Investigation of Primordial Black Hole Bursts using Interplanetary Network Gamma-ray Bursts
T. N. Ukwatta, K. Hurley, J. H MacGibbon, D. S Svinkin, R. L Aptekar, S. V Golenetskii, D. D Frederiks, V. D Pal'shin, J. Goldsten, W. Boynton, A. S Kozyrev, A. Rau, A. von Kienlin, X. Zhang, V. Connaughton, K. Yamaoka, M. Ohno, N. Ohmori, M. Feroci, F. Frontera, C. Guidorzi, T. Cline, N. Gehrels, H. A Krimm, J. McTiernan
IInvestigation of Primordial Black Hole Bursts using Interplanetary NetworkGamma-ray Bursts
T. N. Ukwatta
Director’s Postdoctoral Fellow, Space and Remote Sensing (ISR-2), Los Alamos NationalLaboratory, Los Alamos, NM 87545, USA. [email protected]
K. Hurley
University of California, Berkeley, Space Sciences Laboratory, 7 Gauss Way, Berkeley, CA94720-7450, USA
J. H. MacGibbon
Department of Physics, University of North Florida, Jacksonville, FL 32224, USA the following authors are in order of the number of times their experiments were used in the paperD. S. Svinkin, R. L. Aptekar, S. V. Golenetskii, D. D. Frederiks, V. D. Pal’shin
Ioffe Physical Technical Institute, St. Petersburg, 194021, Russian Federation
J. Goldsten
Applied Physics Laboratory, Johns Hopkins University, Laurel, MD 20723, U.S.A.
W. Boynton
University of Arizona, Department of Planetary Sciences, Tucson, Arizona 85721, U.S.A.
A. S. Kozyrev
Space Research Institute, 84/32, Profsoyuznaya, Moscow 117997, Russian Federation
A. Rau, A. von Kienlin, X. Zhang
Max-Planck-Institut f¨ur extraterrestrische Physik, Giessenbachstrasse, Postfach 1312, Garching,85748 Germany
V. Connaughton
University of Alabama in Huntsville, NSSTC, 320 Sparkman Drive, Huntsville, AL 35805, USA
K. Yamaoka a r X i v : . [ a s t r o - ph . H E ] A p r Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 Fuchinobe,Sagamihara, Kanagawa 229-8558, Japan
M. Ohno
Department of Physics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima739-8526, Japan
N. Ohmori
Department of Applied Physics, University of Miyazaki, 1-1 Gakuen kibanadai-nishi,Miyazaki-shi, Miyazaki 889-2192, Japan
M. Feroci
INAF/IAPS-Roma, via Fosso del Cavaliere 100, 00133, Roma, Italy
F. Frontera , C. Guidorzi University of Ferrara, Dept. of Physics and Earth Science, via Saragat 1, 44122 Ferrara, Italy
T. Cline , N. Gehrels NASA Goddard Space Flight Center, Code 661, Greenbelt, MD 20771, U.S.A.
H. A. Krimm USRA/CRESST/NASA Goddard Space Flight Center, Code 661, Greenbelt, MD 20771, U.S.A.
J. McTiernan
University of California, Berkeley, Space Sciences Laboratory, 7 Gauss Way, Berkeley, CA94720-7450, USA
ABSTRACT
The detection of a gamma-ray burst (GRB) in the solar neighborhood would havevery important implications for GRB phenomenology. The leading theories for cosmo-logical GRBs would not be able to explain such events. The final bursts of evaporating INAF/Istituto di Astrofisica Spaziale e Fisica Cosmica di Bologna, via Gobetti 101, I-40129 Bologna, Italy Emeritus Joint Center for Astrophysics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD21250 Universities Space Research Association, 10211 Wincopin Circle, Suite 500, Columbia, MD 21044 –10 cm(7–10 AU) range, consistent with the expected PBH energetics and with a possibleorigin in the solar neighborhood, although none of the bursts can be unambiguouslydemonstrated to be local. Assuming these bursts are real PBH events, we estimatelower limits on the PBH burst evaporation rate in the solar neighborhood.
Subject headings: primordial black holes; black hole physics; gamma-ray bursts: general
1. Introduction
The composition of the short-duration, hard-spectrum gamma-ray burst (GRB) population isnot yet fully understood. It is believed that most of the bursts are generated in compact binarymergers (Eichler et al. 1989) and while the handful of optical counterparts and host galaxiesdiscovered to date does not contradict this view, it is also thought that the population probablycontains up to 8% extragalactic giant magnetar flares as well (Hurley et al. 2010; Mazets et al.2008; Svinkin et al. 2015a). For the majority of the short-duration GRB population, however,there is simply not enough evidence to determine their origin unambiguously. Hawking radiationfrom primordial black holes (hereafter PBH) was one of the very first explanations proposed forcosmic gamma-ray bursts (Hawking 1974), and it continues to be proposed today (Cline and Hong1996; Czerny et al. 1996; Cline et al. 1997, 1999, 2003, 2005; Czerny et al. 2011). The PBHlifetime and burst duration depend on its mass, so PBHs bursting today have similar masses anddurations, and release similar energies, making them in essence ‘standard candles’. The typicalPBH gamma-ray burst is not expected to be accompanied by detectable intrinsically-generatedextended emission or have an afterglow, although accompanying bursts at other wavelengths orafterglows may arise if, for example, the PBH is embedded in a high density magnetic field orplasma (MacGibbon et al. 2008; Rees 1977; Jelley, Baird, and O’Mongain 1977). In the standardemission scenario, so-called because it uses the Standard Model of particle physics (MacGibbonand Webber 1990), the PBH gamma-ray burst is strongest in the final second of the burst lifetime,has a hard energy spectrum, and should be detectable in the vicinity of the Earth. For a typicalinterplanetary network detector sensitive to bursts of fluence 10 − erg cm − and above, PBH eventscould in principle be detected out to a distance of a few parsecs, depending on the emission model.PBHs evaporating today do not have enough luminosity to be detected at cosmological distanceseven by the most sensitive current instruments, so searching for them locally is a logical step. 4 –When observed by a single detector, the properties of a PBH burst might not appear to besignificantly different from those of other short bursts; instruments with localization or imagingcapabilities would obtain their arrival directions as they would for an infinitely distant source.Indeed many attempts to find evidence for the existence of PBH bursts have to date been basedmainly on the spatial distribution and time histories of a subset of short bursts (Cline and Hong1996; Czerny et al. 1996; Cline et al. 1997, 1999, 2003, 2005; Czerny et al. 2011). Other searchmethods have employed atmospheric Cherenkov detectors (Porter and Weekes 1977, 1978, 1979;Linton et al. 2006; Tesic 2012; Glicenstein et al. 2013), air shower detectors (Fegan et al. 1978;Bhat et al. 1980; Alexandreas et al. 1993; Amenomori et al. 1995; Abdo et al. 2015), radio pulsedetection (Phinney and Taylor 1979; Keane et al. 2012), spark chamber detection (Fichtel et al.1994), and GRB femtolensing (Barnacka et al. 2012). Table 1 gives a comparison of these variousmethods.To widely spaced interplanetary network (IPN) detectors, however, a local PBH burst couldlook significantly different when compared with bursts from distant sources, due to the curvatureof the received wavefront. In this paper, we use this fact to explore the possibility that someshort bursts may originate in the solar neighborhood, and estimate lower limits to the PBH burstevaporation rate assuming these bursts are real PBH bursts. This paper is organized as follows.In Section 2 we derive the fluence expected in the detector from a PBH burst using the standardemission model and, as a maximal alternative, the Hagedorn emission model. In Section 3 weexplain how we localized the detected bursts in 3D relaxing the assumption that they are at infinitedistances. The detailed discussion of the methodology is given in Appendix A. Our data selectioncriteria are described in Section 4. Our results and PBH burst rate limit calculation are given inSection 5. In Section 6, we discuss implications and limitations of our results.
