The dominant origin of diffuse Ly$α$ halos around LAEs explored by SED fitting and clustering analysis
Haruka Kusakabe Kazuhiro Shimasaku Rieko Momose Masami Ouchi Kimihiko Nakajima Takuya Hashimoto Yuichi Harikane John D. Silverman Peter L. Capak

 



1Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
kusakabe@astron.s.u-tokyo.ac.jp


2Research Center for the Early Universe, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan


3Department of Physics, National Tsing Hua University, 101 Section 2 Kuang Fu Road, Hsinchu 30013, Taiwan


4Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
5Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan


6Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
7Cosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, 2100 Copenhagen Ø, Denmark
8DTU-Space, Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby,Denmark



9College of General Education, Osaka Sangyo University, 3-1-1 Nakagaito, Daito, Osaka 574-8530, Japan
10National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan


11Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan





12California Institute of Technology, MC 105-24, 1200 East California Blvd., Pasadena, CA 91125, USA
13Infrared Processing and Analysis Center, California Institute of Technology, MC 100-22, 770 South Wilson Ave., Pasadena, CA 91125, USA

galaxies: high-redshift  — galaxies: star formation — galaxies: halos —intergalactic medium

The dominant origin of diffuse Lyα halos around LAEs explored by SED fitting and clustering analysis
    Peter L. Capak12, 13
    
====================================================================================================




The physical origin of diffuse Lyα halos (LAHs) around star-forming galaxies is still a matter of debate. We present the dependence of LAH luminosity (L( Lyα)_H) on the stellar mass (M_⋆), SFR, color excess (E(B-V)_⋆), and dark matter halo mass (M_ h) of the parent galaxy for ∼ 900 Lyα emitters (LAEs) at z∼2 divided into ten subsamples. We calculate L( Lyα)_H using the stacked observational relation between L( Lyα)_H and central Lyα luminosity of Momose et al. (2016, MNRAS, 457, 2318), which we find agrees with the average trend of VLT/MUSE-detected individual LAEs. We find that our LAEs have relatively high L( Lyα)_H despite low M_⋆ and M_ h, and that L( Lyα)_H remains almost unchanged with M_⋆ and perhaps with M_ h. These results are incompatible with the cold stream (cooling radiation) scenario and the satellite-galaxy star-formation scenario, because the former predicts fainter L( Lyα)_H and both predict steeper L( Lyα)_H vs. M_⋆ slopes. We argue that LAHs are mainly caused by Lyα photons escaping from the main body and then scattered in the circum-galactic medium. This argument is supported by LAH observations of Hα emitters (HAEs). When LAHs are taken into account, the Lyα escape fractions of our LAEs are about ten times higher than those of HAEs with similar M_⋆ or E(B-V)_⋆, which may partly arise from lower HI gas masses implied from lower M_ h at fixed M_⋆, or from another Lyα source in the central part.





§ INTRODUCTION

A Lyα halo (LAH) is a diffuse, spatially extended structure of Lyα emission seen around star-forming galaxies. LAHs around local galaxies, as well as around active galactic nuclei (AGNs) and quasi-stellar objects (QSOs), can be detected individually because they are relatively bright <cit.>. LAHs around high-redshift (z) galaxies are much fainter, but they have been detected in stacked narrow-band images (tuned to redshifted Lyα emission) of 100 – 4000 star-forming galaxies at z∼2–6 <cit.>. Very recently, LAHs around ∼ 170 star forming galaxies at z∼3–6 have been detected individually by deep integral field spectroscopy with VLT/MUSE <cit.>. Since the existence of LAHs has now been established, the next question is what is their physical origin(s).

Theoretical studies have proposed several physical origins of LAHs: resonant scattering in the CGM, cold streams (gravitational cooling radiation), star formation in satellite galaxies (one-halo term), fluorescence (photo-ionization), shock heating by gas outflows, and major mergers <cit.>. The former three are generally considered for high-z star-forming galaxies <cit.>, while the latter three are preferred for giant Lyα nebulae (Lyα blobs; LABs) and/or bright QSOs <cit.>. 

Understanding the origin of LAHs provides crucial information on the circum-galactic medium (CGM), which is closely linked to galaxy formation and evolution. It also enables us to estimate the escape fraction of Lyα emission from central galaxies correctly. If resonant scattering mainly drives LAHs, the Lyα luminosity of LAHs should be included in the calculation of the Lyα escape fraction. LAHs are also important for studies of cosmic reionization because their spatial extent can be used as a probe of the intergalactic medium (IGM) ionization fraction.

Lyman α emitters (LAEs) are suitable objects for studying the nature of LAHs because a large sample of LAEs at a fixed redshift as needed for a stacking analysis can be constructed relatively easily from a narrow-band imaging survey <cit.>. LAEs are typically low-stellar-mass young galaxies with low metallicities and low-dust contents hosted in low-mass dark matter halos <cit.>. They are detected owing to efficient Lyα escapes, which are suggested to stem partly from these physical properties such as low-dust attenuation <cit.>. 

<cit.> have found that LAEs in a large-scale overdense region at z=3.1 have large (∼ 100–200 Å) EWs if LAH components are included. They suggest that those LAHs may partly originate from shock heating due to gas outflows or cold streams, although they have not ruled out other possibilities. On the other hand, <cit.> have stacked ∼3600 LAEs in field regions at z∼2 to find that some subsamples have relatively small Lyα EWs fully consistent with pop II star formation, suggesting that the cold stream scenario is not preferred. Finding no correlation between  the spatial extent (the scale length, r_s) and the surface number density for LAEs at z∼ 3–4, <cit.> have suggested that star formation in satellite galaxies is not the dominant contributor to LAHs <cit.>. They have also found that the radial profile of LAHs is very close to that predicted by models of resonant scattering in <cit.>, leaving only little room for the contribution from satellites galaxies and cold streams modeled by <cit.>. Note, however, that <cit.>'s model reproduces the radial profile of LAHs seen in LAEs at z∼3 in <cit.>. More recently, <cit.> have measured LAH properties of ∼150 individual LAEs at z∼3–6 using VLT/MUSE. They argue that a significant contribution from star formation in satellite galaxies is somewhat unlikely since the UV component of LAEs is compact and not spatially offset from the center of their LAHs, while having not given a firm conclusion on other origins. 

To summarize, although there are a number of observational studies on the origin of LAHs, their results are not very conclusive, nor consistent with each other <cit.>. This is partly because correlations of LAH properties with properties of central galaxies have not been fully studied. Especially important may be correlations with the dark matter halo mass and stellar mass of central galaxies, because they can be directly compared with theoretical predictions <cit.>. Although <cit.> have discussed a correlation between the Lyα luminosity of LAHs and the UV luminosity of central galaxies, they have not estimated those masses. SFRs and dust attenuation are also important quantities to discuss the scattering origin of LAHs. 

Another problem is that r_s, the scale-length of LAHs that is often used to discuss the origin of LAHs in previous studies, is not robust against measurement errors. Indeed, the dependence of r_s on Lyα luminosity found in individually detected MUSE LAEs is not consistent with the average dependence obtained by <cit.> from stacked images. In contrast, as we will see later, relations between the Lyα luminosity of central galaxies and that of LAHs found in <cit.> is in good agreement with those seen in individual MUSE-LAEs in <cit.>. This suggests that Lyα luminosity is more robust against systematic errors from stacking. 

In this paper, we study the dependence of LAH luminosity on stellar properties and dark matter halo mass using ∼900 star-forming LAEs at z∼2 to identify the dominant origin of LAHs around LAEs. Section 2 summarizes the data and sample used in this study. In section 3, we construct subdivided samples based on UV, Lyα, and K-band properties. We present methods to derive the Lyα luminosities of LAHs as well as the stellar properties and dark matter halo masses of subdivided LAEs in section 4. After showing results in section 5, we discuss the origin of LAHs and high Lyα escape fractions in section 6. 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 H_0 = 70  km s^-1Mpc^-1 (h_100=0.7), the power-law index of the primordial power spectrum n_ s = 1, and the linear amplitude of mass fluctuations σ_8=0.8, which are consistent with the latest Planck results <cit.>. We assume a Salpeter initial mass function <cit.> with a mass range of 0.1–100 M_⊙[To rescale stellar masses in previous studies assuming a Chabrier or Kroupa IMF <cit.>, 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.]. Magnitudes are given in the AB system <cit.> and coordinates are given in J2000. Distances are expressed in comoving units. We use “log” to denote a logarithm with a base 10 (log_10). 












§ DATA AND SAMPLE



 §.§ Sample selection

<cit.> have constructed large samples of z=2.2 LAEs in four deep fields: the Subaru/XMM-Newton Deep Survey (SXDS) field <cit.>, the Cosmic Evolution Survey (COSMOS) field <cit.>, the Hubble Deep Field North <cit.>, and the Chandra Deep Field South <cit.>.In this study, we only use their SXDS and COSMOS samples. We do not use the HDFN sample because the R-band image of this field is not deep enough to derive the UV slope for faint LAEs. We also do not use the CDFS sample because the i, z, and H data are too shallow to perform reliable SED fitting as has been pointed out by <cit.>.

We summarize the sample selection and the estimation of the contamination fraction detailed in <cit.>. LAEs at z=2.14–2.22 are selected using the narrow band NB387 <cit.> as described in selection papers <cit.>. The threshold of the rest-frame equivalent width, EW_0, of emission is EW_0( Lyα) ≥ 20–30Å <cit.>. The NB387 limiting magnitude is 25.7 mag for the SXDS sample and 26.1 mag for the COSMOS sample (2” diameter aperture, 5σ). We only use LAEs with NB387 total (i.e., aperture-corrected; see table <ref>) magnitude brighter than 25.5 mag. All sources detected in either X-ray, UV, or radio have been removed since they are regarded as AGNs. Our entire sample consists of 897 LAEs from ≃ 1980 square arcminutes. The survey area of each field is shown in table <ref>. 

<cit.> have conservatively estimated the fraction of possible interlopers in their LAE samples to be 10 ±  10%, where interlopers are categorized into spurious sources,  AGNs without an X-ray, UV, or radio counterpart, foreground/background galaxies, and z=2.2 LAEs with low EW_0( Lyα) which happen to meet the color selection due to photometric errors. See sections 2.2 and 3.2 of <cit.> for details. We use this contamination fraction to obtain true clustering amplitudes from observed ones in section <ref>.



 §.§ Imaging data for SED fitting

Most of the data used in this work are the same as those used in <cit.>, except that the NIR imaging data are replaced to new ones in this work. We overview the data used in SED fitting in the two fields below. 

We use ten 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 Ks); and two mid-infrared (MIR) bands – IRAC ch1 and ch2. The PSFs of the images are matched in each field. The aperture corrections to convert 3” MIR aperture magnitudes to total magnitudes are taken from <cit.>. For each field, a K-band or NIR detected catalog is used to obtain secure IRAC photometry in section <ref> and to divide the LAEs into subsamples in section <ref>. 



  SXDS field
The images used for SED fitting are as follows: B, V, R, i', and z' images with Subaru/Suprime-Cam from the Subaru/XMM-Newton Deep Survey project <cit.>; J, H, and K images from the data release 11 of the UKIRT/WFCAM UKIDSS/UDS project <cit.>; Spitzer/IRAC 3.6 μm (ch1) and 4.5 μm (ch2) images from the Spitzer Large Area Survey with Hyper-Suprime-Cam (SPLASH) project <cit.>. All images are publicly available except the SPLASH data. The aperture corrections for optical and NIR images are given in <cit.>. The catalog used to clean IRAC photometry and to obtain K-band counterparts is constructed from the K-band image.


