Evolution of Stellar-to-Halo Mass Ratio at Identified by Clustering Analysis with the Hubble Legacy Imaging and Early Subaru/Hyper Suprime-Cam Survey Data
We present clustering analysis results from 10,381 Lyman break galaxies (LBGs) at , identified in the Hubble legacy deep imaging and new complimentary large-area Subaru/Hyper Suprime-Cam data. We measure the angular correlation functions (ACFs) of these LBGs at , , , and , and fit these measurements using halo occupation distribution (HOD) models that provide an estimate of halo masses, . Our estimates agree with those obtained by previous clustering studies in a UV-magnitude vs. plane, and allow us to calculate stellar-to-halo mass ratios (SHMRs) of LBGs. By comparison with the SHMR, we identify evolution of the SHMR from to , and to at the confidence levels. The SHMR decreases by a factor of from to , and increases by a factor of from to at the dark matter halo mass of . We compare our SHMRs with results of a hydrodynamic simulation and a semi-analytic model, and find that these theoretical studies do not predict the SHMR increase from to . We obtain the baryon conversion efficiency (BCE) of LBGs at , and find that the BCE increases with increasing dark matter halo mass. Finally, we compare our clustering+HOD estimates with results from abundance matching techniques, and conclude that the estimates of the clustering+HOD analyses agree with those of the simple abundance matching within a factor of 3, and that the agreement improves when using more sophisticated abundance matching techniques that include subhalos, incompleteness, and/or evolution in the star formation and stellar mass functions.
Subject headings:galaxies: formation — galaxies: evolution — galaxies: high-redshift
Dark matter halos play an important role in galaxy formation and evolution in the framework of Lambda cold dark matter (CDM) structure formation models. Such halos can regulate processes such as gas cooling necessary for star formation. Gas cooling is efficient in halos with masses of , where the gas cooling time scale is shorter than the gas infall time scale (Rees & Ostriker 1977; Sutherland & Dopita 1993; Silk & Wyse 1993). In low and high mass haloes, feedback from supernova (SN), radiation pressure, and active galactic nucleus (AGN) are thought to suppress star formation by thermal and kinetic energy input (e.g., Murray et al. 2005; Dekel et al. 2009; Kereš et al. 2009; Harikane et al. 2014). The connection between galaxies and their dark matter halos is essential for understanding galaxy formation, and specifically the stellar-to-halo mass ratio (SHMR), which is defined as the ratio of a galaxy’s stellar mass to its halo mass, is one of the key quantities. The SHMR comprises the integrated efficiency of the past stellar mass assembly (i.e., star formation and mergers). The SHMR has been theoretically investigated with the help of hydrodynamic simulations or semi-analytic models (e.g., Hopkins et al. 2014; Thompson et al. 2014; Birrer et al. 2014; Okamoto et al. 2014; Mitchell et al. 2015; Somerville et al. 2015). From observational studies, the SHMR is measured by analyses of galaxy clustering, weak lensing, satellite kinematics, and rotation curves at low-redshift (e.g., Mandelbaum et al. 2006; More et al. 2011; Leauthaud et al. 2012; Hudson et al. 2015; Shan et al. 2015; Rodríguez-Puebla et al. 2015; Coupon et al. 2015; Skibba et al. 2015; Sofue 2015). These low-redshift studies find a SHMR with a peak at a dark matter halo mass of , independent of redshift, and referred to as a pivot halo mass. Leauthaud et al. (2012) claim a redshift evolution of SHMRs (and pivot halo masses) from to . While some studies estimate the SHMR with clustering analysis at (e.g., Foucaud et al. 2010; Durkalec et al. 2015; McCracken et al. 2015; Hatfield et al. 2015; Ishikawa et al. 2016), it is difficult to investigate the evolution of SHMR at these high redshift due to poor statistics based on small galaxy samples available to date (c.f., McCracken et al. 2015; Hatfield et al. 2015).
