ner_tags: O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Dataset I-Dataset O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: paper, as explained below. 3.1 Peebles–Hauser The Peebles–Hauser (PH) estimator is based on the correlation function proposed by Peebles Hauser ( 1974 ) and can be expressed as where DD (ρ) is defined as the pair of observed quasars counts within a separation radius ρ normalized by the the total number of pairs, N obs ( N obs − 1)/2 and RR (ρ) follows the same definition for the random catalogues. 3.2 Landy–Szalay The Landy–Szalay (LS) estimator is based on the correlation function proposed by Landy Szalay ( 1993 ). In addition to the quantities DD (ρ) and RR (ρ) defined above, DR (ρ) denotes the pair of quasar-random counts normalised by the available number of pairs. Hence, the LS estimator is given by For the PH and LS estimators, the quantity D 2 ( r ) is calculated by replacing equations ( 4 ) and ( 5 ) into equation ( 3 ), respectively. 4 THE HOMOGENEITY SCALE In order to estimate the homogeneity scale from the SDSS-IV DR14 quasars (henceforth ), we first obtain the values of D 2 ( r ) for each redshift slice using both estimators above. Then, a polynomial fitting is used to obtain the corresponding scale where the transition to homogeneity is identified (we refer the reader to Gonçalves et al. 2018 for more details). Following previous analyses (Scrimgeour et al. 2012 ; Laurent et al. 2016 ; Ntelis 2017 ), we define the homogeneity scale as the characteristic scale where the Universe can be considered homogeneous within 1 per cent of the expected fractal dimension for a homogeneous distribution, D 2 = 3 – thus, the scale where the spatial distribution of QSOs reaches D 2 = 2.97. Although arbitrary, the 1 per cent criterion is widely used in the literature, and is justified given some observational issues, such as the survey geometry and incompleteness, as well as the sample noise. Since we use 20 random catalogues to obtain D 2 , we also perform a bootstrap analysis over the 20 values of , quoting its mean and standard deviation as our measurement of the homogeneity scale and its corresponding uncertainty, respectively. 1 Our first measurements of are presented in the second column of Tables 2 and 3 . For the sake of example, Fig. 1 shows the results for both and D 2 obtained at the redshift using the PH estimator (similar results are also obtained for the LS estimator). The red points stand for the observational data with the respective errors. In the right-hand panel of Fig. 1 , the horizontal lines correspond to the D 2 = 2.97 and D 2 = 3 thresholds. From our analysis, we find that the PH estimator provides smaller error bars than the LS estimator. A possible explanation may be attributed to the fact that our approach does not use mock catalogues. Instead, we relied on an accurate bootstrap analysis of the 20 random catalogues we have produced. A full assessment of the homogeneity scale using mock catalogues that reflects the matter power spectrum, redshift distribution, QSO clustering, and angular selection function of the