Quantum-Gravity Based Photon Dispersion in GRBs: The Detection Problem Jay Norris and Jeff Scargle Some mathematical

1. Quantum-Gravity Based Photon Dispersion  in GRBs:  The Detection Problem • Jay Norris and Jeff Scargle • Some mathematical approaches to Quantum Gravity • Why short GRBs are the best probes of QG dispersion: redshift distribution; narrow, energy-independent pulses; expectations at LAT energies • Promising Example: extrapolation of Swift’s brightest short GRB to LAT energies • De-dispersion procedure, Metrics for dispersion recovery • Summary

2. Discreteness of Space-Time Sorkin (2002) emphasizes Riemann’s recognition that a discrete manifold inherently contains dynamical relationships that define its metric, whereas continuous manifolds are incomplete and must have the metric specified ad hoc. Nature of the Problem — Planck distance and temporal scales. Adler and Santiago (1999) demonstrate by simple argument a generalized uncertainty principle, deriving a minimum uncertainty for the measured position of any particle, and for the time to which the measurement refers: LPlanck = (Gℏ/c3)½ ~ 1.6  10–35 m TPlanck = (Gℏ/c5)½ ~ 0.54  10–43 s

3. Mathematical Approaches to Unification (QG), 1 String Theory — Starts with quantum theory, adds GR as a (not very small) perturbation. The forms developed make no specific predictions of photon dispersion, although Lorentz Invariance Violation (LIV) is part of noncommutative field theories invoked in string theory. UL’s on dispersion from astrophysical sources used to fix parameters of string theory (Ellis et al. 2004). Loop Quantum Gravity — Starts with GR, adds the condition that space consists of quanta on the order of LP3. The quanta have properties similar to those of loops, with knots and intersections. The spatial quantization produces photon dispersion at the Planck scale (Alfaro et al. 2002).

4. Mathematical Approaches to Unification (QG), 2 Causal Set — In QG based on causal sets there is no underlying continuum: Space consists of points separated by intervals ~ LP. Causal set models do not predict dispersion effects (Sorkin 2002; Dowker, Henson, & Sorkin 2004; and especially, Bombelli, Henson & Sorkin 2006). Regge Calculus — Treats space-time explicitly as a set of discrete structures (4-D polyhedra in which space is flat) — all space-time curvature effects occur discretely at polyhedra boundaries (Regge 1961). This system is a scheme for carrying out numerical GR computations (e.g. Gionti 2006). Explicit consideration of photon dispersion has not been made.

5. Suspect Long Burst if: • Tdur > 2 s • Pulse width: 0.3 - 50 s, and many pulses • Star-forming Host • Redshift mode ~ 2-4 • (Progenitor = massive star) Swift BeppoSAX Suspect Short Burst if: • Tdur < 2 s • Pulse width: 5 - 30 ms, and few pulses • Star-forming & Old Hosts • Redshift ~ 0.1 - 1 • (Progenitor = coalescing • NS-NS or NS-BH binary) There are Two Kinds GRBs — Long & Short From Bloom et al. astro-ph/0505480

6. Virgo Cluster of Galaxies Half Age of Universe Histogram: Long GRBs Diamonds: Short GRBs

7. GRB 051221a Diamonds: Short GRBs Comoving distance is relevant one for actual photon propagation. Median for short bursts: dCM ~ 1 Gpc.

8. “Era of Long GRBs” Era of Short GRBs? Selection effects … The Big Bang (13.7 Gyrs) Trilobytes (500 Myrs)

9. Expectations at GLAST/LAT Energies In the usual expression for LIV photon dispersion, it’s the value of  that we aim to constrain: E2 = k2 [ 1 + (k/Mp) + … ] by measuring tLIV, which corresponds to energy-dependent difference in light travel time for comoving distance dCM tLIV = tH (E/Ep)  h(z)–1 dz * For E = 1 GeV and dCM = 1 Gpc, tLIV 8 ms Timescale ~ to pulse widths in short GRBs * where h(z) = [ + M(1+z)3] ½ Ep = (ℏc5/G)½ = 1.22  1019 GeV H0 = 71 km s–1 Mpc–1 flat universe with  = 0.73, M = 0.27 tH = H0–1 = 4.34  1017 s.

10. Swift/BAT: Brightest in ~ 4 years z = 0.5465 15–350 keV Pulse centroid times in short GRBs are energy-independent, 10-1000 keV. However, like long GRBs, the pulse widths narrow as energy increases …

11. So, fit pulses, extrapolate pulse narrowing trend to LAT energies … Pulse widths: 1/e ~ 20-50 ms Pulse model, I(t) = A λ/ [exp{1/t} exp{t/ 2}]

12. At GLAST/LAT energies, pulses will be narrower. Here, trend from BATSE energies, wid ~ E-1/3 is extrapolated to GeV regime. Then, “presumed” QG dispersion is added, ~ 10 ms / GeV / Gpc. Apply trial de-dispersions. The process bootstraps using only photon times & recon’ed energies. Several metrics were explored for best dispersion recovery — information or entropy related

13. De-dispersion formula: t’i = tobsi –  Eobsi Measure: Min { Avg [ Intra-pulse photon interval ] } Typical de-dispersion run for synthetic LAT burst. True value = 20 ms. Analysis of 100 realizations of the synthetic burst shows that dispersion recovery for this measure has accuracy of ~ 6% (1 ) …

14. … However, accuracy of the Min{ Avg [T] } measure (left panel) shows large dependence on pulse width parameter, . Shannon Information measure (right panel) shows almost negligible dependence on . min = Min {ti / nintra} I (Shannon) = –  pi log(pi) , pi  ti–1 Fig. 11 – Histogram of 100 estimates of linear dispersion constant for minfrom realizations of randomly generated burst data sets, for three values of pulse width model parameter . Fig. 6 – Similar to Fig. 11, but for Shannon information. Accuracy of recovery is better, and less dependent on width parameter, .

15. Some Caveats • The putative QG time dispersion may be ~ 10 ms / GeV / Gpc. The distance dependence would have to be observed — using several bursts — before attribution to QG would have credibility. • So far, there are not dependable “pseudo-redshift indicators” for short GRBs — only spectroscopic redshifts reveal their distances. • The redshifts are gotten from the short bursts’ afterglows, which are dimmer than long bursts’ afterglows; and when optical afterglows are present, they tend to fade more quickly. • Sufficiently bright short bursts will have ~ GeV photons, affording relatively small error regions. Larger telescopes participate in afterglow searches when small error boxes are quickly available, since spectroscopic redshifts are then often obtainable. • Thus, one additional motivation for sending LAT alerts to the ground — containing a few high energy ’s ID’ed on-board — for accurate ground reconstruction, and rapid GCN dissemination.

16. Summary • Narrow pulses in extremely bright, short bursts offer a unqiue tool for constraining some QG scenarios — those that suggest energy-dependent time dispersion of high energy gammas. Pulses in short bursts have FWHM ~ 10-30 msat LAT energies, comparable to the dispersion in time for gamma rays observable by the LAT, ~ 10 ms / GeV / Gpc, given the nascent redshift distribution of short bursts, z ~ 0.1-1. • We would need to observe the distance dependence, to have confidence that such an effect may be attributable to QG. One short burst in 2-3 years may be bright enough to provide useful constraints. The full duration of the GLAST mission, 5+5 years, may be necessary to detect enough bright short bursts, and thereby perform a definitive experiment. References follow

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