Forward-modelling Milky Way Cepheids: selection effects and physical priors in the Gaia-HST calibration

Abstract

The advent of high-precision Gaia parallaxes for Milky Way Cepheids enables per cent-level calibration of the local distance ladder and $H_0$. We revisit the Milky Way Cepheid calibration from Gaia EDR3 parallaxes using a fully forward-modelled Bayesian framework that simultaneously infers the period--luminosity relation, the Gaia parallax zero-point offset, and individual stellar distances while explicitly incorporating the disk geometry of the Galaxy through the distance prior and the selection functions specified in two distinct HST SH0ES campaigns. We derive an analytic treatment of the detection probability that accounts for magnitude, parallax, period, and extinction cuts and reduces the selection treatment to a tractable integral over distance and sky position. Posterior predictive checks show that this generative model matches well the observed distributions of parallaxes, magnitudes, and periods. Modelling Galactic structure and survey truncation self-consistently in a Bayesian framework yields period--luminosity parameters that agree with the SH0ES maximum-likelihood values at the ${<}0.5\,σ$ level, a consequence, we show, of the small intrinsic scatter of the Cepheid period--luminosity relation. Adopting, as recently advocated, a uniform-in-volume prior without simultaneously accounting for selection leads to a ${\sim}\,0.05~\mathrm{mag}$ bias in the period--luminosity zero-point and posterior predictive distributions incompatible with the observed data; this shift is mostly driven by the omission of the selection model. A consistent Bayesian treatment of Galactic structure and selection effects reinforces the local distance-ladder determination of $H_0$, and hence the Hubble tension with early-Universe inferences.

Figures (10)

• Figure 1: Distributions of observed period ($\log P$), metallicity ($[{\rm O/H}]$), Gaia EDR3 parallax ($\varpi_{\rm obs}$), and Wesenheit apparent magnitude ($m^W_H$) for the C22 (red) and C27 (green) Cepheid samples. Dashed lines in the parallax panel show the approximate expected $\varpi_{\rm obs}$ distribution for a uniform-in-volume prior $\pi(d) \propto d^2$ between the minimum and maximum distances inferred from each campaign's observed parallaxes (assuming $\delta_\varpi = -0.01~\mathrm{mas}$). The dashed curves are normalised to the observed sample size of each campaign. The black dotted line shows the same prior applied to the combined C22 + C27 sample treated as a single population, as done by Hogas_2026 (shown with arbitrary normalisation).
• Figure 2: Galactocentric distribution of the MW Cepheids (C22: dark red circles; C27: olive triangles), with distances from parallax inversion assuming $\delta_\varpi = -0.01~\mathrm{mas}$. Left: face-on projection with Drimmel_2025 spiral arm traces (grey); the Sun and Galactic Centre (GC) positions are marked. The sample spans $R_{\rm GC} \approx 5$--$11~\mathrm{kpc}$, concentrated in the solar neighbourhood. Right: edge-on view, confirming that the Cepheids trace the Galactic thin disk with height above the midplane $|z_{\rm GC}| \lesssim 0.3~\mathrm{kpc}$.
• Figure 3: Directed acyclic graph of the forward model for a single MW campaign. Blue nodes denote parameters drawn from priors, grey nodes observed data, teal nodes deterministic functions of their parents, and purple nodes the likelihoods. The period hyperprior $(\mu_{\log P},\, \sigma_{\log P})$ connects only to the period likelihood, which evaluates the Gaussian period prior at the observed $\log P_i$ (second factor of Eq. \ref{['eq:C27_perstar']}); it does not enter the magnitude likelihood, since the delta-function marginalisation over the true period replaces it with the observed value. The metallicity hyperprior $(\mu_{[{\rm O/H}]},\, \sigma_{[{\rm O/H}]})$ connects to both a standalone metallicity likelihood and the magnitude likelihood. The observed $\ell_i$ and $b_i$ enter as fixed inputs with negligible measurement uncertainty. When multiple campaigns are modelled jointly, the period--luminosity parameters $(M^W_{H,1},\, b_W,\, Z_W)$ and parallax offset $\delta_\varpi$ are shared across all populations, while the period and metallicity hyperparameters and intrinsic scatter are independent per campaign. Selection modelling, described in \ref{['sec:selection_model']}, is not shown here.
• Figure 4: Marginalised posterior of the period--luminosity parameters $(M^W_{H,1},\, b_W,\, Z_W)$ and the parallax offset $\delta_\varpi$ for three model configurations: C22 and C27 (MW only) with a MW disk distance prior and selection (blue, filled), the fiducial model including the LMC and N4258 with the same MW disk prior and selection (red, filled), and all four populations with a uniform-in-volume MW distance prior, no selection, and a wide $\delta_\varpi$ prior, emulating Hogas_2026 (orange). The $0.7\sigma$ agreement in $M^W_{H,1}$ with the baseline SH0ES value translates to a shift in $H_0$ of less than $0.2~\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ (\ref{['sec:discussion_H0']}). The baseline SH0ES contours from Riess_2022 are shown in black, and the Riess_2021$\chi^2$ contours for C22 + C27 are shown in green. The baseline SH0ES (Riess_2022) constraints derive from the full three-rung fit (geometric anchors, Cepheids, and Type Ia supernovae), while the Riess_2021$\chi^2$ uses only the MW Cepheid parallaxes. For visual clarity, the SH0ES constraints are dashed in the panels along the diagonal. All contours show the $1\sigma$ and $2\sigma$ credible regions.
• Figure 5: Marginalised posterior of the intrinsic scatter $\sigma_{\rm int}$ for C22 (solid) and C27 (dashed), comparing the MW-only model with separate per-campaign scatters (blue), the fiducial model including the LMC and N4258 with separate scatters (red), and the MW-only model with a single shared $\sigma_{\rm int}$ for both campaigns (green dotted; $0.051 \pm 0.015~\mathrm{mag}$). Including the LMC and N4258 and inferring their joint scatter yields $\sigma_{\rm int} = 0.063 \pm 0.009~\mathrm{mag}$.