Limits on the Lorentz Invariance Violation from UHECR Astrophysics

Astroparticle physics has recently reached the status of precision science due to (a) the construction of new observatories operating innovative technologies, (b) the detection of large numbers of events and sources, and (c) the development of clever theoretical interpretations of the data. Two observational windows have produced very important results in the last decade. The ultra-high energy cosmic rays (E > EeV) studied by the Pierre Auger and the Telescope Array Observatories (The Pierre Auger Collaboration 2015; Tinyakov 2014) improved our knowledge of the most extreme phenomena known in nature. The GeV–TeV gamma-ray experiments FERMI/LAT (Atwood et al. 2009), H.E.S.S. (The H.E.S.S. Collaboration 2006), MAGIC (The MAGIC Collaboration 2016), and VERITAS (J. Holder for the VERITAS Collaboration 2011) gave a new perspective on gamma-ray production and propagation in the universe. The operation of the current instruments and the construction of future ones (Zhen 2010; The CTA Consortium 2011; Haungs et al. 2015) guarantee the production of even more precise information in the decades to come.

Lorentz invariance (LI) is one of the pillars of modern physics and it has been tested in several experimental approaches (Mattingly 2005). Astroparticle physics has been proposed as an appropriate test environment for possible Lorentz invariance violation (LIV) given the large energy of the particles, the large propagation distances, the accumulation of small interaction effects, and recently the precision of the measurements (Amelino-Camelia et al. 1998; Jacobson et al. 2003; Stecker & Scully 2005, 2009; Ellis et al. 2006, 2008; Galaverni & Sigl 2008a, 2008b; The MAGIC Collaboration 2008; Liberati & Maccione 2009; Ellis & Mavromatos 2013; Fairbairn et al. 2014; Biteau & Williams 2015; Chang et al. 2016; Tavecchio & Bonnoli 2016; Xu & Ma 2016; Rubtsov et al. 2017).

Effective field theories with some Lorentz violation can derive in measurable effects in the data taking by astroparticle physics experiments; nonetheless, in this paper, LIV is introduced in the astroparticle physics phenomenology through the polynomial correction of the dispersion relation in the photon sector and is focused on the gamma-ray propagation and pair production effects with LIV. Other phenomena like vacuum birefringence, photon decay, vacuum Cherenkov radiation, photon splitting, synchrotron radiation, and helicity decay have also been used to set limits on LIV effects on the photon sector but are beyond the scope of this paper; for a review, see Liberati & Maccione (2009), Bluhm (2014), and Rubtsov et al. (2017).

Lorentz invariant gamma-ray propagation in the intergalactic photon background was studied previously in detail by De Angelis et al. (2013), a similar approach is followed in Section 2, but LIV is allowed in the interaction of high energy photons with the background light and their consequences are studied. The process $\gamma \gamma \to {e}^{+}{e}^{-}$ is the only one considered to violate LI, and, as a similar approach used in Galaverni & Sigl (2008a), such LIV correction can lead to a correction of the LI energy threshold of the production process. The latter phenomena modifies the mean-free path of the interaction and therefore the survival probability of a photon propagating through the background light, which depends on the LIV coefficients. This dependence is calculated in Section 2 and the mean-free path and the photon horizon are shown for several LIV coefficients and different orders of the LIV expansion in the photon energy dispersion relation.

In Section 3, the mean-free path of the photo-production process considering LIV is implemented in a Monte Carlo propagation code in order to calculate the effect of the derived LIV in the flux of ultra-high energy photons arriving on Earth due to the GZK effect (Greisen 1966; Zatsepin & Kuz'min 1966) and considering several models for the sources of cosmic rays. Section 3 quantifies the influence of the astrophysical models concerning mass composition, energy spectra shape, and source distribution. These dependencies have been largely neglected in previous studies and it is shown here that they influence the GZK photon flux by as much as four orders of magnitude.

In Section 4, the propagated GZK photon flux for each model is compared to recent upper limits on the flux of photons obtained by the Pierre Auger Observatory. For some astrophysical models, the Auger data is used to set restrictive limits on the LIV coefficients. The astrophysical model used to describe the primary cosmic-ray flux has a very large influence on the flux of GZK photons and therefore on the LIV limits imposed. Finally, in Section 5, the conclusions are presented.

