Gravitational Lensing of Supernova Neutrino Bursts

Round 1

Reviewer 1 Report

The basic idea underlying this paper, that neutrinos emitted in a supernova explosion can be focussed by a gravitational lens between the SN and us, is indeed an interesting one.  That SN1987A provides a bona fide example of this possibility is less clear, as the author indicates.  Relatedly, there is the statement near the bottom of page 9:  "There is no known star cluster in the direction of the LMC to provide the mass"  for such a gravitational lens.  This seems to me to be a key point and deserves expansion.  What in fact is the observational limit and how much more sensitive a search could be made with now, or soon to exist, equipment and what would be the consequences for this paper?  If the paper is going to be linked to SN1987A, as now, I would strongly suggest that the author should look into, and determine, the prospects for detecting such a lens in this case and examine the prospects for finding it in the relevant mass range.  If its presence could either be ruled out or confirmed  by such a search would be very important for the interpretation of this paper and would be important knowledge to include in this paper.  I'm not suggesting that the author undertake a search for such a lens for this paper only that he examine the prospects for this paper.  If the detection looks feasible with existing, or soon to exist, observational capability then he should encourage the undertaking of such a search.  If the existence of a lens within the relevant mass range can be ruled out by a previous search or searches, then he should completely drop the tying of his idea to SN1987A.

I did not check any of the derivations.

Small point:  If there is good reason to believe (pages 7 & 8) that the Russian clock was exactly 30 seconds off, why not consider explicitly the consequences of such an error?

There are a number of small "English" errors that should be corrected.  Here is a nearly complete list:

1. Pages 1 and 2: "principal" should be "principle."
2. Define alpha and beta on page 2 (or at least refer there to the definitions on page 3), and consider including derivation of L1 + L2.  Also, "Where" should be replaced with "where."
3. Page 4: "Which" should be replaced with "which" in two places.
4. Page 5: "Where" should be replaced by "where."
5. Page 6:  Re-word statement "careful in overestimating" to reflect what is really meant.

Author Response

Referee 1

I have addressed the question of "the author should look into, and determine,
the prospects for detecting such a lens".  I have expanded the text in the
second paragraph of the conclusions to consider how the lensing hypothesis
might be tested.  Text has been added after "optical confirmation of such a
lens is still possible."  One goal of writing this paper was to encourage
discussion and to communicate with those who have access to tools and records
that could be used to confirm the hypothesis.  I thank the referee for this
suggestion.

My understanding of the 30 second shift in the Soviet clock, was a Cold War
philosophy to deny outsiders access to precise timing information which has
implications for defense.  This is just speculation.  While the paper is
speculative I would prefer to restrict speculation to scientific subjects that
can be verified.  The fit to the lens hypothesis does not depend on the
synchronization of the experiments.

    Pages 1 and 2: "principal" should be "principle." DONE
    Define alpha and beta on page 2 (or at least refer there to the definitions 
on page 3), and consider including derivation of L1 + L2.  Also, "Where" should be replaced with "where."

Ans.  The definitions are mentioned in figure 1.  A sentence has been added to
refer to figure 1 for definitions.
The derivations were all taken from Refsdal's 1964 papers with minor changes
for modern notation suggested by a colleague.  The derivation of
L1 and L2 is in Refsdal's and two other papers from 1964, all cited.  Copying
these into the current manuscript will lengthen it and not add any substance.
I did do the caculation of (L1+L2)/(L1-L2) since it gives access to a very
important parameter alpha/beta.
    Page 4: "Which" should be replaced with "which" in two places. DONE
    Page 5: "Where" should be replaced by "where." DONE
    Page 6:  Re-word statement "careful in overestimating" to reflect what is re
ally meant.
Ans.  "Quality" has been changed to "precision" in this sentence.
I realy meant quality, which includes precision and other things, such as
backgrounds.  But the example was aimed at precision so I have changed it.
Can I please change it back?  The paragraph is needed to point out that the
differences in the caculated values of alpha/beta and M are all consistent.

Reviewer 2 Report

The author describes how neutrino events observed on the earth from a supernova burst could be affected by gravitational interactions in transit.  The author's motivation for discussing gravitational lensing of neutrinos is the possible distortion of the expected observed signal.  The author suggests, cautiously and not strongly, this may be the reason for fluctuations in the SN1987 neutrino signal through there are no known lensing sources and that not all observations support the lensing interpretation. 

