Skip to main content
Sandbox Physics

Supernova Distances and the Distance Ladder

How does a small distance zero-point error reach the whole universe? A farther adopted anchor makes the calibrated stars and supernovae intrinsically brighter. Their inferred distances grow together, so the expansion rate inferred from the same redshifts falls. A zero-point shift common to both ends can cancel; a difference between them propagates.

Drag the adopted geometric anchor and watch inferred galaxy distances move together. Fit a frozen correlated SH0ES subsystem, or switch to forty Pantheon+ supernovae to study expansion with a free brightness zero point.

Interactive modelSupernova Distances and the Distance Ladder
Conditional Hubble constant  [km s−1 Mpc−1]\;[\mathrm{km\,s^{-1}\,Mpc^{-1}}]H0=66.3H_0=66.3
Central 68% interval[62.2, 70.6][62.2,\,70.6]
Shared host-distance correlation0.6350.635
Identified parameter directions8/88/8

The same observations, a different adopted scale

Measured source vectors remain frozen. Points: adopted Wesenheit values restored from the frozen prepared vector, with the fitted metallicity term removed. Error bars: marginal measurement standard deviations. Lines: the joint fit. The fit uses the full selected covariance.

NGC4258M101Mrk1337
A common stellar ruler in three galaxiesW−γZ=μ+MW+b[log⁡10(P/day)−1]W-\gamma Z=\mu+M_W+b[\log_{10}(P/\mathrm{day})-1]
Conditional Hubble-constant distributionu=5log⁡10H0u=5\log_{10}H_0

One geometric anchor identifies all eight parameter directions. The interval belongs to this small conditional subsystem.

Covariance and fitted stellar parameters

Fitted physical period–luminosity slope:−3.52 mag dex−1-3.52\,\mathrm{mag\,dex^{-1}}

Shared host-modulus covariance matrix:

Cov⁡(μ1,μ2)=(0.01320.009950.009950.0186) mag2\operatorname{Cov}(\mu_1,\mu_2)=\begin{pmatrix}0.0132&0.00995\\0.00995&0.0186\end{pmatrix}\,\mathrm{mag^2}

The released metallicity summary is retained by default; disabling it is a sensitivity test. The broad slope-correction regularizer is also retained. Cepheid Table2 provides IDs and periods here, while all fit uncertainties come from the released matrix covariance. A fresh selection or photometric reformulation would require its own uncertainty study.

Pinned source matrices and source warning · Riess et al. 2022, v3

How a calibration reaches the Hubble flow

BackgroundA geometric distance calibrates the luminosity of Cepheids in one galaxy. Cepheids in supernova hosts then calibrate supernova luminosity, which supplies distances in a farther expansion sample.

Why it mattersAll galaxies that use the same anchor inherit its uncertainty. Distinguishing a common brightness shift from a difference between samples tells us which errors affect the absolute distance scale.

Focus question
Which zero-point shift changes the inferred expansion rate?
One-sentence intuition
A differential calibration shift changes distances; a common Cepheid shift can be absorbed by the common Cepheid luminosity zero point.

Core mathematical model

Distance modulus converts brightness to distance

μ=m−M=5log⁡10(D/10 pc)\mu=m-M=5\log_{10}(D/10\,\mathrm{pc})

At a fixed observed brightness, assuming a brighter intrinsic source places it farther away. The same change propagates through the calibrated supernova luminosity.

A zero-point change rescales expansion

H0′/H0=10−Δμ/5H_0^{\prime}/H_0=10^{-\Delta\mu/5}

Increasing the adopted anchor modulus by 0.10 magnitude grows the distance scale by about 4.71 percent and lowers the Hubble value by about 4.50 percent. The observed redshifts stay fixed.

Fit the correlated brightnesses together

χ2=(y−Aq)TC−1(y−Aq)\chi^2=(\boldsymbol y-A\boldsymbol q)^{\mathsf T}C^{-1}(\boldsymbol y-A\boldsymbol q)

The selected full covariance describes which fluctuations are shared. The linear parameters include host distances, Cepheid and supernova luminosities, slope, metallicity and the logarithmic Hubble value.

Common difficulties

More stars do not supply a missing absolute anchor

Typical misconceptionEnough relative brightness measurements must eventually fix the absolute scale.

Better mental modelDistance and intrinsic luminosity can change together without changing any observed brightness. A geometric distance or another explicit absolute calibration breaks this exact degeneracy.

Run the experiment

  1. 01

    Predict the anchor response

    Keep Distance ladder selected. Drag the gold NGC4258 anchor toward a larger adopted distance; predict the Hubble change first.

    What to observe: The inferred galaxy distances increase together and the Hubble value falls. Recorded Cepheid periods and brightnesses remain fixed.
  2. 02

    Compare common and differential shifts

    Compare A host-only zero-point shift with A common shift cancels. Both apply 0.10 magnitude, but to different Cepheid groups.

    What to observe: The host-only shift changes the inferred distance scale. The common shift moves the Cepheid zero point and preserves host distances and the Hubble value.
  3. 03

    Remove the absolute scale

    Choose Remove the absolute anchor, then compare the relative host-distance ratio with the baseline.

    What to observe: The rank falls from eight to seven. Equally fitting distance scales move in the scene; relative distance remains measurable and the absolute Hubble interval is free.

Related experiments and articles

What this page links to directly

You are hereSupernova Distances and the Distance Ladder

Experiments

Articles

Continue exploring

Measuring the sky

What do positions, colors, interference and radio delays reveal about the sky?

  1. LabObserving Cadence and AliasesMove exposures and compare the rhythms that survive the sampling.
  2. You are hereSupernova Distances and the Distance LadderMove a geometric distance anchor and follow its shared calibration error.
  3. LabStellar Parallax and Proper MotionChange the stellar distance and follow its annual motion.
  4. LabCrowded-star PhotometryMove the aperture and compare the light admitted from neighbouring stars.
  5. LabStellar Absorption LinesChange the gas temperature and velocity, then compare the absorption lines.
  6. LabGalaxy Colors and Photometric RedshiftCompare redshift, dust and stellar-population effects in the same five bands.
  7. LabStellar Interferometry and Sparse ImagingMove a telescope and compare images fitted to the same sparse signals.
  8. LabInterstellar Radio PropagationChange the electron column and field, then follow the arriving radio pulse.

Calibrating the cosmic distance ladder · Reading path

Calibrating the cosmic distance ladder

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleWhy a nearby distance changes the scale of the universe
  2. ArticleFrom inverse-square light to a calibrated stellar distance
  3. ArticleJoin Cepheids and supernovae in one distance model
  4. ArticleFit shared distance errors with a covariance matrix
  5. ArticleReproduce a correlated SH0ES distance-ladder subsystem
  6. ArticleRead the Hubble calibration and its shared uncertainties

Cosmology from expansion to inference · Reading path

Expansion, sound, sky statistics and inference

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleFollowing light through an expanding universe
1 more group

Astronomy and gravity · Reading path

Measuring the history of expansion

Articles on this path that use this experiment, followed by the other experiments they suggest.

  1. ArticleDistant supernovae showed that cosmic expansion had accelerated
Something broken, confusing, or missing on this page?Tell us
What kind of feedback?

Sent with this page’s address, its Lab ID, the site language and your device type. Please leave out personal details. Feedback is kept for up to 12 months.Privacy policy