Reading the BBN evidence on extra radiation and the lithium problem

Four kinds of statement

Before the first table, set up four labels and use them on every number you copy.

• Measurement: an abundance inferred from spectra of real objects, with its error. Example: D/H from quasar absorption systems.
• Prediction: the output of a BBN calculation for given inputs. Example: D/H at the microwave-background baryon density.
• Conditional constraint: a bound on a parameter that holds only if the rest of the standard picture holds. Example: the number of neutrino species from helium and deuterium, assuming standard BBN apart from the expansion rate.
• Open question: something the authors themselves say is unresolved. Example: why lithium in old stars is low.

The review belongs to the Particle Data Group's Review of Particle Physics, a regularly revised compilation of published results. The arXiv paper is a preprint of a published article. Results from the tutorial network on this site are a third kind of thing, a teaching model; they appear here only where they help to read a table.

Review Section 24.3: the three measured abundances

Section 24.3 gives one recommended value per element.

• Deuterium, Eq. 24.2: $\mathrm{D/H}=(25.08\pm0.29)\times10^{-6}$, the weighted mean of the twelve absorption systems in Table 24.1. The raw error $\pm0.27$ is multiplied by $S=1.08$ because the twelve values scatter slightly more than their errors allow. Check this in the last row of Table 24.1.
• Helium-4, Eq. 24.3: $Y_p=0.245\pm0.003$, from metal-poor ionised gas regions in small galaxies, extrapolated to zero metal content. Table 24.2 lists seven recent determinations. The review states that the error is dominated by systematics, estimated at $\pm0.002$, and warns that the partly overlapping data sets are not straightforward to combine.
• Lithium-7, Eq. 24.4: $\mathrm{Li/H}=(1.45\pm0.25)\times10^{-10}$, from halo stars with $-3.5<[\mathrm{Fe/H}]<-1.5$. Here $[\mathrm{Fe/H}]$ is the logarithm of a star's iron-to-hydrogen ratio relative to the Sun's.

The most common misreading is to treat Eq. 24.4 as a primordial measurement like the other two. The review says directly after it that the value should be regarded as a lower bound, because stars may have destroyed part of their lithium. Helium-3 gets no recommended value at all; the review considers it inappropriate as a cosmological probe.

Review Figure 24.1: the abundance plot

Figure 24.1 plots the predicted abundances against $\eta_{10}$ on the bottom axis and $\Omega_bh^2$ on the top axis. Read the band widths carefully. For helium the width mainly reflects the two-sigma neutron-lifetime uncertainty; for D, helium-3 and lithium it is the two-sigma nuclear-rate uncertainty from Monte Carlo. The boxes are the measurements of Section 24.3, and the narrow vertical band is the microwave-background baryon density. When you compare with Figure 2 of the arXiv paper, remember that its curve widths are one sigma.

Review Section 24.4: which baryon density?

The review quotes several baryon densities, and they rest on different inputs.

Only the third row is an independent test of BBN. A frequent mistake is to compare BBN with Eq. 24.8 and call the agreement independent.

The same section bounds the expansion rate. Helium grows as $\Delta Y_p\simeq0.013\,\Delta N_\nu$. A joint likelihood of $\eta$ and $N_\nu$ from helium and deuterium gives

with $\eta_{10}=6.088\pm0.054$ at the same time. Label it a conditional constraint: it assumes standard BBN in every respect except the energy density of light particles. The review also points out that the central value of $N_\nu$ depends on $\eta$, and that for $\eta_{10}\sim6$ a helium value of 0.249 would put it very close to 3. The bound moves with the helium systematics.

Two newer microwave-background results are quoted in one sentence: $N_{\mathrm{eff}}=2.86\pm0.13$ from the Atacama Cosmology Telescope and $2.83\pm0.13$ from the South Pole Telescope. This guide has not opened those papers, the review's references 88 and 89, so read them only as the review reports them. Note also that the Standard Model value is $N_{\mathrm{eff}}=3.044$, not 3, because neutrinos are slightly heated by electron-positron annihilation. The review's helium curves include that heating.

The arXiv paper: the predictions and their errors

Turn to the arXiv paper. Figure 2 is its abundance plot, with $N_\nu=3$ and $\tau_n=879.4\pm0.6\,\mathrm s$. Table II lists logarithmic sensitivities: how much each abundance changes, in percent, per percent change in $\eta$, $N_\nu$, the gravitational constant, $\tau_n$ and each nuclear rate. Figure 3 gives the fractional error of each prediction against $\eta$; the helium error is drawn ten times larger so that it shows.

Equations 22 to 25 are the predictions at the microwave-background baryon density:

Label all three predictions. Compare the error on D/H, 5 percent, with the 1.2 percent of the measurement in Eq. 24.2: the prediction is now less precise than the observation. That is why the nuclear rates matter so much in the rest of the paper.

Figure 4 overlays predicted and measured likelihoods. The caption states the result in one line: excellent agreement for D/H, good agreement for helium, strong discrepancy for lithium. Table III compares three codes at nearly the same baryon density. Their helium agrees to $0.0001$, while D/H differs by about 4 percent and lithium by 20 percent between this paper and the PRIMAT code of Pitrou et al., reference 36. Appendix D attributes a large part of the lithium difference to the choice of $d(p,\gamma){}^3\mathrm{He}$ rate.

