The stellar mass, star formation rate and dark matter halo properties of LAEs at z ∼ 2

Abstract

We present average stellar population properties and dark matter halo masses of z ∼ 2 Lyα emitters (LAEs) from spectral energy distribution fitting and clustering analysis, respectively, using ≃ 1250 objects (⁠|$\mathit {NB387}\le 25.5$|⁠) in four separate fields of ≃ 1 deg² in total. With an average stellar mass of 10.2 ± 1.8 × 10⁸ M_⊙ and star formation rate of 3.4 ± 0.4 M_⊙ yr⁻¹, the LAEs lie on an extrapolation of the star-formation main sequence (MS) to low stellar mass. Their effective dark matter halo mass is estimated to be |$4.0_{-2.9}^{+5.1} \times 10^{10}{\,\,}M_{\odot }$| with an effective bias of |$1.22^{+0.16}_{-0.18}$|⁠, which is lower than that of z ∼ 2 LAEs (1.8 ± 0.3) obtained by a previous study based on a three times smaller survey area, with a probability of 96%. However, the difference in the bias values can be explained if cosmic variance is taken into account. If such a low halo mass implies a low H i gas mass, this result appears to be consistent with the observations of a high Lyα escape fraction. With the low halo masses and ongoing star formation, our LAEs have a relatively high stellar-to-halo mass ratio (SHMR) and a high efficiency of converting baryons into stars. The extended Press–Schechter formalism predicts that at z = 0 our LAEs are typically embedded in halos with masses similar to that of the Large Magellanic Cloud (LMC); they will also have similar SHMRs to the LMC, if their star formation rates are largely suppressed after z ∼ 2 as some previous studies have reported for the LMC itself.

1 Introduction

Galaxies assemble their stellar mass through star formation and galaxy merging under the gravitational influence of their host dark matter halos, which also grow through mass accretion and merging (e.g., Somerville & Davé 2015). Hence, observations of the intrinsic properties of galaxies and their dependence on halo mass in the past are key to tracing the history of the mass growth of galaxies and constraining the physical processes that control star formation (SF).

Low-mass galaxies at high redshift are “building blocks” of present-day galaxies over a wide mass range. Nebular emission lines are useful to detect faint (or low-mass) galaxies at high redshift (z), among which the Lyα line has been used most commonly. Tens of thousands of Lyα emitters (LAEs) have been selected so far by narrow-band (NB) imaging observations (z ∼ 2–7: e.g., Malhotra & Rhoads 2002; Taniguchi et al. 2005; Shimasaku et al. 2006; Gronwall et al. 2007; Ota et al. 2008, 2017; Ouchi et al. 2008, 2010; Guaita et al. 2010; Hayes et al. 2010; Hu et al. 2010; Ciardullo et al. 2012; Nakajima et al. 2012; Yamada et al. 2012; Konno 2014; Sandberg et al. 2015; Shimakawa et al. 2017; Shibuya et al. 2018a) and/or spectroscopically identified (z ∼ 0–7: e.g., Shapley et al. 2003; Kashikawa et al. 2006, 2011; Reddy et al. 2008; Cowie et al. 2010; Blanc et al. 2011; Dressler et al. 2011; Curtis-Lake et al. 2012; Mallery et al. 2012; Nakajima et al. 2013; Erb et al. 2014; Hayes et al. 2014; Hashimoto et al. 2013; Hathi et al. 2016; Karman et al. 2017; Shibuya et al. 2018b) and they are one of the important populations of high-z star-forming galaxies.

Typical LAEs at high redshifts have low stellar masses (M_⋆ ≲ 10⁹ M_⊙: Ono et al. 2010b; Guaita et al. 2011; Kusakabe et al. 2015; Hagen et al. 2016; Shimakawa et al. 2017). They are also dust-poor (Lai et al. 2008; Blanc et al. 2011; Kusakabe et al. 2015) and metal-poor (Nakajima et al. 2012, 2013; Nakajima & Ouchi 2014; Kojima et al. 2017), and have young stellar populations (Pirzkal et al. 2007; Gawiser et al. 2007; Hagen et al. 2014), although a small fraction of them are attributed to dusty galaxies with high stellar masses (Nilsson et al. 2009; Ono et al. 2010a; Pentericci et al. 2010; Oteo et al. 2012).

Since their dust emission is typically too faint to be detected by current infrared (IR) telescopes without gravitational lensing, estimates of their SF rates (SFRs) vary greatly depending on the method of measurement, making it difficult to determine their mode of SF [i.e., starburst or more typical of main-sequence (MS) galaxies] (Finkelstein et al. 2015; Hagen et al. 2016; Hashimoto et al. 2017; Shimakawa et al. 2017). Only at z ∼ 2 has the average SFR of LAEs been estimated from ultraviolet (UV) and dust emission, by means of stacking, from which they are found to lie on the SF main-sequence (SFMS: e.g., Daddi et al. 2007), although the analysis is limited to only a single survey field (Kusakabe et al. 2015). Recent observations have revealed that the stellar properties of LAEs are similar to those of other emission line galaxies at z ∼ 2 (Hagen et al. 2016). Shimakawa et al. (2017) have also found that LAEs at M_⋆ ≲ 10¹⁰ M_⊙ obey the same M_⋆–SFR and M_⋆–size relations as Hα emitters (HAEs) at z = 2.5. Thus, there is a possibility that LAEs are normal star-forming galaxies in the low stellar mass regime at high redshift.

With regard to their dark matter halos, LAEs have been found to reside in low-mass halos from clustering analysis (M_h ∼ 10¹⁰–10¹² M_⊙ over z ∼ 2–7: e.g., Ouchi et al. 2005, 2010, 2018; Kovač et al. 2007; Gawiser et al. 2007; Shioya et al. 2009; Guaita et al. 2010; Bielby et al. 2016; Diener et al. 2017). These results imply that LAEs at z ∼ 4–7 and z ∼ 2–3 evolve into massive elliptical galaxies and L_⋆ galaxies at z = 0, respectively. For both cases, high-z LAEs are likely candidates of the “building blocks” of mature galaxies in the local Universe (see also Rauch et al. 2008; Dressler et al. 2011) because they are embedded in the lowest-mass halos among all the high-z galaxy populations.

With stellar masses, SFRs, and halo masses in hand, one can obtain the stellar-to-halo-mass ratios (≡ M_⋆/M_h: SHMR) and baryon conversion efficiencies (⁠|$\equiv \mathit {SFR}/\mbox{baryon\ accretion\ rate}$|⁠: BCE) to quantify the SF efficiency in dark matter halos. The SHMR measures the time-integrated (time-averaged) efficiency of SF up to the observed epoch, while the BCE measures the efficiency at the observed epoch. Previous studies show tight relations of the SHMR and BCE of galaxies as a function of M_h over a wide redshift range (e.g., Behroozi et al. 2013; Moster et al. 2013; Rodríguez-Puebla et al. 2017). These relation are usually given as the average relations in the literature, thus they are presented here as such. The SF mode also tells us the nature of SF in terms of stellar mass growth.

For LAEs, these parameters are most reliably measured at z ∼ 2, because this redshift is high enough that the Lyα line is redshifted into the optical regime where a wide-field ground-based Lyα survey, critical for clustering analysis, is possible, and low enough that deep rest-frame near-infrared (NIR) photometry, critical for spectral energy distribution (SED) fitting of faint galaxies like LAEs, is still possible with Spitzer/IRAC. This redshift is also scientifically interesting because SF activity in the universe is at a global maximum (Madau & Dickinson 2014).

To date, there is only one clustering study carried out at z ∼ 2, by Guaita et al. (2010), for which they obtain a relatively high halo mass of |${\rm log} (M_{\rm h}/ {M_{\odot }}) \sim 11.5^{+0.4}_{-0.5}$|⁠, which implies an SHMR comparable to or lower than the average relations by Behroozi, Wechsler, and Conroy (2013) and Moster, Naab, and White (2013) at the same dark halo mass. Their LAEs are estimated to have a comparable BCE with the average relation by Behroozi, Wechsler, and Conroy (2013), but its uncertainty is as large as ∼1 dex. However, this halo mass estimate may suffer from statistical uncertainties due to a small sample size (N ∼ 250 objects) and systematic uncertainties from cosmic variance due to a small survey area (∼0.3 deg²). A larger number of sources from a larger survey area with deep multi-wavelength data is needed to obtain SHMRs and BCEs accurately and to overcome these uncertainties.

In this paper, we study star-forming activity and its dependence on halo mass for z ∼ 2 LAEs using ∼1250 NB-selected LAEs from four deep survey fields with a total area of ≃ 1 deg². Section 2 summarizes the data and sample used in this study. In section 3 we estimate halo masses from clustering analysis. In section 4 we perform SED fitting to stacked imaging data to measure stellar population parameters. The SHMR and BCE are calculated and compared with literature results in section 5. Section 6 is devoted to discuss the results obtained in the previous sections. Conclusions are given in section 7.

Throughout this paper, we adopt a flat cosmological model with the matter density Ω_m = 0.3, the cosmological constant Ω_Λ = 0.7, the baryon density Ω_b = 0.045, the Hubble constant H₀ = 70 km s⁻¹Mpc⁻¹ (h₁₀₀ = 0.7), the power-law index of the primordial power spectrum n_s = 1, and the linear amplitude of mass fluctuations in the universe σ₈ = 0.8, which are consistent with the latest Planck results (Plank Collaboration 2016). We assume a Salpeter initial mass function (IMF: Salpeter 1955)¹. Magnitudes are given in the AB system (Oke & Gunn 1983) and coordinates are given in J2000.0. Distances are expressed in comoving units. We use “log” to denote a logarithm with a base 10 (log₁₀).

2 Data and sample

2.1 Sample selection

Our LAE samples are constructed in four deep survey fields: the Subaru/XMM-Newton Deep Survey (SXDS) field (Furusawa et al. 2008), the Cosmic Evolution Survey (COSMOS) field (Scoville et al. 2007), the Hubble Deep Field North (HDFN: Capak et al. 2004), and the Chandra Deep Field South (CDFS: Giacconi et al. 2001). We select LAEs at z = 2.14–2.22 using the narrow band |$\mathit {NB387}$| (Nakajima et al. 2012) as described in Nakajima et al. (2012, 2013), Kusakabe et al. (2015), and Konno et al. (2016). The threshold of rest-frame equivalent width, EW₀, of Lyα emission is EW₀(Lyα) ≥ 20–30 Å (Konno et al. 2016).² While the SXDS field consists of five sub-fields, we use the three regions (SXDS-C, -N, and -S) with deeper |$\mathit {NB387}$| images. The 5σ depths in a 2″ diameter aperture are ≃ 25.7 (SXDS-C, -N, -S), 26.1 (COSMOS), 26.4 (HDFN), and 26.6 (CDFS). For accurate clustering analysis, we remove LAEs in regions with short net exposure times, which result from the dither pattern. In the SXDS field (SXDS-C, -N, -S), we use the overlapping regions to examine if there exists an offset in the |$\mathit {NB387}$| zero-point. A non-negligible offset of 0.06 mag is found in SXDS-N and appropriately corrected. In the other three fields, we examine the |$\mathit {NB387}$| zero-point using the colors of the Galactic stars from Gunn and Stryker (1983) and apply a 0.1 mag correction to LAEs in CDFS. Note that such a correction value changes the Lyα luminosities only slightly. Our entire sample consists of 2441 LAEs from ≃ 1 square degree (each survey area size is shown in table 1). Of these, we use 1937 LAEs with |$\mathit {NB387}_{\rm tot} \le 26.3$|⁠, where |$\mathit {NB387}_{\rm tot}$| is the NB387 total magnitude, for the clustering analysis to examine the halo mass dependence on |$\mathit {NB387}_{\rm tot}$| (see figure 1, table 2 and subsection 3.1). Note that 1248 LAEs with |$\mathit {NB387}_{\rm tot} \le 25.5$| are used to calculate a four-field average effective bias (see subsection 3.3) and derive the SHMR and BCE of our LAEs.

Fig. 1.

|$B-\mathit {NB387}$| (⁠|$\mathit {NB387}$| excess) plotted against |$\mathit {NB387}$| total magnitude. Orange, green, magenta, and blue points show LAEs in SXDS, COSMOS, HDFN, and CDFS, respectively. LAEs are divided into cumulative subsamples with different limiting magnitudes shown by gray solid lines: |$\mathit {NB387}_{\rm tot} \le 25.0\:$|mag, 25.3 mag, 25.5 mag, 25.8 mag, and 26.3 mag. (Color online)

Table 1.

Details of the data.*

Band	SXDS (∼1240 arcmin2)			COSMOS (∼740 arcmin2)			HDFN (∼780 arcmin2)			CDFS (∼580 arcmin2)
	PSF	Aperture	Aperture	PSF	Aperture	Aperture	PSF	Aperture	Aperture	PSF	Aperture	Aperture
	(″)	diameter (″)	correction (mag)	(″)	diameter (″)	correction (mag)	(″)	diameter (″)	correction (mag)	(″)	diameter (″)	correction (mag)
|$\mathit {NB387}$|	0.88	2.0	0.17	0.95	2.0	0.25	0.89	2.0	0.14	0.85	2.0	0.13
B	0.84	2.0	0.17	0.95	2.0	0.12	0.77	2.0	0.15	1.0	2.0	0.20
V	0.8	2.0	0.15	1.32	2.0	0.33	1.24	2.0	0.20	0.94	2.0	0.18
R(r΄)	0.82	2.0	0.16	1.04	2.0	0.19	1.18	2.0	0.22	0.83	2.0	0.16
i΄(I)	0.8	2.0	0.16	0.95	2.0	0.12	0.80	2.0	0.13	0.95	2.0	0.22
z΄	0.81	2.0	0.16	1.14	2.0	0.25	0.81	2.0	0.15	1.1	2.0	0.24
J	0.85	2.0	0.15	0.79	2.0	0.3	0.84	2.0	0.17	0.80	2.0	0.22
H	0.85	2.0	0.15	0.76	2.0	0.2	0.84	2.0	0.17	1.5	2.0	0.55
K(Ks)	0.85	2.0	0.16	0.75	2.0	0.2	0.84	2.0	0.18	0.70	2.0	0.18
IRAC ch1	1.7	3.0	0.52	1.7	3.0	0.52	1.7	3.0	0.52	1.7	3.0	0.52
IRAC ch2	1.7	3.0	0.55	1.7	3.0	0.55	1.7	3.0	0.55	1.7	3.0	0.55

*The FWHM of PSF, aperture diameter, and aperture correction are shown. The value in parentheses shows the area used in clustering analysis.

Table 2.

Number of objects in each subsample.*

Field	|$\mathit {NB387}_{\rm tot}$| magnitude limit (mag)
	25.0	25.3	25.5	25.8	26.3
SXDS	161	368	601 (93)	—	—
COSMOS	119	205	297 (21)	526	—
HDFN	119	200	299 (56)	588	—
CDFS	27	41	51 (4)	92	222

*The value in parentheses shows the number of objects used for SED fitting.

2.2 Contamination fraction

Possible interlopers in our LAE samples are categorized into (i) spurious sources without continuum, (ii) active galactic nuclei (AGNs), (iii) low-z line emitters whose line emission (not Lyα) is strong enough to meet our color selection, (iv) low-z line emitters with weaker emission lines which happen to meet the color selection owing to photometric errors in the selection bands, (v) low-EW (≲ 20–30 Å) LAEs at our target redshift selected owing to photometric errors in the selection bands, and (vi) continuum sources at any redshifts selected as LAEs owing to photometric errors in the selection bands. We describe each in further detail here.

• (i) Spurious sources without continuum are possibly included in our LAE sample even after visual inspection was performed as described in the original papers based on selection. About 1.6% of all 2441 LAEs have neither U- (or u*) nor B-band detection at more than 2σ, and this fraction reduces to 0.2% for the 1248 objects with |$\mathit {NB387} \le 25.5$|⁠.

