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

This paper presents a fully forward-modelled Bayesian framework for calibrating Milky Way Cepheids using Gaia EDR3 data, demonstrating that explicitly accounting for Galactic geometry and survey selection effects is essential to avoid biases in the period-luminosity relation and robustly support the local distance-ladder determination of $H_0$.

The Gist

Here is an explanation of the paper, translated from complex astrophysics into everyday language using analogies.

The Big Picture: The Cosmic Ruler is Broken

Imagine you are trying to measure the size of the entire universe. To do this, astronomers use a "Cosmic Distance Ladder." You start with a ruler you can hold in your hand (nearby stars), use that to measure a room (nearby galaxies), and then use that to measure the whole house (the entire universe).

The problem is that the two ends of this ladder don't agree.

• The "Local" measurement (using stars and supernovae) says the universe is expanding fast.
• The "Early Universe" measurement (using the leftover radiation from the Big Bang) says it's expanding slower.

This disagreement is called the Hubble Tension. It's like two surveyors measuring the same field and getting different results. One says it's 100 acres; the other says 120. Someone is making a mistake, or the universe is playing a trick on us.

The Suspect: The "Local" Ruler (Cepheid Stars)

The "Local" measurement relies heavily on a specific type of star called a Cepheid. Think of these stars as cosmic lighthouses. They pulse (blink) at a rate that tells us exactly how bright they should be. By comparing how bright they should be to how dim they look to us, we can calculate their distance.

The paper focuses on the first rung of the ladder: measuring the distances to Cepheids right here in our own Milky Way galaxy using the Gaia satellite (a space telescope that maps star positions).

The Problem: The "Selection" Bias

The authors argue that previous studies made a mistake in how they counted these stars. They treated the sample of stars they found as if it were a perfectly random bucket of marbles drawn from a giant jar.

But it wasn't random. It was more like fishing with a net.

• The Net: The Hubble Space Telescope and Gaia satellite have limits. They can't see stars that are too dim, too bright (they get "saturated"), or hidden behind too much cosmic dust.
• The Bias: Because of these limits, the "net" caught mostly certain types of stars and missed others. If you try to calculate the average size of all fish in the ocean by only looking at the ones that fit through your net, you will get the wrong answer.

Previous studies (specifically one by Högas & Mörtsell, or HM26) tried to fix the math by assuming the stars were distributed evenly in space (like a uniform cloud). But they forgot to account for the net (the selection criteria). They assumed they could see every star up to a certain distance, but in reality, the "net" cut off the sample in a very specific, messy way.

The Solution: "Forward-Modelling"

The authors built a new, smarter way to do the math. Instead of just looking at the stars they found and guessing, they built a simulation (a "forward model").

The Analogy: The Detective's Simulation
Imagine a detective trying to figure out how many people live in a city.

1. Old Way: The detective stands on a street corner, counts the people walking by, and assumes that's a fair sample of the whole city.
2. New Way (This Paper): The detective builds a computer simulation of the entire city. They program the simulation with the rules of the city (where people live, how tall the buildings are). Then, they program a "virtual camera" into the simulation that has the exact same blind spots and limits as the real telescope (it can't see through fog, it can't see things too far away).
3. The Result: The detective runs the simulation. If the "virtual camera" sees the same pattern of people as the real camera, the simulation is correct. If the simulation sees a different pattern, the detective knows their assumptions about the city are wrong.

The authors did this with the stars. They simulated the Milky Way, applied the exact "rules" of the Hubble and Gaia telescopes (the selection effects), and saw what stars should have been caught. Then they compared that to the real data.

The Key Findings

1. The "Net" Matters: When they ignored the "net" (the selection limits) and just assumed a uniform distribution, they got a result that suggested the universe is expanding slower (reducing the tension). But this was a fake result caused by bad math.
2. The Real Result: When they correctly accounted for the "net," their results matched the original, high-precision measurements from the SH0ES team.
3. The Tension Remains: Because their new, rigorous math agrees with the old "Local" measurement, the Hubble Tension is still there. The universe is still expanding faster than the Big Bang theory predicts. The "mistake" wasn't in the data; it was in the previous attempt to "fix" the data by ignoring the telescope's limitations.

Why This Matters

This paper is a lesson in honesty with data. It shows that when you are measuring the universe, you cannot just look at the numbers you have; you have to understand why you have those numbers and why you are missing the others.

By building a model that respects the "rules of the game" (the telescope's limits and the shape of our galaxy), the authors confirmed that the mystery of the expanding universe is real, not a mathematical glitch. The universe is still behaving strangely, and we need new physics to explain it.

Technical Summary

Here is a detailed technical summary of the paper "Forward-modelling Milky Way Cepheids: selection effects and physical priors in the Gaia–HST calibration."

1. Problem Statement

The paper addresses the Hubble tension, the $\sim5\sigma$ discrepancy between the local distance-ladder determination of the Hubble constant ($H_0$) and early-Universe inferences from the Cosmic Microwave Background (CMB). The sharpest local constraint on $H_0$ relies on the calibration of Cepheid variable stars in the Milky Way (MW) using Gaia EDR3 parallaxes and Hubble Space Telescope (HST) photometry.

Recent work by Högas & Mörtsell (2026, HM26) claimed that adopting a uniform-in-volume distance prior ($\pi(d) \propto d^2$) for MW Cepheids reduces the Hubble tension. However, this approach neglected the sample selection function (the specific criteria used to select stars for observation). The authors argue that ignoring selection effects while specifying a physical prior leads to biased inferences, generating mock populations inconsistent with the observed data. The paper aims to rigorously test these assumptions by constructing a fully forward-modeled Bayesian framework that self-consistently incorporates Galactic geometry, selection effects, and physical priors.