2. PBH Burst Signatures
As a PBH Hawking-radiates, its mass is decreased by the total energy carried off by theemission, and the black hole temperature, which is inversely proportional to its mass, increases.In the standard emission model (SEM) (MacGibbon and Webber 1990), the black hole directlyHawking-radiates those particles which appear fundamental on the scale of the black hole. Once theblack hole temperature reaches the QCD transition scale ( ∼ −
300 MeV), quarks and gluons aredirectly Hawking-radiated. The PBH gamma-ray burst spectrum is the combination of the directlyHawking-radiated photons and those produced by the decay of other directly Hawking-radiatedparticles. An SEM PBH with a remaining emission lifetime of τ (cid:46) M BH ( τ ) ≈ . × (cid:16) τ (cid:17) / g (1)(Ukwatta et al. 2015) and a remaining rest mass energy of E BH ( τ ) ≈ . × (cid:16) τ (cid:17) / erg . (2) 5 –The expected fluence arriving at the detector from a PBH at a distance d from Earth is then F D = E γBH πd (3)where E γBH = η γD E BH and η γD is the fraction of the PBH energy that arrives in the energy bandof the detector, and the maximum distance from which the SEM PBH is detectable is d max (cid:39) . (cid:16) η γD − (cid:17) / (cid:16) τ (cid:17) / (cid:18) F D min − erg cm − (cid:19) − / pc (4)where F D min is the sensitivity of the detector.The SEM analysis is consistent with high energy accelerator experiments (MacGibbon et al.2008). However, an alternative class of PBH evaporation models was proposed before the existenceof quarks and gluons was confirmed in accelerator experiments and these models continue to bediscussed in the PBH burst literature. In such models (which we label HM scenarios), a Hagedorn-type exponentially increasing number of degrees of freedom become available as radiation modesonce the black hole temperature reaches a specific threshold such as the QCD transition scale. Inthe HM scenarios, we assume that the remaining PBH mass is emitted quasi-instantaneously asa burst of energy E (cid:48) BH = M (cid:48) BH c once the black hole mass reaches some threshold M (cid:48) BH ; for theQCD transition scale, M (cid:48) BH ∼ g and E (cid:48) BH ∼ erg. Proceeding as above, the maximumdistance from which the HM PBH burst is detectable is d (cid:48) max (cid:39) (cid:32) η (cid:48) γD − (cid:33) / (cid:18) M (cid:48) BH g (cid:19) / (cid:18) F D min − erg cm − (cid:19) − / pc (5)where η (cid:48) γD is the fraction of the HM PBH energy that arrives in the energy band of the detector.
3. Gamma-ray burst localization for a source at a finite distance
When a pair of IPN spacecraft detects a burst, if the distance to the source is taken to be a freeparameter, the event is localized to one sheet of a hyperboloid of revolution about the axis definedby the line between the spacecraft. If the burst is assumed to be at a distance which is much greaterthan the interspacecraft distance, the hyperboloid intersects the celestial sphere to form the usuallocalization circle (or annulus, when uncertainties are taken into account). Another widely spacedspacecraft would produces a second hyperboloid which intersects the first one to define a locus ofpoints which is a simple hyperbola. Note that both hyperboloids have a common focus. Again,if the burst is assumed to be at a large distance from the spacecraft, the hyperbola intersects thecelestial sphere at two points to define two possible error boxes. A fourth, non-coplanar spacecrafteven at a moderate distance from Earth, such as
Konus-WIND, can often be used to eliminateone branch of the hyperbola and part of the second branch. A terrestrial analogue to this methodis Time Difference of Arrival (TDOA), with the important exception that GRB sources can be at 6 –distances which are effectively infinite. Further details are given in Appendix A. While a singleinstrument with imaging or localization capability would obtain the correct sky position for a PBHburst regardless of its distance, the same is not true of an IPN localization, for which the derivedarrival direction depends on the source distance.In this paper, we relax the assumption that bursts are at infinite distances. If a burst isdetected by three widely spaced spacecraft, then according to the previous discussion, the possiblelocation of the burst traces a simple hyperbola in space as illustrated in Figure 3. In an Earth-centered coordinate system, this hyperbola has a closest distance to the Earth, that is, a distancelower limit. As we explain in Section 5.2, this fact can be used to calculate a lower limit to thePBH burst density rate in the Solar neighborhood assuming that the bursts that we consider areactual PBH bursts. In principle, detections by three spacecraft can rule out a local origin for aburst, but it is impossible to prove a local origin with only three non-imaging spacecraft.In the case where a burst is observed by four widely spaced non-imaging spacecraft, the burstcan be localized to a single point in space (or a region in space if uncertainties are taken intoconsideration). This scenario is illustrated in Figure 4. Thus in order to prove the local origin ofa burst using non-imaging spacecraft, one needs detections from at least four satellites that are atinterplanetary distances from each other.As mentioned in Appendix A, in the special case of two widely separated spacecraft, where onespacecraft has precise imaging capability, it is in principle possible to demonstrate a local origin.We have explored this case in detail and defer treatment of it to a future paper. None of the eventsin this paper are in that category.