The reviewer still thinks that this is a valid note that may having interesting effects in future observed neutrino bursts from relatively close supernova events in our galactic neighborhood. The physics is quite interesting and thus should be considered for publication. 

There are a number of small editorial issues that must be addressed before the manuscript could be accepted for publication:
    1. The references need to be in order.  The first reference is [2] and reference [1] appears on page 10.
    2. The manuscript is riddled with the need for commas.  The lack of commas makes the reading of the paper somewhat distracting.
    3. Equations need to be embedded within sentences.  Often equations are just written on the page.  Again this is distracting to just see a column of equations not separated by commas and conjunctions.  Often there is no punctuation ending a sentence with an equation.
    4. Sometimes variables are not defined (but I understand them from context).
    5. Sometimes I think there may be units that are omitted on some quoted numbers. 
    6.  I think 'figure' and 'table' may need to be capitalized in the body of the manuscript.
    7. The equation on page 4 may need to be broken up onto two lines and embedded in a sentence.

Author Response

Referee 2

1) The reference to Yuichi's work has been moved from item 1 to item 19.
Griest's reference has also been moved.

2) comas, Several comas have been added.

3) It is not unusual to have equations set in an equation environment.
The spacing of things such as fractions and exponents are more easily spread
out in an equation environment.  Many of the equations involve ratios and
powers which would be hard to read in a sentence.  The first equation in
section 3 is in a sentence which involves ratios and exponents.  It is barely
readable, but does not play and important part in the subsequent calculations.
The last sentence of this section also has some fractions in it that are not
easy to read.  I would like to leave the equation made since it improves
readability for these equations.

4) I have added a mention to Figure 1 where many of the variables are labeled.
Everything is defined but often after it has been used.  Figure 1 had been
introduced two paragraphs earlier.

5) CHECK units   The L's are all proportional to L_n.  The angles are all
dimensionless radians, which are converted to seconds of arc for figure 3.   Lengths are all in kpc.  Delta t is in the form of a ratio of a length,
the Schwarzschild radius, divided by a speed (C).  The units look good.

6) Figure and Table have been capitalized.

7) I have shortened the long equation on page 4 by removing the last term.
(-2+1+1)=0 is not needed.

Round 2

Reviewer 1 Report

This paper has definitely been improved.  However, I believe further improvements are needed and could be easily implemented.  I list them here in chronological order:

1. The abstract should be changed to indicate clearly that the gravitational lensing possibility is not a sure thing here, as it is by no means consistent with all of the data examined.
2. Clarify what is the basis for stating on page 1: “Unlike particle like interactions, which are considered unlikely”
3. Clarify the meaning (intended antecedent) and add a reference for the “4.8 months” statement near the top of page 2.
4. The value of the speed of light is close to universally symbolized by “c,” not as in this paper, by “C.”
5. The apparent nearly 5 hour error in the time recorded at the UNO is left unexplained; this error is just not understandable and, to me, puts the UNO’s entire data set under suspicion.
6. It might be well to reference or to show basis/generality on page 2, near bottom, of statement there that beta squared is much less than alpha subscript zero squared.
7. My previous critique of line 5, page 6, was apparently not understood. To be clearer:  I do not think that the author meant what he wrote; I think he meant that, “One should also be careful not to overestimate the precision…”
8. At the end of the third complete paragraph on page6, the author writes “observation of the angle alpha between multiple images unlikely.” This should be changed with “any two” substituted for “multiple.”  There isn’t just one angle when there are multiple (= more than two) images.
9. The last sentence of Section 5, on page 9, should be more quantitative, not wholly qualitative, as now.
10. In the second sentence of Section 6, at the bottom of page 9, there is the statement, ”reported by other observations.” This should be re-phrased; observations don’t report.
11. Finally, I’d like to have added a clear discussion on how likely it is to have a lens mass of about 400,000 solar masses, between us and the LMC, be unobserved and, at present, how, if at all, it might be detected optically.

Author Response

Referee 1 Report 2 September 1, 2021

This paper has definitely been improved.  However, I believe further improvement
s are needed and could be easily implemented.  I list them here in chronological
 order:

1.    The abstract should be changed to indicate clearly that the gravitational 
lensing possibility is not a sure thing here, as it is by no means consistent wi
th all of the data examined.

