The 2024 BBN baryon abundance update

We revisit the state of the light element abundances from big bang nucleosynthesis in early 2024 with particular focus on the derived baryon abundance. We find that the largest differences between the final baryon abundances are typically driven by the assumed Deuterium burning rates, characterized in this work by the underlying code. The rates from theoretical ab-initio calculations favor smaller baryon abundances, while experimentally-determined rates prefer higher abundances. Through robust marginalization over a wide range of nuclear rates, the recently released PRyMordial code allows for a conservative estimate of the baryon abundance at Ω bh 2 = 0.02218 ± 0.00055 (using PDG-recommended light element abundances) in ΛCDM and Ω bh 2 = 0.02196 ± 0.00063 when additional ultra-relativistic relics are considered (ΛCDM + N eff). These additional relics themselves are constrained to ΔN eff = -0.10 ± 0.21 by light element abundances alone.


Introduction
The abundance of baryons in the universe is one of the most important but least discussed ingredients of the ΛCDM model.Indeed, it is critical for almost any discussion of its thermal history, playing a role throughout the big bang nucleosynthesis (BBN), the primordial baryonic acoustic oscillations (BAO), the release of the cosmic microwave background (CMB), and the formation of the earliest stars and galaxies.As such, a precise knowledge of this fundamental parameter of any cosmological model is absolutely crucial for almost all predictions.Yet, compared to more exotic parameters such as the dark matter (or total matter) abundance or the dark energy characteristics, this parameter is comparatively less discussed.
While the parameter is well constrained from the CMB observations [1], in view of the emerging tensions such as the Hubble tension (now reaching beyond 5σ significance) [2][3][4], it is crucial to find independent internal consistency checks for this model parameter.The light element abundances that were generated during BBN offer such a crucial consistency check [5][6][7].As such, this short update/review paper is dedicated to presenting an approximate snapshot from early 2024 of the current state of the baryon abundance determinations from BBN (both from the theoretical and the experimental side), as well as a brief pedagogic review of the importance of the baryon abundance in cosmology.
The issue of the baryon abundance is particularly timely now due to the advent of high-precision surveys of the CMB (ACT, SPT, CMB-S4) and of the large-scale structure (Rubin, Euclid, DESI, etc.).Both of these cosmological probes are fundamentally connected to the acoustic oscillations of the baryons, which imprint both as the main oscillations of the CMB and as oscillations and shape of the large scale structure power spectrum. 1The speed and the corresponding wavelength of the BAO (the sound horizon) is directly related to the baryon abundance.As such, the measurements of the baryon abundance play a crucial role in calibrating the sound horizon standard ruler, allowing the BAO to be used as a probe of the Hubble constant (see for example [8][9][10][11][12][13][14]).
Similarly, the suppression of the power spectrum related to the ratio of abundances of baryonic and cold dark matter can be used as a further probe of cosmology once calibrated.This is what gives the approaches fitting the full power spectrum (full-modeling [15][16][17][18][19][20][21][22][23][24]) as well as the ShapeFit approach (see [25][26][27]) their constraining power.While in future surveys there is a distinct possibility of determining the baryon abundance from these large-scale structure surveys themselves, the abundance as derived from the BBN serves now and will continue to serve as a crucial cross-check.Furthermore, with recent theoretical and experimental advances, predictions of the light element abundances from BBN are now more accurate than ever before.
We give a short pedagogic summary of a few theoretical aspects in Section 2, describe the selection of codes and data used in this work in Section 3, summarize our results in Section 4, and conclude in Section 5. Any reader versed in the theoretical aspects of BBN can safely skip to Section 3.