The abundance matching technique is another indirect probe of the SHMR. The abundance matching technique connects galaxies to their host dark matter haloes by matching the cumulative stellar mass function (or the cumulative luminosity function) and the cumulative halo mass function. Because this technique only requires one-point statistics that are easily measured, many recent studies apply this method from low-redshift to high redshift galaxies (Moster et al. 2013; Behroozi et al. 2013a; Finkelstein et al. 2015; Mashian et al. 2015; Trac et al. 2015; Saito et al. 2015). Behroozi et al. (2013a) investigate the SHMR with abundance matching by using stellar mass functions as well as specific star formation rates and cosmic star formation rate densities; they find that the SHMR evolves from to . While abundance matching is a useful and less expensive method to connect galaxies to their dark matter haloes, there are two major systematic uncertainties with respect to the application to high redshift galaxies. One uncertainty is the star-formation duty cycle (DC) that is defined as the probability of a halo of given mass to host an observable star forming galaxy. Most abundance matching studies of Lyman break galaxies (LBGs) use the UV luminosity function, assuming their star-formation (i.e. UV-bright phase) DC is unity (c.f., Behroozi et al. 2013a). However, star formation activity can be episodic. Moreover, populations of passive and dusty star-forming galaxies are expected to exist that may be missed in LBG samples. In fact, Lee et al. (2009) claim that the DC is at , and Ouchi et al. (2001) indicate a halo mass-dependent DC based on clustering analysis. The other uncertainty, with respect to abundance matching, is the subhalo-galaxy relation. While the majority of abundance matching studies include subhalos (subhalo abundance matching; e.g., Moster et al. 2013; Behroozi et al. 2013a; Finkelstein et al. 2015; Mashian et al. 2015), the subhalo-galaxy relation is poorly constrained. For example, it is unclear which subhalo property best correlates with the stellar mass (or luminosity; Reddick et al. 2013; Guo et al. 2015). Preferably, one needs information independent from abundance to understand these systematics. Because lensing analysis is not feasible for galaxies at due to the limited number of background galaxies and their lower image quality, clustering analysis is a promising technique to test the abundance matching results and to extend our understanding of the connection between galaxies and dark matter halos to high redshift.
The clustering analysis of the high redshift galaxies has been conducted with large survey data. Ouchi et al. (2001, 2004b, 2005) obtained wide area data taken with the Subaru deep survey, and studied the clustering of LBGs at and . As well, Hildebrandt et al. (2009) estimated the angular correlation functions (ACFs) of LBGs at with high accuracy using the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) data. With the LBT Boötes field survey data, Bian et al. (2013) studied the clustering properties of LBGs at . Recently, the deep data of the Hubble Space Telescope legacy survey allowed us to study LBGs at (Barone-Nugent et al. 2014). Furthermore, Ishikawa et al. (2015) investigated the clustering properties of star forming galaxies using the wide area data taken by the United Kingdom Infra-Red Telescope (UKIRT), Subaru telescope, and CFHT.
Recently a wide-field mosaic CCD camera, Hyper Suprime-Cam (HSC; Miyazaki et al. 2012), has been installed at the prime focus of the Subaru telescope (Iye et al. 2004). HSC has a field-of-view (FoV) of with a high sensitivity accomplished with the Subaru 8m primary mirror. An HSC legacy survey under the Subaru Strategic Program (SSP; PI: S. Miyazaki) has been allocated 300 nights over 5 years, and has been ongoing since March 2014.111http://www.naoj.org/Projects/HSC/surveyplan.html The HSC SSP has three survey layers of Wide, Deep, and Ultradeep that will cover the sky areas of , , and with the point-source limiting magnitudes of , , and , respectively. It is expected that full HSC SSP data sets will provide us with LBGs at , which are times larger than current samples identified in the deep fields of the CFHTLS (Hildebrandt et al. 2009), and allow us to investigate statistical properties of LBGs down to . Complementing these HSC SSP efforts, recent deep Hubble Space Telescope observations with Advanced Camera for Surveys (ACS) and Wide Field Camera 3 (WFC3) provide samples of LBGs whose luminosities reach below (e.g., Bouwens et al. 2015; Ishigaki et al. 2015). In this study, we use the unique combined data sets of Subaru/HSC and Hubble/ACS+WFC3 to investigate the galaxy-dark matter connection using LBGs over a wide luminosity range, and investigate SHMRs at , for the first time, using clustering analyses.
