Supernova Neutrino Detection

Supernova Neutrino Detection

Available online at www.sciencedirect.com Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 395–401 www.elsevier.com/locate/npbps Supernova Neutrino D...

491KB Sizes 1 Downloads 125 Views

Available online at www.sciencedirect.com

Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 395–401 www.elsevier.com/locate/npbps

Supernova Neutrino Detection John F. Beacoma,b,c c The author maintains copyright on the full content of this article, except for figures as noted.  a Department

of Physics, Ohio State University, Columbus, Ohio, USA of Astronomy, Ohio State University, Columbus, Ohio, USA c Center for Cosmology and AstroParticle Physics (CCAPP), Ohio State University, Columbus, Ohio, USA b Department

Abstract Detecting neutrinos is the key to understanding core-collapse supernovae, but this is notoriously difficult due to the small interaction cross section of neutrinos and the low frequency of supernovae in galaxies. The revolutionary implications of the detection of about 20 neutrinos from SN 1987A tell us that this quest is worthy. However, there is the sobering fact that there have been no other detections, before or since. Now, after decades of effort and patience, we have good reasons to anticipate that detecting supernova neutrinos again is within reach, in particular for the Diffuse Supernova Neutrino Background (DSNB). A first detection of the DSNB in a short time is possible if SuperKamiokande is upgraded with the proposed modification of dissolved gadolinium to allow neutron tagging. Longerterm, a comprehensive understanding of core-collapse supernovae will require something like the possible HyperKamiokande detector, eventually also with dissolved gadolinium. This systematic path towards increasing sensitivity will surely lead to further revolutionary discoveries in astrophysics. Keywords: Supernova, neutrino, Diffuse Supernova Neutrino Background, Super-Kamiokande, Hyper-Kamiokande

1. Preface and Outline This is the beginning of an age of precision particle astrophysics, echoing the transition to the age of precision cosmology that we entered around a decade ago. What does precision mean? Obviously, it means that some things are measured well. More importantly, it means that multiple things are measured well, in different ways, and that we have an opportunity to compare them to each other and to theoretical expectations. The age of precision is thus really the age of synthesis, and this provides special opportunities to learn surprising new things. The progress from this synthesis in cosmology has been tremendous, and similar benefits are expected in particle astrophysics. One of the most exciting of the several frontiers in particle astrophysics is associated with the detection of astrophysical neutrinos. So far, only the Sun and the nearby Supernova 1987A have clearly been detected. 0920-5632/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysbps.2013.04.038

But there are now strong reasons to believe that detections in three broad energy ranges are imminent: the MeV range (e.g., the Super-Kamiokande detector and neutrinos from core-collapse supernovae), the TeV range (e.g., the IceCube detector and neutrinos from observed gamma-ray sources), and the EeV range (e.g., the ANITA detector and neutrinos produced in the energy losses of the highest-energy cosmic rays). Optimism has always been required in neutrino astrophysics, but now it is objectively warranted. The reasons for optimism are based on improved auxiliary data, better theoretical modeling, and vast recent or pending increases in detector sensitivity. This talk focuses on the prospects in the MeV range; please see other talks at this meeting for the prospects at higher energies. Progress in all three energy ranges is needed to exploit the promise of neutrino astrophysics. Initially, the numbers of detected events will be small, so the measurements themselves will not be precise; however,