One of the most commonly used mechanisms to introduce LIV in particle physics phenomenology is based on the polynomial correction in the dispersion relation of a free propagating particle, mainly motivated by an extra term in the Lagrangian density that explicitly breaks Lorentz symmetry, see, for instance, Amelino-Camelia et al. (1998), Coleman & Glashow (1999), Ahluwalia (1999), Amelino-Camelia (2001), Jacobson et al. (2003), Galaverni & Sigl (2008a, 2008b), Maccione & Liberati (2008), Liberati & Maccione (2011), Jacob & Piran (2008), and Zou et al. (2017). In these models, the corrected expression for the dispersion relation is given by the following equation:

$$\begin{eqnarray}&&{E}_{a}^{2}-{p}_{a}^{2}={m}_{a}^{2}+{\delta }_{a,n}{E}_{a}^{n+2},\end{eqnarray} \tag{ 1 }$$

where a denotes the particle with mass m_a and four-momenta (E_a, p_a). For simplicity, natural units are used in this work. The LIV coefficient, ${\delta }_{a,n}$, parametrizes the particle dependent LIV correction, where n expresses the correction order, which can be derived from the series expansion or from a particular model for such an order, see, for instance, the case of n = 0 (Coleman & Glashow 1997, 1999; Klinkhamer & Schreck 2008), n = 1 (Myers & Pospelov 2003), or a generic n (Vasileiou et al. 2013). The LIV parameter of the order of n, δ_n, is frequently considered to be inversely proportional to some LIV energy scale ${E}_{\mathrm{LIV}}^{(n)}$. Different techniques have been implemented in the search of LIV signatures in astroparticle physics and some of them have been used to derive strong constraints to the LIV energy scale (Amelino-Camelia et al. 1998; Maccione & Liberati 2008; Bi et al. 2009; The H.E.S.S. Collaboration 2011; Otte 2012; Vasileiou et al. 2013; Benjamin Zitzer for the VERITAS Collaboration 2014; Schreck 2014; Biteau & Williams 2015; Martínez-Huerta & Pérez-Lorenzana 2017; Rubtsov et al. 2017).

The threshold analysis of the pair production process, considering the LIV corrections from Equation (1) on the photon sector is discussed in the Appendix and leads to corrections of the LI energy threshold of the process. In the following, ${\epsilon }_{\mathrm{th}}^{\mathrm{LIV}}$ stands for the minimum energy of the cosmic background (CB) photon in the pair production process with LIV. The latter effect can lead to changes in the optical depth, τ_γ (E_γ, z), which quantifies how opaque to photons the universe is. The survival probability, i.e., the probability that a photon, γ, emitted with a given energy, E_γ, and at a given redshift, z, reaches Earth without interacting with the background, is given by

$$\begin{eqnarray}&&{P}_{\gamma \to \gamma }({E}_{\gamma },z)={e}^{-{\tau }_{\gamma }({E}_{\gamma },z)}.\end{eqnarray} \tag{ 2 }$$

The photon horizon is the distance (z_h) for which ${\tau }_{\gamma }({E}_{\gamma },{z}_{h})=1$. z_h defines, as a function of the energy of the photon, the redshift at which an emitted photon will have probability ${P}_{\gamma \to \gamma }=1/e$ of reaching Earth. The evaluation of the photon horizon is of extreme importance because it summarizes the visible universe as a function of the energy of the emitted photon. In this section, the photon horizon is calculated including LIV effects. The argument presented in De Angelis et al. (2013) is followed here.

In the intergalactic medium, the $\gamma {\gamma }_{\mathrm{CB}}\to {e}^{+}{e}^{-}$ interaction is the main contribution to determine the photon horizon. In the approximation where cosmological effects are negligible, the mean-free path, λ (E_γ), of this interaction is given by

$$\begin{eqnarray}&&\lambda ({E}_{\gamma })=\displaystyle \frac{{cz}}{{H}_{0}{\tau }_{\gamma }({E}_{\gamma },z)},\end{eqnarray} \tag{ 3 }$$

where H₀ = 70 km s⁻¹ Mpc⁻¹ is the Hubble constant and c is the speed of light in a vacuum. The optical depth is obtained by

