Integrated Sachs-Wolfe effect in Cross-Correlation: The Observer's Manual

TL;DR

This work analyzes the Integrated Sachs-Wolfe (ISW) effect as revealed through cross-correlation between the CMB and a low-redshift galaxy tracer in a flat universe, focusing on how survey geometry and systematics affect detectability. It develops a Limber-based framework for the cross-power spectrum $C_{gT}(\ell)$, links it to the matter power spectrum via the growth factor and its derivative, and derives the dominant noise contributions from the primary CMB. The study quantifies the best- and worst-case signal-to-noise scenarios, showing that a nearly full-sky survey of $\sim 10^7$ galaxies in $0<z<1$ could reach $\sim 5\sigma$ ISW detection, but reaching a cosmic-variance-limited $\sim 7.5\sigma$ requires controlling systematics to $\lesssim 0.1\%$ on $\sim 10^{\circ}$ scales and redshift biases $\Delta z_{sys} \lesssim 0.05$. It also discusses the cosmological utility of ISW measurements, indicating modest tightness on dark-energy parameters but highlighting ISW’s value as a clean test of gravity and large-scale consistency tests for the concordance model.

Abstract

The Integrated Sachs-Wolfe (ISW) effect is a direct signature of the presence of dark energy in the universe, in the absence of spatial curvature. A powerful method for observing the ISW effect is through cross-correlation of the Cosmic Microwave Background (CMB) with a tracer of the matter in the low redshift universe. In this paper, we describe the dependence of the obtained cross-correlation signal on the geometry and other properties of a survey of the low redshift universe. We show that an all-sky survey with about 10 million galaxies, almost uniformly distributed within 0<z<1 should yield a near optimal ISW detection, at ~ 5σlevel. In order to achieve this level of signal-to-noise, the systematic anisotropies in the survey must be below ~ 0.1 %, on the scale of ~ 10 degrees on the sky, while the systematic error in redshift estimates must be less than 0.05. Then, we argue that, while an ISW detection will not be a good way of constraining the conventional properties of dark energy, it could be a valuable means of testing alternative theories of gravity on large physical scales.

Figures (7)

• Figure 1: The expected cross-power spectrum of the ISW effect, for an all-sky survey with $b^2_g dN/dz =10^7$, and $z_{\rm max} =1$
• Figure 2: Figures show the expected $(S/N)^2$ distribution for a redshift or resolution limited full-sky ISW cross-correlation signal ($\ell_{\rm max}$ refers to the scale at which the (white) detector noise is equal to the true signal). In either figures, the enclosed area for the region covered by a survey, multiplied by its sky coverage, gives the optimum $(S/N)^2$ for the cross-correlation signal. The spikiness of the distribution in $\ell$-space is due to the use of the actual observed WMAP power spectrum in Eq. (\ref{['dsn21']}). Note that for partial sky coverage, the low-$\ell$ multipoles that are not covered by the survey should also be excluded from the area under the curves.
• Figure 3: The distribution of $(S/N)^2$ for different galaxy redshift distributions. For partial sky coverage, the result should be multiplied by $f_{\rm sky}$.
• Figure 4: The integrated (total) $(S/N)^2$ for a constant $dN/dz$ up to $z_{\rm max}$, corresponding to the area under the curves in Fig. \ref{['figdsnp']}.
• Figure 5: The graph shows the maximum allowed amplitude of anisotropies, $\Delta^{sys}_g$ introduced by possible systematics of the galaxy survey. The solid line shows the level of systematics that reduces the $(S/N)^2$ for an ISW detection by 50%. The dotted line shows the level of correlated contamination that introduces a systematic error in an ISW detection comparable to the cosmic variance. Here, we assume that the correlated amplitude of CMB contamination is $\sim 1 \mu K$, consistent with the expected leftover Galactic contamination in WMAP foreground-cleaned W-band map Bennett:2003ca.