4. Data Selection
The IPN database contains information on over 25,000 cosmic, solar, and magnetar eventswhich occurred between 1990 and the present ( http://ssl.berkeley.edu/ipn3/index.html ).During this period, a total of 18 spacecraft participated in the network. Some were dedicatedGRB detectors, while others were primarily gamma-ray detectors with GRB detection capability.Indeed, the composition of the IPN changed regularly during this time, as old missions were retiredand new ones were launched. However, all the instruments were sensitive to bursts with fluencesaround 10 − erg cm − or peak fluxes 1 photon cm − s − and above, resulting in a roughly constantdetection rate. All known bursts, regardless of their intensity or duration, or the instruments whichdetected them, are included in this list. We have searched it for gamma-ray bursts with the followingproperties. • Confirmed cosmic bursts which occurred between 1990 October and 2014 December (24.25 y;10795 GRBs survived this cut). • Bursts observed by three or more spacecraft, of which two were at interplanetary distances; 7 –839 GRBs survived this cut. This small number is due firstly to the relatively high sensitivitythresholds of the distant IPN detectors (roughly 10 − erg cm − or 1 photon cm − s − ), andtheir somewhat coarser time resolutions, and secondly to a 2.5 year period between 1993August and 1996 February when there was only one interplanetary spacecraft in the network. • Bursts with no X-ray or optical afterglow, either because there were no follow-up observations,or because searches were negative. In addition, as discussed before, the arrival directionderived from IPN localization depends on its assumed distance, and the bursts were initiallytriangulated assuming an infinite distance. Thus even if a simultaneous search had takenplace, it might not have identified an event within the error box if the burst was local. Otherselection effects come into play starting with the launch of the HETE spacecraft in 2000October, and later with the launch of Swift in 2004 November, namely that X-ray and opticalobservations were often done rapidly, leading to more X-ray and optical detections and theelimination of the bursts from further consideration here. On the other hand, the launchesof Suzaku in 2005 July and Fermi in 2008 June resulted in an increase in the short burstdetection rate which more than compensated for the previous effect. • Bursts with durations < <
5. Results5.1. Distance Limits and Localizations of PBH Burst Candidates
According the methodology described in Section 3 and Appendix A, we have calculated theminimum possible distances to the sample of 36 bursts selected in section 4. This burst sample isshown in Table 2 and the 12 columns give:1. the date of the burst, in yymmdd format, with suffix A or B where appropriate,2. the Universal Time of the burst at Earth, in seconds of day,3. the spacecraft which were used for the triangulation; a complete list of the spacecraft which de-tected the burst may be found on the IPN website ( http://ssl.berkeley.edu/ipn3/index.html ),4. the burst duration, in seconds,5. the fluence of the burst in erg cm − ,6. the energy range over which the duration and fluence were measured, in keV,7. the lower limit to the burst distance, obtained by triangulation, in cm,8. the distance to which this burst could have been detected if it were a PBH burst of energy10 erg, assuming that all the energy went into the measured fluence (this is essentially themaximum possible detectable distance),9. the maximum detectable distance assuming the SEM model (Equation 4) in terms of theundetermined parameter ( η γD ) . ,10. the maximum detectable distance assuming the HM model (Equation 5) in terms of theundetermined parameter ( η (cid:48) γD ) . ,11. whether or not counterpart searches took place and if so, their references,12. references to the duration, peak flux, and/or fluence measurements, and/or to the localization.The shortest burst in Table 2 has a duration of 60 ms. Due to the relatively coarse timeresolutions of interplanetary detectors, bursts with shorter durations must have greater intensitiesto be detected, effectively setting a higher detection threshold for very short events. The weakestevent has a fluence of 4 . × − erg cm − . The bursts in Table 2 could not have come fromdistances less than the distance lower limits in column 7; however, all of them have time delayswhich are also consistent with infinite distances. Figure 5 shows a histogram of these minimumdistances. The detector-dependent distance upper limits in column 8 are calculated assuming theextreme case that these events are caused by ∼ erg HM-type bursts from primordial black 9 –holes of mass ∼ g and that all of the emitted energy spectrum is contained within the detectormeasurement limits. Table 3 gives the coordinates of the centers and corners of the error boxes forthe events in Table 2, assuming that the sources are at infinity. If in fact the sources are local, thearrival directions are distance-dependent, and different from the ones in Table 3. These coordinatesrepresent the intersections of annuli, and in some cases the curvature of the annuli would make itinaccurate to construct an error box by connecting the coordinates with straight-line segments. All previous direct PBH burst searches resulted in null detections (Abdo et al. 2015). In thiscase, one can derive an upper limit on the local PBH burst rate density, that is, an upper limit onthe number of PBH bursts per unit volume per unit time in the local solar neighborhood.However, in our case, we have PBH burst candidates with short duration, no known afterglowdetection and minimum distances that are sub-light-years. Since we have PBH candidates, weshould be able to derive an actual measurement of the PBH burst rate density under the assumptionthat the candidates are actual PBH bursts. Thus, the actual PBH burst rate density is R = nV S(cid:15) (6)where n is the