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Sandbox Physics
How the cosmic distance ladder acquires a scale

L2 · Cosmic distance ladder

Join Cepheids and supernovae in one distance model

Sandbox Physics ·

A distance ladder can be fitted as one system because every brightness equation contains a few shared quantities. A host distance appears in both its Cepheid and supernova equations. The common supernova luminosity then joins those hosts to the expansion sample. Writing these links explicitly reveals which measurement supplies the absolute scale.

Choose the eight quantities before forming a matrix

Use distance modulus in magnitudes. Our two supernova hosts are M101 and Mrk 1337, abbreviated M1337 in the released table. NGC4258 supplies the geometric anchor. Its fitted distance parameter is an offset from the fixed reference modulus 29.398; the host parameters are full distance moduli. Keeping this distinction avoids adding the reference distance twice.

Add a ten-day Cepheid absolute zero point, a period-slope correction, a standardized supernova absolute magnitude, a metallicity coefficient and a logarithmic expansion parameter. The slope correction is added to the released base slope−3.285. We express the Hubble constant numerically in kilometres per second per megaparsec before taking a logarithm.

q=(μ1,μ2,Δμa,M10W,δb,MB,γ,u)T,u=5log⁡10 ⁣[H0km s−1 Mpc−1]\boldsymbol q=(\mu_1,\mu_2,\Delta\mu_a,M^W_{10},\delta b,M_B,\gamma,u)^{\mathsf T},\qquad u=5\log_{10}\!\left[\frac{H_0}{\mathrm{km\,s^{-1}\,Mpc^{-1}}}\right]
The superscript T turns a row listing into a column. All eight quantities appear linearly in the released prepared observations. The physical Cepheid slope is the base slope plus its correction.

Reading: PantheonPlusSH0ES DataRelease · Riess et al.

Restore the period convention before interpreting a row

For a Cepheid define its period coordinate as the base-ten logarithm of period in days minus one. The released observation has already had the base-slope term removed. For NGC4258, it also has the reference anchor modulus removed. The remaining coefficients multiply a host modulus or anchor offset, the common zero point, the slope correction and relative metallicity.

An observed period is treated as fixed here, matching this numerical release. The compact subsystem retains the supplied covariance of prepared magnitudes. It does not redo photometry, selection, extinction-law fitting or the artificial-star crowding tests that produced those measurements. Those operations would require the original imaging analysis.

yij=mijW−b0xij−1i=aμa,ref=μi∗+M10W+δb xij+γZij,xij=log⁡10(Pij/day)−1y_{ij}=m^W_{ij}-b_0x_{ij}-\mathbf1_{i=a}\mu_{a,\mathrm{ref}}=\mu_i^*+M^W_{10}+\delta b\,x_{ij}+\gamma Z_{ij},\quad x_{ij}=\log_{10}(P_{ij}/\mathrm{day})-1
Use the full host modulus for mu-star in M101 or Mrk 1337, and the anchor offset for NGC4258. The indicator equals one only for anchor Cepheids. Add a noise term when calculating the likelihood.

Reading: PantheonPlusSH0ES DataRelease · Yuan et al.

A supernova carries a host distance to the Hubble flow

A calibrated supernova brightness is the host distance modulus plus the common standardized absolute magnitude. The selected five calibration rows come from two supernovae, SN2011fe in M101 and SN2006D in Mrk 1337. Multiple rows describe separate light-curve measurements; their covariance makes clear that five rows are not five independent supernovae.

For distant low-redshift supernovae write luminosity distance as a redshift-dependent velocity scale divided by the Hubble constant. The velocity scale includes the adopted low-redshift expansion correction and redshift treatment. Moving that known part to the observation side leaves a linear combination of supernova absolute magnitude and the logarithmic expansion parameter. The source matrix already provides these prepared Hubble-flow values. We keep their source row IDs and do not invent a catalogue-name match.

ycal=μi+MB,yHF=mB0−5log⁡10 ⁣[V(z)km s−1]−25=MB−uy_{\mathrm{cal}}=\mu_i+M_B,\qquad y_{\mathrm{HF}}=m_B^0-5\log_{10}\!\left[\frac{\mathcal V(z)}{\mathrm{km\,s^{-1}}}\right]-25=M_B-u
The standardized apparent peak magnitude is mB-zero. Luminosity distance in Mpc is the numerical velocity scale in km/s divided by the numerical Hubble constant in km/s/Mpc. This prepared relation uses the source authors’ redshift conventions.

Reading: Riess et al. · PantheonPlusSH0ES DataRelease

Worked example of a change in the adopted calibration

Suppose the adopted anchor modulus increases by 0.10 magnitude at fixed measured Cepheid brightness. The Cepheid zero point must decrease by 0.10. The two host moduli then increase by 0.10 to preserve their observed Cepheid values. Their calibrated supernova absolute magnitude decreases by 0.10. The fixed Hubble-flow combination requires the logarithmic Hubble parameter to decrease by 0.10 as well.