396

J.F. Beacom / Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 395–401

these measurements will be greatly leveraged by the high precision of other data and by theory. Eventually, we will have bigger detectors and large numbers of neutrinos. (Maybe someday like for gamma rays, for which the precision era is already here.) The data from SN 1987A provide a Rosetta Stone for relating the different languages used to describe core-collapse supernovae. With this event, three important lines of inquiry were finally joined (e.g., see Ref. [1]). First, observational evidence from Astronomy: that the progenitor stars of optical Type II supernovae were shown to be massive stars, in this case a blue ∼ 20M supergiant. Second, experimental evidence from Physics: that an optical Type II supernova was shown to be preceded by a short (∼ 10 s), energetic (∼ 1053 erg) burst of low-energy (∼ 10 MeV) neutrinos. Third, a long-ago-proposed unifying framework from Theory, now confirmed: that the core collapse of massive stars at the ends of their lives leads to the production of a proto-neutron star, which must be accompanied by a burst of neutrinos to balance the change in binding energy. Now, 25 years later, where do things stand? There has been tremendous progress in astronomy on understanding the massive stars that lead to core collapses, the optical supernovae they produce, and the mapping between the different varieties of each. As an example of the latter, it has now been confirmed that the minimum progenitor mass to eventually lead to core collapse is ∼ 8M , as expected from theory. And there has been tremendous progress in the theoretical work of modeling core collapses, the neutrino signals they produce, and if they lead to visible explosions. As an example of the latter, explosions are now seen in some simulations, modeling with two dimensions has nearly complete physics, and modeling in full three dimensions is beginning (see Ref. [2] and references therein). However, there have been no new experimental detections of supernova neutrinos, and we cannot make progress on the synthesis of the above three lines of inquiry until we have new, better Rosetta Stones, ideally a well-observed Milky Way supernova combined with data from a variety of more distant supernovae. Without new neutrino data, we will never be able to satisfactorily answer many important questions. How do core-collapse supernovae explode? How do they form neutron stars and black holes? What are the nucleosynthesis products of supernovae? What are the actions and properties of neutrinos there? What is the cosmic rate of black hole formation? Which supernovalike events make neutrinos? What else is out there that makes neutrinos?

Those are only a few examples of pressing questions. Why is neutrino detection so crucial? Fundamentally, because only neutrinos can reveal the physical conditions deep inside collapsing stars. (Neutrino emission depends on temperature and density, which are known to be large enough; gravitational wave emission requires significant deviations from spherical symmetry, which may not be present.) In many cases, detecting even small numbers of neutrinos could give decisive answers about supernovae. Finally, the next detection of supernova neutrinos will open new frontiers in the nascent field of observational neutrino astrophysics. Please see other talks at this meeting for a more complete perspective on supernova neutrinos. In the following, I first discuss the three detection modes for supernova neutrinos, for nearby, semi-nearby, and distant sources. After explaining why the most likely next detection will come from the most distant supernovae, I focus the discussion there, on the Diffuse Supernova Neutrino Background (DSNB). In the three subsequent sections, I review the theoretical predictions, the present experimental limits, and the future prospects for the DSNB. Finally, I offer some concluding remarks, including on more general perspectives. My remarks in these proceedings are abbreviated and simplified. Please see the posted slides of the talk for more details, as well as the papers cited there and the few that are cited below. I use the term supernova a bit sloppily. I always mean a core-collapse, and I usually mean to include those failed explosions that don’t produce an optical supernova at all, instead forming a stellar-mass black hole, but still do produce a neutrino burst. 2. Three Detection Modes for Supernova Neutrinos There are three frontiers in the detection of supernova neutrinos, all with different advantages and challenges, but with a common goal: to provide unique insights into what happens when massive stars die, that rate at which that occurs, and the variety in the outcomes. For more details, see Ref. [3] and references therein. • Burst mode: The high-statistics detection of a supernova in the Milky Way will probe all details of the neutrino emission, including the full time and flavor information. The supernova rate in our Galaxy is low, R ∼ 0.01 yr−1 , but the number of detected neutrinos per supernova is large, e.g., N ∼ 104 in Super-Kamiokande. Other existing detectors will also play important roles, e.g., oil-based detectors can measure the fluxes of all