• (ii) All sources detected in either X-ray, UV, or radio are regarded as AGNs and have been removed as described in the selection papers. Their fraction of the entire sample is about 2%. Obscured faint AGNs at these wavelengths may contaminate our sample, although heavily obscured AGNs are unlikely to have emission lines strong enough to pass our color selection. Following Guaita et al. (2010), we estimate the possible fraction of obscured AGNs in our LAE sample to be ∼2%, i.e. similar to that of X-ray, UV, or radio-detected AGNs (i.e., Xue et al. 2010; Stern et al. 2012; Heckman & Best 2014; Aird et al. 2018; Ricci et al. 2017).

• (iii) Candidate emitters are [O ii] λ 3727 emitters at z ≃ 0.04, Mg ii λ 2798 emitters at z ≃ 0.4, and C iv λ 1550 and C iii] λ 1909 emitters at z ≃ 1.5. However, the survey volume of [O ii] emitters at z ≃ 0.04 is three orders of magnitude smaller than that of LAEs at z = 2.2. Moreover, the EW₀([O ii]) of the vast majority of [O ii] emitters is too small (∼8 Å) to meet our color selection of EW₀([O ii]) ≥ 70Å (see Konno et al. 2016; Ciardullo et al. 2013). [O ii] emitters with such a large EW₀([O ii]) should be AGNs. Mg ii, Civ and Ciii] emitters which satisfy our selection criteria are also likely to be AGNs. X-ray, UV, or radio-detected AGNs have been removed. Therefore, the fraction of contaminants (iii) is expected to be negligibly small and is included in the possible fraction of obscured AGNs as described in category (ii).

• (iv), (v), (vi) We evaluate the contamination fraction contributed by (iv), (v), and (vi) sources that do not satisfy the selection criteria if they have no photometric error (hereafter, intrinsically unselected sources), using Monte Carlo simulations. We use bright sources with |$\mathit {NB387} \le 24.0\:$|mag where photometric errors are negligible in all three selection bands of U (or u*), B, and |$\mathit {NB387}$| in the four fields. Assuming that the relative distribution of |$\mathit {NB387}$|-detected objects in the two-color selection plane, U (or u*)–|$\mathit {NB387}$| vs. |$B\mbox{--}\mathit {NB387}$|⁠, is unchanged with |$\mathit {NB387}$| magnitude intrinsically, we create a mock catalog by adding photometric errors to the three selection bands. Here, the distribution of |$\mathit {NB387}$| magnitudes of simulated sources is set equal to that of real |$\mathit {NB387}$|-detected objects down to the σ limiting magnitude of |$\mathit {NB387}$| in each of the four fields as described in subsection 2.1.

  We then apply the same selection as for the real catalog to obtain the number of objects passing the selection. The contamination fraction is calculated by dividing the number of intrinsically unselected sources passing the selection by the number of all sources passing the selection. The latter are a mixture of real LAEs with EW₀(Lyα) ≥ 20–30 Å and intrinsically unselected sources passing the selection, i.e., (iv), (v), and (vi). We find that the contamination fraction at |$\mathit {NB387} \le 25.5$| is 10%–20% for all four fields. This contamination fraction is conservative in the sense that (v) real LAEs with EW₀(Lyα) ≤ 20–30 Å are categorized as intrinsically unselected sources, the fraction of which is expected to be significantly higher than that of (iv).

To summarize, the fractions of possible interlopers (i), (ii), and (iii) are negligibly small, and those of (iv), (v), and (vi) are estimated to be 10%–20% in total for all four fields.

Spectroscopic follow-up observations of Lyα emission of bright LAEs in our sample (⁠|$\mathit {NB387} \le 24.5\:$|mag) have also been carried out with Magellan/IMACS, MagE, and Keck/LRIS by Nakajima et al. (2012), Hashimoto et al. (2013, 2015, 2017), Shibuya et al. (2014), and M. Rauch et al. (in preparation). In total, more than 40 LAEs are spectroscopically confirmed and no foreground interlopers such as [O ii] emitters at z = 0.04 are found (Nakajima et al. 2012). Although faint LAEs cannot be confirmed spectroscopically, the contamination fraction is probably not high. Indeed, Konno et al. (2016) have not applied contamination correction in deriving luminosity functions. On the basis of the results of the Monte Carlo simulations and the spectroscopic follow-up observations, 0%–20%, we conservatively adopt 10 ± 10% for the contamination fraction. This value is similar to a previous result for NB-selected LAEs at z ∼ 2, 7 ± 7%, which is a sum of (i), (ii), (iii) and (vi) (Guaita et al. 2010). The effect of contamination sources is taken into account in clustering analysis (see subsection 3.2). On the other hand, it is negligible in SED fitting for median-stacked subsamples as discussed in section 4.

2.3 Imaging data for SED fitting

We use 10 broadband images for SED fitting: five optical bands [B, V, R (or r), i (or i΄), and z (or z΄)], three NIR bands [J, H, and K (or K_s)], and two mid-infrared (MIR) bands (IRAC ch1 and ch2). The point spread functions (PSFs) of the images are matched in each field (not in each sub-field). The aperture corrections for converting 3″ MIR aperture magnitudes to total magnitudes are taken from Ono et al. (2010b, see table1). For each field, a K-band or NIR-detected catalog is used to obtain secure IRAC photometry in subsection 4.1. Here we summarize the data used in SED fitting and IRAC cleaning in the four fields.

• SXDS fields. The images used for SED fitting are as follows: B, V, R, i΄, and z΄ images taken with the Subaru/Suprime-Cam from the SXDS project (Furusawa et al. 2008); J, H, and K images from the data release 8 of the UKIRT/WFCAM UKIDSS/UDS project (Lawrence et al. 2007, Almaini et al. in preparation); Spitzer/IRAC 3.6 μm (ch1) and 4.5 μm (ch2) images from the Spitzer Large Area Survey with Hyper-Suprime-Cam (SPLASH) project (SPLASH: PI: P. Capak; Laigle et al. 2016). All images are publicly available except the SPLASH data. The aperture corrections for optical and NIR images are given in Nakajima et al. (2013). The catalog used to clean IRAC photometry is constructed from the K-band image of the UKIDSS/UDS data release 11 (O. Almaini et al. in preparation).

• COSMOS field. We use the publicly available B, V, r΄, i΄, and z΄ images taken with the Subaru/Suprime-Cam from COSMOS (Capak et al. 2007; Taniguchi et al. 2007) and J, H, and K_s images with the VISTA/VIRCAM from the first data release of the UltraVISTA survey (McCracken et al. 2012). We also use Spitzer/IRAC ch1 and ch2 images from the SPLASH project. The aperture corrections for the optical images are derived in Nakajima et al. (2013) and those for the NIR images follow McCracken et al. (2012). The catalog used to clean IRAC photometry is from Laigle et al. (2016), for which sources have been detected in the |$\mathit {z^{\prime }YJHK_{\rm s}}$| images.

• HDFN field. The images used for SED fitting are: B, V, R, I, and z΄ images taken with the Subaru/Suprime-Cam from HDFN (Capak et al. 2004); J (Lin et al. 2012), H (L. Hsu et al. in preparation), and K_s (Wang et al. 2010) images with CFHT/WIRCAm (PI of the J and H imaging observations: L. Lin); Spitzer/IRAC ch1 and ch2 images from the Spitzer Extended Deep Survey (SEDS: Ashby et al. 2013). We use reduced J-band and K_s-band images given in Lin et al. (2012). All images are publicly available. The aperture corrections for the optical images are given in Nakajima et al. (2013). Those of the NIR images with a 2″ radius aperture are evaluated using bright and isolated point sources in each band. We measure fluxes for 20 bright point sources in a series of apertures from 2″ with an interval of 0|${^{\prime\prime}_{.}}$|1 and find that the fluxes level off for >7|${^{\prime\prime}_{.}}$|8 apertures. We measure the difference in magnitude between the 2″ and 7|${^{\prime\prime}_{.}}$|8 apertures of 100 bright and isolated sources and perform Gaussian fitting to the histogram of differences. We adopt the best-fitting mean as the aperture correction term. The catalog used to clean IRAC photometry is constructed from the K-band image (Wang et al. 2010).

• CDFS fields. We use the publicly available B, V, R, and I images taken with the MPG 2.2 m telescope/WFI from the Garching–Bonn Deep Survey (GaBoDS: Hildebrandt et al. 2006; Cardamone et al. 2010), the z΄ image taken with the CTIO 4 m Blanco telescope/Mosaic-II camera from the MUltiwavelength Survey by Yale-Chile (MUSYC: Taylor et al. 2009; Cardamone et al. 2010), the H image taken with the ESO-NTT telescope/SofI camera by the MUSYC (Moy et al. 2003; Cardamone et al. 2010), and the J and K_s images from the Taiwan ECDFS Near-Infrared Survey (TENIS: Hsieh et al. 2012). We also use the Spitzer/IRAC ch1 and ch2 images from the Spitzer IRAC/MUSYC Public Legacy Survey in the Extended CDF-South (SIMPLE: Damen et al. 2011). The aperture corrections for optical and NIR photometry are derived in a similar manner to those in HDFN. The catalog used to clean IRAC photometry is from Hsieh et al. (2012), for which sources have been detected in the J image.

The FWHM of the PSF, aperture diameters, and aperture corrections are summarized in table 1.

3 Clustering analysis

3.1 Subsamples divided by NB387 magnitude

The distribution of |$B-\mathit {NB387}$| as a function of total |$\mathit {NB387}$| magnitude, |$\mathit {NB387}_{\rm tot}$|⁠, is shown in figure 1. To examine the dependence of halo mass on the total |$\mathit {NB387}$| magnitude, we divide our LAE sample of each field into up to five cumulative subsamples with different limiting magnitudes, as shown in table 2 and figure 1. There are 1937 LAEs with |$\mathit {NB387}_{\rm tot} \le 26.3$| used in the clustering analysis.

3.2 Angular correlation function

Angular correlation functions of our LAEs are derived from clustering analysis. The sky distributions of the LAEs in the four fields are shown in figure 2.³ We measure the two-point angular correlation function (ACF), ω_obs(θ), for a given (sub) sample using the calculator given in Landy and Szalay (1993):

\begin{equation} \omega _{\rm obs}(\theta )=\frac{DD(\theta )-2DR(\theta )+RR(\theta )}{RR(\theta )}, \end{equation}

(1)

where DD(θ), RR(θ), and DR(θ) are the normalized numbers of galaxy–galaxy, galaxy–random, and random–random pairs, respectively:

\begin{equation} DD(\theta ) = \frac{DD_0(\theta )\times 2}{N_{\,\rm D}(N_{\rm D}-1)}, \end{equation}

(2)

\begin{equation} RR(\theta ) = \frac{RR_0(\theta )\times 2}{N_{\,\rm R}(N_{\rm R}-1)}, \end{equation}

(3)

\begin{equation} DR(\theta ) = \frac{DR_0(\theta )}{N_{\,\rm D}{\times }N_{\rm R}}, \end{equation}

(4)

Here, N is the total number of pairs, with subscripts “D” and “R” indicating galaxies and random points, respectively, and subscript “0” indicates the raw number of pairs. We use a random sample composed of 100000 sources with the same geometrical constraints as the data sample (see figure 2). The 1σ uncertainties in ACF measurements are estimated as

\begin{equation} \Delta \omega _{\rm obs}(\theta )=\frac{1+\omega (\theta )}{\sqrt{DD_0(\theta )}} \end{equation}

(5)

following Guaita et al. (2010). While Norberg et al. (2009) find that Poisson errors underestimate the 1σ uncertainties in ACF measurements and that bootstrapping errors overestimate them by 40% using a large number of sources (∼10⁵–10⁶), Khostovan et al. (2017) show that Poisson errors and bootstrapping errors are comparable in the case of a small sample size using ∼200 Hβ + [O iii]emitters at z ∼ 3.2 (see also our footnote 5 and figure 5b).

Fig. 2.

Sky distribution of LAEs in SXDS (a), COSMOS (b), HDFN (c), and CDFS (d). Filled and open black circles represent objects with NB_tot ≤ 25.5 mag and NB_tot > 25.5 mag, respectively. Gray points indicate 100000 random sources used in the clustering analysis. Masked regions are shown in white.

We approximate the spatial correlation function of LAEs using a power law:

\begin{equation} \xi (r)=\left(\frac{r}{r_0}\right)^{-\gamma }, \end{equation}

(6)

where r, r₀, and γ are the spatial separation between two objects in comoving scale, the correlation length, and the slope of the power law, respectively (Totsuji & Kihara 1969; Zehavi et al. 2004). We then convert ξ(r) into the ACF, ω_model(θ), following Simon (2007), and describe it as

\begin{equation} \omega _{\rm model}(\theta ) = C{\,\,} \omega _{\rm model,{\,\,} 0}(\theta ), \end{equation}

(7)

where ω_{model, 0}(θ) is the ACF in the case of |$r_0= 1 \ h^{-1}_{100}\:$|Mpc and C is a normalization constant:

\begin{equation} C = \left(\frac{r_0 \: h^{-1}_{100}\:\mbox{Mpc}}{1\: h^{-1}_{100}\:\mbox{Mpc}}\right)^{\gamma }. \end{equation}

(8)

The correlation amplitude of the ACF at θ = 1″, A_ω, is

\begin{eqnarray} A_{\omega } &=& C{\,\,}\omega _{\rm model,{\,\,} 0}(\theta =1^{\prime \prime }). \end{eqnarray}

(9)

An observationally obtained ACF, ω_obs(θ), includes an offset due to the fact that the measurements are made over a limited area. This offset is given by the integral constraint (IC):

\begin{equation} \omega (\theta )=\omega _{\rm obs}(\theta )+IC, \end{equation}

(10)

\begin{equation} IC= \frac{\Sigma _{\theta } RR(\theta ){\,\,}C{\,\,}\omega _{\rm model,{\,\,} 0}(\theta )}{\Sigma _{\theta } RR(\theta )}, \end{equation}

(11)

where ω(θ) is the true ACF. We fit the ω_model(θ) to this ω(θ) over ∼40″–1000″ by minimizing χ²:

\begin{eqnarray} \chi ^2&=&\Sigma _{\theta }\left[ \frac{\omega _{\rm obs} (\theta ) +IC-\omega _{\rm model}(\theta )}{\Delta \omega _{\rm obs}(\theta )} \right]^2 \nonumber \\ &=&\Sigma _{\theta }\left\lbrace \frac{\omega _{\rm obs}(\theta )+C{\,\,}[IC_0-\omega _{\rm model,0}(\theta )]}{\Delta \omega _{\rm obs}(\theta )} \right\rbrace ^2, \end{eqnarray}

(12)

where IC₀ = IC/C. This θ range is determined conservatively, avoiding the one-halo term at small scales and large sampling noise at large scales. We fix γ to the fiducial value 1.8 following previous clustering analyses (e.g., Ouchi et al. 2003). The analytic solution of the best-fitting correlation amplitude is

\begin{equation} A_{\omega }=\frac{ \Sigma _{\theta }\left\lbrace \frac{\omega _{\rm obs}(\theta )[\omega _{\rm model,0}(\theta )-IC_0]}{\Delta \omega _{\rm obs}(\theta )^2} \right\rbrace }{\Sigma _{\theta }\left[ \frac{IC_0-\omega _{\rm model,0}(\theta )}{\Delta \omega _{\rm obs}(\theta )} \right]^2}{\,\,}\omega _{\rm model,{\,\,} 0}(\theta =1^{\prime \prime }). \end{equation}

(13)

The 1σ fitting error in A_ω, ΔA_ω, is estimated from |$\chi ^2_{\rm min} +1$|⁠, where |$\chi ^2_{\rm min}$| is the minimum χ² value. We also derive, for each limiting magnitude, the field-average correlation amplitude over the four survey fields by minimizing the summation of χ² over the four fields:

\begin{eqnarray} A_{\omega , ave} =\frac{ \Sigma _{\theta , i=field}\left\lbrace \frac{\omega _{{\rm obs},i}(\theta )[\omega _{\rm model,0}(\theta )-IC_{0,i}]}{\Delta \omega _{{\rm obs}, i}(\theta )^2} \right\rbrace }{\Sigma _{\theta , i=field}\left[ \frac{IC_{0,i}-\omega _{\rm model,0}(\theta )}{\Delta \omega _{{\rm obs},i}(\theta )} \right]^2} \omega _{\rm model,{\,\,} 0}(\theta =1^{\prime \prime }).\nonumber\\ \end{eqnarray}

(14)

The best-fitting ACFs are shown in figure 3.

