Anisotropies in the diffuse gamma-ray background measured by the Fermi LAT

The contribution of unresolved sources to the diffuse gamma-ray background could induce anisotropies in this emission on small angular scales. We analyze the angular power spectrum of the diffuse emission measured by the Fermi Large Area Telescope at Galactic latitudes $|b|>30°$ in four energy bins spanning 1–50 GeV. At multipoles $\ell \ge 155$, corresponding to angular scales $\lesssim 2°$, angular power above the photon noise level is detected at $>99.99\%$ confidence level in the 1–2 GeV, 2–5 GeV, and 5–10 GeV energy bins, and at $>99\%$ confidence level at 10–50 GeV. Within each energy bin the measured angular power takes approximately the same value at all multipoles $\ell \ge 155$, suggesting that it originates from the contribution of one or more unclustered source populations. The amplitude of the angular power normalized to the mean intensity in each energy bin is consistent with a constant value at all energies, $C_{\mathrm{P}}/⟨I{⟩}^2=9.05\pm 0.84\times {10}^{-6}\mathrm{sr}$, while the energy dependence of $C_{\mathrm{P}}$ is consistent with the anisotropy arising from one or more source populations with power-law photon spectra with spectral index ${\Gamma }_{\mathrm{s}}=2.40\pm 0.07$. We discuss the implications of the measured angular power for gamma-ray source populations that may provide a contribution to the diffuse gamma-ray background.

Figure 1

All-sky intensity maps of the data in the four energy bins used in this analysis, in Galactic coordinates; the map projection is Mollweide. The data shown are the average of the maps of the front- and back-converting events, and are shown unmasked (left panels) and with the default mask applied (right panels). The mask excludes Galactic latitudes $|b|<30°$ and a 2° angular radius around each source in the 1FGL catalog. The map images shown have been downgraded in resolution to $N_{\mathrm{side}}=128$ to improve the visual quality of the images; however, the analysis was performed on the higher resolution maps as described in the text.

Figure 2

No-anisotropy sky map created by summing 20 000 shuffled maps using front- and back-converting events with $E>1\mathrm{GeV}$, binned into a HEALPix map with $N_{\mathrm{side}}=256$. The map projection is Hammer-Aitoff. The features in the no-anisotropy sky map result from the fact that the sky was not observed with uniform exposure.

Figure 3

Comparison of intensity angular power spectra of the data and Galactic-foreground-cleaned data. For $\ell \ge 155$ the measured power at all energies is approximately constant in multipole, suggesting that it originates from one or more unclustered source populations. The large increase in angular power in the default data at $\ell <155$ in the 1–2 and 2–5 GeV bins is likely attributable largely to contamination from Galactic diffuse emission. In these two energy bins, foreground cleaning primarily reduces angular power at $\ell <155$, with the most significant reductions at $\ell <105$. At energies greater than 5 GeV the effect of foreground cleaning is small for $\ell \ge 55$. Expanded versions of the top panels are shown in Fig. 4.

Figure 4

Expanded versions of top panels of Fig. 3, focusing on the high-multipole angular power.

Figure 5

Predicted amplitude of the source angular power spectrum, $C_{\mathrm{P}}$ [see Eq. (9)], for energies of 1.04–10.4 GeV as a function of a source detection threshold flux, $S_c$.

Figure 6

Intensity angular power spectrum of a simulated observation of the source population model, compared with the theoretical prediction (shaded band). The angular power spectrum of the simulated population is in excellent agreement with the prediction over a large multipole range.

Figure 7

Fluctuation angular power spectra $C_{\ell }/⟨I{⟩}^2$ calculated using the default analysis pipeline compared with those obtained using the exposure map from the event shuffling technique described in Sec. 4. Angular power is measured in all four energy bins by both analysis methods. The lack of significant differences at the multipoles of interest between the angular power spectra yielded by the two methods demonstrates that any inaccuracies in the exposure map have a negligible impact on the measured angular power spectra. Expanded versions of the top panels are shown in Fig. 8.

Figure 8

Expanded versions of top panels of Fig. 7.

