Reading the DESI 2024 modified-gravity paper

Read Modified gravity in four functions, and when it is stable first. This guide assumes you know $\mu$, $\Sigma$, the braiding $\alpha_B$ and the running $\alpha_M$. It goes through the sections in reading order, then sorts the paper's claims into four kinds of statement.

What the paper uses

The new data are the full-shape clustering measurements from the first DESI data release. The galaxy power spectrum is modelled in full, redshift-space distortions included, so it carries the growth rate as well as the acoustic scale. They are combined with the Planck CMB, CMB lensing from Planck and ACT, a nucleosynthesis prior on the baryon density, Dark Energy Survey year-3 weak lensing and clustering, and Dark Energy Survey year-5 supernovae. The paper never uses CMB lensing and the DES weak-lensing data together, because they are correlated.

Three codes do the work. ISiTGR, built on CAMB, handles $\mu$ and $\Sigma$. EFTCAMB handles the effective-field-theory basis. mochi_class, a branch of hi_class, handles the four-function basis. All three run inside the cobaya sampler.

Section 3: two numbers that test general relativity

Section 3.1 writes the two potential equations of the earlier chapters. The paper also defines the slip $\eta=\Phi/\Psi$, related by $\Sigma=\mu(1+\eta)/2$ at late times. Section 3.2 makes both functions follow dark energy:

in their time-only form, so general relativity is $\mu_0=\Sigma_0=0$. Note the hard prior $\mu_0<2\Sigma_0+1$. The authors impose it to avoid numerical failures in the CAMB-based code and argue that it does not affect the interpretation.

Figure 2 is the quickest summary: whisker plots of $\mu_0$ and $\Sigma_0$ across data combinations. DESI alone gives $\mu_0=0.11^{+0.44}_{-0.54}$ and no constraint on $\Sigma_0$. Galaxy velocities feel only $\mu$, so Figure 1 shows DESI alone as a horizontal band in $\Sigma_0$. The tightest combination, DESI with the CMB without lensing, DES year-3 and DES year-5 supernovae, gives $\mu_0=0.05\pm0.22$ and $\Sigma_0=0.008\pm0.045$. Both are consistent with zero, and $\Sigma_0$ is about five times better constrained than $\mu_0$.

Read the passage on the Planck PR3 likelihood carefully. With PR3 and no CMB lensing, $\Sigma_0$ departs from zero, and adding DESI pushes that departure above three standard deviations. The authors trace it to the known lensing anomaly of PR3, the excess smoothing usually expressed through a non-physical parameter $A_{\mathrm{lens}}$. It weakens with CamSpec and largely disappears with the newer LoLLiPoP and HiLLiPoP likelihoods. A reader who quotes the PR3 number without this context misreports the paper.

On an evolving dark-energy background, with the newer CMB likelihood, the same section finds $w_0=-0.784\pm0.061$ and $w_a=-0.82^{+0.28}_{-0.24}$, with $\mu_0$ and $\Sigma_0$ still consistent with zero. The preference for evolving dark energy survives the extra freedom in gravity.

Section 4.2: braiding and running

Section 4.2 is where this course's Lab lives. The paper takes $\alpha_i=c_i\Omega_{\mathrm{DE}}(a)$, sets $\alpha_T=0$ because of GW170817, and fixes $c_K=0.01$ because observations barely constrain it. It prints the quasi-static $\mu$ and $\Sigma$ that the Lab uses. Three classes are fitted: braiding and running both free; running alone, called only-run gravity; and no-slip gravity with $\alpha_B=-2\alpha_M$, which gives $\mu=\Sigma=1/M_*^2$.

On a cosmological-constant background with DESI full shape, DES year-5 supernovae and the CMB, the text reports $c_M=1.05\pm0.96$ and $c_B=0.92\pm0.33$.

Figure 10, left panel, plots $c_M$ upward against $c_B$ to the right. All contours sit at positive $c_B$ and are sharply cut along a diagonal edge at their lower left. The caption attributes the cut to gradient instabilities. The right panel shows the one-dimensional $c_M$ for the three classes. The only-run curve rises from a wall at $c_M=0$, because stability forbids negative running without braiding. The no-slip curve peaks below zero, and the text quotes $c_M=-0.51\pm0.21$. The only-run bound is $c_M<1.14$ at 95%.

