Cluster-void degeneracy breaking: Modified gravity in the balance

Combining galaxy cluster and void abundances is a novel, powerful way to constrain deviations from general relativity and the $\mathrm{\Lambda }\mathrm{CDM}$ model. For a flat $w\mathrm{CDM}$ model with growth of large-scale structure parametrized by the redshift-dependent growth index $\gamma (z)={\gamma }_0+{\gamma }_{1}z/(1+z)$ of linear matter perturbations, combining void and cluster abundances in future surveys with Euclid and the four-meter multiobject spectroscopic telescope could improve the figure of merit for ($w,{\gamma }_0,{\gamma }_1$) by a factor of 20 compared to individual abundances. In an ideal case, improvement on current cosmological data is a figure of merit factor 600 or more.

Figure 1

Forecast marginalized PDFs for the growth index $\gamma (z)$ from cluster and void abundances in Euclid and 4MOST surveys, for a flat $w\mathrm{CDM}$ model with separate growth index $\gamma (z)$. Two different parametrizations are shown, with the two-parameter case $({\gamma }_0,{\gamma }_1)$ displaying 68% confidence contours. A PDF representative of current data precision [23, 24] is included as a dashed line. The fiducial value of ${\gamma }_0=0.545$, ${\gamma }_1=0$ for $\mathrm{GR}+\mathrm{\Lambda }\mathrm{CDM}$ is also indicated.

Figure 2

Forecast 68% parameter contours, and marginal PDFs, from cluster and void abundances in future Euclid and 4MOST surveys, for a flat $w\mathrm{CDM}$ model with separate growth index $\gamma (z)={\gamma }_0+{\gamma }_{1}z/(1+z)$.

Figure 3

Void distribution parameter sensitivity in the redshift-dependent growth index model. We assume the same fiducial cosmological and galaxy bias models as elsewhere, but consider a generic survey of deep voids with $R_{\lim }=10h^{-1}\mathrm{Mpc}$ and $z=0.05–2.05,\mathrm{\Delta }{\log }_{10}(R/h^{-1}\mathrm{Mpc})=0.1,\mathrm{\Delta }z=0.2$. For each parameter, the figure shows $\mathrm{\Delta }{\chi }_{i,j}^{\mathrm{rel}}$ when that parameter only is varied (hence, ${\sigma }_8$ is kept normalized to the fiducial value at $z=0$ when other parameters are varied). The indicated turnover radius $R_{\mathrm{to}}(z)$ (black, dotted lines) is clearly visible in the sensitivity to power-spectrum and growth parameters. Scales related to the cosmological parameters are in brown, dotted lines ($k_{\text{ql}}=0.13h{\text{Mpc}}^{-1}$, $k_{\text{BAO}}=0.06h{\text{Mpc}}^{-1}$, $k_{n_s}=0.05h{\text{Mpc}}^{-1}$, $k_{\text{eq}}=0.012h{\text{Mpc}}^{-1}$). Note the distinct baryon acoustic oscillation signature in the ${\mathrm{\Omega }}_{\mathrm{b}}$ panels. The radius-redshift coverage of the surveys is also shown (4MOST, yellow solid lines; Euclid, red dashed lines). Some $\theta$’s are defined with negative sign to aid comparison. Note also that $\mathrm{\Delta }{\chi }_{i,j}^{\mathrm{rel}}$ does not depend on $f_{\mathrm{sky}}$. The normalization $\mathrm{\Delta }\theta /\sqrt{\mathrm{\Delta }{\chi }^2/f_{\mathrm{sky}}}$ is also given for each parameter. Quantities are described further in the text.