Lorentz Invariance Violation Effects on Gamma-Gamma Absorption and Compton Scattering
aa r X i v : . [ a s t r o - ph . H E ] S e p Draft version September 5, 2018
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LORENTZ INVARIANCE VIOLATION EFFECTS ON GAMMA-GAMMA ABSORPTION AND COMPTONSCATTERING
Hassan Abdalla a and Markus B¨ottcher b Centre for Space Research, North-West University, Potchefstroom 2520, South Africa Department of Astronomy and Meteorology, Omdurman Islamic University, Omdurman 382, Sudan
ABSTRACTIn this paper, we consider the impact of Lorentz Invariance Violation (LIV) on the γ − γ opacityof the Universe to VHE-gamma rays, compared to the effect of local under-densities (voids) of theExtragalactic Background Light, and on the Compton scattering process. Both subluminal and super-luminal modifications of the photon dispersion relation are considered. In the subluminal case, LIVeffects may result in a significant reduction of the γ − γ opacity for photons with energies &
10 TeV.However, the effect is not expected to be sufficient to explain the apparent spectral hardening ofseveral observed VHE γ -ray sources in the energy range from 100 GeV – a few TeV, even when in-cluding effects of plausible inhomogeneities in the cosmic structure. Superluminal modifications ofthe photon dispersion relation lead to a further enhancement of the EBL γγ opacity. We consider,for the first time, the influence of LIV on the Compton scattering process. We find that this effectbecomes relevant only for photons at ultra-high energies, E & Keywords: radiation mechanisms: non-thermal — galaxies: active — galaxies: jets — cosmology:miscellaneous — quantum-gravity: Lorentz Invariance Violation (LIV) INTRODUCTIONRecent astronomical observations and laboratory experiments appear to show hints that several phenomena inphysics, astrophysics and cosmology oppose a traditional view of standard-model physics (e.g., Riess et al. 1998;Furniss et al. 2013). This has motivated developments of modified or alternative theories of quantum physics andgravitation (e.g., Capozziello et al. 2013; Wanas & Hassan 2014; Nashed & El Hanafy 2014; Arbab 2015; Sami et al.2018), generally termed physics beyond the standard model (e.g., Sushkov 2011; Abdallah et al. 2013; El-Zant 2015).The special theory of relativity postulates that physical phenomena are identical in all inertial frames. Lorentzinvariance is one of the pillars of modern physics and is considered to be a fundamental symmetry in Quantum FieldTheory. However, several quantum-gravity theories postulate that familiar concepts such as Lorentz invariance may bebroken at energies approaching the Planck energy scale, E P ∼ . × GeV (e.g., Amelino et al. 1998; Jacob & Piran2008; Liberati & Maccione 2009; Amelino 2013; Tavecchio & Bonnoli 2016). Currently such extreme energies areunreachable by experiments on Earth, but for photons traveling over cosmological distances the accumulated deviationsfrom Lorentz invariance may be measurable using Imaging Atmospheric Cherenkov Telescope facilities, in particularthe future Cherenkov Telescope Array (CTA) (e.g., Fairbairn et al. 2014; Lorentz & Brun 2017).A deviation from Lorentz invariance can be described by a modification of the dispersion relation of photons andelementary particles such as electrons (e.g., Amelino et al. 1998; Tavecchio & Bonnoli 2016). It is well known thatthe speed of light in a refractive medium depends on its wavelength, with shorter wavelength (high momentum) modestraveling more slowly than long wavelength (low momentum) photons. This effect is due to the sensitivity of lightwaves to the microscopic structure of the refractive medium. Similarly, in quantum gravity theories, very high energyphotons could be sensitive to the microscopic structure of spacetime, leading to a violation of strict Lorentz symmetry. a [email protected] b [email protected]