  COSMOS field
We use the publicly available B, V, r', i', and z' images with Subaru/Suprime-Cam by the Cosmic Evolution Survey <cit.> and J, H, and Ks images with the VISTA/VIRCAM from the third data release of the UltraVISTA survey <cit.>. We also use Spitzer/IRAC ch1 and ch2 images from the SPLASH project <cit.>. The aperture corrections for the optical images are taken from <cit.> and those for the NIR images follow <cit.>. The catalog used to clean IRAC photometry and to obtain K-band counterparts is the one given by <cit.>, for which sources have been detected in a combined z'YJHKs image.








§ SUBSAMPLES



A vast majority of our LAEs are too faint to estimate stellar masses on individual basis.
To study how LAH luminosity depends on stellar and dark matter halo masses, we therefore divide the entire sample into subsamples in accordance with the following five quantities which are expected to correlate with stellar mass, and perform a stacking analysis on each subsample. 
(i) K-band apparent magnitude, m_ K, known as a good tracer of stellar mass <cit.>. 
(ii) Rest-frame UV absolute magnitude, M_ UV, which is related to SFR and hence expected to trace stellar mass through the star formation main sequence <cit.>. 
(iii) UV spectral slope β (f_λ∝λ^β), an indicator of dust attenuation and may correlate with stellar mass <cit.>. 
(iv) Lyα luminosity L( Lyα) and (v) rest-frame Lyα equivalent width EW_0( Lyα), both of which possibly anti-correlate with stellar mass according to Ando relation <cit.>. 



While only 30–40% of our LAEs are detected in the K band with m_ K≲25.0 (see section <ref>), the other four quantities can be measured for almost all objects because they need only optical imaging data, which are deep enough as shown in table <ref>. We divide the whole sample of each field into two subsamples in accordance with each of m_ K, M_ UV, β, L( Lyα), and EW_0( Lyα); further division makes stacked SEDs too noisy to do reliable SED fitting. Among the five quantities, m_K and M_ UV are expected to correlate with M_⋆ most tightly. The subsamples by β, L( Lyα), and EW_0( Lyα) are useful to check the results obtained for the m_K and M_ UV subsamples, because these three quantities are affected by the NB selection bias differently from m_K and M_ UV as discussed in appendix <ref> (see figure <ref>). As shown later, all five subsample pairs give similar results.




 §.§ UV and Lyα properties

For each object, we measure M_ UV, β, L( Lyα), and EW_0( Lyα) from NB387, B, V, and R magnitudes in the following manner. First, we approximate the UV SED of the object by a simple SED composed of a power-law continuum and a Lyα line centered at rest-frame 1216 Å:

    f_ν (erg s^-1  cm^-2  Hz^-1)   =    A   10^-0.4(m_ UV(1+z)+48.60) ( ν_λ=UV(1+z)/ν) ^β+2
       +    F_ Lyα δ(ν-ν_λ=1216(1+z)),

where A, m_ UV(1+z), and F_ Lyα are the IGM attenuation factor from <cit.>, the apparent UV magnitude (corresponding to M_ UV), and the Lyα flux (erg s^-1  cm^-2), respectively. 
The apparent magnitude of the model SED in a given band i is calculated from its transfer function T_i(λ) as below: 

    m_i, model= -2.5log_10(  ∫ f_ν c/λ^2 T_i(λ) dλ/∫ c/λ^2 T_i(λ) dλ)-48.6,

where c is the speed of light. 

We fit this model SED to the apparent magnitudes of the object with M_ UV, β, and F_ Lyα as free parameters. We search for the best-fit parameter values that minimize

    χ^2 = Σ_i=NB, B, V, R (m_i - m_i, model/σ m_i)^2,

where m_i and σ m_i are the i-th band apparent magnitude and its 1 σ error, respectively. We calculate apparent magnitudes from 2” diameter aperture magnitudes <cit.> assuming that our LAEs are point sources in all four bands including NB387 which detects Lyα emission. We also assume that their Lyα lines are located at the peak of the response function of NB387 and do not correct for flux loss. The best-fit F_ Lyα is obtained by solving ∂χ^2/∂ F_ Lyα=0. Hereafter, we refer to the L( Lyα) and EW_0( Lyα) obtained with the assumption of point sources as L( Lyα)_ps and EW_0, ps( Lyα). Since the best-fit EW_0( Lyα) is derived from the other three parameters, the degree of freedom is one.

Among the 897 LAEs, six sources are undetected in at least one of the three broad bands. We do not use these objects in the following analyses because the four quantities derived from the SED fitting are highly uncertain. 



 §.§ Subsample construction

Since we divide LAEs into two subsamples in accordance with each of the five quantities, we have a total of ten subsamples for each field. The boundaries of the subsamples are defined from the distribution of the five quantities, which is shown in figure <ref>. 

 
Our LAEs are widely distributed over the four UV and Lyα properties as shown in figures <ref> (a) – (d). The distribution of M_ UV, β, L( Lyα)_ps, and EW_0, ps( Lyα) is different between the two fields. This is possibly because of systematic offsets of the zero-point magnitudes (ZPs) of the optical images adopted in the original papers[ZP offsets of optical broad bands can shift the relation between M_ UV and β (figure <ref> [f]). They have a larger effect on smaller-EW_0, ps( Lyα) objects in the EW_0, ps( Lyα) vs. M_ UV plot (figure <ref> [e]), since the contribution of the UV continuum flux in NB387 is larger for such objects (see Appendix <ref> for more details). Because of the NB-selection bias (see also Appendix <ref>), small-EW_0, ps( Lyα) objects tend to have bright M_ UV.] as has been discussed in both the COSMOS <cit.> and SXDS <cit.> fields. However, these papers often claim opposite error directions (see Appendix <ref> for more details). Another possible reason for the different distribution is field-to-field variance from large scale structure (cosmic variance). In this paper, we use the original ZPs following <cit.> and include ZP uncertainties in the flux-density errors in the calculations given in sections <ref> and <ref>.  Although the causes of the different distributions and the correct ZPs remain to be unclear, a pair of subsamples (with the same definition) from the two fields give consistent 
SED fitting results and Lyα luminosities in most cases (see figure <ref>(b), figure <ref>(b), and Table <ref> in Appendix <ref>). 

We define the boundary for the four UV and Lyα quantities so that the two subsamples have roughly comparable sizes:

    M_ UV=-19.2  mag,


    β=-1.6,


    L( Lyα)_ps=1.2×10^42  erg s^-1,

and 

    EW_0, ps( Lyα)=34 A

as indicated by black lines in figure <ref> (a) – (d). The numbers of the LAEs in the eight subsamples are shown in table <ref>. 


For each field, we also construct two subsamples divided by m_ K.  The K-band catalog mentioned in section <ref> effectively include sources with m_ K≲25 mag. Indeed, the 5σ limiting magnitude of the SXDS K-band image is 25.3 mag and the detection image for the COSMOS catalog, a combined z'YJHKs image, reaches deeper than 25.3 mag (5σ). As a result, about 30–40% of the LAEs in each field have a K-band counterpart with m_ K< 25.0 as shown in figure <ref>(e). Therefore, we define the K-magnitude boundary as:

    m_ K=25.0  mag.
 
Note that the COSMOS K image is composed of Deep and Ultradeep stripes. Since this could add an artificial pattern in the sky distribution of K-divided subsamples, we do not use the K-divided subsamples for clustering analysis.

We derive the four UV and Lyα quantities for each subsample from a median-stacked SED (see section <ref>) in the same manner as in section <ref>. We then calculate average values over the two field, e.g., the average M_ UV of the two faint-M_ UV subsamples, as shown by red symbols in figures <ref> (f) – (k). They are located in the middle of the distribution of individual sources (orange and green points), implying that the average SEDs of the subsamples represent well individual LAEs. We find that the subsamples with red β, faint L( Lyα)_ps, small EW_0, ps( Lyα), and bright m_ K as well as bright M_ UV have bright M_ UV as shown by red open symbols. Note that the lower left part in figures <ref> (g) and (h) and the upper left part in figure <ref>(k) show a selection bias: LAEs with faint M_ UV can be detected only if they have bright L( Lyα)_ps. 


 
  
  








§ METHODS

The Lyα luminosities of LAHs are estimated from a stacked observational relation obtained by <cit.>.  We do not perform a stacking analysis of LAHs on our own subsamples since their sample sizes, which are one ninth to one half of the subsample sizes (∼700 each) in <cit.>, are not large enough to obtain reliable results. Parameters that characterize stellar populations and the mass of dark matter halos are derived from SED fitting and clustering analysis, respectively, in the same manner as in <cit.>. 




 §.§ LAH luminosities

The LAHs of LAEs have been studied either by a stacking analysis of large samples or using individually detected objects.
<cit.> have used stacked images of ∼ 700 LAEs in each subsample (in total ∼3600) at z ∼ 2 to compare Lyα luminosities within r=40 kpc (∼5”) to those within r=1” (∼8 kpc). They have estimated an empirical relation between the two Lyα luminosities from ∼3000 LAEs that are the parent sample of our ∼ 900 LAEs.  On the other hand, <cit.> have measured Lyα luminosities for 3≤z≤6 LAEs with an individually detected LAH by fitting a two component model consisting of halo and continuum-like components. We define three kinds of Lyα luminosities as below.




  L( Lyα)_ C
Lyα luminosity at the central part, i.e., the main body of the object where stars are being formed. In <cit.>, it corresponds to the continuum-like component of Lyα luminosities. We assume that the Lyα luminosities within r=1” in 2D images in <cit.> are approximately equal to L( Lyα)_ C. The aperture size r=1” (∼8 kpc) is often used in photometry with ground-based telescopes for point sources, since it is comparable to their typical PSF size and hence r=1” fluxes are nearly equal to total fluxes. <cit.> show that the scale length (r_s) of the continuum-like component of LAEs is typically smaller than 1 kpc, ensuring our assumption that LAEs are point sources.


  L( Lyα)_ H
Lyα luminosity of the LAH. In <cit.>, it approximately corresponds to the halo component of Lyα luminosity. We assume that the Lyα luminosities falling in the annulus of 8 ≤ r ≤ 40 kpc in <cit.> approximately equal to L( Lyα)_ H.  In <cit.>, the typical r_s of the stacked Lyα emission including the LAH component is ∼10 kpc, and LAHs are found to extend up to r∼40 kpc. 


  L( Lyα)_ tot
Total Lyα luminosity. In <cit.>, it corresponds to a sum of L( Lyα)_ C and L( Lyα)_ H. we assume that the Lyα luminosities within 40 kpc in <cit.> approximately equal to L( Lyα)_ tot. 


<cit.> have found that LAEs with fainter L( Lyα)_ C have a higher L( Lyα)_ tot to L( Lyα)_ C ratio, X(L_ Lyα)_ tot/C, as shown in their figure 14. This means that the relative contribution of the halo component to the total Lyα luminosity increases with decreasing L( Lyα)_ C. The best-fitting linear function between X(L_ Lyα)_ tot/C and L( Lyα)_ C, shown as their equation 2 is: 

    X(L_ Lyα)_ tot/C=103.6-2.4 log_10[L( Lyα)_ C (erg/s)].
 
This equation is calculated over 41.5 < log_10(L( Lyα)_ C) < 42.7[They use images with the PSF matched to 1”.32 in FWHM. Here we have corrected a typo in their equation 2 and revised the range of log_10(L( Lyα)_ C). We conclude that this equation is valid over 41.7 < log_10(L( Lyα)_ C) < 42.3 from the discussions below. ] and is shown in figure <ref>(b). 


<cit.> have used the MUSE Hubble Ultra Deep Field survey data to detect LAHs for 145 star forming galaxies (essentially all are LAEs) at 3≤z≤6 individually. They have measured the size and L( Lyα)_ H of Lyα halos as well as L( Lyα)_ C. They do not find a significant evolution of the LAH size with redshift. This result is consistent with that obtained by <cit.> with stacked LAEs at z≃2.2–6.6, implying that the difference in redshift can be ignored in a comparison of the two studies. Indeed, there is no clear redshift evolution in the relations of MUSE LAEs shown by gray filled circles (z<=4.5) and gray open circles (z>4.5) in figure <ref> described below.

