Issue 
A&A
Volume 554, June 2013



Article Number  A31  
Number of page(s)  5  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201321294  
Published online  30 May 2013 
Measuring the correlation length of intergalactic magnetic fields from observations of gammaray induced cascades
^{1} ISDC Data Centre for Astrophysics, Department of AstronomyUniversity of Geneva, Ch. d’Ecogia 16, 1290 Versoix, Switzerland
^{2} Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin 2, Ireland
email: taylora@cp.dias.ie
Received: 14 February 2013
Accepted: 25 April 2013
Context. The imaging and timing properties of γray emission from electromagnetic cascades initiated by veryhighenergy (VHE) γrays in the intergalactic medium depend on the strength B and correlation length λ_{B} of intergalactic magnetic fields (IGMF).
Aims. We study the possibility of measuring both B and λ_{B} via observations of the cascade emission with γray telescopes.
Methods. For each measurement method, we find two characteristics of the cascade signal, which are sensitive to the IGMF B and λ_{B} values in different combinations. For the case of IGMF measurement using the observation of extended emission around extragalactic VHE γray sources, the two characteristics are the slope of the surface brightness profile and the overall size of the cascade source. For the case of IGMF measurement from the time delayed emission, these two characteristics are the initial slope of the cascade emission light curve and the overall duration of the cascade signal.
Results. We show that measurement of the slope of the cascade induced extended emission and/or light curve can both potentially provide measure of the IGMF correlation length, provided it lies within the range 10 kpc ≲ λ_{B} ≲ 1 Mpc. For correlation lengths outside this range, gammaray observations can provide an upper or lower bound on λ_{B}. The latter of the two methods holds great promise in the near future for providing a measurement/constraint using measurements from present/nextgeneration γraytelescopes.
Conclusions. Measurement of the IGMF correlation length will provide an important constraint on its origin. In particular, it will enable to distinguish between an IGMF of galactic wind origin from an IGMF of cosmological origin.
Key words: gamma rays: galaxies / BL Lacertae objects: general
© ESO, 2013
1. Introduction
Observations of the absorbed γray component from high energy blazars have in recent times been used to provide constraints on the physical parameters of the intergalactic medium (IGM) such as the density of optical/infrared radiation backgrounds (Franceschini et al. 2008; Orr et al. 2011). Furthermore, the subsequent electromagnetic cascade produced in the IGM (Aharonian et al. 1994) provides the opportunity to probe the intergalactic magnetic field (IGMF) strength (Plaga 1995; Neronov & Semikoz 2007; Ichiki et al. 2008; Murase et al. 2008; Takahashi et al. 2008). Recent negative results on the search of the secondary γray emission from the γray induced electromagnetic cascades in the GeV energy band have been used to derive lower bounds on the IGMF strength and correlation length (Neronov & Vovk 2010; Tavecchio et al. 2011; Dermer et al. 2011; Taylor et al. 2011; Vovk et al. 2012). Furthermore, such fields must necessarily permeate a significant fraction (>60%) of the column depth to the blazar in order to exert a sufficient effect on the electromagnetic cascade development (Dolag et al. 2011). Thus, combining the obtained lower bounds with the known upper bounds from radio data and theoretical considerations, one finds that the allowed range of IGMF parameters spans several decades in both strength (10^{17} < B < 10^{9} G) and correlation length (10^{13} < λ_{B} < 10^{28} cm) parameter space.
The allowed region of (B,λ_{B}) parameter space is consistent with various intergalactic magnetic field generation scenarios, from phase transitions in the Early Universe (Hogan 1983; Quashnock et al. 1989; Vachaspati 1991; Sigl et al. 1997) to supernova and AGN generated outflows from the galaxies during the recent Cosmological epoch (Bertone et al. 2006). In general, relic cosmological magnetic fields from phase transitions in the Early Universe are expected to have short correlation lengths which depend on the magnetic field strength (1)which corresponds to the largest processed eddy scalelength at the end of the radiation dominated epoch (Banerjee & Jedamzik 2004). An exception to this rule are magnetic fields produced during the epoch of inflation. In this case the initial correlation length of magnetic field could be arbitrarily large, so that the correlation length of the remaining magnetic field at zero cosmological redshift might also be arbitrarily large. However, the largest processed scalelength still limits the correlation length of the relic inflationary magnetic field to be .