In that case, γ -rays with higher energy are expected to propagate more slowly than their lower-energy counterparts(e.g., Amelino et al. 1998; Fairbairn et al. 2014; Tavecchio & Bonnoli 2016; Lorentz & Brun 2017). This wouldlead to an energy-dependent refractive index for light in vacuum. Therefore, the deviation from Lorentz symmetry canbe measured by comparing the arrival time of photons at different energies originating from the same astrophysicalsource (e.g., Amelino et al. 1998; Azzam 2009; Tavecchio & Bonnoli 2016; Wei 2017; Lorentz & Brun 2017).Gamma rays from objects located at a cosmological distance with energies greater than the threshold energy forelectron-positron pair production can be annihilated due to γ − γ absorption by low-energy extragalactic backgroundphotons (Nikishov 1962). The intergalactic γ − γ absorption signatures have attracted great interest in astrophysicsand cosmology due to their potential to indirectly measure the Extragalactic Background Light (EBL) and therebyprobe the cosmic star-formation history (e.g., Biteau & Williams 2015). The predicted γ − γ absorption imprintshave been studied employing a variety of theoretical and empirical methods (e.g., Stecker 1969, 1902; Hauser & Dwek2001; Primack et al. 2005; Aharonian et al. 2006; Franceschini et al. 2008; Razzaque et al. 2009; Finke et al. 2010;Dominguez et al. 2011a; Gilmore et al. 2012).Recent observations indicate that the very-high-energy (VHE; E >
100 GeV) spectra of some distant ( z & . γ − γ absorption by the EBL, appear harder than physically plausible (e.g., Furniss et al.2013), although systematic studies of the residuals of spectral fits with standard EBL absorption on large samples ofVHE blazars (e.g., Biteau & Williams 2015; Mazin et al. 2017) reveal no significant, systematic anomalies on the entiresamples. Nevertheless, the unexpected VHE- γ -ray signatures seen in a few individual blazars are currently the subjectof intensive research. Possible explanations of this spectral hardening include the hypothesis that the EBL densitycould be lower than expected from current EBL models (Furniss et al. 2013), an additional γ -ray emission componentdue to interactions along the line of sight of extragalactic Ultra-high-Energy Cosmic Rays (UHECRs) originatingfrom the blazar (e.g., Essey & Kusenko 2010; Dzhatdoev 2015), the existence of exotic Axion-Like Particles (ALPs)into which VHE γ -rays can oscillate in the presence of a magnetic field, thus enabling VHE-photons to avoid γ − γ absorption (e.g., Dominguez et al. 2011b; Dzhatdoev et al. 2017), EBL inhomogeneities (e.g., Furniss et al. 2015;Kudoda & Faltenbacher 2016; Abdalla & B¨ottcher 2017) and the impact of LIV, which can lead to an increase of the γγ interaction threshold and thus to a reduction of cosmic opacity (especially at energies beyond ∼
10 TeV), thusallowing high-energy photons to avoid γ − γ absorption (e.g., Tavecchio & Bonnoli 2016; Abdalla & B¨ottcher 2018).In this paper, we discuss the reduction of the EBL γ − γ opacity due to the existence of underdense regions along theline of sight to VHE gamma-ray sources (including contributions of both the direct star light and re-processed emissionto the EBL) and compare the results with the LIV effect on cosmological photon propagation. We consider the LIVeffect only for photons, but not for electrons, since the high-energy synchrotron spectrum of the Crab Nebula imposesa stringent constraint on any deviation of the electron dispersion relation from Lorentz invariance (e.g., Jacobson et al.2003).LIV may also effect the process of Compton scattering, which is likely to be an important γ -ray production processin many astrophysical high-energy sources, such as accreting black hole binaries, pulsar wind nebulae, the jets fromactive galactic nuclei, and supernova remnants. In this paper, we discuss, to our knowledge for the first time, theimpact of LIV on the Compton scattering process, both on energy-momentum conservation and on the Compton crosssection.In Section 2 we investigate the impact of the existence of cosmic voids along the line of sight to a distant VHE γ -ray source, by using the EBL model devoloped by Finke et al. (2010). In Section 3 we review the impact of LIVon the EBL γγ opacity. In Section 4 we investigate LIV effects on the Compton scattering process, starting withbasic conservation of energy and momentum, using the LIV-deformed dispersion relation for photons. The results arepresented in Section 5, where we compare our results with predictions from standard quantum electrodynamics (QED).We summarize and discuss our results in Section 6. Throughout this paper the following cosmological parameters areassumed: H = 70 km s − Mpc − , Ω m = 0 .