In figure <ref>, we compare the stacked observational relation of LAEs at z=2.2 in <cit.> (black lines and red stars) with the individual results by <cit.> (gray and black circles), where X( Lyα)_x/y indicates the Lyα luminosity ratio of the component x to the component y. Figure <ref>(a) is originally discussed in <cit.>, while figure <ref>(b) is used to determine the best-fit linear relation (equation<ref>) in <cit.>. Black lines in figures <ref> (a), (c), and (d) are converted from one in figure <ref>(b). It is notable that the y-axis depends on the x-axis by construction [We regard L( Lyα)_ C and L( Lyα)_ H as two independent parameters in the measurements in <cit.> and <cit.>. Even if objects are randomly distributed in the L( Lyα)_ C and L( Lyα)_ H plane (panel [c]), we will see a 'correlation' in the other three panels because the y axis of these panels is a combination of L( Lyα)_ C and L( Lyα)_ H. Instead, the y-axis of panels (a) and (b) are not affected by a flux-limited detection bias for a sample with a wide range of redshift.  ] in figures <ref> (a), (b) and (d). 
We find that all five stacked data points (red stars) lie in the middle of the distribution of individual MUSE-LAEs (grey circles) over a range of log_10[ L( Lyα)_ C (erg/s)]≃41.7–42.6 or log_10[ L( Lyα)_ tot (erg/s)]≃42.3–42.8. It is also found that the median values of individual MUSE-LAEs (black filled circles in figures <ref> [a] and [c]) are located near the stacked values. 

This means that the stacked results represent the average halo luminosities of LAEs despite the fact that there is a great variation in halo luminosity among objects. The best-fit relation shown by a black line traces well the stacked points except for the brightest one. This is because the brightest point already deviates from the best-fit linear relation determined in figure <ref>(b) while the other four are on the relation. Based on figure <ref>(a), <cit.> have concluded that there is no significant correlation between L( Lyα)_ tot and X( Lyα)_ H/tot on the basis of a Spearman rank correlation coefficient of -0.05 (see their figure 7 and their section 5.3.1). Although the existence of a correlation is not clear, and a further test is needed, figures <ref> (a) and (c) indicate that the stacked results (red stars) also trace the median trend of individual MUSE LAEs (black filled circles). 



In this work, we estimate average L( Lyα)_ H and L( Lyα)_ tot for each subsample from the stacked relation (equation<ref>) as well as average L( Lyα)_ C by multiplying average L( Lyα)_ps (in section <ref>) by 0.77 as an inverse aperture correction of 1”.32 PSF (see table <ref> in appendix). The L( Lyα)_ C values of our subsamples are found to be within the range shown by skyblue inverted triangles in figures <ref> (c) and (d) where the stacked relation traces well the stacked points.  The typical 1σ uncertainties in the individual data points in <cit.> are propagated to uncertainties in L( Lyα)_ H and L( Lyα)_ tot of ∼22% and ∼16%, respectively.  <cit.> also present a stacked relation (anti-correlation) between X(L_ Lyα)_ tot/C and EW_0,ps(L_ Lyα). Using this relation instead of equation <ref> gives nearly the same L( Lyα)_ H and L( Lyα)_ tot values (see appendices <ref> and <ref>).






 §.§ SED fitting

We derive parameters that characterize the stellar populations of our subsamples in the two fields by fitting SEDs based on stacked multiband images. We use 810 LAEs (∼91% of the entire sample, 891) that have data in all ten broadband filters (B, V, R, i, z, J, H, K,  ch1, and ch2). To obtain secure IRAC photometry, some prescriptions are adopted in previous studies <cit.>. In this paper, we follow <cit.> and only use LAEs that are not contaminated by other objects in the ch1 and ch2 images. To do so, we exclude LAEs that have either one or more neighbors or a high sky background through a two-step cleaning process. We are thus left with 121 LAEs for stacking of ch1 and ch2 images <cit.>. We briefly describe stacking analysis, photometry, and SED models below. A detailed description can be found in <cit.>. 



  §.§.§ Stacking Analysis and Photometry

For each band, we use the task IRAF/imcombine to create a median-stacked image at the NB387 source positions from images of size 50” × 50” that are cut out with IRAF/imcopy task. While a stacked SED is not necessarily a good representation of individual objects <cit.>, stacking is still useful for our faint objects to obtain a rest-frame UV to NIR SED. 

An aperture flux is measured for each stacked image using the task PyRAF/phot with the same parameters in <cit.>. We use an aperture diameter of 2” for the NB387, optical, and NIR band images and 3” for the MIR (IRAC) images following <cit.>. For each of the ch1 and ch2 images, we obtain the net 3”-aperture flux density by subtracting an offset of the sky background as described in section 4.2 of <cit.>. All aperture magnitudes are corrected for Galactic extinction, E(B-V)_b, of 0.020 and 0.018 for the SXDS and COSMOS fields, respectively <cit.>. 

The aperture magnitudes are converted into total magnitudes using the aperture correction values summarized in table <ref>. The 1σ uncertainty in the total magnitudes is the sum of the errors in photometry, aperture correction, and the ZP. For the ch1 and ch2 data, the errors in sky subtraction are also included. The photometric errors are determined in the same procedure as <cit.>. The aperture correction errors in the NB387, optical, and NIR bands are estimated to be 0.03 mag, and those in the ch1 and ch2 bands are set to 0.05 mag. The ZP errors for all bands are set to be 0.1 mag. The stacked SEDs thus obtained for individual subsamples are shown in figures <ref> and <ref> in appendix.



  §.§.§ SED models

We perform SED fitting on the stacked SEDs with model SEDs in the same manner as in <cit.>. The model SEDs are constructed by adding nebular emission (lines and continuum) to the stellar population synthesis model GALAXEV <cit.>. We assume constant star formation history, 0.2Z_⊙ stellar metallicity, and E(B-V)_ gas=E(B-V)_⋆ <cit.> following previous SED studies of LAEs <cit.>. We also assume an SMC-like dust extinction model for the attenuation curve <cit.> since it is suggested to be more appropriate for LAEs at z∼2 and low-mass star forming galaxies at z≥2 than the Calzetti curve <cit.> in <cit.> and <cit.>, respectively. The Lyman continuum escape fraction, f^ ion_ esc, is fixed to 0.2 <cit.>. We also examine the case of the Calzetti attenuation curve for comparison with previous studies and conservative discussion. The case without nebular emission (f^ ion_ esc=1) has been examined and discussed in <cit.>.

We search for the best-fitting model SED to the stacked SED of each subsample that minimizes χ^2 and derive the following stellar parameters: stellar mass (M_⋆), color excess (E(B-V)_⋆), age, and SFR. The stellar mass is calculated by solving ∂χ^2/∂ M_⋆ = 0, while the SFR is determined from M_⋆ and age. Thus, the degree of freedom is 7. The 1σ confidence interval in each stellar parameter is obtained from the range of the values giving  χ^2≤χ^2_ min +1, where χ^2_ min is the minimum χ^2 value. Figures <ref> and <ref> in appendix shows the best-fit SEDs and table <ref> in appendix summarize the results of the best-fit parameters in the two fields. The field-average values are shown in table <ref>.














 §.§ Clustering analysis

We derive the angular two-point correlation functions (ACFs) of our subsamples from clustering analysis and convert the correlation lengths into bias factors and then into dark matter halo masses in the same manner as in <cit.>. We briefly describe our methods below. 



  §.§.§ Angular correlation function

The ACF, , for a given subsample is measured by the calculator given in <cit.>: 

    ω_ obs(θ)=DD(θ)-2DR(θ)+RR(θ)/RR(θ),

where DD(θ), RR(θ), and DR(θ) are the normalized numbers of galaxy-galaxy, galaxy-random, and random-random pairs, respectively. We use a random sample composed of 100,000 sources with the same geometrical constraints as the data sample. The sky distributions of the LAEs and the random sources in the two fields are shown in figure 2 of <cit.>. Following <cit.>, the 1 σ uncertainties in ACF measurements are estimated as: 

    Δω_ obs(θ)=1+ω(θ)/√(DD_0(θ)),

where DD_0(θ) is the raw number of galaxy-galaxy pairs. 

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

    ξ(r)=(r/r_0)^-γ,

where r, r_0, and γ are the spatial separation between two objects in comoving scale, the correlation length, and the slope of the power law, respectively <cit.>. We then convert ξ(r) into the ACF, ω_ model(θ), following <cit.>, and describe it as:

    ω_ model(θ) 
    = (r_0  h^-1_100 Mpc/1 h^-1_100 Mpc)^γ ω_ model,  0(θ),

where ω_ model,  0(θ) is the ACF in the case of r_0= 1  h^-1_100 Mpc. The correlation amplitude of the ACF at θ=1”, A_ω, is ω_ model(θ=1”).

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), 

    ω(θ)=ω_ obs(θ)+IC,


    IC= Σ_θ RR(θ) ω_ model,  0(θ)/Σ_θ RR(θ)(r_0  h^-1_100 Mpc/1 h^-1_100 Mpc)^γ ,

where ω(θ) is the true ACF. We fix γ to the fiducial value 1.8 following previous studies <cit.> and fit the ω_ model(θ) to this ω(θ) by minimizing χ^2 over ∼40” -1000”, where we avoid the one-halo term at small scales and large sampling noise at large scales. The best-fit field-average correlation amplitude, , is calculated analytically by minimizing the summation of χ^2 over the two fields in the same manner as in <cit.>. The 1 σ fitting error in A_ω, Δ, is estimated from χ^2_ min +1, where χ^2_ min is the minimum χ^2 value. 

The correlation amplitude corrected for randomly distributed foreground and background interlopers, , is given by

    = A_ω/(1-f_ c)^2,

where  f_ c is the contamination fraction. The contamination fraction of our LAEs is estimated to be 10 ± 10% (0–20%) conservatively (see section <ref>) and the error range in  includes both the no correction case and the maximum correction case <cit.>. The 1 σ error in the contamination-corrected correlation amplitude, Δ, is derived from error propagation of  and f_ c:

    Δ/≃√((Δ/)^2 + (2Δ f_ c/f_ c) ^2  ),

where Δ f_ c(= 0.1) is the uncertainty in the contamination estimate. The value of the contamination-corrected correlation length, r_ 0, corr and its 1 σ error are calculated from  and Δ. Figure <ref> in appendix shows the best-fit ACFs and table <ref> summarizes the results of the clustering analysis.




  §.§.§ Bias factor and dark matter halo mass

The galaxy-matter bias, b_ g, is defined as 
 
    b_ g(r)=√(ξ(r)/ξ_ DM(r,z)),

 where ξ_ DM(r,z) is the spatial correlation function of underlying dark matter calculated with the linear dark matter power spectrum <cit.>. We estimate the effective galaxy-matter bias, b_ g,  eff, at r=8  h^-1_100 Mpc following previous clustering analyses <cit.> using a suite of cosmological codes called Colossus <cit.>. The obtained b_ g,  eff is converted into the peak height in the linear density field, ν, by the formula given in <cit.>. The effective dark matter halo mass is derived from ν with the top-hat window function and the linear dark matter power spectrum <cit.> using a cosmological package for Python called CosmoloPy[http://roban.github.com/CosmoloPy/]. The effective bias and the effective halo mass of each subsample is listed in table <ref>. 






§ RESULTS










The field-average results of the SED fitting and clustering analysis are shown in tables <ref> and <ref>, respectively. In sections <ref> and <ref>, we focus on their LAH luminosities and Lyα escape fractions, respectively. In section <ref>, we compare the infrared excess (IRX) and star formation mode of our subsamples with the average relations of star forming galaxies and examine whether they are normal galaxies in terms of these two properties, which will be employed in the discussion of the origin of LAHs in section <ref>.




