arXiv:0712.2781v1 [astro-ph] 17 Dec 2007 e words: Key orcinlast lgtices ftelmto h scal the on limit r the a of obtain th increase to on slight > linearly [1,2] a M depends to refs. that leads in invariance correction used Lorentz formula of fitting violation the correct We Abstract eateto hsc,TxsA&MUiest,CleeStatio College University, M & A Physics, of Department PACS: transform rpitsbitdt leirSine2 coe 2018 October 27 Science Elsevier to submitted Preprint Violation Lorentz on Limits Robust Erratum: hoyDvso,PyisDprmn,CR,C-21Gnv 23, CH-1211 CERN, Department, Physics Division, Theory hoyDvso,PyisDprmn,CR,C-21Gnv 23, Geneva CH-1211 CERN, Department, Physics Division, Theory eateto hsc,Kn’ olg odn nvriyof University London, College King’s Physics, of Department cdm fAhn,Dvso fNtrlSine,2 Panepis 28 Sciences, Natural of Division , of Academy PDvso,PyisDprmn,CR,C-21Gnv 3 Swi 23, Geneva CH-1211 CERN, Department, Physics Division, EP srpril hsc ru,HutnAvne eerhCen Research Advanced Group, Physics Astroparticle 1 hsc eatet nvrietAtepn -60Wilri B-2610 Antwerpen, Universiteit Department, Physics ws nttt fTcnlg,EHZuih 03Z¨urich, Sw ETH-Z¨urich, 8093 Technology, of Institute Swiss . 4 45.h 46.v 98.80.Cq 04.62.+v, 04.50.+h, × 10 oet naine am a us,qatmgaiy wave gravity, quantum burst, ray gamma invariance, Lorentz 16 GeV. icelCmu,Wolns X731 USA 77381, TX Woodlands, Campus, Mitchell rmGmaRyBursts Gamma-Ray from odnW2 L,UK 2LS, WC2R London ..Mavromatos N.E. ...Sarkisyan E.K.G. ..Nanopoulos D.V. tes169 10679, Athens ..Sakharov A.S. onEllis John ∗ htneeg.The energy. photon e ftevoain to violation, the of e ,T 74,USA 77843, TX n, odn Strand, London, iiuAvenue, timiou k Belgium jk, bs ii na on limit obust e (HARC), ter itzerland Switzerland Switzerland tzerland let It has been resently pointed out in [3] that, due to the fact that the comoving distance that light travels while coming from an object at redshift z in the expanding Universe is bigger by a factor (1 + z) than the proper distance [4], formula (1) in [1] (see also formula (13) in [2]) for the difference in the arrival times of two photons with energies differing by ∆E in the case of a linear violation of Lorentz invariance should be corrected to read: z −1 ∆E (1 + z)dz ∆tLV = H , (1) 0 M Z h z 0 ( ) where H0 is the Hubble expansion rate,

3 h(z)= ΩΛ + ΩM (1 + z) , (2) q and we assume a spatially-flat Universe: Ωtotal = ΩΛ + ΩM = 1 with cosmo- logical constant ΩΛ ≃ 0.7.

As a result of this correction, the arrival time delays calculated in [1] should be fitted by a linear function, as in equation (4) of [1] but in terms of the variable: z 1 (1 + z)dz K ≡ . (3) z Z h z 1+ 0 ( )

The fit replacing the left panel of Fig. 2 in [1] is presented in Fig. 1. The linear fit corresponds to ∆tdw obs = (0.0068 ± 0.0067) K − (0.0065 ± 0.0046), (4) 1+ z and the likelihood function for the slope parameter analyzed in equation (14) of [1] is presented in Fig. 2 and, in fact, reflects better the sensitivity of the fit and in this sense replaces Fig. 4 of [1].

The 95% confidence-level lower limit obtained by solving equation (14) of [1] is

M ≥ 1.4 × 1016 GeV, (5)

∗ Corresponding author. Email addresses: John.Ellis@.ch (), [email protected] (N.E. Mavromatos), [email protected] (D.V. Nanopoulos), [email protected] (A.S. Sakharov), [email protected] (E.K.G. Sarkisyan).

2 0.3

0.25

0.2

0.15

0.1

0.05 t/(1+z) ∆ 0

−0.05

−0.1

−0.15

−0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 K

Fig. 1. The rescaled spectral time-lags between the arrival times of pairs of genuine high-intensity sharp features detected in the light curves of the full set of 35 GRBs with measured redshifts observed by BATSE (closed circles), HETE (open circles) and SWIFT (triangles). compared with our previous limit M ≥ 0.9 × 1016 GeV.

We thank Tsvi Piran for his communication on the subject.

References

[1] J. R. Ellis, N. E. Mavromatos, D. V. Nanopoulos, A. S. Sakharov and E. K. G. Sarkisyan, Astropart. Phys. 25 (2006) 402 [arXiv:astro-ph/0510172]. [2] J. R. Ellis, N. E. Mavromatos, D. V. Nanopoulos and A. S. Sakharov, Astron. Astrophys. 402 (2003) 409 [arXiv:astro-ph/0210124]. [3] U. Jacob and T. Piran, arXiv:0712.2170 [astro-ph]. [4] M. Rodriguez Martinez and T. Piran, JCAP 0604 (2006) 006 [arXiv:astro-ph/0601219].

3 1

0.9

0.8

0.7

0.6

L 0.5

0.4

0.3

0.2

0.1

0 −0.01 −0.005 0 0.005 0.01 0.015 a LV

Fig. 2. The likelihood function for the slope parameter.

4