This paper is organized as follows. We present the observational data sets of Subaru/HSC and Hubble/ACS+WFC3 in Section 2. We describe the photometry and sample selection of LBGs in Section 3. The clustering analysis is presented in Section 4. Sections 5 and 6 detail our results on the dark matter halo mass and SHMR, respectively. We discuss the implications of the SHMR evolution and differences between our results and those from abundance matching in Section 7. Section 8 summarizes our findings. Throughout this paper we use the following cosmological model: , , , , and . We use that is the radius in which the mean enclosed density is 200 times higher than the mean cosmic density. To define the halo mass, we use that is the total mass enclosed in . We assume a Chabrier (2003) initial mass function (IMF). All magnitudes are in the AB system (Oke & Gunn 1983).
|Field||()||coaddaaCoadd image of -bands.|
|PSF FWHMbbMean PSF FWHM values.||0.12||0.11||0.10||0.11||0.21||0.20||0.20||0.20||0.20||0.21||0.8|
Note. – Columns: (1) Field. (2) Effective Area in . (3)-(12) Limiting magnitudes which correspond to variations in the sky flux measured in a circular aperture of ″-diameter in PSF-matched images. (13) Limiting magnitude defined by a sky noise in a ″-diameter aperture. See Skelton et al. (2014) for limiting magnitudes in other bands.
2. Observational Data Sets
2.1. Hubble Data
We use 10 deep optical-NIR imaging data sets of the Hubble Ultra Deep Field (HUDF), Great Observatories Origins Deep Survey (GOODS)-North-Deep, GOODS-North-Wide, GOODS-South-Deep, GOODS-South-Wide, Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS)-All-Wavelength Extended Groth Strip International Survey (AEGIS), CANDELS-Cosmological Evolution Survey (COSMOS), CANDELS-Ultra Deep Survey (UDS), Hubble Frontier Field (HFF)-Abell2744P, and HFF-MACS0416P that are taken with ACS and WFC3 on the Hubble Space Telescope. The total area of the Hubble data is . We mask regions that are contaminated by the halos of bright stars or diffraction spikes by visual inspection, and measure limiting magnitudes in a ″-diameter circular aperture with sdfred (Yagi et al. 2002; Ouchi et al. 2004a), after homogenizations of the point-spread functions (PSFs; see Section 3.1.1 for more details). The typical FWHMs of the PSFs of ACS and WFC3 images are ″ and ″, respectively. The limiting magnitudes, PSF FWHMs, and effective areas of these images are summarized in Table 1.
HUDF has the deepest ACS and WFC3 imaging data, ever taken, from the combination of the three surveys, HUDF (Beckwith et al. 2006), HUDF09 (GO 11563; PI: G. Illingworth; e.g., Bouwens et al. 2010), and HUDF12 (GO 12498; PI: R. Ellis; e.g., Ellis et al. 2013; Koekemoer et al. 2013). We use the combined HUDF data set compiled by the eXtreme Deep Field (XDF) team222https://archive.stsci.edu/prepds/xdf/ (Illingworth et al. 2013). The HUDF data consist of 9-band images of , and cover sky area. The limiting magnitudes are over these 9 bands.
2.1.2 GOODS-North and GOODS-South
We use the data sets of the GOODS-North and GOODS-South fields, available from the CANDELS and 3D-HST teams (Brammer et al. 2012; Skelton et al. 2014) ,333http://candels.ucolick.org/ 444http://3dhst.research.yale.edu/Home.html that are obtained by CANDELS (PIs: S. Faber and H. Ferguson; Grogin et al. 2011; Koekemoer et al. 2011). The GOODS fields are comprised of deep and wide survey data whose limiting magnitudes are typically and , respectively. About half of the GOODS-North and GOODS-South fields are deep survey areas, while the remaining half of GOODS-North and quarter of the GOODS-South are wide survey areas. The GOODS-North and GOODS-South data are observed with bands of with effective areas of and , respectively.
2.1.3 CANDELS-AEGIS, CANDELS-COSMOS, and CANDELS-UDS
The largest area Hubble data sets in our study come from CANDELS-AEGIS, CANDELS-COSMOS, and CANDELS-UDS imaging data (Grogin et al. 2011; Koekemoer et al. 2011) available from the 3D-HST (Brammer et al. 2012; Skelton et al. 2014). These imaging regions are covered by ACS and WFC3 observations with the typical limiting magnitude of . Ground-based optical images taken with the CFHT and Subaru telescope are also available for these fields. We use the CFHT band images of the CANDELS-AEGIS field, the CFHT and Subaru band images of the CANDELS-COSMOS field, and the CFHT and Subaru band images of the CANDELS-UDS field.