 §.§ Halo and total Lyα luminosities

Figure <ref> plots L( Lyα)_ H and L( Lyα)_ tot against SFR, E(B-V)_⋆, M_⋆, and M_ h. The ten subsamples have similar L( Lyα)_ H of ∼ 2×10^42  erg s^-1, and similar L( Lyα)_ tot of ∼ 2×10^42–4×10^42  erg s^-1 within a factor of 1.5 (see also table <ref> in appendix). This is expected from the small difference in L( Lyα)_ps between the subsamples as described in the next paragraph. What we newly find is that L( Lyα)_ H and L( Lyα)_ tot remain almost unchanged when M_⋆ increases by factor 2–5. This has not been confirmed with SED fitting (including nebular emission in models). 

The nearly constant (or even slightly decreasing) L( Lyα)_ H against M_⋆ is a result of two competing trends. One is that L( Lyα)_ C is constant or decreases with M_⋆ as expected from the L( Lyα)_ps vs. M_ UV plot (figure <ref> [g]), and the other is that L( Lyα)_ H/L( Lyα)_ C decreases with L( Lyα)_ C as found from equation (<ref>). Let us take the L( Lyα)–divided and K–divided subsamples as two examples. For the former subsamples, the L( Lyα)_ C of the massive subsample is factor 2.5 lower than that of the less massive one, but the difference is reduced to factor 1.5 in L( Lyα)_ H because objects with lower L( Lyα)_ C have higher L( Lyα)_ H/L( Lyα)_ C. For the latter, the two subsamples have almost the same L( Lyα)_ C and hence almost the same L( Lyα)_ H. The slightly decreasing trend of L( Lyα)_ tot with mass is due to the fact that L( Lyα)_ tot/L( Lyα)_ C decreases with L( Lyα)_ C more mildly than L( Lyα)_ H/L( Lyα)_ C does.


Figure <ref> shows that L( Lyα)_ H and L( Lyα)_ tot are also nearly independent of SFR, E(B-V)_⋆, and M_ h, although the uncertainties in M_ h are relatively large. The fact that differently defined subsamples follow a common trend in each panel indicates that the nearly constant L( Lyα)_ H and L( Lyα)_ tot against M_⋆ and the other three parameters are real; it is unlikely that grouping the LAEs into two by the five quantities has erased strong mass dependence which otherwise would be visible. We discuss the physical origins of diffuse Lyα halos from these results in section <ref>. 



 §.§ Escape fraction of Lyα photons

Following previous studies, we define the escape fraction of Lyα photons, f_ esc( Lyα), as the ratio of observed Lyα luminosity, L( Lyα)_obs, to intrinsic Lyα luminosity, L( Lyα)_int, produced in the galaxy due to star formation <cit.>:

    f_ esc( Lyα)=L( Lyα)_obs/L( Lyα)_int= SFR_Lyα/SFR_ tot,

where SFR_ tot is the total (i.e., dust-corrected) star formation rate and SFR_Lyα is the star formation rate converted from L( Lyα)_obs as below: 

    SFR_Lyα ( M_⊙ yr^-1)=9.1×10^-43 L( Lyα)_obs (erg s^-1)

<cit.>. 
In this work, we derive f_ esc( Lyα) from L( Lyα)_ tot (total Lyα escape fraction,  f_ esc( Lyα)_ tot; see table <ref>) unlike previous studies which have ignored the contribution from the LAH <cit.>. For SFR_ tot we use the one obtained from the SED fitting.  This definition of f_ esc( Lyα) thus assumes that all Lyα photons including those of the LAH are produced from star formation in the central galaxy. We discuss the possibility of the existence of additional Lyα sources later. 

Figure <ref> shows f_ esc( Lyα)_ tot as a functions of M_⋆, SFR, and E(B-V) for the ten subsamples. All values are field-average values. For a thorough discussion, results with a Calzetti curve are also shown (figures <ref> [b], [d], and [f]) as well as those with an SMC curve (the other panels). Two interesting features are seen in these figures. 

First, f_ esc( Lyα)_ tot anti-correlates with M_⋆, SFR, and E(B-V) regardless of the assumed curve. Similar anti-correlations have been found for HAEs by <cit.> who have measured total Lyα luminosities on a 6” diameter aperture, corresponding to 24 kpc in radius (blue crosses in the Calzetti-curve panels; see also footnote <ref>). Any galaxy population may have such anti-correlations. Indeed, an anti-correlation between f_ esc( Lyα) and E(B-V) is found for star forming galaxies at z∼0–3 <cit.>. 
Although Lyα halos are not included in their calculations, these results imply an anti-correlation between f_ esc( Lyα)_ tot and E(B-V) since L( Lyα)_ tot increases with L( Lyα)_ C as seen in figure <ref>(d).  

Second, our LAEs have very high f_ esc( Lyα)_ tot values. For an SMC-like curve, they are higher than ∼ 30%, with some exceeding 100%. Using a Calzetti curve makes f_ esc( Lyα)_ tot lower but still in a range of ∼ 10–100%. The typical f_ esc( Lyα)_ tot of the LAE sample is ∼1 dex higher than that of the HAE sample, which is similar to the result obtained in <cit.>. More importantly, a large f_ esc( Lyα)_ tot difference is found even in comparison at a fixed M_⋆, SFR, and E(B-V)_⋆. We discuss mechanisms by which LAEs can achieve such high escape fractions in section <ref>. 



 §.§ IRX and star formation mode

Star-forming galaxies have a positive correlation that more massive ones have higher IRXs. The IRX ≡ L_ IR/L_ UV is an indicator of dustiness, where L_ IR and L_ UV are IR (8–1000μm) and UV (1530Å) luminosities, respectively <cit.>. Average M_⋆-IRX relations have been obtained by several studies at z∼2 <cit.>. Another important correlation seen in star-forming galaxies is that more massive galaxies have higher SFRs, i.e., the star formation main sequence <cit.>. Outliers above the SFMS are starburst galaxies <cit.>. 
We use these two correlations to test whether or not our subdivided LAEs are outliers in terms of dustiness and star-formation activity. Here, we include nebular emission in SED fitting unlike previous work for subdivided LAEs at z∼2 <cit.> following our previous work for whole LAE sample <cit.>.



  §.§.§ IRX

The IRX can be calculated from the UV attenuation A_1530 <cit.>. <cit.> have found that high-z galaxies (z≃0.95-2.2) follow the relation given in <cit.>:  

    log_10IRX = log_10(10^0.4A_1530-1 ) - log_10(0.595),

as shown in their figure 14 [This formula is derived with the total IR luminosity (3–1000μm, TIR) for local galaxies. According to the result in <cit.>, we do not correct IRXs to those with IR luminosity (8–1000μm) in the relation, unlike our previous work <cit.>. ]. We convert the E(B-V)_⋆ of our subsamples into IRX and compare them with two average relations at z∼2 <cit.>[<cit.> have obtained a consensus relation from previous analyses for galaxies at z ∼ 2–3 <cit.>, which is consistent with their result using ALMA data. On the other hand, <cit.> derives a relation for UV-selected galaxies at z ∼ 1.5 giving higher IRXs than the consensus relation at low-stellar masses regime, however it is consistent wit a new result of star forming galaxies at 2<z<3 with ALMA data <cit.>.] as shown in figure <ref>. At low-stellar masses with M_⋆≲3–5×10^9  M_⊙, the average relation has not been defined well but it is probably located between the two.   

Figure <ref> (a) shows the field-average values of our subsamples with the assumption of an SMC-like attenuation curve (red symbols), which are calculated from the results for the two fields shown in figure <ref> (b) (orange and green symbols). The field-average results lie on an extrapolation of the relation for UV-selected galaxies at z∼1.5 in <cit.>. Considering the relatively large uncertainties remaining in the two average relations, we conclude that our subdivided LAEs are not outliers but have normal dustinesses. This result is consistent with those obtained for all LAEs using Spitzer/MIPS 24μm data by <cit.> and from SED fitting by <cit.>. Note, however, that if we assume a Calzetti-like attenuation curve instead, our LAEs are expected to be dustier galaxies than ordinary galaxies at the same stellar masses as shown by pink symbols in figure <ref> (c). In section <ref>, we use the relation in <cit.> for the discussion of the origin of LAHs.



  §.§.§ Star formation mode

At z∼2, the SFMS has been determined well down to M_⋆∼10^10  M_⊙ <cit.>
since SFRs can be accurately measured from either rest-frame UV and FIR (or MIR) fluxes or Hα and Hβ emission-line fluxes.　Although these results are not consistent with each other as shown in figure <ref>, the true SFMS probably lies somewhere between the <cit.> and <cit.>'s results. Below M_⋆∼10^10  M_⊙, <cit.> suggest that the SFMS continues down to M_⋆∼10^8  M_⊙ without changing its power-law slope. We compare the results for our LAEs with the extrapolated SFMS shown in <cit.> and <cit.> below.

Figure <ref> (a) shows the field-average values for the ten subsamples with an SMC-like attenuation curve (red symbols) while figure <ref> (b) the separate results for the two fields (orange and green symbols). All the field-average data points lie on the extrapolation of the SFMS in <cit.>, being only slightly above the Shivaei et al. relation. This result is also consistent with those obtained for all LAEs by <cit.> and <cit.>. We conclude that the majority of our subdivided LAEs are in a moderate star formation mode even after divided into two subsamples by various properties. 
In section <ref>, we use the relation in <cit.> for the discussion of the origin of LAHs.


We also compare our results to previous studies on individual LAEs and Hα emitters (HAEs) at similar redshifts. For this comparison, we use the results based on a Calzetti attenuation curve (figure <ref> [c]) following these previous studies. We find in figure <ref> (d) that our ten subsamples (pink symbols) are distributed in the middle of individual LAEs with M_⋆ and SFR measurements <cit.>[ In <cit.> and <cit.>, M_⋆ are derived from SED fitting with the Calzetti curve and SFR from the IRX–β relation in <cit.>. On the other hand, <cit.> and <cit.> derive both quantities from SED fitting with the Calzetti curve.]. 
In figure <ref> (e), our LAEs are found to be located at the lower-mass regime of NB-detected HAEs <cit.>. While the HAEs in <cit.> (open cyan hexagons)
[They derive M_⋆ from SED fitting with the Calzetti curve <cit.>, while deriving SFRs from Hα luminosities except for MIPS 24μm detected objects whose SFRs are estimated from UV and MIPS photometry <cit.>. 
Note that SFRs calculated from PACS data are not plotted here. ]
lie on the SFMS, those in <cit.> (filled blue hexagons)

[When analyzing individual galaxies, they assume the Calzetti curve to derive M_⋆ and assume E(B-V)_⋆=E(B-V)_g to correct Hα luminosities (and hence SFRs) for dust extinction<cit.>. However, when stacking, they use A( Hα)=1 mag to correct Hα luminosities for all subsamples.] are widely scattered along the horizontal direction around the SFMS because they are essentially Hα luminosity selected. Some HAEs in <cit.> have similarly low stellar masses to our LAEs but with higher SFRs due to this selection bias.



§ DISCUSSION






 §.§ The origin of LAHs

As described in section <ref>, theoretical studies have suggested three physical origins of LAHs around high–z star-forming galaxies: (a) cold streams (gravitational cooling), (b) star formation in satellite galaxies, and  (c) resonant scattering of Lyα photons in the CGM which have escaped from the central galaxy. 
In origins (a) and (b), the Lyα photons of LAHs are produced in situ, while in origin (c) they come from central galaxies. The difference between (a) and (b) is how to produce Lyα photons.　A flow chart and an illustration of these origins are shown in figure 6 in <cit.> and figure 15 in <cit.>, respectively. So far, observations have not yet identified the dominant origin(s) as explained below.