number of PBH bursts, V is the effective PBH detectable volume and S is theobserved duration. The selection efficiency of the IPN is (cid:15) .If all the candidates identified in Section 4 are real PBH bursts, then we have 36 PBH bursts,i.e., n =36. Hence, our PBH burst rate density estimate is, R = 36 V S(cid:15) . (7)Next we need to estimate the values of S , V , and (cid:15) . Because we have studied IPN burstscollected over 21.75 years (the 2.5 year IPN non–sensitivity period is excluded), our observedduration is S =21.75 years. The effective PBH detectable volume, V , calculation for this 21.75year period, however, is not obvious. Each PBH candidate has a distance consistent with someminimum distance up to infinity. We also know that PBH bursts are not bright enough to bedetected from large distances. The maximum possible detectable distance of a PBH burst dependson the high-energy physics model used to calculate the final PBH burst spectrum (Ukwatta et al.2015). Currently there are no accurate calculations for final PBH burst photon spectra in the keV-MeV energy range. Thus as a conservative maximum possible detectable PBH burst distance, wecan take the maximum value of the minimum distances in our candidate PBH burst sample. Thiscorresponds to a distance of 0.47 parsecs (1 . × cm). Because all the PBH burst candidates inthe sample are actual IPN detections, this distance value is model-independent. On the other hand,it is important to note that the IPN is not capable of detecting all the bursts within this distance 10 –over the entire observation duration due to various factors such as the orientation of the satellites,and/or instrument duty cycles. Thus the effective volume calculated from the above maximumpossible detectable PBH burst distance is an overestimate. Hence the PBH burst rate calculatedin Equation 7 is in reality a lower limit on the PBH burst rate density, R LL = 36 V S(cid:15) . (8)In Section 4, we made a rough estimate of the selection efficiency of IPN, (cid:15) , for PBH bursts.However, we note that it is very challenging to calculate (cid:15) accurately due to a number of unknownfactors such as the fraction of bursts without EE, the fraction of bursts without afterglows, thefraction of bursts to which the IPN is not sensitive (for example due to orientation or deadtime),etc. Using the estimated effective PBH detectable volume V , observed duration S , and selectionefficiency (cid:15) , we can now estimate the lower limit of the PBH burst rate in the best case scenariowhere all the PBH burst candidates are actual PBH bursts. In this case, our PBH burst lowerlimit is ∼ − yr − . If we assumed 100% efficiency then the PBH burst lower limitis ∼ − yr − . If only one of the candidates is an actual PBH burst then the valueof the rate density lower limit depends on the minimum distance to that particular burst. If theburst with the largest minimum distance (GRB 140807) is the PBH burst, then the PBH burstrate density lower limit is ∼ − yr − . If the burst with the smallest minimum distance(GRB 970902) is real then the PBH burst rate density lower limit is ∼ × bursts pc − yr − (this value is excluded by other high-energy experiments, however). All these estimates assumethat PBHs are distributed uniformly in the solar neighborhood. The IPN PBH burst rate densitylower limit values are shown in Figure 6. PBH burst upper limits from various other searches arealso shown in the figure.In the worst case scenario where none of our candidates is a real PBH burst, we cannot estimatea lower limit to the PBH burst rate density and instead consider to estimate an upper limit to thePBH burst rate density. However, the assumption that none of the bursts in the sample is real butstill we have candidates implies that our criteria to identify PBH bursts defined in section 4 is notsufficient. This means our method is not capable of setting an upper limit on the PBH burst ratedensity.
6. Discussion
The detection of gamma-ray transients points to very high energy explosive phenomena inthe Universe. Their detection in the solar neighborhood would indicate a previously unrecognizedand potentially exotic phenomenon in our cosmic backyard. The sample of bursts identified in thispaper are candidates for such explosions. They have short durations, no known afterglow detections,and have distance limits consistent with the solar neighborhood. In principle our methodology is 11 –capable of proving the local origin of bursts. However in order to do that we need either four widelyseparated non-imaging spacecraft or two spacecraft that include one with imaging capability (seeSection 3 and Appendix A). This is not the case for any of the bursts we considered in our study.While some events were indeed detected by four or more spacecraft, the spacecraft were not widelyseparated, i.e. at interplanetary distances. With four widely separated non-imaging spacecraftdetections or detections by one imaging spacecraft and one non-imaging spacecraft, it would bepossible to prove that some of these bursts are in the solar neighborhood and this would definitelypoint to an exotic origin for these bursts. Lacking that however, we can look at other properties ofthese bursts and discuss how likely it is that they may have a PBH origin.Firstly, it is of interest to investigate the sky distribution of our PBH burst candidates. Forexample, Cline et al. (1997) have argued that, due to the fact that very short duration GRBs(i.e., GRBs with duration ≤
100 msec) have a non-isotropic sky distribution, they may be drawnfrom a different GRB population, possibly from PBH bursts. In order to investigate this we havecalculated burst density maps using the Gaussian kernel density methodology described in Ukwatta& Wo´zniak (2016). Since the sky locations of the PBH candidates depends on their distance fromearth, we started by assuming all the bursts are at their minimum distances and calculated theirsky density map. This map is shown in Figure 7. The map is presented in Galactic coordinateswith a 25 degree smoothing radius. It shows some relatively high density areas, but the probabilityof generating this density contrast by chance, in the case when the true sky distribution is uniform,is ∼ ∼
