Statistical Analysis of Galaxy Surveys-II. The 3-point galaxy correlation function measured from the 2dFGRS

TL;DR

Using the final 2dFGRS, the study measures the reduced three-point function $Q_3(r_1,r_2,r_3)=\zeta/\xi^2$ across triangle shapes and scales to test gravitational instability and quantify galaxy bias. By comparing galaxy $Q_3$ to dark-matter predictions and modeling bias with $Q_3 \simeq \frac{1}{B}[Q_3^{\rm DM}+C]$, the authors extract linear and quadratic bias parameters, finding $b_1\simeq0.93$ and $c_2\simeq-0.34$ on weakly nonlinear scales, with a strong rejection of unbiased tracers. They observe scale-, colour-, and luminosity-dependent deviations in the nonlinear regime and estimate $\sigma_8\simeq0.88$, confirming LCDM gravity on large scales while highlighting nontrivial galaxy-formation effects and the influence of large structures. The analysis uses a grid-based $Q_3$ estimator, full covariance from mock catalogs, and singular-value decomposition to constrain bias in a two-parameter space, providing robust insights into galaxy bias and the growth of structure. The results reconcile prior S3 measurements with a nonzero quadratic bias and establish a framework for future higher-order clustering tests with larger surveys.

Abstract

We present new results for the 3-point correlation function, ζ, measured as a function of scale, luminosity and colour from the final version of the two-degree field galaxy redshift survey (2dFGRS). The reduced three point correlation function, Q_3 is estimated for different triangle shapes and sizes, employing a full covariance analysis. The form of Q_3 is consistent with the expectations for the Λ-cold dark matter model, confirming that the primary influence shaping the distribution of galaxies is gravitational instability acting on Gaussian primordial fluctuations. However, we find a clear offset in amplitude between Q_3 for galaxies and the predictions for the dark matter. We are able to rule out the scenario in which galaxies are unbiased tracers of the mass at the 9-sigma level. On weakly non-linear scales, we can interpret our results in terms of galaxy bias parameters. We find a linear bias term that is consistent with unity, b_1 = 0.93^{+0.10}_{-0.08} and a quadratic bias c_2 = b_2 /b_1 = -0.34^{+0.11}_{-0.08}. This is the first significant detection of a non-zero quadratic bias, indicating a small but important non-gravitational contribution to the three point function. Our estimate of the linear bias from the three point function is independent of the normalisation of underlying density fluctuations, so we can combine this with the measurement of the power spectrum of 2dFGRS galaxies to constrain the amplitude of matter fluctuations. We find that the rms linear theory variance in spheres of radius 8Mpc/h is σ_8 = 0.88^{+0.12}_{-0.10}, providing an independent confirmation of values derived from other techniques. On non-linear scales, where ξ>1, we find that Q_3 has a strong dependence on scale, colour and luminosity.

Figures (10)

• Figure 1: 3-points define a triangle, which is characterized here by the two sides $r_{12}$ and $r_{13}$ and the interior angle $\alpha$.
• Figure 2: $Q_3$ in the weakly nonlinear regime. The upper row shows measurements from the 2dFGRS $-21<$$\rm M_{\rm b_{\rm J}}^{\rm h}$$<-20$ volume limited sample. Different columns show the results for different triangle sizes, as indicated by the legend. The squares show the mean value of $Q_3$ as a function of $\alpha$ and the error bars are derived from the scatter in the mock catalogues with the same magnitude limits. The thick solid curves in the upper two rows show the predictions for $Q_3$ in the $\Lambda$CDM model. The thick dashed curves in these panels shows the effect of applying a transformation (Eq. \ref{['eq:Q3G']}) to this prediction corresponding to $B=1$ and $C=-0.3$. The thin solid lines in the middle row show the mean $Q_3$ measured in individual mocks. In the bottom row, we show the constraints on $B$ and $C$, derived from an eigenmode analysis. The four contours shown from outside in correspond to $\chi^2=11,8,6.17,2.3$ (ie 99.7%, 95.4% and 68.3% confidence interval for 2 parameters) and $\chi^2=1$ (ie 68.3% for one of the parameters). The cross point shows $B=1$, $C=0$ for reference.
• Figure 3: Same as Fig. \ref{['fig:q3_210_wnl']}, but for different volume limited samples: $-21<$$\rm M_{\rm b_{\rm J}}^{\rm h}$$<-20$ (left) and $-19<$$\rm M_{\rm b_{\rm J}}^{\rm h}$$<-18$ (right). In all panels, we consider the case where $r_{13}=2~r_{12}=16~{\,h^{-1}\,{\rm Mpc}}$.
• Figure 4: The combined constraints on $B$ and $C$, in the weakly non-linear regime, using different triangle configurations and the joint covariance of $Q_3(\alpha)$ for the 2dFGRS volume limited sample $-21<$$\rm M_{\rm b_{\rm J}}^{\rm h}$$<-20$. Each panel uses different triangle configurations (as indicated by each legend). For isosceles triangles with $r_{12}=r_{13}=9~{\,h^{-1}\,{\rm Mpc}}$, we fit only for $\alpha>60$ degrees, to ensure that only weakly non-linear scales (9 to $18~{\,h^{-1}\,{\rm Mpc}}$) are considered.
• Figure 5: $Q_3(\alpha)$ in the non-linear regime for two different luminosities (increasing from bottom to top) and different scales (decreasing from left to right), as indicated in the legend in each panel. The biased model (dashed lines) has $B=1$ and $C=-0.3$, whereas $Q_3$ for $\Lambda$CDM is plotted with a solid line.