Our study also includes imaging data from HFF (PI J. Lotz; e.g., Ishigaki et al. 2015; Kawamata et al. 2015). These data are parallel-field observations of Abell2744 and MACS0416 galaxy clusters that are taken from the HFF team.555http://www.stsci.edu/hst/campaigns/frontier-fields/ Because lensing effects such as magnification and survey volume distortion are negligibly weak in HFF parallel fields (see, e.g., Ishigaki et al. 2015), we regard these HFF parallel images as blank field data. These two HFF parallel fields are observed with 7 bands of over a total effective area of . The typical limiting magnitude is .
2.2. Subaru Data
Our study includes early data of the HSC SSP survey taken from March to November of 2014 (S14A_0b). We use the HSC SSP Wide layer data of the XMM field (, [J2000]) and GAMA09h field (, [J2000]).666These are the central coordinates of the early HSC data that we use. While the HSC data is magnitudes shallower than the Hubble data, the HSC data cover times larger effective area: and , in XMM and GAMA09h, respectively. As a result, the HSC data can provide clustering measurements at the bright end. The HSC data are reduced by the HSC SSP collaboration with hscPipe (version 3.4.1) that is the HSC data reduction pipeline based on the Large Synoptic Survey Telescope (LSST) software pipeline (Ivezic et al. 2008; Axelrod et al. 2010). The HSC data reduction pipeline performs CCD-by-CCD reduction and calibration for astrometry, warping, coadding, and photometric zeropoint measurements. The astrometric and photometric calibration are based on the data of Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) 1 imaging survey (Magnier et al. 2013; Schlafly et al. 2012; Tonry et al. 2012). We mask imaging regions contaminated with diffraction spikes and halos of bright stars using the mask extension outputs from the HSC data reduction pipeline and information of bright stars from our source catalogs (Section 3.2.1) and the Sloan Digital Sky Survey (SDSS) DR12 (Alam et al. 2015).777http://www.sdss.org/dr12/ We use the PSF outputs from the pipeline (Jee & Tyson 2011), and typical PSF FWHMs are ″″. The limiting magnitudes measured with sdfred are (Table 2).
|PSF FWHMaaMean PSF FWHM values.||0.82||0.85||0.62||0.67|
Note. – Columns: (1) Field. (2) Effective Area in . (3)-(6) Limiting magnitudes defined by a sky noise in a PSF--flux-radius circular aperture in PSF-matched images.
3. Photometric Samples and LBG Selections
3.1. Hubble Samples
3.1.1 Multi-band Photometric Catalogs
We construct multi-band source catalogs from the Hubble data. To measure object colors, we match the image PSFs to the WFC3 -band images whose typical FWHM of the PSF is ″, the largest of the Hubble multi-band images. We use SWarp (Bertin et al. 2002) to produce our detection images that are the co-added data of , , and -band images. The limiting magnitudes of the detection images are typically deeper than those of the single-band images (Table 1).
We perform source detection and photometry with SExtractor (Bertin & Arnouts 1996). We run SExtractor (version 2.8.6) in dual-image mode for each multi-band image with its detection image, having the parameter set as follows: , , , , and . The total number of the objects detected is 130,655. We measure the object colors with MAG_APER magnitudes defined in a ″-diameter circular aperture. We use the MAG_AUTO measurements of SExtractor for total magnitudes. In the CANDELS-AEGIS, CANDELS-COSMOS, and CANDELS-UDS fields, we use the CFHT and Subaru imaging data to reduce low- interlopers from high- galaxy samples. Because we only need magnitude upper limits of high- galaxy candidates for this purpose, we do not homogenize the PSFs of the CFHT and Subaru images. We obtain aperture magnitudes of SExtractor MAG_APER with a ″-diameter circular aperture. If a source is not detected either in a Hubble or CFHT/Subaru band, we replace the source flux with the -upper limit flux.
3.1.2 Lyman Break Galaxy Selection
We select LBGs from our source catalogs using color information. From the HUDF, GOODS-North, and GOODS-South source catalogs, we select LBGs at , , , and with the following LBG color criteria as given in Bouwens et al. (2015):
We select galaxies that have a Lyman break according to the criteria of Equations (1), (4), (7), and (10), and exclude intrinsically-red galaxies by the additional constraints of Equations (2), (3), (5), (6), (8), (9), (11), and (12). Figure 1 presents these color selection criteria, together with all sources from the HUDF catalog. These LBG color selection criteria are extensively tested by simulations, and used to study evolution of the UV luminosity functions (e.g., Bouwens et al. 2015).
