First constraints on the strength of the extragalactic magnetic ﬁeld from 𝛾 -ray observations of GRB 221009A

The extragalactic magnetic ﬁeld (EGMF) could be probed with 𝛾 -ray observations of distant sources. Primary very high energy (VHE) 𝛾 -rays from these sources absorb on extragalactic background light photons, and secondary electrons/positrons from the pair production acts create cascade 𝛾 -rays. These cascade 𝛾 -rays could be detected with space 𝛾 -ray telescopes such as Fermi -LAT. The 𝛾 -ray burst GRB 221009A was an exceptionally bright transient well suited for intergalactic 𝛾 -ray propagation studies. Using publicly-available Fermi -LAT data, we obtain upper limits on the spectrum of delayed emission from GRB 221009A during the time window of 30 days after the burst, and compare these with model spectra calculated for various EGMF strengths 𝐵 , obtaining lower limits on 𝐵 . We show that the values of 𝐵 ≤ 10 − 18 G are excluded. For some optimistic models of the VHE spectrum of GRB 221009A, the values of 𝐵 ≤ 10 − 17 G are excluded.


INTRODUCTION
GRB 221009A, an exceptionally bright (Williams et al. 2023;Lesage et al. 2023;Burns et al. 2023) and relatively nearby (redshift = 0.1505 (de Ugarte Postigo et al. 2022;Castro-Tirado et al. 2022)) -ray burst, has been detected with the WCDA and KM2A arrays of the LHAASO experiment in the energy range > 200 GeV (Cao et al. 2023;Huang et al. 2022).In particular, the detection of -rays above the energy of 10 TeV from GRB 221009A was reported.
Several GRBs were detected in the very high energy (VHE, >  Blanch et al. 2020).For one of them, GRB 190114C, it was shown that the intensity of the cascade -ray echo is below the sensitivity of the operating -ray telescopes ★ E-mail: timur1606@gmail.com 1 hereafter collectively called "electrons" even for the EGMF strength = 0 (Dzhatdoev et al. 2020) (hereafter D20), and this conclusion was confirmed by Da Vela et al. (2023).Veres et al. (2017) show that for GRB 130427A the cascade echo is detectable with the existing -ray telescopes for > 10 −18 G under certain optimistic assumptions on the high intensity of this GRB in the VHE domain.Unfortunately, the GRB 130427A was not detected at TeV energies and thus the latter constraints on remain conjectural.
In this Letter, we report on the constraints on the EGMF strength from -ray observations of GRB 221009A with LHAASO (Ma et al. 2022) and Fermi-LAT (Atwood et al. 2009).We describe our analysis of Fermi-LAT data in Section 2. Section 3 is devoted to the primary -ray spectrum reconstruction.The intergalactic cascade pair echo calculation procedure is outlined in Section 4. The main results are presented in Section 5; then follows a brief discussion (Section 6) and conclusions (Section 7).Appendices contain some details on the data analysis, simulations and results.All graphs in this paper were produced with the ROOT software (Brun & Rademakers 1997).

FERMI-LAT DATA ANALYSIS
We select Fermi-LAT data within 90 days of observation, starting at the Fermi-GBM trigger time 0 (Lesage et al. 2023).We reconstruct the Fermi-LAT spectral energy distribution (SED = 2 / ) of GRB 221009A in the time window from 0 to 0 + , where = 2 × 10 3 s is the duration of the LHAASO observation of the source according to Cao et al. (2023); Huang et al. (2022) over which the LHAASO -ray spectrum was measured.The Fermi-LAT SED obtained by us is shown in Fig. 1 as red circles with statistical uncertainties.Some details of this analysis are presented in Appendix A. We derive upper limits (95 % C.L.) on the SED of the cascade -ray echo from GRB 221009A, starting at 0 + , where = 2×10 5 s is an approximate duration of the -ray afterglow of GRB 221009A visible with Fermi-LAT (Stern & Tkachev 2023), and ending at 0 + + , with three options for = 10, 30, and 90 days.The results for the upper limits are shown in Fig. 2, 3, 4, and 5 (red horizontal bars with downward arrows).Some details of this analysis are presented in Appendix A as well.