10 parsecs (which is the maximum possible distancethey can be detected assuming the optimistic Hagedorn-type model). We then also calculated thesky locations of the PBH candidates assuming they are at 10 parsecs and derived the sky densitymap as shown in Figure 8. This map is also consistent with a uniform source distribution. This isthe behaviour one would expect if these PBH burst are local with maximum detectable distance ∼
10 parsecs.According to the standard model for Hawking radiation (MacGibbon and Webber 1990),PBH bursts are standard candles, that is, all PBH bursts are intrinsically identical at the source.However, the way the burst appears at large distances may vary depending on its host environment.The final burst properties of the PBH burst depend on its mass and the number of fundamentalparticle degrees of freedom available at various energies (Ukwatta et al. 2015). In principle, bymeasuring the photon flux arriving from a PBH burst candidate in a given energy range, we cancalculate the distance to that burst. However, during the last second of the PBH lifetime, itstemperature is well above ∼ ∼ . ∼ . ∼ < (cid:29) yrand observationally would be a stable source not a burst. This low energy photon component, inturn, is predominantly generated by other higher energy Hawking-radiated species via decays or theinner bremsstrahlung effect (Page et al. 2008) and is not the directly Hawking-radiated photon fluxwhich decreases in the keV/MeV band as the burst progresses. Acknowledging the uncertainty inthe PBH light curves in the keV/MeV range, it is also possible that the PBH burst signal may havea longer or shorter duration in the keV/MeV range than in the GeV/TeV range due to differencesboth in production at the source and in detector sensitivity, and that the duration difference varieswith the distance to the PBH. Figure 9 shows the light curves of the 36 IPN bursts in our sample.Some are clearly single-peaked, others are clearly multi-peaked, and some were not recorded withsufficient statistics to determine the true number of peaks. It is interesting to note that bursts suchas GRB 970921, GRB 080222, GRB000607, GRB 101009, GRB 121127, GRB 131126A, and GRB141011A display a keV/MeV time profile that resembles the PBH light curve profile calculatedby Ukwatta et al. (2015) for the GeV/TeV energy range.In this paper, we introduced a novel method to constrain the distances to GRBs using detec-tions from multiple spacecraft. Utilizing detections from three non-imaging spacecraft we couldonly constrain the minimum distances to our current sample of bursts. The maximum distance isconstrained by the energy available during the final second of the PBH burst. However, the amountof energy released in the keV/MeV energy band is not known and may be highly model-dependent.On the other hand, with detections by four widely separated ( ∼ AU distances) non-imaging space-craft or one non-imaging spacecraft and one imaging spacecraft, we can constrain burst distancesindependent of any high energy physics model, and potentially show that some bursts are local.Such a detection will not only prove the existence of PBH bursts, by fitting light curves and spectraderived using various beyond the standard model physics theories, we can also identify which theorydescribes nature. 13 –
7. Acknowledgments
Support for the IPN was provided by NASA grants NNX09AU03G, NNX10AU34G, NNX11AP96G,and NNX13AP09G (
Fermi ); NNG04GM50G, NNG06GE69G, NNX07AQ22G, NNX08AC90G, NNX08AX95G,and NNX09AR28G (INTEGRAL); NNX08AN23G, NNX09AO97G, and NNX12AD68G(
Swift ); NNX06AI36G,NNX08AB84G, NNX08AZ85G, NNX09AV61G, and NNX10AR12G (
Suzaku ); NNX07AR71G (MES-SENGER); NAG5-3500, and JPL Contracts 1282043 and Y503559 (
Odyssey ); NNX12AE41G,NNX13AI54G, and NNX15AE60G (ADA); NNX07AH52G (Konus); NAG5-13080 (RHESSI); NAG5-7766, NAG5-9126, and NAG5-10710, (
BeppoSAX ); and NNG06GI89G. TNU acknowledges supportfrom the Laboratory Directed Research and Development program at the Los Alamos National Lab-oratory (LANL). The Konus-Wind experiment is partially supported by a Russian Space Agencycontract and RFBR grants 15-02-00532 and 13-02-12017-ofi-m. We would also like to thank JimLinnemann (MSU), Dan Stump (MSU), Brenda Dingus (LANL), and Pat Harding (LANL) foruseful conversations on the analysis.
A. GRB triangulation when the source distance is allowed to vary
Assume two spacecraft, SC and SC , separated by a distance d , observe a GRB. For anyassumed distance between the GRB and the spacecraft, the difference in arrival times δ t must beconstant. Let x , y , z and x , y , z be the coordinates of the two spacecraft. Then the locus ofpoints x, y, z which describes the possible source locations is given by (cid:112) ( x − x ) + ( y − y ) + ( z − z ) − (cid:112) ( x − x ) + ( y − y ) + ( z − z ) − c ∗ δt = 0where c is the speed of light.Consider first the two-dimensional problem for simplicity. Let SC and SC define the z-axisof a coordinate system whose origin is halfway between the spacecraft. The positions of SC and SC are the foci of the hyperbola: z /a − x /b = 1This is shown in Figure 1. Here 2 a is the difference between the distances of any point on thehyperbola from the foci, so 2 a = c δt , and b = ( d / − a ) / . For every point on this hyperbola,the difference in the arrival times is δt . If we assume an infinite distance for the source, theasymptotes of the hyperbola define the two possible arrival directions of the GRB.Now consider the three-dimensional case. If we rotate the hyperbola of Figure 1 about the zaxis, we obtain one sheet of a hyperboloid of rotation of two sheets. Its formula is 14 – − x /b + z /a − y /b = 1 . Here, the x axis is perpendicular to the y and z axes, and