Primordial nucleosynthesis in the era of cosmological tensions

1- An extremely clear presentation of BBN. Very concise and excellently explained. 2- I really like the explanation of the importance of $N_\rm{eff}$ in the formation of the light elements and the extent to which it is constrained by data.

1- Needs correcting for some spelling and grammar mistakes. 2- Figures could be improved by being more consistent with each other (larger fonts, labels, right-hand plot of Fig.1 in particular). 3- I think the lectures would benefit from more explanation in the observational status section.

1- Spelling and grammar corrections: - Synthetized -> Synthesized (abstract, p1) - big bang -> Big Bang (abstract, p1) - 'I also recommend the reviews [5-9] ' should have a comma after the references - tempature -> temperature (p.2) - Commas after Eq.(2) and Eq.(15) - equilbrium -> equilibrium (above Eq.(8) on p. 3) - sastisfied -> satisfied (top of p. 5) - efficients ->  efficient (top of p.5) - Flavour -> Flavor if using US spelling (which was used elsewhere) (3 times on p.2) - 'The mass of nucleons must be taken into account (m_p=...)' -> 'The mass of nucleons must be taken into account (m_p=...),' (should have a comma) - When referring to equations they should be ' in Eq.(1)' ... Not 'in (1)'

2- Typo at bottom of page 6. It currently reads ${}^3\rm{Li}$ but should be ${}^7\rm{Li}$.

3- In Figure 1: Make the labels and legend in the right-hand plot larger. The formatting across the two plots could be more consistent in general. It looks like Latex wasn't used in the labels (in comparison to Fig.2).

4- In Figure 2: The Label for the Planck constraints could be better placed to avoid the line passing through it. The tick marks could be larger.

5- Above Eq.(2) it would be good to be clear that $N_\rm{eff}$ is the effective number of neutrinos: 'All these effects are characterized by an effective number of neutrinos, $N_\rm{eff}$, defined by'.

6- I think you should mention in more detail the Lithium problem. I assume this is what you are touching upon with the sentence 'The abundance of lithium is indirectly constrained via the observation of the surface of very old stars but it crucially depends on the assumption that it is not destroyed in sizeable amounts.' Maybe refer to the figure after this?

7- In Figure 2: make it clear that we are looking at the abundance of the light elements ie abundance of ${}^7\rm{Li}$ with respect to H for example. eg 'observational constraints on the abundance of light elements as a fraction of the abundance of hydrogen. In the top panel we plot the helium fraction defined in Eq.(17).'

8- In Eq.(17) it would be nice to be reminded of what $X_{{}^{4}\rm{He}}$ means. ', where $X_{{}^{4}\rm{He}}=n_{{}^{4}\rm{He}}/n_b$' .

9- How ${}^{4}\rm{He}$ and deuterium are measured could be mentioned in passing in the observational status section (as was mentioned in the abstract eg quasar absorption lines etc).

10- It might be nice to include a sentence or two on how the CMB is used to provide observational evidence.

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Minor corrections:
``sastisfied'' $\to$ ``satisfied'' (p. 5).
``equilbrium'' $\to$ ``equilibrium'' (p. 3).
On p. 6, ``$^3$Li'' is meant to be $^7$Li or ``lithium-7''???
Eq.~(2): briefly remind the reader that $\rho_\gamma$ is the photon energy density.
Eq.~(15): the approximation step could be explained more clearly (mention dropping order-one factors).
Eq.~(17): clarify that $Y_P = 4X_{^4\mathrm{He}}$ corresponds to the helium mass fraction.
Fig.~1: axis labels are a bit small compared to the text; increasing font size would improve readability.
The phrase ``up to unimportant order one numerical factors'' could be rephrased as ``up to factors of order unity, which do not significantly affect the estimate''.

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1- The author provides a very systematic and clear summary of the complex topic of Big Bang Nucleosynthesis (BBN). The article logically progresses from the thermodynamic background of the early universe to the key weak interaction processes that determine the neutron-to-proton ratio, and finally to the nuclear reaction network that forms the light elements. The entire logical chain is very smooth and reader-friendly. 2- It successfully captures the core physics of BBN, such as the neutron "freeze-out" process, the deuterium bottleneck, and the sensitivity of the final elemental abundances to key cosmological parameters like the baryon-to-photon ratio and the effective number of neutrino species . These are the most important takeaways in this field.

1- As a lecture note, the target audience likely includes graduate students and researchers from other fields. Some key equations, such as the integral expression for the weak interaction rates, are presented without a brief explanation of their physical origin. This could be confusing for newcomers to the subject.

1- This manuscript provides an excellent, concise, and accurate review of the fundamental principles, calculation procedures, and observational tests of BBN. The author presents a clear narrative, guiding the reader through the key physical steps of how the first light elements were formed in the first few seconds to minutes after the Big Bang. The article not only correctly presents the framework of the standard BBN theory but also includes discussions on high-precision corrections like neutrino decoupling and finite-temperature QED effects. By integrating the latest experimental data, it highlights the remarkable consistency between theory and observation, which stands as a powerful pillar of the standard cosmological model. 2- Overall, this manuscript is a perfect fit for the SciPost Physics Lecture Notes series. It is systematic, accurate, easy to understand, and covers a topic of ongoing interest to the cosmology and particle physics communities. It serves as an excellent introductory material for students or researchers entering the field.

1- It would be beneficial to add a sentence or two around key physical equations (such as Equations 5 and 11) to briefly explain the physical origin or meaning of the terms. This would significantly enhance the pedagogical value of the lecture note, making the content more accessible to non-specialist readers. 2- Fig. 2 is one of the core figures of the article, and its inset is very useful. It is suggested to slightly increase the font size of the axis labels and numbers for both the main plot and the inset to improve readability.

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