Measuring the growth of matter fluctuations with third-order galaxy correlations

TL;DR

This study tests how to measure the growth of matter fluctuations by combining second- and third-order galaxy statistics, addressing the degeneracy between growth and galaxy bias. It compares two third-order approaches—the reduced three-point amplitude $Q$ and the skewness/zero-lag correlator combination $\tau = 3C_{12}-2S_3$—and introduces a bias-ratio method that yields $D(z)$ directly from observable galaxy clustering without modeling the dark-matter field. Using the MICE-GC simulation, the authors quantify biases in $b_1$ from $Q$ and $\tau$, showing $b_Q$ overestimates the linear bias by ~20–30% while $b_\tau$ tracks $b_ξ$ more closely but with larger errors, especially for high-mass halos. They demonstrate a new, model-independent route to measure $D(z)$ and $f(z)$ via bias ratios across redshifts, achieving growth measurements with competitive precision and highlighting practical considerations for applying these methods to real redshift surveys, including redshift-space distortions and shot-noise effects.

Abstract

Measurements of the linear growth factor $D$ at different redshifts $z$ are key to distinguish among cosmological models. One can estimate the derivative $dD(z)/d\ln(1+z)$ from redshift space measurements of the 3D anisotropic galaxy two-point correlation $ξ(z)$, but the degeneracy of its transverse (or projected) component with galaxy bias $b$, i.e. $ξ_{\perp}(z) \propto\ D^2(z) b^2(z)$, introduces large errors in the growth measurement. Here we present a comparison between two methods which break this degeneracy by combining second- and third-order statistics. One uses the shape of the reduced three-point correlation and the other a combination of third-order one- and two-point cumulants. These methods use the fact that, for Gaussian initial conditions and scales larger than $20$ $h^{-1}$Mpc, the reduced third-order matter correlations are independent of redshift (and therefore of the growth factor) while the third-order galaxy correlations depend on $b$. We use matter and halo catalogs from the MICE-GC simulation to test how well we can recover $b(z)$ and therefore $D(z)$ with these methods in 3D real space. We also present a new approach, which enables us to measure $D$ directly from the redshift evolution of second- and third-order galaxy correlations without the need of modelling matter correlations. For haloes with masses lower than $10^{14}$ $h^{-1}$M$_\odot$, we find $10%$ deviations between the different estimates of $D$, which are comparable to current observational errors. At higher masses we find larger differences that can probably be attributed to the breakdown of the bias model and non-Poissonian shot noise.

Figures (20)

• Figure 1: Top: Number of haloes in the four mass samples M0-M3 as a function of redshift in the two comoving outputs at z=0.0 and z=0.5 (symbols) and the seven redshift bins in the light cone (lines). Bottom: number density of the same halo mass samples as in the top panel.
• Figure 2: Top: two-point correlation $\xi$ of the MICE-GC dark matter field measured in the comoving outputs at redshift $z=0.0$, $0.5$, $1.0$ and $1.5$ (blue circles, green crosses, orange squares and red triangles respectively) as a function of scale $r_{12}$. Dotted Lines show a fit of the amplitude of $\xi$ at $z=0$ to those from other redshifts between $40-60$$h^{-1}$Mpc, via equation (\ref{['ximgrow']}). Bottom: growth factor $D=\sqrt{\xi(r_{12},z)/\xi(r_{12},0)}$ obtained from the ratio of the above correlations together with the fits displayed as dotted lines with the same colour code as the upper panel.
• Figure 3: Comparison between the linear growth of matter $D$ as a function of redshift $z$ measured in the MICE-GC comoving outputs (symbols) and the corresponding theoretical predictions from equation (\ref{['eq:growth_cosmology']}) (dashed line). The MICE-GC measurements are the best fit values obtained considering the scale range $40$-$60h^{-1}$Mpc, shown as lines with the same colour coding in Fig. \ref{['fig:D_from_2pc']}.
• Figure 4: Top: two-point correlation $\xi$ of the MICE-GC dark matter field (continuous lines) and the four halo mass samples M0-M3 (blue circles, green crosses, orange squares and red triangles respectively) in the comoving outputs at redshift $z=0.0$ (left) and $z=0.5$ (right) as a function of scale $r_{12}$. Bottom: linear bias parameter $b_{\xi}$ derived from the two-point correlations via equation (\ref{['bxi']}). Dotted lines are $\chi^2$-fits between $20-60$$h^{-1}$Mpc. The minimum $\chi^2$ values per degree of freedom are $1.05, 2.02, 0.37, 0.70$ for M0, M1, M2, M3 respectively at $z=0.0$ and $0.42, 0.78, 0.12, 0.82$ for M0, M1, M2, M3 respectively at $z=0.5$.
• Figure 5: Top: reduced three-point correlation $Q$ measured from the MICE-GC dark matter field in the comoving outputs at redshift $z=0.0,0.5,1.5$ (blue squares, green circles, red triangles respectively) for different triangle opening angles $\alpha$ using $r_{12}=r_{13}/2=12$$h^{-1}$Mpc (open symbols) and $r_{12}=r_{13}/2=24$$h^{-1}$Mpc (filled symbols) compared with predictions from second-order perturbation theory (PT) using a linear power spectrum. Bottom: Deviations between $Q$ from PT and measurements divided by the $1 \sigma$ errors of the measurements (dashed lines correspond to $\pm 2\sigma$ discrepancies).