$$\begin{eqnarray}{\tau }_{\gamma }({E}_{\gamma },z) & = & {\displaystyle \int }_{0}^{z}{dz}\displaystyle \frac{c}{{H}_{0}(1+z)\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{M}{(1+z)}^{3}}}\\ & & \times {\displaystyle \int }_{-1}^{1}d(\cos \theta )\displaystyle \frac{1-\cos \theta }{2}\\ & & \times {\displaystyle \int }_{{\epsilon }_{\mathrm{th}}^{\mathrm{LIV}}}^{\infty }d\epsilon {n}_{\gamma }(\epsilon ,z)\sigma ({E}_{\gamma },\epsilon ,z),\end{eqnarray} \tag{ 4 }$$

where θ is the angle between the direction of propagation of both photons θ = [−π, +π], Ω_Λ = 0.7 is the dark energy density, Ω_M = 0.3 is the matter density, σ is the cross section of the interaction, and ${\epsilon }_{\mathrm{th}}^{\mathrm{LIV}}$ is the threshold energy of the interaction as given by Equation (13).

${n}_{{\gamma }_{\mathrm{CB}}}$ is the background photon density. The dominant backgrounds are the extra-galactic background light (EBL) for E_γ < 10^14.5 eV, the cosmic background microwave radiation (CMB) for 10^14.5 eV < E_γ < 10¹⁹ eV and the radio background (RB) for E_γ > 10¹⁹ eV. In the calculations presented here, the Gilmore model (Gilmore & Ramirez-Ruiz 2010) was used for the EBL. Since LIV effects in the photon horizon are expected only at the highest energies (E_γ > 10¹⁶ eV), using different models of EBL would not change the results. For the RB, the data from Gervasi et al. (2008) with a cutoff at 1 MHz were used. Different cutoffs in the RB data lead to different photon horizons as shown in De Angelis et al. (2013). Since no new effect shows up in the LIV calculation due to the RB cutoff, only the 1 MHz cutoff will be presented.

It is usual for studies such as the one presented here, in which the threshold of an interaction is shifted, causing a modification of the mean-free path, to neglect direct effects in the cross section, σ, when solving Equation (4). However, an implicit change of the cross section is taken into account given its dependence on the energy threshold ${\epsilon }_{\mathrm{th}}^{\mathrm{LIV}}$ (Breit & Wheeler 1934).

Figures 1–3 show the mean-free path for $\gamma {\gamma }_{\mathrm{CB}}\to {e}^{+}{e}^{-}$ as a function of the energy of the photon, E_γ, for several LIV coefficients with n = 0, n = 1, and n = 2, respectively. The main effect is an increase in the mean-free path that becomes stronger the larger the photon energy, E_γ, and the LIV coefficient are. Consequently, fewer interactions happen and the photon, γ, will have a higher probability of traveling farther than it would have in an LI scenario. Similar effects due to LIV are seen for n = 0, n = 1, and n = 2. The LIV coefficients are treated as free parameters; therefore, there is no way to compare the importance of the effect between the orders n = 0, n = 1, and n = 2, each order must be limited independently. Note that ${\delta }_{\gamma ,n}$ units depend on n.

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The LIV effect becomes more tangible in Figure 4 in which the photon horizon (z_h) is shown as a function of E_γ for n = 0. For energies above E_γ > 10^16.5 eV and the given LIV values, the photon horizon increases when LIV is taken into account, increasing the probability that a distant source emitting high energy photons produces a detectable flux at Earth. Similar results are found for n = 1 and n = 2.

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Even though the effects of LIV on the propagation of high energy photons are strong, they cannot be directly measured and, therefore, used to probe LIV models. In order to do that, in this section, the flux of GZK photons on Earth considering LIV is obtained and compared to the upper limits on the photon flux from the Pierre Auger Observatory (Carla Bleve for the Pierre Auger Collaboration 2015; The Pierre Auger Collaboration 2017a).

UHECRs interact with the photon background producing pions (photo-pion production). Pions decay shortly after production, generating EeV photons among other particles. The effect of this interaction chain suppresses the primary UHECR flux and generates a secondary flux of photons (Gelmini et al. 2007). The effect was named GZK after the authors of the original papers (Greisen 1966; Zatsepin & Kuz'min 1966). The EeV photons (GZK photons) also interact with the background photons as described in the previous sections.

