Correlation of CMB with large-scale structure. I. Integrated Sachs-Wolfe tomography and cosmological implications

Abstract

We cross correlate large-scale structure (LSS) observations from a number of surveys with cosmic microwave background (CMB) anisotropies from the Wilkinson Microwave Anisotropy Probe (WMAP) to investigate the integrated Sachs-Wolfe (ISW) effect as a function of redshift, covering $z\sim 0.1–2.5$. Our main goal is to go beyond reporting detections towards developing a reliable likelihood analysis that allows one to determine cosmological constraints from ISW observations. With this in mind we spend a considerable amount of effort in determining the redshift-dependent bias and redshift distribution ($b(z)\times dN/dz$) of these samples by matching with spectroscopic observations where available, and analyzing autopower spectra and cross-power spectra between the samples. Because of wide redshift distributions of some of the data sets we do not assume a constant-bias model, in contrast to previous work on this subject. We only use the LSS data sets for which we can extract such information reliably and as a result the data sets we use are 2-Micron All Sky Survey (2MASS) samples, Sloan Digital Sky Survey (SDSS) photometric Luminous Red Galaxies, SDSS photometric quasars, and NRAO VLA Sky Survey (NVSS) radio sources. We make a joint analysis of all samples constructing a full covariance matrix, which we subsequently use for cosmological parameter fitting. We report a $3.7\sigma$ detection of ISW combining all the data sets. We do not find significant evidence for an ISW signal at $z>1$, in agreement with theoretical expectation in the $\Lambda \mathrm{CDM}$ model. We combine the ISW likelihood function with weak lensing of CMB (hereafter Paper II [C. M. Hirata, S. Ho, N. Padmanabhan, U. Seljak, and N. A. Bahcall, arXiv:0801.0644.]) and CMB power spectrum to constrain the equation of state of dark energy and the curvature of the Universe. While ISW does not significantly improve the constraints in the simplest six-parameter flat $\Lambda \mathrm{CDM}$ model, it improves constraints on seven-parameter models with curvature by a factor of 3.2 (relative to WMAP alone) to ${\Omega }_K=-{0.004}_{-0.020}^{+0.014}$, and with dark energy equation of state by 15% to $w=-{1.01}_{-0.40}^{+0.30}$ [posterior median with “$1\sigma$” (16th–84th percentile) range]. A software package for calculating the ISW likelihood function can be downloaded at http://www.astro.princeton.edu/~shirley/ISW_WL.html.