A short theory summary
The aim of this section is not to review all of BBN -we leave this task to excellent reviews such as [28,29] or the works describing the respective codes such as [5,30,31].Instead, this section aims to give a short description of the most important aspects allowing the reader to quickly develop a physical intuition of how the different parts of BBN work and how the abundances are connected.
After the primordial quark-gluon plasma condensed into protons and neutrons (the only relatively stable baryons), the universe is dominated by a bath of photons and neutrinos, electrons and positrons, and protons and neutrons.Importantly, due to baryogenesis, the number of photons is around 10 9 times larger than the number of baryons ( At around T ≃ 0.7MeV almost simultaneously the electrons and positrons begin their annihilation (heating up the photons and neutrinos) and the weak reactions freeze out.The freeze-out of the weak reactions decouples the neutrinos from the primordial plasma and prevents further proton-neutron conversion reactions (such as ν e + n ↔ e − + p + or n + e + ↔ νe + p + ), causing a freeze-out also of the neutron and proton abundance -up to the residually allowed (but much slower) β − decay (n → p + + e − + νe ).As such, after this threshold is reached, the neutrons start slowly decaying with an initial abundance given approximately by their Boltzmann factor (n n /n p ≃ exp(−(m n − m p )/T ) ≃ 0.15) until they are fused into the heavier elements.
One could expect that any heavier element could easily be fused by just combining enough protons and neutrons, but this is not true.The fusion of heavier elements is suppressed by higher powers of the tiny baryon-to-photon ratio (η b ), and thus the elements always have to be built up in subsequent fusions from already existing lighter elements.Given that the fusion into Deuterium only happens relatively late (see below), this 'Deuterium bottleneck' is the fundamental reason why so much primordial matter remains in relatively light elements.
Because the Deuterium binding energy is around 2.22MeV it may seem contradictory that the fusion of such Deuterium nuclei begins only long after the weak freeze-out threshold at 0.7MeV.The solution to this puzzle is that the fusion of the light nuclei is constantly opposed by photodisintegration, and since the photons outnumber the baryons one to 10 9 , the Deuterium nuclei can still be disintegrated by high-energy photons in the high-frequency tail of the photon phase-space distribution, even when the bulk of photons carry an energy much too small to do that.This fact is enough to delay the fusion of Deuterium nuclei until the mean photon energy is around T ≃ 0.07MeV.
As such, Deuterium fusion only begins relatively late.At that point further fusion to heavier elements (like Tritium or Helium) is highly energetically beneficial, and the Deuterium is almost immediately fused away. 2 This immediate process of converting Deuterium to heavier elements like Tritium and Helium-3 (often dubbed 'Deuterium-burning') is the reason why the remaining abundance of Deuterium in the primordial plasma is only around 10 −5 relative to the Hydrogen.This represents a small enough abundance such that further fusion reactions are too inefficient to happen frequently.
The efficiency of heavier fusion reactions means that almost all neutrons that are initially used up for Deuterium are further fused into 4 He.Indeed, a simple estimate can be made that all neutrons are used up in the Helium production, leading to a fraction of around Y p ≃ 0.24 (which is very close to actual predictions).
We can thus summarize as follows: • The formation of the elements is delayed to T ≃ 0.07MeV due to the small baryon-to-photon ratio η b and begins with the formation of Deuterium.
• Deuterium is almost immediately burnt up, with the reaction rates of these fusion reactions being fundamentally important for the final deuterium abundance (see Section 3).Since the start of the Deuterium burning is tightly connected to the baryon-to-photon ratio, the Deuterium abundance is also very tightly connected to the baryon abundance (more baryons → higher temperature Deuterium fusion → stronger Deuterium burning → lower final Deuterium abundance).
• The final Helium abundance is mostly determined by the time that the neutrons have to decay between the freeze-out of weak reactions and the start of nucleosynthesis (as well as the nuclear rates).As such, the abundance is very tightly connected to the overall Hubble expansion rate (connecting physical time to temperature energy scales), which at this early time is given directly by the neutrino number ∆N eff (higher N eff → higher Hubble rate → younger universe → more neutrons → more Helium).These relationships are also summarized in Fig. 1.It should be noted that a higher Hubble rate will also cause a faster freeze-out of the Deuterium burning, leading to a higher Deuterium abundance.Thus the green line and the grey strips in Fig. 1 are not vertical but slanted.Instead, the final Helium abundance (blue line and orange strips) does not strongly depend on the baryon abundance for the aforementioned reasons and is almost horizontal.0.2448 ± 0.0033 Aver+2021 [37] Table 1.A comparison of different data sets used in the community and employed within this work.The data titled "BAO+BBN papers" was used in [13,14], the data titled "PDG Aug 2023" is the most recent recommendation of the particle data group (PDG) at the time of writing, and the data titled "Yeh+2022" is the one that is employed by the [36], which the PDG uses as a reference for their Ω b h 2 constraint.Notes: The YP data of Aver+2021 [37] includes LeoP (already in [38]) and AGC 198691 compared to Aver+2015 [35].The DH data from [36] cites references [34,[39][40][41][42][43][44][45] for their value, but the given result of their eq.(1.4) is neither the weighted nor unweighted average of these measurements.One possible explanation would be obtained by rounding the measurement of [42] (2.547 ± 0.033) to the nearest digit of precision, but the authors of [36] do not describe their procedure of summarizing the D/H data in more detail.