Pre-publication estimates
Even before the publication of Cao et al. (2023), it was possible to obtain some estimates of the primary -ray spectrum for GRB 221009A.We fit the Fermi-LAT spectrum shown in Fig. 1 with a power-law function, obtaining the best-fit index 1 = 1.56.The broadband -ray SED of GRB 221009A could be characterised with the following smoothly broken power law (SBPL) function: where = 5.38 × 10 −8 erg cm −2 s −1 is the normalization factor, 2 = 2, = 10 GeV, = 1, and = 422 MeV is the reference energy.The corresponding SED is shown in Fig. 1 as solid black curve.Two additional options for the primary -ray SED, namely, the primary power law (PWL) and log-parabolic (LP) SEDs, are covered in Appendix B.

Post-publication estimates
After the release of the LHAASO-WCDA dataset on GRB 221009A Cao et al. (2023), it became possible to constrain the primaryray spectrum of GRB 221009A more tightly than before the publication of this paper.Here we use the same approach as in D20 except that Cao et al. (2023) presented five partial spectra over different time intervals, and not one spectrum as was the case for GRB 190114C.After fitting all five partial SEDs with the ∝ − (− / ) functional form and obtaining tables for the corresponding 1 ( ), 2 ( ), (...), 5 ( ) functions, the average primary -ray SED over 2000 s of observation is computed: where is the duration of the th observation episode, = 1, 2,...,5.Finally, we utilized the LHAASO-KM2A dataset on GRB 221009A (Wang, X. Y. (on behalf of the LHAASO collaboration) 2023) to correct the SED 4 ( ) for the observation period of 326-900 s.Some details of this procedure are presented in Appendix C.
The result of this procedure -the time-averaged primary SED (i.e. the SED before the absorption on EBL photons) in the energy range > 100 -is shown in Fig. 1 as green solid curve.The same SED, but after the EBL absorption effect was applied, is shown in the same Figure as long-dashed blue curve.An approximation of the Fermi-LAT SED below 10 GeV is shown for consistency. 2

SIMULATION OF THE PAIR ECHO FROM GRB 221009A
We calculate the observable SED of the intergalactic cascade pair echo using the publicly available code ELMAG 3.03 (Blytt et al. 2020;Kachelrieß et al. 2012) in the time window from 0 + to 0 + + .The maximal energy of the primary -rays is set to 100 TeV.The general scheme of calculations follows D20.
We assume the fiducial EBL model of Gilmore et al. (2012).As in D20, the EGMF was modeled as isotropic random nonhelical turbulent field with a Kolmogorov spectrum and Gaussian variance RMS (hereafter simply ) following the approach of Giacalone & Jokipii (1994, 1999).The minimal and maximal EGMF spatial scales were set as min = 5 × 10 −4 Mpc and max = 5 Mpc3 , respectively, with 200 field modes in total.Full three-dimensional simulation was employed.The jet opening angle jet was set to 1 degree, not far from the value obtained in Cao et al. (2023); however, the conclusions of this paper are almost independent on the value of jet (see Discussion below).We neglect collective (plasma) energy losses for cascade electrons (Broderick et al. 2012;Schlickeiser et al. 2012Schlickeiser et al. , 2013;;Miniati & Elyiv 2013;Chang et al. 2014;Sironi & Giannios 2014;Menzler & Schlickeiser 2015;Kempf et al. 2016;Vafin et al. 2018Vafin et al. , 2019;;Perry & Lyubarsky 2021).For sufficiently large values of , the width of the pair echo's observable angular distribution obs is comparable to or larger than the width of the point spread function (PSF) of Fermi-LAT PSF .This could affect the reconstructed point-like spectrum of the source.In what follows we neglect the latter effect since obs ≪ PSF for the values of 10 −18 G (Neronov & Semikoz 2009).