3, Ω Λ = 0 . THE IMPACT OF A COSMIC VOID ON THE EBL ENERGY DENSITY DISTRIBUTIONA generic study of the effects of cosmic voids along the line of sight to a distant astronomical object (e.g. blazar)on the EBL γ − γ opacity has been done in (Abdalla & B¨ottcher 2017). In that paper, the EBL was representedusing the prescription of Razzaque et al. (2009), taking into account only the direct starlight contribution to the EBL.Assuming that a spherical cosmic void with raduis R is located with its center at redshift z v , between an observer anda VHE γ -ray source located at redshift z s , the angle- and photon-energy-dependent EBL energy density at each pointbetween the observer at redshift zero and the source was calculated. The cosmic void was represented by setting thestar formation rate to 0 within the volume of the void. We found that the EBL deficit is proportional to the size ofthe void. Therefore, the effect of a number n of voids of radius R is aproximately the same as the effect of a largevoid with radius R n = nR .Since in the Razzaque et al. (2009) prescription, only the direct starlight contribution to the EBL is considered, thework of (Abdalla & B¨ottcher 2017) under-predicts the EBL γ − γ opacity for VHE γ -rays with energies of E & − h − Mpc. Also, there is evidencefor a 300 h − Mpc under-dense region in the local galaxy distribution (e.g., Keenan et al. 2013). Recent measurementsof optical and NIR anisotropies (e.g., Matsuura et al. 2017), at 1 . . µm , indicate that the resulting amplitudeof relative EBL fluctuations is typically in the range of 10 to 30% Zemcov et al. (2014).The impact of an accumulation of cosmic voids amounting to a total size of radius R = 1 h − Gpc (where h = H / (100 km s − Mpc − )) centered at redshift z v = 0 .
3, is illustrated in Figure 1. The EBL energy density spectrumin the presence of voids (dashed lines) is compared to the homogeneous case (solid lines) at different points (redshifts,as indicated by the labels) along the line of sight in the left panel of Figure 1. The fractional difference between thehomogeneous and the inhomogeneous case as a function of photon energy for various redshifts along the line of sightis presented in the right panel of Figure 1. We notice that the EBL deficit is smaller for low-energy (IR) photons thanfor optical – ultraviolet photons. This is because the UV EBL is dominated by hot, young stars, thus more stronglyreflecting the local effect of the void. Since in this work we set only the star formation rate inside the void equal tozero, dust re-processing of star light produced outside the void, still takes place inside the void. As can be seen fromFigure 1, with our choice of a void configuration, the impact of underdense regions is comparable to the measuredoptical and NIR anisotropy Zemcov et al. (2014). The impact of the EBL deficit due to the cosmic voids on the EBL γ − γ opacity will be presented in Section 5.1. −3 −2 −1 Energy (eV)10 −15 −14 E B L d e n s i t y ( e r g s c m − ) StarsDust z = 0.0z = 0.2z = 0.4z =0.6 z = 0.8 10 −3 −2 −1 Energy (eV)0.000.050.100.150.200.250.300.350.40 − u v o i d E B L / u h o m . E B L z = 0.0z = 0.2z = 0.4z =0.6 z = 0.8 Figure 1 . Left panel: Differential EBL photon energy density as a function of distance (redshift) along the line of sight. Thesolid lines represent the homogeneous case ( R = 0), and the dashed lines represent the EBL energy density considering anaccumulation of about 10 voids of typical sizes with radius R = 100 h − Mpc centered at redshift z v = 0 .
3. The EBL energydensity increases with redshift because of the star formation rate increasing with redshift at low redshifts (e.g., Cole et al. 2001)Right panel: Relativie EBL energy density deficit due to the presence of a void for the same cases as represented in the leftpanel. U voidEBL and U homEBL are the EBL energy densities considering the cosmic void case and the homogeneous case, respectively.As expected, the maximum EBL energy density deficit occurs around the center of the voids ( z = 0 . LORENTZ INVARIANCE VIOLATION: COSMIC OPACITYIn this Section we review the imprints of LIV on the cosmic γ − γ opacity, primarily based on the work byTavecchio & Bonnoli (2016). The results will be compared to the imprints of EBL inhomogeneities discussed inthe previous section. The deviation from Lorentz symmetry can be described by a modification of the dispersionrelation of photons and electrons (e.g., Amelino et al. 1998; Tavecchio & Bonnoli 2016): E = p c + m c + S E (cid:18) EE LIV (cid:19) n , (1)where c is the conventional speed of light in vacuum, “ S = −
