Gravitational waves from the early universe

The authors present a set of compact lecture notes containing several relevant topics to understand cosmic sources of gravitational waves and their detection. I appreciate the effort that the authors have put into writing the lecture notes. But the target readership is not clear to me. Sometimes the explanations are very heuristic, sometimes they become very technical. I list some comments/questions below that will improve the lecture notes.

General comments: 1. Please clearly specify the target audience. Is the target audience advanced graduate students familiar with some cosmology? What is the necessary background? 2. Sometimes the explanations (I say more below) are not rigorous. Although at some point they encourage the reader to check other references, I recommend writing a small paragraph at the end of the Motivation section (which I would call “Introduction and Motivation”), explaining that the aim of these lecture notes is to provide an “entrance door” for the graduate student to cosmic gravitational waves but that for rigorous and detailed explanations the reader should read instead certain references. Then, below that paragraph, please add a list of references on which the notes are based, sorted by topic. This will be very helpful to have clear expectations for the lecture notes, and it will be very useful for students to delve into more specific literature. 3. In a similar direction, I encourage the authors to add a short paragraph at the beginning of each section specifying the main literature used. Then, the readers can directly find the relevant references in case of doubt. 4. The motivation section has technical terms that are not defined, such as cosmic inflation and stochastic GWB. This could be solved by explaining Figure 1 earlier with a short introduction to the standard model of cosmology. 5. Below Eq.(2.13) the authors say that the Riemann tensor is a physical quantity. This is not correct as it depends on the coordinate system. In flat spacetime, it is true, though, that the Riemann tensor is invariant under linear gauge transformations as it has vanishing background value. Then, it can be related to the effects of GWs through the geodesic deviation equation. But in general, the Riemann tensor is not “physical” per se. 6. The time dependence of h_{ij} in Section 3.1 needs better explanation. If I understand correctly, most of the equations in section 3.1 make sense in the far zone regime (i.e. x>>y) where one can drop the y dependence in the t-r/c argument inside T_{ij}. 7. I am confused by the statement below Eq.(3.13) “the question arises whether the curvature is a GW or part of the background”. What does this mean? Also, they say “In the latter case, it can locally be gauged away.” But this is always true. Locally, there is no effect from gravity as one can go to the local inertial frame. Since they are focusing on Minkowski spacetime, it maybe good to add a short paragraph explaining the set up. For example, if we try to measure the GWs in a box of length L, etc. This gives a notion of scale. The background in Minkowski has no scale. If they are thinking about a cosmological setup. Then, they need more explanation. I understand that one can do the calculation in general by splitting background and perturbations, but one must be a bit careful. In a general setup, with no symmetries, it is difficult to think of GWs. It’d be better to specify that they are working on Minkowski or Cosmological spacetimes. 8. Below Eq. (3.15), there is some analogy with renormalization group approaches. Although it is a curious analogy, I do not think it helps much. Are the students supposed to be familiar with renormalization group approaches? The setup is quite different, though, in my opinion. We are not integrating out anything but rather looking at the average effect of the GWs. There is no renormalization involved. If the authors want to keep this paragraph, they must state that this is only an analogy. However, I recommend putting it as a footnote. 9. The energy momentum tensor of GWs, Eq. (3.16), is a Pseudo-tensor. It would be better to call it (Pseudo) Energy momentum tensor. It is only well defined in linearized gravity but becomes ambiguous at higher orders. It is in general not conserved, only at linear order. 10. The discussion in Section 3.4 is a bit confusing. It would be better to state that Eqs. (3.27) and (3.28) are equivalent but that one usually chooses one for convenience. 11. Above Eq. (3.29), it is mentioned that “In turn, we can show that the quantum theory of cosmological perturbations allows us to determine […]”. Why so technical suddenly? Eq.(3.29) follows from demanding Homogeneity and Isotropy of the background. Nothing to do with quantum fluctuations. 12. Maybe it would be good to mention that the GW background from inflation is, in fact, non-stationary (see https://arxiv.org/abs/gr-qc/9906054). But becomes stationary for all practical purposes. 13. In section 4.1, the authors say, “It is then very similar to the cosmic microwave background (CMB) from the electromagnetic spectrum.” Qualitatively, this is more or less correct. But they are very different in nature. The CMB is more of a snapshot, while the GWB is actually found using time correlations. The way they are studied is very different. While sometimes the CMB analogy helps, sometimes it does not. 14. The authors list PBHs as early universe sources that emitted GWs in the past. This is not correct. PBHs, if not evaporated, merge today and emit GWs in the nearby universe. If they mean the induced GW background associated with PBH formation, then say explicitly so. 15. This may be my ignorance, but in section 4.3.1, the authors say “which list all the events that have at least 50% of probability of being real astrophysical events rather than noise”. Is this correct? A 50% probability of being noise is just a fifty-fifty chance we get things wrong. 