How growth and lensing feel gravity

This chapter follows How distances record the 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

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

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:

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

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

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

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

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,

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

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 In linear theory it reduces to

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

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.