cuts in the plane z=constant givecircles. This is illustrated in Figure 2.In practice, we will have two or more hyperboloids generated by three or more spacecraft, andwe will want to work in Earth-centered Cartesian coordinates with one axis oriented towards rightascension zero, declination zero, and another axis oriented towards declination 90 ◦ . Consider thethree-spacecraft case. A spacecraft pair will define two foci of a hyperboloid; the line joining thetwo spacecraft, which defines the axis of rotation of the hyperboloid, will be oriented with respectto Earth-centered coordinates such that it represents a rotation and a translation. We want toexpress the formula for the hyperboloid in the Earth-centered system.The coordinate rotation can be described by three sets of direction cosines: z (cid:48) = x ∗ cos( αz ) + y ∗ cos( βz ) + z ∗ cos( γz ) y (cid:48) = x ∗ cos( αy ) + y ∗ cos( βy ) + z ∗ cos( γy ) x (cid:48) = x ∗ cos( αx ) + y ∗ cos( βx ) + z ∗ cos( γx )Here the primed coordinate system is the one defined by the foci of the hyperboloid; its origin isthe same as that of the unprimed, Earth-centered system, and it is rotated, but not translated, withrespect to it. Now perform a translation of the primed system so that its origin is at the midpointof the two foci. If the coordinates of the two spacecraft, expressed in the unprimed system, are x , y , z and x , y , z , the origin of the translated system will be at ( x + x )/2, ( y + y )/2, ( z + z )/2. The formula for the hyperboloid, expressed in the Earth-centered system, becomes( x ∗ cos( αz ) + x ∗ cos( αz ) + y ∗ cos( βz ) + y ∗ cos( βz ) + z ∗ cos( γz ) + z ∗ cos( γz ) − ∗ x ∗ cos( αz ) − ∗ y ∗ cos( βz ) − ∗ z ∗ cos( γz )) / (4 ∗ a ) − ( x ∗ cos( αx ) + x ∗ cos( αx ) + y ∗ cos( βx ) + y ∗ cos( βx ) + z ∗ cos( γx ) + z ∗ cos( γx ) − ∗ x ∗ cos( αx ) − ∗ y ∗ cos( βx ) − ∗ z ∗ cos( γx )) / (4 ∗ b ) − ( x ∗ cos( αy ) + x ∗ cos( αy ) + y ∗ cos( βy ) + y ∗ cos( βy ) + z ∗ cos( γy ) + z ∗ cos( γy ) − ∗ x ∗ cos( αy ) − ∗ y ∗ cos( βy ) − ∗ z ∗ cos( γy )) / (4 ∗ b ) − a and b refer to the hyperboloid for spacecraft 1 and 2. A similar equation describesthe hyperboloid for spacecraft 1 and 3. Although a third equation can be derived for spacecraft 2and 3, it is not independent of the other two, because it is constrained by the condition δt + δt + δt = 0. 15 –The locus of points describing the intersection of two hyperboloids is a simple hyperbola,contained in a plane. This is shown in Figure 3. This hyperbola contains all the points satisfyingthe time delays for the two spacecraft pairs, δt and δt , when the GRB distance is allowedto vary. The two branches of the hyperbola intersect the celestial sphere at two points; if thedistance is taken to be infinite, the two points are the possible source locations. It follows thata GRB observed by three, and only three, widely separated non-imaging spacecraft, cannot beunambiguously proven to originate at a local distance; on the other hand, in the case where thehyperbola degenerates to a single point, that point must be at an infinite distance, and a localorigin can be ruled out. None of the bursts in this sample were in this category.In the simplest case, we have one spacecraft near Earth, and two spacecraft in interplanetaryspace. So (cid:112) ( x − x ) + ( y − y ) + ( z − z ) − (cid:112) ( x − x ) + ( y − y ) + ( z − z ) − c ∗ δt = 0 (A1) (cid:112) ( x − x ) + ( y − y ) + ( z − z ) − (cid:112) ( x − x ) + ( y − y ) + ( z − z ) − c ∗ δt = 0 (A2)The lower limit to the source distance is the point on the hyperbola ( x, y, z ) which is closestto Earth. This can be found by solving for the minimum of the expression (cid:112) x + y + z (theEarth distance) subject to the constraints imposed by equations A1 and A2. In practice there areuncertainties associated with δt and δt , and we have used the most probable values to derivethe lower limits. Since x , y , and z vary along the hyperbola, the apparent arrival direction for anobserver at Earth depends on the assumed distance; if the source distance is assumed to be infinite,the derived right ascension and declination will not be correct if the source is actually local. Forexample, if GRB 140807 were at its minimum allowable distance (1 . × cm, or 9 . × AU),the angle between its true coordinates and the coordinates for an infinitely distant source would be0 . ◦ . But for GRB 101129, whose minimum allowable distance is only 3 . × cm, or 2.6 AU,the angle would be 54.2 ◦ .In a number of cases, a fourth non-coplanar experiment, in this case Konus-Wind , at up to 7light-seconds from Earth, can be used to constrain the lower limits further. Adding the constraint (cid:112) ( x − x ) + ( y − y ) + ( z − z ) − (cid:112) ( x − x ) + ( y − y ) + ( z − z ) − c ∗ δt = 0eliminates one branch of the hyperbola and part of the second branch, leading to a largerdistance lower limit. Thus, in the case of four widely separated spacecraft, it is in principle possibleto rule out an infinite distance and prove that the origin is local as illustrated in Figure 4. However,this was not the case for any of the bursts in this study; they are all consistent with both local andinfinite distances. 16 –Note that this method does not depend on the properties of the GRB itself, such as dura-tion or intensity; the lower limit is determined by the IPN configuration (through the spacecraftcoordinates) and the direction of the burst (through the time delays). Thus, for example, it canbe applied equally to long- and short-duration bursts, and a sample of long-duration events wouldyield a distribution of lower limits which was comparable to a sample of short-duration events.One special case should be noted here. With just two widely separated spacecraft, if one hasprecise imaging capability, the problem reduces to finding the intersection of a hyperboloid andthe vector defined by the precise localization from the imager. In principle, a local origin canbe demonstrated, or a distance lower limit can be obtained. We have studied approximately twohundred GRBs which are in this category, and analysis of the results is ongoing. 17 – REFERENCES
