How baryon density and extra radiation move the light elements

This chapter builds on When neutrons and protons stop turning into each other and Why deuterium waits for the universe to cool. You need logarithms and the idea of a derivative as the ratio of two small changes. Standard deviations and adding independent errors in quadrature are explained where they are used. By the end you should be able to compute a logarithmic sensitivity from two runs, invert an observed abundance into a baryon density with an error bar, turn a helium measurement into a bound on extra radiation, and build an error budget from rate uncertainties.

Logarithmic sensitivity

Let $X$ be an abundance and $p$ an input: the baryon-to-photon ratio $\eta$, the number of neutrino-like species $N_\nu$, the neutron lifetime or a reaction rate. The logarithmic sensitivity

is the percentage change in $X$ per percent change in $p$, all other inputs fixed. If $S$ is constant over a range, $X\propto p^{S}$ there. Sensitivities are convenient because they do not depend on units, and because they turn small fractional changes into a linear sum:

The method guide computes each $S$ with a central difference. It runs the network with the input multiplied by $1.01$ and by $0.99$, and divides the change in $\ln X$ by $\ln(1.01/0.99)$.

Worked example: the neutron lifetime from two runs

The saved runs at $\tau_n=860\,\mathrm s$ and $900\,\mathrm s$ give $Y_p=0.23861$ and $0.24675$. Then

The 1 percent central difference gives $0.738$. Fields, Olive, Yeh and Young list $0.729$ in Table II of their 2019 preprint, computed with their full code at $\eta_{10}=6.129$.

The agreement to within $0.04$ is a check of the tutorial network, not new evidence: both calculations share the same physics, and the tutorial's rates are older. The table already says what each element is good for. Deuterium has the largest $\eta$ exponent. Helium has a tiny $\eta$ exponent but a sizeable one for $N_\nu$. Lithium-7 rises with $\eta$, because at high density it is made as beryllium-7, which later captures an electron.

Weighing the baryons with deuterium

The 2025 Review of Particle Physics recommends, from twelve quasar absorption systems,

Worked example: invert the deuterium abundance

The network gives $\mathrm{D/H}=2.548\times10^{-5}$ at $\eta_{10}=6.12$. Treat the sensitivity as constant over the small step:

A direct bisection on the network gives $6.180$ as well. The error has two independent parts. The observational error is $0.29/25.08=1.16$ percent of D/H, or $1.16/1.618=0.71$ percent of $\eta$. The rate error of the prediction, derived below, is $0.69$ percent of D/H, or $0.43$ percent of $\eta$. Independent errors add in quadrature, as the square root of the sum of squares, because their variances add:

The review gives $(\eta_{10})_{\mathrm{SBBN}}=6.040\pm0.118$ (Eq. 24.5), a range it describes as essentially determined by D/H, and $6.104\pm0.058$ from the microwave background without BBN. The tutorial value sits 1.2 standard deviations above the review's BBN value. The difference comes from the predicted D/H at a given $\eta$, so from the nuclear rates: the tutorial keeps the 2004 Serpico fits, while the review's value comes from a later analysis that this chapter has not reproduced. Table 4.1 shows why the rates matter so much here: a 1 percent change in $d(p,\gamma){}^3\mathrm{He}$ moves D/H by a third of a percent, and so the inferred $\eta$ by a fifth of a percent.

Bounding extra radiation with helium

Any extra relativistic energy, counted as $\Delta N_\nu$ additional neutrino species, speeds up the expansion. The network's $S_{Y_p,N_\nu}=0.165$ at $N_\nu=3$ converts into an absolute slope

in line with $\Delta Y_p\simeq0.013\,\Delta N_\nu$ in the review. The direct run with $\Delta N_\nu=1$ gives $0.25485$, a rise of $0.0125$; the slope falls slowly because the sensitivity is logarithmic in $N_\nu$.

