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Dear Editor,

We thank the referee for a very detailed report. We feel that his/her comments have dramatically improved the clarity and readability of our paper.

We have addressed all the points raised by the referee. We include in this reply an answer to each comment. We have included the file diff_cosmic_voids.pdf showing the differences in the body of the paper.

The Authors.

Conceptual:

  1. It seems that part of the motivation for using FA is that it is very well correlated with lambda_1, at low lambda_1. FA is a rather interesting quantity in itself, differentiating haloes and voids from filaments and walls, but it is not clear why it should be superior to lambda_1 for void finding. The authors should more clearly articulate why this might be, if there is a reason. The FA being "conceived to quantify the anisotropy of the diffusivity of water molecules through cerebral tissue in nuclear magnetic resonance imaging" should be elaborated on; this does not clarify the usefulness of the FA for cosmologists. Conceptually, I also wonder: do filaments and walls behave differently from each other in FA, or do they both have FA ~ 1?

    Response:

    • Although lambda_1 is the defining property of voids, the FA has a key advantage that makes it a better void tracer, namely that it is possible to define a boundary in regions with FA~1. Boudary setting becomes arbitrary in the case of using lambda_1 as a tracer because lambda_1 is not bound. We make this clarification in the 3rd paragraph of subsec. 3.2 of the new version.
    • Now it is explicitly explained why the FA, as defined in the context of MRI, is useful to cosmologist
      (6th paragraph of introduction and 1st paragraph of subsec 3.1).
    • We prepare a new Figure (Fig. 3) to shows that filaments and sheets have different behavior in FA. Only sheets are associated with high values of FA~1.

  2. I did not understand the sentence at the end of Sect 2, "In this work we propose an optimal value of the threshold based on the maximization of the fractional anisotropy field in the locations label[ed] as filaments and walls," and this seems to be an important point. Is the threshold mentioned in this sentence a threshold in lambda_1, or FA? Is this the 0.95 threshold in FA? It seems to be a reasonable choice, but I do not see a sense in which it is "optimal," except that it seems to be the boundary between the FA being well-correlated with lambda_1 and not-so-well. Also, if this sentence is about the 0.95
    threshold, I do not see why it "does not enter into our computations," since the boundary of many voids will depend on this threshold. It would be useful to know for what fraction of voids the limit FA = 0.95 actually affects the boundary (in the case of the largest voids, I suppose).

    Response: We have rewritten this paragraph. The threshold that is irrelevant to our computations is lambda_th, not FA=0.95.

  3. I do not see how the value of FA=0.65 (corresponding to lambda_1 = 0) is used "to remove ridges."

    Response:

    • We have improved our explanation about the ridge removing procedure in the 3rd paragraph of subsection 3.3.

  4. In Fig. 6, I only see a clear ridge in FA at rr_eff in a few cases. the sentence in Sect. 5.5 "The difference between the radius where the density ridge is reached ([r/]reff = 3) and the radius of the FA ridge ([r/]reff = 1 ) ..." implies that I should see a ridge in FA at rr_eff. Is this because the ridge is actually a 2D surface that is spherically averaged? It would be helpful to point this (or any other reasons) out. Also, the statement "This makes the FA ridge a reasonable boundary for voids compared to the traditional definition that puts the boundary at the density ridge" implies (perhaps unintentionally) that the FA ridge is a superior marker of the edge of a void than a density ridge. It is a physically different sort of ridge, and I agree that it is reasonable, but I do not see why it should be superior.

    Response:

    • We now talk about "maxima" instead of "ridge" in the FA profiles. We have also removed the references of the FA as a "superior" characterization.

  5. I am not convinced that the "tiny white bubbles located inside sheets" are embedded in overdense regions. Based just on Fig. 1, they could be physical, but small, voids. We also no nothing of their extent perpendicular to the plane of Fig. 1; we could just be looking of corners of much larger objects.

    Response:

    • In the right panels of Fig. 1, the color of each void refers now the effective radius. The size could be inferred from the figure. We have checked by visual inspection some cases and small voids are indeed located inside sheets and filaments. This was already found and explained in Hoffman et al. 2012.

  6. 2D histograms of (lambda_1, delta) and (lambda_1, FA) are shown in Fig. 2, and I would appreciate also seeing a (FA, delta) 2D histogram, maybe with 1+delta on a log scale, which would help to clarify the physical meaning of FA.

    Response:

    • We have included the new plot in Fig. 2. This shows that indeed the overdensity of zero is found at the highest Fractional Anisotropy.

More minor things:

  1. I'm confused a bit about the middle panels of Fig. 1. The title, "Visual impression for lambda_th=..." is strange. Aren't there several thresholds here? Also, what are the blue dots? Also "impresion" is misspelled.

    Response:

    • We have rewritten the caption. There are two different thresholds, one for each web finder. Blue dots were dark matter halos (FOF). We removed them for clarity. The typo ("impresion") is also corrected.

  2. Sect 3.3, 2nd paragraph: I do not see why "the analysis of [y]our results" shows that a fixed Cartesian mesh does not induce spurious results". It's true, that the results look reasonable, but I think it would be better to state something about the level of discreteness noise, e.g. the number of zero-density cells, or the minimum if there are none.

    Response:

    • We have rephrased this in terms of the number of particles per cell in the lowest density regions.

  3. Sect. 5.4: The sentence "subcompensated voids have outflowing ve- locity profiles all the way up to the effective radius where the average radial density reaches the average value" is a bit confusing; this is not r_eff, right? If so, this statement seems to be an understatement (v_r >0 out to several r_eff, typically). If not, "effective" should be removed for clarity.

    Response:

    • We removed "effective" for clarity.

  4. In Sect. 5.5, there are a few instances where I think you meant "r/r_eff =" instead of "r_eff =".

    Response:

    • Indeed. We have revised all the instances and replaced r_eff by r/r_eff wherever it was appropriate.

  5. All quantities are computed at the pixel scale, ~1 Mpc/h, correct? If the authors have any insights about how the results would change if this scale were changed, it would be helpful to state them. A quantitative discussion of how some results would change with a different scale would be great, but not necessary.

    Response:

    • We now mention in the first paragraph of the Results section the expected results for larger smoothing scales. Namely, that the statistics for the cosmic web tend to be those of a random Gaussian field.

  6. Finally, about the necessity of color: Figs 4-6, as they currently are, require color, but it would be easy for the authors to use different linestyles, as well, to make color unnecessary.

    Response:

    • These figures have already different linewidths for each bin. We think it might be easier to have 6 different widths than 6 different linestyles.