Method and Data
There are several different combinations of Deuterium and Helium measurements commonly used in the literature.In this work we make use of the data described in Table 1, representing some of the most up-to-date measurements of light element abundances.Note that we discuss the anomalous Helium measurement from [33] in Section 4.3.
As far as the theoretical predictions for the light element abundances are concerned, there are two main approaches that modern codes usually adapt, related to the treatment of the underlying nuclear interaction cross sections.Either the energy/temperature-dependence of these cross sections are computed from theoretical ab-initio computations (such as in the PRIMAT code [7,30]), or are interpolated from databases of measured values (such as in the PArthENoPE code [31,46,47]). 3For the cosmological abundances of Helium and Deuterium the most important differences between these two approaches are in the Deuterium burning reactions, namely the radiative capture (dpγ) and the two transfer reactions (ddn, ddp) where n is a neutron, p is a proton, γ a photon, 2 H is Deuterium, 3 H is Tritium, and 3 He is Helium-3.
Very recently the LUNA experiment has measured the dpγ [54] radiative capture cross section at BBN energies.This allowed both methods/codes to update this crucial rate, differences remaining now mostly with the two transfer reactions ddn and ddp (see also [55]).
Even more recently, a new code has been released (PRyMordial [56]), which can represent both calculation types and has a very simple interface that can be used to marginalize over uncertainties related to the reaction rates. 4Note that this is also in principle possible for PArthENoPE, though that would require more severe modifications of the underlying likelihood code structure.
As such, in terms of codes used to predict light element abundances, we use the following: • The old PArthENoPE v2.0 [31] code, which does not incorporate the radiative capture (dpγ) measurements from the LUNA experiment.
• The new PArthENoPE v3.0 [57] code, which does take into account these new LUNA measurements (as well as somewhat more recent measurements of ddp, ddn from [58]).
• The recently released PRyMordial [56] code, which can be run in two states: -In its first state, it uses the NACRE II database [59] of thermonuclear rates for obtaining the light element abundances (updating the radiative capture with LUNA results, and the Berillium7-neutron to Lithium7-proton cross section with the fit from [50]) -In its second state, it uses the reaction rates tabulated in the PRIMAT code (which themselves are based on theoretical ab-initio calculations like [60][61][62][63][64][65], most notably [61] for the key Deuterium burning reactions) This choice of codes allows us to compare the predictions amongst all different modern BBN codes and to test the impact of the various assumptions going into the measurement.In particular, we represent the theoretical ab-initio calculations with the PRyMordial code in its PRIMAT-driven mode, and the calculations based on experimental fits will be represented either by PArthENoPE v3.0 (without explicit marginalization over reaction rate uncertainties) or by PRyMordial in its NACRE II state (with explicit and more conservative marginalization over reaction rate uncertainties).It should be noted that in [13,14] and here for PArthENoPE there is no explicit marginalization over the uncertainties over the nuclear rates.Instead, an approximate theoretical uncertainty on the final abundance was derived from the main results reported in [47] (6 • 10 −7 for D/H).In the results derived from the PRyMordial code the marginalization is instead taken into account explicitly, by marginalizing over log-normally distributed rate uncertainties as described in [56].If this marginalization was not performed, the uncertainty would be dramatically smaller (dominated instead only by the experimental uncertainty), as we show in Fig. 2. It should be also noted that this marginalization performed in the PRyMordial code is overall more conservative compared to the rate uncertainties adopted in [47], which are derived from those discussed in [5].
We put flat priors on the baryon abundance and the additional neutrino number (where applicable).The neutron lifetime is varied freely between 876.4s and 882.4s with a flat prior, conservatively encompassing the currently measured value from bottle experiments (877.5 +0.5 −0.44 s [66]). 6

