---
title: "Modified gravity in four functions, and when it is stable"
description: "Braiding and a running Planck mass set the growth and lensing response, and two stability conditions decide which models a fit may use."
canonical_url: "https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/modified-gravity-in-four-functions-and-stability/"
language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/stories/dark-energy-or-modified-gravity/modified-gravity-in-four-functions-and-stability/"
last_modified: "2026-11-12"
---

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

# Modified gravity in four functions, and when it is stable

Sandbox Physics · 2026-11-12

A broad family of scalar-field theories of gravity affects linear cosmology only through the expansion history and four functions of time. Two of them, braiding and the running of the Planck mass, set the growth and lensing responses $\mu$ and $\Sigma$. Not every choice is allowed. The scalar perturbation must have positive kinetic energy and a positive squared sound speed, and these two conditions decide which parts of the parameter space a fit may use.

This chapter follows [How growth and lensing feel gravity](https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/how-growth-and-lensing-feel-gravity/). It assumes you are comfortable with the growth equation in $\ln a$ and with algebra involving several symbols. The aims are to state what each function does, derive the stability conditions in two limits that can be done by hand, and evaluate $\mu$ and $\Sigma$ for the Lab's default universe.

## One scalar field, four functions

Horndeski theories are the most general theories with one scalar field and the metric whose equations of motion stay second order. On a cosmological background, Bellini and Sawicki showed that their linear perturbations depend on the expansion rate plus four dimensionless functions of time. [Bellini and Sawicki 2014](https://arxiv.org/abs/1404.3713) In the notation used by the DESI analysis and the hi\_class code:

- the kineticity $\alpha_K$ is the kinetic energy of the scalar perturbation itself;
- the braiding $\alpha_B$ mixes the scalar field with the metric, so that matter sources the scalar;
- the running $\alpha_M=\mathrm d\ln M_*^2/\mathrm d\ln a$ is the rate at which the effective Planck mass $M_*^2$ changes, that is, how fast the strength of gravity itself drifts;
- the tensor excess $\alpha_T$ shifts the speed of gravitational waves to $c_T^2=1+\alpha_T$.

General relativity with any dark-energy fluid has $\alpha_B=\alpha_M=\alpha_T=0$. Braiding and running are the two that change how matter clumps.

## Why the tensor speed is set to zero

In 2017 the gravitational waves of the neutron-star merger GW170817 and the gamma-ray burst GRB 170817A arrived 1.74 seconds apart. That delay limits the difference between the speed of gravity and the speed of light to between $-3\times10^{-15}$ and $+7\times10^{-16}$ of the speed of light. [LIGO, Virgo, Fermi and INTEGRAL 2017](https://arxiv.org/abs/1710.05834) Since $\alpha_T=c_T^2-1\simeq2(c_T-1)$, it must vanish to about one part in $10^{15}$ today. The Lab, like the DESI analysis, sets $\alpha_T=0$.

## A time dependence that follows dark energy

A fit needs a concrete form for the remaining functions. A common choice makes each one proportional to the dark-energy fraction:

$$
\alpha_i(a)=c_i\,\Omega_{\mathrm{DE}}(a),\qquad i\in\{K,B,M\}.
$$

The modification then switches on only when dark energy matters. The Planck mass follows by integrating its running. The Lab starts it at one at $z=30$:

$$
\ln M_*^2(a)=\int_{a_{\mathrm{ini}}}^{a}c_M\,\Omega_{\mathrm{DE}}(a')\,\frac{\mathrm da'}{a'}.
$$

The DESI analysis fixes $c_K=0.01$. The Lab uses $0.1$ by default and keeps it in the advanced settings, because the quasi-static responses below are independent of it. [Ishak et al. 2024, DESI 2024 modified gravity, §4.2](https://arxiv.org/abs/2411.12026v3)

## The two stability conditions

The scalar perturbation obeys a wave equation with a kinetic coefficient and a sound speed. A healthy theory needs both to be positive. Pogosian and Silvestri give both in the same basis; the form below writes the matter term with the effective Planck mass explicit, as in hi\_class. [Pogosian and Silvestri 2016](https://arxiv.org/abs/1606.05339) No ghost requires

$$
D\equiv\alpha_K+\frac32\alpha_B^2>0,
$$

and no gradient instability requires $c_s^2>0$, with

$$
Dc_s^2=(2-\alpha_B)\Bigl(\alpha_M+\frac{\alpha_B}{2}-\frac{\dot H}{H^2}\Bigr)+\frac{\mathrm d\alpha_B}{\mathrm d\ln a}-\frac{3\,\Omega_{\mathrm m}(a)}{M_*^2}.
$$

A negative $c_s^2$ is not a small correction: short-wavelength scalar ripples then grow exponentially, and the cosmology stops making sense. A fit must reject such parameters, which is why the Lab marks them in red instead of drawing a growth curve. The expression contains $\alpha_K$ only through $D$, and $D$ multiplies $c_s^2$, so the product $Dc_s^2$ is independent of $\alpha_K$.

