Abstract
Dark energy is one of the biggest puzzles in modern astronomy, and its nature is still unknown. The Integrated Sachs-Wolfe (ISW) effect provides us with a direct approach to dark energy through the temperature anisotropy in the Cosmic Microwave Background (CMB). When passing through the gravitational wells, the CMB photons first become blueshifted while falling in the wells and then become redshifted while climbing out the wells. In a matter-dominated universe, the blueshifts and redshifts compensate each other, and there is no net change on the CMB photons. However, in a dark-energy-dominated universe, the gravitational wells would decay with time due to the accelerated expansion of the Universe caused by dark energy, and hence the redshifts could not cancel the blueshifts anymore, which makes the CMB photons eventually become hotter due to the remained blueshifts. The ISW effect can be detected through the cross-correlation between the temperature anisotropies of the CMB and the large-scale structures, and in the literature, most significant detections are at $2-3\sigma$ via the cross-correlation of the CMB temperature maps from the Wilkinson Microwave Anisotropy Probe (WMAP) or Planck with various survey of galaxies. The Wide-field Infrared Survey Explorer (WISE) is an ideal data-set of galaxies for the ISW studies, because WISE has a survey area of the whole sky and is deeper than other all-sky surveys. In spite of WISE's great advantage and Planck's improvement over WMAP, few works use WISE with Planck to study the ISW effect. Most previous works, including the one using both WISE and Planck, perform simple analysis on the selection of the galaxy samples, the treatment of foreground contamination, and the statistical inference, while the detection level of the ISW effect varies among papers because of those aspects. Therefore, using the galaxies from WISE and the CMB temperature maps from Planck, we present a comprehensive analysis of the ISW effect with a careful examination on the galaxy selection, the foreground treatment, and the statistical inference. We optimise the galaxy sample by developing a quantitative method to evaluate the uniformness of a galaxy sample. With the selected WISE galaxy sample, which is more uniform than those in the previous studies, we obtain a positive ISW signal. We analyse the cross-correlation of WISE galaxies with different foreground-cleaned CMB maps in Planck. We further investigate the influence on the ISW signal from different treatments of foreground contamination. We also check the influence on the detection level of the ISW effect from various utilisation of the cross-power spectra used in previous studies. Among the different foreground-cleaned CMB maps in Planck, we obtain $1.2-2.5\sigma$ detections of the ISW effect with amplitudes consistent with the $\Lambda$CDM prediction, providing us with an alternative evidence for the accelerated expansion driven by dark energy.