The text explains why the growth data matter. Without them, braiding and running are constrained mainly by the late integrated Sachs-Wolfe effect on the CMB, and the two can cancel each other there. Full-shape growth breaks that cancellation by pinning down $\alpha_M$.

On the evolving background, Figure 11 plots the same plane without CMB lensing. The text gives $c_M=0.33^{+0.63}_{-0.91}$ and $c_B=1.25\pm0.33$, again consistent with no running but preferring positive braiding. The authors add that at least one $\alpha_i$ should be nonzero to stabilise the phantom crossing of $w$. That expectation is Vikman's no-go result read in reverse. Vikman 2005

Check the numbers against Table 8

Table 8 collects the four-function results, and it does not always match the text digit for digit. For braiding and running on a cosmological-constant background, the table lists $c_M=0.98\pm0.89$ and $c_B=0.91\pm0.31$, against the text's $1.05\pm0.96$ and $0.92\pm0.33$. For the evolving background, the text prints $w_0=-0.801\pm0.65$, while the table lists $-0.801\pm0.065$. The table's error matches every other $w_0$ in the paper, so the text has almost certainly dropped a zero. These are small differences, but quote the table and say so when you cite a value.

Sorting the claims

A careful summary keeps four kinds of statement apart.

Published results are the posteriors in the paper: the $\mu_0$ and $\Sigma_0$ values, the braiding and running constraints, and the Table 8 entries. They are conditional on the stated data, likelihood, functional form and priors.

Later published work changes the context, not these numbers. The DESI second data release found a 2.8 to 4.2 standard-deviation preference for evolving dark energy when its acoustic scale is combined with the CMB and different supernova samples. DESI DR2 BAO results, 2025, §VII.1 Ye and collaborators reconstructed the equation of state and the effective-field-theory functions without a fixed form. They argue that a crossing below $z\simeq1$ rules out quintessence and requires nonminimally coupled gravity within Horndeski theories. Ye et al. 2025 This course read only that paper's abstract. The DESI extended dark-energy analysis cited by the paper for the crossing argument was not read here at all.

Model predictions are calculations in this course, not data. The Lab's stability grid reproduces the shape of the Figure 10 cut-off. On the cosmological-constant background, the lowest stable $c_M$ falls from $0$ at $c_B=0$ to $-0.5$ at $c_B=1$ and $-1$ at $c_B=2$. By eye, the 95% contours of Figure 10 reach somewhat lower, to about $c_M\simeq-0.9$ near $c_B\simeq1.1$. Possible causes include the smoothing of contours across a sharp edge, the different backgrounds and the different stability windows; this course did not resolve it. On the DESI-like crossing background, the grid finds that braiding of about $c_B\ge0.6$ allows negative running, and that running alone with $c_M\ge1$ also stabilises the crossing.

Open problems are the ones the authors list. They did not repeat the four-function fit with the PR4 CMB likelihoods, which could move braiding toward zero. Projection effects in the full-shape analysis could produce part of the preference. The integrated Sachs-Wolfe effect and its cross-correlation with galaxies were not included. The time dependence $\alpha_i\propto\Omega_{\mathrm{DE}}$ is one choice among many. And whether the crossing of minus one is physical at all remains a question for the dark-energy data.

Checklist for citing this paper

1. Say which version you read and quote numbers from Table 8.
2. Give the data combination with every number; the $\Sigma_0$ result changes meaning with the CMB likelihood.
3. Keep phenomenological results ($\mu_0$, $\Sigma_0$) separate from the four-function results ($c_B$, $c_M$).
4. Report the braiding preference with the authors' own caveats: projection effects, systematics, the PR3 likelihood.
5. Remember the stability cut is a prior; it shapes every contour in Figures 10 and 11.
6. Label anything you compute yourself, such as a stability map, as a model calculation.