Results
In this section we find the results for various combinations of codes and data, in order to find conservative and reliable estimates of the current baryon abundance (and the effective number of neutrinos).
For this, we investigate in Section 4.1 the various codes and in Section 4.2 the different available datasets, discussing the anomalous Helium measurement in Section 4.3.Finally, we discuss the impact of varying the effective number of neutrinos in Section 4.4.A summary of all results may be found in Table 2.

Sensitivity to employed code
As a first step we compare the old results from the PArthENoPE v2.0 and PArthENoPE v3.0 codes to highlight the impact of the deuterium radiative capture rate (dpγ) as measured by LUNA [54].For this purpose we use the data from the BAO+BBN papers (see Table 1).This comparison is shown in Fig. 3, comparing the red versus blue lines.
It is immediately evident that there is a large impact of the new measurements of the nuclear rate -the different deuterium burning rates lead to a decreased deuterium abundance for a given fixed Ω b h 2 , requiring lower baryon abundances to produce the same deuterium abundance due to their anti-correlation -which had also been pointed out in [14].The result is a shift in the mean of Ω b h 2 by around −0.9σ from Ω b h 2 = 0.02271 ± 0.00038 with the old code to Ω b h 2 = 0.02236 ± 0.00034 with the new code without a significant change in the uncertainty (see [14] for a discussion on the uncertainties). 7  There are two important questions to answer at this point: 1. Is there a large difference between the different nuclear rate computations?This question is answered by comparing the blue and yellow lines in Fig. 3 (comparing PArthENoPE v3.0 as a stand-in for the experimental rates and PRyMordial in its PRIMAT-driven mode as a stand-in for the theoretical rates).One can observe that there is indeed a decent shift, from Ω b h 2 = 0.02236 ± 0.00034 to Ω b h 2 = 0.02195 ± 0.00021, 8 the two results differing in mean by around 1.0σ (and the uncertainty being smaller by around 40%).
2. Can the conservative marginalization over larger experimental uncertainties bring the two results back into agreement?This question is answered by comparing the blue, green, and yellow lines in Fig. 3 (the green line represents the PRyMordial code in its experimentally driven mode with the additional marginalization over the uncertainties).Indeed, we find that the corresponding constraint of Ω b h 2 = 0.02231 ± 0.00055 covers all of these results nicely.The corresponding shifts are +0.08σ for PArthENoPE v3.0 and −0.6σ for PRyMordial in its PRIMAT-driven mode (and +0.6σ for PArthENoPE v2.0). 6In any case we do not find a strong correlation of the neutron lifetime with the final derived baryon abundance.Even when the lifetime is fixed to 879.4s (PDG 2019 [67, 2019 update]), the baryon abundance is very similar.A degeneracy is expected if much larger variations of the lifetime of around O(100s) would be taken into account (see for example [68, Fig. 1]) 7 These numbers differ marginally from those reported in [14] due to the method used to derive them from the MCMC chains.In [14] the maximum-posterior intervals as derived by MontePython [69] are reported, whereas here we report the mean and square root of the variance for the given chains.Additionally, for the PArthENoPE v3.0 run, new interpolation tables have been derived including also the neutron lifetime (τn). 8This almost exactly reproduces the published PRIMAT results of [7] and the published PArthENoPE results of [47].To summarize, without marginalization over experimental uncertainties there is a somewhat significant difference between the theoretical ab-initio calculations and the experimental calculations, but when the marginalization feature of PRyMordial is used, the results largely agree between the codes.