In comparison, magnetic fields generated by galaxy outflows at the late stages of the Universe’s evolution are expected to have correlation lengths of the order of typical galaxy sizes (2)Different possible scenarios for the generation of intergalactic magnetic field can potentially be distinguished through the measurement of λ_{B}. In what follows we show that gammaray measurements of emission from electromagnetic cascades in the intergalactic medium can provide a measurement of magnetic field correlation length if it lies within the range 10 kpc ≲ λ_{B} ≲ 1 Mpc and provide upper or lower bounds on λ_{B} outside this range for all measurable values of magnetic field strength B. We note that the effects of plasma physics on cascade development within the voids, the importance of which remains unresolved (Broderick et al. 2012; Schlickeiser et al. 2012; Miniati & Elyiv 2012), are neglected in this work.
2. IGMF coherence length from the source angular profile
Pencil beam model – In order to derive simplified analytic expressions for cascade quantities, we start with the twogeneration model depicted in Fig. 1. Considering a narrow beam of γrays emitted by a source S in a direction SJ misaligned with the line of sight SO (Fig. 1). For simplicity, we suppose that the primary veryhighenergy (VHE) γray beam consists of photons with the same energy E_{γ0} emitted by the source S at constant rate of N_{0}γrays per unit time.
Interaction of the VHE γrays with extragalactic background light (EBL) leads to their absorption. As a result, the number of photons along the primary γray beam decreases with distance as N(r) = N_{0}exp^{(} − r/D_{γ0}^{)}, where D_{γ0} is the mean free path of γrays through the EBL. Each primary photon produces two electrons, so that the rate of injection of e^{+}e^{−} pairs along the γray beam is (3)The subsequent cooling of these e^{+}e^{−} pairs due to the inverse Compton (IC) scattering of the cosmic microwave background (CMB) photons leads to emission of the secondary γray photons with energies . We suppose that the corresponding cooling distance (4)is much smaller than distance to the source D. This means that the energy absorbed from the initial beam is emitted in situ, immediately after absorption.
Fig. 1 Geometry of cascade emission from a blazar jet SJ misaligned with the line of sight SO. 

Open with DEXTER 
The conservation of energy demands that the total power of IC emission from the e^{+}e^{−} pairs is equal to the power removed from the primary γray beam: (5)In the presence of IGMF, electrons and positrons are deflected from their original directions. The deflection angle of these pairs depends on the correlation length of magnetic field, λ_{B}. Two different deflection regimes can be identified:
“One cell” regime – (D_{e} ≪ λ_{B}). In this case, the e^{+}e^{−} pairs move in nearly homogeneous magnetic field. The deflection angle changes as (6)where x is the coordinate along the electron trajectory counted from the pair production point and R_{L} = E_{e}/eB is the Larmor radius. As the characteristic value of x is D_{e}, the previous equation can be rewritten as: (7)
“Many cells” regime – (D_{e} ≫ λ_{B}). In this case particles move through many regions with different orientations of magnetic field. Under such conditions electrons and positrons experience random walks in angle, so that the average deflection angle is given by: (8)or, substituting x = D_{e}: (9)The deflections of electrons^{1} by the IGMF determine the angular pattern of the IC emission. The angular distribution of electrons at each point along the beam can be written in the following way:(10)The factor f_{δ}dδ here denotes the fraction of the total number of particles, that were deflected within the range [δ:δ + dδ], and N_{e}(δ) is the corresponding number of electrons. We assume here, that the probability density function f_{x} for the electron to emit secondary γ ray is constant over its trajectory and drops to zero at x = D_{e}. The dependence of this density function on the deflection angle can then be written in the following way: (11)The constant here is defined through the requirement that . We thus can rewrite Eq. (10) in the following way: (12)The energy of electrons also changes with the distance x as E_{e}(x) = E_{0}/(1 + x/D_{e}), where E_{0} ≃ E_{γ0}/2 is the initial energy at the injection point. As the power of the IC emission scales as , we can write: (13)where the normalization of is fixed in such a way that with δ_{max} = D_{e}/R_{L} in the case λ_{B} ≫ D_{e} and in the case λ_{B} ≪ D_{e}.