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21 –Fig. 1.— A two-dimensional example of GRB triangulation when the source distance is allowed tovary. The two spacecraft, 1 and 2, are aligned along the z-axis, and are the foci of a hyperbola.The hyperbola defines the loci of possible source distances. If the distance is assumed to be infinite,the two possible GRB arrival directions are along the asymptotes (dashed lines). 22 –Fig. 2.— The hyperbola of Figure 1 rotated to obtain one sheet of a hyperboloid of rotation of twosheets. The two spacecraft are aligned along the z-axis, and are the foci of the hyperboloids. Eachhyperboloid defines the loci of possible source distances in the three-dimensional problem. If thedistance is assumed to be infinite, the circle defined by the intersection of the hyperboloid with thecelestial sphere gives the possible GRB arrival directions. 23 –
Hyperboloid HyperboloidSpacecraft 02 Spacecraft 03Spacecraft 01 Intersection is a hyperbola
Fig. 3.— The intersection of two hyperboloids. With detections with three spacecraft, each space-craft pair constrain the location of the source to a surface of a hyperboloid. The intersection of thetwo hyperboloids is a simple hyperbola contained in a plane. The point on this hyperbola which isclosest to Earth gives the lower limit to the source distance. 24 –
Intersection of the three hyperboloids is a pointHyperboloid HyperboloidHyperboloid Spacecraft 01 Spacecraft 04Spacecraft 03Spacecraft 02
Fig. 4.— The intersection of three hyperboloids. With detections with four widely separatedspacecraft, each spacecraft pair constrain the location of the source to a surface of a hyperboloid.The intersection of the three hyperboloids is a point in space as illustrated in the figure. In thiscase one can determine the source distance purely by timing measurements. 25 – N u m b e r o f B u r s t s Fig. 5.— Histogram of the minimum distances to the PBH burst candidates in the IPN sample. 26 – -3 -2 -1 Search Duration (sec)10 -2 -1 P B H B u r s t R a t e ( p c − y r − ) All limits assume PBHs are distributed uniformly in the solar neighborhood. Whipple Upper Limits (1, 3, 5 sec)CYGNUS Upper Limit (1 sec)Tibet Air Shower Array Upper LimitVERITAS Upper Limit (1 sec)Milagro Upper LimitHESS Upper Limit (1, 30 sec)IPN Best Case Lower LimitIPN Best Case Lower Limit (with 2.3 % efficiency)
Fig. 6.— IPN PBH burst rate lower limit estimates assuming all the candidates are real PBHbursts. The horizontal green line gives the IPN PBH burst rate lower limit considering the selectionefficiency of 2.3%. The blue horizontal line shows the lower limit if we assume 100% selectionefficiency. Published PBH burst rate upper limits for various other burst search experiments areshown for comparison (Alexandreas et al. 1993; Amenomori et al. 1995; Linton et al. 2006; Tesic2012; Glicenstein et al. 2013; Abdo et al. 2015). 27 – -75°-60°-45°-30°-15°0°+15° +30° +45° +60° +75° 210°240°270°300°330°0°30°60°90°120°150°
Fig. 7.— GRB density map in Galactic coordinates for the PBH burst candidates assuming mini-mum distances given in Table 2. This map is normalized to represent a probability density function(PDF) that integrates to 1 over the entire sphere. The smoothing parameter is taken to be 25 de-grees. The circles indicate the locations of the individual bursts. The maximum and the minimumdensity values in this map are 0.166 and 0.041, respectively. The probability of generating thisdensity contrast by chance in the case when the true sky distribution is uniform, is ∼ -75°-60°-45°-30°-15°0°+15° +30° +45° +60° +75° 210°240°270°300°330°0°30°60°90°120°150° Fig. 8.— GRB density map in Galactic coordinates for the PBH burst candidates assuming constantdistances of 10 parsecs to sources. The map has only 24 candidates with four spacecraft detections.Remaining 8 bursts have only three spacecraft detections, so they don’t have a single localization.This map is normalized to represent a probability density function (PDF) that integrates to 1 overthe entire sphere. The smoothing parameter is taken to be 25 degrees. The circles indicate thelocations of the individual bursts. The maximum and the minimum density values in this map are0.202 and 0.002, respectively. The probability of generating this density contrast by chance in thecase when the true sky distribution is uniform, is ∼ Fig. 9.— Normalized light curves of all the PBH burst candidates in the IPN sample. Black labelsshow the burst name with parenthesis showing the instrument. The blue labels give the minimumdistance to the bursts based on our analysis. Each light curve shows a time range of 4 secondscentered on the brightest peak. 30 – T a b l e . C o m p a r i s o n o f m e t h o d s u s e d t o d e t ec t P B H e v a p o r a t i o n s . M e t h o d B u r s t du r a t i o n , s E n e r g y o r f r e q u e n c y R a t e upp e r li m i t s , p c − y − R e f e r e n ce s A t m o s ph e r i c C h e r e n k o v − − . M e v - T e V . − . × A i r Sh o w e r − − . × − (cid:38) × e V . × − × R a d i o P u l s e < × − - M H z × − Sp a r k C h a m b e r − M e V - G e V × − Sp a t i a l D i s t r i bu t i o n < . k e V - M e V ··· R e f e r e n ce s . — ( ) P o r t e r a nd W ee k e s ( , , ) ; A bd o e t a l. ( ) ; ( ) B h a t e t a l. ( ) ; F e ga n e t a l. ( ) ; ( ) P h i nn e y a nd T a y l o r ( ) ; K e a n ee t a l. ( ) ; ( ) F i c h t e l e t a l. ( ) ; ( ) C li n e a nd H o n g ( ) ; C ze r n y e t a l. ( ) ; C li n ee t a l. ( , , , ) ; C ze r n y e t a l. ( )
