Fundamental physics from future weak-lensing calibrated Sunyaev-Zel’dovich galaxy cluster counts

Future high-resolution measurements of the cosmic microwave background (CMB) will produce catalogs of tens of thousands of galaxy clusters through the thermal Sunyaev-Zel’dovich (tSZ) effect. We forecast how well different configurations of a CMB Stage-4 experiment can constrain cosmological parameters, in particular, the amplitude of structure as a function of redshift ${\sigma }_8(z)$, the sum of neutrino masses $\mathrm{\Sigma }{m}_{\nu }$, and the dark energy equation of state $w(z)$. A key element of this effort is calibrating the tSZ scaling relation by measuring the lensing signal around clusters. We examine how the mass calibration from future optical surveys like the Large Synoptic Survey Telescope (LSST) compares with a purely internal calibration using lensing of the CMB itself. We find that, due to its high-redshift leverage, internal calibration gives constraints on cosmological parameters comparable to the optical calibration, and can be used as a cross-check of systematics in the optical measurement. We also show that in contrast to the constraints using the CMB lensing power spectrum, lensing-calibrated tSZ cluster counts can detect a minimal $\mathrm{\Sigma }{m}_{\nu }$ at the $3–5\sigma$ level even when the dark energy equation of state is freed up.

Figure 1

Top: The number of clusters that can be detected through tSZ emission by CMB Stage-4 in each redshift bin of width $\mathrm{\Delta }z=0.1$. The different colors correspond to different beam FWHMs in the 150 GHz channel. Bottom: The ratio of the number of clusters at various resolutions to that for a telescope with $3^{\prime }$ resolution at 150 GHz.

Figure 2

The number of cluster detections as a function of the ${\ell }_{\mathrm{knee}}$ atmospheric noise parameter for each resolution considered and for three different values of the $\alpha$ atmospheric noise parameter. The solid lines correspond to $\alpha =-4.5$, dashed to $\alpha =-4$ and dot-dashed to $\alpha =-5$. The vertical dashed line corresponds to the ${\ell }_{\mathrm{knee}}$ used in the rest of the analysis.

Figure 3

The dependence of signal-to-noise ratio of the lensing signal per cluster as a function of redshift for an LSST-like optical survey (dashed lines) and internal CMB lensing calibration using both temperature and polarization (solid lines) with a telescope that has beam FWHM of $1^{\prime }$ at 150 GHz. The different colors correspond to specific $M_{500{\rho }_c}$ masses of clusters. The increase in S/N at low redshift is partly due to clusters becoming more concentrated at lower redshifts, since we assume a $c_{500}$ dictated by the relation in Duffy et al. [45]. For clusters with $M_{500{\rho }_c}\gtrsim 5\times {10}^{14}{M}_{\odot }$ CMB lensing offers higher S/N per cluster at some redshift $z>1$ and for all clusters above $z=2$ where the ability to measure shapes of background galaxies degrades quickly.

Figure 4

The mass sensitivity (uncertainty on $M_{500}$ over $M_{500}$) for $N=1000$ clusters with $M_{500}=2\times {10}^{14}{M}_{\odot }/h$, $z=0.7$, $c_{500}=1.18$ as a function of the beam FWHM at 150 GHz when masses are estimated using a matched filter on lensing reconstructions from CMB Stage-4. The solid curves use both CMB temperature and polarization data while the dashed curves only use polarization data. The gray curve assumes that there is no noise from astrophysical foregrounds in the temperature maps. Polarization maps are always assumed to be foreground free. The blue curves incorporate miscentering by convolving the convergence profile with a beam-dependent Rayleigh distribution.

Figure 5

The uncertainty on ${\sigma }_8(z)$ as a function of redshift for tSZ clusters from a CMB Stage-4 experiment calibrated internally using CMB halo lensing (including temperature and polarization data). The dark blocks have redshift bin edges of [0., 0.5, 1.0, 1.5, 2.0, 3.0] chosen to roughly produce the same relative constraint in each bin, while the light blocks illustrate the ${\sigma }_8(z)$ constraints for bins of width $\mathrm{\Delta }z=0.1$. The different colors correspond to different beam FWHMs at 150 GHz. With a 1 arcminute beam at 150 GHz it is possible to constrain ${\sigma }_8(z)$ at $\sim 1\%$ in the redshift bins chosen above.

Figure 6

Resolution dependence of CMB Stage-4 parameter constraints for internal CMB lensing calibration with temperature and polarization data for a $\mathrm{\Lambda }\mathrm{CDM}+m_{\nu }+w_0+w_a$ cosmology. The contours correspond to 68% C.L. For clarity, not all parameters varied are shown: $\{{\mathrm{\Omega }}_c{h}^2,{\mathrm{\Omega }}_b{h}^2,A_s,n_s\}$ are excluded.

Figure 7

Constraint in the $m_{\nu }-w_0$ plane for a CMB Stage-4 telescope with $1^{\prime }$ beam FWHM at 150 GHz shown as 68% C.L. Left: For a $\mathrm{\Lambda }\mathrm{CDM}+m_{\nu }+w_0$ cosmology with $w_a=0$ held fixed. Right: For a $\mathrm{\Lambda }\mathrm{CDM}+m_{\nu }+w_0+w_a$ cosmology. The blue contours are for mass calibration using CMB lensing (temperature and polarization in solid and polarization only in dashed). The orange contours use mass calibration from an LSST-like optical survey. The blue dashed contour uses CMB polarization only, and the dashed orange contour only uses $z<1$ source galaxies.

Figure 8

The $1-\sigma$ uncertainty on neutrino mass obtained when marginalizing over $\mathrm{\Lambda }\mathrm{CDM}$, $\mathrm{\Lambda }\mathrm{CDM}+w_0$ and $\mathrm{\Lambda }\mathrm{CDM}+w_0+w_a$, from tSZ clusters detected using a CMB Stage-4 telescope with $1^{\prime }$ beam FWHM at 150 GHz. Unlike in the case where only the CMB lensing auto spectrum is used see [63], the constraints do not degrade significantly when freeing up dark energy equation of state parameters owing to the redshift resolution of growth of structure with tSZ clusters. Left: Constraints when the mass calibration is from internal CMB lensing reconstruction with temperature and polarization data. Middle: Constraints when the mass calibration is from an LSST-like optical survey using clusters up to $z=2$. Right: Constraints when the mass calibration is a combination of internal CMB and optical weak lensing. Note that “DESI” corresponds to adding BAO measurements from the DESI survey. We show how constraints improve when the prior on the optical depth $\tau$ is tightened from the fiducial width 0.01 to the Planck Blue Book value [67] of 0.006 and further to the cosmic-variance limit value of 0.002. The grey solid line shows the value of the minimal neutrino mass in the normal hierarchy of 58 meV, and the dashed and dot-dashed lines show levels required for a $3\sigma$ and $5\sigma$ detection, respectively.

Figure 9

The constraint on the dark energy equation of state and sum of neutrino masses as we increase prior information on each parameter for CMB Stage-4 (1 arcminute at 150 GHz) with internal CMB halo lensing ($\mathrm{T}+\mathrm{P}$) calibration. With each step in the forward-$x$ direction, we decrease the 68% C.L. width of the prior on the respective parameter on a logarithmic scale until it saturates the constraint (determined as the point when the change in constraint is less than 0.1%). Once this saturation is seen, we retain the final value of the prior for that parameter and proceed to repeat the same procedure on the next parameter of interest (represented by a different color). This makes clear which parameter degeneracies are primarily limiting the constraint. The dashed curve includes DESI BAO information.