Integrated approach to cosmology: Combining CMB, large-scale structure, and weak lensing

Recent observational progress has led to the establishment of the standard $\mathrm{\Lambda }\mathrm{CDM}$ model for cosmology. This development is based on different cosmological probes that are usually combined through their likelihoods at the latest stage in the analysis. We implement here an integrated scheme for cosmological probes, which are combined in a common framework starting at the map level. This treatment is necessary as the probes are generally derived from overlapping maps and are thus not independent. It also allows for a thorough test of the cosmological model and of systematics through the consistency of different physical tracers. As a first application, we combine current measurements of the cosmic microwave background (CMB) from the Planck satellite, and galaxy clustering and weak lensing from SDSS. We consider the spherical harmonic power spectra of these probes including all six auto- and cross-correlations along with the associated full Gaussian covariance matrix. This provides an integrated treatment of different analyses usually performed separately including CMB anisotropies, cosmic shear, galaxy clustering, galaxy-galaxy lensing and the integrated Sachs-Wolfe effect with galaxy and shear tracers. We derive constraints on $\mathrm{\Lambda }\mathrm{CDM}$ parameters that are compatible with existing constraints and highlight tensions between data sets, which become apparent in this integrated treatment. We discuss how this approach provides a complete and powerful integrated framework for probe combination and how it can be extended to include other tracers in the context of current and future wide-field cosmological surveys.

Figure 1

Synopsis of the framework for integrated probe combination employed in this work.

Figure 2

Summary of the three maps in Galactic coordinates used in this analysis. The all-sky maps are in Mollweide projection while the zoom-in versions are in Gnomonic projection. The HMHS map of CMB temperature anisotropies as derived using Commander is shown in the top panel. It is masked using the $UT78$ mask. The middle panel shows the systematics-corrected (see text) galaxy overdensity map for CMASS1-4 galaxies. Grey areas have been masked either because they lie outside the survey footprint or are potentially contaminated by systematics. The lower panel shows the map of the SDSS Stripe 82 shear modulus $|\widehat{\gamma }|$. Grey areas have been masked because they are either unobserved or do not contain galaxies for shear measurement. The zoom-in figures (left) are enlarged versions of the $5\times 5{\mathrm{deg}}^2$ region centred on $(\mathtt{l},\mathtt{b})=(53°,-33.5°)$ shown in the maps. The zoom-in for the galaxy shear map is overlaid with a whisker plot of the galaxy shears. All three maps have resolution $\mathtt{NSIDE}=1024$.

Figure 3

Redshift distribution for the LSS probes. The figure shows the redshift selection function of SDSS CMASS1-4 galaxies, the redshift selection function for the SDSS Stripe 82 galaxies as well as the weak lensing shear window function defined in Eq. (3). The redshift selection function for CMASS1-4 galaxies as well as the weak lensing shear window function have been rescaled relative to the Stripe 82 redshift selection function.

Figure 4

Spherical harmonic power spectra for all probes used in the cosmological analysis. The top left panel shows the power spectrum of CMB anisotropies computed from the Commander CMB temperature map at resolution of $\mathtt{NSIDE}=2048$. The middle left panel shows the cross-power spectrum between CMB temperature anisotropies and galaxy overdensity computed from the systematics-reduced SDSS CMASS1-4 map and the Commander map at resolution $\mathtt{NSIDE}=1024$. The middle right panel shows the spherical harmonic power spectrum of the galaxy overdensity computed from the systematics-reduced SDSS CMASS1-4 map at $\mathtt{NSIDE}=1024$. The bottom left panel shows the spherical harmonic power spectrum between CMB temperature anisotropies and weak lensing shear measured from the Commander CMB map and the SDSS Stripe 82 weak lensing maps at resolution $\mathtt{NSIDE}=1024$. The bottom-middle panel shows the spherical harmonic power spectrum between galaxy overdensity and galaxy weak lensing shear computed from the systematics-reduced SDSS CMASS1-4 map and the SDSS Stripe 82 galaxy weak lensing shear map at resolution $\mathtt{NSIDE}=1024$. The bottom right panel shows the spherical harmonic power spectrum of cosmic shear E modes computed from the SDSS Stripe 82 weak lensing shear maps. The angular multipole ranges and binning schemes for all power spectra are summarized in Table 2. All power spectra are derived from the maps in Galactic coordinates. The solid lines show the theoretical predictions for the best-fit cosmological model determined from the joint analysis which is summarized in Table 3. The theoretical predictions have been convolved with the PolSpice kernels as described in Sec. 5. The error bars are derived from the Gaussian simulations described in Sec. 6b and Appendix pp6.