1” represents a subluminal scenario (decreasing photonspeed with increasing energy), and “ S = +1” represents the superluminal case (increasing photon speed with increasingenergy). The characteristic energy E LIV is parameterized as a fraction of the Planck energy, E LIV = E P /ξ n , where thedimensionless parameter ξ n and the order of the leading correction n depend on particle type and theoretical framework(e.g., Amelino et al. 1998; Tavecchio & Bonnoli 2016). A value of E LIV ∼ E P (i.e., ξ = 1) has been considered tobe the physically best motivated choice (e.g., Liberati & Maccione 2009; Fairbairn et al. 2014; Tavecchio & Bonnoli2016) This is consistent with the results of Biteau & Williams (2015) which constrained E LIV > . E P . Someauthors (e.g., Schaefer 1998; Billers et al. 1999) argue that the best constraint from current data is ξ ≤ O (1000).In the literature (e.g., Tavecchio & Bonnoli 2016), usually only the subluminal case is considered for the LIV effecton γγ absorption, as this is the case that could lead to an increase of the γγ interaction threshold and consequently,a decrease of the opacity. In this work, for completeness, we consider both the subluminal and superluminal cases.Based on the revised dispersion relation (1) with n = 1, the modified pair-production threshold energy ǫ min can bewritten as (e.g., Tavecchio & Bonnoli 2016): ǫ min = m c E γ − S E γ E LIV . (2)Using equation (2), the target photon energy threshold for pair-production as a function of the γ -ray photon energyfor the subluminal and the superluminal cases is illustrated in Figure 2. -1 E(TeV) -3 -2 -1 ǫ ( e V ) S= −1Standard E
LIV = E P E LIV =5 E P E LIV =20 E P E LIV =100 E P E LIV =400 E P -1 E(TeV) -3 -2 -1 ǫ ( e V ) S= +1Standard E
LIV = E P E LIV =5 E P E LIV =20 E P E LIV =100 E P E LIV =400 E P Figure 2 . Left panel: Photon target energy at threshold for pair-production as a function of γ -ray photon energy, for thesubluminal case. The black solid line represents the case of standard QED and the dashed lines show the LIV-modifiedthreshold for different values of E LIV . Right panel: Same as the left panel, but for the superluminal case.
Also from equation (1), an effective mass term for photons can be defined as (e.g., Liberati & Maccione 2009;Tavecchio & Bonnoli 2016): ( m γ c ) ≡ S E (1 + z ) E LIV . (3)Following (Fairbairn et al. 2014; Tavecchio & Bonnoli 2016), we assume that the functional form of the γ − γ crosssection (as a function of the center-of-momentum energy squared s ) remains unchanged by the LIV effect, and onlythe expression for s is modified. The optical depth at the energy E γ and for γ -ray photons from a source at redshift z s can thus be evaluated as (Fairbairn et al. 2014; Tavecchio & Bonnoli 2016): τ γγ ( E γ , z s ) = c E γ Z z s dzH ( z )(1 + z ) Z ∞ ǫ min n ( ǫ, z ) ǫ Z s ( z ) max s min [ s − ( m γ c ) ] σ γγ ( s ) ds, (4)where H ( z ) = H p [Ω m (1 + z ) + Ω Λ ], s min = 4( m e c ) and s ( z ) max = 4 ǫE γ (1 + z ) + ( m γ c ) . n ( ǫ, z ) is the EBLphoton energy density as a function of redshift z and energy ǫ , and σ γγ ( s ) is the total pair production cross-section asa function of the modified square of the center of mass energy s = ( m γ c ) + 2 ǫE γ (1 − cos( θ )), where θ is the anglebetween the soft EBL photon of energy ǫ and the VHE γ -ray photon. Obviously, when E LIV −→ ∞ , the standardrelations are recovered.By using equation (4) with the EBL model by Finke et al. (2010), we calculate the optical depth for VHE γ -rayphotons from a source at redshift 0 .