J.F. Beacom / Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 395–401

flavors through the neutral-current neutrino-proton elastic scattering process, liquid argon-based detectors may be excellent for detecting electron neutrinos, and IceCube can measure the time profile of the burst with high precision. New large detectors and a wider variety of targets are needed to gather more complete information. • Mini-burst mode: The low-statistics detections of several supernovae in semi-nearby galaxies will probe the connections between optical and neutrino transients, as well as possibly detecting individual black-hole forming failed supernovae. The supernova rate within several Mpc is reasonable, R ∼ 1 yr−1 , and the number of detected neutrinos per supernova is moderate, e.g., N ∼ 1 in Super-Kamiokande for a supernova in Andromeda (M31), the nearest large galaxy. A larger detector like Hyper-Kamiokande or something even larger is needed to take advantage of the frequent bursts in semi-nearby galaxies. • Steady mode: The detection of the DSNB from all core-collapses in the universe will probe the neutrino emission per average supernova, as well as the cosmic core-collapse rate. The supernova rate in the universe is huge, R ∼ 108 yr−1 , but the number of detected neutrinos per supernova has a Poisson expectation value of N  1, such that the average neutrino detection rate is ∼ 1 − 10 yr−1 in Super-Kamiokande. So far only SuperKamiokande is large enough and (nearly) has a low enough background rate; an upgrade with gadolinium to reduce backgrounds is needed in the short term, and a much larger detector like HyperKamiokande is needed in the long term. Because the best immediate prospects are for the DSNB, this talk focuses on that. More details and references are given in my review [4]; see also Refs. [5, 6]. Even if a Milky Way supernova is detected tomorrow, it could be a century until the next one. To answer questions about things like the variation in neutrino emission between different core collapses, we will need data from other supernovae. 3. DSNB: Theoretical Predictions The signal rate spectrum of the DSNB is expressed as   dNe (Ee ) = NT σ(Eν ) dEe  ∞    c dt    dz . (1 + z) ϕ[Eν (1 + z)] RS N (z)  dz  0

397

The left-hand side (units MeV−1 s−1 ) is the measurable positron or electron signal rate spectrum that can be compared directly to measured data, which also include detector backgrounds. The first factor on the right-hand side is the detector effective area, which depends on the number of targets and the cross section per target; these are precisely known. For Super-Kamiokande, electron antineutrinos interact with free (hydrogen) proton targets, of number NT , via the inverse-beta-decay process, which has a well-known cross section σ(Eν ) (units cm2 ). The second factor is based on ϕ(Eν ), the time-integrated neutrino spectrum per average supernova (units MeV−1 ), appropriately redshifted. This is the primary unknown that can only be measured by detecting neutrinos. The third is RS N (z), the comoving core-collapse rate density as a function of redshift (units yr−1 Mpc−3 ), roughly the supernova rate per galaxy in an expanding universe. This is now largely known from astronomical observations, was about one order of magnitude greater in the past than it is today, and its precision is quickly improving. The last is the cosmological line-of-sight factor (units Mpc), which is essentially perfectly known; a more subtle point is that changes in it due to changes in the cosmological parameters cancel in the definition of the supernova rate. Why is there such a focus on the neutrino spectrum? As noted earlier, this is the missing link that would enable a new synthesis between other types of increasingly-precise observational and theoretical information. Even with minimal theory, the spectrum obtained with a measurement of the DSNB could be compared to data from SN 1987A and from an eventual high-statistics Milky Way supernova – are they the same or not? With theory, there is a rich program possible through detailed comparisons to supernova simulations and calculations of the effects of neutrino flavor mixing in supernovae. These comparisons will test properties of neutrinos that cannot be tested elsewhere, including the index of refraction induced by the high densities of matter and the neutrinos themselves (see Ref. [7] and references therein). In the above equation, the electron antineutrino emission spectrum is assumed to be that just outside the supernova, after all neutrino mixing effects in the supernova have occurred; there are no mixing effects en route, because the neutrinos leave the dense supernova as incoherent mass eigenstates. Eventually, it will be possible to work backwards from the spectra in vacuum at the surface of the supernova to the spectra that are emitted from the proto-neutron star. This will require complete knowledge of the neutrino mixing parameters,

398

J.F. Beacom / Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 395–401

Figure 1: Predicted DSNB ν¯ e event rates in Super-Kamiokande as a function of positron energy. The hatched region shows the 2003 upper limit from the Super-Kamiokande Collaboration, which applies to all spectra. Figure taken from Ref. [8].

which terrestrial neutrino experiments have nearly provided, and theoretical work to model the complicated mixing effects induced at high densities. All of the above has been combined into reasonably precise predictions for Super-Kamiokande, as shown in Fig. 1. Four examples of possible emission spectra are shown. For each, the band shows the full width of the astrophysical uncertainties, which are decreasing quickly. (The central value for each example, which would be intermediate between the upper and lower curves, is not shown, to avoid cluttering the figure.) The different examples indicate how a measurement of the DSNB spectrum would constrain the emission spectrum. In this figure, backgrounds above 10 MeV are neglected because it is assumed that neutron tagging with dissolved gadolinium and other techniques will strongly suppress the presently-significant backgrounds in this energy range. Below 10 MeV, the background from nuclear reactors is overwhelming (even if all of the Japanese reactors are turned off). Reading from top to bottom in the legend in Fig. 1, the numbers of signal events in the energy range 10 − 26 MeV for the examples shown are 4.2, 3.5, 1.8, and 1.7 per year, all for the (not shown) central values of each example [8]. This is the most promising energy range for reducing backgrounds; the total numbers of events at all energies are larger, as is evident. An interesting question is what happens to the neu-

