Notes on Host matching

Finding host around pointing

Host are matched to a pointing if they have an angular separation of less than the LSST fied radius (\(\Theta_\mathrm{Rubin} =\) 1.75 deg). Mathematically we can write, for a host of angular coordinates (RA, Dec) = \((\theta,\phi)\) at an angular separation \(\Theta_\mathrm{Rubin}\) from the field of angular coordinates \((\theta_f,\phi_f)\)

\[\cos\Theta_\mathrm{Rubin} = \sin\phi_f \sin\phi + \cos\theta_f\cos\theta \cos\left(\theta - \theta_f\right).\]

At second order (first order is null) this expression can be write

\[1 = \frac{\Delta\phi^2}{\Theta_\mathrm{Rubin}^2} + \frac{\Delta\theta^2}{\Theta_\mathrm{Rubin}^2}\frac{1 + \cos2\phi_f}{2}\]

where \(\Delta\phi = \phi - \phi_f\) and \(\Delta\theta = \theta - \theta_f\). It corresponds to an ellipse where the axes along the RA angle is changing with Dec angle. All host that are lying inside this ellipse are matched with the corresponding field.

Applying a SNANA WGT MAP

Host catalog can be huge and over-use memory when openned in a code. For example, a Uchuu mock up to a redshift \(z \sim 0.17\) contains more than 10 milions hosts. However, their mass distribution doesn’t match the one of SN Ia hosts. As we can see on Fig 1, there is a lot of small masses host that have few probability to be a SN Ia host.

Figure 1: Distribution of host masses in an example Uchuu mock.

If we define \(H\) the event beeing a SN Ia host, the target distribution that we want to match is the mass distribution among SN Ia host \(\mathcal{P}(M | H)\) that can be written

\[\mathcal{P}(M | H) \propto \mathcal{P}(H | M) \mathcal{P}(M),\]

where \(\mathcal{P}(M)\) is the distribution of mass in the host catalog and \(\mathcal{P}(H | M)\) is given by the WGTMAP. We plot in Fig 2 these 3 distributions for the WGTMAP of DES using the Wiseman 2021 distribution for SN Ia host masses.

Figure 2: Distribution of host masses, probability of being a SN Ia host knowing the host mass and distribution of probability to be a SN Ia host.

In order to match the targeted distribution OpSimSummaryV2, the host are binned and then the count in the bin that corresponds to the peak of the target distribution \(P_\mathrm{peak}\) is take as the reference \(N_\mathrm{peak}\) and the number of host to draw from each bin \(b\) is computed as

\[N_b = \frac{P_b}{P_\mathrm{peak}} N_\mathrm{peak},\]

where \(P_b\) is the targeted distribution evalueted at the center of bin \(b\).

It can happens that \(N_b\) is larger than the available number of host \(N_\mathrm{available}\) in the bin \(b\). A possible way to handle that is to force \(N_b = N_\mathrm{available}\), but it can lead to transform the shape of the distribution.

To correct for that the \(N_b\) are multiplied by a factor \(C\). This factor is computed as the minimum of the ratio \(N_\mathrm{available} / N_b\) for the bins that satisfy the criterium to be inside the interval [5%, 95%] of the target distribution’ cdf.