## Limit one: running alone

Set $\alpha_B=0$ on a cosmological-constant background. There, $-\dot H/H^2=\frac32\Omega_{\mathrm m}(a)$, and the condition becomes

$$
Dc_s^2=2\alpha_M+3\,\Omega_{\mathrm m}(a)\Bigl(1-\frac{1}{M_*^2}\Bigr).
$$

Both terms have the sign of $c_M$. A positive running gives a stable theory; a negative one is unstable at every time after the start. Today, with $c_M=0.1$, $\alpha_M=0.069$ and $M_*^2=1.0398$, so $Dc_s^2=0.1736$. With $c_M=-0.1$ it is $-0.1750$. The DESI paper states the same result for its cosmological-constant fits: with no braiding, stability requires $\alpha_M\ge0$. [Ishak et al. 2024, §4.2.1](https://arxiv.org/abs/2411.12026v3)

## Limit two: dark energy alone crossing minus one

Now set $\alpha_B=\alpha_M=0$, so that $M_*^2=1$, and let dark energy have any $w$. The slope of the expansion rate is $\dot H/H^2=-\frac32[\Omega_{\mathrm m}(a)+(1+w)\Omega_{\mathrm{DE}}(a)]$. Substituting, the matter terms cancel:

$$
Dc_s^2=3(1+w)\,\Omega_{\mathrm{DE}}(a).
$$

The sign is the sign of $1+w$, whatever the kineticity. A single field without braiding has a negative squared sound speed whenever $w<-1$, and the crossing of $w=-1$ itself cannot be stable. This is Vikman's no-go result in the present language. [Vikman 2005](https://arxiv.org/abs/astro-ph/0407107)

For the DESI-like background $w_0=-0.752$ and $w_a=-0.86$, $w$ equals minus one at $a=1-0.248/0.86$, that is $z=0.405$, and lies below minus one before that. The formula gives its most negative value $Dc_s^2=-0.121$ near $z=0.94$. That is the red band in the Lab's response chart.

## How braiding rescues the crossing

With braiding switched on, the terms $(2-\alpha_B)\alpha_B/2$ and $\mathrm d\alpha_B/\mathrm d\ln a$ add to the sound speed. For moderate positive $c_B$ both are positive while dark energy is growing in importance, and they can outweigh the negative $3(1+w)\Omega_{\mathrm{DE}}$. Deffayet and collaborators built explicit braided theories that cross minus one without instability. [Deffayet, Pujolàs, Sawicki and Vikman 2010](https://arxiv.org/abs/1008.0048)

Take $c_B=1.25$ and $c_M=0.33$, the central values of the DESI fit on its evolving background, and place them on the Lab's DESI-like background. At $z=0.94$ the condition becomes $Dc_s^2=0.797$, and it stays positive from $z=30$ to today. The DESI result is the pair of coefficients; their stability on this slightly different background is a model prediction of this course.

## The quasi-static responses

On scales well inside the sound horizon of the scalar, its time derivatives can be dropped next to its spatial gradients. In that quasi-static limit the potentials respond to matter with

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

These are the expressions printed in the DESI paper. With $\alpha_T=0$ they also match the small-scale limit of Pogosian and Silvestri and the effective Newton constant coded in hi\_class. Read them term by term. The prefactor $1/M_*^2$ is a changed Newton constant. The second term is the scalar's own pull: matter sources the scalar through $\alpha_B+2\alpha_M$, and the scalar then pulls on matter and light. Stability enters through the denominator. As $Dc_s^2$ approaches zero, the responses grow without bound, a warning that the theory is near its instability.

Two special cases check the formulas. If $\alpha_B=-2\alpha_M$ the scalar decouples from matter, and $\mu=\Sigma=1/M_*^2$: there is no slip between the potentials. If $\alpha_B=-\alpha_M$, the relation that holds for $f(R)$ theories, then $\Sigma=1/M_*^2$ exactly while $\mu$ differs, so lensing feels only the changed Planck mass. With $c_B=-0.5$ and $c_M=0.5$ the Lab's model gives $\Sigma=0.8227=1/1.2155$ today. On this background that particular choice fails the stability test at early times, where the growth of the negative braiding dominates, so it serves only as an algebra check.