Fig. 3.

ACF measurements for LAEs with |$\mathit {NB387}_{\rm tot} \le 25.0$| (a), |$\mathit {NB387}_{\rm tot} \le 25.3$| (b), |$\mathit {NB387}_{\rm tot} \le 25.5$| (c), |$\mathit {NB387}_{\rm tot} \le 25.8$| (d), and |$\mathit {NB387}_{\rm tot} \le 26.3$| (e). For each panel, colored symbols (orange squares, green circles, magenta inverted triangles, and blue triangles) represent measurements in SXDS, COSMOS, HDFN, and CDFS, respectively. Colored lines, as labeled in the lower right-hand panel, indicate the best-fitting ACFs with fixed β = 0.8 in SXDS, COSMOS, HDFN, and CDFS, respectively. A dotted black line shows the average of the best-fitting ACFs over the four fields. In panels (a)–(d), we slightly shift all data points along the abscissa by a value depending on the field for presentation purposes. (Color online)

Contaminations by randomly-distributed foreground and background interlopers dilute the apparent clustering amplitude. The correlation amplitude corrected for randomly distributed interlopers, A_{ω, corr}, is given by

\begin{equation} A_{\rm \omega ,corr}= \frac{A_{\omega }}{(1-f_{\rm c})^2}, \end{equation}

(15)

where f_c is the contamination fraction. The contamination fraction of our LAEs is estimated to be 10 ± 10% (0%–20%) conservatively from the Monte Carlo simulations and the spectroscopic follow-up observations (see subsection 2.2). This A_{ω, corr} is the maximum permitted value because interlopers themselves are also clustered in reality. Indeed, some previous clustering studies (e.g., Khostovan et al. 2017) have not applied any contamination correction. In this study, we apply this equation assuming f_c = 10 ± 10% so that the error range in A_{ω, corr} include both the no correction case and the maximum correction case. The 1σ error in the contamination-corrected correlation amplitude, ΔA_{ω, corr}, is derived by summing the 1σ error in the ACF fitting, ΔA_ω, and the uncertainty in the contamination estimate, Δf_c = 0.1, in quadrature (error propagation):

\begin{equation} \frac{\Delta A_{\rm \omega ,corr}}{A_{\rm \omega ,corr}} \simeq \sqrt{\left(\frac{\Delta A_{\omega }}{A_{\omega }}\right)^2 + \left(\frac{2\Delta f_{\rm c}}{f_{\rm c}}\right) ^2 }. \end{equation}

(16)

The value of the contamination-corrected correlation length, r_{0, corr}, and its 1σ error are calculated from A_{ω, corr} and ΔA_{ω, corr}. Table 3 summarizes the results of the clustering analysis.

Table 3.

Clustering measurements of our LAEs.*

Field	Aω	Aω, corr	r0, corr	bg, eff	Mh	Reduced |$\chi ^2_{\nu }$|	IC
|$\mathit {NB387}_{\rm tot}$| (mag)			|$({h^{-1}_{100}\:\mbox{Mpc}})$|		( × 1010 M⊙)
	(1)	(2)	(3)	(4)	(5)	(6)	(7)
SXDS
≤25.0	4.70 ± 2.86	5.80 ± 3.75	|$2.78^{+ 0.89}_{- 1.22}$|	|$1.40^{+ 0.40}_{- 0.57}$|	|$10.1^{+28.8}_{-10.1}$|	1.74	0.0137
≤25.3	2.07 ± 1.27	2.56 ± 1.67	|$1.77^{+ 0.57}_{- 0.78}$|	|$0.93^{+ 0.27}_{- 0.38}$|	|$0.4^{+ 3.2}_{- 0.4}$|	5.40	0.0060
≤25.5	3.35 ± 0.78	4.14 ± 1.33	|$2.31^{+ 0.39}_{- 0.45}$|	|$1.18^{+ 0.18}_{- 0.21}$|	|$3.3^{+ 5.2}_{- 2.7}$|	3.02	0.0097
COSMOS
≤25.0	3.88 ± 3.03	4.79 ± 3.88	|$2.50^{+ 0.98}_{- 1.51}$|	|$1.27^{+ 0.44}_{- 0.72}$|	|$5.5^{+25.3}_{- 5.5}$|	0.89	0.0176
≤25.3	4.44 ± 1.81	5.48 ± 2.54	|$2.70^{+ 0.64}_{- 0.79}$|	|$1.36^{+ 0.29}_{- 0.36}$|	|$8.5^{+16.6}_{- 7.7}$|	1.11	0.0201
≤25.5	3.32 ± 1.25	4.10 ± 1.79	|$2.29^{+ 0.51}_{- 0.63}$|	|$1.18^{+ 0.23}_{- 0.29}$|	|$3.1^{+ 7.5}_{- 2.9}$|	0.62	0.0150
≤25.8	3.70 ± 0.70	4.57 ± 1.33	|$2.44^{+ 0.37}_{- 0.42}$|	|$1.24^{+ 0.17}_{- 0.20}$|	|$4.7^{+ 6.0}_{- 3.5}$|	0.95	0.0168
HDFN
≤25.0	6.89 ± 3.77	8.51 ± 5.03	|$3.44^{+ 1.01}_{- 1.35}$|	|$1.70^{+ 0.44}_{- 0.61}$|	|$29.3^{+55.5}_{-27.6}$|	0.81	0.0319
≤25.3	9.55 ± 2.28	11.79 ± 3.84	|$4.13^{+ 0.70}_{- 0.81}$|	|$2.00^{+ 0.30}_{- 0.36}$|	|$62.9^{+52.0}_{-38.3}$|	1.33	0.0441
≤25.5	5.18 ± 1.51	6.40 ± 2.34	|$2.94^{+ 0.56}_{- 0.66}$|	|$1.47^{+ 0.25}_{- 0.30}$|	|$13.6^{+17.7}_{-10.5}$|	0.95	0.0240
≤25.8	2.52 ± 0.75	3.11 ± 1.15	|$1.97^{+ 0.38}_{- 0.45}$|	|$1.03^{+ 0.18}_{- 0.21}$|	|$1.0^{+ 2.6}_{- 0.9}$|	1.12	0.0116
CDFS
≤25.0	3.78 ± 11.89	4.67 ± 14.72	|$2.47^{+ 2.97}_{- 2.47}$|	|$1.26^{+ 1.30}_{- 1.26}$|	|$5.0^{+170.0}_{- 5.0}$|	0.71	0.0215
≤25.3	5.43 ± 8.12	6.70 ± 10.14	|$3.02^{+ 2.02}_{- 3.02}$|	|$1.51^{+ 0.88}_{- 1.51}$|	|$15.5^{+117.8}_{-15.5}$|	0.61	0.0309
≤25.5	5.47 ± 6.34	6.75 ± 7.97	|$3.03^{+ 1.64}_{- 3.03}$|	|$1.51^{+ 0.72}_{- 1.51}$|	|$15.8^{+85.5}_{-15.8}$|	1.07	0.0311
≤25.8	2.61 ± 3.43	3.22 ± 4.29	|$2.01^{+ 1.21}_{- 2.01}$|	|$1.04^{+ 0.55}_{- 1.04}$|	|$1.2^{+20.0}_{- 1.2}$|	0.94	0.0148
≤26.3	8.62 ± 1.49	10.64 ± 2.99	|$3.90^{+ 0.58}_{- 0.65}$|	|$1.90^{+ 0.25}_{- 0.29}$|	|$50.2^{+35.9}_{-28.0}$|	1.66	0.0490
Field average (number of fields)
≤25.0 (4)	4.69 ± 1.70	5.80 ± 2.46	|$2.78^{+ 0.60}_{- 0.74}$|	|$1.40^{+ 0.27}_{- 0.34}$|	|$10.1^{+17.0}_{- 8.8}$|	0.75
≤25.3 (4)	4.04 ± 0.90	4.99 ± 1.57	|$2.56^{+ 0.42}_{- 0.48}$|	|$1.30^{+ 0.19}_{- 0.22}$|	|$6.3^{+ 8.3}_{- 4.8}$|	2.04
≤25.5 (4)	3.55 ± 0.58	4.39 ± 1.21	|$2.38^{+ 0.34}_{- 0.39}$|	|$1.22^{+ 0.16}_{- 0.18}$|	|$4.0^{+ 5.1}_{- 2.9}$|	1.01
≤25.8 (3)	2.75 ± 0.45	3.40 ± 0.94	|$2.07^{+ 0.30}_{- 0.34}$|	|$1.07^{+ 0.14}_{- 0.16}$|	|$1.5^{+ 2.4}_{- 1.2}$|	1.08
≤26.3 (1)	8.62 ± 1.49	10.64 ± 2.99	|$3.90^{+ 0.58}_{- 0.65}$|	|$1.90^{+ 0.25}_{- 0.29}$|	|$50.2^{+35.9}_{-28.0}$|	1.66

*(1) The best-fitting correlation amplitude without f_c correction; (2) the best-fitting correlation amplitude with f_c correction used to derive (3)–(5); (3) the best-fitting (contamination-corrected) correlation length; (4) the best-fitting effective bias factor (contamination-corrected); (5) the best-fitting effective dark matter halo mass (contamination-corrected); (6) reduced chi-squared value; (7) the best-fitting integral constant; the value in parentheses shows the number of fields used to calculate the field-average correlation amplitude using equation (14).

3.3 Bias factor

The galaxy-matter bias, b_g, is defined as

\begin{equation} b_{\rm g}(r)=\sqrt{\frac{\xi (r)}{\xi _{\rm DM}(r,z)}}, \end{equation}

(17)

where ξ_DM(r, z) is the spatial correlation function of underlying dark matter,

\begin{equation} \xi _{\rm DM}(r,z)= \int \frac{k^2 {dk}}{2\pi ^2} \frac{\sin (k{r})}{k{r}}P_{m}(k,z), \end{equation}

(18)

where P_m(k, z) is the linear dark matter power spectrum as a function of wavenumber, k, at redshift z (Eisenstein & Hu 1999) with the Eisenstein and Hu (1998) transfer function. We estimate the effective galaxy-matter bias, b_{g, eff}, at |$r=8{\,\,} h^{-1}_{100}\:$|Mpc following previous clustering analyses (e.g., Ouchi et al. 2003) using a suite of cosmological codes called Colossus (Diemer & Kravtsov 2015).

Figure 4a shows b_{g, eff} for the cumulative subsamples in the four fields, where Lyα luminosity limits are calculated from the limiting |$\mathit {NB387}$| magnitudes of the subsamples. We find that the average bias value of our LAEs [represented by black stars in panel (a) and also by red stars in panel (b)] does not significantly change with the Lyα luminosity limit. A possible change in b_{g, eff} over L_Lyα ≃ (3–10) × 10⁴¹ erg s⁻¹ is less than 20% since the uncertainties in the average biases are ∼10%–20%.

Fig. 4.

Bias value plotted against Lyα limiting luminosity for the four fields. (a) Orange squares, green circles, magenta inverted triangles, and blue triangles represent the SXDS, COSMOS, HDFN, and CDFS fields, respectively. Black stars indicate the average (weighted mean) over available fields at each limiting luminosity [also shown by red stars in panel (b)]. For presentation purposes, we slightly shift all of the points except for black stars along the abscissa. (b) The measurements shown by small black stars in panel (a) are plotted by small red stars except for the value at Lyα_limit ≃ 6 × 10⁴¹ erg s⁻¹ (or |$\mathit {NB387}_{\rm tot} \le 25.5\:$|mag) shown by a large red star. Guaita et al.’s (2010) measurement is also plotted by a blue circle. (Color online)

This weak dependence may be partly due to radiative transfer effects on Lyα photons. Star-forming galaxies in more massive (i.e., larger-bias) halos are thought to have higher SFRs and thus brighter nebular emission lines. Indeed, Cochrane et al. (2017) have found a significant positive correlation between Hα luminosity and bias for bright z = 2.23 HAEs, indicating a similarly strong correlation between intrinsic Lyα luminosity and bias for bright galaxies. However, such a strong correlation, if any, weakens when observed (escaped) Lyα luminosity is used in place, because brighter (i.e., more massive) galaxies have lower Lyα escape fractions, |$f_{\rm esc}^{\rm Ly\alpha }$| (e.g., Vanzella et al. 2009; Matthee et al. 2016). Indeed, our cumulative subsamples do not show a significant correlation between the observed Lyα luminosity and the total SFR (derived from SED fitting in the same manner as described in section 4), but rather show a positive correlation between the observed Lyα luminosity and the Lyα escape fraction, where the intrinsic Lyα luminosity is calculated from the total SFR (Brocklehurst 1971; Kennicutt 1998).

Moreover, some previous studies have found that high-redshift UV-selected galaxies with comparably faint UV luminosities (L_UV) to our LAEs (the average absolute magnitude of our LAEs is M_UV ∼ −19 mag) have weak dependence of b_g on UV luminosity [z ∼ 3–4 Lyman break galaxies (LBGs): Ouchi et al. 2004, 2005; Harikane et al. 2016; Bielby et al. 2016; see, however, Lee et al. 2006, who find significant dependence for z ∼ 4–5 LBGs], suggesting that the correlation between intrinsic Lyα luminosity and bias is not so strong for typical LAEs with modest Lyα luminosities.

The faintest limiting Lyα luminosity at which b_{g, eff} measurements are available for all four fields is L_Lyα = 6.2 × 10⁴¹ erg s⁻¹ (corresponding to 25.5 mag in |$\mathit {NB387}$|⁠). In order to reduce the uncertainty due to cosmic variance as much as possible, we adopt the average b_{g, eff} at this limiting luminosity, |$b_{\rm g,{\,\,} eff}^{\rm ave}=1.22^{+0.16}_{-0.18}$|⁠, as the average b_{g, eff} of our entire sample.

This average bias is lower than that of the previous work on narrow-band-selected LAEs at z ∼ 2.1, b_{g, eff} = 1.8 ± 0.3 [Guaita et al. 2010: see the blue po-int in panel (b) of figure 4], with a probability of 96%. The median Lyα luminosity of their sample is L_Lyα = 1.3 × 10⁴² erg s⁻¹ and their 5σ detection limit in Lyα luminosity is L_Lyα = 6.3 × 10⁴¹ erg s⁻¹, which is similar to the luminosity limit of our |$\mathit {NB387}\le 25.5$| samples. Our clustering method is essentially the same as of Guaita et al. (2010), and in both studies the bias value is calculated at |$r=8{\,\,} h^{-1}_{100}\:$|Mpc. Although we use a slightly different cosmological parameter set, (Ω_m, Ω_Λ, h, σ₈) = (0.3, 0.7, 0.7, 0.8), from Guaita et al.’s (2010), (Ω_m, Ω_Λ, h, σ₈) = (0.26, 0.74, 0.7, 0.8), using their set changes b_{g, eff} only negligibly. Our contamination fraction, f_c = 10 ± 10%, is comparable to or slightly more conservative than theirs, f_c = 7 ± 7%. The error in Guaita et al.’s (2010) b_{g, eff} is a quadrature sum of the uncertainty in f_c and the fitting error (statistical error), with the latter dominating because of the small sample size (250 objects). As discussed in subsection 3.4, their high b_{g, eff} value is attributable to cosmic variance since their survey area is approximately one third of ours (see figure 5b). Indeed, the sky distribution of their LAEs has a large-scale excess at the northwest part, and the ACF measurements seem to deviate to higher values from the best-fitting power law at large scales because of it.⁴

Fig. 5.

