A stack of AGN as a test source

As no individual gamma-ray source can be considered unpolarized1_{, we use a stack of AGN as a test} source to estimate the magnitude of the systematic uncertainties afecting the measurement. Even if the emission of individual AGN is polarized, the polarization angles of the individual sources are expected to be uncorrelated. The total lux of a stack of many sources will have low residual polariza- tion. Another reason for using the AGN stack is that, by not looking directly at individual sources, we do not add signiicant trials to any future analysis.

We set up a simple model of the sources in order to decide the best composition of the stack. The basic idea is to represent the total emission of the stack (Nstack_{(ψ)) as a sum of partially polarized} signals ni(ψ) from the Nsrcssources of the stack:

Nstack(ψ) = Nsrcs



i

ni(ψ) (5.11)

where each ni(ψ) has the form of Eq. 5.3. The contribution ni(ψ) of each source to the global az- imuthal distribution of the stack is made up of two components: a point source emission assumed to be polarized in a random direction with a high degree of polarization, ΠAGN = 60%, and the inte- grated Galactic and isotropic difuse emission in the vicinity of the source. This difuse component, ndif_i is assumed to be unpolarized, therefore:

ni(ψ) = Ki1 − A100ΠAGNcos2(ψ − ψi) + ndifi (5.12) The point source contribution is normalized to the total number of counts expected from the source Ni: Ki = Ni/π[1 − (A100ΠAGN/2)]. To compute Niwe proceed as in the previous section, using the 3FGL spectra of the sources and the IRFs for silicon-converted events. For the rest of the analysis, we will assume an observation time corresponding to 5 years of the mission.

All 3FGL AGN are ranked in descending order of Fsrc/Fdifin order to maximize the signal-to-noise ratio in the global emission. As seen in Section 5.3, the bright blazar 3C 454.3 is a good candidate for a real measurement, and is therefore excluded from the stack. The two FSRQs 3FGL J1833.6−2103 and 3FGL J0109.8+6132 are also removed from the stack. The simulation of the sky region around these sources, located close the Galactic plane (Galactic latitudes of 5.71◦_{and −1.24}◦_{respectively) was not}

1_{The only exception is the Moon, whose gamma-ray glow is mostly due to the decay of π}0

produced by inelastic colli- sions of CRs on its surface (Ackermann et al.,2016b). Unfortunately our Moon is an extended source and is not so bright in gamma rays, both attributes that disqualify it as a good test source.

Chapter 5 Estimate of the Fermi-LAT sensitivity to gamma-ray polarization

in a satisfactory agreement with the data. The stack is then constructed by adding one source at a time, but excluding candidates within 20◦_{of any source already included in the stack. For each new} source that is added to the stack, we estimate the amplitude of the azimuthal modulation of the stack using a MC procedure: all the signals ni(ψ) from the sources of the stack are summed, each with a random phase, and the amplitude of the azimuthal modulation of the stack (A_stack) is measured1_{. This} procedure is repeated 103_{times, randomizing the phases at each new realization. The irst moments} of the distribution of Astackare used to characterize the residual polarization of the stack. We continue to add sources to the stack until our sample satisies the following requirements:

• Brightness: the total lux from the stacked sources must be greater than 10−5_{ph cm}−2_s−1 • Low residual polarization: The mean residual amplitude modulation ⟨Astack⟩ is at least 2σ

below 2%.

nearby stack

Figure 5.11: Positions, in galactic coordinates and in Aitof projection, of the AGN constituting the stack (red dots) and of the nearby sources which are also included in the simulation (blue crosses). For 3 AGN of the stack, visible towards the center of the norther hemisphere, less than 10 neighboring sources have been considered. Although there are more 3FGL sources in the vicinity of these AGN, only a few of them are bright enough to be included in the analysis. Details on how the neighboring sources are selected are given in Section 5.4.2.

The resulting 24 sources in the stack are listed in Table 5.2 and their position on the sky is shown

1_{The sum of sinusoids with equal frequencies is still a sinusoid with the same frequency. To see this we can think of}

taking the Fourier transform F[Nstack_{(ψ)] of the stack emission. The Fourier transform is linear: F (}_{x) =}_{F (x) and}

F[sin(ω0t)] ∼ δ(ω − ω0). It follows that F [Nstack(ψ)] ∼ δ(ω − ω0); the reverse transformation returns a sinusoid at the

5.4 Estimating systematic uncertainties

in Fig. 5.11. The total lux of these sources between 50 and 200 MeV is ∼ 1.3 × 10−5_{ph cm}−2_s−1_, while the total difuse emission in the same energy range and within 3◦_{of the sources is ≈ 7.9×10}−6 ph cm−2_s−1_.

Table 5.2: Composition of the AGN stack. As is customary in the LAT catalogs, uppercase (lowercase) class designations indicate identiied (associated) sources. Γ is the source photon index computed with a power-law spectrum over the 0.1−300 GeV energy range. The integral source and difuse luxes Fsrc_{and F}dif_{are computed}

by integrating in energy between 50 and 200 MeV, and within 4◦_{from the source.}

Source Name l b Fsrc Γ Fdif class

[deg.] [deg.] [10−7_{ph cm}−2_s−1_{] [10}−7_{ph cm}−2_s−1_] 3FGL J1512.8−0906 351.29 40.13 19.61 2.36 2.48 FSRQ 3FGL J1229.1+0202 289.98 64.35 12.90 2.66 0.61 FSRQ 3FGL J1522.1+3144 50.16 57.02 8.19 2.35 0.75 fsrq 3FGL J2329.3−4955 332.01 -62.31 4.87 2.27 0.55 fsrq 3FGL J0403.9−3604 237.73 -48.47 4.41 2.49 0.65 FSRQ 3FGL J0237.9+2848 149.48 -28.52 4.24 2.35 2.10 FSRQ 3FGL J2151.8−3025 17.13 -50.76 3.32 2.97 1.05 fsrq 3FGL J0841.4+7053 143.54 34.43 3.39 2.84 1.14 FSRQ 3FGL J0730.2−1141 227.76 3.14 6.08 2.28 6.64 fsrq 3FGL J2202.7+4217 92.60 -10.44 5.32 2.25 5.17 BLL 3FGL J1427.9−4206 321.45 17.26 4.69 2.21 5.27 FSRQ 3FGL J2147.3−7536 315.79 -36.53 3.51 2.47 2.14 FSRQ 3FGL J0108.7+0134 131.87 -60.99 2.75 2.39 0.60 fsrq 3FGL J1246.7−2547 301.61 37.07 3.18 2.28 2.25 fsrq 3FGL J1159.5+2914 199.45 78.37 2.20 2.21 0.20 fsrq 3FGL J2025.6−0736 36.90 -24.39 2.46 2.33 3.08 fsrq 3FGL J0920.9+4442 175.69 44.81 2.04 2.29 0.69 fsrq 3FGL J0442.6−0017 197.20 -28.46 2.63 2.50 1.58 fsrq 3FGL J2345.2−1554 65.70 -70.99 1.92 2.09 0.37 FSRQ 3FGL J0909.1+0121 228.95 30.93 2.16 2.41 1.19 fsrq 3FGL J0725.2+1425 203.64 13.92 2.09 2.23 1.95 FSRQ 3FGL J1700.1+6829 99.54 35.20 1.69 2.40 1.31 fsrq 3FGL J0145.1−2732 217.96 -78.07 1.22 2.57 0.20 fsrq 3FGL J1208.7+5442 135.82 61.34 1.30 2.54 0.47 fsrq