{\it Fermi}-LAT Stacking Analysis Technique: An Application to Extreme Blazars and Prospects for their CTA Detection

We present a likelihood profile stacking technique based on the {\it Fermi}-Large Area Telescope (LAT) data to explore the $\gamma$-ray characteristics of {\it Fermi}-LAT undetected astrophysical populations. The pipeline is applied to a sample of $\gamma$-ray unresolved extreme blazars, i.e., sources with the highest synchrotron peak frequencies ($\nu_{\rm Syn}^{\rm peak}\geqslant 10^{17}$ Hz), and we report a cumulative $\gamma$-ray detection with more than 32$\sigma$ confidence for 2 degrees of freedom. Comparing the generated stacked $\gamma$-ray spectrum with the sensitivity limits of the upcoming Cherenkov Telescope Array (CTA), we find that the {\it Fermi}-LAT undetected population of such extreme blazars, on average, may remain well below the CTA detection threshold due to their faintness and extragalactic background light (EBL) absorption. However, $\gamma$-ray detected blazars belonging to the same class are promising candidates for CTA observations. The EBL corrected stacked spectra of these sources do not show any softening up to 1 TeV. This finding suggests the inverse Compton peak of extreme blazars to lie above 1 TeV, thus indicating a hard intrinsic TeV spectrum. Our analysis also predicts that at 100 GeV, at least $\sim$10\% of the diffuse extragalactic $\gamma$-ray background originates from the $\gamma$-ray undetected extreme blazars. These results highlight the effectiveness of the developed stacking technique to explore the uncharted territory of $\gamma$-ray undetected astrophysical objects.