Effect of cosmic variance on clustering analysis. (a) Uncertainties in the amplitude of the dark matter ACF as a function of survey area. Green squares and a light-green dashed line denote the empirical measurements at z ∼ 0.8 and power law that best fits them, respectively, by Sobral et al. (2010: Δω_gal/ω_gal). Other lines show our analytic calculations for four NB surveys: green solid line for Sobral et al. (2010), light-gray thick solid line for this study (Suprime-Cam/|$\mathit {NB387}$|⁠), blue solid line for Guaita et al. (2010), and black dashed line for an on-going Hyper Suprime-Cam/|$\mathit {NB387}$| survey (see subsection 6.5). (b) Effective bias factor as a function of survey area. The cosmic variance on |$b_{\rm g,{\,\,} eff}^{\rm ave}$|⁠, which is indicated by a light-gray thick solid line in panel (a), is shown by a light-gray filled region around |$b_{\rm g,{\,\,} eff}^{\rm ave}$| (fixed) shown by a dark gray dashed line. A red star and a blue circle indicate the |$b_{\rm g,{\,\,} eff}^{\rm ave}$| in this work and the b_{g, eff} in Guaita et al. (2010), respectively, where colored error bars include the uncertainty due to cosmic variance while black bars next to them do not. A black circle corresponds to the expected HSC/|$\mathit {NB387}$| survey area when completed. A small orange square, green circle, magenta inverted triangle, and blue triangle represent b_{g, eff} with |$\mathit {NB387}\le 25.5\:$|mag from SXDS, COMOS, HDFN, and CDFS, respectively. (Color online)

3.4 Cosmic variance on bias factor

Our average effective bias value and that of Guaita et al. (2010) are not consistent within the 1σ uncertainties, in spite of the similar limiting Lyα luminosities. Biases derived from limited survey areas possibly suffer from cosmic variance due to spatial variations in the ACF of dark matter. We analytically estimate cosmic variance in the bias value derived from clustering analysis for the first time. With the ACF the galaxy-matter bias can be expressed as |$b(\theta ) = \sqrt{\omega _{\rm gal}(\theta ) / \omega _{\rm DM}(\theta )}.$| Assuming that the cosmic variance in b originates solely from the spatial variation of the dark matter ACF, we can express the b of a given galaxy sample in a given survey field as

\begin{equation} b({\rm field}) = \sqrt{ \frac{\omega _{\rm DM}({\rm field})}{\langle \omega _{\rm DM}\rangle } \frac{\omega _{\rm gal}({\rm field})}{\omega _{\rm DM}({\rm field)}}} = \sqrt{ \frac{\omega _{\rm DM}({\rm field})}{\langle \omega _{\rm DM}\rangle }} b_{\rm int},\!\!\!\!\! \end{equation}

(19)

where 〈ω_DM〉 is the cosmic average of the dark matter ACF, ω_DM(field) is the dark matter ACF in the field, ω_gal(field) is the observed galaxy ACF in the field, and

\begin{equation} b_{\rm int} \equiv \sqrt{\frac{\omega _{\rm gal}({\rm field})}{\omega _{\rm DM}({\rm field)}}} \end{equation}

(20)

is the intrinsic bias of this galaxy population, which we assume to be unchanged from field to field (parameter θ is omitted for clarity). This assumption is the same as the one adopted to predict cosmic variance in number density (e.g., Moster et al. 2011), as explained below. Field-to-field fluctuations of number density, |$\sigma _{\rm ND,{\,\,} g^{\prime }}$|⁠, are assumed to come from field-to-field fluctuations of dark matter distribution (i.e., cosmic variance in the density of dark matter), σ_{ND, DM}, as

\begin{equation} \sigma _{\rm ND,{\,\,} g^{\prime }}=b_{\rm g^{\prime }}\ \sigma _{\rm ND,{\,\,} DM}, \end{equation}

(21)

where the intrinsic galaxy bias, |$b_{\rm g^{\prime }}$|⁠, is uniform and independent of fields by definition. We also assume that |$\omega _{\rm gal}^2({\rm field})$| is proportional to |$\omega _{\rm DM}^2({\rm field)}$| by a factor of b_int.

The covariance in ω_DM between two angular separations for area Ω_s is given by the first term of equation (19) of Cohn (2006):⁵

\begin{eqnarray} {\rm Cov}(\omega _{\rm DM}(\theta ),\ \omega _{\rm DM}(\theta ^{\prime })) = \frac{1}{\pi \Omega _{\rm s}}\int \ {\rm K}\ {\rm dK} J_0({\rm K}\theta ) _0({\rm K}\theta ^{\prime })\ {\rm P^2_2({\rm K})},\!\!\!\!\!\!\! \nonumber\\ \end{eqnarray}

(22)

where K, P₂(K) and J₀(Kθ) are the Fourier transform of θ, the projected power spectrum calculated using the redshift distribution defined by the filter, and the zeroth-order Bessel function of the first kind, respectively. With this equation we calculate ω_DM and its standard deviation, σ_DM, for the three angular bins used to determine the A_ω of our LAEs. We then fit a power-law correlation function to those values in the same manner as for observed data but also considering the intrinsic covariance given in equation (22), and obtain the relative uncertainty in A_ω due to the variation in ω_DM, |$\frac{\Delta {\omega _{\rm DM}}}{{\omega _{\rm DM}}}$|⁠. According to equation (22), the relative uncertainty in A_ω depends on Ω_s as:

\begin{equation} \frac{\Delta {\omega _{\rm DM}}}{{\omega _{\rm DM}}} \propto \Omega _{\rm s}^{-0.5}, \end{equation}

(23)

as shown by the light-gray solid line in figure 5a.

We find Δω_DM/ω_DM ≃ 53% for Ω_s = 0.25 deg², a typical area of the four survey fields, and ≃ 26% for the entire survey area (≃ 1 deg²).

Sobral et al. (2010) have empirically estimated relative uncertainties in ACF measurements for NB-selected z = 0.85 HAEs as a function of area by dividing their survey regions, ≃ 1.3 deg² in total, into sub-regions with different sizes (green squares in figure 5a). This empirical relation has been used to estimate cosmic variance in ACF measurements in a ≃ 2 deg² survey area of emission line galaxies at z ∼ 0.8–4.7 in Khostovan et al. (2017). Our analytic method applied to the Sobral et al. (2010) survey with their own NB filter (over the same fitting range of θ as that for our LAEs, for simplicity), however, gives larger uncertainties, as shown by the green solid line in figure 5a. This may be partly because the area of Sobral et al.’s (2010) survey is not large enough to catch the total variance. Our analytic estimation seems to be more conservative than theirs.

We expect that Guaita et al.’s (2010) b_{g, eff} obtained from a ∼0.28 deg² area also has a ≃ 51% uncertainty using their NB3727 filter (solid blue line in figure 5a). The 1σ uncertainty in an observed bias including cosmic variance, Δb_{g, eff, CV}, is given by

\begin{eqnarray} \frac{\Delta b_{\rm g,{\,\,}eff,{\,\,} CV}}{ b_{\rm g,{\,\,}eff}} &\simeq & \frac{1}{2}\sqrt{\left(\frac{\Delta A_{\omega }}{A_{\omega }}\right)^2 + \left(\frac{2\Delta {f_c}}{f_c}\right)^2 + \left(\frac{\Delta {\omega _{\rm DM}}}{{\omega _{\rm DM}}}\right)^2 } \end{eqnarray}

(24)

\begin{eqnarray} &\simeq &\frac{1}{2}\sqrt{ \left(\frac{2\Delta {b_{\rm g,{\,\,}eff}}}{b_{\rm g,{\,\,} eff}}\right)^2+ \left(\frac{\Delta {\omega _{\rm DM}}}{{\omega _{\rm DM}}}\right)^2 }, \end{eqnarray}

(25)

where Δb_{g, eff} is the 1σ error in b_{g, eff}.

By updating the errors using this equation (where for our b_{g, eff} the plus and minus errors are treated separately), our average effective bias and that of Guaita et al. (2010) are written as |$b_{\rm g,{\,\,} eff}^{\rm ave}=1.22^{+0.23}_{-0.26}$| and b_{g, eff} = 1.8 ± 0.55, respectively, thus becoming consistent with each other within the errors (see figure 5b). We also note that the relatively large scatter of b_{g, eff} among the four fields at each limiting Lyα luminosity seen in figure 4a may be partly due to cosmic variance although the observational errors are too large to confirm it (see figure 5b). All the best-fitting b_{g, eff} values for the four fields fall within the 1σ uncertainty range from cosmic variance, as shown by the shaded light-gray region in figure5b.

3.5 Dark matter halo mass

We estimate the effective dark matter halo masses from b_{g, eff} directly assuming that each halo hosts only one galaxy and that our sample has a narrow range of dark matter halo mass. We use the formula of bias and peak height in the linear density field, ν, given in Tinker et al. (2010), which is based on a large set of collisionless cosmological simulations in flat ΛCDM cosmology. The obtained ν is converted to the effective dark matter halo mass with the top-hat window function and the linear dark matter power spectrum (Eisenstein & Hu 1998, 1999) using a cosmological package for Python called CosmoloPy.⁶

The effective halo mass of each sub-sample is listed in table 3. The field average of effective halo masses corresponding to the field average of effective biases of our LAEs with |$\mathit {NB387}_{\rm tot}\le 25.5\:$|mag, |$b_{\rm g,{\,\,} eff}^{\rm ave}=1.22^{+0.16}_{-0.18}$|⁠, is |$4.0_{-2.9}^{+5.1}\times 10^{10}{\,\,}M_{\odot }$|⁠. This value is roughly comparable to previous measurements for z ∼ 3–7 LAEs with similar Lyα luminosities, M_h ≃ 10¹⁰–10¹² M_⊙ (e.g., Ouchi et al. 2005, 2010, 2018; Kovač et al. 2007; Gawiser et al. 2007; Shioya et al. 2009; Bielby et al. 2016; Diener et al. 2017), suggesting that the mass of dark haloes which can host typical LAEs is roughly unchanged with time.

The average M_h of our LAEs is smaller than those of HAEs at z ∼ 1.6 (Kashino et al. 2017), M_h ∼ 7 × 10¹² M_⊙, and at z ∼ 2.2, a few times 10¹² M_⊙ (Cochrane et al. 2017). The typical dust-corrected Hα luminosity, L_{Hα, corr}, of our LAEs is estimated to be 4.3 ± 0.9 × 10⁴¹ erg s⁻¹ from the SFR obtained by SED fitting in section 4 using the conversion formula given in Kennicutt (1998) on the assumption of case B recombination. This Hα luminosity corresponds to an effective halo mass of |$M_{\rm h, eff}=5.2^{+4.8}_{-2.7}\times 10^{10}{\,\,}M_{\odot }$| according to the redshift-independent relation between the normalized luminosity |$L_{\rm H\alpha ,{\,\,} corr}/L_{\rm H\alpha }^{\star }(z)$| and M_{h, eff} found by Cochrane et al. (2017). The estimated halo mass of our LAEs, |$M_{\rm h}=4.0_{-2.9}^{+5.1} \times 10^{10}{\,\,}M_{\odot }$|⁠, is thus consistent with this relation. This result supports the result by Shimakawa et al. (2017) and Hagen et al. (2016) that the stellar properties of LAEs at z ∼ 2–3 do not significantly differ from those of other emission galaxies, such as HAEs and [O iii] emitters. However, Cochrane et al. (2017) assume a constant dust attenuation against Hα luminosity, A_Hα = 1.0 mag, for all HAEs, which is larger than that of our LAEs, A_Hα ∼ 0.13 ± 0.04 mag, derived from the average E(B − V) in section 4. If the (extrapolated) relation overestimates L_{Hα, corr} at low halo masses owing to overestimation of A_Hα, then the true log–log slope of L_{Hα, corr} as a function of M_h would be steeper, implying that our LAEs would lie above the relation (see also subsection 5.2 and figure 10).

4 SED fitting

We derive parameters that characterize the stellar populations of LAEs with NB_tot ≤ 25.5 mag in each of the four fields by fitting SEDs based on stacked multiband images. This threshold magnitude is the same as that adopted in the clustering analysis to determine the average halo masses. We only use 170 objects (∼14% of the entire sample, 1248) that have data in 10 broadband filters (B, V, R, i, z, J, H, K, ch1, and ch2) and are not contaminated by other objects in the IRAC images (section 2.1 and table 2). The procedure to select “IRAC-clean” objects is described in the next subsection.

4.1 Selection of IRAC-clean objects

The IRAC images have lower spatial resolution (i.e., larger FWHMs of the PSF) compared with images in other bands. Moreover, they have large-scale residual backgrounds (contaminated sky regions) around bright objects and in crowded regions due to the extended profile of the IRAC PSF. Contamination by nearby objects and large-scale sky residuals can give significant systematic errors in the photometry of stacked images because our LAEs are expected to have very low stellar masses, or very faint IRAC magnitudes. To minimize such contamination, we select clean LAEs through a two-step process.

First, we exclude all LAEs which have one or more neighbors. Assuming that objects bright in IRAC are similarly bright in the K band, we exclude all LAEs which have one or more K-detected objects with a separation between 0.85″ and 4.5″; an object within 0.85″ separation is conservatively considered to be the counterpart to the LAE (the typical separation is ∼0.2″; see subsection 2.3 for the K-detected catalogs).⁷ 4.5″ is 2.5 times larger than the PSF size of IRAC ch1.

Secondly, we exclude all LAEs with a high sky background, as determined in the following manner. For each field, we randomly select 5000 positions with no K-band objects within 4|${^{\prime\prime}_{.}}$|5 (i.e., passing the first step) and measure the sky background in an annular region of 3|${^{\prime\prime}_{.}}$|5 radius centered at these positions. We then make a histogram of the sky background values, which is skewed toward higher values because of contamination by bright or crowded objects outside of the 4|${^{\prime\prime}_{.}}$|5 radius. We fit a Gaussian to the low-flux side (including the peak) of the histogram and obtain its average, μ_rand, which we consider to be the true sky background. If cutout images at all the random positions are median-stacked, the annular-region sky background will be brighter than μ_rand. A similar systematic sky-background difference will also be seen when all LAEs are stacked, possibly introducing some systematic errors in photometry. The sky background of the median-stacked random image becomes equal to μ_rand if positions whose sky background is higher than a certain threshold, sky_thres, are removed, where sky_thres can be determined so that the total number of the remaining positions (i.e., positions with faint sky background below sky_thres) is twice as large as the number of positions below μ_rand. Thus, we conservatively remove LAEs with a higher annular-region sky background than sky_thres, and are left with 93, 21, 56, and 4 IRAC-clean LAEs in SXDS, COSMOS, HSFN and CDFS, respectively. The stacked flux densities of the IRAC-clean LAEs in the B to K bands are mostly consistent with those of the all LAEs before cleaning.

4.2 Stacking analysis and photometry

We perform a stacking analysis for each subsample in almost the same manner as Nakajima et al. (2012) and Kusakabe et al. (2015). Images of size 50″ × 50″ are cut out at the position of LAEs in the |$\mathit {NB387}$| image with IRAF/imcopy task. For each of the B to K bands of the SXDS field, PSFs are matched to the largest among the SXDS-C, -N, and -S sub-fields using the IRAF/gauss task (see table 1). We use the IRAF/imcombine task to create a |$\mathit {NB387}$|-centered median image. While a stacked SED is not necessarily a good representation of individual objects (Vargas et al. 2014), stacking is still useful for our faint objects to obtain a SED covering rest-frame ∼1000–10000 Å.

An aperture flux is measured for each stacked image using the task PyRAF/phot. Following Ono et al. (2010b), we use an aperture diameter of 2″ for the |$\mathit {NB387}$|-, optical-, and NIR-band images and 3″ for the MIR (IRAC) images. For the |$\mathit {NB387}$|- to K-band images, the inner radius of the annulus to measure the sky flux is set to twice the FWHM of the largest PSF among these images, and the area of the annulus is set to five times larger than that of the aperture.⁸ For each of the ch1 and ch2 images, we obtain the net 3″-aperture flux density of LAEs by subtracting the offset between the annular-region and the 3″-aperture flux densities of the stacked image of IRAC-clean random positions generated in the previous subsection from the 3″-aperture flux density of the LAE image (output of the PyRAF/phot task).⁹

We use the original zero-point magnitudes (ZPs) from references given in subsection 2.3, although some previous work argues that some ZPs need to be corrected (e.g., Yagi et al. 2013; Skelton et al. 2014), especially since the direction of the correction given by Yagi et al. (2013) is opposite to that by Skelton et al. (2014) for optical bands of the SXDS field. All aperture magnitudes are corrected for Galactic extinction, E(B − V)_b, of 0.020, 0.018, 0.012, and 0.008 for the SXDS, COSMOS, HDFN, and CDFS fields, respectively (Schlegel et al. 1998).