I had been meaning to update the abstract to emphasize the application to
SN1987A.  I have used a conditional tone since the hypothesis has not been
proven.  But the referee's assertion is too strong.  The statistical errors
on L2/L1 are large due to the fact that there are only 2 or 3 events in L2.
So the estimated error on alpha_0/beta is large.  This gives a large error
on the estimated lens mass even if the time difference is well known.
The masses and angles are consistent between the 3 observations, Kamioka, IMB
and Baksan.  It would be a mistake to dump this and almost everything else we
know about SN1987A neutrinos because UNO is strange.

2.    Clarify what is the basis for stating on page 1: “Unlike particle like int
eractions, which are considered unlikely”

The statement is true.  Why is it there?  I have spent several years designing
neutrino beams at Brookhaven and Fermilab.  This is hard since one can not
focus neutrinos with any conventional technology so one must make do with
focusing the source of the neutrinos, mostly charged pions.  I am proud to
report, that with this paper, I have found a physical mechanism that can focus neutrinos to provide a brighter source.

3.    Clarify the meaning (intended antecedent) and add a reference for the “4.8  months” statement near the top of page 2.

References 1 and 2 do this and reference 3, Shapiro, explains the physics.
I have added a paranthetical definition of the Shapiro delay.

4.    The value of the speed of light is close to universally symbolized by “c,”
 not as in this paper, by “C.”

OK, I have found all C and made them c.  Sorry.

5.    The apparent nearly 5 hour error in the time recorded at the UNO is left unexplained; this error is just not understandable and, to me, puts the UNO’s entire data set under suspicion.

This is part of the last paragraph of section 6.  I have no need to
antagonize the UNO collaboration.  I have called it "unexplained".
That data has not been used in the lens analysis.

6.    It might be well to reference or to show basis/generality on page 2, near 
bottom, of statement there that beta squared is much less than alpha subscript z
ero squared.

This is done in Refsdal 1964 and shown as the second equation on page 4.
The reason it is put here is that double images are indicative of lensing
but they can be missed if the second image is very dim.  One basically has
a selection effect that forces alignment.

7.    My previous critique of line 5, page 6, was apparently not understood. To 
be clearer:  I do not think that the author meant what he wrote; I think he mean
t that, “One should also be careful not to overestimate the precision…”

Yes, that change to "precission" was made in the second draft.  One might also
worry about the presence of background, producing a fake event, that would
distort the statistics limited bightness ratio, which is why I used a more
general term, "quality", in the first draft.  I would like to go back to
"quality" since it covers these cases and even calibration errors.

8.    At the end of the third complete paragraph on page6, the author writes “ob
servation of the angle alpha between multiple images unlikely.” This should be c
hanged with “any two” substituted for “multiple.”  There isn’t just one angle wh
en there are multiple (= more than two) images.

OK

9.    The last sentence of Section 5, on page 9, should be more quantitative, no
t wholly qualitative, as now.

This is about as quantitative as I want to be.  The previous sentence gives it
as about roughly a factor of 2.  I estimate it to be 2 +/- 0.7 so approximately
2 seems the right way to quote it.

10.    In the second sentence of Section 6, at the bottom of page 9, there is th
e statement, ”reported by other observations.” This should be re-phrased; observ
ations don’t report.

Changed "by" to "for" and added the word "neutrino" before observations.

11.    Finally, I’d like to have added a clear discussion on how likely it is to
 have a lens mass of about 400,000 solar masses, between us and the LMC, be unobserved and, at present, how, if at all, it might be detected optically.

This discussion was added.  A static distortion of a background field of stars
by a large mass can not be distinguished from a different distribution of stars.
One might hope to look at the distorted shape of the stellar images or a time
dependent brightness variation as the alignment changes as is used for
microlensing.  I have added a comment on distortion or brightness variation.

If this is a lensed system, it would be of a class of weak lenses, which can
be constructed as the sum of Newtonian gravitational potentials which may help hide the lens or at least make it harder to pin down.  Also neutrinos can go through a mass distribution that light can not.

Do I believe there is a lens there?  I doubt it but my beliefs are not
important.  I don't think enough is known at present to confirm or refute the hypothesis.  Publishing this is one way to get others to take a look.  With the quality of the neutrino data available the hypothesis fits.