Figure 5

Correlation matrix for the spherical harmonic power spectra derived from the sample variance of the Gaussian simulations. The binning scheme and angular multipole range for each probe follow those outlined in Table 2.

Figure 6

Cosmological parameter constraints derived from the joint analysis, marginalized over ${\tau }_{\text{reion}}$, $b$ and $m$ and from the single probes. The single probe constraints have been marginalized over the respective nuisance parameters i.e. ${\tau }_{\text{reion}}$ for the CMB temperature anisotropies, $b$ for galaxy clustering and $m$ for the weak lensing shear. In each case the inner (outer) contour shows the 68% C.L. (95% C.L.). For clarity the cosmic shear 68% C.L.s are solid while the 95% C.L.s are dashed.

Figure 7

Comparison between the parameter constraints derived from the joint analysis, marginalized over $b$ and $m$ and the constraints from Ref. [69] using only CMB data ($\mathrm{TT}+\mathrm{lowP}$) or adding external data (TT, TE, $\mathrm{EE}+\mathrm{lowP}+\text{lensing}+\mathrm{ext}$). The Planck constraints are marginalized over all nuisance parameters. In each case the inner (outer) contour shows the 68% C.L. (95% C.L.).

Figure 8

Galaxy number density $n_{\mathrm{gal}}$ relative to mean galaxy density ${\overline{n}}_{\mathrm{gal}}$ as a function of potential systematic value. The figures show both the uncorrected data and the data corrected using a third-order polynomial fit to the uncorrected relation. The error bars assume Poisson noise and are thus likely underestimated due to the correlations between galaxy positions.

Figure 9

Residuals between the ellipticity component $e_1$ of a random sample of 10000 bright stars measured on the co-add images and PSF models for these objects as a function of declination (DEC) before (left panel) and after (right panel) applying the correction described in the text.

Figure 10

Histograms of the ellipticity components $e_1$ and $e_2$ averaged over HEALPix pixels of resolution $\mathtt{NSIDE}=512$. The left panels show the distributions before correction for the PSF and subtraction of a camcol-dependent additive bias, while the right panels show the distributions after the application of these corrections.

Figure 11

Real space correlation function of the SDSS Stripe 82 mask. The dashed line denotes the value chosen for ${\theta }_{\max }$.

Figure 12

Comparison between input power spectra and mean recovered power spectra as estimated from $N_{\mathrm{sim}}=1000$ Gaussian realizations generated using the algorithm outlined in Appendix pp6. The noise level of the Gaussian realizations is tuned to the data and the spherical harmonic power spectra have been computed using the same methodology and PolSpice settings as applied on the data. The angular multipole ranges and binning schemes for all power spectra are summarized in Table 2. Dashed lines denote negative spherical harmonic power spectrum values.

Figure 13

The cosmic shear B-mode power spectrum reconstructed from $N_{\mathrm{sim}}=1000$ Gaussian realizations generated using the algorithm outlined in Appendix pp6. The angular multipole range and binning scheme is summarized in Table 2.

Figure 14

Comparison between spherical harmonic power spectra computed from the maps in Galactic and equatorial coordinates.

Figure 15

Comparison between cosmic shear spherical harmonic power spectra prior to noise removal measured from the maps in Galactic and equatorial coordinates. The left-hand panel shows the results when applying the correction for additive bias in equatorial coordinates and then rotating the shears to Galactic coordinates. The right-hand panel shows the results when no additive bias correction is applied.

Figure 16

Comparison between spherical harmonic power spectra $C_{\ell }^{\mathrm{TT}}$, $C_{\ell }^{{\delta }_g\mathrm{T}}$, $C_{\ell }^{\gamma \mathrm{T}}$ derived using the four different foreground-reduced CMB maps from Commander, NILC, SEVEM and SMICA.

Figure 17

Comparison between galaxy overdensity power spectra computed before and after systematics removal.

Figure 18

HMHD Commander CMB temperature anisotropy map. This map contains only noise and potential residual systematics.

Figure 19

Spherical harmonic power spectrum of cosmic shear B modes computed from the SDSS Stripe 82 maps in Galactic coordinates. The angular multipole range and binning scheme is summarized in Table 2.