The aperture magnitudes are then converted into total magnitudes using the aperture correction values summarized in table 1 (see also subsection 2.3). The stacked SEDs thus obtained for individual subsamples are shown in figure 6. The errors include photometric errors and errors in aperture correction and the ZP. For the ch1 and ch2 data, errors in sky subtraction, ∼0.02–0.17 mag, are also included. The photometric errors are determined following the procedure of Kusakabe et al. (2015). The aperture correction errors in the |$\mathit {NB387}$|⁠, optical, and NIR bands are estimated to be less than 0.03 mag, and those in the ch1 and ch2 bands are set to 0.05 mag. We adopt 0.1 mag as the ZP error for all bands, which is the typical value of the offsets of the images used in this paper (e.g., Yagi et al. 2013; Skelton et al. 2014) and is twice as large as those adopted in previous studies (e.g., Nakajima et al. 2012).

Fig. 6.

Results of SED fitting to stacked LAEs with |$\mathit {NB387}_{\rm tot} \le 25.5\:$|mag in the SXDS, COSMOS, HDFN, and CDFS fields from panels (a) to (d). For each panel, a gray solid line and a light-gray dotted line show the best-fitting model spectrum and its stellar continuum component, respectively. The difference of these two lines shows a contribution of its nebular continuum component. Red filled circles and black filled triangles represent the observed flux densities and the flux densities calculated from the best-fitting spectrum, respectively. (Color online)

4.3 SED models

We perform SED fitting on the stacked SEDs to derive stellar population parameters in a similar manner to Kusakabe et al. (2015). Nebular emission (lines and continuum) is added to the stellar population synthesis model of GALAXEV with constant SF history and 0.2 Z_⊙ stellar metallicity, following previous SED studies of LAEs (Bruzual & Charlot 2003; Ono et al. 2010b; Vargas et al. 2014). We assume an SMC-like dust extinction model for the attenuation curve (hereafter an SMC-like attenuation curve; Gordon et al. 2003), which is suggested to be more appropriate for LAEs at z ∼ 2 than the Calzetti curve (Calzetti et al. 2000) used by Kusakabe et al. (2015) and at z ≥ 2 by Reddy et al. (2017) for star-forming galaxies.¹⁰ We also examine the case of the Calzetti attenuation curve for comparison (see appendix 1.1). We also assume E(B − V)_gas = E(B − V)_⋆ (Erb et al. 2006). The Lyman continuum escape fraction, |$f^{\rm ion}_{\rm esc}$|⁠, is fixed to 0.2 considering recent observations of |$f^{\rm ion}_{\rm esc} \sim 0.1\mbox{--}0.3$| for z ∼ 3 LAEs by Nestor et al. (2013).¹¹ This means that 80% of ionizing photons produced are converted into nebular emission (see Ono et al. 2010b).

For each field’s stacked SED we search for the best-fitting model SED that minimizes χ² and derive the following stellar parameters: stellar mass (M_⋆), color excess [E(B − V)_⋆ or UV attenuation of A₁₆₀₀], age, and SFR. Stellar masses are calculated by solving ∂χ²/∂M_⋆ = 0 since it is the amplitude of the model SED. SFR is not a free parameter in the fit but determined from M_⋆ and age, and thus the degree of freedom is 7. The 1σ confidence interval in these stellar parameters is estimated from |$\chi ^2_{\rm min} +1$|⁠, where |$\chi ^2_{\rm min}$| is the minimum χ² value.

4.4 Results of SED fitting

Table 4 summarizes the best-fitting parameters and figure 6 compares the best-fitting SEDs with the observed SEDs.¹² The mean value for each parameter over the four fields is M_⋆ = 10.2 ± 1.8 × 10⁸ M_⊙, A₁₆₀₀ = 0.6 ± 0.1 mag, age =3.8 ± 0.3 × 10⁸ yr, and |$\mathit {SFR} = 3.4\pm 0.4{\,\,}M_{\odot }\:$|yr⁻¹. We discuss the IR excess and the SF mode in the following subsections using the results with an SMC-like curve.

Table 4.

Results of SED fitting.*

Field	M⋆	E(B − V)⋆ [A1600]	Age	SFR	|$\chi ^2_r$|
	(108 M⊙)	(mag)	(108 yr)	(M⊙ yr−1)
	(1)	(2)	(3)	(4)	(5)
SXDS	|$9.7^{+3.6}_{-1.7}$|	|$0.05^{+0.01}_{-0.02}\ [0.6^{+0.1}_{-0.2}]$|	|$3.6^{+2.8}_{-1.1}$|	|$3.3^{+ 0.5}_{- 0.7}$|	0.604
COSMOS	|$14.0^{+3.4}_{-3.6}$|	|$0.07^{+0.02}_{-0.02}\ [0.8^{+0.2}_{-0.2}]$|	|$4.1^{+2.4}_{-1.8}$|	|$4.2^{+ 1.2}_{- 0.8}$|	0.473
HDFN	|$7.6^{+4.0}_{-1.9}$|	|$0.06^{+0.02}_{-0.03}\ [0.7^{+0.2}_{-0.4}]$|	|$3.2^{+4.0}_{-1.4}$|	|$2.9^{+ 0.8}_{- 0.8}$|	1.298
CDFS	|$10.3^{+11.1}_{-9.7}$|	|$0.02^{+0.07}_{-0.01}\ [0.2^{+0.8}_{-0.1}]$|	|$5.7^{+8.6}_{-5.7}$|	|$2.2^{+534}_{- 0.4}$|	0.120
Average	10.2 ± 1.8	0.06 ± 0.01 [0.6 ± 0.1]	3.8 ± 0.3	3.4 ± 0.4

*(1) The best-fitting stellar mass; (2) the best-fitting color excess [UV attenuation]; (3) the best-fitting age; (4) the best-fitting SFR; (5) reduced chi-squared value. The UV attenuation is derived from an SMC-like attenuation curve. Metallicity, redshift, and |$f_{\rm esc}^{\rm ion}$| are fixed to 0.2 Z_⊙, 2.18, and 0.2, respectively.

While the SMC-like and Calzetti attenuation curves fit the data equally well, the resulting parameter values are different (see appendix 1.1 and figure 13). The Calzetti curve tends to give a smaller stellar mass, a higher attenuation, a younger age, and a higher SFR as the best-fitting value compared with an SMC-like curve. The difference in the average stellar mass is a factor of ∼3 but that in the average SFR reaches a factor of ∼4.

4.4.1 M_⋆–IRX relation

As shown in figure 7, galaxies with higher stellar masses tend to have higher infrared excesses, IRX ≡ L_IR/L_UV, where L_IR is the IR luminosity (see also footnote 13), which is an indicator of dustiness (the consensus relation: Reddy et al. 2010; Whitaker et al. 2014; Bouwens et al. 2016). The dust emission of typical LAEs with M_⋆ ∼ 10⁹ M_⊙ is too faint to be detected, although a few LAEs at z ∼ 2–3 are detected by Herschel/PACS and Spitzer/MIPS (e.g., Pentericci et al. 2010; Oteo et al. 2012). In order to compare IRXs and stellar masses of LAEs with the consensus relation, we convert the A₁₆₀₀ of our LAEs obtained above to IRXs using equation (1) in Overzier et al. (2011).¹³ We find that our LAEs are located near an extrapolation of the consensus relation (see filled color symbols in figure 7). Their IRX values are also consistent with that [≲ 2.0 (3σ)] of typical LAEs obtained by Kusakabe et al. (2015), who constrain the upper limit of the IR luminosity from stacked Spitzer/MIPS 24 μm images.¹⁴ While unlikely, for our LAEs to require a Calzetti attenuation curve, they would be dusty galaxies whose values of IRX are more than 10 times higher than expected from the extrapolated consensus relation (see open colored symbols in figure 7) and comparable to those of 10 times more massive average galaxies.

Fig. 7.

 IRX vs M_⋆. Dim gray squares, dark gray circles, a black square, and a light-gray solid band represent, respectively, 3D-HST galaxies at z ∼ 2 in Whitaker et al. (2014), UV-selected galaxies at z ∼ 2 in Reddy et al. (2010), LBGs at z ∼ 2–3 in Bouwens et al. (2016), and the consensus relation of them determined by Bouwens et al. (2016), with its extrapolation indicated by a gray striped band (see also footnote 1). A filled (open) orange square, green circle, magenta inverted triangle, and blue triangle indicate the SXDS, COSMOS, HDFN, and CDFS fields, respectively, on the assumption of an SMC-like attenuation curve (the Calzetti curve). An open blue square represents the 3σ upper limit of stacked LAEs at z ∼ 2 with IR observations in Kusakabe et al. (2015). All data are rescaled to a Salpeter IMF according to footnote 1. (Color online)

4.4.2 M_⋆–SFR relation

The mode of SF in star-forming galaxies can be divided into two categories: the main-sequence (MS) mode, where galaxies form stars at moderate rates, making a well-defined sequence in the SFR–M_⋆ plane (SFMS: e.g., Elbaz et al. 2007; Speagle et al. 2014), and the burst mode, where galaxies have much higher specific SF rates, sSFRs|$(=\mathit {SFR}/M_\star )$|⁠, than MS galaxies with similar masses (e.g., Rodighiero et al. 2011). While it is well established that LAEs are mostly low-mass galaxies, which mode they typically have is still under some debate because of differences in SFR estimates.

The SFMS itself at z ∼ 2 has been determined well using rest UV to far-infrared (FIR) data at M_⋆ ≳ 10¹⁰ M_⊙ (e.g., Whitaker et al. 2014; Tomczak et al. 2016). Below this stellar mass, the SFMS is suggested to continue at least down to M_⋆ ∼ 10⁸–10⁹ M_⊙, keeping its power-law slope unchanged [e.g., by Santini et al. 2017, using gravitationally-lensed galaxies in the Hubble Space Telecope (HST) Frontier Fields], although SFRs have large uncertainties since without FIR data. In this paper, we simply extrapolate the SFMS given in the literature (Daddi et al. 2007; Tomczak et al. 2016; Shivaei et al. 2017) towards lower masses without changing the power-law slope.

Figure 8b shows previous results for LAEs at z ∼ 2–2.5. Hagen et al. (2016) have found that bright, individually detected LAEs lie along or above the SFMS, while Shimakawa et al. (2017) have found that fainter individually detected LAEs lie on the SFMS. Guaita et al.’s (2010) estimates based on stacking analysis have errors that are too large to distinguish the SF mode, although they are consistent with the MS mode. Kusakabe et al. (2015) have stacked IR and UV images of z ∼ 2 LAEs to show that they are MS galaxies on average.

Fig. 8.

 SFR plotted against M_⋆. (a) An orange square, green circle, magenta inverted triangle, and blue triangle represent stacked LAEs with |$\mathit {NB387}_{\rm tot} \le 25.5\:$|mag in the SXDS, COSMOS, HDFN, and CDFS fields, respectively, and a red star shows the average over the four fields. The orange square and the red star overlap with each other. A blue open rectangle denotes the permitted range for stacked LAEs from L_UV and L_IR in Kusakabe et al. (2015). Light-gray dots, dim gray squares, and dim gray circles indicate BzKs from Rodighiero et al. (2011), BzKs from Lin et al. (2012), and 3D-HST galaxies from Whitaker et al. (2014), respectively. Black thin, medium, and thick solid lines represent the SF main sequence at z ∼ 2 in Tomczak et al. (2016, hererafter T16), Shivaei et al. (2017, hereafter S17), and Daddi et al. (2007), respectively (determined well using L_UV and L_IR), with extrapolated parts shown by dashed lines. (b) Same as panel (a) but LAEs taken from the literature are also plotted. Cyan squares and light green pentagons show individual LAEs at z ∼ 2 from Hagen et al. (2016) and Shimakawa et al. (2017), respectively. A blue circle indicates stacked LAEs at z ∼ 2 from Guaita et al. (2011). SFRs in Hagen et al. (2016) and Shimakawa et al. (2017) are derived from the IRX–β relation with the Calzetti curve (Meurer et al. 1999) and SFRs in Guaita et al. (2011) are derived from SED fitting with the Calzetti curve, while SFRs in this work are derived from SED fitting with an SMC-like curve. We also show our results with the IRX–β and SED fitting with the Calzetti curve in figure 15. All data are rescaled to a Salpeter IMF according to footnote 1. (Color online)

The M_⋆ and SFR of our LAEs averaged over the four fields are M_⋆ = 10.2 ± 1.8 × 10⁸ M_⊙ and SFR =3.4 ± 0.4 M_⊙ yr⁻¹. Thus, our LAEs are on average placed near a lower-mass extrapolation of the SFMS, as shown by a red star in figure 8b, confirming the result obtained by Kusakabe et al. (2015) with a 6 times larger survey area and using deep IRAC data. We also see in figure 8a that the LAEs in individual fields also lie on the extrapolated SFMS, although that in the CDFS has large uncertainties (blue triangle in figure 8a). This result is unchanged even when we stack all objects including those with |$\mathit {NB387}_{\rm tot}\ge 25.5\:$|mag.

Hagen et al.’s (2016) sample is a mixture of two samples: bright, spectroscopically selected LAEs at z = 1.90–2.35 from the HETDEX survey (L_Lyα > 10⁴³ erg s⁻¹: Hagen et al. 2014) and bright, NB selected LAEs at z ≃ 2.1 from Guaita et al. (2010) and Vargas et al. (2014) with a counterpart in the 3D-HST catalog. They derive SFRs from the IRX–β relation with the Calzetti curve. Note that we also find our LAEs to have higher sSFRs similar to theirs if we use the Calzetti curve, as shown later in figures 15a–15c.¹⁵ They also expect that their objects would move downward toward the SFMS in the M_⋆–SFR plane if they adopt an SMC-like curve. Shimakawa et al. (2017) select LAEs using a narrow-band [NB ≤ 26.55 mag (5σ)] and only include those with a counterpart in the 3D-HST catalogue (Skelton et al. 2014). They also derive SFRs from the IRX–β with the Calzetti curve, while stellar masses are derived from SED fitting without IRAC photometry. Since their LAEs have blue β (∼−1.9 on average), their SFRs and stellar masses do not change so much if an SMC-like curve is used instead. Hashimoto et al. (2017) have also examined six LAEs with EW₀(Lyα) ≃ 200–400 Å selected from the same sample as ours and found that they are star-burst galaxies with M_⋆ ∼ 10⁷–10⁸ M_⊙. However, as suggested in Hashimoto et al. (2017), their high sSFRs are probably a consequence of high EW₀(Lyα)s [because younger galaxies have a larger EW₀(Lyα)] and the stellar population properties of these six LAEs do not represent those of our LAE sample.

We infer that our sample better represents the majority of z ∼ 2 LAEs because of the wide luminosity coverage (⁠|$\sim 0.1\mbox{--}2\times L_{{\rm Ly}\alpha }^\star$|⁠: see Konno et al. 2016) and a simple selection based only on EW₀(Lyα) ≥ 20-30 Å, being less biased toward/against other quantities such as UV luminosity. The majority of z ∼ 2 LAEs are probably normal star-forming galaxies with low stellar masses in terms of SF mode.

5 Stellar and halo properties

In this section, we combine the stellar masses, SFRs, and halo masses derived in the previous sections (summarized in tables 3 and 4) to evaluate the SF efficiency in dark matter halos.

5.1 Relation between M_⋆ and M_h

The stellar to halo mass ratio (=M_⋆/M_h: SHMR) indicates the efficiency of SF in dark matter halos integrated over time from the onset of SF to the observed epoch, which we refer to as the integrated SF efficiency. The SHMR as a function of halo mass is known to have a peak and the halo mass at the peak (pivot mass) is ≃2–3 × 10¹² M_⊙ at z ∼ 2 (e.g., Behroozi et al. 2013; Moster et al. 2013). The shape of the average relation show almost no evolution at z ∼ 0–5, although the behavior of the z ∼ 2 SHMR below M_h ∼ 10¹¹ M_⊙ has not been constrained well. We plot the SHMRs of LAEs at z ∼ 2, comparing them with the average relations for the first time, and discuss the typical SHMR of our LAEs with largest survey area so far.