In order to consider LIV in the GZK photon calculation, the CRPropa3/EleCa (Settimo & Domenico 2015; Batista et al. 2016) codes were modified. The mean-free paths calculated in Section 2 were implemented in these codes and the propagation of the particles was simulated. The resulting flux of GZK photons is, however, extremely dependent on the assumptions about the sources of cosmic rays, such as the injected energy spectra, mass composition, and the distribution of sources in the universe. Therefore, four different models for the injected spectra of cosmic rays at the sources and five different models for the evolution of sources with redshift are considered in the calculations presented below.

3.1. Models of UHECR Sources

No source of UHECR was ever identified and correlation studies with types of sources are not conclusive. Several source types and mechanisms of particle production have been proposed. The amount of GZK photons produced in the propagation of the particles depends significantly on the source model used. In this paper, four UHECR source models are used to calculate the corresponding GZK photons. The models are used as illustrations of the differences in the production of GZK photons; an analysis of the validity of the models and their compatibility with experimental data is beyond the scope of this paper. However, it is important to note that strong constrains to the source models can be set by new measurements (The Pierre Auger Collaboration 2017b). The models used here are labeled as follows:

• 1.  
  ${{\boldsymbol{C}}}_{{\bf{1}}}$: Aloisio et al. (2014);
• 2.  
  ${{\boldsymbol{C}}}_{{\bf{2}}}$: Unger, Farrar, & Anchordoqui (2015)—Fiducial model (Unger et al. 2015);
• 3.  
  ${{\boldsymbol{C}}}_{{\bf{3}}}$: Unger et al. (2015) with the abundance of galactic nuclei from (Olive & Group 2014);
• 4.  
  ${{\boldsymbol{C}}}_{{\bf{4}}}$: Berezinsky, Gazizov, & Grigorieva (2007)—Dip model (Berezinsky et al. 2006).

All four models propose the energy spectrum at the source to be a power-law distribution of the energy with a rigidity cutoff:

$$\begin{eqnarray}\displaystyle \frac{{dN}}{{{dE}}_{s}}=\left\{\begin{array}{ll}{E}_{s}^{-{\rm{\Gamma }}}, & \mathrm{for}\,{R}_{s}\lt {R}_{\mathrm{cut}}\\ {E}_{s}^{-{\rm{\Gamma }}}{e}^{1-{R}_{s}/{R}_{\mathrm{cut}}}, & \mathrm{for}\,{R}_{s}\geqslant {R}_{\mathrm{cut}}\end{array}\right.,\end{eqnarray} \tag{ 5 }$$

where the spectral index, Γ, and the rigidity cutoff, R_cut, are parameters given by each model. Five different species of nuclei (H, He, N, Si, and Fe) are considered in these models and their fraction (fH, fHe, fN, fSi, and fFe) are given in Table 1.

Table 1.  Parameters of the Four Source Models Used in This Paper

Model	Γ	${\mathrm{log}}_{10}({R}_{\mathrm{cut}}/V)$	fH	fHe	fN	fSi	fFe
C1	1	18.699	0.7692	0.1538	0.0461	0.0231	0.00759
C2	1	18.5	0	0	0	1	0
C3	1.25	18.5	0.365	0.309	0.121	0.1066	0.098
C4	2.7	$\infty$	1	0	0	0	0

Note. Γ is the spectral index, R_cut is the rigidity cutoff and fH, fHe, fN, fSi, and fFe are the fractions of each nuclei.

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The composition of UHECR has a strong influence on the generated flux GZK photons and, therefore, on the possibility to set limits on LIV effects. The models chosen in this study range from very light (C₄) to very heavy (C₂) passing by intermediate compositions C₁ and C₃. Heavier compositions produce less GZK photons and therefore are less prone to reveal LIV effects.

Figure 5 shows the dependence of the GZK photon flux on the source model used. The integral of the GZK photon fluxes for the LIV case of ${\delta }_{\gamma ,0}={10}^{-20}$ are shown as a function of energy. The use of different LIV coefficients results in a shift up and down in the integral flux for each source model, having negligible changes in each ratio. The dependence on the model is of several orders of magnitude and should be considered in studies trying to impose limits on LIV coefficients. The capability to restrict LIV effects is proportional to the GZK photon flux generated in each model assumption.

