We know that intrinsic scatter has a color dependence!
In this work we will consider 4 models of intrinsic scatter:
Random coherent (achromatic) scatter
Unrealistic
The G10 model (Guy et al. 2010): $\sim 70\%$ achromatic / $\sim 30\%$ chromatic
Historically used (Pantheon, Pantheon+)
The C11 model (Chotard et al. 2011): $\sim 30\%$ achromatic / $\sim 70\%$ chromatic
Historically used (Pantheon, Pantheon+)
The BS21 model (Brout et al. 2021) with its parameters fitted in Popovic et al. 2023 (P23): dust-based model
Currently favored by data (DES 5 year)
Why is SNe Ia intrinsic scatter a concern?
Intrinsic scatter is the most important systematic for the DES 5 year analysis of the dark energy equation of state parameter $w$ (Vincenzi et al. 2024).
Rubin-LSST Simulations
We used the SNANA software (Kessler et al. 2009) to simulate the 10 years of the Rubin-LSST survey!
Survey parameters from the LSST survey simulation OpSim
SN Ia model: Spectra model (SALT) + intrinsic scattering model + SALT parameter distributions
Host catalog: The Uchuu UniverseMachine N-body simulation
Simulation of the SN Ia hosts
8 mocks cut from the (2 Gpc $h^{-1}$)$^3$ box of the Uchuu UniverseMachine galaxy catalog (Ishiyama et al. 2021,
Aung et al. 2023) $\Rightarrow$ Density and velocity field corresponding to Planck15 cosmology
We simulated SN Ia parameters correlations with their host properties:
SN Ia rate - Host mass correlation from Wiseman et al. 2021
SN Ia parameters - Host mass correlation from Popovic et al. 2021
${\color{red}M_0}$, ${\color{red}\alpha}$, ${\color{red}\beta}$, ${\color{red}\gamma}$ and ${\color{red}\sigma_\mathrm{int}}$ will be
fitted along $f\sigma_8$
Running an extra-large simulation ($\sim40\times$ LSST) and fitting the hubble diagram
Binning over the parameters $p=\left\{z_\mathrm{obs}, x_1, c, M_\mathrm{host}\right\}$
Computing the correction in each cell $\delta_\mathrm{corr.} = \left<\mu_\mathrm{obs} - \mu_\mathrm{fid}\right>_\mathrm{cell}$
Interpolate over the cells to obtain $\delta_\text{corr.}(p)$
${\color{red}\alpha}$, ${\color{red}\beta}$, ${\color{red}\gamma}$ and ${\color{red}\sigma_\mathrm{int}}$ are fitted prior to
$f\sigma_8$
The maximum likelihood method
The maximum likelihood mehod
The Maximum likelihood method is implemented within the
package (Ravoux, Carreres et al.
2025)
We want to maximize the likelihood function: $$\mathcal{L}(f\sigma_8; \hat{v}) \propto
\left(2\pi\right)^{-\frac{N}{2}}\left|\text{C}(f\sigma_8)\right|^{-\frac{1}{2}}\exp\left(-\frac{1}{2}\boldsymbol{\hat{v}}^T\text{C}(f\sigma_8)^{-1}\boldsymbol{\hat{v}}\right)$$
The maximum likelihood mehod: estimated velocities
The data vector are the estimated velocities: $$\hat{v} = -\frac{c\ln10}{5}\left(\frac{(1 + z)c}{H(z)r(z)} - 1\right)^{-1}\Delta\mu \
\text{ where } \ \Delta\mu = \mu_\mathrm{obs} - \mu_\mathrm{model}(z_\mathrm{obs})$$
The velocity field is approximated as a Gaussian random field
The covariance of the velocity field is: $$\langle v_i(x_i) v_j(x_j)\rangle = \text{C}_{ij}^{vv}\propto ({\color{red}f\sigma_8})^2
\int_{k_\mathrm{min}}^{k_\mathrm{max}} P(k) W_{ij}(k; \mathbf{r}_i, \mathbf{r}_j) {\rm d}k $$
The observational covariance is: $$C^{vv, \mathrm{obs}} = \left[\frac{c\ln10}{5}\left(\frac{(1 + z)c}{H(z)r(z)} -
1\right)^{-1}\right]^2\text{diag}\left[\sigma_\mu^2\right]$$
The total covariance is: $\text{C} = \text{C}^{vv}({\color{red}f\sigma_8}) + C^{vv, \mathrm{obs}} + {\color{red}\sigma_v}^2
\mathbf{I}$
The $\sigma_u$ redshift space parameter
Position are evaluated using $z_\mathrm{obs} \ \Rightarrow$ Redshift Space Distorsion
Empirical damping introduced in Koda et al. 2014: $D_u = \text{sinc}(k\sigma_u)$