The arXiv paper: bounding the number of neutrino species

Figure 8 repeats the abundance plot for $N_\nu=2$, 3 and 4. The text beneath makes the key point: without a lower bound on $\eta$, no upper limit on $N_\nu$ follows, even from a firm upper bound on helium. Helium alone cannot separate faster expansion from a higher baryon density.

Figure 11 shows one-dimensional likelihoods for $N_\nu$ and Figure 12 the two-dimensional likelihood in $N_\nu$ and $\eta$. Table V gives the numbers:

The abstract quotes the last row as $N_\nu=2.86\pm0.15$, explicitly setting the lithium problem aside. Read it as a conditional constraint: it holds if lithium's discrepancy has nothing to do with the expansion rate. The BBN-only and CMB-only rows probe different epochs, minutes and hundreds of thousands of years, so their agreement limits any change in radiation between the two. The review's Section 24.6 makes the same point.

Lithium: what both sources say

The review's Section 24.5 starts from Eq. 24.10, $\mathrm{Li/H}=(4.72\pm0.7)\times10^{-10}$, attributed to the arXiv paper, and states a factor of 3.3, a 4.4-sigma discrepancy. The printed sentence says that Eq. 24.10 exceeds "the prediction of eq. 24.4". Read the labels from the equations themselves: Eq. 24.4 is the stellar value of Section 24.3, and Eq. 24.10 equals the CMB-plus-BBN prediction in Eq. 25 of the arXiv paper. The prediction exceeds the observation.

The review then sorts the explanations, and each deserves its own label.

• Nuclear physics: missed resonances that would destroy more beryllium-7 have been searched for and closed off, making a nuclear fix increasingly unlikely. The arXiv paper's Appendix D adds a test of its own. Switching to a theory-based $d(p,\gamma){}^3\mathrm{He}$ rate shifts D/H, helium-3 and lithium-7 by $-6.1$, $+7.6$ and $+11.8$ percent at $\eta_{10}=6.129$. Lithium gets slightly worse, and deuterium falls out of agreement. The caption of Figure 22 says the change "increases D/H"; the text and the sign of the D/H sensitivity in Table II both say it decreases. Trust the text.
• Stellar depletion: lithium destroyed inside stars over billions of years. The review calls it the leading resolution, citing the GALAH survey of more than 100,000 stars, new upper limits on lithium-6 in plateau stars, and interstellar lithium in the Small Magellanic Cloud. Label it the leading explanation, not a settled one; the review says future lithium-6 work will be critical.
• New physics during BBN: decaying particles that inject energetic nucleons, or changing fundamental constants such as a weaker deuteron binding. The review stresses that the precise D/H now forces particle-injection scenarios to change lithium far less than needed, with one recent exception. Label these conditional and increasingly constrained.

The arXiv paper's Section VII calls lithium "the elephant in the room" and leaves its resolution open among nuclear physics, new particle physics, the observations and stellar history. Both sources agree: open question.

Review Section 24.6: worked examples, not current limits

Section 24.6 shows how a bound on $N_\nu$ becomes a bound on new particles. It imposes $N_\nu<4$ as an illustrative constraint and derives that three right-handed neutrinos must have decoupled above $140\,\mathrm{MeV}$, that their interaction strength must satisfy $G_R\lesssim10^{-2}G_F$, and that $m_{\nu_\tau}\lesssim1\,\mathrm{MeV}$. These are worked examples with a deliberately loose input. Do not quote them as today's limits. The same section quotes the Planck microwave-background value $N_\nu^{\mathrm{CMB}}=2.92^{+0.36}_{-0.37}$ at 95 percent confidence and an earlier South Pole Telescope value of $3.85\pm0.62$, which had prompted discussion of extra radiation before Planck.

Using the tutorial network alongside

The tutorial behind the companion Lab can reproduce the sensitivities of Table II to within $0.04$ and the helium slope per neutrino species. It cannot reproduce the absolute helium prediction, because it stops at the Born approximation for the weak rates and sits about $0.005$ low; a bound on $N_\nu$ computed from it would be biased by about a third of a species. Use it to see why each table looks the way it does, and the published analyses for any number you quote. The details are in Recompute a small network from a frozen rate grid and propagate rate errors and How baryon density and extra radiation move the light elements.

Checklist for the next BBN paper

• For every number, write down: measurement, prediction, conditional constraint or open question.
• For every baryon density, check whether BBN was used to obtain it.
• For every bound on $N_\nu$ or $N_{\mathrm{eff}}$, find where $\eta$ came from and whether lithium was excluded.
• Check whether bands are one sigma or two sigma.
• When predicted D/H or lithium differ between papers, look first at the nuclear rates, especially $d(p,\gamma){}^3\mathrm{He}$.
• When a caption and the text disagree, check against an independent part of the paper, such as a sensitivity table.
• Note the version you read, and whether you compared it with the published text.