Integrated Sachs-Wolfe map reconstruction in the presence of systematic errors

The decay of gravitational potentials in the presence of dark energy leads to an additional, late-time contribution to anisotropies in the cosmic microwave background (CMB) at large angular scales. The imprint of this so-called integrated Sachs-Wolfe (ISW) effect to the CMB angular power spectrum has been detected and studied in detail, but reconstructing its spatial contributions to the CMB map, which would offer the tantalizing possibility of separating the early- from the late-time contributions to CMB temperature fluctuations, is more challenging. Here, we study the technique for reconstructing the ISW map based on information from galaxy surveys and focus in particular on how its accuracy is impacted by the presence of photometric calibration errors in input galaxy maps, which were previously found to be a dominant contaminant for ISW signal estimation. We find that both including tomographic information from a single survey and using data from multiple, complementary galaxy surveys improve the reconstruction by mitigating the impact of spurious power contributions from calibration errors. A high-fidelity reconstruction further requires one to account for the contribution of calibration errors to the observed galaxy power spectrum in the model used to construct the ISW estimator. We find that if the photometric calibration errors in galaxy surveys can be independently controlled at the level required to obtain unbiased dark energy constraints, then it is possible to reconstruct ISW maps with excellent accuracy using a combination of maps from two galaxy surveys with properties similar to Euclid and SPHEREx.

Figure 1

Effects of calibration errors on galaxy power. The solid curve shows the theoretical angular power spectrum for the NVSS survey [17, 18]. The colored dashed curves show the theoretical spectrum with two representative levels of calibration error. The shaded region is the $1\sigma$ uncertainty from the survey’s sample variance, and the dotted curves indicate the ideal, all-sky cosmic variance.

Figure 2

Same as Fig. 1, but for the SDSS MphG catalog, following Refs. [17, 18]. In this case, the sample variance is driven by sky coverage ($f_{\text{sky}}=0.22$) as opposed to number density as for NVSS.

Figure 3

Quality of map reconstruction $\rho$ vs the calibration error variance ${\sigma }_{\mathrm{cal}}^2$ for our fiducial, Euclid-like survey. The colors of the lines indicate how tomographic information is handled, showing that splitting the survey into six redshift bins (red) improves the reconstruction compared to the single-bin case (blue). Solid curves indicate cases when the calibration error is included in the ISW estimator [$R_{\ell }(C_{\ell }^{\text{sky}})$], while the dashed curves show the reconstructions in which the effects of the calibration errors are not included [$R_{\ell }(C_{\ell }^{\mathrm{th}})$] (see Sec. 2b for details). Points (offset horizontally for clarity) show the mean ($\overline{\rho }$) of 2,000 realizations, with error bars indicating the 68% spread across realizations. The corresponding smooth curves are $\widehat{\rho }$, the analytical estimate of $\overline{\rho }$ from Eq. (17). The inset illustrates the redshift distribution across bins overlaid with the ISW kernel in gray (reproduced from Ref. [29]). The vertical, shaded regions show the approximate current and projected levels of control over residual calibration errors. Calibration errors between redshift bins are modeled as uncorrelated.

Figure 4

Reconstruction quality when using binned Euclid-like survey and CMB intensity data separately and in combination. The purple curve and shaded band show the mean and 68% spread of $\rho$ from simulations. As in Fig. 3, red curves are results when using the binned Euclid-like survey alone, whereas blue curves are the result of using both the fiducial survey and CMB intensity map. Like in Fig. 3, solid curves are for the case where calibration error power is correctly modeled in the estimator, and dashed curves are for when they are not modeled at all. Neglecting the presence of calibration errors in a LSS map can actually degrade the quality of the ISW reconstruction compared to using the CMB temperature alone.

Figure 5

Comparison of ISW reconstruction quality using the LSS surveys and CMB temperature individually and in combination, for various levels of calibration error in the Euclid-like and SPHEREx-like surveys. Colors are the same as those of Fig. 4. Both of the LSS surveys are split into six redshift bins (see Sec. 3a), with calibration errors uncorrelated between bins and surveys. The dashed curve shows the combined reconstruction if calibration errors are not included in the estimator. Using LSS surveys to improve the ISW map reconstruction from the CMB temperature-only case requires calibration errors to be controlled to ${\sigma }_{\mathrm{cal}}^2\lesssim {10}^{-4}$.

Figure 6

Left: Effect of cross-correlation between calibration errors in different bins of the fiducial Euclid-like survey, given by $C_{\ell }^{\mathrm{cal},ij}=r_{\mathrm{cc}}\sqrt{C_{\ell }^{\mathrm{cal},ii}{C}_{\ell }^{\mathrm{cal},jj}}$, for bins $i\ne j$. Solid curves have calibration errors accounted for in the estimator [$R_{\ell }(C_{\ell }^{\text{sky}})$]. The dashed curve indicates the case where only the autopower contributions of the calibration errors are accounted for in the estimator [$R_{\ell }(C_{\ell }^{\mathrm{XY},\mathrm{th}})$], and the dotted curve indicates the case where calibration errors are not accounted for at all [$R_{\ell }(C_{\ell }^{\mathrm{th}})$]. Cross-correlation of the errors results in mild degradation of the reconstruction for ${\sigma }_{\mathrm{cal}}^2\lesssim {10}^{-4}$ but otherwise has little effect as long as the autocorrelation is correctly modeled in the estimator. Right: Dependence of $\rho$ on the shape of $C_{\ell }^{\text{cal}}$. Solid curves indicate $C_{\ell }^{\text{cal}}$ of the same form as Eq. (6) but with width $w^{\mathrm{cal}}$ varied. The dashed curve indicates the case where the error spectrum takes the form $C_{\ell }^{\text{cal}}\propto (\ell +1{)}^{-2}$. Reconstruction fares worse when calibration error power contributions are more concentrated at the largest angles, where the ISW kernel is largest. In all cases, the estimator uses the true observed LSS power spectrum ($C_{\ell }^{\text{sky}}$).

Figure 7

Contribution to squared signal-to-noise ratio per multipole for our fiducial Euclid-like survey with varying levels of calibration error. Total combined $S/N$ for each level of calibration error ${\sigma }_{\mathrm{cal}}^2$ is given in the legend.