The VIMOS Public Extragalactic Redshift Survey (VIPERS): Hierarchical scaling and biasing

Aims. Building on the two-point correlation function analyses of the VIMOS Public Extragalactic Redshift Survey (VIPERS), we investigate the higher-order correlation properties of the same galaxy samples to test the hierarchical scaling hypothesis at z ∼ 1 and the dependence on galaxy luminosity, stellar mass, and redshift. With this work we also aim to assess possible deviations from the linearity of galaxy bias independently from a previously performed analysis of our survey. Methods. We have measured the count probability distribution function in spherical cells of varying radii (3 ≤ R ≤ 10h ^-1 Mpc), deriving σ <inf>8g</inf> (the galaxy rms at 8 h ^-1 Mpc), the volume-averaged two-, three-, and four-point correlation functions and the normalized skewness S<inf>3g</inf> and kurtosis S<inf>4g</inf> for different volume-limited subsamples, covering the following ranges: -19.5 ≤ MB(z = 1.1) - 5log(h) ≤ -21.0 in absolute magnitude, 9.0 ≤ log(M<inf>∗</inf>/M<inf>⊙</inf>h^-2) ≤ 11.0 in stellar mass, and 0.5 ≤ z ≤ 1.1 in redshift. Results. We have performed the first measurement of high-order correlation functions at z ∼ 1 in a spectroscopic redshift survey. Our main results are the following. 1) The hierarchical scaling between the volume-averaged two- and three-point and two- and four-point correlation functions holds throughout the whole range of scale and redshift we could test. 2) We do not find a significant dependence of S <inf>3g</inf> on luminosity (below z = 0.9 the value of S<inf>3g</inf> decreases with luminosity, but only at 1σ-level). 3) We do not detect a significant dependence of S<inf>3g</inf> and S<inf>4g</inf> on scale, except beyond z ∼ 0.9, where S<inf>3g</inf> and S4<inf>g</inf> have higher values on large scales (R 10 h ^-1 Mpc): this increase is mainly due to one of the two CFHTLS Wide Fields observed by VIPERS and can be explained as a consequence of sample variance, consistently with our analysis of mock catalogs. 4) We do not detect a significant evolution of S <inf>3g</inf> and S4<inf>g</inf> with redshift (apart from the increase of their values with scale in the last redshift bin). 5) σ <inf>8g</inf> increases with luminosity, but does not show significant evolution with redshift. As a consequence, the linear bias factor b = σ <inf>8g</inf>/σ <inf>8m</inf>, where σ<inf>8m</inf> is the rms of matter at a scale of 8 h ^-1 Mpc, increases with redshift, in agreement with the independent analysis of VIPERS and of other surveys such as the VIMOS-VLT Deep Survey (VVDS). We measure the lowest bias b = 1.47 ± 0.18 for galaxies with M<inf>B</inf>(z = 1.1) - 5 log(h) ≤ -19.5 in the first redshift bin (0.5 ≤ z ≤ 0.7) and the highest bias b = 2.12 ± 0.28 for galaxies with MB(z = 1.1) - 5 log(h) ≤ -21.0 in the last redshift bin (0.9 ≤ z ≤ 1.1).6) We quantify deviations from the linear bias by means of the Taylor expansion parameter b<inf>2</inf>. We obtain b<inf>2</inf> = -0.20 ± 0.49 for 0.5 ≤ z 0.7 and b<inf>2</inf> = -0.24 ± 0.35 for 0.7 ≤ z ≤ 0.9, while for the redshift range 0.9 ≤ z < 1.1 we find b<inf>2</inf> = +0.78 ± 0.82. These results are compatible with a null non-linear bias term, but taking into account another analysis for VIPERS and the analysis of other surveys, we argue that there is evidence for a small but non-zero non-linear bias term.