Calibrating cluster number counts with CMB lensing

CMB Stage-4 experiments aim to reduce the uncertainties on the gravitational lensing potential by an order of magnitude compared to current measurements, and to also produce a Sunyaev-Zel’dovich (SZ) cluster catalog containing $\sim 10^5$ objects, two orders of magnitudes higher than what is currently available. In this paper we propose to combine these two observables and show that it is possible to calibrate the masses of the full Stage-4 cluster catalog internally owing to the high signal to noise measurement of the cosmic microwave background (CMB) lensing convergence field. We find that a CMB Stage-4 experiment will constrain the hydrostatic bias parameter to sub-percent accuracy. We also show constraints on a nonparametric $Y-M$ relationship which could be used to study its evolution with mass and redshift. Finally we present a joint likelihood for thermal SZ (tSZ) flux and mass measurements, and show that it could lead to a $\sim 5\sigma$ detection of the lower limit on the sum of the neutrino masses in the normal hierarchy ($\sum m_{\nu }=60\mathrm{meV}$) once combined with measurements of the primordial CMB and CMB lensing power spectra assuming a cosmic-variance-limited measurement of the optical depth to reionization.

Figure 1

Noise for the different estimators of the lensing field. We show the two minimum variance combinations highlighted in the text, containing all two-point combinations (solid green) and polarization-only combinations (solid red). The CMB lensing convergence power spectrum is shown as a black solid line. The noise curves correspond to the survey specifications in Table 1.

Figure 2

Mass and redshift distribution of tSZ detected clusters for a CMB S4 experiment, the contour lines correspond to the signal to noise on the mass measurement of each individual cluster using CMB lensing estimated from polarization data.

Figure 3

Constraints on the parameters of the $Y-M$ scaling relationship for CMB S4 experiment. The 1 and $2\sigma$ contours are displayed. In green we show the result for the full reconstruction of the convergence field and in red the result based on polarization data only.

Figure 4

Nonparametric reconstruction of the $\mathcal{Y}-M$ relationship using estimate of the convergence field from the polarized CMB. The left panel shows the error bars on the scaling relation, the right panel shows the S/N for each of the 44 mass bins. Note that the S/N will depend on the exact choice of model that is assumed for the relationship. Results are shown in green for the minimum variance lensing reconstruction and in red for reconstruction using only polarization-based estimators.

Figure 5

$S/N$ on the reconstruction of the $\mathcal{Y}-M$ in seven bins of redshift. A S4 CMB experiment will be able to measure the redshift evolution of this scaling relation to good accuracy in a wide range of masses. The number of clusters found in each redshift bin is indicated in the legend. The masses are estimated using only polarization-based estimators.

Figure 6

Uncertainties on the cluster nuisance parameters marginalized over the cosmological parameters for an SZ survey carried out with S4 without lensing information (black ellipses), with lensing masses reconstructed using only polarization information (red ellipses) and using also temperature (green ellipses). In all cases we include prior information on the cosmological parameters from Planck.

Figure 7

Uncertainties ${\mathrm{\Omega }}_M$, $A_s$ and the sum of neutrino masses from an SZ catalog carried out with S4 in combination with constraints from S4 primary and lensing power spectra, as well as Planck temperature and polarization on $\ell <30$. Results are shown in the absence of lensing mass estimates (black ellipses), and for lensing masses computed using only polarization (red ellipses) and temperature and polarization (green ellipses). The results are marginalized over all other cosmological parameters as well as the cluster nuisance parameters.

Figure 8

Posterior distribution for the sum of neutrino masses for different experiment configurations: S4 primary and lensing power spectra (gray curve), S4 power spectra and Planck primary on $\ell <30$ (orange curve), S4 and Planck power spectra together with SZ number counts from S4 in the absence of lensing masses (green curve) and for lensing masses measured using only polarization information (blue curve).