Can the CIB constrain the dark energy?

Galaxies are often used as tracers of the large scale structure (LSS) to measure the Integrated Sachs-Wolfe effect (ISW) by cross-correlating the galaxy survey maps with the Cosmic Microwave Background (CMB) map. We use the Cosmic Infrared Background (CIB) as a tracer of the LSS to perform a theoretical CIB-CMB cross-correlation to measure the ISW for different Planck HFI frequencies. We discuss the detectability of this ISW signal using a Signal-to-noise ratio analysis and find that the ISW detected this way can provide us with the highest SNR for a single tracer ranging from 5 to 6.7 (maximum being for 857 GHz) with the CIB and CMB maps extracted over the whole sky. A Fisher matrix analysis showed that this measurement of the ISW can improve the constraints on the cosmological parameters; especially the equation of state of the dark energy $w$ by $\sim 47\%$. Performing a more realistic analysis including the galactic dust residuals in the CIB maps over realistic sky fractions shows that the dust power spectra dominate over the CIB power spectra at $\ell < 100$ and ISW can't be detected with high SNR. We perform the cross-correlation on the existing CIB-CMB maps over $\sim 11\%$ of the sky in the southern hemisphere and find that the ISW is not detected with the existing CIB maps over such small sky fractions.