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Recent studies have shown that the observed colour distributions of Type IaSNe (SNIa) are well-described by a combination of distributions from dust andintrinsic colour. Here we present a new forward-modeling fitting method(Dust2Dust) to measure the parent dust and colour distributions, includingtheir dependence on host-galaxy mass. At each fit step, the SNIa selectionefficiency is determined from a large simulated sample that is re-weighted toreflect the proposed distributions. We use five separate metrics to constrainthe Dust2Dust parameters: distribution of fitted light-curve colour $c$,cosmological residual trends with $c$, cosmological residual scatter with $c$,fitted colour-luminosity relationship $\beta_{\rm SALT2}$, and intrinsicscatter $\sigma_{\rm int}$. Using the Pantheon+ data sample, we present resultsfor a Dust2Dust fit that includes 4 parameters describing intrinsic colourvariations and 8 parameters describing dust. Furthermore, we propagate theDust2Dust parameter uncertainties and covariance to the dark energyequation-of-state $w$ and Hubble constant H$_0$: we find $\sigma_w = 0.005$ and$\sigma_{\textrm{H}_0} = 0.145~$km/s/Mpc. The Dust2Dust code is publicallyavailable.

The Pantheon+ Analysis: Forward-Modeling the Dust and Intrinsic Colour  Distributions of Type Ia Supernovae, and Quantifying their Impact on  Cosmological Inferences