6. The comparison with the standard case (homogeneous EBL, no LIV) and withthe effect of EBL inhomogeneities (as discused in Section 2) is presented in Section 5.1. LORENTZ INVARIANCE VIOLATION: COMPTON SCATTERINGOne of the most important fundamental high-energy radiation mechanisms is Compton scattering, the process bywhich photons gain or lose energy from collisions with electrons. In the Compton scattering processes, the energy ofa scattered photon E γf follows from momentum and energy conservation: (cid:16) E γi /c, −→ P γi (cid:17) + (cid:16) E ei /c, −→ P ei (cid:17) = (cid:16) E γf /c, −→ P γf (cid:17) + (cid:16) E ef /c, −→ P ef (cid:17) , (5)which is assumed to still hold even in a Lorentz-invariance violating framework. In Equ. (5), E γi , E γf and E ei , E ef are initial and final energies for the photon and electron respectively, and −→ P γi , −→ P γf and −→ P ei , −→ P ef are initial and finalmomenta for the photon and electron, respectively. To consider the LIV effect, we consider the first order correction n = 1 in the modified dispersion relation (1): E γ = p γ c + S E γ E LIV , (6)As motivated in the introduction, and consistent with our treatment of LIV on the EBL opacity in Section 3, weconsider LIV only for photons, not for electrons. Substituting for E ef using the standard electron dispersion relationand momentum conservation (considering that in the electron rest frame, p e,i = 0), the energy conservation part ofEqu. (5) can be written as: E γf = E γi + E ei − q c ( p γi − p γf ) + ( m e c ) . (7)Squaring and rearranging Equ. (7), expressing all photon momenta in terms of energies using the dispersion relation(6) yields 2 E γi E γf + 2( E γf − E γi ) m e c = S E γi E LIV + E γf E LIV ! + 2 µ s E γi − S E γi E LIV s E γf − S E γf E LIV . (8)where µ = cos θ is the cosine of the scattering angle in the electron rest frame. In the limit E LIV ≫ E γ , the square-rootexpressions in Equ. (8) can be simplified to s E γ − S E γ E LIV ≈ E γ (cid:18) − S E γ E LIV (cid:19) . (9)Thus, to lowest order in E γ /E LIV , Equ. (8) can be written as:2 E γi E γf + 2( E γf − E γi ) m e c = S E γi E LIV + E γf E LIV ! + 2 µE γi E γf (cid:18) − S E γi E LIV − S E γf E LIV (cid:19) . (10)Equ. (10) is solved numerically to find the scattered photon energy E γf as a function of initial photon energy E γi andscattering angle θ = cos − µ . Results are presented in Section 5.2.From QED, the Klein-Nishina cross-section σ KN can be written as: σ KN = Z dσ KN d Ω d Ω = Z r e (cid:18) E γf E γi (cid:19) (cid:18) E γi E γf + E γf E γi − sin θ (cid:19) d Ω , (11)where dσ KN d Ω is the differential Klein-Nishina cross section and d Ω is the solid angle, and r e is the classical electronradius.As for our considerations of the LIV effect on the γ − γ opacity, we assume that the functional dependence of theKlein-Nishina cross section on the incoming and scattered photon energies remains unaffected. Thus, in order tomodify the Klein-Nishina cross-section considering the LIV effect, we use the scattered photon E γf from the solutionof Equ. (10) in the Klein-Nishina formula (11) and integrate numerically. The results of this integration comparedwith the standard QED case are presented in Section 5.2. RESULTS AND DISCUSSIONIn this section, we present the results for representative test cases for the LIV effect on the cosmic γ − γ opacity,compared standard Lorentz-invariance case and the suppression of the opacity due to EBL inhomogeneities, and onthe Compton scattering process, compared to the standard-model case.5.1. EBL Absorption
To study the opacity or transparency of the Universe to VHE γ -ray photons from distant sources (e.g. blazars) dueto their interaction with intergalactic EBL photons, we compare the effects of the EBL inhomogeneities due to thepresence of cosmic voids to those of the LIV effect. Figure (3) shows the absorption coefficient exp( − τ γγ ) as a functionof energy for VHE-gamma rays from a source at redshift z s = 0 .
6. The standard-model QED case is representedby the black solid line. The impact of an EBL underdensity (for parameters as used in Fig. 1) is illustrated bydot-dashed lines and the LIV effect is represented by dashed lines for different values of the chracteristic LIV energyscale E LIV = E P /ξ . Note that the standard case without LIV is recovered for E LIV . E(TeV) -10 -8 -6 -4 -2 e x p ( − τ γγ ) z= 0.6S= −1 standard E LIV = E P E LIV = 5 E P E LIV = 10 E P void E(TeV) -1 R e l a t i v e o p t i c a l d e p t h d e f i c i t s = − 1z = 0.6 E LIV = E P E LIV = 5 E P E LIV = 10 E P voidvoid + (E LIV = E P ) E(TeV) -10 -8 -6 -4 -2 e x p ( − τ γγ ) z= 0.6S= +1 standard E LIV = E P E LIV = 5 E P E LIV = 10 E P void E(TeV) -2 -1 R e l a t i v e o p t i c a l d e p t h d e f i c i t z= 0.6S= +1 E LIV = E P E LIV = 5 E P E LIV = 10 E P voidvoid + (E LIV = E P ) Figure 3 . Left panels: Absorption coefficient exp( − τ γγ ) as a function of energy for VHE γ -rays from a source at redshift z s = 0 .