Sensitivity to employed datasets
The second kind of difference to investigate is that based on the adopted Deuterium (and Helium) abundance data.In Fig. 4 we show the baryon abundances derived for the different datasets of Table 1 for each of the three adopted codes.The corresponding baryon abundance shifts only very slightly for the different light element datasets, typically by less than 0.3σ.
However, the abundance of the PRyMordial code used in its PRIMAT mode does shift more drastically.There we do see a shift up to 1.4σ towards lower values of baryon abundance when different light element abundances are used.For the datasets, see Table 1.In black and blue we show the parameters compatible with the PDG-recommended measurements for the Deuterium and Helium abundances (mean+1σ), respectively (see Table 1).In green we show the parameters compatible with the EMPRESS Helium abundance measurement (mean+1σ).The BBN computations were performed here with PArthENoPE v3.0.Finally, the red contours show the baryon abundance (1σ and 2σ) from Planck [1] as a comparison.

The Helium anomaly
Recently the Helium abundance has been measured by the EMPRESS survey performed on the Subaru telescope [33], giving results that are quite a bit lower (Y P = 0.2370 ± 0.0034, at −1.8σ compared to the PDG recommended value [32]) than any other recent measurement in the literature [32].It is thus important to clarify if a much lower Helium abundance has any drastic impact on the derived baryon abundance.
Depending on whether or not marginalization over the nuclear rates is taken into account, the results are somewhat different.To gain a better understanding, we first look at the case without marginalization, where the shift is rather minor (−0.1σ for PArthENoPE v3.0).However, there is an incompatibility of the data and the model which becomes evident in the maximum likelihood.For example, while the minimum χ 2 = −2 ln L reached for the run assuming the PDG-recommended abundances is 0 to the numerical precision of around ∆χ 2 ∼ 0.1 of the MCMC chains, it is around χ 2 min ≃ 4.6 if the EMPRESS Helium abundance is forced.This incompatibility is most easily visualized when extending the parameter space to include additional free-streaming neutrinos through the ∆N eff parameter as in Fig. 5.The only intersection of the Helium abundance and the Deuterium abundance (black contour) that is compatible with the assumption of no additional dark neutrinos (the thin black line at ∆N eff = 0) is for the PDG recommended value of the Helium abundance (blue contour).Instead, the EMPRESS-derived value of the Helium abundance is not compatible with the measured Deuterium abundance and simultaneously ∆N eff = 0 in this extended parameter space.
Instead, if the PRyMordial code is used in either of its modes, the shift is around −1.2σ (see Table 2).This is because the broad simultaneous marginalization over nuclear rates and the neutron lifetime can put such low Helium abundances back into accordance at the cost of putting the various rates at the edges of their experimentally/theoretically allowed regions.If instead we fix the neutron lifetime to a central value of 879.4s in these cases, for example, the results are again within 0.3σ to their baseline values.
As such, in this work we will treat the Helium abundance measurement of [33] as an 'outlier' until further evidence emerges.

Additional relativistic degrees of freedom
The inclusion of additional neutrino-like species through the parameter ∆N eff has a large impact on BBN and can correspondingly be constrained by the light element abundances, providing a constraint of similar strength as from the CMB.However, this constraint is instead much more dependent on the assumed Helium abundance, as visible in Fig. 5 and as described in Section 2. In Fig. 6 we show the two-dimensional constraints obtained from the various codes for the PDG recommended abundances from Table 1.It is immediately evident that the inclusion of additional relativistic degrees of freedom during BBN does not significantly bias the constraints on the baryon abundance, typically leading to a mild widening of the contours as well as a slight downward shift of the mean value (see also Table 2).
The constraint on Ω b h 2 degrades to around Ω b h 2 = 0.02196 ± 0.00063 with PDG-recommended light element abundances once marginalizing over ∆N eff , though a more conservative estimate is given by the older Helium abundance measurements from [35] that were used in the BAO+BBN papers, leading to Ω b h 2 = 0.02212 ± 0.00072.In any case, almost all variations are well within one sigma of each other, and there is no obvious bias.
The constraint on the ∆N eff parameter depends on the tightness of the assumed Helium abundance (as expected from Section 2), with the constraints ranging from −0.10 ± 0.21 (for PDGrecommended abundances and the PArthENoPE v3.0 code) up to −0.09 ± 0.28 (for the abundances used in the BAO+BBN papers and the PArthENoPE v3.0 code).See Table 2 for further results.

