F-X2 · Forty distant explosions

Supernova Distance and Cosmic Expansion

How do distant supernovae reveal the history of cosmic expansion? After correcting their light curves and colours, Type Ia supernovae let astronomers compare distance with redshift. The distant events appeared fainter than expected in a decelerating universe, pointing to a recent period of accelerated expansion.

Select a real supernova on the sky and compare its standardized brightness with an expanding-universe model. Change the matter density, then fit a small public sample with its correlated uncertainties.

Present deceleration parameter $0$

Covariance-weighted mismatch $0$

Selected redshift $0$

Brightness residual $0$

The forty observations are a deterministic, redshift-spaced subset of the public Pantheon+ release, with one light curve per supernova and no Cepheid calibrators. Fits use the corresponding full statistical-plus-systematic covariance submatrix. The model assumes a spatially flat universe with pressureless matter and a cosmological constant; radiation is omitted. The Hubble constant is fixed to seventy kilometres per second per megaparsec, and a free brightness offset absorbs the absolute-luminosity calibration. This subset is for teaching and does not reproduce the collaboration’s full cosmological analysis or measure the Hubble constant. Sky coordinates are measured catalogue positions. The selected ring areas compare model and observed standardized brightness, normalized per source; they do not show angular size. Background stars are visual guides.

Pantheon+ · measured catalogue subset

The distance curve bends with the expansion history

Points: forty standardized observed magnitudes. Teal line: chosen flat-universe model. Amber line: matter-only comparison with its own fitted brightness offset. Lines join predictions at the catalogue’s two redshifts for each source. Error bars show marginal uncertainties; fitting uses all covariance terms.

z,\quad m_{B,\mathrm{corr}}/\mathrm{mag}

Residuals make a small bend visible

\Delta m=m_{B,\mathrm{corr}}-m_{B,\mathrm{model}}

Read the model and original data

d_L=(1+z_{\mathrm{hel}})\frac{c}{H_0}\int_0^{z_{\mathrm{HD}}}\frac{\mathrm dz}{\sqrt{\Omega_m(1+z)^3+1-\Omega_m}}

$\chi^2=\boldsymbol r^{\mathsf T}C^{-1}\boldsymbol r$

The forty-source subset is selected for redshift coverage. Its best fit is a teaching example, with a freely profiled brightness offset. It supplies no published confidence interval or full-sample cosmological result.

Pantheon+ data and covariance Brout et al. (2022)Hogg (1999), distance measures

Read the discoveries behind these models →

Start with the essentials

Focus question: Can a change in the brightness zero point imitate every expansion history?
One-sentence intuition: The zero point shifts the whole distance curve; the matter fraction changes its shape.

Core mathematical model

Distance turns into apparent brightness

m_{B,\mathrm{corr}}=5\log_{10}(d_L/\mathrm{Mpc})+25+\mathcal M

The corrected apparent magnitude increases when an object is fainter. Luminosity distance is in megaparsecs; the offset absorbs the unknown absolute brightness and reference distance scale.

The sign of present acceleration

q_0=\frac32\Omega_m-1

In this flat matter-plus-cosmological-constant model, a negative deceleration parameter means acceleration today.

Common difficulties

One number does not locate dark energy

Typical misconceptionA dim supernova directly measures the amount of dark energy.

Better mental modelThe fit compares many objects with a specified expansion model. Light-curve calibration, dust, selection effects and correlated errors all enter the inference.

Run the experiment

01
Change the expansion history
Choose Matter-only universe, then Accelerating universe.
What to observe: The difference grows with redshift; one vertical shift cannot make the curves identical.
02
Choose a distant explosion
Tap a coloured source on the sky or a point in the Hubble diagram.
What to observe: The same object lights up in the sky, distance plot and residual plot.
03
Fit the selected public sample
Press Fit this sample and compare the matter fraction with the two presets.
What to observe: The calculation profiles the brightness offset using the full covariance submatrix. The result belongs to these forty observations.