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

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.

Figure 1

The overdensity maps of various tracer samples in Galactic coordinates. The scale runs from $g=-1$ (black, no galaxies) to $g=-0.25$ (blue), $g=0$ (green), $g=+0.25$ (red), and $g=+1$ (white, $\ge 2\times$ mean density).

Figure 2

LRG and QSO overdensity vs various quantities such as reddening, PSF FWHM ($r$ band), observing time (MJD), red star density, and star density. In each panel the circles show the low-redshift sample and the squares show the high-redshift sample. The Modified Julian Date (MJD) of the DR3 ending date is 52821. Note that there are very few accepted pixels at the extremes of reddening, PSF FWHM, and stellar density, resulting in the large fluctuations seen in the figure.

Figure 3

Galaxy density correlations with WMAP temperatures (4 bands: Ka (crosses), Q (triangles), V (squares), W (empty triangles), error bars are from the correlations with V-band. This contains 2MASS galaxy density correlations with WMAP, starting from (from left to right, top to bottom) the brightest sample, to the bottom the dimmest sample. We shift the points on x-axis for clarity. The dotted line shows the predicted signal for the sample with WMAP 3-year parameters and $bdN/dz$ estimated in Sec. 5.

Figure 4

Same as Fig. 3 except for the SDSS density maps from (from left to right, top to bottom): low-z LRG, high-z LRG, low-z QSO, high-z QSO.

Figure 5

Same as Fig. 3 except for the NVSS cross-correlation.

Figure 6

The 2MASS redshift distribution, binned in units of $\Delta z=0.01$. The top panel shows the raw measured distribution, $\Pi (z)$, and the bottom panel is corrected for relative bias $b_{\mathrm{rel}}(z)\Pi (z)$.

Figure 7

The galaxy power spectra for the four 2MASS and two SDSS LRG samples, and the $Q$-model fits. The solid lines show the range of multipoles used in the fit, the dashed lines are extrapolations. Note that at very small scales the $Q$-model is not a good description of the power spectrum.

Figure 8

The redshift distributions of the two LRG samples. The dashed lines show the probability distribution for the photo-$z$s, where as the solid (“deconvolved”) lines show the smoothed true redshift distribution based on the reconstruction method of Padmanabhan et al. [55].

Figure 9

(a) The preliminary quasar redshift distribution, constructed from the successful matches to spectroscopic data. (b) The best-fit $f(z)$ for the two quasar samples as described in the text for the WMAP cosmology.

Figure 10

The model fits to the power spectra of the quasars and their cross-correlation with the LRGs. The low- and high-$z$ quasar slices are denoted “QSO0” and “QSO1” respectively, and a similar nomenclature is used for the LRGs. The model fits using linear theory are shown with the solid lines over the range of multipoles used in the fit. The dashed lines show the extension of the model across the remaining range of multipoles. Note that for the highest multipoles the linear theory is expected to break down.

Figure 11

The cross-spectra of NVSS with the other samples. The solid lines show the linear theory predictions in the region used for the fits, and the dashed lines show the extension to higher or lower multipoles. Note that for the highest multipoles linear theory is not valid.

Figure 12

The redshift histogram of NVSS sources matched to COSMOS using the Mobasher et al. [80] photometric redshifts. The dashed line is the fit three-parameter $f_{\mathrm{NVSS}}(z)$, normalized to unity (i.e. the redshift distribution assuming constant bias and negligible effect from magnification).

Figure 13

The constraints on the NVSS redshift distribution from the cross-correlations with the other eight samples. The horizontal error bars show the redshift window functions as described in the text. The dashed line shows the result of using the redshift distribution based on the Dunlop & Peacock [77] luminosity function assuming constant bias and neglecting magnification, as has been done in most ISW studies.

Figure 14

The ISW amplitude (A) and errorbars $\sigma (A)$ for all samples plotted along the redshifts compared with predictions of WMAP-3 year parameters. The fitting and errors for this figure used the Fisher matrices from the correlation of the tracer samples with the various WMAP maps. We also show the expected amplitude for 3 model Universes along the angular diameter distance degeneracy curve. We calculate the expected amplitude by substituting our observed correlations with predicted correlations for each of the model Universe and proceed in the same manner as described in Eq.(45). The three model Universes are: $\Lambda \mathrm{CDM}$ model with the WMAP-3 year parameters (open triangle with dotted line); closed Universe (open pentagons with short dashed line) ${\Omega }_b$=$0.215$, ${\Omega }_m$ = $1.25$, ${\Omega }_K$ = $-0.29$, $h=0.32$, ${\sigma }_8=0.61$; and Open Universe (open squares with long dashed line) ${\Omega }_b$=$0.015$, ${\Omega }_m$ = $0.089$, ${\Omega }_K$ = $0.03$, $h=1.20$, ${\sigma }_8=0.73$. Note that the redshift distribution is very broad for NVSS, giving rise to the jump in the open model prediction, even though the effective redshift of the sample is nearly the same as for low redshift QSO sample.

Figure 15

The predicted ISW signal for the low-z LRGs (above) and high-z QSOs (below) sample for sample open, closed, and flat $\Lambda \mathrm{CDM}$ models. Parameters are the same as in Fig. 14.

Figure 16

$\Lambda \mathrm{CDM}$ + ${\Omega }_K$ model: the 1-D distributions of ${\Omega }_{\Lambda }$ and ${\Omega }_m$. The solid (dashed) line represents constraints from using WMAP+ISW+WL (WMAP alone).

Figure 17

$\Lambda \mathrm{CDM}$ + ${\Omega }_K$ model: the 1-D distribution of ${\Omega }_K$ and the 2-D distribution of ${\Omega }_{\Lambda }$ and ${\Omega }_K$ (68% and 95% confidence contours shown). The solid (dot-dashed) line represents constraints from using WMAP+ISW+WL (WMAP alone).

Figure 18

$\Lambda \mathrm{CDM}$ + $w$ model: the 2-D distribution of ${\Omega }_{\Lambda }$ and $w$ (68% and 95% confidence contours shown). The solid (dashed) line represents constraints from using WMAP+ISW+WL (WMAP alone).