Pre-publication estimates
Here we report our results for the case of the SBPL primary spectrum option (solid black curve in Fig. 1) and = 30 days.The simulated -ray spectra of the cascade echo are shown in Fig. 2 for = 10 −19 G (black curves), = 10 −18 G (green curves) and = 10 −17 G (blue curves).An additional step-function cutoff in the primaryray spectrum was introduced; solid curves correspond to the cutoff

Post-publication estimates
Here we report our results for the case of the post-publication estimate of the primary -ray SED (solid green curve in Fig. 1).Fig. 3  < 10 −20 G is already excluded as stems from the negative results of the search for cascade echo from blazars (Dermer et al. 2011;Taylor et al. 2011;Finke et al. 2015;Podlesnyi et al. 2022).
Finally, the comparison of cascade echo SEDs for the case of = 90 days and two different values of = 2 × 10 5 s and = 0 is presented in Appendix E. For = 0, the SED curves for = 10 −18 G and = 10 −17 G successively branch down from the SED curve for = 10 −19 G.This behaviour of the SEDs was reported in D20 (see discussion of Fig. 3 in D20).

DISCUSSION
The obtained results are directly relevant for a large-scale EGMF with the coherence length > 10 − 100 kpc.In this case the typical electron energy loss length − < (Neronov & Semikoz 2009).In the opposite case of a "turbulent" EGMF ( − > ) the resulting limits on become even stronger (Neronov & Semikoz 2009).
The obtained constraints depend on the assumed EBL model.Similar to D20, we performed calculations of the pair echo spectrum for a modified EBL model with the intensity normalization factor EBL = 0.7.The resulting borderline values of typically change only slightly, within ≈ 20 %.

Dependence of results on the jet profile
The obtained results may also depend on the exact shape of the GRB 221009A jet profile, the opening angle of the jet jet and the value of the angle view at which the observer looks into the jet.The typical deflection angle of cascade electrons (Neronov & Semikoz 2009); (Dzhatdoev et al. 2020, see eq. ( 4)) for 10 −18 G ≪ jet , thus the dependence of our results on jet is small.4

Comparison of results with other authors
After the first version of our paper was sent to arXiv and to the journal, two other papers on the EGMF constraints from GRB 221009A appeared (Huang et al. 2023;Vovk et al. 2023).The approach of Huang et al. (2023) is similar to the one of D20 and the present work; however, they assumed a slightly different EBL model of Saldana-Lopez et al. (2021).Huang et al. (2023) find that the values of ≤ 10 −18.5 G are excluded.The difference of EBL models and time windows assumed may account for the slightly different results of Huang et al. (2023) compared to the ones obtained in the present work.Vovk et al. (2023), on the other hand, utilize the Fermi-LAT light curve of GRB 221009A in the time range of 10 −3 − 10 days to obtain constraints on .They exclude ≤ 10 −19 G. Given the different approach of Vovk et al. (2023) compared to our work, such a difference of the results could be expected.

Observational prospects
We note that the advent of the next-generation space -ray telescopes such as MAST (Dzhatdoev & Podlesnyi 2019) could dramatically improve the pair echo detectability prospects.The improved sensitivity of the Cherenkov Telescope Array (CTA) (Actis et al. 2011;Acharya et al. 2013) in the energy range of 100 GeV -10 TeV could significantly facilitate the measurement of the intrinsic spectrum, reducing the uncertainty of the pair echo characteristics.

CONCLUSIONS
Using the -ray observations of GRB 221009A with LHAASO and Fermi-LAT, we were able, for the first time, to obtain constraints on the EGMF strength from GRB emission.We show that the values of ≤ 10 −18 G are excluded for all values of the EGMF coherence length and for all considered models of the primary -ray spectrum.

