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Sandbox Physics

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 modelDark Energy or Modified Gravity
Distance to the shell, relative to ΛCDM\Lambda\mathrm{CDM}11
Clustering growth fσ8f\sigma_8 over Einstein gravity, same expansion1.061.06
Lensing over growth, EGE_G0.428  (GR 0.409)0.428\;(\mathrm{GR}\ 0.409)
Scalar perturbationsstable\text{stable}

Three probes of the same universe

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

Comoving distance in units of the Hubble length DM H0/cD_M\,H_0/c
Clustering growth fσ8(z)f\sigma_8(z)
Pull on matter and on light μ(z), Σ(z)\mu(z),\ \Sigma(z)

Amber: this modelTeal dashed: Einstein gravity, same expansionGray dashed: cosmological constantRed 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

BackgroundSupernovae 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 mattersDESI 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

H2(a)H02=Ωma−3+(1−Ωm) a−3(1+w0+wa)e−3wa(1−a),w(a)=w0+wa(1−a)\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

δ′′+(2+dln⁡Hdln⁡a)δ′=32 Ωm(a) μ(a) δ,fσ8=dln⁡δdln⁡a σ8(a)\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 μ=1\mu=1; a larger μ\mu pulls clumps together faster on the same expansion.

Quasi-static response in the alpha parametrisation

μ=1M∗2[1+(αB+2αM)22 D cs2],Σ=1M∗2[1+(αB+2αM)(αB+αM)2 D cs2]\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 μ\mu, which drives growth, and Σ\Sigma, which bends light. When cs2<0c_s^2<0 the scalar perturbation grows without bound, so such a choice is rejected.

Common difficulties

Distances do not measure gravity

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

Better mental modelDistances 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, cs2c_s^2 turns negative once w<−1w<-1. Braiding removes the instability.

Related experiments and articles

What this page links to directly

You are hereDark Energy or Modified Gravity

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. ArticleCan the same expansion come with different gravity?
  2. ArticleHow distances record the expansion
  3. ArticleHow growth and lensing feel gravity
  4. ArticleModified gravity in four functions, and when it is stable
  5. ArticleRecompute the expansion and growth grid
  6. ArticleReading the DESI 2024 modified-gravity paper
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