trino signal when a core collapse explosion fails and there is little or perhaps no optical signal. In short, the neutrino signal remains, and seems to be even somewhat enhanced as compared to the successful case, due to the formation of a large, hot proto-neutron star with a high accretion rate that must radiate a great deal of energy in neutrinos to fully collapse to a black hole, which terminates the neutrino emission. Importantly, both the total energy carried away in neutrinos (with a higher luminosity for a shorter time) and the average energy per neutrino can be enhanced. The fraction of core collapses that lead to failed explosions and stellar-mass black holes is unknown, and is a very important question with broad implications. Observational data allow a failed fraction as large as ∼ 50%, and simple theoretical models predict a fraction ∼ 10%. If the failed fraction is indeed large, it would improve detection prospects for the DSNB, especially at high neutrino energies, which provides a way to potentially isolate the failed-supernova contribution to the DSNB. More generally, neutrinos would be a powerful tool for detecting individual failed collapses (along with stellar disappearance observations). 4. DSNB: Experimental Limits The above predictions show that there are definitely a small but nonzero number of DSNB events in SuperKamiokande every year. (Super-Kamiokande has a fiducial volume of 22.5 kton, which shows that large detectors like it are required.) Finding these events is a question of rejecting detector backgrounds. Confirming even a small number of events would have immediate value for revealing the normalization of the DSNB flux. Already in 2003, the Super-Kamiokande Collaboration showed that they could achieve dramatic background rejection and detector performance, obtaining sensitivity in reasonable proximity to the predictions [9]. In the energy range above 18 MeV, the remaining dominant backgrounds are due to atmospheric neutrinos, as shown in Fig. 2. In 2012, the Super-Kamiokande Collaboration made many improvements to the analysis, which required a lot of challenging new work [10]. In the first time period (Super-Kamiokande-I data, as used in the previous analysis), the best fit for the DSNB flux is slightly negative. In the new time periods (Super-Kamiokande-II and -III), the best fit for the DSNB flux is slightly positive. Both facts lead to a somewhat weaker upper limit on the DSNB flux than before. Following Ref. [11], this upper limit can be interpreted in terms of the two most important supernova

J.F. Beacom / Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 395–401

399

-1

-1

dN / dEe [ (22.5 kton yr) MeV ]

3.5 3

2.5 2

1.5 1

0.5 0

20

30

40

50

60

70

Visible Energy Ee [MeV]

80

Figure 2: Results of the Super-Kamiokande 2003 DSNB ν¯ e search. The points and error bars indicate the efficiency-corrected measured data. The thin solid lines indicate components of the atmospheric neutrino background, and the thick solid line the total. The shaded region on the left indicates the largest DSNB signal consistent with the data. Figure adapted from Ref. [9] by changing from 4-MeV to 1MeV steps, to allow direct comparison to the dN/dEe results in Fig. 1 above. (To recover the dN results shown in Ref. [9], multiply the bin values by 4 MeV.) Figure taken from Ref. [4].

neutrino emission parameters: the total energy in neutrinos and the average energy per neutrino (or temperature). This is shown in Fig. 3. This shows that the Super-Kamiokande sensitivity is near the required level, but that improvements are needed. Besides accumulating more data, the challenge now is to further reduce detector backgrounds, especially in the energy range ∼ 10 − 20 MeV, where the DSNB signal is larger. In that energy range, muon-spallation-induced beta radioactivities are a dominant background. 5. DSNB: Future Prospects The DSNB detection process in Super-Kamiokande is inverse beta decay, ν¯ e + p → e+ + n. At present, only the positron, which carries nearly the full neutrino energy, can be detected. These signal events cannot be separated from several important detector backgrounds. However, if the neutron were detectable, this could help isolate the signal reaction from detector backgrounds with electrons or positrons, which generally do not also have neutrons. A true coincidence detection would be a positron signal followed by a neutron capture signal, with small but nonzero separations in time

