The two-point correlation function of the galaxy distribution is a key cosmological observable that allows us to constrain the dynamical and geometrical state of our Universe. To measure the correlation function we need to know both the galaxy positions and the expected galaxy density field. The expected field is commonly specified using a Monte-Carlo sampling of the volume covered by the survey and, to minimize additional sampling errors, this random catalog has to be much larger than the data catalog. Correlation function estimators compare data-data pair counts to data-random and random-random pair counts, where random-random pairs usually dominate the computational cost. Future redshift surveys will deliver spectroscopic catalogs of tens of millions of galaxies. Given the large number of random objects required to guarantee sub-percent accuracy, it is of paramount importance to improve the efficiency of the algorithm without degrading its precision. We show both analytically and numerically that splitting the random catalog into a number of subcatalogs of the same size as the data catalog when calculating random-random pairs, and excluding pairs across different subcatalogs provides the optimal error at fixed computational cost. For a random catalog fifty times larger than the data catalog, this reduces the computation time by a factor of more than ten without affecting estimator variance or bias.
Aims. We perform clustering measurements of 800 X-ray selected Chandra COSMOS Legacy (CCL) Type 2 AGN with known spectroscopic redshift to probe the halo mass dependence on AGN host galaxy properties, such as galaxy stellar mass M star , star formation rate (SFR) and specific black hole accretion rate λ BHAR , in the redshift range z = [0 -3]. Methods. We split the sample of AGN with known spectroscopic redshits according to M star , SFR and λ BHAR , while matching the distributions in terms of the other parameters, including redshift. We measure the projected two-point correlation function w p (r p ) and model it with the 2-halo term to derive the large-scale bias b and the corresponding typical mass of the hosting halo, for the different subsamples.Results. We found no significant dependence of the large-scale bias and typical halo mass on galaxy stellar mass and specific BHAR for CCL Type 2 AGN at mean z∼1, while a negative dependence on SFR is observed, with lower SFR AGN residing in richer environment. Mock catalogs of AGN matched to have the same X-ray luminosity, stellar mass, λ BHAR and SFR of CCL Type 2 AGN, almost reproduce the observed M star − M h , λ BHAR − M h and SFR-M h relations, when assuming a fraction of satellite AGN f sat AGN ∼ 0.15, which corresponds to a ratio between the probabilities of satellite and central AGN of being active Q ∼ 2. Mock matched normal galaxies follow a slightly steeper M star − M h relation -with low mass mock galaxies residing in less massive halos than mock AGN of similar mass, and are less biased than mock AGN with similar specific BHAR and SFR, at least for Q > 1.
Aims. We study the spatial clustering of 632 (1130) XMM-COSMOS Active Galactic Nuclei (AGNs) with known spectroscopic (spectroscopic or photometric) redshifts in the range z = [0.1 − 2.5] in order to measure the AGN bias and estimate the typical mass of the hosting dark matter (DM) halo as a function of AGN host galaxy properties. We create AGN subsamples in terms of stellar mass M * and specific black hole accretion rate L X /M * , to probe how AGN environment depends on these quantities. Further, we derive the M * − M halo relation for our sample of XMM-COSMOS AGNs and compare it to results in literature for normal non-active galaxies. Methods. We measure the projected two-point correlation function w p (r p ) using both the classic and the generalized clustering estimator based on photometric redshifts as probability distribution functions in addition to any available spectroscopic redshifts. We measure the large-scale (r p 1 h −1 Mpc) linear bias b by comparing the clustering signal to that expected of the underlying DM distribution. The bias is then related to the typical mass of the hosting halo M halo of our AGN subsamples. Since M * and L X /M * are correlated, we match the distribution in terms of one quantity, while split the distribution in the other. Results. For the full spectroscopic AGN sample, we measure a typical DM halo mass of log(M halo /h −1 M ⊙ ) = 12.79 +0.26 −0.43 , similar to galaxy group environments and in line with previous studies for moderate-luminosity X-ray selected AGN. We find no significant dependence on specific accretion rate L X /M * , with log(M halo /h −1 M ⊙ ) = 13.06 +0.23 −0.38 and log(M halo /h −1 M ⊙ ) = 12.97 +0.39 −1.26 for low and high L X /M * subsamples, respectively. We also find no difference in the hosting halos in terms of M * with log(M halo /h −1 M ⊙ ) = 12.93 +0.31 −0.62 (low) and log(M halo /h −1 M ⊙ ) = 12.90 +0.30 −0.62 (high). By comparing the M * − M halo relation derived for XMM-COSMOS AGN subsamples with what is expected for normal non-active galaxies by abundance matching and clustering results, we find that the typical DM halo mass of our high M * AGN subsample is similar to that of non-active galaxies. However, AGNs in our low M * subsample are found in more massive halos than non-active galaxies. By excluding AGNs in galaxy groups from the clustering analysis, we find evidence that the result for low M * may be due a larger fraction of AGNs as satellites in massive halos.
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