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3.2. Models of Source Distribution

Figure 4 shows how the photon horizon increases significantly when LIV is considered. Therefore, the source distribution in the universe is an important input in GZK photon calculations usually neglected in previous studies. Five different models of source evolution(R_n) are considered here.

• 1.  
  ${{\boldsymbol{R}}}_{{\bf{1}}}$: sources are uniformly distributed in a comoving volume;
• 2.  
  ${{\boldsymbol{R}}}_{{\bf{2}}}$: sources follow the star formation distribution given in Hopkins & Beacom (2006). The evolution is proportional to (1 + z)^3.4 for z < 1, to (1 + z)^−0.26 for 1 ≤ z < 4 and to (1 + z)^−7.8 for z ≥ 4;
• 3.  
  ${{\boldsymbol{R}}}_{{\bf{3}}}$: sources follow the star formation distribution given in Yüksel et al. (2008). The evolution is proportional to (1 + z)^3.4 for z < 1, to (1 + z)^−0.3 for 1 ≤ z < 4 and to (1 + z)^−3.5 for z ≥ 4;
• 4.  
  ${{\boldsymbol{R}}}_{{\bf{4}}}$: sources follow the GRB rate evolution from Le & Dermer (2007). The evolution is proportional to ${(1+8z)/[1+(z/3)}^{1.3}]$;
• 5.  
  ${{\boldsymbol{R}}}_{{\bf{5}}}$: sources follow the GRB rate evolution from Le & Dermer (2007). The evolution is proportional to ${(1+11z)/[1+(z/3)}^{0.5}]$.

Figure 6 shows the ratio of sources as a function of redshift for the five source distributions considered. The source evolution uniformly distributed in a comoving volume is shown only for comparison. It is clear that even astrophysical motivated evolutions are different for redshifts larger than two. Charged particles produced in sources farther than redshifts equal to one have a negligible probability of reaching Earth; however, the GZK photons produced in their propagation could travel farther if LIV is considered.

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Figure 7 shows the effect of the source evolution in the prediction of GZK photons including LIV effects. Once more, the use of different LIV coefficients results in a shift up an down in the integral flux for each source evolution model, having negligible changes in each ratio. The differences for each source evolution model are as large as 500% at E = 10¹⁸ eV. The capability to restrict LIV effects is proportional to the GZK photon flux generated in each model assumption.

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The GZK photon flux of the five astrophysical models shown above are considered together with the upper limits on the photon flux imposed by the Pierre Auger Observatory to set limits on the LIV coefficients. The simulations considered sources up to 9500 Mpc ($z\approx 8.88$). The reference results are for model ${C}_{3}{R}_{5}$, as this is the model that best describes current UHECR data. The three orders of LIV (n = 0, 1, and 2) are considered for each astrophysical model C_i. Two limiting cases are also considered: LI and maximum LIV, labeled as δ_γ = 0 and ${\delta }_{\gamma }\to -\infty$, respectively. The Lorentz invariant case (LI) is shown for comparison. The maximum LIV case (${\delta }_{\gamma }\to -\infty$) represents the limit in which the mean-free path of the photon–photon interaction goes to infinity at all energies and therefore no interaction happens. These two cases bracket the possible LIV solutions. The UHECR flux reaching Earth was normalized to the flux measured by the Pierre Auger Observatory (Inés Valiño for the Pierre Auger Collaboration 2015) at E = 10^18.75 eV, which sets the normalization of the GZK photon flux produced in the propagation of these particles.

Figures 8–10 show the results of the calculations. For some LIV coefficients, models C₁R₅, C₃R₅, and C₄R₅ produces more GZK photons than the upper limits imposed by Auger, therefore, upper limits on the LIV coefficients can be imposed. Model C₂R₅ produces less GZK photons than the upper limits imposed by Auger even for the extreme scenario ${\delta }_{\gamma }\to -\infty$; therefore, no limits on the LIV coefficients could be imposed. Table 2 shows the limits imposed in this work for each source model and LIV order.