Worked example: a one-measurement bound

The review's helium abundance is $Y_p=0.245\pm0.003$ (Eq. 24.3). The full-physics prediction with three species and the microwave-background baryon density is $Y_p=0.24691\pm0.00018$ (Fields et al., Eq. 22). Then

The two-sided 95 percent range is roughly $-0.59<\Delta N_\nu<0.30$. One extra fully populated neutrino species lies about five standard deviations away. The review's joint analysis of helium and deuterium gives $N_\nu=2.898\pm0.141$ (Eq. 24.9), which is $\Delta N_\nu=-0.10\pm0.14$.

Notice what would happen with the tutorial's own prediction, $Y_p=0.2424$. The same arithmetic gives $\Delta N_\nu=+0.20$. The shift of $0.34$ species is entirely the weak-rate corrections the tutorial omits, discussed at the end of the first chapter. A bound on new physics is only as good as the absolute prediction it is compared with.

Propagating rate errors

Each nuclear rate fit comes with a one-standard-deviation fractional error $\sigma_i=\delta f_i/f_i$ that depends on temperature. To first order, and if the rate errors are independent,

The method guide does slightly better than multiplying one number by one sensitivity: it shifts each rate by its full temperature-dependent error and reruns the network. The neutron lifetime is included with an assumed error of $0.5\,\mathrm s$.

As a check that does not rely on the linear formula, the guide also draws 200 random sets of rates and repeats the calculation. The spread is 0.72 percent for D/H and 1.78 percent for lithium-7, close to the quadrature values.

The lithium column answers a question raised in the first article. The review quotes $\mathrm{Li/H}=(1.45\pm0.25)\times10^{-10}$ for metal-poor halo stars (Eq. 24.4) and a BBN-plus-CMB prediction of $(4.72\pm0.7)\times10^{-10}$ (Eq. 24.10), a factor of 3.3 apart. With $S=0.963$, closing that gap through ${}^3\mathrm{He}(\alpha,\gamma){}^7\mathrm{Be}$ alone would need its rate to fall by a factor of about $3.3^{1/0.963}\approx3.4$. The fit's quoted error is 1 to 2 percent at the temperatures where beryllium-7 forms. A drop of about 70 percent is 35 to 70 times that error.

Lab task: weigh the baryons, then test the radiation

1. Predict. Before moving anything, estimate from Table 4.1 how far you must move the baryon slider to change D/H by 10 percent, and how far the radiation slider to change $Y_p$ by $0.003$.
2. Change. Drag the baryon slider until the deuterium curve crosses the centre of the observed band. Then return it to the default and drag the radiation slider until $Y_p$ reaches $0.245$.
3. Compare. The first value should be close to $\eta_{10}=6.18$; the Lab's coarser time step can move it by about $0.01$. For the second, compare your $\Delta N_{\mathrm{eff}}$ with the $+0.20$ from the worked example.
4. Explain. Explain why the radiation slider moves D/H as well as helium, and why the second reading cannot be quoted as a measurement of extra radiation.

Exercises

1. Use Table 4.1 to predict the change in ${}^7\mathrm{Li/H}$ when $\eta_{10}$ goes from $6.12$ to $10$. The saved run gives $1.09\times10^{-9}$ from $4.51\times10^{-10}$. Why does the prediction miss?
2. A future survey halves the error on D/H to $0.145\times10^{-6}$ while the rate errors stay the same. What is the new error on $\eta_{10}$, and which source of error would then dominate?
3. Extra radiation also raises D/H. Use Table 4.1 to find the $\Delta\eta/\eta$ that cancels the D/H change from $\Delta N_\nu=0.5$. Then find the resulting helium change. Explain why deuterium alone cannot measure both $\eta$ and $N_\nu$.
4. Transfer. A colleague's code gives $Y_p=0.2440$ for the standard case. They compare it with $Y_p=0.245\pm0.003$ and report evidence for $\Delta N_\nu\approx0.08$. List what you would check before believing the number.

What these numbers do and do not show

All model numbers in this chapter come from the tutorial network and are saved in teaching-values.json and expected.json. They show how sensitivities and error budgets are built; they are not a new determination of $\eta$ or $N_\nu$. Published determinations use a fuller weak-rate treatment, newer rates and joint likelihoods with the microwave background. The guide Reading the BBN evidence on extra radiation and the lithium problem walks through those analyses table by table.