An observer looking at the jet SJ with angle α will be able to observe the cascade emission from the e^{+}e^{−}pairs deposited along the jet as long as the offaxis angle of IC γrays emitted in the direction of the observer is δ < δ_{max} (see Fig. 1). The flux of IC emission detected by an observer at point O depends on the angular distance from the source θ. Taking into account that r = Dsinθ/sinδ (Fig. 1), one can calculate the jet brightness profile (14)where δ = α + θ.
Jet opening angle effects – If the blazar jet is aligned to the line of sight, the cascade emission appears as an extended “halolike” emission around the primary γray source, rather than as a onesided jetlike extension. In this case the measurable characteristic of the cascade emission is the slope of the cascade source’s surface brightness profile (rather than the linear brightness profile of the jetlike extension).
The cascade emission surface brightness profile of a γray beam with an opening angle α_{jet} aligned along the line of sight can be found by summing the linear profiles (14) of all the narrow γray beams forming the jet: (15)where α_{min} is determined by the condition α_{min} = θ(τ − 1), which can be understood from Fig. 1. Indeed, in our calculations we assume that the characteristic distance r, at which e^{+}e^{−}pairs are produced, is D_{γ0}. Making this substitution in the limit of small α and θ one finds a condition α_{min}/(D − D_{γ0}) = θ/D_{γ0}, which then transforms in the above lower bound of the integral in Eq. (15). Substituting dF/dθ from (14) and taking the integral one finds in the limit of small θ(16)Thus, in the case λ_{B} ≪ D_{e}, the cascade emission is disklike with a flat surface brightness profile. To the contrary, in the case λ_{B} ≫ D_{e}, the cascade emission has a steep brightness profile peaked at the central source.
The dependence of the extended emission’s surface brightness profile on λ_{B} can therefore potentially be used for the measurement of the IGMF correlation length λ_{B}. Indeed, measurement of a nonzero slope in the surface brightness profile at energy E_{γ} would imply the constraint λ_{B} > D_{e}. To the contrary, measurement of a flat profile would impose the constraint λ_{B} < D_{e}. If λ_{B} is larger than the IC cooling distance of the highest energy electrons, but shorter than the cooling distance of the lowest energy electrons contributing to the cascade γray emission detectable by a γray telescope, one may hope to detect a change in the slope of the brightness profile of the cascade emission at the energy E_{γ,br} where D_{e} ~ λ_{B}. In this case, measurement of the break energy E_{γ,br} would provide a measurement of the IGMF correlation length (17)It should be noted, however, that a measure of the coherence length employing this method would require considerable improvement in angular resolution relative to that of the ~0.1° present day limit for γray telescopes. In the following section we describe an alternative method for measuring the coherence length, potentially employable using present/next generation γray instruments.
3. IGMF coherence length from the source flare light curve
A different regime of deflection of electrons by IGMF affects not only the slope of the profile of cascade emission but also the time delay of the cascade photons. Calculation of the temporal characteristics of the cascade emission signal can be done in a similar way to the calculation of the linear and/or surface brightness profiles, performed in the previous section. As was done there, we consider two model situations: a narrow jet misaligned with the line of sight and a finite opening angle jet aligned with the line of sight.
Pencil beam model – We consider again a jet SJ misaligned by an angle α with respect to the line of sight SO (Fig. 1). However, instead of constant in time injection of the primary γrays at the source, we consider an instantaneous injection of N_{0}γrays. The γrays propagate along the jet and deposit dN_{e}/dr pairs on the time scale of the light crossing time of the distance r. At any given time, only electrons injected in the cascade over the time interval D_{e}/c contribute to the IC radiation in the cascade, so that the total number of the highest energy electrons is N_{e}(r) = D_{e}dN_{e}/dr. The portion of the γray beam which produces cascade emission detectable with the time delay t_{d} compared to the direct signal from the source is situated at distance(18)on the SJ line. Following the calculation in Sect. 2, the amount primary photon energy converted into the cascade γray emission within distance interval dr and angular distribution of the cascade emission are given by Eqs. (5) and (13), so we can now write the amount of the cascade emission which reaches the observer per time interval dt_{d} as (19)where r and δ = α + 2ct_{d}/Dα are expressed as functions of t_{d}.