These shifts satisfy every unchanged brightness equation exactly. Only the adopted anchor value changes. The full-covariance numerical fit returns the same chi-squared and the following saved values. A constant chi-squared is expected for this coordinated calibration shift, so residual quality alone cannot discover an incorrect external zero point.

D′D=10Δμ/5,H0′H0=10−Δμ/5=0.9549926(Δμ=0.10)\frac{D^{\prime}}{D}=10^{\Delta\mu/5},\qquad \frac{H_0^{\prime}}{H_0}=10^{-\Delta\mu/5}=0.9549926\quad(\Delta\mu=0.10)
The distance grows by about 4.71 percent and the expansion rate falls by about 4.50 percent. The finite changes are reciprocal, so their percentages need not have exactly the same magnitude.
FitBaselineAnchor shifted by 0.10 mag
Hubble constant, km/s/Mpc66.304963.3207
Central 68% interval62.2356–70.640359.4345–67.4610
Chi-squared75.1994975.19949

Reading: PantheonPlusSH0ES DataRelease · Riess et al.

Without an anchor, an entire direction is invisible

Remove the geometric-anchor observation. Add the same arbitrary magnitude to both host moduli and the anchor modulus offset, subtract it from both stellar luminosity zero points, and subtract it from the logarithmic expansion parameter. The period and metallicity coefficients stay unchanged. Each Cepheid sum, each calibrated supernova sum and each Hubble-flow difference is preserved.

This is a scale degeneracy: several different absolute distance scales predict identical observations. The design matrix has rank seven for eight parameters. Choosing one anchor offset to be zero can draw a scene or solve relative distances, but it supplies an arbitrary convention. It does not turn the missing absolute distance into a measurement.

v=(1,1,1,−1,0,−1,0,−1)T,Av=0,q′=q+δv\boldsymbol v=(1,1,1,-1,0,-1,0,-1)^{\mathsf T},\qquad A\boldsymbol v=\boldsymbol0,\qquad \boldsymbol q^{\prime}=\boldsymbol q+\delta\boldsymbol v
A matrix row lists the coefficients multiplying the unknowns. The null direction v changes the parameters but gives zero change in every active predicted observation. The geometric-anchor row alone has a nonzero response to this direction.

Reading: Riess et al. · PantheonPlusSH0ES DataRelease

Try the observation

Find the missing absolute scale

  1. Choose Remove the absolute anchor. Read rank seven and the free Hubble scale, then watch the equally fitting inferred distances move.
  2. Compare the relative distance ratio with the baseline. Explain why an arbitrary absolute scale still leaves this ratio measurable.

Exercises and explained answers

Apply a 0.10 magnitude shift to every Cepheid, including the anchor Cepheids, with the geometric distance unchanged. Which parameters absorb it?

The common Cepheid zero point increases by 0.10. The host distances, supernova zero point and Hubble parameter stay fixed. A differential host shift instead changes host distances and then the supernova calibration.

An analyst removes the anchor but reports a narrow Hubble error after fixing its distance to an arbitrary number. What result should replace that claim?

Report rank seven, the free common-scale direction and identifiable relative quantities such as the difference of host moduli. A finite absolute Hubble interval would depend on the added distance convention or an explicitly stated external prior.

Why does increasing the Cepheid count not repair the missing scale direction?

Every added Cepheid equation has the same invariance: a distance increase can be canceled by a luminosity-zero-point decrease. More such equations improve relative distances and slopes, but only an absolute calibration responds to the null direction.

Sources and calculation scope

  1. Riess et al., A Comprehensive Measurement of the Local Value of the Hubble Constant, arXiv:2112.04510 v 3 (2022)

    Sections 2.1 and 4 define the joint model and calibrations. Section 5, Table 5 and Figures 14–16 report the full fit and its correlations; sections 6–7 discuss variants and comparisons. All paper-specific claims here refer to this fixed version.

  2. PantheonPlusSH0ES DataRelease, commit c447f0fea703fcd0fff57de5000947b5ca81286b

    The ally, alll and allc FITS arrays contain 3492 prepared observations,47 parameter columns and the full covariance. The public tutorial retains 87 observations and every nonzero model column they use. The table 2.README explicitly warns against refitting from its diagonal errors.

  3. Yuan et al., A New Cepheid Distance to the Maser-Host Galaxy NGC4258, arXiv:2203.06681 v 1 (2022)

    Sections 2–4 describe HST observations, Cepheid selection and period–luminosity fitting. The discovery catalogue and the infrared-selected calibration sample have different sizes; the matrix release contains 443 NGC4258 Cepheids.

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