There are two observational studies on the origin of LAHs around star-forming galaxies. <cit.> use 166 LAEs at z ∼ 3–5 detected with the MUSE, while <cit.> are based on a stacking analysis of ∼ 3600 z≃2.2 LAEs from a narrow-band survey, the same parent sample as we use in this study. <cit.> have argued that a significant contribution from (b) star formation in satellite galaxies is somewhat unlikely since the UV component of MUSE-LAEs is compact and not spatially offset from the center of their LAH. However, they have not given a firm conclusion on the contributions from the remaining two origins. This is because while they have found a scaling relation of L( Lyα)_ H∝ L_ UV^0.45 which is not dissimilar to the scaling predicted from hydrodynamical simulations of cold streams by <cit.>, resonant scattering also prefers such a positive scaling relation if f_ esc( Lyα)_ tot is constant. Moreover, they have also found that ∼ 80% of their sample have a not-so-large total EW of Lyα emission, EW_0,tot( Lyα)≲200Å, not exceeding the maximum dust-free EW_0( Lyα) of population II star formation, ∼50–240Å,  with a solar metallicity and a Salpeter IMF <cit.>. If EW_0( Lyα) is larger than ∼ 200Å, Lyα radiation from cold streams would be responsible for LAHs. 

<cit.> have also found relatively low EW_0,tot( Lyα) and marginally ruled out the cold-stream origin based on a similar discussion to <cit.>'s. In these two observational studies, EW_0,tot( Lyα) are calculated by dividing the total Lyα luminosity by the UV luminosity of the central part. Therefore, the relatively low EW_0,tot( Lyα) values do not necessarily mean that the net EW_0 of LAHs are also low; they would even be extremely high if LAHs do not have UV emission. Thus, the cold-stream scenario cannot be ruled out from the low EW_0,tot( Lyα) values alone. The discussion using the L( Lyα)_ H–L_ UV relation assumes L_ UV∝ M_ h^0.5 because the simulations have calculated L( Lyα)_ H against M_ h. Since L_ UV may not be a perfect tracer of M_ h, it is more desirable to use directly the L( Lyα)_ H-M_ h relation, or the L( Lyα)_ H–M_⋆ relation as a better substitute. In addition, comparing the normalization of the relation as well as its power-law slope can better constrain this scenario. With regard to (b) satellite star formation, independent observations are desirable to strengthen the conclusion by <cit.> since <cit.> have not been able to rule out this origin. Finally, if resonant scattering is the dominant origin, LAH luminosities have to be explained by the properties of the main body of galaxies such as SFR and E(B-V).

In section <ref>, we find that the L( Lyα)_ H and L( Lyα)_ tot of our LAEs remain unchanged with increasing stellar mass. We also obtain a constant or increasing X( Lyα)_ H/tot with M_⋆ (see figure <ref>[g]). In the following subsections, we use these relations to discuss the three origins  with figure <ref>. We also use the results on HAEs obtained by <cit.>[They discuss the escape fraction using L( Lyα) on r=12 kpc (3” diameter) and 24 kpc (6”) apertures. Although the average profile of their LAHs extends to r=40 kpc, we refer to 6” aperture luminosity as L( Lyα)_ tot and to the difference in 3” and 6” aperture luminosities as L( Lyα)_ H. ] to strengthen the discussion. We also briefly examine the fluorescence scenario in appendix <ref>, following the very recent study on fluorescence emission for star-forming LAEs by <cit.>. 　





  §.§.§ (a) Cold streams


Theoretical studies and simulations suggest that high-z (z≳2) galaxies obtain baryons through the accretion of relatively dense and cold (∼ 10^4 K) gas known as cold streams <cit.>. The accreting gas releases the gravitational energy and emits Lyα photons, thus producing an extended Lyα halo without (extended) UV continuum emission <cit.>. 

The Lyα luminosity due to cold streams is suggested to increase with the M_ h of host galaxies. A scaling of L( Lyα)_ H∝M_ h^1.1-M_ h^1.25 at M_=10^10–10^13 M_⊙ has been predicted by (zoom-in) cosmological hydrodynamical simulations in <cit.> and <cit.>. <cit.> have obtained a similar correlation to <cit.>'s from an analytic model which reproduces the Lyα luminosities, Lyα line widths, and number densities of observed LABs at M_ h≳10^11M_⊙. On the other hand, <cit.> have derived a shallower power law slope ∼0.8 for LAB-hosting massive (M_ h∼ 10^12–10^13 M_⊙) halos from high-resolution cosmological hydrodynamical adaptive mesh refinement simulations.


We examine if our subsamples are consistent with these theoretical predictions by comparing the power-law slope and amplitude of the L( Lyα)_ H-M_ h relation. For a conservative discussion, we use <cit.>'s relation which gives the steepest slope and <cit.>'s relation giving the shallowest slope as shown in figure <ref>(b) [We shift the relation shown in figure 8 in <cit.> at z=3 to z=2 by multiplying redshift-evolution term, (1+z)^1.3, given in figure 12 and equation 21 in <cit.>. We also note that the relation at z∼3 predicted in <cit.> has a lower amplitude than that in <cit.> typically about a factor of two <cit.>.]:　

    L( Lyα)_ H∼ 8×10^42( M_ h/10^12M_⊙)^1.25(1+z/1+3)^1.3,



    L( Lyα)_ H=9.72×10^42(M_ h/10^12M_⊙)^0.8 (1+z)^1.3.

In figure <ref>(a), we convert M_ h to M_⋆ using the average relation between M_⋆ and M_ h at z∼2 in <cit.>[<cit.> have found that our LAEs are on average slightly offset from the average relation to lower M_ h values. Our discussion is unchanged if we instead use M_ h reduced by this offset.]. The constant  L( Lyα)_ H with M_⋆ and M_ h seen in the LAEs is inconsistent with the increasing L( Lyα)_ H predicted by the theoretical models, although the uncertainties in our M_ h estimates are large. The HAEs have also non-increasing L( Lyα)_ H over two orders of magnitude in M_⋆, highlighting the inconsistency found for the LAEs. As for amplitude, the LAEs shown by red filled (open) symbols have ∼2–4 (∼1–2) times higher L( Lyα)_ H than the two model predictions at the same M_⋆ (figure <ref>[a]), and at least ∼1–10 (∼1–10) times higher at the same M_ h (figure <ref>[b]). Even when the individual distribution of <cit.>'s galaxies is considered, low-M_⋆ LAEs (red filled symbols) have more than 10 σ brighter L( Lyα)_ H than the simulated galaxies with similar M_⋆ (a gray shaded region). In other words, cold streams cannot produce as many Lyα photons in the CGM as observed. 

 Note that as mentioned in appendix <ref>, the L( Lyα)_ H values of the faint m_K and M_UV subsamples are possibly overestimated since they miss small EW( Lyα) (faint L( Lyα)_ C) sources due to the NB-selection bias. If we derive L( Lyα)_ H conservatively from the M_UV–L( Lyα)_ H relation for individual MUSE-LAEs without such a selection bias in <cit.>, we obtain ∼1.5 times smaller L( Lyα)_ H, which results in a slightly positive correlation between M_⋆ and L( Lyα)_ H. However, the power law index and the amplitude of the M_⋆–L( Lyα)_ H correlation of the m_K subsamples is still shallower and higher than theoretical results at more than the 2 σ and 10 σ confidence levels, respectively (see more details in appendix <ref>). Consequently, our study suggests that (a) cold streams are not the dominant origin of LAHs.





  §.§.§ (b) Satellite star formation

Satellite galaxies emit Lyα photons through star formation. If satellite star formation significantly contributes to LAHs, they will involve an extended UV emission from the star formation <cit.>. Unfortunately, this emission is expected to be too diffuse to detect even by stacking of some 10^3 objects as mentioned in <cit.>. 


The Lyα luminosity from satellite star formation can be interpreted as a function of the M_ h and M_⋆ of the central galaxy. In the local universe, the number of disk (i.e., star-forming) satellite galaxies is found to be described by a power law of the host halo mass of the central galaxy with a slope of 0.91±0.11 for galaxies with M_ h∼10^12–10^14 M_⊙ <cit.>. At high redshifts, at least for massive central galaxies (M_⋆∼10^11 M_⊙ at z∼1.4), the radial number density profile of satellite galaxies is not significantly different from that at z∼0 <cit.>. These local properties are reproduced by theoretical models <cit.>. With an assumption that the total Lyα luminosity from satellite galaxies is proportional to the sum of their SFRs of satellite galaxies, L( Lyα)_ H can be calculated from cosmological galaxy formation models. 

The “New Numerical Galaxy Catalogue” (ν^2GC) is a cosmological galaxy formation model with semi-analytic approach <cit.>, which is based on a state-of-the-art N body simulation performed by <cit.>. It can reproduce not only the present-day luminosity functions (LF) and HI mass function but also the evolution of the LFs and the cosmic star formation history <cit.>. We use model galaxies at z∼2.2 in the ν^2GC-S with a box size of 280  h^-1cMpc (LAE NB selection is not applied). The number of central galaxies is ∼6×10^6. For each central galaxy, we calculate L( Lyα)_ H by summing the SFRs of the satellites with an assumption of case B recombination. We find that the average L( Lyα)_ H can be approximated as

    L( Lyα)_ H∼ 8.3×10^42( M_ h/10^12M_⊙)^2.58

at M_ h∼10^10–10^12 M_⊙ and 

    L( Lyα)_ H∼ 1.9×10^42( M_⋆/10^10M_⊙)^1.36

at M_⋆∼10^8–10^10 M_⊙ as shown with a thin black dashed line in figures <ref> (c) and (d). The power law of M_ h for L( Lyα)_ H is steeper than that for the observed number of disk satellite galaxies.

We focus on the amplitude and slope of the L( Lyα)_ H – mass relations. The LAEs shown by red symbols have more than ∼1 dex higher L( Lyα)_ H than the mean of the model galaxies at the same M_⋆ and M_ h. 
However, observations show that LAEs occupy only ∼10% (∼2%) of all galaxies with the same M_⋆ (M_ h) <cit.>. For a conservative comparison, we limit the model galaxies to those with the top 10% (2%) L( Lyα)_ H at a fixed M_⋆ (M_ h). We find that the mean L( Lyα)_ H of these L( Lyα)_ H-bright model galaxies (thick dashed lines in figures <ref> [c] and [d]) 
is still about three times lower than the observed values. Moreover, the positive correlations of L( Lyα)_ H with M_⋆ and M_ h seen for the model galaxies are incompatible with the constant L( Lyα)_ H of our LAEs and with the decreasing L( Lyα)_ H of the HAEs in <cit.>. These LAEs and HAEs span two orders of magnitude in M_⋆. A non-increasing L( Lyα)_ H over this wide mass range may be achieved if the Lyα photons from satellites of massive galaxies are heavily absorbed in the CGM, but the offset of L( Lyα)_ H from our LAEs becomes larger. Such a heavy dust pollution in the CGM is probably unlikely. 



 As described in the previous subsection, using <cit.>'s M_ UV–L( Lyα)_ H relation results in a slightly positive correlation. However, the power law index determined by the m_K subsamples is still shallower than that of the model (see appendix <ref> for detalis). In addition, it remains difficult for the model to explain the results of LAEs and HAEs in a unfied manner.  
From these results, we conclude that satellite star formation is unlikely to be the dominant origin.