Jet opening angle effects – The light curve of cascade emission from a jet of finite opening angle aligned with the line of sight can be obtained by summation of the light curves of all the beams forming the jet, i.e. via integration over the angle of the beam with respect to the line of sight α: (20)where α_{jet} is the jet opening angle and α_{min} is found from the condition D/sinδ = D_{γ0}/sinα_{min}, which gives (21)Substituting the expression for dF_{IC}/dt_{d} (19) into (20) and taking the integral one finds at the limit of small t_{d}: (22)The slope of the cascade emission light curve depends on the relation between the IGMF correlation length and electron cooling distance. This fact can be used for the measurement of λ_{B}. At a fixed cascade photon energy, measurement of a flat cascade emission light curve would impose an upper bound on the IGMF correlation length, λ_{B} ≪ D_{e}. For the opposite case, a lower bound on λ_{B} would be set. If λ_{B} is larger than the IC cooling distance of the highest energy electrons, but shorter than the cooling distance of the lowest energy electrons contributing to the cascade γray emission detectable by a γray telescope, one expects to find a change in the slope of the cascade emission light curve at the energy E_{γ,br} where D_{e} ~ λ_{B}. In this case, measurement of the break energy E_{γ,br} would provide a measurement of the IGMF correlation length, given by Eq. (17). The range of length scales probable by this method is therefore dictated by the dynamic energy range of the instrument. For the example case of Fermi LAT, with a dynamic range of approximately 4 decades (20 MeV to 300 GeV Atwood et al. 2009), the corresponding coherence scale range probable by this method is 10 kpc ≲ λ_{B} ≲ 1 Mpc.
4. Verification with Monte Carlo simulation
Since relations (16) and (22) have been obtained using a simplified twogeneration model, we here compare these results against those obtained with a complete multigeneration numerical description. Using the Monte Carlo simulation described in Taylor et al. (2011) in which the full cascade development is carried out and the spatial deflection of the electrons tracked a comparison of the twogeneration results was carried out.
Adopting a deltatype injection spectrum with dN/dE_{γ} = δ(10^{13} eV), for a blazar redshift of z = 0.13, we compare in Figs. 2 and 3 the Monte Carlo obtained with expressions (16) and (22). Such comparisons confirm that the simplified analytic expressions obtained can indeed provide reasonably accurate descriptions of these distributions, particularly in the asymptotic regions. However, we do note that some degree of divergence is found in the intermediate region between the asymptotic zones for the small correlation length case (L_{B}/D_{e} ≪ 1).
Fig. 2 Angular profile of the arriving γray flux following a flaring episode obstained with both Monte Carlo and analytic (Eq. (16)) methods. The angles shown are measured relative to the center of the blazar. For this plot, the angular profile of 1−3 GeV photons in a cascade from a source at z = 0.13 for a 10^{15} G IGMF with coherence lengths 10 kpc (black lines) and 10 Mpc (red lines) are shown. 

Open with DEXTER 
Fig. 3 Timedelay in the arriving γray flux following a flaring episode obstained with both Monte Carlo and analytic (Eq. (22)) methods. The time’s shown are measured relative to the straight line (SO in Fig. 1) arrival time. For this plot, the timedelay of 1−3 GeV photons in a cascade from a source at z = 0.13 for a 10^{15} G IGMF with coherence lengths 10 kpc (black lines) and 10 Mpc (red lines) are shown. 

Open with DEXTER 
5. Conclusion
Present generation γray observational results have recently been used to provide challenging new bounds on the IGMF strength. The coherence length for this field, however, remains largely unconstrained. We here consider what handle future γray observations may be able to provide with regards a measurement of both the IGMF strength and its coherence length. We show that measuring either the initial slope of the time delayed emission or the slope of the surface brightness profile of extended emission can be used to provide a measure of the IGMF correlation length.
Through the application of a simplified analytic twogeneration model we describe two possible methods for probing this coherence length. Though both of these methods are potentially viable, the employment of the first of these methods would require considerable improvement in angular resolution above that achieved by present day γray telescopes. The second of the methods put forward, however, does have the potential to be applied using forthcoming γray observational data. A subsequent comparison of the twogeneration model results with those obtained using the full Monte Carlo description confirms that this signature is still expected to survive once the full cascade physical description is added back into the picture.