Figure 9a shows M_⋆ and M_h of our LAEs in each of the four fields (pink symbols) and those values averaged over the four fields: M_⋆ = 10.2 ± 1.8 × 10⁸ M_⊙ and |$M_{\rm h}=4.0_{-2.9}^{+5.1}\times 10^{10}{\,\,}M_{\odot }$| (a red star). Those of LAEs at z ∼ 2.1(Guaita et al. 2010),¹⁶ star-forming galaxies based on clustering analysis (Lin et al. 2012; Ishikawa et al. 2016; Ishikawa 2017),¹⁷ and the average relation based on abundance matching (Behroozi et al. 2013; Moster et al. 2013)¹⁸ at z ∼ 2 are shown in figure 9 for comparison. In contrast to Guaita et al.’s result (a blue circle), our LAEs averaged over the four fields (a red star) lie above a simple lower-mass extrapolation (without changing the slope in the log–log space) of the M_⋆–M_h relation of star-forming galaxies and the average relation. Due to the high stellar mass and low halo mass, our LAEs have a SHMR of |$0.02^{+0.07}_{-0.01}$| as high as galaxies at the pivot mass, M_h ≃ 2–3 × 10¹² M_⊙. Here, the errors in this SHMR value indicate the ± 1σ (68%) range. The inset of figure 9b shows the two-dimensional probability distribution of our four-field average M_h and SHMR values calculated from a Monte Carlo simulation with 500000 trials. A magenta contour presents the 68% confidence interval, while brown dots indicate randomly selected 150000 trials. Although the contour touches the +1σ limit of the average relation, only ∼ 2.5% of the entire trials reach the +1σ limit (an orange dashed line).

Fig. 9.

(a) M_⋆ vs M_h and (b) SHMR vs M_h. For each panel, a filled pink square, circle, inverted triangle, and triangle represent average (stacked) LAEs with |$\mathit {NB387}_{\rm tot} \le 25.5\:$|mag in the SXDS, COSMOS, HDFN, and CDFS fields, respectively, and a large red star shows the average over the four fields. A blue circle indicates median (stacked) LAEs at z ∼ 2 in Guaita et al. (2011). Black thick and thin solid lines represent the average relation of galaxies at z ∼ 2 in Behroozi, Wechsler, and Conroy (2013) and Moster, Naab, and White (2013), respectively; their extrapolations are shown by dotted black lines. A gray shaded region indicates the 1σ uncertainty in M_⋆ in the relation in Behroozi, Wechsler, and Conroy (2013). Gray circles and gray triangles denote BzK galaxies in Lin et al. (2012) and gzK galaxies in Ishikawa et al. (2016) and Ishikawa (2017), respectively. For each data point, the horizontal error bars indicate the ± 1σ (68%) range of the M_h measurement, and the vertical error bars the ± 1σ (68%) range of the M_⋆ (a) and SHMR (b) measurement. The inset of the panel (b) shows the two-dimensional probability distribution of our four-field average Mh and SHMR values calculated from a Monte Carlo simulation with 500000 trials. A magenta contour presents the 68% confidence interval while brown dots indicate randomly selected 150000 trials for the presentation purpose. An orange dashed line indicates the +1σ limit of the average relation. All data are rescaled to a Salpeter IMF according to footnote 1. See also footnotes 16–19. (Color online)

We discuss whether there are any systematic differences in M_⋆ and/or M_h between our LAEs and the average relation which result in the departure of our results from the relations. The average relation by Moster, Naab, and White (2013) expresses the mean stellar mass of the central galaxy as a function of halo mass and has a double power-law form, while that by Behroozi, Wechsler, and Conroy (2013) uses the median stellar mass and has five fitting parameters, the functional form of which at low halo masses is approximated by a power law.¹⁹ Although the definitions of stellar masses of the two relations are different, the relations are similar to one another. Our average stellar mass is a field-average median stellar mass since stellar masses are derived from SED fitting for median-stacked SEDs, which are commonly used to prevent contamination (see section 4). The field-average mean stellar mass of our sample is possibly higher than the field-average median. In fact, the mean value of K-band flux densities, which is an approximation of stellar mass, is approximately twice as high as the median one in the SXDS field, the field with the deepest K data. We derive effective halo masses of our LAEs from effective biases directly (see subsection 3.5) assuming a one-to-one correspondence between galaxies and dark matter halos with a narrow range of halo mass. Our field-average effective halo mass probably corresponds to the true mean and/or median within the large uncertainty, the permitted 1σ range of which is ∼1 dex. Even though the uncertainty by cosmic variance discussed in subsection 3.4 is added to the total uncertainty in the field-average halo mass, by which the halo mass and SHMR are written as |$M_{\rm h, {\,\,}cv}=4.0_{-3.5}^{+8.4} \times 10^{10}{\,\,}M_{\odot }$| and SHMR|$=0.02^{+0.18}_{-0.01}$|⁠, respectively, our result is not consistent with the extrapolated average relations within 1σ. Therefore, the departure of our field-average LAEs (a red star) from the average relation are caused neither by a systematic difference of the definition of M_⋆ nor a 1σ cosmic variance on M_h.

On the other hand, if LAEs represent average galaxies, the average M_h–SHMR relation must have an upturn at M_h ≲ 10¹¹ M_⊙. This, however, appears to be unphysical because no such upturn is seen at z ∼ 0, the only epoch at which the average relation below M_h ∼ 10¹¹ M_⊙ has been constrained well (Behroozi et al. 2013), unless the low-mass slope of the average relation evolves drastically from z ∼ 2 to ∼0. Another possibility is that the scatter of the average relation becomes significantly larger at lower halo masses and the SHMR of our LAEs is within the scatter.

Note that the SHMRs in the HDFN and CDFS are consistent with the average relations, although with large uncertainties. We obtain consistent stellar masses between the four fields and it is just the halo masses that are different. The difference in M_h, and hence in b_{g, eff}, among the four fields seen in figure 4 (see also subsections 3.3 and 3.5) is not due to a difference in the limiting magnitude because all four fields have the same limit, |$\mathit {NB387}_{\rm tot} = 25.5$|⁠. As shown in figure 9, fitting errors and contamination fraction errors possibly drive the offsets of M_h in the two fields to the average values. The difference is also explained by cosmic variance, as shown in figure 5b (see also subsection 3.4), and averaging over the four fields reduces the effect of cosmic variance.

5.2 Baryon conversion efficiency

The baryon conversion efficiency (BCE), defined as

\begin{equation} {\it BCE} = \frac{{\dot{M_{\star }}}}{\dot{M_{\rm b}}}, \end{equation}

(26)

measures the efficiency of SF in dark matter halos at the observed time, where |$\dot{M_{\rm b}}$| is the baryon accretion rate (BAR). Here we assume that most of the accreting baryons are in a (cold) gas phase (i.e., the BAR is equal to the inflow rate of cold gas). The average BAR at a fixed halo mass is proportional to the halo mass accretion rate, |$\dot{M_{\rm h}}(z, M_{\rm h})$|⁠, which is estimated as a function of redshift and halo mass from cosmological simulations (Dekel et al. 2009):

\begin{eqnarray} BAR = f_{\rm b}\times \dot{M_{\rm h}}(z,\ M_{\rm h}) \end{eqnarray}

(27)

\begin{eqnarray} &\sim & 6\times \left( \frac{M_{\rm h}}{10^{12}{{\,\,}M_{\odot }}}\right)^{1.15}\times (1+z)^{2.25}{\,\,}M_{\odot }\:\mathrm{yr}^{-1}, \end{eqnarray}

(28)

where f_b ≡ Ω_b/Ω_m = 0.15.

Figure 10 shows the BCE against halo mass. Our LAEs have |$\mathit {BCE}=1.6^{+6.0}_{-1.0}$| and, as shown by a red star, lie above an extrapolation (keeping the slope unchanged) of the average relation by Behroozi, Wechsler, and Conroy (2013) and most of the BzK galaxies in Lin et al. (2012). Here, the errors in our BCE value indicate the ± 1σ (68%) range. The inset of figure 10 shows the two-dimensional probability distribution of our four-field average M_h and BCE values calculated from a Monte Carlo simulation with 500000 trials. A magenta contour presents the 68% confidence interval, while brown dots indicate the 500000 trials. Only ∼ 0.3% of the entire trials reach the +1σ limit of the average relation (an orange dashed line). On the other hand, Guaita et al.’s (2010, 2011) LAEs at z ∼ 2 have a moderate BCE, although with large uncertainties, which is consistent with the average relation as shown by a blue circle. The average SFRs of both samples are nearly equivalent and it is the clustering measurements that differ and drive our BCE up. Thus the difference in the clustering affects the discrepancy in both axes in figure 10, making the offset worse.

Fig. 10.

Baryon conversion efficiency (BCE) as a function of M_h. A filled pink square, circle, inverted triangle, and triangle represent average (stacked) LAEs with |$\mathit {NB387}_{\rm tot} \le 25.5\:$|mag in the SXDS, COSMOS, HDFN, and CDFS fields, respectively, and a red star shows the average over the four fields. A blue circle indicates median (stacked) LAEs at z ∼ 2 in Guaita et al. (2011). A black thick solid and gray circles show the average relation of galaxies at z ∼ 2 in Behroozi, Wechsler, and Conroy (2013) and measurements for BzK galaxies in Lin et al. (2012), respectively. For each data point, the horizontal (vertical) error bars indicate the ±1σ (68%) range of the M_h (BCE) measurement. Extrapolations and 1σ scatter of BCE at fixed M_h are shown by a dotted black line and vertical gray bands, respectively. The scatter of BCE is estimated from the scatter of SFRs at M_h = 1 × 10¹¹, 1 × 10¹², and 1 × 10¹³. The inset shows the two-dimensional probability distribution of our four-field average Mh and BCE values calculated from a Monte Carlo simulation with 500000 trials. A magenta contour presents the 68% confidence interval while brown dots indicate the entire trials. An orange dashed line indicates the +1σ limit of the average relation. All data are rescaled to a Salpeter IMF according to footnote 1. See also footnotes 16–19. (Color online)

We discuss whether there are any systematic differences in SFR and/or M_h between our LAEs and the average relation, which result in the departure of our results from the relations. The average relation by Behroozi, Wechsler, and Conroy (2013) expresses the mean SFR as a function of halo mass. Our field-average SFR is derived from SED fitting for median-stacked SEDs and probably does not overestimate the true average SFR, since the median of B-band flux densities, which traces rest-frame UV, is similar to the average B-band flux density. Even when we neglect dust attenuation at UV, A₁₆₀₀ = 0.6 ± 0.1 mag, the field-averaged SFR (=3.4 ± 0.4 M_⊙ yr⁻¹) decreases by only a factor of ∼2. Moreover, even when the uncertainty by cosmic variance discussed in subsection 3.4 is added to the measured value, |$\mathit {BCE}=1.6^{+6.0}_{-1.0}$|⁠, the 1σ lower limit of the field-averaged BCE is still larger than 0.4. Thus, it seems difficult for our LAEs to fall on the average relation shown in figure 10.

As described in subsection 5.1, logically we cannot rule out the possibilities that our LAEs do lie on or near the average relation which changes the slope and/or scatter below M_h ∼ 1 × 10¹¹ M_⊙ for some reason.

6 Discussion

In this section, we interpret our results on LAEs in terms of the general evolution of galaxies and discuss the physical origin of their high SHMR and BCE, as well as predicting their present-day descendants. We assume that the three average relations shown in figures 8, 9, and 10 do not change either the slope (in log–log plane) or the scatter at low masses. We also assume that our LAEs are central galaxies. If they are satellite galaxies, their dark matter halo (sub halo) masses will be overestimated and their true SHMR and BCE would be higher than reported in this study.

6.1 Duty cycle

The duty cycle of LAEs, |$f_{\rm duty}^{\rm LAEs}$|⁠, is defined as the fraction of dark mater halos hosting LAEs. Previous studies find that |$f_{\rm duty}^{\rm LAEs}$| at z ∼ 3 is a few tenths to a few percent (Ouchi et al. 2010; Chiang et al. 2015). We estimate the duty cycle of our LAEs to be:

\begin{equation} f_{\rm duty}^{\rm LAEs}= \frac{ND_{\rm LAE}}{ND_{\rm DMH}}\sim 2\%, \end{equation}

(29)

where ND_LAE and ND_DMH are the number density of LAEs with NB_tot ≤ 25.5 mag and that of dark matter halos estimated from the halo mass function at z ∼ 2 using the calculator provided by Murray, Power, and Robotham (2013), respectively. For this calculation, we assume that dark matter halos hosting our LAEs have a 1-dex range of mass, 10¹⁰–10¹¹ M_⊙, since the K-band magnitudes, an approximation of stellar mass, of our LAEs are distributed with a FWHM of ∼3.2 mag, or ∼1.3 dex. Our result is comparable with those of previous studies.

We also estimate the fraction of galaxies in a given stellar mass range classified as LAEs (LAE fraction), |$f_{\rm gals}^{\rm LAEs}$|⁠. Assuming that our LAEs have a 1-dex range of stellar mass, 10^8.5–10^9.5 M_⊙, we obtain:

\begin{equation} f_{\rm gals}^{\rm LAEs}= \frac{ND_{\rm LAE}}{ND_{\rm gal}}\sim 10\%, \end{equation}

(30)

where ND_gal is the number density of galaxies estimated by extrapolating Tomczak, Quadri, and Tran’s (2013) stellar mass function at z ∼ 2–2.5 below 10⁹ M_⊙. This result is comparable with those of previous spectroscopic observations of star-forming galaxies at z ∼ 2–2.5 (∼10%, Hathi et al. 2016) and BX galaxies at z ∼ 1.9–2.7 (∼12% with EW_Lyα ≧ 20 Å; Reddy et al. 2008). Note that typical galaxies embedded in dark matter halos with M_h = 10¹⁰–10¹¹ M_⊙ have lower stellar masses than M_⋆ = 10^8.5–10^9.5 M_⊙ because of the high SHMR of our LAEs. The low fractions obtained above imply that only a few percent of galaxies within these mass ranges studied here can evolve into LAEs and/or that galaxies within these mass ranges can experience the LAE phase only for a very short time.

6.2 Physical origin of Lyα emission

The result that our LAEs have a higher SHMR than average galaxies with the same stellar mass may explain why they have strong Lyα emission. A higher SHMR at a fixed M_⋆ means a lower M_h and hence a lower gas mass (M_gas), since the M_gas of a galaxy is written as M_gas ≃ f_bM_h − M_⋆. Galaxies with a low M_gas likely have a low H i column density, thus making it easier for Lyα photons to escape because of a reduced number of resonant scatterings. Indeed, Pardy et al. (2014) have found a tentative anticorrelation of H i gas mass with the Lyα escape fraction and the Lyα equivalent width using 14 local galaxies (LyαReference Sample: Hayes et al. 2013; Östlin et al. 2014).

Furthermore, our LAEs may have high outflow velocities because a high BCE means a high SFR at a fixed M_h (recall BAR |$\propto M_{\rm h}^{1.15}$|⁠) and hence a high kinetic energy from SF at a fixed gravitational binding energy of dark mater halos. In high-velocity outflowing H i gas, the probability of the resonant scattering of Lyα photons is reduced because of reduced cross-sections of H i atoms due to large relative velocities (e.g., Kunth et al. 1998; Verhamme et al. 2006; Hashimoto et al. 2015). Note also that our LAEs have absolutely low dust attenuation due probably to a low stellar mass as shown in figure 7, which also helps Lyα photons survive in galaxies. To summarize, the high SHMR, high BCE, and moderate SFR obtained for our LAEs are in concord with the strong Lyα emission observed.

6.3 Physical origin of moderate SF mode, high SHMR, and high BCE

Our LAEs have a higher SHMR and a higher BCE than average galaxies but have a moderate SFR, being located on the (extrapolated) SFMS defined by average galaxies. Indeed, it is not trivial for galaxy formation models to reproduce these three properties simultaneously.