In the five fields of CANDELS-AEGIS, CANDELS-COSMOS, CANDELS-UDS, HFF-Abell2744P, and HFF-MACS0416P, the number of the available multi-bands are smaller than those in HUDF and GOODS. We use different color criteria, and select LBGs at , , and in these five fields (Bouwens et al. 2015) as follows:
We select galaxies that have a Lyman break by the criteria given in Equations (13), (16), and (19), and exclude intrinsically-red galaxies by the criteria of Equations (14), (15), (17), (18), (20), and (21).
In addition, we also adopt the following four criteria that are similar to those in Bouwens et al. (2015). First, to identify secure sources, we apply detection limits of and levels in the detection images in HUDF and the other fields, respectively. Since the HUDF data are deep and clean, the detection criterion of HUDF is moderately loosened than the other fields. Second, for reducing foreground interlopers, we remove sources with continuum detected at wavelengths shortward of the Lyman breaks of the target LBGs. In all of the Hubble fields except CANDELS-AEGIS, CANDELS-COSMOS, and CANDELS-UDS, we apply the criterion of no-detection in the band for candidate LBGs at and , if data are available. Additionally, we require non-detection in band or for the LBG candidates. For the LBG candidates, we calculate an optical value for each source with the flux measurements, if available, in the same manner as Bouwens et al. (2011). The optical value is defined by , where , , and are the flux in each band, its uncertainty, and its sign, respectively. We remove LBG candidates whose values are larger than 4. For the rest of the fields, CANDELS-AEGIS, CANDELS-COSMOS, and CANDELS-UDS, we calculate values using the ground-based data whose wavelength are shorter than the redshifted Lyman break for the target LBGs at , , and . We use a threshold value of 2, 3, or 4 that corresponds to the number of the ground-based bands of , , or , respectively (Bouwens et al. 2015), and remove LBG candidates whose value is larger than the threshold. Third, to isolate LBGs from foreground Galactic stars, the LBG candidates should have an SExtractor stellarity parameter, CLASS_STAR, less than (Hildebrandt et al. 2009; Bouwens et al. 2015), if the candidates are brighter than the detection limit. Finally, to avoid multiple identifications of a source satisfying two sets of selection criteria at different redshifts, we keep the source in a catalog of LBGs at a redshift higher than the other, and remove the source from the low- catalog. For example, if a source meets the criteria of Equations (4)-(6) and (7)-(9), the source is not included in the LBG catalog of , but . After adopting these criteria, the estimated contamination fractions by foreground galaxies are estimated to be , , , and for the , , , and LBG samples, respectively, based on the Monte-Carlo simulations in Bouwens et al. (2015).
We construct a total sample of , , , and LBGs at , , , and , respectively, based on the Hubble data. Table 3 shows magnitudes of the LBGs. For conservative estimates of the clustering signals, we use the LBGs whose aperture magnitudes in the rest-frame UV band, , are brighter than the limiting magnitudes. The rest-frame UV band is defined by the observed band whose central wavelength is nearest to the rest-frame wavelength of for the Hubble data as well as the Subaru data (Section 3.2). Table 4 summarizes the numbers of LBGs for each field. We compare our sample with the sample of Bouwens et al. (2015), and find that our sample is consistent with that of Bouwens et al. (2015). In the deep fields used in our comparison, more than of the galaxies in our sample are included in the sample of Bouwens et al. (2015) at magnitudes brighter than the limiting magnitude. Similarly, more than of the galaxies of the Bouwens et al. (2015) sample are included in our sample. The remaining galaxies are located near the border of the color selection window, and are missed due to photometric errors. We also compare the surface number densities of our LBGs with those of Bouwens et al. (2015) in Figure 2; we confirm that the surface number densities of our LBGs are consistent. The mean redshifts of the , , , and LBGs are , , , and , respectively, and the redshift distributions are the same as those shown in Figures 1 and 19 of Bouwens et al. (2015).
|Catalog ID||coaddaaCoadd image of -bands.