  §.§.§ (c) Resonant scattering of Lyα photons in the CGM which are produced in central galaxies

HI gas in the CGM can resonantly scatter Lyα photons which have escaped from the main body of the galaxy <cit.>. However, there is no theoretical study that predicts L( Lyα)_ H and its dependence on galaxy properties by solving the radiative transfer of Lyα photons in the CGM. In this subsection, we first describe the LAH luminosity of a galaxy assuming that all Lyα photons come from the main body. To do so, we introduce two parameters: the escape fraction out to the CGM and the scattering efficiency in the CGM. Then, we examine if resonant scattering can explain reasonably well the behavior of LAEs and HAEs shown in the previous section. Let L( Lyα)_int be the total luminosity of Lyα photons produced in the main body. Some fraction of L( Lyα)_int is absorbed by dust in the interstellar medium (ISM) and the rest escapes out into the CGM. With an assumption that dust absorption in the CGM is negligibly small, the escaping luminosity is equal to L( Lyα)_ tot (= L( Lyα)_ C + L( Lyα)_ H), and the escape fraction into the CGM is calculated as f_ esc( Lyα)_ tot = L( Lyα)_ tot / L( Lyα)_int.

Then, a fraction, X( Lyα)_ H/tot, of the escaping  photons are scattered in the CGM, being extended as a LAH with L( Lyα)_ H.
Thus, L( Lyα)_ H can be written as: 

    L( Lyα)_ H   =   L( Lyα)_int f_ esc( Lyα)_ tot  X( Lyα)_ H/tot
       =   L( Lyα)_ tot  X( Lyα)_ H/tot.


In the following modeling, we assume that L( Lyα)_int originates only from star formation, and express it as a function of M_⋆ using the SFMS:

    L( Lyα)_int (erg s^-1)= SFR_ MS ( M_⊙ yr^-1)/9.1 × 10^-43.

We then describe f_ esc( Lyα)_ tot as a function of M_⋆ using the M_⋆–IRX relation discussed in section <ref>. The dust attenuation for 1216 Å continuum, A_ 1216con, at a fixed M_⋆ is calculated from IRX (M_⋆):    

    A_ 1216con(M_⋆)=2.5log_10(0.595IRX(M_⋆)+1.0) (κ_1216/κ_1500),

where κ_1216 and κ_1500 are the coefficients of the attenuation curve at λ=1216 Å and 1500 Å, respectively. Introducing the relative efficiency of the attenuation of Lyα emission to the continuum at the same wavelength, q=A_ Lyα/A_ 1216con <cit.>, we can write f_ esc( Lyα)_ tot as:

    f_ esc( Lyα)_ tot=10^-0.4 q A_ 1216con(M_⋆),

where q=0 and q=1 correspond to the case without attenuation of Lyα emission and with the same attenuation as that of continuum. 
We thus obtain: 

    L( Lyα)_ tot(M_⋆)=(SFR_ MS (M_⋆)/9.1×10^-43)10^-0.4 q A_1216(M_⋆).

We use <cit.>'s SFMS and <cit.>'s IRX-M_⋆ relation because our LAEs are on these relations (see section <ref>). We also assume an SMC-like attenuation curve.

Shown in figure <ref>(e) are three calculations with q=0.0,  1.0, and 1.3 (gray (thick), dark gray, and black (thin) dotted lines, respectively). The constant L( Lyα)_ tot with increasing M_⋆ seen in the LAEs is achieved if q increases with M_⋆. We note that all LAEs require q<1, with the less massive subsamples suggesting q=0, meaning that Lyα photons escape much more efficiently than UV photons. We do not compare the HAEs with these models since they do not follow well the SFMS and the IRX-M_⋆ relation (see section <ref>). As we show later, the HAEs can be explained by large q values. Further discussion of f_ esc( Lyα)_ tot and q for our LAEs and the HAEs is given in section <ref>. We also find that this result is unchanged even if we instead use a Calzetti attenuation curve, <cit.>'s SFMS, and/or <cit.>'s IRX-M_⋆ relation.

The term X( Lyα)_ H/tot can be interpreted as the efficiency of resonant scattering in the CGM. More massive galaxies may have a larger amount of HI gas in the halo and thus have a higher X( Lyα)_ H/tot value. Figure <ref>(g) shows that this picture is consistent with our LAEs and <cit.>'s HAEs, because these two populations appear to follow a common, positive (although very shallow) correlation between X( Lyα)_ H/tot and M_⋆. This picture is also consistent with the X( Lyα)_ H/tot – M_ h plot for our LAEs (figure <ref>[h]) within the large uncertainties in M_ h. In this case, the LAHs of our LAEs (≲40 kpc in radius) are caused by HI gas roughly within the virial radius of hosting dark matter halos, ∼20–50 kpc, whose mass is estimated to be in the range M_ h∼10^10–10^11 M_⊙.
This relative extent of LAHs is close to those inferred for the LAHs of MUSE-LAEs by <cit.>, typically 60–90% of the virial radius, where they predict M_ h from observed UV luminosities using the semi-analytic model of <cit.>.　

Thus, in the resonant scattering scenario, the constant (or decreasing) L( Lyα)_ H observed is achieved by a combination of increasing L( Lyα)_int, decreasing f_ esc( Lyα)_ tot, and (slightly) increasing X( Lyα)_ H/tot with mass, and all three trends are explained reasonably well. Our study suggests that (c) resonant scattering is the dominant origin of the LAHs. 




  §.§.§ Summary of the three comparisons

It is found that resonant scattering most naturally explains the L( Lyα)_ H and its dependence on galaxy properties seen in our LAEs and <cit.>'s HAEs. We, however, note that hydrodynamic cosmological simulations in <cit.> show that scattered Lyα in the CGM can reach only out to ∼15 kpc, suggesting that cold streams or satellite star formation are also needed, although they slightly overestimate the observed radial Lyα profile at 15 kpc (by a factor of 2). On the other hand, <cit.> have found for LAEs at z ∼ 4 that the radial profile of LAHs is very close to a predicted profile by <cit.> who have only considered resonant scattering. Theoretical models discussing the contribution of scattering to f_ esc( Lyα)_ tot and X( Lyα)_ H/tot as a function of M_⋆ and M_ h are needed for a more detailed comparison.  <cit.> show that different origins give different spatial profiles of Lyα, UV, and Hα emission. According to the best-effort observations of Lyα and Hα emission of LAEs in <cit.>, Lyα photons of LAEs at z∼2 are found to escape over two times larger radii than Hα photons, which implies (a) cold stream scenario or (c) resonant scattering scenario, although their results are based on
images with the PSF as large as ∼2 arcsecond (FWHM). Deep, spatially resolved observations of Hα emission with James Webb Space Telescope (JWST) would provide us with important clues to the origin of LAHs.





 §.§ The origin high Lyα escape fractions




By including L( Lyα)_ H in the total Lyα luminosity, we obtain very high f_ esc( Lyα)_ tot values for our LAEs as shown in section <ref>. These values are systematically higher than those obtained for LAEs in previous studies which have not considered L( Lyα)_ H <cit.>.
They are also about one order of magnitude higher than those of HAEs with the same M_⋆ and E(B-V) (figure <ref>), suggesting a large scatter in f_ esc( Lyα)_ tot among galaxies.

It is helpful to discuss f_ esc( Lyα) using E(B-V), since additional mechanisms are needed to make f_ esc( Lyα) higher or lower than that expected from E(B-V).  The attenuation of Lyα emission relative to that of continuum emission is evaluated by the q-parameter

[The q-parameter can be rewritten as:q=-log(f_ esc( Lyα))/0.4E(B-V)κ_1216 =κ/kκ_1216 - log C/0.4E(B-V)κ_1216, where κ and C are two parameters of a commonly used fitting formula of f_ esc( Lyα) = C 10^-0.4E(B-V)κ <cit.>. The two parameters are difficult to interpret physically, especially for a case with C<1.  <cit.> and <cit.> do not include L( Lyα)_ H to calculate the f_ esc( Lyα) and obtain C=0.445 with κ=13.6 and C=0.22 with κ=6.67, respectively. 

Although <cit.> include L( Lyα)_ H to calculate f_ esc( Lyα)_ tot, their C is less than 1 (C=0.08^+0.02_-0.01 with κ=7.64^+1.38_-1.36), which is slightly larger than the value derived without L( Lyα)_ H ( C=0.03^+0.01_-0.01 with κ=10.71^+0.89_-1.01). Note that <cit.> uses Balmer decrements to estimate E(B-V)_gas, while other studies use SED fitting. ] <cit.>, 
as discussed in section <ref>. Figure <ref> shows q as a function of E(B-V) for our LAEs and <cit.>'s HAEs, which are divided into subsamples in accordance with E(B-V). Regardless of the attenuation curve, the LAEs have small q less than unity, which increases with E(B-V). Remarkably, about a half of the subsamples, shown by red filled symbols, have q<0, meaning that the observed Lyα luminosity exceeds the one calculated from the SFR. On the other hand, the HAEs have larger q (>1) decreasing with E(B-V). The difference in q between these two galaxy populations becomes larger at smaller E(B-V). Note that if we calculate q of our LAEs from L( Lyα)_ps instead of including L( Lyα)_ H, we obtain higher values, q∼1, 
being closer to the values found in previous studies <cit.>. 

Below, we discuss how LAEs can have low q and hence high f_ esc( Lyα)_ tot than HAEs with the same E(B-V), by grouping possible origins into three categories: 
(i) less efficient resonant scattering in a uniform ISM,
(ii) less efficient resonant scattering in a clumpy ISM,
and (iii) additional Lyα sources. We then discuss the difference in q and f_ esc( Lyα)_ tot between the LAEs and HAEs. In this discussion, we assume that the contribution from cold streams and satellite galaxies to L( Lyα)_ H is negligible.



  §.§.§ (i) Less efficient resonant scattering in a uniform ISM
 

In a uniform ISM where dust and gas are well mixed, Lyα photons have a higher chance of dust absorption than continuum photons because of resonant scattering.  To reduce the efficiency of resonant scattering in a uniform ISM, one needs to reduce the column density of HI gas (N_ HI) or the scattering cross section (σ_ Lyα) <cit.>. 

First, it appears that LAEs indeed have lower N_ HI than average galaxies with the same M_⋆ (and hence the same E(B-V) since average galaxies are expected to follow a common IRX-M_⋆ relation). This is because <cit.> suggest that LAEs at z∼ 2 have lower M_ h than expected from the average M_⋆-M_ h relation. At a fixed M_⋆, a lower M_ h means a lower baryon mass and hence a lower gas mass, and it is reasonable to expect that galaxies with a lower gas mass have a lower N_ HI. The N_ HI of LAEs is further reduced if they have a high ionizing parameter as suggested by e.g., <cit.>, <cit.>, and <cit.> or have a relatively face-on inclination <cit.>.

The idea that LAEs have lower N_ HI than average galaxies appears to be consistent with results based on observed Lyα profiles that LAEs have lower N_ HI than LBGs <cit.>. This idea is also consistent with an anti-correlation between M_ HI and f_ esc( Lyα) found for local galaxies, although their f_ esc( Lyα) values at a fixed E(B-V) are lower than those of our LAEs <cit.>.

The probability of the resonant scattering of Lyα photons is also reduced if the ISM is outflowing, 
because the gas sees redshifted Lyα photons <cit.>. This mechanism should work in LAEs because most LAEs have outflows <cit.>. Outflowing gas is also needed to reproduce observed Lyα profiles characterized by a relatively broad, asymmetric shape with a redshifted peak. Note, however, that it is not clear whether LAEs have higher outflow velocities than average galaxies with the same M_⋆ and E(B-V).

To summarize, low HI column densities combined with some other mechanisms such as outflows appear to contribute to the high f_ esc( Lyα)_ tot seen in LAEs. However, none of these mechanisms can reduce q below unity as long as a uniform ISM is assumed.