Figure 3: Implications of the Super-Kamiokande 2012 DSNB ν¯ e search. For the excluded region in the upper right, the neutrino emission per average supernova would be too large to be compatible with the measured data. The allowed regions in the lower left are examples of fits to the SN 1987A data. Figure taken from Ref. [10].

and space. An accidental coincidence would simply be two unrelated events in close proximity. At present, most neutrons thermalize and then capture on free (hydrogen) protons, producing 2.2 MeV gamma rays. While the gamma ray may Compton-scatter electrons and produce detectable signals in Super-Kamiokande, the accidental coincidence rate is very high, making this technique impractical [12]. Beacom and Vagins proposed that dissolved gadolinium in Super-Kamiokande would solve these problems, create a new tool for particle identification, and lead to several new physics programs [13]. When neutrons capture on gadolinium, the gamma-ray signal is more energetic and happens more promptly, removing accidental coincidences. Though a similar technique has been used in oil-based detectors for decades, this was the first proposal to use dissolved gadolinium in large light water detectors showing that it could be practical and effective. Some of the prospects are illustrated in Fig. 4. Since that time, there have been extensive research and development efforts in the United States and Japan. To give a sense of these efforts, since 2009 Masayuki Nakahata, Mark Vagins, and other SuperKamiokande collaborators have been constructing a new, multimillion-dollar test facility called EGADS (Evaluating Gadolinium’s Action on Detector Systems) [14]. This is a ∼ 1% scale model of SuperKamiokande, located deep underground in a new cavern next to the original detector. EGADS is designed to be an end-to-end test of using gadolinium in Super-Kamiokande. One of the complex new tech-

400

J.F. Beacom / Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 395–401

transition from a background-dominated to a ratedominated experiment, further running time would be much more valuable. An extensive research and development program, now at the level of a scale model of Super-Kamiokande with gadolinium, has so far yielded very promising results. • First detection of the DSNB will be important for understanding supernovae. When the DSNB is detected, we will learn new things about the neutrino emission per average supernova that cannot be tested in other ways. The average neutrino energy is of special interest, and the signal rate depends on this exponentially, so even a small number of events could be decisive.

Figure 4: Detection prospects for Super-Kamiokande with gadolinium. With the ability to tag neutrons, detector backgrounds would be dramatically reduced, isolating the DSNB signal, as well as the reactor signal at lower energies. (Even if all the reactors in Japan are turned off, the total reactor signal is only reduced by a factor ∼ 100.) Figure taken from Ref. [13].

nologies required was an elaborate system to purify the gadolinium-loaded water without removing the gadolinium itself from solution. Virtually all aspects of the testing indicate positive results. The completed study will be the basis of a decision – likely during 2014 – by the Super-Kamiokande Collaboration on whether to add gadolinium. If they do, then significant new physics capabilities will be available immediately, and major discoveries could follow quickly. 6. Concluding Perspectives The prospects for a first detection of the DSNB in the near future are very encouraging. • The DSNB is a guaranteed signal that is within reach. Super-Kamiokande is large enough that there are at least a few DSNB signal events per year in the detector, and the detector backgrounds are already nearly low enough. The facts that the astrophysical uncertainties are low and quickly decreasing are crucial to these statements. • An upgrade to Super-Kamiokande would improve sensitivity. With dissolved gadolinium, Super-Kamiokande would be able to tag neutrons, which would dramatically reduce detector backgrounds and isolate the DSNB signal. With the