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Table 2.  Limits on the LIV Coefficients Imposed by This Work for Each Source Model and LIV Order (n)

Model	${\delta }_{\gamma ,0}^{\mathrm{limit}}$	${\delta }_{\gamma ,1}^{\mathrm{limit}}({\mathrm{eV}}^{-1})$	${\delta }_{\gamma ,2}^{\mathrm{limit}}({\mathrm{eV}}^{-2})$
${C}_{1}{R}_{5}$	∼−10−20	∼−10−38	∼−10−56
${C}_{2}{R}_{5}$	⋯	⋯	⋯
C3R5	∼−10−20	∼−10−38	∼−10−56
${C}_{4}{R}_{5}$	∼−10−22	∼−10−42	∼−10−60

Note. Model ${C}_{3}{R}_{5}$ is pointed out as containing the reference values of this paper because it describes better the current UHECR data.

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Table 3 shows the limits imposed by other works for the photons sector for comparison. The direct comparison of the results obtained here (${C}_{3}{R}_{5}$) is only possible with Galaverni & Sigl (2008a; first line in Table 3) because of the similar technique based on GZK photons. The differences between the calculations presented here and the limits imposed in Galaverni & Sigl (2008a) can be explained by (a) the different assumptions considered in the γγ interactions with LIV, (b) the different astrophysical models used, and (c) the upper limit on the GZK photon flux used. In Galaverni & Sigl (2008a), the limits were obtained by calculating the energy in which the interaction of a high energy photon with a background photon at the peak of the CMB, i.e., with energy  = 6 × 10⁻⁴ eV, becomes kinematically forbidden. In this work, a more complete approach was used, where the energy threshold was calculated, the mean-free path was obtained by integrating the whole background photon spectrum and the propagation was simulated, obtaining the intensity of the flux of GZK photons. The astrophysical scenario used in Galaverni & Sigl (2008a) was a pure proton composition with energy spectrum normalized by the AGASA measurement (The AGASA Collaboration 2006) and index Γ = 2.6. The source distribution was not specified in the study. However, this astrophysical scenario is ruled out by the X_max measurements from the Pierre Auger Observatory (The Pierre Auger Collaboration 2014a, 2014b). In the calculations presented here, the LIV limits were updated using astrophysical scenarios compatible to the Auger X_max data. Finally, in this paper, new GZK photon limits published by Auger are used. The LIV limits presented here are, therefore, more realistic and up to date.

Table 3.  Limits on the LIV Coefficients Imposed by Other Works Based on Gamma-Ray Propagation

Model	${\delta }_{\gamma ,0}^{\mathrm{limit}}$	${\delta }_{\gamma ,1}^{\mathrm{limit}}({\mathrm{eV}}^{-1})$	${\delta }_{\gamma ,2}^{\mathrm{limit}}({\mathrm{eV}}^{-2})$
Galaverni & Sigl (2008a)	⋯	−1.97 × 10−43	−1.61 × 10−63
H.E.S.S.—PKS 2155−304 (2011)	⋯	−4.76 × 10−28	−2.44 × 10−40
Fermi—GRB 090510 (2013)	⋯	−1.08 × 10−29	−5.92 × 10−41
H.E.S.S.—Mrk 501 (2017)	⋯	−9.62 × 10−29	−4.53 × 10−42

Note. First line shows a previous result, which can be compared to the calculations presented here in Table 2. The last three lines are shown for completeness. These limits are based on gamma-ray arrival time and are not directly comparable to the ones in Table 2.

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The other values in Table 3 are shown for completeness. The second and third entries are based on energy dependent arrival time of TeV photons: (a) a PKS 2155−304 flare measured with H.E.S.S. (The H.E.S.S. Collaboration 2011) and (b) GRB 090510 measured with Fermi-LAT (Vasileiou et al. 2013). Entry H.E.S.S.—Mrk 501 (2017) (Lorentz & Brun 2017) in Table 3 is based on the kinematics of the interactions of photons from Mrk 501 with the background. All of the studies shown in Table 3 assume LIV only in the photon sector. However, the systematics of the measurements and the energy of photons (TeV photons versus EeV photons) are very different and a direct comparison between the GZK photon calculations shown here and the time of arrival of TeV photon is not straightforward.

In this paper, the effect of possible LIV in the propagation of photons in the universe is studied. The interaction of a high energy photon traveling in the photon background was solved under LIV in the photon sector hypothesis. The mean-free path of the $\gamma {\gamma }_{\mathrm{CB}}\to {e}^{+}{e}^{-}$ interaction was calculated considering LIV effects. Moderate LIV coefficients introduce a significant change in the mean-free path of the interaction as shown in Section 2 and Figures 1–3. The corresponding LIV photon horizon was calculated as shown in Figure 4.