Dutton, van den Bosch, and Dekel (2010) have used a semi-analytic model to study the evolution of the SFMS and its dependence on several key parameters in the model. As shown in their figure 12 and our figure 11, model galaxies (at z ∼ 2) at a fixed halo mass move along the SFMS upward when the supernova (SN) feedback is weakened or the halo’s spin parameter is reduced, thus having a higher SHMR and a higher BCE on the SFMS. With a lower feedback efficiency, a larger amount of cold gas can be stored, thus resulting in a higher SFR and a higher stellar mass. A lower spin causes the gas density to be higher, thereby the SFR per unit gas mass is elevated. Although these results may not necessarily be applicable to our LAEs that have halo masses 10 times lower, it is interesting to note that there is a relatively simple way to explain MS galaxies with an elevated SHMR and BCE.

Fig. 11.

Changes in the position of model galaxies in the M_⋆–SFR plane (a), M_h–M_⋆ plane (b), and M_h–BCE plane (c) due to variations in the halo spin parameter, λ, and the feedback efficiency, ε_FB, calculated by Dutton, van den Bosch, and Dekel (2010, hereafter D10 in this caption). Pentagons show D10’s model galaxies with a fixed halo mass [M_{h, z = 0} = 4 × 10¹¹ M_⊙, corresponding to ∼2 × 10¹¹ M_⊙ at z = 2 according to figures 7 and 8 in Behroozi, Wechsler, and Conroy (2013)], where black, cyan, and magenta colors denote, respectively, positions with median halo parameters, those with ±2σ variation in λ, and those with ±2σ variation in ε_FB. All model data of M_⋆ and SFR are taken from figure 12 in D10. (In D10 four data points are shown as ±2σ variation in ε_FB.) The BARs of model galaxies are calculated from equation (28). Cyan and magenta arrows indicate the direction in which galaxies move when λ and ε_FB increase. In all panels, red stars represent the average LAEs with |$\mathit {NB387}_{\rm tot} \le 25.5\:$|mag. In panel (a), several SFMS measurements in previous studies are shown by black lines in the same manner as figure 8. The average relations in Behroozi, Wechsler, and Conroy (2013) and Moster, Naab, and White (2013) are plotted by black lines in panels (b) and (c), similar to figures 9 and 10. All data are rescaled to a Salpeter IMF according to footnote 1. See also footnotes 16–19. (Color online)

It is beyond our scope to identify the mechanism(s) by which our LAEs acquire a high SHMR and a high BCE. If, however, the high SHMR and BCE of our LAEs are due to some systematic differences in one or more parameters controlling the SF and/or internal structure of halos similar to Dutton, van den Bosch, and Dekel (2010) study, then it implies that not all, but only a certain fraction of, (low-mass) halos at z ∼ 2 experience the LAE phase.

6.4 Present-day descendants of our LAEs

LAEs are found to reside in low-mass halos with M_h ∼ 10¹⁰–10¹² M_⊙ over the wide redshift range z ∼ 2–7 as found in subsection 3.5 (e.g., Ouchi et al. 2005, 2010, 2018; Kovač et al. 2007; Gawiser et al. 2007; Shioya et al. 2009; Guaita et al. 2010; Bielby et al. 2016; Diener et al. 2017). In other words, the bias value of LAEs tends to decrease with decreasing redshift more rapidly than that of dark matter halos (see figure 7 in Ouchi et al. 2018). Although this trend may be biased because faint LAEs in lower-mass halos are missed at high redshifts, it implies that at lower redshifts only galaxies with relatively lower masses in the halo mass function can be LAEs, which is analogous to and/or maybe related to downsizing (Cowie et al. 1996).

A roughly constant halo mass with redshift also implies that local descendants of LAEs vary depending on their redshift. The growth of dark matter halos is statistically predicted by the extended Press–Schechter (EPS: Press & Schechter 1974; Bond et al. 1991; Bower 1991) model. An application of the EPS model to distant galaxies can be found in, e.g., Hamana et al. (2006). Previous studies suggest that LAEs at z ∼ 4–7 evolve into massive elliptical galaxies at z = 0 (Ouchi et al. 2005, 2010; Kovač et al. 2007), while LAEs at z ∼ 3 are expected to be progenitors of present-day L_⋆ galaxies (Gawiser et al. 2007; Ouchi et al. 2010). Guaita et al. (2010) show that LAEs at z ∼ 2 could be progenitors of present-day L_⋆ galaxies like the Milky Way (MW) and that they could also be descendants of z ∼ 3 LAEs, depending on SF and dust formation histories (see also Acquaviva et al. 2012).

With the EPS model, we find that at z = 0 our LAEs are embedded in dark matter halos with a median mass similar to the mass of the Large Magellanic Cloud (LMC: M_h ∼ 0.2–3 × 10¹¹ M_⊙; van der Marel & Kallivayalil 2014; Peñarrubia et al. 2016 and references therein), not in MW-like halos (M_h ∼ 8 × 10¹¹–2 × 10¹² M_⊙; e.g., Wilkinson & Evans 1999; Kafle et al. 2014; Eadie et al. 2015, summarized in figure 1 in Wang et al. 2015), as shown in figure 12.²⁰ This is consistent with the prediction by Acquaviva et al. (2012) from SED fitting that LAEs at z ∼ 3, which are progenitors of present-day L_⋆ galaxies, do not evolve into LAEs at z ∼ 2. Combined with the previous studies, our results imply that the mass of present-day descendants of halos hosting LAEs depends on the redshift at which they are observed, with higher-z LAEs evolving into more massive halos.

Fig. 12.

Dark matter halo mass evolution as a function of redshift predicted by the EPS formalism. A red (blue) curve indicates the evolution of the mode of the M_h distribution starting from the mass of our z = 2.2 LAEs shown by a red star [z = 2.1 LAEs from Guaita et al. 2010 are shown by a blue circle], with a shaded region indicating the 68% confidence interval of the distribution. Black and gray rectangles represent the measured halo mass ranges of the MW and the LMC, respectively (e.g., Wilkinson & Evans 1999; Kafle et al. 2014; van der Marel & Kallivayalil 2014; Eadie et al. 2015; Peñarrubia et al. 2016, see also Wang et al. 2015). (Color online)

Since the stellar mass of our LAEs, 10.2 ± 1.8 × 10⁸ M_⊙, is comparable to that of the LMC within only a factor of ∼3 (M_⋆ ∼ 2.9 × 10⁹ M_⊙: van der Marel et al. 2002), their star formation has to be largely suppressed over most of the cosmic time until z = 0, or even be quenched, if they really become LMC-like galaxies. The SF history of the LMC has been inferred to have multiple components, i.e., an initial burst and subsequent periods with moderate or quiescent SF (e.g., Harris & Zaritsky 2009). For example, Rezaeikh et al. (2014) argue that it consists of two components: an initial burst ∼10 Gyr ago, or at z ∼ 2, with a |$\mathit {SFR} \sim 2.4{\,\,}M_{\odot }\:$|yr⁻¹ assembling ∼90% of the total mass, and a much milder SF with |$\mathit {SFR} \sim 0.3{\,\,}M_{\odot }\:$|yr⁻¹ after that as shown in their figure 4 [see however Weisz et al. (2013), who obtained a much lower SFR]. If our LAEs follow such a history with suppressed SF over ∼5–10 × 10⁹ Gyr, they will grow to be LMC-like galaxies at z = 0. In this case, if at z ∼ 2 they lie above the average M_h–SHMR relation, they will evolve into galaxies with an SHMR consistent with the average relation at z ∼ 0 (Behroozi et al. 2013; Moster et al. 2013).

6.5 Future survey

In the near future, we will extend this work using new |$\mathit {NB387}$| data from ≃ 25 deg² taken with Hyper Suprime-Cam (HSC) as part of a large imaging survey program (Aihara et al. 2018). This program uses five broadband and four NB filters, among which the new |$\mathit {NB387}$| is included. We call the LAE surveys with the four NB filters SILVERRUSH (Ouchi et al. 2018; Shibuya et al. 2018a). The survey volume for |$\mathit {NB387}$| (z ∼ 2) LAEs is |$6\times 10^6\ {\rm (h^{-1}_{100}\ Mpc )^3}$| with an expected number of ∼9000 objects. As shown in figures 5a and 5b, the uncertainty from cosmic variance is expected to be negligibly small, ∼3%, compared with other uncertainties. With the HSC data, we will be able to determine the SHMR and BCE of z ∼ 2 LAEs without suffering from cosmic variance.

7 Conclusions

We have investigated stellar populations and halo masses of LAEs at z ∼ 2, low-mass galaxies at cosmic noon, using ∼1250 |$\mathit {NB387}$|-selected LAEs from four separate fields with ∼1 deg² in total. In particular, we have derived the average SF mode, SHMR, and BCE of objects with |$\mathit {NB387} \le 25.5$| for which measurements for all four fields are available, and discussed SF activity and its dependence on halo mass. Our main results are as follows.

1. The bias parameter of |$\mathit {NB387} \le 25.5$| objects averaged over the four fields is |$b_{\rm g,{\,\,} eff}^{\rm ave}=1.22^{+0.16}_{-0.18}$|⁠, which is lower than that in Guaita et al. (2010) from 0.3 deg² with a probability of 96%. We estimate an external error from cosmic variance which inversely scales with the square root of the survey area. The high bias value obtained by Guaita et al. (2010) becomes consistent with our value if the uncertainties from cosmic variance, ± 26% and ± 51% for this work and Guaita et al. (2010), are considered. We have also found that b_{g, eff} does not significantly change with limiting NB387 magnitude, or limiting Lyα luminosity, which may be partly due to two trends canceling each other out: galaxies in more massive halos have brighter intrinsic Lyα luminosities but lower Lyα escape fractions.

2. The halo mass corresponding to the above |$b_{\rm g,{\,\,} eff}^{\rm ave}$| value is |$4.0^{+5.1}_{-2.9}\times 10^{10}{\,\,}M_{\odot }$|⁠. This value is roughly comparable to previous measurements for z ∼ 3–7 LAEs with similar Lyα luminosities, M_h ∼ 10¹⁰–10¹² M_⊙ (e.g., Ouchi et al. 2010), suggesting that the mass of dark halos which can host typical LAEs is roughly unchanged with time.

3. The mean of each stellar parameter over the four fields is: M_⋆ = 10.2 ± 1.8 × 10⁸ M_⊙, A₁₆₀₀ = 0.6 ± 0.1 mag, Age=3.8 ± 0.3 × 10⁸ yr, and SFR = 3.4 ± 0.4 M_⊙ yr⁻¹. Our LAEs are thus located near an extrapolation of the consensus relation of IRX against stellar mass with an assumption of an SMC-like attenuation curve (see figure 7). We have also found that our LAEs are on average placed near a lower-mass extrapolation of the SFMS, confirming the results obtained by Kusakabe et al. (2015) with a ∼6 times larger survey area (shown in figure 8).

4. With SHMR|$= 0.02^{+0.07}_{-0.01}$|⁠, our LAEs lie above a simple lower-mass extrapolation of the average M_⋆–M_h relation (figure 9). The higher SHMR than average galaxies with the same M_⋆ may make it easy for Lyα photons to escape since they are expected to have lower gas masses (baryon mass) and thus lower H i column densities. Our LAEs also have a high |$\mathit {BCE}=1.6^{+6.0}_{-1.0}$|⁠, lying above the average BCE–M_h relation (figure 10). Thus, our LAEs have been converting baryons into stars more efficiently than average galaxies with similar M_h both in the past and at the observed epoch but with a moderate SF similar to average galaxies. Galaxies with weak SN feedback and small halo spin parameters possibly have such properties according to the semi-analytic model by Dutton et al. (2010).

5. The duty cycle of LAEs (fraction of M_h ∼ 3 × 10¹⁰ M_⊙ halos hosting LAEs) is estimated to be ∼2%, and the LAE fraction (fraction of M_⋆ ∼ 1 × 10⁹ M_⊙ galaxies classified as LAEs) is found to be ∼10%. These low fractions imply either that only a small fraction of all galaxies can evolve into LAEs and/or that even low-mass galaxies can emit Lyα only for a very short time.

6. We have calculated the halo mass evolution of our LAEs with the EPS model, to find that at z = 0 our LAEs are embedded in dark matter halos with a median halo mass similar to the mass of the LMC. If their star-formation is largely suppressed after the observed time until z = 0 similar to the star-formation history of the LMC, they would have a similar SHMR to the present-day LMC. This result, combined with the previous studies, implies that the mass of present-day descendant halos of LAEs depends on the redshift at which the LAEs are observed, with higher-z LAEs evolving into more massive halos.

Acknowledgements

We thank the anonymous referee for his/her helpful comments and suggestions. We are grateful to Lihwai Lin and Li-Ting Hsu for kindly providing us with J, H, and K_s images of the HDFN field and data in Lin et al. (2012) plotted in figures 8, 9, and 10. We are also grateful to Yoshiaki Ono for giving insightful comments and suggestions on SED fitting. We would like to show our appreciation to Takashi Hamana for helpful comments on cosmic variance and computer programs of the covariance of dark matter angular correlation function and the EPS model. We would like to express our gratitude to David Sobral, Naveen A. Reddy, Giulia Rodighiero, and Shogo Ishikawa for kindly providing their data plotted in figures 5a, 7, 8, and 9, respectively. We would like to thank Alex Hagen, James E. Rhoads, Jorryt Matthee, and Peter S. Behroozi for useful comments on their results. We also would like to thank Akio K. Inoue, Cai-Na Hao, Hidenobu Yajima, Ikkoh Shimizu, Ken Mawatari, Kotaro Kohno, Kyoung-Soo Lee, Tsutomu T. Takeuchi, Mana Niida and Yuki Yoshimura for insightful discussion. We acknowledge Ryota Kawamata, Taku Okamura, and Kazushi Irikura for constructive discussions at weekly meetings. This work is based on observations taken by the Subaru Telescope which is operated by the National Astronomical Observatory of Japan. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. Based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under ESO programme ID 179.A-2005 and on data products produced by TERAPIX and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium. This research made use of IRAF, which is distributed by NOAO, which is operated by AURA under a cooperative agreement with the National Science Foundation and of Python packages for Astronomy: Astropy (Astropy Collaboration et al. 2013), Colossus, CosmoloPy and PyRAF, which is produced by the Space Telescope Science Institute, which is operated by AURA for NASA. H.K acknowledges support from the JSPS through the JSPS Research Fellowship for Young Scientists. This work is supported in part by KAKENHI (16K05286) Grant-in-Aid for Scientific Research (C) through the JSPS.

Appendix 1. Result of SED fitting with different assumptions

We show the SED fitting results with the Calzetti curve and without nebular emission below.

A.1.1 The Calzetti curve

We also examine the cases of the Calzetti curve for comparison. The best-fitting parameters with an SMC-like curve and the Calzetti curve are listed in table 5. Figures 6 and 13 show the best-fitting SEDs with the observed ones in the case with an SMC-like curve and the Calzetti curve, respectively. We compare the best-fitting parameters in subsection 4.4.

Fig. 13.

Same as figure 6 but with the Calzetti curve. Panels (a) to (d) show results for SXDS, COSMOS, HDFN, and CDFS, respectively. (Color online)

Table 5.