6, using the EBL model of Finke et al. (2010). The black solid line represents the case of standard QED; the dashedlines show the LIV-modified coefficient for different values of E LIV , for the subluminal case (top panel) and the superluminalcase (bottom panel). The blue dot-dashed line represents the case of standard QED and EBL energy density calculated byconsidering an accumulation of 10 voids of typical sizes with radius R = 100 h − Mpc along the line of sight, centered at redshift z v = 0 .
3. Right panels: Relative optical depth deficit as a function of energy for VHE γ -rays for the same cases as in the leftpanel. The Relative optical depth deficit is defined as (1 − τ DF Sγγ /τ Stand.γγ ), where τ Stand.γγ represents the optical depth calculatedin standard QED and using the homogeneous EBL energy density distribution, and τ DF Sγγ represents the optical depth calculatedincluding the effects of cosmic voids (blue dashed-dot line) or of LIV (dashed lines). The black dot-dashed line represents therelativie optical depth deficit due to the combined effect of LIV and EBL inhomogeneities.
The reduction of the EBL γ − γ opacity due to plausible EBL inhomogeneities is only of the order of .
10 % anddecreases with energy. The LIV effect is negligibly small for energies below about 5 TeV, but the cosmic opacity forVHE γ -rays with energies &
10 TeV can be strongly reduced for the subluminal case and increased for the superluminalcase. Therefore, if LIV is described by the subluminal dispersion relation ( S = − γ -ray photonsbeyond 10 TeV to be observable even from distant astrophysical sources.However, the spectral hardening of several observed VHE gamma-ray sources with energy from 100 GeV up to a fewTeV (e.g. PKS 1424+240) still remains puzzling. Compared to the Finke et al. (2010) EBL absorption model for anobject at a redshift of z s ∼ .
6, the opacity would have to be reduced by &
60 % in order to explain the spectralhardening of the VHE spectrum of PKS 1424+240 with standard emission mechanisms. Even if we consider thecombined effects of EBL underdensities and LIV, as represented by the solid line in the right panel of Figure (3), therelative optical depth τ γγ deficit is only around 10 % in the energy range from hundred of GeV to a few TeV.5.2.
Compton scattering -7 -5 -3 -1 E i (TeV) -7 -6 -5 -4 -3 -2 E f ( T e V ) S= −1 θ= 1 degree standardE
LIV = 0.001 E P E LIV = 0.01 E P E LIV = 0.1 E P E LIV = 1 E P E LIV = 10 E P E LIV = 100 E P -7 -5 -3 -1 E i (TeV) -7 -5 -3 -1 E f ( T e V ) S= +1 θ= 1 degree standardE
LIV = 0.001 E P E LIV = 0.01 E P E LIV = 0.1 E P E LIV = 1 E P E LIV = 10 E P E LIV = 100 E P -7 -5 -3 -1 E i (TeV) -7 -6 E f ( T e V ) S= −1 θ= 180 degree standardE
LIV = 0.001 E P E LIV = 0.01 E P E LIV = 0.1 E P E LIV = 1 E P E LIV = 10 E P E LIV = 100 E P -7 -5 -3 -1 E i (TeV) -7 -6 -5 -4 -3 -2 -1 E f ( T e V ) S= +1 θ= 180 degree standardE
LIV = 0.001 E P E LIV = 0.01 E P E LIV = 0.1 E P E LIV = 1 E P E LIV = 10 E P E LIV = 100 E P Scattering angle (degrees) -9 -8 -7 -6 -5 -4 -3 -2 -1 E f ( T e V ) S= −1E i =10 TeV standardE
LIV = 0.001 E P E LIV = 0.01 E P E LIV = 0.1 E P E LIV = 1 E P E LIV = 10 E P E LIV = 100 E P Scattering angle (degrees) -6 -5 -4 -3 -2 -1 E f ( T e V ) S= +1E i =10 TeV standardE