The dependence of the integral flux of GZK photons on the model for the sources of UHECRs is discussed in Section 3 and shown in Figures 5 and 7. The flux changes several orders of magnitude for different injection spectra models. A difference of about 500% is also found for different source evolution models. Previous LIV limits were calculated using GZK photons generated by source models currently excluded by the data (Galaverni & Sigl 2008a). The calculations presented here show LIV limits based on source models compatible with current UHECR data. In particular, model ${C}_{3}{R}_{5}$ was shown to describe the energy spectrum, composition, and arrival direction of UHECR (Unger et al. 2015) and therefore is chosen as our reference result.

The calculated GZK photon fluxes were compared to most updated upper limits from the Pierre Auger Observatory and are shown in Figures 8–10. For some of the models, it was possible to impose limits on the LIV coefficients, as shown in Table 2. It is important to note that the LIV limits shown in Table 2 were derived from astrophysical models of UHECR, compatible to the most updated data. The limits presented here are several orders of magnitude more restrictive than previous calculations based on the arrival time of TeV photons (The H.E.S.S. Collaboration 2011; Vasileiou et al. 2013); however, the comparison is not straightforward due to different systematics of the measurements and energy of the photons.

R.G.L. is supported by FAPESP (2014/26816-0, 2016/24943-0). H.M.H. acknowledges IFSC/USP for their hospitality during the developments of this work, Abdel Pérez Lorenzana for enlightening discussions, and the support from Conacyt Mexico under grant 237004 and the Brazilian agency FAPESP (2017/03680-3). V.d.S. thanks the Brazilian population support via FAPESP (2015/15897-1) and CNPq. This work has partially made use of the computing facilities of the Laboratory of Astroinformatics (IAG/USP, NAT/Unicsul), whose purchase was made possible by the Brazilian agency FAPESP (2009/54006-4) and the INCT-A. The authors acknowledge the National Laboratory for Scientific Computing (LNCC/MCTI, Brazil) for providing HPC resources of the SDumont supercomputer, which have contributed to the research results reported within this paper (http://sdumont.lncc.br).

Software: CRPropa3 (Batista et al. 2016, as developed on https://github.com/CRPropa/CRPropa3, EleCa (Settimo & Domenico 2015).

Equation (1) leads to unconventional solutions of the energy threshold in particle production processes of the type ${AB}\to {CD}$. In this paper, the $\gamma {\gamma }_{\mathrm{CB}}\to {e}^{+}{e}^{-}$ interaction is considered. From now on, the symbol γ refers to a high energy gamma-ray with energy E_γ = [10⁹, 10²²] eV that propagates in the universe and interacts with the CB photons, γ_CB, with energy  = [10⁻¹¹, 10] eV.

Considering LIV in the photon sector, the specific dispersion relations can be written as

$$\begin{eqnarray}{E}_{\gamma }^{2}-{p}_{\gamma }^{2} & = & {\delta }_{\gamma ,n}\,{E}_{\gamma }^{n+2},\\ {\epsilon }^{2}-{p}_{{\gamma }_{\mathrm{CB}}}^{2} & = & {\delta }_{\gamma ,n}\,{\epsilon }^{n+2},\end{eqnarray} \tag{ 6 }$$

where ${\delta }_{\gamma ,n}$ is the n-order LIV coefficient in the photon sector and therefore taken to be the same in both dispersion relations. The standard LI dispersion relation for the electron–positron pair follows: ${E}_{{e}^{\pm }}^{2}-{p}_{{e}^{\pm }}^{2}={m}_{e}^{2}.$

Taking into account the inelasticity (K) of the process (${E}_{{e}^{-}}={{KE}}_{\gamma }$) and imposing energy–momentum conservation in the interaction, the following expression for a head-on collision with collinear final momenta can be written to leading order in ${\delta }_{\gamma ,n}$

$$\begin{eqnarray}&&4\epsilon {E}_{\gamma }-{m}_{e}^{2}\left(\displaystyle \frac{1}{K(1-K)}-\displaystyle \frac{{m}_{e}^{2}}{2K(1-K){({E}_{\gamma }+\epsilon )}^{2}}\right)\\ &&\quad =\,-{\delta }_{\gamma ,n}{E}_{\gamma }^{n+2}\left[1+\displaystyle \frac{{\epsilon }^{n+2}}{{E}_{\gamma }^{n+2}}-\displaystyle \frac{\epsilon }{{E}_{\gamma }}\left(1+\displaystyle \frac{{\epsilon }^{n}}{{E}_{\gamma }^{n}}\right)\right].\end{eqnarray} \tag{ 7 }$$