APPENDIX B: THE CASE OF LP AND PWL SPECTRA (PRE-PUBLICATION ESTIMATES)
Huang et al. ( 2022) reported the observation of more than = 5 × 10 3 -rays from GRB 221009A in the energy range between 500 GeV and 18 TeV.Assuming = 5 × 10 3 , ≥ 0 and taking the effective areas of the WCDA and KM2A arrays according to the Supplementary Information for LHAASO Collaboration (2021), we estimate the parameters of the LP SED as follows: where = 8.85 × 10 −8 erg cm −2 s −1 , = 1.57, = 0, and the reference energy = 1.33 GeV.In this case the best fit is a pure power-law function10 shown in Fig. B1 as green line and denoted as PWL.Finally, we consider another option of the primary spectrum (denoted as LP) with = 9.50 × 10 −8 erg cm −2 s −1 , = 1.30, = 4 × 10 −2 , and = 1.33 GeV (shown in Fig. B1 as blue curve).While performing the fitting for both the PWL and LP cases we added the following additional constraint on the fit: , where −fit is the estimated number of -ray events with the energy > 500 GeV that would be registered with the LHAASO detector.
Results similar to those presented in Fig. 2  and Fig. B3 for the case of the power-law and log-parabolic primary spectra (green line and blue curve in Fig. B1, respectively).In these cases, the values of (approximately) ≤ 10 −16 − 10 −17 G could be excluded depending on the value of .

APPENDIX C: RECONSTRUCTION OF THE PRIMARY SED FROM THE LHAASO DATASET
The partial spectra of GRB 221009A for five time intervals measured with the LHAASO-WCDA detector were taken from Supplementary Information of Cao et al. (2023) (see their Figure S4C).The fitting of the partial spectra was done in the same manner as in D20 for the case of GRB 190114C.
For the case of the fourth time interval (the 326-900 s time range) the dataset of LHAASO-KM2A (Wang, X. Y. (on behalf of the LHAASO collaboration) 2023) for the time range 300-900 s was utilized as well.These time ranges are slightly different.To account for these conditions, a re-normalization of the LHAASO-KM2A SED was performed as follows.At first, the fluence correction factor was calculated:

Figure 1 .
Figure1.Fermi-LAT SED of GRB 221009A over the first 2 × 10 3 s after the Fermi-GBM trigger (red circles with statistical uncertainties) together with two options for the possible -ray spectrum of the source in the 100 MeV -100 TeV energy range (solid curves; see the text for more details).Blue dashed curve accounts for the effect of -ray absorption on EBL photons vs. green solid curve.

Figure 2 .
Figure 2. Fermi-LAT upper limits on the SED of GRB 221009A in the time interval 2×10 5 s -30 days + 2×10 5 s after the Fermi-GBM trigger (red horizontal bars with downward arrows) together with model SEDs for various values of (curves; see the text for more details).
shows the results for = 30 days, Fig. 4 -for = 10 days, Fig. 5for = 90 days.A plot including additional EGMF strength values in range from = 10 −21 G to = 3 × 10 −17 G for the case of = 90 days is presented in Appendix D (see Fig. D1), leading to the exclusion of the range of values from = 10 −20 G to = 10 −18 G.The option of

Figure 3 .Figure 4 .Figure 5 .
Figure 3. Same as in Fig. 2, but for the case of the post-publication primary SED estimate.

Figure B1 .Figure B2 .
Figure B1.Fermi-LAT SED of GRB 221009A over the first 2 × 10 3 s after the Fermi-GBM trigger (red circles with statistical uncertainties) together with three options for the possible -ray spectrum of the source in the 100 MeV -100 TeV energy range (curves; see the text for more details).

EFigure B3 .
Figure B3.Same as in Fig.2, but for the case of the primary log-parabolic spectrum.

Figure D1 .Figure E1 .
Figure C1.SEDs observed by LHAASO-WCDA and LHAASO-KM2A in different time intervals after 0 and the fitting curves to them.LHAASO-KM2A data points were rescaled to match the LHAASO-WCDA time intervals (see the text for more details.)