(The complete table is available in a machine-readable form in the online journal.)
Note. – All magnitudes listed are measured in ″-diameter circular apertures. Upper limits are .
|HUDF||3.7||30.6||290 (348)||48 (130)||0 (86)||0 (50)|
|GOODS-N-Deep||57.4||28.6||1411 (1655)||431 (630)||43 (136)||81 (113)|
|GOODS-N-Wide||58.2||28.1||788 (800)||193 (223)||63 (69)||27 (31)|
|GOODS-S-Deep||52.7||29.0||1139 (1872)||205 (696)||142 (311)||66 (203)|
|GOODS-S-Wide||30.4||28.3||461 (510)||92 (142)||28 (51)||13 (31)|
|CANDELS-AEGIS||174.9||28.0||304 (381)||73 (101)||0 (28)|
|CANDELS-COSMOS||122.0||27.9||314 (348)||76 (80)||0 (27)|
|CANDELS-UDS||129.3||27.9||268 (310)||54 (65)||0 (25)|
|HFF-Abell2744P||3.1||29.3||30 (37)||0 (26)||0 (7)|
|HFF-MACS0416P||3.8||29.5||56 (67)||0 (53)||0 (9)|
|4089 (5185)||2671 (3694)||585 (978)||291 (524)|
Note. – Columns: (1) Field. (2) Effective area in . (3) limiting magnitude in the coadd image. (3)-(7) Number of the LBGs for our analysis at each redshift that are brighter than the limiting magnitude in the rest-frame UV band whose central wavelength is nearest to rest-frame Å. The value in parentheses is the number of LBGs in the parent sample.
3.2. Subaru Samples
3.2.1 Multi-band Photometric Catalogs
We make HSC source catalogs from the reduced images in the same manner as the Hubble source catalogs. First, we homogenize the PSFs of the HSC images to ″ in FWHM by convolving images with a Gaussian, matching a PSF’s -flux circular radius that includes 75% of a total flux for a PSF profile source. We then run SExtractor (version 2.8.6) in dual-image mode to detect sources in the detection image, and to carry out photometry in the HSC images for MAG_APER in a circular aperture of the PSF’s -flux radius. The total magnitude is estimated from the aperture magnitude with an aperture correction. The aperture correction is estimated to be under the assumption of the PSF profile. We use the parameter set of , , , , and .
3.2.2 Lyman Break Galaxy Selection
We select LBGs at with the HSC data, because we can conduct secure selections with the -band image. We apply color criteria similar to those of the CFHT study (Hildebrandt et al. 2009) that uses the photometric system almost identical to the one of our HSC data. The color selection criteria for the HSC sources are as follows:
We select galaxies that have a Lyman break by the criterion of Equation (22), and exclude intrinsically-red galaxies by the criteria of Equations (23) and (24). These LBG color selection criteria are used for the study of clustering evolution (e.g., Hildebrandt et al. 2009).
In addition to the selection criteria above, we require sources to be detected at the level in the -band image, and to be undetected at the level in the -band image. We also apply a criterion of SExtractor stellarity parameter, CLASS_STAR, of . We obtain 730 LBGs at . The surface number densities of our HSC LBGs are presented in Figure 2, which agree with the previous results of Hildebrandt et al. (2009). More details of the data reduction and the LBG selection are presented in Y. Ono et al. (in preparation).
4. Clustering Analysis
We derive the ACFs, , with our LBG samples. We calculate the observed ACFs, , using the estimator presented in Landy & Szalay (1993),
where , , and are numbers of galaxy-galaxy, galaxy-random, and random-random pairs normalized by the total number of pairs. We create a random sample composed of 10,000 (100,000) sources for each Hubble (Subaru) field with the geometrical shape same as the observational data including the mask positions. The errors are estimated by the bootstrap technique of Ling et al. (1986) with 100 resamples replacing individual galaxies for each field. It is known that this bootstrap technique tends to overestimate the errors of the correlation function (Fisher et al. 1994; Mo et al. 1992). Although we cannot quantitatively evaluate this trend with our data that are not large enough, the forthcoming data of the HSC survey will enable us to investigate this effect.
Due to the finite size of our survey fields, the observed ACF is underestimated by a constant value known as the integral constraint, (Groth & Peebles 1977). Including the correction for the number of objects in the sample, (Peebles 1980), the true ACF is given by
We estimate the integral constraint with
where is the best-fit model ACF, and refers the angular bin.