## Worked example: the default universe today

The default Lab universe has a cosmological-constant background, $c_K=0.1$, $c_B=0.9$ and $c_M=0.6$. Today $\Omega_{\mathrm{DE}}=0.69$, so $\alpha_B=0.621$, $\alpha_M=0.414$ and $\alpha_K=0.069$. Integrating the running gives $M_*^2=1.264$.

For the sound speed, $-\dot H/H^2=1.5\times0.31=0.465$ and $\mathrm d\alpha_B/\mathrm d\ln a=c_B\times3\times0.31\times0.69=0.5775$. Then

$$
Dc_s^2=1.379\times(0.414+0.3105+0.465)+0.5775-\frac{0.93}{1.264}=1.482.
$$

It is positive. The coupling is $\alpha_B+2\alpha_M=1.449$, so

$$
\mu=\frac{1}{1.264}\Bigl[1+\frac{1.449^2}{2.964}\Bigr]=1.352,\qquad \Sigma=\frac{1}{1.264}\Bigl[1+\frac{1.449\times1.035}{2.964}\Bigr]=1.192.
$$

Matter today feels 35% more pull than in Einstein gravity, and light bends 19% more. Integrated over the past, that is what made $f\sigma_8$ at $z=0.5$ 6.4% higher in the previous chapter.

## Lab tasks

1. Choose Einstein gravity with a constant, then drag the running strength below zero while braiding stays at zero. The scalar-perturbation readout should switch to unstable at once, and the growth readout to rejected. Drag it above zero and confirm that the model returns.
2. Choose the crossing preset that fails alone. Find the red band in the response chart and read the redshift range. Compare it with the redshift where $w$ falls below minus one.
3. Switch to the modified-gravity preset that allows $w<-1$. Lower braiding slowly and find roughly where the red band returns.

## Exercises

1. Use the expression for $Dc_s^2$ to show that with $\alpha_B=\alpha_M=0$ the kineticity cannot rescue a model with $w<-1$.
2. For running alone with $c_M=0.1$, today's values are $\alpha_M=0.069$, $M_*^2=1.0398$ and $Dc_s^2=0.1736$. Compute $\mu$ and $\Sigma$.
3. Explain why setting $\alpha_T=0$ is a measurement-driven choice, not a convenience.
4. A fit finds a best-fitting point with $Dc_s^2$ slightly negative at $z=1$. Should it be reported as the best fit?

Answers

For the first exercise, $Dc_s^2=3(1+w)\Omega_{\mathrm{DE}}$ does not contain $\alpha_K$. Since $D=\alpha_K>0$ only rescales, $c_s^2=3(1+w)\Omega_{\mathrm{DE}}/\alpha_K$ keeps the sign of $1+w$. A larger kineticity makes the instability slower, not absent.

For the second, the coupling is $\alpha_B+2\alpha_M=0.138$ and $\alpha_B+\alpha_M=0.069$. Then $\mu=(1+0.138^2/0.3472)/1.0398=1.0145$ and $\Sigma=(1+0.138\times0.069/0.3472)/1.0398=0.9881$. Running alone strengthens the pull on matter slightly and weakens lensing slightly, because the larger Planck mass dominates $\Sigma$.

For the third, the gravitational-wave and gamma-ray arrival times bound the speed difference to about one part in $10^{15}$. Any cosmologically interesting $\alpha_T$ today would be of order $0.1$, more than ten orders of magnitude above the bound.

For the fourth, no. A negative squared sound speed means exponential growth of short-wavelength scalar ripples, and the linear predictions used in the fit are invalid there. The point must be excluded, as the DESI analysis does by imposing stability, and the best fit is taken from the stable region. Report the boundary as a prior, because it shapes the posterior.

## Sources

1. [Bellini and Sawicki 2014](https://arxiv.org/abs/1404.3713)
2. [LIGO, Virgo, Fermi and INTEGRAL 2017](https://arxiv.org/abs/1710.05834)
3. [Ishak et al. 2024, DESI 2024 modified gravity, §4.2](https://arxiv.org/abs/2411.12026v3)
4. [Pogosian and Silvestri 2016](https://arxiv.org/abs/1606.05339)
5. [Vikman 2005](https://arxiv.org/abs/astro-ph/0407107)
6. [Deffayet, Pujolàs, Sawicki and Vikman 2010](https://arxiv.org/abs/1008.0048)

Cross minus one with and without braiding and watch the red unstable band. [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. [Article **How growth and lensing feel gravity**](https://sandboxphysics.com/stories/dark-energy-or-modified-gravity/how-growth-and-lensing-feel-gravity/)
4. [You are here **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/)