Results of SED fitting with an SMC-like curve and the Calzetti curve (⁠|$f_{\rm esc}^{\rm ion}=0.2$|⁠).*

Attenuation curve	M⋆	E(B − V)⋆ [A1600]	Age	SFR	|$\chi ^2_r$|
	(108 M⊙)	(mag)	(108 yr)	(M⊙ yr−1)
	(1)	(2)	(3)	(4)	(5)
SXDS
SMC	|$9.7^{+3.6}_{-1.7}$|	|$0.05^{+0.01}_{-0.02}\ [0.6^{+0.1}_{-0.2}]$|	|$3.6^{+2.8}_{-1.1}$|	|$3.3^{+ 0.5}_{- 0.7}$|	0.604
Calzetti	|$7.8^{+3.4}_{-1.9}$|	|$0.11^{+0.02}_{-0.05}\ [1.1^{+0.2}_{-0.5}]$|	|$1.6^{+2.4}_{-0.7}$|	|$5.7^{+ 1.7}_{- 2.3}$|	0.665
COSMOS
SMC	|$14.0^{+3.4}_{-3.6}$|	|$0.07^{+0.02}_{-0.02}\ [0.8^{+0.2}_{-0.2}]$|	|$4.1^{+2.4}_{-1.8}$|	|$4.2^{+ 1.2}_{- 0.8}$|	0.473
Calzetti	|$7.9^{+5.1}_{-2.5}$|	|$0.18^{+0.03}_{-0.05}\ [1.8^{+0.3}_{-0.5}]$|	|$0.7^{+1.6}_{-0.4}$|	|$12.3^{+ 6.4}_{- 5.5}$|	0.648
HDFN
SMC	|$7.6^{+4.0}_{-1.9}$|	|$0.06^{+0.02}_{-0.03}\ [0.7^{+0.2}_{-0.4}]$|	|$3.2^{+4.0}_{-1.4}$|	|$2.9^{+ 0.8}_{- 0.8}$|	1.298
Calzetti	|$3.2^{+0.6}_{-0.8}$|	|$0.20^{+0.02}_{-0.03}\ [2.0^{+0.2}_{-0.3}]$|	|$0.3^{+0.2}_{-0.1}$|	|$13.3^{+ 5.1}_{- 3.9}$|	0.866
CDFS
SMC	|$10.3^{+11.1}_{-9.7}$|	|$0.02^{+0.07}_{-0.01}\ [0.2^{+0.8}_{-0.1}]$|	|$5.7^{+8.6}_{-5.7}$|	|$2.2^{+534}_{- 0.4}$|	0.120
Calzetti	|$3.1^{+17.5}_{-2.4}$|	|$0.14^{+0.05}_{-0.13}\ [1.4^{+0.5}_{-1.3}]$|	|$0.4^{+14.0}_{-0.3}$|	|$9.0^{+23.4}_{- 7.1}$|	0.101
Average
SMC	10.2 ± 1.8	0.06 ± 0.01 [0.6 ± 0.1]	3.8 ± 0.3	3.4 ± 0.4
Calzetti	3.4 ± 0.4	0.19 ± 0.01 [1.9 ± 0.1]	0.3 ± 0.04	12.7 ± 0.6

*(1) The best-fitting stellar mass; (2) the best-fitting color excess [UV attenuation]; (3) the best-fitting age; (4) the best-fitting SFR; (5) reduced chi-squared value. The UV attenuation is derived from the attenuation curve listed in the first column. Metallicity, redshift, and |$f_{\rm esc}^{\rm ion}$| are fixed to 0.2 Z_⊙, 2.18, and 0.2, respectively.

A.1.2 Without nebular emission

It is well known that considering nebular emission generally leads to a lower stellar mass (e.g., de Barros et al. 2014). To obtain upper limits of stellar mass and determine the SF mode of our LAEs, we also examine the case without nebular emission, |$f^{\rm ion}_{\rm esc}=1$|⁠. The best-fitting parameters with an SMC-like curve and the Calzetti curve are listed in table 6. Figure 14 shows the best-fitting SEDs with the observed ones in the case with an SMC-like curve and the Calzetti curve.

Fig. 14.

Same as figure 6 but without nebular emission, |$f^{\rm ion}_{\rm esc}=1$|⁠. Panels (a) to (d) show results with an SMC-like curve for SXDS, COSMOS, HDFN, and CDFS, respectively. Panels (e) to (h) show results with the Calzetti curve for SXDS, COSMOS, HDFN, and CDFS, respectively. (Color online)

Table 6.

Results of SED fitting without nebular emission, |$f_{\rm esc}^{\rm ion}=1.^{*}$|

Attenuation	M⋆	E(B − V)⋆ [A1600]	Age	SFR	|$\chi ^2_r$|
curve	(108 M⊙)	(mag)	(108 yr)	(M⊙ yr−1)
	(1)	(2)	(3)	(4)	(5)
SXDS
SMC	|$11.4^{+2.7}_{-1.3}$|	|$0.06^{+0.02}_{-0.02}\ [0.7^{+0.2}_{-0.2}]$|	|$3.6^{+2.1}_{-1.1}$|	|$3.9^{+ 0.9}_{- 0.8}$|	0.350
Calzetti	|$5.1^{+7.1}_{-0.4}$|	|$0.27^{+0.02}_{-0.16}\ [2.7^{+0.2}_{-1.6}]$|	|$0.1^{+2.4}_{-0.0}$|	|$45.3^{+12.0}_{-40.0}$|	0.586
COSMOS
SMC	|$14.6^{+5.2}_{-2.7}$|	|$0.08^{+0.02}_{-0.02}\ [1.0^{+0.2}_{-0.2}]$|	|$3.6^{+2.8}_{-1.3}$|	|$4.9^{+ 1.3}_{- 1.1}$|	0.611
Calzetti	|$6.6^{+1.5}_{-0.7}$|	|$0.29^{+0.01}_{-0.04}\ [2.9^{+0.1}_{-0.4}]$|	|$0.1^{+0.2}_{-0.0}$|	|$56.2^{+12.8}_{-26.4}$|	0.821
HDFN
SMC	|$9.8^{+2.4}_{-2.5}$|	|$0.05^{+0.03}_{-0.02}\ [0.6^{+0.4}_{-0.2}]$|	|$4.5^{+2.7}_{-2.0}$|	|$2.7^{+ 1.0}_{- 0.5}$|	1.865
Calzetti	|$4.4^{+0.0}_{-0.8}$|	|$0.30^{+0.00}_{-0.04}\ [3.0^{+0.0}_{-0.4}]$|	|$0.09^{+ 0.03}_{-0.01}$|	|$51.8^{+ 5.9}_{-18.9}$|	1.653
CDFS
SMC	|$13.1^{+10.9}_{-8.9}$|	|$0.02^{+0.06}_{-0.01}\ [0.2^{+0.7}_{-0.1}]$|	|$7.1^{+8.9}_{-6.2}$|	|$2.3^{+ 3.0}_{- 0.3}$|	0.148
Calzetti	|$12.1^{+12.7}_{-10.0}$|	|$0.05^{+0.25}_{-0.04}\ [0.5^{+2.5}_{-0.4}]$|	|$5.1^{+11.9}_{-5.1}$|	|$2.9^{+135.8}_{- 1.0}$|	0.157
Average
SMC	11.2 ± 1.2	0.06 ± 0.01 [0.6 ± 0.1]	4.1 ± 0.5	3.2 ± 0.6
Calzetti	4.7 ± 0.4	0.29 ± 0.02 [2.9 ± 0.2]	0.09 ± 0.01	51.8 ± 4.5

*(1) The best-fitting stellar mass; (2) the best-fitting color excess [UV attenuation]; (3) the best-fitting age; (4) the best-fitting SFR; (5) reduced chi-squared value. The UV attenuation is derived from the attenuation curve listed in the first column. Metallicity, redshift, and |$f_{\rm esc}^{\rm ion}$| are fixed to 0.2Z_⊙, 2.18, and 1, respectively.

When we assume an SMC-like curve, the average stellar mass and SFR without nebular emission, M_⋆ = 11.2 ± 1.2 × 10⁸ M_⊙ and |$\mathit {SFR} =3.2{\,\,}\pm {\,\,}0.6{\,\,}M_{\odot }\:$|yr⁻¹, are consistent with those with nebular emission, M_⋆ = 10.2 ± 1.8 × 10⁸ M_⊙ and |$\mathit {SFR} =3.4{\,\,}\pm {\,\,}0.4{\,\,}M_{\odot }\:$|yr⁻¹. This means that the average stellar mass and SF mode of our LAEs are insensitive to |$f^{\rm ion}_{\rm esc}$| when an SMC-like curve is used. On the other hand, if we assume the Calzetti curve, the average SFR without nebular emission, |$\mathit {SFR} =51.8{\,\,}\pm {\,\,}4.5{\,\,}M_{\odot }\:$|yr⁻¹, is about four times higher than that with nebular emission, |$\mathit {SFR} =12.7{\,\,}\pm {\,\,}1.0{\,\,}M_{\odot }\:$|yr⁻¹. Their average stellar mass without nebular emission, M_⋆ = 4.7 ± 0.7 × 10⁸ M_⊙, is slightly higher than that with nebular emission, M_⋆ = 3.4 ± 0.8 × 10⁸ M_⊙. With this high SFR, our LAEs lie above the SFMS at z ∼ 2. However, this case seems unrealistic because our LAEs have Lyα emission, one of nebular emission lines. Indeed, the reduced χ-square values in the case without nebular emission are larger than those with nebular emission in all the fields except SXDS. In addition, results with |$f^{\rm ion}_{\rm esc}=1$| give a high UV attenuation of A₁₆₀₀ = 2.9 ± 0.2 mag and hence a high IRX (⁠|$=22^{+5}_{-4}$|⁠), which is significantly higher than predicted by the consensus relation (see figure 7).

Appendix 2. SFMS based on the IRX–β relation with the Calzetti curve

In the discussion of the SF mode of LAEs at z ∼ 2 in sub-subsection 4.4.2, we derive the average SFR of our LAEs using SED fitting with an SMC-like curve, while Hagen et al. (2016) and Shimakawa et al. (2017) derive SFRs using the IRX–β relation with the Calzetti curve. For a fair comparison, figure 15c shows our results with the IRX–β relation with the Calzetti curve (Meurer et al. 1999). We find our LAEs to have higher sSFRs similar to LAEs in Hagen et al. (2016). Note that the selections of these three samples are different as described in sub-subsection 4.4.2. We also compare our results by the three different methods discussed in appendix 1.1 and in this section (see figures 15a and 15b).

Fig. 15.

 SFR plotted against M_⋆. Panels (a) and (b) compare different SFR calculation methods for our LAEs; in panel (a) SFRs calculated from SED fitting with two different attenuation curves are compared; in panel (b) SFRs from SED fitting are compared with those from the IRX–β relation, where the Calzetti curve is used in both calculations. Panel (c) uses the IRX–β relation with the Calzetti curve and compares our LAEs with Hagen et al. (2016)’s and Shimakawa et al. (2017)’s. In panel (a), orange squares, green circles, magenta inverted triangles, and blue triangles represent stacked LAEs with |$\mathit {NB387}_{\rm tot} \le 25.5\:$|mag in the SXDS, COSMOS, HDFN, and CDFS fields, respectively; filled and open symbols are for an SMC-like curve and the Calzetti curve, respectively. In panel (b), encircled symbols indicate that SFRs are derived from the IRX–β relation with the Calzetti curve (Meurer et al. 1999). In panel (c), cyan squares and light green pentagons show individual LAEs at z ∼ 2 in Hagen et al. (2016) and Shimakawa et al. (2017), respectively; in both studies, SFRs are derived from the IRX–β relation with the Calzetti curve (Meurer et al. 1999). Our results based on the IRX–β relation with the Calzetti curve are also plotted (encircled symbols). In all panels, several SFMS measurements in previous studies are shown by black lines in the same manner as figure 8. All data are rescaled to a Salpeter IMF according to footnote 1. (Color online)

Footnotes

1

To rescale stellar masses in previous studies assuming a Chabrier or Kroupa IMF (Kroupa 2001; Chabrier & Chabrier 2003), we divide them by a constant factor of 0.61 or 0.66, respectively. Similarly, to convert SFRs in the literature with a Chabrier or Kroupa IMF, we divide them by a constant factor of 0.63 or 0.67, respectively.

2

The threshold varies from 20 to 30 Å because the response curves of the selection bands U (or u*) and B are slightly different among the four fields. Two-color diagrams of U (or u^⋆)–|$\mathit {NB387}$| and |$B\mbox{--}\mathit {NB387}$| for selection in each of the four fields are shown in figure 1 in Konno et al. (2016).

3

In the COSMOS field, Matthee et al. (2016, hereafter M16 in this footnote) find an overdense region in their HAE sample at z = 2.231 ± 0.016 (see their figure 2) and a part of their survey region overlaps with that of our LAEs at z = 2.14–2.22. In their overdense region, two X-ray sources at z = 2.219 and z = 2.232 have bright Lyα emission. The first one is roughly at the center of the overdense region but just outside of our |$\mathit {NB387}$| image coverage (ID:1139; see figure 2 and table 2 in M16). The second one is included in our coverage but not selected by our color–color criteria, probably because its redshift is too large (ID: 1037). Indeed, we do not find, by eye inspection, any overdense region in figure 2d as significant as the one discovered in M16.

4

We do not include the result of Guaita et al. (2010) when calculating the average bias.

5

Cohn’s (2006) equation (19) corresponds to the full covariance including those due to a discrete sampling with a finite number of objects; the second term is proportional to P₂(K)/NΩ_S, where N is the number density of objects, and the subsequent terms correspond to the uncertainty shown in our equation (5). Inclusion of the second term in our equation (22) increases Δω_DM by ∼30% for our LAE survey, although in this study we neglect this term and only consider cosmic variance not dependent on N.

6

(http://roban.github.com/CosmoloPy/).

7

0.85″ is the largest PSF FWHM among the K (or K_s) bands shown in table 1.

8

The PSF size of the CDFS H-band image is exceptionally large and we determine the radius of the annulus for this image independently.

9

The sky background value on a 3|${^{\prime\prime}_{.}}$|5-radius annulus placed at the image center is consistent between the stacked LAE images and the stacked images of IRAC-clean random positions. For stacked images of random positions, annular-region sky flux densities are brighter than aperture-region sky flux densities with differences corresponding to ∼7%–28% of the aperture fluxes of median-stacked LAEs.

10

While Hagen et al. (2017) have found that the SMC indeed has a flatter extinction curve on average than the classical (Pei 1992; Gordon et al. 2003) curve, we adopt the classical curve which is consistent with recent observations of high-z galaxies including LAEs. Reddy et al. (2017) find that galaxies at z = 1.5–2.5 prefer an SMC-like attenuation curve combined with sub-solar metallicity stellar population models.

11

We also perform SED fitting with models without nebular emission, |$f^{\rm ion}_{\rm esc}=1$|⁠, to examine to what extent SFRs and M_⋆ change, in appendix 1.2.

12

The uncertainties in the best-fitting parameters in the CDFS are large since the number of LAEs used in stacking analysis is smaller than those in the other fields as shown in table 2. Moreover, the i, z, and H band images in this field are ∼0.5–2 mag shallower than those in the other fields.

13

We shift the derived IRXs downward by 10% because the L_IR of the consensus relation is defined as L_IR ≡ L_{8–1000 μm} instead of L_IR ≡ L_{3–1000 μm}

14

This IRX has also been 10% corrected from the original value in Kusakabe et al. (2015, see our footnote 13).

15

Hagen et al. (2016) suggest that their LAEs are undergoing starbursts, that the SFMS becomes shallower at low stellar masses and their LAEs are distributed around it, or that their LAEs are biased towards high Lyα luminosities, not representing typical LAEs.

16

The SFR and stellar mass in Guaita et al. (2010, 2011)are derived from SED-fitting to a median-stacked SED and their halo mass is a median halo mass. We plot the values without any correction (see also subsection 3.3).

17

We recalculate halo masses in Lin et al. (2012) from the effective biases given in their table, using the same method as ours.

18

The values of cosmological parameters adopted in Behroozi, Wechsler, and Conroy (2013) and Moster, Naab, and White (2013) are slightly different from ours, but we have not corrected for those differences in this study. The M_h value in Behroozi, Wechsler, and Conroy (2013) becomes ∼0.15 dex higher at M_h ≤ 10¹² M_⊙ when our values are used (P. Behroozi 2017, private communication).

19

The Behroozi, Wechsler, and Conroy (2013) relations including extrapolated parts in figures 9–11 are taken from the website of P. Behroozi: 〈http://www.peterbehroozi.com/data.html〉. see also footnote 18.

20

We use a publicly released code by T. Hamana 〈http://th.nao.ac.jp/MEMBER/hamanatk/OPENPRO/index.html〉.