Modified gravity in four functions, and when it is stable

This chapter follows 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 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 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:

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 $z=30$:

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

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 No ghost requires

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

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

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

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:

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

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

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

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

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

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?