Figure 9

Comparison of the beam window functions for the P6_V3 and P6_V11 IRFs; the P6_V3 IRFs are the default used in this analysis. The quantity $W_{\ell }^2$, which is the factor by which the angular power is suppressed due to the finite angular resolution of the instrument, is shown for the front-converting (left panel) and back-converting (right panel) events, evaluated at the log-center of each energy bin used in this analysis. The differences between the $W_{\ell }^2$ of these two IRFs are small ($\lesssim \mathrm{\text{few percent}}$) at all energies considered, indicating that our results are insensitive to the differences between the PSF models implemented in these IRFs.

Figure 10

Intensity angular power spectra of the data calculated with different latitude cuts. The point source mask was applied in addition to the latitude mask in all cases. The differences between the results masking $|b|<30°$ (the default latitude cut) and $|b|<40°$ are small for $\ell \ge 155$ for all four energy bins, demonstrating that the power observed in the data at these multipoles is not strongly correlated with a component that has a strong latitude dependence in the range $30°<|b|<40°$, such as the Galactic diffuse emission. At energies above 5 GeV convergence is seen for multipoles $\ell \ge 155$ even when masking only $|b|<20°$. Points from different data sets are offset slightly in multipole for clarity. The lowest multipole data point for the $|b|<20°$ mask in each panel is above the range shown in the figure. Expanded versions of the top panels are shown in Fig. 11.

Figure 11

Expanded versions of top panels of Fig. 10.

Figure 12

Intensity angular power spectra of the data calculated with a mask excluding a 1° or 2° angular radius around each source; excluding a 2° angular radius is the default in this analysis. The default latitude mask excluding $|b|<30°$ was applied in addition to the source mask in all cases. At all energies the angular power spectra obtained using the different source mask radii are consistent at $\ell \ge 155$ (the multipole range of interest), and above 2 GeV the results are consistent at $\ell \ge 55$. Expanded versions of the top panels are shown in Fig. 13.

Figure 13

Expanded versions of top panels of Fig. 12.

Figure 14

Intensity angular power spectra of the data calculated using the source mask defined by the 2FGL catalog compared with the results using the 1FGL catalog; the source mask defined by the 1FGL catalog is the default used in this analysis. The angular power at $\ell \ge 155$ is smaller in the 2FGL case by $\sim 20–30\%$ in the bins spanning 1–10 GeV and by $\sim 70\%$ at 10–50 GeV. Expanded versions of the top panels are shown in Fig. 15.

Figure 15

Expanded versions of top panels of Fig. 14.

Figure 16

Angular power spectra of the data, the default simulated model (MODEL), and the alternate simulated model (ALT MODEL). The angular power spectra of the two models are in good agreement in all energy bins. The smaller amplitude angular power at $\ell \ge 155$ measured at lower significance in both models is inconsistent with the angular power observed in the data at all energies. Points from different data sets are offset slightly in multipole for clarity. Expanded versions of the top panels are shown in Fig. 17.

Figure 17

Expanded versions of top panels of Fig. 16.

Figure 18

Angular power spectra of the components of the default simulated model (MODEL). As expected, most of the total angular power at all multipoles (TOTAL MODEL) is due to the GAL component. By construction, the isotropic component (ISO) component contributes no significant angular power; likewise, the source component (CAT) provides no contribution because all sources were masked. Points corresponding to the TOTAL MODEL are offset slightly in multipole for clarity. Expanded versions of the top panels are shown in Fig. 19.

Figure 19

Expanded versions of top panels of Fig. 18.

Figure 20

Anisotropy energy spectra of the data. Top: Fluctuation anisotropy energy spectrum. The data are consistent with no energy dependence over the energy range considered, although a mild energy dependence is not excluded. Bottom: Differential intensity anisotropy energy spectrum. The energy dependence is consistent with that arising from a single source population with a power-law intensity energy spectrum with spectral index ${\Gamma }_{\mathrm{s}}=2.40\pm 0.07$ for the default data ($2.33\pm 0.08$ for the cleaned data).