LIV = 0.001 E P E LIV = 0.01 E P E LIV = 0.1 E P E LIV = 1 E P E LIV = 10 E P E LIV = 100 E P Figure 4 . Top and middle panels: Scattered photon energies E γ,f as a function of incoming photon energy E γ,i , for scatteringangles of 1 and 180 degrees, respectively. The black solid line represents the case of standard QED; the dashed lines show theLIV effect for different values of E LIV , for a subluminal case (left) and superluminal case (right). Bottom panels: Scatteredphotons energies E f vs. scattering angle, for an incoming photon energy of E i = 1 PeV in the subluminal case (left) andsuperluminal case (right). The black solid line represents the case QED; the dashed lines illustrate the LIV effect for differentvalues of E LIV . The LIV effect on the Compton scattering process has been evaluated as described in Section 4. To assess theimportance of LIV signatures, we have evaluated this effect for a large range of values of E LIV . All calculations aredone in the electron rest frame.Figure 4 illustrates the effect of LIV on the scattered photon energies as a function of the incoming photon energies E i for two representative scattering angles (1 and 180 degrees — top and middle panels) for different values of E LIV ,as well as the scattered photon energies as a function of the scattering angle θ for one representative incoming photonenergy (10 TeV — bottom panels). The subluminal cases are illustrated in the left, the superluminal cases in theright panels. In the standard QED case (black solid curves), the kinematic constraints (recoil) lead to the well-knownlevelling-off of the scattered photon energies at a value of E γ,f ∼ m e c / (1 − cos θ ). In Figure 5 we illustrate the LIV -9 -7 -5 -3 -1 E i (TeV) -12 -10 -8 -6 -4 -2 σ K N / σ T S= −1 standardE
LIV = 0.001 E P E LIV = 0.01 E P E LIV = 0.1 E P E LIV = 1 E P E LIV = 10 E P E LIV = 100 E P -9 -7 -5 -3 -1 E i (TeV) -12 -10 -8 -6 -4 -2 σ K N / σ T S= +1 standardE
LIV = 0.001 E P E LIV = 0.01 E P E LIV = 0.1 E P E LIV = 1 E P E LIV = 10 E P E LIV = 100 E P Figure 5 . Total Klein-Nishina cross-section σ KN (in the units of σ T ) as a function of the incoming photon energy E γ,i . Theblack solid line represents the case of QED; the dashed lines show the LIV-modified Klein-Nishina cross-section for differentvalues of E LIV , for the subluminal (left panel) and superluminal case (right panel). effect on the total Klein-Nishina cross-section σ KN (in units of σ T ), plotted as a function of the incoming photon energy E γ,i . The black solid line represents the case of standard QED and the dashed lines show the modified Klein-Nishinacross-section for different values of E LIV , calculated as described in Section 4. Again, the subluminal and superluminalcases are illustrated in the left and right panel, respectively.From Figures (4) and (5) we see that LIV signatures in the Compton scattering processes are expected to beimportant only for very large incoming photon energies, E γ,i & E γ,i >
10 PeV, even in the superluminal case the scattered photon energy E γ,f is still much smaller thanthe incoming photon energy E γ,i . This indicates that the electron recoil effect is still substantial, as expected, butstrongly reduced/increased compared to standard-model kinematics, in the superluminal/subluminal case, respectively.Equally, at energies E γ,i & E γ,i ∼ m e c ) in the superluminal case, but is expected to remain suppressed to σ KN . − σ T for photon energies below ∼ E LIV . Inthe subluminal case, Compton scattering of photons at energies E γ,i & SUMMARY AND CONCLUSIONSWe have presented calculations of the modification of the EBL γ − γ opacity for VHE γ -ray photons from sources atcosmological distances, by considering two effects: the impact of under-densities (voids) along the line of sight to thesource and the LIV effect. For the LIV effect, we considered both a subluminal and a superluminal modification of thedispersion relation for photons. We found that the reduction of the optical depth due to the existence of cosmic voidsis insignificant for realistic parameters of the void and is thus insufficient to explain the unexpected spectral hardeningof the VHE spectra of several blazars. The effect of LIV becomes important only at γ -ray energies above ∼