16. In Section 4.3.1, the authors use the common analogy that the strain changes the length of the interferometer’s arms. Note that this is a heuristic statement. What is being measured is the time delay of the electromagnetic waves, which causes the interference. It would be better to be clear about this. This is the same effect (time delay) that is used in PTAs, although in a different setup. For example, in Section 4.3.3 the authors state that PTAs do not work like interferometers because they rely on the time delay. It’d be better to be clear that both measure time delays but use different methods to measure it. 17. End of page 22, please mention that one can do the calculations in the different frames (TT frame and detector frame) for convenience, but they are equivalent. 18. Section 5.2.1 says that the maximal accessible frequency at PTAs is the cadence time. This is more or less correct. However, the cadence time could, in principle, be arbitrarily high and reach arbitrarily high frequencies. But the truth is that one will hit a limit of Poisson noise before that. 19. In footnote 22, the authors say that “we would like to integrate overall non-vanishing GW solutions hA( f ,ˆΩ)”. I do not understand what the problem is. Maybe it comes from the renormalization group analogy that the “random SGWB realizations” should contain all possible realizations. But this is a classical system, if there is no background, then there is no background. Section 5.2.3, is about measuring the background and correlations describing it. Footnote 22 adds unnecessary confusion. 20. In page 31, the authors talk about a divergence when f_a=f_b. Then they say “so it is sufficient for us to keep the discussion in the Fourier space (where this divergence is swept behind the carpet if the reader wishes) without integrating over time.” This sounds like we are cheating, and we do not know what we are doing. This is just an artifact of impossing f_a=f_b beforehand in Eq. (5.30). Equation (5.31) is totally fine. Please do not confuse the reader. 21. Equation (5.35) has a \delta_{ab} term. If I am correct, this is necessary because they have dropped the Pulsar term. But, if one does the calculations properly, the discontinuous jump due to “ \delta_{ab} “ is not needed. 22. Paragraph before section 5.2.5, why anisotropies are expected to be so small in the astrophysical set up? For cosmic GWs, I agree that the expectation is that they are like the CMB (although we do not know, they could be larger in some models). But for astrophysics, it should be quite anisotropic and dominated by nearby sources, right? 23. Eq.(6.5) is not the Friedmann equation. It is the fluid equation coming from the conservation of the energy momentum tensor. 24. It is not 100% correct to think of the Hubble radius as the radius of a 2-sphere. This gives a notion of “instantaneous” Hubble radius. I understand that the authors want to give some intuition by saying that “everything beyond its radius cannot be causally connected to the interior of the 2-sphere.” But it is not causally disconnected from the interior, but the center. This misinterpretation comes from thinking of a 2-sphere. The Hubble radius gives a rough idea of the causally connected region from the center, but it is a spacetime quantity. It roughly says how far a photon or a GW can travel in a Hubble time. Please be a bit more precise. 25. Below Eq. (6.20), the authors say “This mechanism is the same as the one occurring in inflation, discussed in” The reader is not familiar with inflation, so why say this? Also, this is no mechanism. What the authors are describing are properties of the solutions for tensor modes. It is a general fact that in the absence of a source, they are constant on super-Hubble scales. Nothing to do with inflation. When the authors say “The re-entry time is the time when the GWs are produced.” this assumes super-Hubble initial conditions from inflation. Please extend the explanation or move it when inflation is discussed. 26. Beginning of page 41, the authors say “Why is ΩGW(k,τ0) relevant?” but the answer is not really correct. We look at \Omega_GW because it is related to the noise strain power spectrum measured <h_{ij}h_{ij}>. GW detectors do not care about the energy density of GWs in the universe. But on the strain at the detector. We simply use \Omega_GW for convenience. 27. Section 7 becomes very technical. Here, it is important to give useful references at the start of the section, sorted by topic. Otherwise, any student will get lost very easily. 28. The title of section 7.1 is confusing. Are the authors discussing the GWs from the thermal plasma? If so, it has nothing to do with CMB scales. I understand Eq.(7.1) is useful, but \Omega_CMB probably means \Omega_{radiation} today. 29. Section 7.4, induced GWs are sourced by first-order scalar modes at second order in perturbation theory. That is, the source is squared in first-order scalar modes. Also, the authors use Fig. 19 as analogy with quantum field theory but usually the calculations of induced GWs are a classical system. Namely, one has density waves propagating in the universe which induce GWs. So it may be a good idea to say that the Feynman diagrams are only used for calculations, but that in standard situations there are no quantum effects.