To test the dependence of the clustering strength on the luminosity of the galaxies, we make subsamples that are brighter than the threshold UV magnitudes, , that are listed in Table 5. In each subsample, we obtain the best-estimate of the ACF that is the weighted mean of the ACFs of the different fields in an angular bin. Table 5 shows the numbers of LBGs in the subsamples. Note that the numbers of the faint-magnitude subsamples are smaller than those of bright-magnitude subsamples (e.g. and ). This is because the faint-magnitude subsamples are only composed of LBGs in very deep data covering a small field (e.g. HUDF). Using the UV luminosity functions of Bouwens et al. (2015), we calculate the the number density of LBGs for each subsample and associated errors corrected for incompleteness. We estimate the cosmic variance in the number densities using the bias values obtained in Section 4.2, following the procedures in Somerville et al. (2004). We include the uncertainty from cosmic variance in our estimate of the error on the number density. The LBG number densities and the errors are presented in Table 5.
We fit the ACFs with a simple power law model,
Because we obtain no meaningful constraints on for most of the subsamples, we fix the value of to that is used in previous clustering analyses (e.g., Ouchi et al. 2001, 2004b, 2010; Foucaud et al. 2003, 2010). We use Equation (28) for to determine (Equation 27), and obtain the best-fit values with Equation (26).
Note. – Columns: (1) Mean redshift. (2) Threshold aperture magnitude in the rest-frame UV band. (3) Threshold absolute total magnitude in the rest-frame UV band. (4) Mean absolute total magnitude in the rest-frame UV band. (5) Threshold SFR in a unit of derived from the threshold total magnitude, . (6) Threshold stellar mass in a unit of derived from via equation (58), (59), (60). (7) Number of galaxies in our subsample. (8) Number density of our subsample derived from a UV luminosity function of Bouwens et al. (2015). (9) Power law amplitude (the power law index is fixed to .). (10) Spatial correlation length. (11) Galaxy-dark matter bias estimated by the power law model. See column (6) in Table 6 for the best estimate from the HOD modeling. (12) Reduced value.
Contaminating sources in a galaxy sample reduce the value of . If contaminants have a homogeneous sky distribution, the true is underestimated by a factor of , where is a contamination fraction. Because contaminants are more or less clustered, a clustering amplitude multiplied by provides the upper limit of the value of ,
The contamination fractions are , , , and for the , , , and LBG samples, respectively (Bouwens et al. 2015). The corresponding values are , , , and that are significantly smaller than the statistical errors. Therefore, we do not apply these contamination corrections to our estimate of . Table 5 presents the best-fit values. In Figure 3, we plot ACFs of our subsamples with the best-fit power law model.
4.2. Correlation Length and Bias
An ACF shows clustering properties of galaxies projected on the sky, and depends upon a combination of a galaxy redshift distribution and a galaxy spatial correlation function . The spatial correlation function is approximated by a single power law,
where is the angular diameter distance and is the redshift distribution of the sample. describes the redshift dependence of . Assuming that the clustering pattern is fixed in comoving coordinates in the redshift range of our sample, we use the functional form for (Roche & Eales 1999), where is the average redshift of the sample LBGs (Section 3.1). The value does not significantly depend on over . The slope of the spatial correlation function, , is related to that of the ACF, , by
We adopt the redshift distribution of LBGs presented in Bouwens et al. (2015, the left panel of Figure 1) and Y. Ono et al. (in preparation) for our Hubble and Subaru samples, respectively. These redshift distributions include the photometric uncertainties based on the Monte-Carlo simulations. Y. Ono et al. obtain the redshift distribution by placing artificial objects randomly in the real images using a method similar to the one in Bouwens et al. (2015). The object colors are calculated with redshifted model spectra (Bruzual & Charlot 2003) and the HSC filter response curves. We check the systematic errors on the halo mass estimates originating from the uncertainties, and find that the errors change the mass estimates negligibly, only by , assuming systematic shift of that is found in the spectroscopic results of Steidel et al. (1999).
We calculate the galaxy-dark matter bias on scale of , which is given by
where is the spatial correlation function of the underlying dark matter calculated with the linear dark matter power spectrum, , which is defined by
Table 5 presents the bias values thus obtained.