10 TeV,where the γγ interaction threshold is increased and consequently, the EBL opacity is reduced in the subluminal case.The opposite effect (reduced pair production threshold and increased EBL opacity) results in the superluminal case.The effect is negligible for VHE spectra in the range ∼
100 GeV – a few TeV. However, these results suggest that,if LIV is manifested by a subluminal modification by the photon dispersion relation, VHE γ -ray sources may bedetectable at cosmological redshifts z & E &
10 TeV, as the EBL opacity at those energies may begreatly reduced compared to standard-model predictions. Observations with the small-size telescopes of the futureCherenkov Telescope Array (CTA: Acharya et al. 2013) — and its predecessors, such as the ASTRI (Astrofisica conSpecchi a Technologia Replicante Italiana: Vercellone 2016) array — will provide excellent opportunities to test thishypothesis.We have presented, to the authors’ knowledge for the first time, detailed calculations of the effect of LIV on theCompton scattering process. As for γγ absorption, we considered both subluminal and superluminal modifications tothe photon dispersion relation. In the superluminal case, we find that for incoming photon energies of E γ,i & >> E e ≫ E e ≫ ACKNOWLEDGMENTSWe thank the anonymous referee for a quick review and helpful suggestions. The work of M.B. is supported throughthe South African Research Chair Initiative of the National Research Foundation and the Department of Science andTechnology of South Africa, under SARChI Chair grant No. 64789.REFERENCES Abdalla, H. & B¨ottcher, M., 2017, ApJ, 835, 237Abdalla, H. & B¨ottcher, M., 2018, PoS(HEASA2017),028Acharya, B. S., et al., 2013, Astropart. Phys., 43, 3Abdallah, W., Delepine, D., Khalil, S., et al. 2013, Phys. Lett.,B725, 361Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., et al.2006, Nature, 440, 1018Amelino-Camelia, G., Ellis, J., Mavromatos, N. E., Nanopoulos,D. V., & Sarkar, S. 1998, Nature, 393, 763Amelino-Camelia, G. 2013, Living Rev. Rel., 16, 05Arbab, A.I. 2015, strophys Space Sci, 355, 343Azzam, W.J., Alothman, M. J., Guessoum, N, 2009, Advancesin Space Research, 44, 1354Biller, S. D., Breslin, A. C, Buckley, J., Catanese, M. & Carson,M. 1999, Phys. Rev. Lett., 83, 2108Biteau, J. & Williams D. A. 2015, ApJ, 812, 1B¨ottcher, M., Reimer, A., Sweeney, K., & Prakash, A., 2013,ApJ, 768, 54Capozziello, S., Gonzlez, P. A., Saridakis, E. N., & Vsquez, Y.2013, JHEP, 039, 02Cerruti, M., Zech, A., Boisson, C., & Inoue, S., 2015, MNRAS,448, 910Cole, S., Norberg, P.& Baugh, C., et al. 2001, MNRAS, 326, 255Dominguez, A., Primack, J. R., Rosario, D. J., et al., 2011a,MNRAS, 410, 2556 Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not acceptany liability in this regard. Dominguez, A., S´anchez-Conde, M. A., & Prada, F., 2011b, J.Cosmol. Astropart. Phys., 11, 020Dzhatdoev, T. A., 2015, J. Phys. Conf. Ser, 632, 01Dzhatdoev, T. A., Khalikov, E. V., Kircheva, A. P. & Lyukshin,A. A., 2017, Astron. Astrophys., 603, A59El-Zant A., Khalil, S. & Sil, A. 2015, Phys. Rev., D91, 03Essey W. & Kusenko, A. 2010, ApJ, 33, 81Fairbairn, M., Nilsson, A., Ellis, J., Hinton, J. & White, R.2014, JCAP, 1406, 005February, S., Clarkson, C. & Maartens, R. 2013, JCAP, 1303, 23Finke, J. D., Razzaque, S., & Dermer, C. D., 2010, ApJ, 712, 238Franceschini, A., Rodigheiro, G., & Vaccari, M., 2008, A&A,487, 837Furniss, A., Williams, D. A., Danforth, C., et al., 2013, ApJ, 768,L31Furniss A., Stutter, P. M., Primack, J. R., & Dominguez, A.,2015, MNRAS, 446, 2267Gilmore, R. C., Sommerville, R. S., Primack, J. R., &Dominguez, A., 2012, MNRAS, 422, 3189Gould, R. J. & Schr´eder, G. P. 1967, Phys. Rev., 155, 1408Hauser, M. G. & Dwek, E. 2001, ARA&A, 39, 249Jacob, U. and Piran, T. 2008, PhRvD, D78, 12Jacobson, T., Liberati. S. & Mattingly, D. 2003, Nature, 424,1019Jones, F. C., 1968, Phys. Rev., 167, 1159Keenan, R., C., Barger, A., J. and Cowie, L., L. 2013, ApJ, 775,620