---
title: "Dark Energy or Modified Gravity Interactive Lab | Sandbox Physics"
description: "Can the same expansion history come with a different law of gravity? Yes. Distances and the expansion rate only fix how fast space stretches."
canonical_url: "https://sandboxphysics.com/labs/dark-energy-modified-gravity/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/dark-energy-modified-gravity/"
last_modified: "2026-11-14"
---

# Dark Energy or Modified Gravity

**Can the same expansion history come with a different law of gravity?** Yes. Distances and the expansion rate only fix how fast space stretches. On that same expansion, gravity can still pull matter into clusters faster or slower and bend light by a different amount. Comparing measured clustering growth and lensing with what Einstein gravity predicts for the measured expansion tells the two ideas apart.

Change dark energy and two gravity parameters, then compare distance, clustering growth and lensing at a redshift you drag along the light cone.

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

Model assumptions

The universe is flat and contains matter and dark energy; radiation is left out. Dark energy has an equation of state that changes linearly with the scale factor, the form used by DESI. Gravity is a Horndeski scalar–tensor theory in the alpha parametrisation with gravitational waves at the speed of light, and each alpha is proportional to the dark-energy fraction, as in the hi\_class propto\_omega option. The kinetic coefficient is fixed at 0.1. Growth and lensing use the quasi-static response on scales well inside the sound horizon, with no screening and no scale dependence. Growth starts at redshift 30 with the same amplitude for every model and is normalised so that Einstein gravity with a cosmological constant has a present-day clustering amplitude of 0.81. A choice is rejected when the scalar kinetic term or its sound speed squared turns negative before today. Curves are model predictions computed in your browser. They are not observations, and no data are fitted.

## Three probes of the same universe

Model predictions computed in your browser; no observations are shown.

Comoving distance in units of the Hubble length $D_M\,H_0/c$

Clustering growth $f\sigma_8(z)$

Pull on matter and on light $\mu(z),\ \Sigma(z)$

Amber: this model Teal dashed: Einstein gravity, same expansion Gray dashed: cosmological constant Red band: unstable

How to read the response chart

The solid amber line is the pull on matter, which drives growth. The dotted amber line is the pull on light, which sets lensing. Both equal one in Einstein gravity.

## Test gravity against the expansion it lives in

**Background** Supernovae and baryon acoustic oscillations measure distances, and distances fix how fast the universe has expanded. The growth of galaxy clustering and the bending of light depend on that expansion and also on how strongly gravity pulls.

**Why it matters** DESI finds hints that dark energy changes with time. The same data could also come from a universe where gravity itself is modified, so the expansion alone cannot decide between the two.

- **Focus question**: If two universes expand identically, how can you tell whether their gravity differs?
- **One-sentence intuition**: Predict growth and lensing from the measured expansion with Einstein gravity; a mismatch points to new gravity, not new dark energy.

### Core mathematical model

#### Expansion rate of a flat universe with evolving dark energy

$$
\frac{H^2(a)}{H_0^2}=\Omega_m a^{-3}+(1-\Omega_m)\,a^{-3(1+w_0+w_a)}e^{-3w_a(1-a)},\qquad w(a)=w_0+w_a(1-a)
$$

Every distance in the Lab comes from this expansion rate. Gravity parameters do not appear here, so a given expansion can carry any of them.

#### Growth of matter clumps with an effective Newton constant

$$
\delta''+\Bigl(2+\frac{d\ln H}{d\ln a}\Bigr)\delta'=\frac{3}{2}\,\Omega_m(a)\,\mu(a)\,\delta,\qquad f\sigma_8=\frac{d\ln\delta}{d\ln a}\,\sigma_8(a)
$$

Primes are derivatives with respect to the logarithm of the scale factor. Einstein gravity has $\mu=1$; a larger $\mu$ pulls clumps together faster on the same expansion.

#### Quasi-static response in the alpha parametrisation

$$
\mu=\frac{1}{M_*^2}\Bigl[1+\frac{(\alpha_B+2\alpha_M)^2}{2\,D\,c_s^2}\Bigr],\qquad \Sigma=\frac{1}{M_*^2}\Bigl[1+\frac{(\alpha_B+2\alpha_M)(\alpha_B+\alpha_M)}{2\,D\,c_s^2}\Bigr]
$$

Braiding and a running Planck mass change both $\mu$, which drives growth, and $\Sigma$, which bends light. When $c_s^2<0$ the scalar perturbation grows without bound, so such a choice is rejected.

### Common difficulties

#### Distances do not measure gravity

**Typical misconception** If the distances match the cosmological-constant model, gravity must be Einstein’s.

**Better mental model** Distances only record the expansion. On the same expansion the braiding and running parameters still speed up or slow down clustering, which only growth and lensing data can reveal.

### Run the experiment

1. 01

   #### Keep the expansion, change gravity

   Choose Same expansion, stronger pull and look at the distance chart and the growth chart.

   **What to observe:** The distance curve sits on the cosmological-constant curve, yet the amber growth curve runs above the teal Einstein-gravity curve.

2. 02

   #### Split growth from lensing

   Choose Faster growth, weaker lensing and drag the probe shell toward low redshift.

   **What to observe:** Clusters pull galaxies in faster while the rays bend less, so the lensing-over-growth ratio drops below its Einstein-gravity value.

3. 03

   #### Let dark energy change instead

   Choose Changing dark energy, Einstein gravity.

   **What to observe:** Distances shift and growth changes too, but the amber and teal growth curves coincide: Einstein gravity on this expansion explains the growth.

4. 04

   #### Cross the phantom line

   Choose the crossing preset that fails alone, then the modified-gravity preset next to it, and watch the red band in the response chart.

   **What to observe:** With dark energy alone, $c_s^2$ turns negative once $w<-1$. Braiding removes the instability.

Physical conclusion

Expansion, growth and lensing must be checked together, and any modified gravity must keep its scalar perturbations stable.

## Related experiments and articles

Dark energy or modified gravity · Reading path

### Distances, growth, lensing and stability

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

1. [Article **Can the same expansion come with different gravity?**](https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/same-expansion-different-gravity/)
2. [Article **How distances record the expansion**](https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/how-distances-record-expansion/)
3. [Article **How growth and lensing feel gravity**](https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/how-growth-and-lensing-feel-gravity/)
4. [Article **Modified gravity in four functions, and when it is stable**](https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/modified-gravity-in-four-functions-and-stability/)
5. [Article **Recompute the expansion and growth grid**](https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/recompute-a-background-growth-grid/)
6. [Article **Reading the DESI 2024 modified-gravity paper**](https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/reading-desi-2024-modified-gravity/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

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