4.3. HOD Model
To connect observed galaxies to their host dark matter halos, we use a halo occupation distribution (HOD) model that is an analytic model of galaxy clustering (e.g., Seljak 2000; Berlind & Weinberg 2002; Cooray & Sheth 2002; Berlind et al. 2003; Kravtsov et al. 2004; Zheng et al. 2005). The HOD model is adopted not only to low-redshift galaxies (e.g., Zehavi et al. 2005; Zheng et al. 2007; van den Bosch et al. 2013; More et al. 2015), but also to high redshift galaxies (e.g., Bullock et al. 2002; Hamana et al. 2004; Ouchi et al. 2005; Lee et al. 2006, 2009; Hildebrandt et al. 2007; Bian et al. 2013). The key assumption of our HOD model is that the number of galaxy, , in a given dark matter halo depends only on the halo mass, . We parameterize the mean number of galaxies in dark matter halos with a mass of , , that is given by
where is the duty cycle of LBG activity (see Section 1 for the definition). , and are the mean number of central and satellite galaxies, respectively. Here the LBG activity for is defined by the properties of galaxies that are UV-bright star-forming galaxies selected as LBGs brighter than . We assume that DC does not depend on the halo mass in each subsample, because the present data are not large enough to investigate the mass dependence of DC that hides in the statistical errors. We adopt functional forms of and that are motivated by N-body simulations, smoothed particle hydrodynamic simulations, and semi-analytic models for low- galaxies and LBGs (e.g., Kravtsov et al. 2004; Zheng et al. 2005; Garel et al. 2015). is approximated as a step function with a smooth transition,
where is a transition width reflecting the scatter in the luminosity-halo mass relation. is the mass scale at which of halos host a central galaxy. Similarly, the mean number of satellite galaxies, , follows a power law with a mass cut,
where is the cut off mass, and () is the amplitude (slope) of the power law.
We calculate galaxy number densities from the HOD model with
where is the halo mass function. We use the halo mass function derived in Behroozi et al. (2013a), which is a modification of the Tinker et al. (2008) halo mass function for the high redshift universe () matching to the Consuelo simulation (McBride et al. 2009; see also Leauthaud et al. 2011; Behroozi et al. 2013b)888http://lss.phy.vanderbilt.edu/lasdamas/. The difference between the Behroozi et al. (2013a) and Tinker et al. (2008) halo mass functions is in the number density of the dark matter halo at . If we use the original mass function of Tinker et al. (2008), we find that none of our conclusions are changed.
In our HOD model, is computed from the galaxy power spectrum, , through the Fourier transformation,
The galaxy power spectrum is described by
where () is the one (two) halo term for pairs of galaxies in one (two different) halo(s).
Note. – Columns: (1) Mean redshift. (2) Threshold magnitude in the rest-frame UV band. (3) Best-fit value of in a unit of . (4) Star formation duty cycle. (5) Best-fit value of in a unit of . The value in parentheses is derived from via equation (55). (6) Effective bias. (7) Mean halo mass in a unit of . (8) Reduced value.
The one-halo term consists of a central-satellite part and a satellite-satellite part ,
The quantities of and are given by
where is the Fourier transform of the dark matter halo density profile normalized by its mass (e.g., Cooray & Sheth 2002). Here we assume that satellite galaxies in halos trace the density profile of the dark matter halo by NFW profile (Navarro et al. 1996, 1997), and adopt the mass-concentration parameter relation by Bullock et al. (2001) with an appropriate correction (see Shimizu et al. 2003). If we assume the mass-concentration parameter relation for the halo mass estimate of the subsample, we find the negligible change of in . The values of and are the mean number of central-satellite and satellite-satellite galaxy pairs, respectively. If we assume the independence of central and satellite galaxies and a Poisson distribution of the satellite galaxy’s distribution, these values are
The two-halo term is expressed as
where is the halo bias factor (Tinker et al. 2010).
To compare with the observational results, we calculate the ACF from the galaxy power spectrum projecting on the redshift distribution using the Limber approximation (see e.g. chapter 2 of Bartelmann & Schneider 2001),
where is the normalized redshift distribution of galaxies and is the zeroth-order Bessel function of the first kind. Here we assume that and do not vary as a function of redshift within the redshift ranges of the subsamples. The quantity is the radial comoving distance given by