31 – T a b l e . D i s t a n ce l o w e r li m i t s f o r I P N ga mm a - r a y bu r s t s . G R B S O D Sp a cec r a f t a D u r ., F l u e n ce E n e r g y M i n i m u m M a x i m u m M a x i m u m M a x i m u m C o un t e r p a r t R e f s . s e r g c m − r a n g e , D i s t a n ce , d e t ec t a b l e d e t ec t a b l e d e t ec t a b l e s e a r c h ? k e V c m . d i s t ., c m . d i s t ., ( η γ D ) . c m . d i s t ., ( η (cid:48) γ D ) . c m . ( S E MM o d e l )( H MM o d e l ) U l y , P V O , P h e . . × − . × . × . × . × N o1 , U l y , B A T , P V O . . × − . × . × . × . × N o2 , , U l y , K o n , N E A R . . × − . × . × . × . × N o4 , U l y , K o n , N E A R . . × − . × . × . × . × N o4 , U l y , K o n , N E A R . . × − . × . × . × . × N o5 ,
13 99071227919 U l y , B A T , K o n , N E A R . . × − . × . × . × . × N o5 , U l y , K o n , N E A R . . × − . × . × . × . × , ,
13 00102571369 U l y , K o n , N E A R . . × − . × . × . × . × , ,
13 00120428869 U l y , K o n , S AX , N E A R . . × − . × . × . × . × , , , ,
13 01010462490 U l y , K o n , S AX , N E A R . . × − . × . × . × . × N o5 , ,
13 01011937177 U l y , K o n , N E A R . . × − . × . × . × . × N o5 , ,
13 08020308456 O d y , K o n , I N T , M E S . . × − . × . × . × . × N o5 ,
13 08022237262 O d y , K o n , M E S . . × − . × . × . × . × N o5 ,
13 08053058296 O d y , I N T , M E S , Su z . . × − . × . × . × . × N o6 09022817600 O d y , K o n , R H E , M E S . . × − . × . × . × . × N o5 , , , ,
19 09052334075 O d y , K o n , M E S , Su z . . × − . × . × . × . × N o5 , ,
13 10022309491 O d y , K o n , M E S , Su z . . × − . × . × . × . × N o5 , , ,
10 10062969243 O d y , K o n , I N T , M E S . . × − . × . × . × . × N o5 , , ,
10 10081109349 O d y , K o n , I N T , M E S . . × − . × . × . × . × N o5 , , ,
11 10100924858 O d y , K o n , I N T , M E S . . × − . × . × . × . × N o5 , ,
13 10112956371 O d y , K o n , I N T , M E S . . × − . × . × . × . × N o5 , ,
11 11051080844 O d y , K o n , S w i, M E S . . × − . × . × . × . × N o6 ,
13 11070513031 O d y , K o n , I N T , M E S . . × − . × . × . × . × N o9 , , , ,
22 11080255157 O d y , K o n , I N T , M E S . . × − . × . × . × . × N o13 , ,
24 12022201776 O d y , K o n , M E S , F e r . . × − . × . × . × . × N o9 ,
11 12051962294 O d y , K o n , M E S , F e r . . × − . × . × . × . × N o9 , , , ,
27 12081101230 O d y , K o n , M E S , F e r . . × − . × . × . × . × N o9 , , , , ,
31 12102327857 O d y , M E S , F e r . . × − . × . × . × . × N o9 ,
11 12112778960 O d y , K o n , I N T , M E S . . × − . × . × . × . × N o11 , , , ,
34 13050100831 O d y , K o n , M E S , Su z . . × − . × . × . × . × N o13 130504 B O d y , K o n , M E S . . × − . × . × . × . × N o35 , , ,
38 131126 A O d y , K o n , M E S , F e r . . × − . × . × . × . × , ,
32 – T a b l e — C o n t i nu e d G R B S O D Sp a cec r a f t a D u r ., F l u e n ce E n e r g y M i n i m u m M a x i m u m M a x i m u m M a x i m u m C o un t e r p a r t R e f s . s e r g c m − r a n g e , D i s t a n ce , d e t ec t a b l e d e t ec t a b l e d e t ec t a b l e s e a r c h ? k e V c m . d i s t ., c m . d i s t ., ( η γ D ) . c m . d i s t ., ( η (cid:48) γ D ) . c m . ( S E MM o d e l )( H MM o d e l ) B O d y , K o n , I N T , M E S . . × − . × . × . × . × N o13 14080743173 O d y , K o n , M E S , F e r . . × − . × . × . × . × C O d y , K o n , I N T , M E S . . × − . × . × . × . × N o44 ,
45 141011 A O d y , K o n , M E S , F e r . . × − . × . × . × . × N o46 , , a F e r : F e r m i , I N T : I n t e r n a t i o n a l G a mm a - R a y L a b o r a t o r y , K o n : K o n u s - W i n d , M E S : M e r c u r y S u r f a ce , S pa ce E n v i r o n m e n t , G e o c h e m i s t r y , a n d R a n g i n g m i ss i o n , N E A R : N e a r E a r t h A s t e r o i d R e n d e z v o u s m i ss i o n , O d y : M a r s O d y ss e y , P h e : P h eb u s , P V O : P i o n ee r V e n u s O r b i t e r , R H E : R a m a t y H i g h E n e r g y S o l a r S p ec t r o s c op i c I m a ge r , S AX : S a t e ll i t e p e r A s t r o n o m i a X ( B e ppo S AX ) , Su z : S u z a k u , S w i: S w if t ( bu r s t w a s o u t s i d e t h ec o d e dfi e l d o f v i e w o f t h e B A T , a ndn o t l o c a li ze db y i t) , U l y : U l y ss e s R e f e r e n ce s . — ( ) T e r e k h o v e t a l. ( ) ; ( ) G o l d s t e i n e t a l. ( ) ; ( ) P a c i e s a s e t a l. ( ) ; ( ) ; ( ) P a l’ s h i n e t a l. ( ) ; ( ) http://ssl.berkeley.edu/ipn3/index.html ; ( ) F r o n t e r a e t a l. ( ) ; ( ) ; ( ) G o l d s t e i n e t a l. ( ) ; ( ) P a c i e s a s e t a l. ( ) ; ( ) v o n K i e n li n e t a l. ( ) ; ( ) G o l d s t e i n e t a l. ( ) ; ( ) S v i n k i n e t a l. ( b ) ; ( ) M a s e tt i e t a l. ( ) ; ( ) H u r l e y e t a l. ( ) ; ( ) P a r k e t a l. ( ) ; ( ) P r i cee t a l. ( ) ; ( ) V r ee s w i j k a nd R o l ( ) ; ( ) G u i r i ece t a l. ( ) ; ( ) G o l e n e t s k ii e t a l. ( ) ; ( ) G o l e n e t s k ii e t a l. ( b ) ; ( ) Y a s ud a e t a l. ( ) ; ( ) H u r l e y e t a l. ( ) ; ( ) G o l e n e t s k ii e t a l. ( c ) ; ( ) G o l e n e t s k ii e t a l. ( ) ; ( ) G o l e n e t s k ii e t a l. ( b ) ; ( ) S a k a m o t o e t a l. ( ) ; ( ) G o l e n e t s k ii e t a l. ( c ) ; ( ) G o l e n e t s k ii e t a l. ( d ) ; ( ) G o l e n e t s k ii e t a l. ( e ) ; ( ) X i o n ga nd M ee ga n ( ) ; ( ) G o l e n e t s k ii e t a l. ( f ) ; ( ) G o l e n e t s k ii e t a l. ( ) ; ( ) I s h i d a e t a l. ( ) ; ( ) V r ee s w i j k a nd R o l ( ) ; ( ) G o l e n e t s k ii e t a l. ( ) ; ( ) G o l e n e t s k ii e t a l. ( b ) ; ( ) Y a s ud a e t a l. ( ) ; ( ) S i n g e r e t a l. ( ) ; ( ) G o l e n e t s k ii e t a l. ( c ) ; ( ) G o l e n e t s k ii e t a l. ( d ) ; ( ) P e l a ss aa nd M ee ga n ( ) ; ( ) S i n g e r ( ) ; ( ) G o l e n e t s k ii e t a l. ( ) ; ( ) G o l e n e t s k ii e t a l. ( b ) ; ( ) v o n K i e n li n ( ) ; ( ) G o l e n e t s k ii e t a l. ( c ) ; ( ) G o l e n e t s k ii e t a l. ( d )
33 –Table 3. Localizations of IPN gamma-ray bursts assuming infinite distances. Some bursts havetwo possible error boxes; GRB080203 has an eight-cornered error box.
GRB α δ
34 –Table 3—Continued
GRB α δ
35 –Table 3—Continued
GRB α δ
36 –Table 3—Continued
GRB α δ
37 –Table 3—Continued
GRB α δ
38 –Table 3—Continued
GRB α δα δ