• Detection of the DSNB is necessary to develop neutrino astrophysics. Besides the Sun and SN 1987A, no astrophysical neutrino sources have clearly been detected yet. The DSNB could be the next signal, helping to usher in the age of observational neutrino astrophysics. If Super-Kamiokande is upgraded, and the DSNB is not discovered, then new physics will be required, which would be even more interesting. New large detectors based on oil or liquid argon could have comparable or somewhat larger signal rates than Super-Kamiokande. Besides increasing the total statistics, these would be useful for confirming the signal with different techniques and allowing sensitivity to DSNB νe . Ultimately, we must go beyond even detectors with masses of a few times 10 kton. In the possible HyperKamiokande detector, the fiducial volume would be 25 times larger than that of Super-Kamiokande, allowing a high-statistics measurement of the DSNB spectrum, e.g., to test for a higher-average-energy component corresponding to failed supernovae. Returning to the other detection modes for supernova neutrinos, all of the present and proposed detectors would be invaluable for improving the physics return from a Milky Way supernova. A very large detector like Hyper-Kamiokande would make it possible to detect neutrinos from supernovae in semi-nearby galaxies, and even larger detectors may make this routine. Without detections in all three modes, we will never be able to convincingly answer all of the many open questions we about supernovae and their neutrinos. Some examples are shown in Fig. 5. A burst from a Milky Way supernova is the only way to measure the precise details of the time profile and the flavor information, which will allow powerful tests of supernova sim-

J.F. Beacom / Nuclear Physics B (Proc. Suppl.) 235–236 (2013) 395–401

Acknowledgments

     

    -$!  '  $ $  ' ! $# ( !%

%    ( &!( *   % (

I am grateful to my many collaborators on these and related topics for discussions, to Shunsaku Horiuchi, Ranjan Laha, and Mark Vagins for helpful comments on the manuscript, and to the organizers for their patience in waiting for it. This research is supported by National Science Foundation Grant No. PHY-1101216.

!$% (

   $  !

-

401

$%'  

$%+ , % !&% $  ! $( (

References

$%  ($   ' ! $ 0./0* ( **$0./0

21

Figure 5: The necessity of neutrino detection for understanding supernovae. Some examples of how theory topics (left column) can only be connected to observational topics (right column) through the detection of supernova neutrinos.

ulations. Mini-bursts from supernovae in semi-nearby galaxies are the only way to test if several types of optical events do emit neutrinos and to individually detect failed supernovae. The steady flux from the DSNB is the only way to measure the average neutrino emission and to test the cosmic core-collapse rate. As is often said, we should expect great surprises whenever a new type of telescope is invented. It would be astonishing if this were not also true for neutrino astrophysics. The detections of neutrinos at a variety of energies will surely comprise an essential element of bringing about the age of precision particle astrophysics.

[1] W. D. Arnett, J. N. Bahcall, R. P. Kirshner and S. E. Woosley, Ann. Rev. Astron. Astrophys. 27, 629 (1989). [2] C. D. Ott, E. P. O’Connor, S. Gossan, E. Abdikamalov, U. C. T. Gamma and S. Drasco, arXiv:1212.4250 [astro-ph.HE]. [3] K. Scholberg, Ann. Rev. Nucl. Part. Sci. 62, 81 (2012) [arXiv:1205.6003 [astro-ph.IM]]. [4] J. F. Beacom, Ann. Rev. Nucl. Part. Sci. 60, 439 (2010) [arXiv:1004.3311 [astro-ph.HE]]. [5] S. ’I. Ando and K. Sato, New J. Phys. 6, 170 (2004) [astroph/0410061]. [6] C. Lunardini, arXiv:1007.3252 [astro-ph.CO]. [7] A. B. Balantekin, arXiv:1209.5894 [astro-ph.SR]. [8] S. Horiuchi, J. F. Beacom and E. Dwek, Phys. Rev. D 79, 083013 (2009) [arXiv:0812.3157 [astro-ph]]. [9] M. Malek et al. [Super-Kamiokande Collaboration], Phys. Rev. Lett. 90, 061101 (2003) [hep-ex/0209028]. [10] K. Bays et al. [Super-Kamiokande Collaboration], Phys. Rev. D 85, 052007 (2012) [arXiv:1111.5031 [hep-ex]]. [11] H. Yuksel, S. ’I. Ando and J. F. Beacom, Phys. Rev. C 74, 015803 (2006) [astro-ph/0509297]. [12] H. Watanabe et al. [Super-Kamiokande Collaboration], Astropart. Phys. 31, 320 (2009) [arXiv:0811.0735 [hep-ex]]. [13] J. F. Beacom and M. R. Vagins, Phys. Rev. Lett. 93, 171101 (2004) [hep-ph/0309300]. [14] M. Vagins, talk given at the Pacific 2012 conference, September 2012, Moorea, French Polynesia, https://hepconf.physics.ucla.edu/pacific/.