---
title: "How growth and lensing feel gravity"
description: "Derive the growth equation in the scale factor, solve it exactly for matter alone, and use growth and lensing to measure the pull of gravity."
canonical_url: "https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/how-growth-and-lensing-feel-gravity/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/dark-energy-or-modified-gravity/how-growth-and-lensing-feel-gravity/"
last_modified: "2026-11-12"
---

[Dark energy or modified gravity](https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/)

# How growth and lensing feel gravity

Sandbox Physics · 2026-11-12

Once distances have fixed the expansion rate, two measurements remain sensitive to gravity itself. The growth of matter clumps responds to the pull on slow matter, written $\mu$, and gravitational lensing responds to the bending of light, written $\Sigma$. Predicting both from the measured expansion with Einstein's values $\mu=\Sigma=1$, and comparing with data, is the test this course is about.

This chapter follows [How distances record the expansion](https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/how-distances-record-expansion/). You need derivatives and a second-order differential equation whose solutions are powers. The aims are to derive the growth equation in the logarithm of the scale factor, solve it exactly in a matter-only universe, define the growth rate and the lensing-to-growth ratio, and read a synthetic record.

## Two potentials and two responses

On large scales the universe is a smooth background plus small ripples. Write the density of matter as $\bar\rho(1+\delta)$, where $\delta$ is the fractional overdensity. The ripples in spacetime are described by two potentials. $\Psi$ perturbs the flow of time and accelerates slow matter; $\Phi$ perturbs space. Light responds to their sum, because a massless particle feels space and time equally.

For a ripple of comoving wavenumber $k$, well inside the horizon and still linear, the two equations that replace Newton's law are

$$
k^2\Psi=-4\pi Ga^2\mu\,\bar\rho\,\delta,\qquad k^2(\Phi+\Psi)=-8\pi Ga^2\Sigma\,\bar\rho\,\delta.
$$

In general relativity, with negligible anisotropic stress, $\Phi=\Psi$ and both functions equal one. A modified theory can change either one, and in general they can depend on time and scale. The DESI modified-gravity analysis defines its parameters exactly this way. [Ishak et al. 2024, DESI 2024 modified gravity, §3.1](https://arxiv.org/abs/2411.12026v3)

The first equation controls how matter falls. The second controls how light bends. They are independent pieces of information, and neither depends on the expansion rate alone.

## The growth equation

On these scales matter behaves as a pressureless fluid. Mass conservation, the Euler equation and the first potential equation combine into a single equation for the overdensity:

$$
\ddot\delta+2H\dot\delta=4\pi G\mu\,\bar\rho\,\delta.
$$

The second term is Hubble drag: expansion carries matter apart and slows the collapse. The right side is the gravitational pull, scaled by $\mu$.

Distances are naturally written in the scale factor, so change variables to $\ln a$. With a prime meaning $\mathrm d/\mathrm d\ln a$, a time derivative is $H$ times a prime. Then

$$
\ddot\delta=H^2\Bigl(\delta''+\frac{\dot H}{H^2}\,\delta'\Bigr),\qquad 4\pi G\bar\rho=\frac32H^2\Omega_{\mathrm m}(a),
$$

where $\Omega_{\mathrm m}(a)$ is the matter fraction at that time. Divide by $H^2$ and use $\dot H/H^2=\mathrm d\ln H/\mathrm d\ln a$:

$$
\delta''+\Bigl(2+\frac{\mathrm d\ln H}{\mathrm d\ln a}\Bigr)\delta'=\frac32\,\Omega_{\mathrm m}(a)\,\mu\,\delta.
$$

The background enters through $\Omega_{\mathrm m}(a)$ and the slope of $H$; gravity enters only through $\mu$. Both background quantities come from the expansion rate of the previous chapter. That is why one expansion history plus one value of $\mu$ fixes the growth.

## Exact growth in a matter-only universe

When $\Omega_{\mathrm m}=1$, the expansion rate falls as $a^{-3/2}$, so its slope is $-3/2$, and $\Omega_{\mathrm m}(a)=1$ at all times. Hold $\mu$ constant. The equation becomes

$$
\delta''+\frac12\delta'=\frac32\mu\,\delta.
$$

Try a power of the scale factor, $\delta\propto a^p$. Each prime brings down a factor $p$, so $p^2+p/2-3\mu/2=0$, and the growing solution is

$$
p=\frac{-1/2+\sqrt{1/4+6\mu}}{2}.
$$

With $\mu=1$ this gives $p=1$: clumps grow in proportion to the scale factor, the classic result. With $\mu=1.1$, a pull 10% stronger, $p=1.0586$. A 10% stronger pull gives a growth exponent only 5.9% larger. Hubble drag absorbs part of the extra pull.

## Growth rate and the number surveys report

The growth rate is the logarithmic slope of the growing solution,

$$
f\equiv\frac{\mathrm d\ln\delta}{\mathrm d\ln a}.
$$

In the matter-only universe it is simply $p$. In the cosmological-constant model with $\Omega_{\mathrm m}=0.31$, the numerical solution gives $f=0.5224$ today, $0.7571$ at $z=0.5$ and $0.8743$ at $z=1$. Growth slows at late times because dark energy takes over the expansion. A famous fit, $f\simeq\Omega_{\mathrm m}(a)^{0.55}$, gives $0.5251$ today and $0.7569$ at $z=0.5$. Linder showed that this growth-index form tracks the full solution to better than a percent for a cosmological constant and for a range of evolving dark energy, with the index shifting slightly with the equation of state. [Linder 2005](https://arxiv.org/abs/astro-ph/0507263v2)

Redshift-space distortions measure the infall velocities of galaxies. Those velocities are proportional to $f$ times the amplitude of the matter ripples, so surveys report the product $f\sigma_8$, where $\sigma_8$ is the root-mean-square matter fluctuation in spheres of $8\,h^{-1}\mathrm{Mpc}$. In the Lab every model starts with the same early amplitude, normalised so that the cosmological-constant model has $\sigma_8=0.81$ today. Its value at $z=0.5$ is $0.624$, so $f\sigma_8=0.4724$ there.

