Conventional propagation of photons

The survival probability (or attenuation factor) of γ rays due to the EBL is described by a decreasing exponential of the optical depth parameter, τγγ(E, z). This model dependent parameter is an increasing function of the photon energy and the distance to the source. The observed γ-ray spectra of sources are then described by,

φ_obs(E, z) =φ_int(E)·exp (−τ(E, z)), (5.2.1) where φobs and φint are the observed and intrinsic spectra, respectively. The intrinsic spectral shapes and fits were discussed in the previous section.

Throughout this work we use the observationally-based Dominguez et al. model [150].

The overall spectrum of the EBL is derived directly from galaxy SED observations over a

wide redshift range. Data from the All-wavelength Extended Groth Strip International Survey (AEGIS) [254] of∼6000 galaxies with redshifts between 0.2 and 1 are combined with the observed evolution of the rest-frame K-band galaxy luminosity function up to redshift 4 [255]. We decided to use the Dominguez et al. model due to its observational method and wide range of redshifts covered, but there is no particular reason to choose any EBL model over others derived under different approaches, since their predictions are compatible with the current constraints. The effects of choosing a different EBL model in our analysis will be discussed in Section 5.5.

Fig. 5.2.1 shows the optical depth parameter for different values of redshift computed with the Dominguez et al. model. Correspondingly, Fig. 5.2.2 displays the photon survival probabilities computed with the optical depths for each case. We can see that the survival probability of a γ-ray photon of energy E propagating through the IGM decreases with energy and traveled distance.

The bumps in the EBL intensity from Fig. 3.1.2 are reflected in the optical depth parameter. The initial rise until optical wavelengths results in a rapid growth of τγγ

until ∼500 GeV. Between 1 and 10 TeV, the slope of τ_γγ becomes smaller due to a decrease in the EBL intensity. This energy dependence causes spectral breaks on γ-ray sources [125].

Figure 5.2.1: Photon survival probabilities for the Dominguez et al. EBL model at different redshifts. Data were taken from Ref. [150].

Figure 5.2.2: Photon survival probabilities for the Dominguez et al. EBL model at different redshifts. Data were taken from Ref. [150].

Figure 5.2.3: Dashed line: power law intrinsic spectrum of a blazar. Solid lines: ob-served spectrum for different redshifts, after EBL attenuation. Computed with the Dominguez et al. model. The flux normalization is taken as unity, the pivot energy is 1 GeV and the spectral index is 1.5.

An example of intrinsic spectrum of a blazar and EBL absorption is given in Fig. 5.2.3 computed with Eq. 5.2.1 and with the power law shape of Eq. 5.1.3. The figure displays the spectral attenuation that increases with energy and distance to the source. The observed spectra cannot be described by a power law over the entire energy range but can be fit to two different power laws in two different energy ranges that depend on the redshift.

Gamma ray telescopes observe mostly the attenuated emission. As we discussed in the previous Section, in order to obtain the intrinsic spectrum of a source, a fit can be done in the region in which EBL effects are negligible. A intrinsic spectral index ΓGeV is obtained from this fit. Assuming that the intrinsic spectral shape of the source does not change for higher energies, an spectral break is expected to happen. The observed spectral index in the TeV range ΓT eV increases due to the EBL absorption. The presence of this spectral break has been confirmed [125] and can be used for studying the EBL with blazar observations.

The cosmic γ-ray horizon (CGRH) is the curve, given by the values of E0 and z, at which the optical depth parameter becomes unity, τ(E0, z) = 1. The solid black line displayed in Fig. 5.2.4 is the CGRH derived with the Dominguez et al. model. Above this curve, the survival probability of HE and VHE photons decreases exponentially as the optical depth parameter increases, therefore the Universe becomes less transparent to γ rays. In the region below the CGRH, the survival probabilities are larger and the Universe is more transparent toγ rays. It is less probable that photon events of energy E0, that come from a source located at z, survive the EBL absorption for large values ofτ and appear in the plot, hence the density of events is larger below the horizon. For a given redshift, the CGRH quantifies the maximum energy of photons that survive the EBL.

If there were modifications of the expected γ-ray propagation, the observed HEP event for each source, shown in Fig. 5.2.4, should change correspondingly, allowing us to search for ALPs with these events. Since the EBL intensity determination is still an open problem, we also expect transparency discrepancies between EBL models. In Section 5.4, we also test how a different EBL model affects our results.

Figure 5.2.4: HEP energy vs redshift. Red points represent the observed energy of the 2FHL catalog sources HEP with the Fermi-LAT. The solid black line is the CGRH computed with the Dominguez et al. EBL model, with its uncertainties as a shaded grey band. Solid lines represent other constant survival probabilities, P. Data were taken from [17].