Conclusions
The state of the baryon abundance as inferred from measurements of the light element abundances from big bang nucleosynthesis is at this point relatively clear.While there is certainly still a spread of the constraints between various codes (and their underlying assumptions about the Deuterium burning rates), the field has come a long way of more systematically characterizing these uncertainties and biases.With relatively conservative assumptions about the reaction rate uncertainties it is possible to derive a baryon abundance estimate that nicely covers all available results in terms of available codes and datasets.For this purpose, we recommend the value derived by the PRyMordial code using the NACRE II database, giving Ω b h 2 = 0.02218 ± 0.00055 , (5.1) with PDG-recommended abundances [32], and Ω b h 2 = 0.02231 ± 0.00055 , ( with the most recent Deuterium determination from [34].Notably, the inflated uncertainties compared to the analyses performed in [7,32,36] and many others is related to a broader marginalization over the uncertainties in the nuclear rates, leading to more conservative estimates that encompass both the theoretical ab-initio calculations and the experimental rates for the ddn and ddp Deuterium burning processes. These values are largely insensitive to the assumed Helium abundance, as long as it remains within a range compatible with the standard BBN codes.The EMPRESS result [33] would correspond to a much lower abundance, but this is an effect of the breakdown of the compatibility with standard rates and assumptions of the BBN codes -which can be circumvented only in more exotic neutrino models (see for example [70,71]).This result is also (mildly) incompatible with the collection of all previous results.As such, until further evidence in this direction emerges, we consider this result an 'outlier'.
As far as more exotic cosmologies with additional neutrinos are concerned, we find that here too results are mostly consistent, pointing to roughly the same baryon abundances.We find in ΛCDM+∆N eff that Ω b h 2 = 0.02196 ± 0.00063 (5.3) with PDG-recommended abundances, and Ω b h 2 = 0.02212 ± 0.00072 (5.4) with the older estimates from the BAO+BBN papers) and the amount of additional neutrino species constrained at the level of around ∆N eff = −0.1 ± 0.2 (see Table 2 for precise values).
While further cosmological measurements of the Deuterium and Helium abundances will also continue to be necessary (especially to cross-check the previous measurements), the largest advances in our understanding of the light element abundances from big bang nucleosynthesis are expected to come from further laboratory measurements of the Deuterium burning rates, which currently represent the most crucial uncertainty in the estimation of the baryon abundance.As such, we expect the future of the baryon abundance as determined from big bang nucleosynthesis to bring even higher accuracy and precision.

Ω b h 2 − 2 Figure 1 .
Figure 1.Light element abundances as a function of baryon abundance Ω b h 2 and number of effective additional neutrino species ∆N eff .The green and blue line and contours represent the mean, and the 1σ and 2σ contours of the Deuterium and Helium recommendation from PDG [32], respectively.Computed from the PArthENoPE v3.0 code.

Figure 2 .
Figure2.Impact of marginalization over nuclear reaction rates for the PRyMordial code, compared to the PArthENoPE v3.0 adopted uncertainties (as described in the text).The remaining width of the PRyMordial code without marginalization directly represents the uncertainty of the deuterium determination.

Figure 4 .
Figure 4. Comparison of the baryon abundance constraints from different data sets when using the PArthENoPE v3.0 code (top left), using the PRyMordial code in NACRE II mode (top right), and using the PRyMordial code in its PRIMAT mode (bottom).For the datasets, see Table1.

Figure 5 .
Figure 5.A comparison of various constraints in the ∆N eff − Ω b h 2 plane.In black and blue we show the parameters compatible with the PDG-recommended measurements for the Deuterium and Helium abundances (mean+1σ), respectively (see Table1).In green we show the parameters compatible with the EMPRESS Helium abundance measurement (mean+1σ).The BBN computations were performed here with PArthENoPE v3.0.Finally, the red contours show the baryon abundance (1σ and 2σ) from Planck [1] as a comparison.
Comparison of the Ω b h 2 constraints in the ΛCDM model between different codes (or settings) for the same underlying data titled "BAO+BBN papers" in Table1.

Table 2 .
The numerical value of the constraints (mean and standard deviation) of the various runs considered in this paper.