In the ultra relativistic limit E_γ ≫ m_e and E_γ ≫ , this equation reduces to

$$\begin{eqnarray}&&{\delta }_{\gamma ,n}{E}_{\gamma }^{n+2}+4{E}_{\gamma }\epsilon -{m}_{e}^{2}\displaystyle \frac{1}{K(1-K)}=0.\end{eqnarray} \tag{ 8 }$$

Equation (8) implies two scenarios: (I) ${\delta }_{\gamma ,n}\gt 0$ the photo-production threshold energy is shifted to lower energies and (II) ${\delta }_{\gamma ,n}\lt 0$ the threshold takes place at higher energies than that expected in an LI regime, except for scenarios below a critical value for delta, where the photo-production process is forbidden. Notice that, if ${\delta }_{\gamma ,n}=0$ in Equation (8) the LI regime is recovered. In the LI regime, it is possible to define ${E}_{\gamma }^{\mathrm{LI}}=\tfrac{{m}_{e}^{2}}{4\epsilon K(1-K)}$. The math can be simplified by the introduction of the dimensionless variables

$$\begin{eqnarray}&&{x}_{\gamma }=\displaystyle \frac{{E}_{\gamma }}{{E}_{\gamma }^{\mathrm{LI}}},\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\gamma ,n}=\displaystyle \frac{{E}_{\gamma }^{\mathrm{LI}\,(n+1)}}{4\epsilon }{\delta }_{\gamma ,n}.\end{eqnarray} \tag{ 10 }$$

Then, Equation (8) takes the form

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{\gamma ,n}{x}_{\gamma }^{n+2}+{x}_{\gamma }-1=0.\end{eqnarray} \tag{ 11 }$$

Studying the values of ${\delta }_{\gamma ,n}$ for which Equation (11) has a solution, one can set the extreme allowed LIV coefficient (Galaverni & Sigl 2008b; Martínez-Huerta & Pérez-Lorenzana 2017). The limit LIV coefficient (${\delta }_{\gamma ,n}^{\mathrm{lim}}$) for which the interaction is kinematically allowed for a given E_γ and is given by

$$\begin{eqnarray}&&{\delta }_{\gamma ,n}^{\mathrm{lim}}=-4\displaystyle \frac{\epsilon }{{E}_{\gamma }^{\mathrm{LI}\,(n+1)}}\displaystyle \frac{{(n+1)}^{n+1}}{{(n+2)}^{n+2}}.\end{eqnarray} \tag{ 12 }$$

Equation (11) has real solutions for x_γ only if ${\delta }_{\gamma ,n}\gt {\delta }_{\gamma ,n}^{\mathrm{lim}}$. Therefore, under the LIV model considered here, if ${\delta }_{\gamma ,n}\lt {\delta }_{\gamma ,n}^{\mathrm{lim}}$, high energy photons would not interact with background photons of energy .

For a given E_γ and ${\delta }_{\gamma ,n}$ the threshold background photon energy (${\epsilon }_{\mathrm{th}}^{\mathrm{LIV}}$) including LIV effects is

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{th}}^{\mathrm{LIV}}=\displaystyle \frac{{m}_{e}^{2}}{4{E}_{\gamma }K(1-K)}-\displaystyle \frac{{\delta }_{\gamma ,n}{E}_{\gamma }^{n+1}}{4}.\end{eqnarray} \tag{ 13 }$$

The superscript LIV is used for emphasis. In the paper, ${\epsilon }_{\mathrm{th}}^{\mathrm{LIV}}$ as given by Equation (13) will be used for the calculations of the mean-free path of the $\gamma {\gamma }_{\mathrm{CB}}\to {e}^{+}{e}^{-}$ interaction. Figure 11 shows the allowed parameter space of E_γ and for different values of ${\delta }_{\gamma ,0}$. The gray areas are cumulative from darker to lighter gray.

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