  §.§.§ (ii) Less efficient resonant scattering in a clumpy ISM
 
Lyα photons are not attenuated by dust if dust is confined in HI clumps <cit.> 
because Lyα photons are scattered on the surface of clumps before being absorbed by dust. <cit.> find that the clumpy dust screen (ISMs) can reproduce observed line ratios of Lyα to Hα (or f_ esc( Lyα)), and Hα to Hβ (orE(B-V)) of local LAEs <cit.>. It is, however, not clear what causes such a clumpy ISM geometry especially for LAEs. Indeed, <cit.> argue that any real ISM is unlikely to give q<1. <cit.> also find that the clumpy ISM model <cit.> can achieve q<1 only under unrealistic conditions: a large covering factor of clumps with high E(B-V), a low HI content in interclump regions, and a uniform, constant, and slow outflow. 



  §.§.§ (iii) Additional Lyα sources 

If galaxies have other Lyα-photon sources in the main body besides star formation, the number of produced Lyα photons is larger than expected from the SFR, resulting in underestimation of q and overestimation of f_ esc( Lyα)_ tot. We discuss three candidate sources: AGNs, cold streams, and hard ionizing spectra.

First, the contribution of AGNs should be modest.
This is because we have removed all objects detected in either X-ray, UV, or radio regarding them as AGNs, and because the fraction of obscured AGNs (AGNs without detection in either X-ray, UV, or radio) in the remaining sample is estimated to be only 2% <cit.>. 

Second, <cit.> have found from hydrodynamical simulations of galaxies with M_ h=10^11.5 M_⊙ at z∼ 3 that the Lyα luminosity from cold streams in the central part of galaxies amounts to as high as ∼45% of that from star formation. 
This result may apply to our LAEs to some degree.　

Third, if our LAEs have a hard ionizing spectrum (in other words, the production efficiency of ionizing photons compared to the UV luminosity, ξ_ ion, is large) as suggested in previous studies on higher-z LAEs <cit.> and brighter LAEs at z∼2.2 <cit.>, the intrinsic number of ionizing photons is larger than that assumed in equation <ref>. A hard ionizing spectrum arises from a young age, a low metallicity, a stellar population with a contribution of massive binary systems, an increasing star formation history, and/or a top-heavy IMF. If our LAEs have ∼0.2 dex harder ξ_ ion than the assumed fiducial value (log_10(ξ_ ion/ Hz  erg^-1)∼25.11), they have f_ esc( Lyα)_ tot lower than unity even in the case of an SMC-like curve. A much harder ξ_ ion by ∼0.4–1 dex would even help to explain the difference in f_ esc( Lyα)_ tot between LAEs and HAEs seen in figure <ref> (right) in the case of the Calzetti curve. To infer ξ_ ion for our sample, we adopt an empirical relation presented by <cit.> in their figure 2[Their f_ esc( Lyα) is derived from Hα luminosity with dust attenuation correction, 0.9 mag <cit.>, and Lyα flux measured as a point source with a 3”-diameter aperture. ]. This relation implies a higher f_ esc( Lyα) and a harder ξ_ ion for LAEs with a larger EW_0,ps( Lyα). Using this relation, we indeed
obtain a harder ξ_ ion of log_10(ξ_ ion/ Hz  erg^-1)∼25.3 for our large-EW LAE subsample whose typical EW_0,ps( Lyα) is ∼70Å. This value is also comparable to those found for z∼3 LAEs in <cit.>. In this case, their total Lyα escape fraction, f_ esc( Lyα)_ tot, would become smaller than unity (∼0.6–0.8) based on equation <ref>, suggesting that an additional Lyα source is not necessarily needed. However, the same relation gives a modest ξ_ ion of log_10(ξ_ ion/ Hz  erg^-1)∼25.1 for the small-EW LAE subsample (EW_0,ps( Lyα)∼25Å ), resulting in f_ esc( Lyα)_ tot∼0.1–0.3 which remains significantly higher than those of HAEs with the same  M_⋆/SFR/E(B-V). These calculations imply that it remains uncertain whether or not LAEs , especially those with a small EW_0( Lyα), typically have a hard ionizing spectrum. They also imply that another mechanism is possibly needed (in addition to hard ionizing spectra) to fully explain the large f_ esc( Lyα)_ tot including the systematic difference from HAEs.



In any case, the very low q values (≲0) seen in about half of our LAEs (red filled objects) indicate a non-neglible contribution from additional Lyα sources. <cit.> have also found several bright LAEs with q<0 as shown in their figure 14, where q would decrease more if they include L( Lyα)_ H in the calculation of f_ esc( Lyα)_ tot. 



  §.§.§ Summary of the mechanisms affecting the q-parameter

The origin of very high f_ esc( Lyα)_ tot and very low q found for LAEs is a long-standing problem. This study makes this problem more serious by including L( Lyα)_ H in the calculation of these parameters. Remarkably, all of our subsamples have q<1 and a half of them reach q<0.

Low N_ HI and small σ_ Lyα should help to increase f_ esc( Lyα)_ tot and reduce q to some degree. However, additional mechanisms are needed to reduce q less than unity, as highlighted by the very low q values, with some being negative, found for our LAE subsamples. Cold streams in the main body of LAEs and hard ionizing spectra are candidate mechanisms while a clumpy ISM may be unlikely.

The q value of galaxies is probably determined by the balance between the efficiency of resonant scattering and additional Lyα-photon sources. Spectroscopic observations of LAEs' Hα luminosities would provide more accurate measurements of f_ esc( Lyα)_ tot (q-parameters). They will also enable us to evaluate the spectral hardness from the UV to Hα luminosity ratio and to constrain the contribution of cold streams from the Lyα to Hα luminosity ratio. 


Our LAEs have much lower q values than the HAEs in the lowest-E(B-V) bin, which indicates that not all galaxies with small E(B-V) (or equivalently, small M_⋆) can be LAEs. A possible reason for this large difference is that our LAEs have lower M_ h and hence lower M_ HI. <cit.>'s HAEs in the lowest-M_⋆ bin (M_⋆∼3×10^9–8×10^9 M_⊙) used for clustering analysis by <cit.> reside in massive dark matter halos of M_ h∼7^+9_-4×10^12 M_⊙ <cit.>, which is one dex larger than the average M_ h of our LAEs

[
We calculate this M_ h value from the correlation length given in <cit.> in the same manner as for our LAEs. Their r_0 and M_ h are higher than those averaged over all the LAEs (r_0=2.30^+ 0.36_- 0.41 h^-1 Mpc, i.e., M_ h=3.2^+ 4.7_- 2.5×10^10 M_⊙), although their median M_⋆ (∼6×10^9 M_⊙) is slightly higher than our average value (∼1×10^9 M_⊙). ].

It would imply that the lowest-E(B-V) HAEs in <cit.> have higher M_ h than our LAEs, since the lowest-E(B-V) HAEs should largely overlap with the lowest-M_⋆ HAEs. Furthermore, a large fraction of low-M_⋆ (M_⋆≲10^9 M_⋆) HAEs are expected to be starburst galaxies as shown in figure <ref>, implying a large amount of gas (including HI) to fuel the starburst. 

However, the higher M_ h also imply that they have brighter L( Lyα) from  cold streams (in the main body). 
If the higher M_ HI can reduce the L( Lyα) produced from both star formation and cold streams sufficiently, the higher q values of the HAEs can be reproduced. 




§ CONCLUSIONS

We have studied the dependence of LAH luminosity on stellar properties and dark matter halo mass using ∼900 star forming LAEs at z∼2 to identify the dominant origin of LAHs. To do so, we have divided the whole sample into ten subsamples in accordance with five physical quantities (m_ K, M_ UV, β, L( Lyα) and EW_0( Lyα)), some of which are expected to correlate with M_⋆ and 
M_ h. We have estimated for each subsample the LAH luminosity from  a stacked observational relation obtained by <cit.>. We have used the obtained dependence of LAH luminosity to test three candidate origins: cold streams, satellite star formation, and resonant scattering. We have also derived 
total Lyα escape fractions and q values by including the halo component, and discussed how LAEs can have high escape fractions. Our main results are as follows. 



  * We compare <cit.>'s observational L( Lyα)_ C–L( Lyα)_ H relation obtained from stacking analysis of essentially the same sample as ours, with the distribution of individual LAEs by VLT/MUSE in <cit.>. We find that their observational relation agrees well with the average trend of individual LAEs as shown in figure <ref>, ensuring the use of the relation for our analysis.


  * Our LAEs are found to lie on an extrapolation of the M_⋆–IRX relation at z∼1.5 in <cit.> and that of the SFMS at z∼2 in <cit.> if an SMC-like attenuation curve is assumed (shown in figures <ref> and <ref>). These results are used in the discussion of the origin of LAHs. 


  * The ten subdivided LAE samples are found to have similar L( Lyα)_ H∼2×10^42  erg s^-1 and L( Lyα)_ tot∼2×10^42–4×10^42  erg s^-1 (shown in figure <ref>). 
Their L( Lyα)_ H and L( Lyα)_ tot remain almost unchanged or even decrease when M_⋆ increases by factor 2–5. They are also nearly independent of SFR, E(B-V)_⋆, and M_ h, although the uncertainties in M_ h are large. The HAEs in <cit.> also have non-increasing L( Lyα)_ H and L( Lyα)_ tot. 
These results are inconsistent with the cold stream scenario and the satellite star formation scenario both of which predict a nearly linear scaling of L( Lyα)_ H with mass (figure <ref>). Specifically, the power law slope of the M_⋆–L( Lyα)_ H relation for the K-divided subsamples, the most stellar-mass sensitive subsamples, is shallower than predictions with more than the 2 σ confidence level.
The former scenario also fails to reproduce the bright L( Lyα)_ H of low-mass subsamples  at, e.g., a more than the 10σ level for the faint m_K subsample. The most likely is the resonant scattering scenario because it can naturally explain these results. 

  * The f_ esc( Lyα)_ tot of all ten subsamples is higher than ∼30%, with some exceeding 100%, with very low q values of -1 ≲ q ≲ 1. Using the Calzetti curve instead of an SMC-like curve makes f_ esc( Lyα)_ tot lower but still in a range of 10–100% with q∼0–1. 
The f_ esc( Lyα)_ tot of the LAEs anti-correlates with M_⋆, SFR, and E(B-V) regardless of the assumed attenuation curve (figure <ref>).
Their f_ esc( Lyα)_ tot and q are higher and lower, respectively, than those of HAEs with similar M_⋆ and E(B-V). 
The very low q values of the LAEs suggest the existence of an additional Lyα source in the main body; Lyα emission from cold streams and hard ionizing spectra are possible candidates.
The difference in q between the LAEs and HAEs is possibly caused by a different balance between resonant scattering and additional Lyα-photon source(s). 


In the near future, we will obtain much better M_ h estimates for ∼9000 LAEs with new NB387 data from ≃ 25 deg^2 taken with Hyper Suprime-Cam <cit.> as part of a large imaging survey program <cit.>. It will enable us to compare observed relations of L( Lyα)_ tot with theoretical predictions more directly. 








§ ACKNOWLEDGMENTS

We thank the anonymous referee for constructive comments and suggestions. We are grateful to Kazuyuki Ogura and Masahiro Nagashima for kindly providing ν^2GC data and giving helpful comments. We are grateful to Yoshiaki Ono for giving insightful comments and suggestions on SED fitting. We would like to express our gratitude to Jorryt Matthee and Ken-ichi Tadaki for kindly providing their data plotted in figures <ref>, and <ref>–<ref> and figure <ref>, respectively. We thank David Sobral for giving insightful comments and suggestions. We acknowledge Ryosuke Goto, Akira Konno, Ryota Kawamata, Taku Okamura, Kazushi Irikura, Ryota Kakuma, and Makoto Ando for constructive discussions at 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 Maunakea has always had within the indigenous Hawaiian community. Based 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 <cit.>, 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.










§ NB SELECTION BIAS


In this Appendix, we first describe the NB-selection bias of our LAE sample, and then discuss the effect of this bias on the obtained M_⋆–L( Lyα)_ H relations. As shown in figures <ref> (g) and (h), our sample misses UV-faint LAEs (M_UV≳-19 mag) with faint L( Lyα)_ps and small EW_0, ps( Lyα). This selection bias has the following effects on subsample properties. 