## Lensing against growth

Lensing measures $\Phi+\Psi$, and velocities measure $f$. Their ratio removes the unknown amplitude of the matter ripples and the unknown bias of galaxies relative to matter. Zhang and collaborators proposed the combination $E_G$ for exactly this purpose. [Zhang et al. 2007](https://arxiv.org/abs/0704.1932) In linear theory it reduces to

$$
E_G=\frac{\Omega_{\mathrm m}\,\Sigma}{f}.
$$

For the cosmological-constant model at $z=0.5$ this is $0.31/0.7571=0.4094$. Reyes and collaborators measured $E_G=0.39\pm0.06$ on scales of tens of megaparsecs, in agreement with the general-relativistic expectation near $0.4$. [Reyes et al. 2010](https://arxiv.org/abs/1003.2185)

## Worked example: the default Lab universe

The Lab opens on the cosmological-constant expansion with a modified response. At $z=0.5$ its gravity gives $\mu=1.170$ and $\Sigma=1.094$. Every distance equals the cosmological-constant value, because the background is identical. The growth equation, solved with the larger pull, gives $f\sigma_8=0.5025$, while Einstein gravity on the same expansion gives $0.4724$: 6.4% more growth. The lensing ratio is $E_G=0.4276$ against $0.4094$, 4.4% higher.

A distance survey would see nothing here. A growth survey would see clusters that are too developed for the measured expansion. Only the two together locate the discrepancy in gravity.

## Reading a synthetic record

The course data include a synthetic record, not an observation. It was generated from the default model with random scatter from seed 81: five values of $f\sigma_8$ between $z=0.3$ and $z=1.3$ with 4% errors, and two values of $E_G$ with 6% errors. Fit it in three ways.

Einstein gravity on the cosmological-constant expansion gives $\chi^2=21.6$ for seven points. The model that generated the record gives $\chi^2=5.0$. A scan over the stable models in the frozen grid finds a smaller value, $\chi^2=3.7$, at braiding $c_B=1.8$ and running $c_M=-1.0$, far from the true $c_B=0.9$, $c_M=0.6$.

The first comparison rejects Einstein gravity for this record. The second and third show what such data cannot do: very different gravity parameters produce nearly the same growth and lensing. Detecting a departure from $\mu=\Sigma=1$ is easier than naming the theory responsible.

## Lab tasks

1. Choose Same expansion, stronger pull and drag the probe shell from $z=0.2$ to $z=1.5$. Note where the amber model growth curve sits furthest above the teal Einstein-gravity curve, and connect it to when dark energy matters.
2. Choose Faster growth, weaker lensing. The growth ratio rises above one while $E_G$ falls below the Einstein value. Read $\mu$ and $\Sigma$ in the response chart and check which one sits below one.
3. Choose Changing dark energy, Einstein gravity. Growth differs from the cosmological-constant model, yet the amber and teal curves coincide. Explain why this is not evidence for modified gravity.

## Exercises

1. In a matter-only universe with constant $\mu=0.9$, find the growth exponent.
2. Why do surveys report $f\sigma_8$ rather than $f$ alone?
3. A theory has no gravitational slip, so $\Sigma=\mu$, with both equal to $0.95$ at $z=0.5$. Without solving the growth equation, say whether $E_G$ must fall below its Einstein value.
4. A survey finds $f\sigma_8$ 5% below the cosmological-constant prediction. Name two explanations that do not involve modified gravity.

Answers

For the first exercise, $p=(-1/2+\sqrt{1/4+5.4})/2=(-0.5+2.377)/2=0.938$. A pull 10% weaker slows the growth exponent by about 6%.

For the second, redshift-space distortions measure the velocity field, which is $f$ times the density ripple. The ripple amplitude and the growth rate enter as a product, so the data constrain $f\sigma_8$ directly. Lensing or the CMB can then supply the amplitude separately.

For the third, probably yes, but by less than 5%. $E_G$ is proportional to $\Sigma/f$. The weaker pull lowers $\Sigma$ by 5% and also lowers $f$. The matter-only example suggests $f$ moves by a little over half the fractional change in $\mu$, so $E_G$ falls by roughly 2%. The growth rate depends on the whole history of $\mu$, so a real answer needs the integration.

For the fourth, an expansion history different from the assumed one, such as evolving dark energy, changes both the drag and $\Omega_{\mathrm m}(a)$. A smaller early amplitude than assumed lowers $\sigma_8$ at every redshift. Survey systematics, such as errors in modelling small-scale velocities, are a third possibility. Each must be ruled out before $\mu$ is blamed.

## Sources

1. [Ishak et al. 2024, DESI 2024 modified gravity, §3.1](https://arxiv.org/abs/2411.12026v3)
2. [Linder 2005](https://arxiv.org/abs/astro-ph/0507263v2)
3. [Zhang et al. 2007](https://arxiv.org/abs/0704.1932)
4. [Reyes et al. 2010](https://arxiv.org/abs/1003.2185)

Choose Faster growth, weaker lensing and read the growth and lensing ratios. [Open the Lab](https://sandboxphysics.com/labs/dark-energy-modified-gravity/)

## Keep reading

Dark energy or modified gravity · Reading path

### Distances, growth, lensing and stability

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. [You are here **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/)