth and large cluster-scale halos are given byφ_cluster,iand the additional subhalos which create minor but non-negligible perturbations are given byφ_subhalo,j. These latter halos are defined as a potential hosting a galaxy. Each set of halos is modelled differently, as discussed below.

5.5.2.1 Cluster-scale halos

The main component of the lens mass model is made up of at least one cluster-scale mass halo to trace the overall/total mass distribution of the cluster. The simplest case is to start from a single cluster-scale halo, and increase the number of halos if the model requires it, i.e. we then look at the different parameter values produced by the model such as theχ² (see§5.5.3.2), the evidence (the normalisation of the posterior), and the rms difference between predicted and observed positions of multiple images.

Each halo is parameterized by its sky position(x_c, y_c), projected ellipticityε_Σ, position angle β, and a set of additional parameters specific to the type of mass profile chosen. The parameters for a Singular Isothermal Sphere (SIS), a S´ersic profile, and a NFW profile are given in J07. Here I will focus on the PIEMD (Pseudo Isothermal Elliptical Mass Distribution) profile, which has

three additional parameters: the halo ellipticity,ε_ϕ, given by ε_ϕ = 1−p

1−ε²_Σ

ε_Σ ; (5.37)

the halo density distribution of the form

ρ(R) = ρ₀

(1 +R²/r²_core) (1 +R²/r_cut² ), (5.38) whereρ₀ is the central density which depends on the central velocity dispersionσ₀:

ρ₀ = σ²₀

2πGr²_core; (5.39)

Here r_core and r_cut are the core radius and cut-off radius respectively. The two-dimensional surface density distribution for the PIEMD profile (Limousin et al., 2005) is given by

Σ(R) = σ²₀r_cut 2G(rcut−rcore)

1

pr_core² +R² − 1 pr_cut² +R²

!

. (5.40)

Note that the brightest cluster galaxy, called the cD galaxy, can be modelled either as part of a cluster-halo, or as its own subhalo. We usually model it independently of the cluster-scale halos as the centre of mass of a cluster-scale halo is not necessarily the same as the centre of the cD galaxy (Smith et al., 2005).

5.5.2.2 Galaxy-scale halos

In addition to the effect caused by the main cluster-scale halos, smaller lensing perturbations are introduced by cluster galaxies themselves (galaxy-galaxy lensing effect). These small-scale perturbers are thus modelled using galaxy-scale halos, and are essential to reproduce the observed patterns of multiple images (Kneib et al., 1996). However, so as not to over-complicate a lensing model, the optimal number of subhalos needs to be quantified. J07 use the following criteria for inclusion of a subhalo:

• First, the strong lensing deviation angleα(as per equation 5.5) is measured and compared to the spatial resolutionδof the observation (for the HST,δ ∼0.1⁰⁰). If the deflection angle is significantly increased at the position of the subhalo galaxy, that subhalo is included.

• For a cluster member, a subhalo is included if its Einstein radius, defined in equation 5.4, satisfiesθ_E > δ/µ, whereµis the magnification at the position of the member; otherwise the lensing contribution of the cluster galaxy is regarded as negligible and ignored.

• Non-cluster members are treated differently according to their position in projection rela- tive to the strong lensing (SL) region:

outside SL region: a subhalo is included at the cluster redshift ifθE > δ/µ, by rescaling its mass so its global lensing effect is preserved.

inside SL region: if its lensing effect is detectable, then the subhalo is included using a multi-plane lensing technique which takes into account gravitational field variations between the lens and the source, and the lens and observer (not covered here, but see e.g. Schneider, 2014, and references therein).

Once the set of subhalos to include in the model has been identified, certain assumptions need to be made in order to ensure the number of subhalo parameters is comparable with the number of available constraints. First, the subhalo position, ellipticity, and orientation are matched to their luminous counterparts. This assumption is based on the strong correlation between the light and mass profiles of elliptical galaxies in the field, observed by Koopmans et al. (2006).

Secondly, the number of subhalo parameters can be reduced by enforcing exact scaling relations between the luminosity of the associated galaxy and the subhalo mass. For PIEMD potentials, the subhalo properties are related to those of the galaxy at the cluster redshift (denoted by^∗) by the following:

σ₀ =σ₀^∗ L

L^∗ 1/4

; r_core=r^∗_core L

L^∗ 1/2

; r_cut=r^∗_cut L

L^∗ η

. (5.41)

The total mass of the subhalo can thus be defined in terms of the properties of the associated galaxy:

M = π

Gσ^∗₀²r_cut^∗ L L^∗

1/2+η

. (5.42)

The scaling of the velocity dispersion with total luminosity agrees with the Tully-Fisher rela- tion for spiral galaxies, and the Faber-Jackson relation for ellipticals. The scaling forr_cutis less

constrained. For example, withη = 0.5, r_cuthas a constant mass-to-light ratio, making it inde- pendent of the galaxy luminosity. However withη = 0.8, the mass-to-light ratio scales asL^0.3, which is similar to the scaling of the fundamental plane (Jørgensen et al., 1996; Natarajan and Kneib, 1997; Halkola et al., 2006).