References
 Aharonian, F. A., Coppi, P. S., & Voelk, H. J. 1994, ApJ, 423, L5 [NASA ADS] [CrossRef] [Google Scholar]
 Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, ApJ, 697, 1071 [NASA ADS] [CrossRef] [Google Scholar]
 Banerjee, R., & Jedamzik, K. 2004, Phys. Rev. D, 70, 3003 [NASA ADS] [Google Scholar]
 Bertone, S., Vogt, C., & Enßlin, T. 2006, MNRAS, 370, 319 [NASA ADS] [Google Scholar]
 Broderick, A. E., Chang, P., & Pfrommer, C. 2012, ApJ, 752, 22 [NASA ADS] [CrossRef] [Google Scholar]
 Dermer, C. D., Cavadini, M., Razzaque, S., et al. 2011, ApJ, 733, L21 [NASA ADS] [CrossRef] [Google Scholar]
 Dolag, K., Kachelriess, M., Ostapchenko, S., & Tomàs, R. 2011, ApJ, 727, L4 [NASA ADS] [CrossRef] [Google Scholar]
 Franceschini, A., Rodighiero, G., & Vaccari, M. 2008, A&A, 487, 837 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Hogan, C. J. 1983, Phys. Rev. Lett., 51, 1488 [NASA ADS] [CrossRef] [Google Scholar]
 Ichiki, K., Inoue, S., & Takahashi, K. 2008, ApJ, 682, 127 [NASA ADS] [CrossRef] [Google Scholar]
 Miniati, F., & Elyiv, A. 2012 [arXiv:1208.1761] [Google Scholar]
 Murase, K., Takahashi, K., Inoue, S., Ichiki, K., & Nagataki, S. 2008, ApJ, 686, L67 [NASA ADS] [CrossRef] [Google Scholar]
 Neronov, A., & Semikoz, D. V. 2007, JETP Lett., 85, 473 [CrossRef] [Google Scholar]
 Neronov, A., & Vovk, I. 2010, Science, 328, 73 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Orr, M. R., Krennrich, F., & Dwek, E. 2011, ApJ, 733, 77 [NASA ADS] [CrossRef] [Google Scholar]
 Plaga, R. 1995, Nature, 374, 430 [NASA ADS] [CrossRef] [Google Scholar]
 Quashnock, J. M., Loeb, A., & Spergel, D. N. 1989, ApJ, 344, L49 [NASA ADS] [CrossRef] [Google Scholar]
 Schlickeiser, R., Ibscher, D., & Supsar, M. 2012, ApJ, 758, 102 [NASA ADS] [CrossRef] [Google Scholar]
 Sigl, G., Olinto, A. V., & Jedamzik, K. 1997, Phys. Rev. D, 55, 4582 [NASA ADS] [CrossRef] [Google Scholar]
 Takahashi, K., Murase, K., Ichiki, K., Inoue, S., & Nagataki, S. 2008, ApJ, 687, L5 [NASA ADS] [CrossRef] [Google Scholar]
 Tavecchio, F., Ghisellini, G., Bonnoli, G., & Foschini, L. 2011, MNRAS, 414, 3566 [NASA ADS] [CrossRef] [Google Scholar]
 Taylor, A. M., Vovk, I., & Neronov, A. 2011, A&A, 529, A144 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Vachaspati, T. 1991, Phys. Lett. B, 265, 258 [NASA ADS] [CrossRef] [Google Scholar]
 Vovk, I., Taylor, A. M., Semikoz, D., & Neronov, A. 2012, ApJ, 747, L14 [NASA ADS] [CrossRef] [Google Scholar]
All Figures
Fig. 1 Geometry of cascade emission from a blazar jet SJ misaligned with the line of sight SO. 

Open with DEXTER  
In the text 
Fig. 2 Angular profile of the arriving γray flux following a flaring episode obstained with both Monte Carlo and analytic (Eq. (16)) methods. The angles shown are measured relative to the center of the blazar. For this plot, the angular profile of 1−3 GeV photons in a cascade from a source at z = 0.13 for a 10^{15} G IGMF with coherence lengths 10 kpc (black lines) and 10 Mpc (red lines) are shown. 

Open with DEXTER  
In the text 
Fig. 3 Timedelay in the arriving γray flux following a flaring episode obstained with both Monte Carlo and analytic (Eq. (22)) methods. The time’s shown are measured relative to the straight line (SO in Fig. 1) arrival time. For this plot, the timedelay of 1−3 GeV photons in a cascade from a source at z = 0.13 for a 10^{15} G IGMF with coherence lengths 10 kpc (black lines) and 10 Mpc (red lines) are shown. 

Open with DEXTER  
In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.