Specific comments: 1. Why do the authors use the word “could” in the abstract? We can and will be able do it, so I would use “can” instead. 2. Page 3, the sentence after "These first photons are still visible today as the Cosmic Microwave Background (CMB), as background noise from all directions" would be better if it starts with “In contrast, before photon decoupling […]. 3. Page 3, the authors say “the GW background is expected to be noise from all directions”. But that’s the definition of background, not an expectation. It’s more correct to say “the GW background would be seen as noise from all directions.” 4. Page 3, “Finally, different early-universe sources can produce GWs.” -> “Lastly, […] 5. Page 3, “the gravitational wave background is a laboratory to probe new physics that is complementary …”, it is not complementary (in the strict sense), it goes beyond current probes. It will probe the unexplored Universe. 6. Section 2, below eq. (2.1), “and the indices are raised with ημν”. I guess the authors mean the indices after the perturbative expansion. Otherwise, they are raised by g_{\mu\nu}. 7. Section 2.1, above eq.(2.2), Einsteins equations —> Einstein’s equations 8. Section 2.2, below eq.(2.9), I am not sure if the notation “gauge symmetry” is used for general covariance or diffeomorphism invariance. It is usually employed after fixing a background metric. Then one talks about gauge transformations and gauge invariance. But I would call it coordinate reparametrization invariance. Besides, general coordinate transformations also have their meaning and can be interpreted as the metric seen by different observers. This is why I think “gauge symmetry” is not appropriate at that stage. 9. Below, eq. (2.9), the authors say “The linearized theory, however, is only invariant under infinitesimal coordinate transformations and finite, global Poincare transformations." What do they mean by that? What is invariant? 10. Also, the “Minkowski background has a lot of rotational symmetries” could be more precise. It has 3 spatial translations and 3 spatial rotations, plus the time translation and 3 Lorentz boosts. 11. Why is there a scale factor “a” in eq. (2.10)? 12. Maybe the author’s want to mention that Eq.(2.13) corresponds to the lie derivative along \xi. 13. Below Eq. (2.14), they say “We can show that the other vector and scalar parts do not dynamically propagate” do they mean “one can show […]”. Also, a citation is needed. This is far from a trivial statement. 14. Above Eq. (2.15), the authors say “The Lorenz gauge is always applicable”, I guess they mean in Minkowski spacetime. In cosmology, there are more degrees of freedom. 15. Below (2.29) it may be a good idea to state that GWs propagate at the speed of light in General Relativity. 16. I am a bit confused by the introduction of the tetrads in Eqs. (2.34)-(2.36). Are they needed? If yes, please define them before. 17. I think Eq.(2.37) should equate to zero. 18. The T_{\mu\nu} in Eq.(3.4) may be missing a second argument. 19. In Eq.(3.18), the average of the second-order Ricci scalar vanishes. Why is that? 20. Below Eq. (3.21) the authors demand \partial_\mu t^{\mu\nu}=0, but this should follow from the definition of the energy momentum tensor of GWs. It’d be better to be precise here. 21. Below Eq. (3.26): “electromagnetic case can be gained in”—>” electromagnetic case can be gained from Ref.”. 22. Why do the authors use the symbol ~ in Eq. (3.30)? It’d be better to be precise. 23. Section 4, it would be better to provide the most relevant references before section 4.1. This is related to my 3rd general comment. 24. Section 4.1, “this background can seem like a random noise” —> “ this background is a random noise”. 25. Section 4.1, “the background behaves like useless random noise” —> Why useless? 26. End of page 18, “On the other hand, they would mask a signal from the cosmological background, which is much weaker, mostly because of the large redshift.” A priori, it does not have to be weaker. That’s the general expectation. 27. Typo in Section 5, “reference fro” -> “reference for” 28. Above eq. (5.11) The residual —> The residual (time delay). 29. Eq. (6.14), why is the minus sign in front of the scale factor? 30. Eq. (6.15) please define the Box operator. In general, the d’Alembertian applied to a tensor has also the riemann tensor and terms proportional to H. In my opinion, it is better to define box as the d’Alembertian applied to a scalar, that is \box_s=1/\sqrt{-g} \partial_\mu(\sqrt{g} \partial^\mu). Then the equation is correct and does not need a’/a.

Ask for major revision

Publish (meets expectations and criteria for this Journal)