Searching for Integrated Sachs-Wolfe Effect from Fermi-LAT Diffuse
  $\gamma$-ray Map
  In this paper, we estimate the cross-correlation power spectra between the
Planck 2018 cosmic microwave background (CMB) temperature anisotropy map and
the unresolved $\gamma$-ray background (UGRB) from the 9-years Fermi-Large Area
Telescope (LAT) data. In this analysis, we use up to nine energy bins over a
wide energy range of [0.631, 1000] GeV from the Fermi-LAT UGRB data. Firstly,
we find that the Fermi data with the energy ranges [1.202, 2.290] GeV and
[17.38, 36.31] GeV show the positive evidence for the Integrated Sachs-Wolfe
(ISW) effect at $1.8\sigma$ confidence level, and the significance would be
increased to $2.7\sigma$ when using these two energy bins together. Secondly,
we apply the single power-law model to normalize the amplitude and use all the
nine Fermi energy bins to measure the significance of the ISW effect, we
obtained ${\rm A_{amp}}=0.95 \pm 0.53$ ($68\%$ C.L.). For the robustness test,
we implement a null hypothesis by randomizing the Fermi mock maps of nine
energy bins and obtain the non-detection of ISW effect, which confirms that the
ISW signal comes from the Fermi-LAT diffuse $\gamma$-ray data and is consistent
with the standard $\Lambda$CDM model prediction essentially. We use a
cross-correlation coefficient to show the relation between different energy
bins. Furthermore, we vary the cut ranges $|b|$ of galactic plane on the mask
of Fermi map and carefully check the consequent influence on the ISW signal
detection.
