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In this paper, we estimate the cross-correlation power spectra between thePlanck 2018 cosmic microwave background (CMB) temperature anisotropy map andthe unresolved $\gamma$-ray background (UGRB) from the 9-years Fermi-Large AreaTelescope (LAT) data. In this analysis, we use up to nine energy bins over awide 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 beincreased 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 thenine Fermi energy bins to measure the significance of the ISW effect, weobtained ${\rm A_{amp}}=0.95 \pm 0.53$ ($68\%$ C.L.). For the robustness test,we implement a null hypothesis by randomizing the Fermi mock maps of nineenergy bins and obtain the non-detection of ISW effect, which confirms that theISW signal comes from the Fermi-LAT diffuse $\gamma$-ray data and is consistentwith the standard $\Lambda$CDM model prediction essentially. We use across-correlation coefficient to show the relation between different energybins. Furthermore, we vary the cut ranges $|b|$ of galactic plane on the maskof Fermi map and carefully check the consequent influence on the ISW signaldetection.

Searching for Integrated Sachs-Wolfe Effect from Fermi-LAT Diffuse $\gamma$-ray Map