2. Methodology

The authors developed a Bayesian forward model that jointly infers the Period-Luminosity (PL) relation parameters, the Gaia parallax zero-point offset ($\delta\varpi$), and individual stellar distances.

• Data: The analysis uses 66 MW Cepheids from two distinct HST campaigns (Cycle 22 and Cycle 27) with Gaia EDR3 parallaxes, supplemented by geometric anchors from the Large Magellanic Cloud (LMC) and the galaxy N4258.
  • C22: Targets long-period Cepheids ($P > 8$ days) up to $\sim6.6$ kpc.
  • C27: Targets nearby, high-parallax Cepheids ($\varpi_{phot} > 0.8$ mas) to break degeneracies.
• Physical Priors (Distance): Instead of a uniform-in-volume prior, the model adopts a thin-disk geometry prior for MW Cepheids:
  $$\pi(d, \ell, b) \propto d^2 \exp\left(-\frac{R_{GC}}{R_d}\right) \exp\left(-\frac{|z|}{z_d}\right)$$
  where $R_{GC}$ is the Galactocentric radius, $z$ is the height above the plane, and $R_d, z_d$ are scale lengths. This reflects the actual distribution of Cepheids in the Galaxy.
• Selection Modelling: The core innovation is the explicit analytic treatment of the selection function. The authors derive the detection probability $p(S=1|\Lambda)$ by integrating over all latent variables (distance, period, metallicity) and observed quantities, accounting for cuts on:
  • Period ($P$)
  • Apparent magnitude ($m_W$)
  • Parallax ($\varpi$)
  • Extinction ($A_H$)
  • The selection function is modeled as a smooth transition (using the cumulative normal distribution $\Phi$) rather than a hard cut, allowing for an analytic reduction of the high-dimensional integral to a tractable 2D integral over distance and sky position.
• Inference Engine: The model is implemented in JAX and sampled using the No-U-Turn Sampler (NUTS) via numpyro. It includes 91 sampled parameters, including global PL parameters, parallax offset, intrinsic scatters, and per-star distances.

3. Key Contributions

1. First Forward Model with Selection: This is the first Bayesian forward model of the MW Cepheid population that explicitly accounts for the sample selection function and Galactic disk geometry simultaneously.
2. Analytic Selection Treatment: The authors derived an analytic method to compute the detection probability for complex, multi-dimensional selection cuts (magnitude, parallax, period, extinction), reducing a 9-dimensional integral to a numerically tractable form involving multivariate normal CDFs.
3. Debunking the "Reduced Tension" Claim: The paper demonstrates that the reduced Hubble tension reported by HM26 is an artefact of neglecting selection effects. The authors show that a uniform-in-volume prior without selection modeling biases the results, whereas a consistent treatment of both geometry and selection yields results consistent with the standard SH0ES analysis.
4. Validation: The model passes rigorous Posterior Predictive Checks (PPC), successfully reproducing the observed distributions of parallaxes, magnitudes, and periods, whereas models ignoring selection fail these checks.

4. Key Results

• Period-Luminosity Zero-Point: The fiducial model (disk prior + selection) yields a zero-point of $M_W^H,1 = -5.909 \pm 0.022$ mag. This agrees with the baseline SH0ES value ($M_W^H,1 \approx -5.894$) within $0.7\sigma$.
• Gaia Parallax Offset: The inferred zero-point offset is $\delta\varpi = -12.4 \pm 5.3$ $\mu$as, consistent with independent validations and previous SH0ES analyses.
• Impact of Selection Modelling:
  • Neglecting Selection: When selection is ignored and a uniform-in-volume prior is used (mimicking HM26), the zero-point shifts to brighter values ($\sim -5.95$ to $-5.98$ mag) and $\delta\varpi$ becomes more negative ($\sim -26$ to $-36$ $\mu$as). This shift corresponds to a lower $H_0$, artificially reducing the Hubble tension.
  • Role of the Prior: Once selection is correctly modeled, the specific choice of distance prior (disk vs. uniform-in-volume) has a negligible effect ($<0.002$ mag shift). The bias arises primarily from the omission of the selection model, not the prior itself.
• Intrinsic Scatter: The model infers distinct intrinsic scatters for the two campaigns: $\sigma_{C22} \approx 0.071$ mag and $\sigma_{C27} \approx 0.040$ mag, likely due to residual extinction uncertainties in the more distant C22 sample.
• Hubble Constant Implications: Replacing the SH0ES Gaia constraint with the authors' inferred zero-point results in $H_0 = 73.12 \pm 1.08$ km s$^{-1}$ Mpc$^{-1}$. This shift is negligible ($\sim0.1\sigma$), confirming that the Hubble tension remains at $\sim5\sigma$.

5. Significance

• Statistical Rigor: The paper establishes that percent-level precision in the cosmic distance ladder requires a principled treatment of selection effects. Ignoring selection leads to biased generative models that produce posterior predictive distributions incompatible with observed data.
• Resolution of Discrepancies: It clarifies that the "mitigation" of the Hubble tension claimed by HM26 was not due to new physical insights but to a statistical error (missing selection modeling).
• Future Framework: This work sets a new standard for distance-ladder analyses. It provides a template for future Bayesian forward modeling of other distance indicators (e.g., Type Ia supernovae, TRGB) and suggests that simulation-based inference or rigorous forward modeling is essential for upcoming high-precision cosmological surveys.
• Validation of SH0ES: The results reinforce the robustness of the SH0ES distance ladder and the persistence of the Hubble tension, suggesting that the tension is likely physical rather than a result of statistical mishandling of the local Cepheid calibration.