---
title: "Supernova Distance and Cosmic Expansion Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/supernova-distance-expansion/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/supernova-distance-expansion/"
last_modified: "2026-10-24"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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](https://github.com/PantheonPlusSH0ES/DataRelease/tree/c447f0fea703fcd0fff57de5000947b5ca81286b/Pantheon%2B_Data/4_DISTANCES_AND_COVAR) [Brout et al. (2022)](https://arxiv.org/abs/2202.04077) [Hogg (1999), distance measures](https://arxiv.org/abs/astro-ph/9905116)

[Read the discoveries behind these models →](https://sandboxphysics.com/stories/astronomy-gravity/)

Physics tutorial

## Compare the shape, then the brightness scale

**Background** A Type Ia supernova brightens and fades over weeks. Astronomers use its light-curve shape and colour to estimate how bright it would appear at a common distance.

**Why it matters** A single faint object could simply be intrinsically dim. Many standardized explosions over a wide range of redshifts let us compare the history of expansion.

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 misconception** A dim supernova directly measures the amount of dark energy.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

Physical conclusion

The evidence lies in how standardized brightness changes across redshift, together with its calibration and uncertainties.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)