 
  M_UV and m_K subsamples
 The UV-faint (M_UV>-19.2 mag) subsample is biased toward  brighter L( Lyα)_ps and larger EW_0, ps( Lyα). The K-faint subsample (m_K>25.0 mag) is probably biased similarly. Although the L( Lyα)_ H of these subsamples is probably overestimated, we find in appendix <ref> that it does not change our results. This selection bias probably does not change M_⋆ values since m_K and M_UV are a good tracer of M_⋆. The bright m_K and M_UV subsamples are almost free from this bias. 

 
  β subsamples
 Galaxies with fainter UV luminosities generally have smaller β <cit.>. Although our β subsamples are probably biased to some degree, it is difficult to evaluate the effects on M_⋆ and L( Lyα)_ H estimates quantitatively. However, the effects should be smaller than those on the UV and K subsamples, since the M_UV–β correlation has a large scatter (see figure <ref>[f]). 


 
  L( Lyα)_ps and EW_0, ps(Lyα) subsamples
 The faint L( Lyα) and small EW( Lyα) subsamples are biased toward bright UV magnitudes. Although their L( Lyα)_ H values are probably not affected by the selection bias, their M_⋆ values are expected to be overestimated to some degree. The bright L( Lyα) and large EW( Lyα) subsamples are not biased. If the M_⋆ of the faint L( Lyα) and small EW( Lyα) subsamples decreases, the power-law slope of the M_⋆–L( Lyα)_ H relation becomes shallower, enlarging the descrepancy from the models of cold streams and satellite star formation.

 


In appendix <ref>, we use the MUSE sample to evaluate the robustness of L( Lyα)_ H estimates for our faint m_K and M_UV subsamples. The MUSE sample is complementary to our sample, because it is essentially UV-limited but contains much fewer objects than ours.



§ ISSUES ON ZP OFFSETS OF BROADBAND IMAGES

In section <ref>, we find that the distribution of M_ UV, β, L( Lyα)_ps, and EW_0, ps( Lyα) is different between the SXDS and COSMOS fields. LAEs in the SXDS field tend to have bluer β, fainter L( Lyα)_ps and weaker EW_0, ps( Lyα) compared with those in the COSMOS fields. This is possibly because of systematic offsets of the ZP magnitudes of the optical broadband images adopted in the original papers <cit.>. According to <cit.> and <cit.>, the amount of ZP offsets of the B, V, and R images in the SXDS field is expected to be ∼0.0–0.2 mag. <cit.>, <cit.>, and <cit.> suggest that the amount of ZP offsets of the B, V, and R images in the COSMOS field is ∼0.0–0.2 mag. If the relative ZP offset of B to V images is 0.04 mag <cit.>, the corresponding shift of β is 0.18. If the case with a larger relative offset of 0.148 mag <cit.>, the corresponding shift of β is as large as 0.66. 

More seriously, papers in a given field often claim opposite shift directions, and we can not give a firm conclusion. For instance, in SXDS field, the ZP corrections based on <cit.> make β redder, while those based on <cit.> make β bluer. In this paper, we use the original ZPs following <cit.> and include ZP uncertainties in the flux-density errors in the calculations given in sections <ref> and <ref>. In the following paragraphs, we roughly estimate possible offsets of β, L( Lyα)_ps, and EW_0, ps( Lyα) due to such ZP offsets. 


The shifts of β have a larger effect on L( Lyα)_ps and EW_0, ps( Lyα) for smaller-EW_0, ps( Lyα) objects, since the contribution of the continuum emission around rest-frame 1216Å in an NB387 image is larger than that for larger-EW_0, ps( Lyα) objects as shown in equations <ref> and <ref>. To evaluate this effect quantitatively, we consider a simple case that the ZPs of the broadband images in one of the two fields are shifted resulting in a difference of ∼0.4 in β, which is found in figure <ref>. 

For instance, a large-EW_0, ps( Lyα) object with M_ UV=-19 mag, β=-2.0, and EW_0, ps(  Lyα)=100 Å has NB387=24.38 mag. If the β is overestimated to be -1.6 due to ZP shifts of broad-bands images (with a correct M_ UV estimation at rest-frame 1600Å of -19 mag), the corresponding flux density at rest-frame 1216Å is underestimated. It results in relatively small shifts of L( Lyα)_ps and EW_0, ps( Lyα), ∼3 % and ∼15 % overestimation, respectively, for the same NB387 magnitude (24.38 mag). On the other hand, the same shift of β causes larger overestimation of L( Lyα)_ps and EW_0, ps( Lyα), ∼14 % and ∼28 %, respectively, for a smaller-EW_0, ps( Lyα) object with M_ UV=-19 mag, β=-2.0, and EW_0, ps(  Lyα)=20 Å (and with a fixed NB387, 25.44 mag). In an extreme case of a very-small-EW_0, ps( Lyα) object with EW_0, ps( Lyα)=10Å[Our LAEs are selected from narrow band selection criteria with 20–30 Å cut for galaxy SEDs with 30 Myr without dust attenuation
(i.e. a fixed β) as shown in figure <ref> in <cit.>. The limiting EW depends on β as is common in narrow band selections of LAEs. Therefore, EWs of some of
our LAEs are derived to be smaller than 20–30 Å.], a large overestimation of β=-1.34 (0.66 offset) can overestimate L( Lyα)_ps and EW_0, ps( Lyα) by ∼46 % and ∼75%, respectively. This L( Lyα)_ps offset reaches ∼0.2 dex, which corresponds to the difference in the minimum L( Lyα)_ps in figure <ref>.


 
As shown in figure <ref>, since the EW_0, ps( Lyα) of a stacked LAE for each subsample is larger than 20 Å, such ZP offsets do not change our results, though L( Lyα)_ps and EW_0, ps( Lyα) of some of our individual LAEs (before stacking) with a small EW_0, ps( Lyα) are possibly under/overestimated. 





§ ROBUSTNESS OF L( LYΑ)_H ESTIMATES








We first examine the robustness of the stacked relation (equation <ref>) in <cit.>. We then evaluate the effects of the NB-selection bias on the m_K and M_UV subsamples. 

 To test the robustness of L( Lyα)_ H values derived from equation <ref>, we calculate L( Lyα)_ H from EW_0,ps(L_ Lyα) using another stacked relation presented in <cit.>, an anti-correlation between X(L_ Lyα)_ tot/C and EW_0,ps(L_ Lyα). We find that using this relation gives nearly the same L( Lyα)_ H values as those derived from equation <ref>, with differences being at most 0.09 dex. 

To evaluate the effects of the NB-selection bias on L( Lyα)_ H for the faint m_K and M_UV subsamples, we re-estimate L( Lyα)_ H with a complementary result of the MUSE-LAEs in <cit.> which is essentially free from this kind of bias: the relation between M_UV and L( Lyα)_ H. They have found L( Lyα)_ H anti-correlates with L_ UV (see their figure 16). As shown in figure <ref>, our high-M_⋆ LAEs (red filled objects), which are not affected by the NB-bias, are consistent with the best-fit relation of MUSE-LAEs (black solid line), while the faint m_K and M_UV subsamples are found to lie alightly above the relation. As a result, the power-law slopes of the m_K and M_UV subsamples become positive as shown in figure <ref>. However, they are still shallow. For example, the m_K-divided subsamples give a power-law index of 0.26±0.05, which is more than 2 σ shallower than those of the cold streams models in <cit.> and <cit.>, ∼0.38 and ∼0.75, respectively. This slope is also more than 20 σ shallower than that of the satellite star formation model, ∼1.36. Moreover, the L( Lyα)_ H values of the faint m_K and M_UV subsamples also remain higher than predicted from the cold streams models at a >10 σ level. We conclude that the conclusions obtained in section 6 are robust.




§ ESTIMATED LYΑ LUMINOSITIES



In table <ref>, we show the three kinds of Lyα luminosities for individual subsamples. Note that the typical 1σ uncertainties in the individual data points in <cit.>'s L( Lyα)_ C - L( Lyα)_ H relation are propagated to uncertainties in L( Lyα)_ H and L( Lyα)_ tot of ∼22% and ∼16%, respectively (see figure <ref>)















§ RESULTS OF THE SED FITTING

Figures <ref> and <ref> show the best-fit SEDs for the ten subsamples for each field,  and table <ref> the best-fit stellar parameters.



§ THE BEST FIT ACFS

We show the best-fit ACFs for the two fields and their field-average values in figure <ref>. We do not perform clustering analysis for the K-divided subsamples as described in section <ref>. We do not plot the M_ h of the UV bright and blue β subsamples (figures <ref> and <ref>), since they are not constrained well. This is partly because at small r_0 values like those of these two subsamples, M_ h depends very sensitively on r_0 according to the bias model <cit.>. The differences in the ACF measurement between the two fields have been discussed in <cit.>.



§ THE FLUORESCENCE SCENARIO
 
Some of the ionizing photons produced in central galaxies are converted to fluorescence Lyα emission due to recombination of neutral hydrogen gas in the CGM. We do not include the fluorescence scenario in the discussion of LAHs (section <ref>), since this scenario has been favored for QSOs' LAHs <cit.>. Recently, however, the fluorescence emission of star forming LAEs has been discussed with MUSE data <cit.>. 
In this appendix, we briefly examine this scenario on the basis of
the minimum amount of ionizing photons, N( ion) (photon s^-1), and  hense ξ_ ion, required to maintain the LAH of LAEs with fluorescence while reproducing the nebular emission of the main bodies. 

Our LAEs in a sub region in the SXDS field have been observed with a narrow band targeting Hα emission <cit.>. The stacked Hα luminosity as a point source is L( Hα)_ps, tot∼ 8.4×10^41 L_⊙ with dust attenuation correction <cit.>. The number of ionizing photons that are produced by star formation and then converted to nebular emission in the LAEs is N( ion)_Hα, corr=L( Hα)_ps, tot/1.36×10^-12∼6.1×10^53 photon s^-1 following <cit.>. 

The LAH luminosity is calculated as L( Lyα)_ H∼ 2.0×10^42 L_⊙ from the stacked L( Lyα)_ps∼ 1.8×10^42 L_⊙. It is converted to the number of Lyα photons, N( Lyα)_ LAH=L( Lyα)_ H/hν_ Lyα∼1.3×10^53 photon s^-1 with the Planck constant (h) and the frequency of Lyα (ν_ Lyα). The fraction of recombinations which results in Lyα photons in the optically thick case (case B) is η_ thick=0.66, which is larger than the fraction for the optically thin limit, η_ thin=0.42 <cit.>. The minimum number of ionizing photons that escape from the ISM to the CGM required to maintain the observed LAHs is N( ion)_ LAH=N( Lyα)_ LAH/η_ thick∼1.8×10^53 photon s^-1.  It is notable that the LAH luminosity (surface brightness, more accurately) is independent of the luminosity of ionizing radiation in the highly-ionized, optical thin regime <cit.> which requires larger number of N( ion)_ LAH than that in optically thick case at a fixed hydrogen gas distribution. 
Ionizing radiation is attenuated by dust in the ISM before escaping out to the CGM. The dust-attenuation corrected N( ion)_ LAH, N( ion)_ LAH, corr, is estimated roughly to be 1.1×10^54 photon s^-1 with an underestimated correction with κ∼20 from an SMC-like attenuation curve at ∼1000Å  (>912 Å).  This part gives the largest uncertainty in the whole calculation. 

The minimum value of the intrinsic N( ion) produced in the galaxy is N( ion)_Hα, corr+N( ion)_ LAH, corr∼1.8×10^54 photon s^-1. Here, we do not consider ionizing photons escaping out to the IGM, although LAEs (at z∼3) are found to have high escape fractions of ∼10–30% <cit.>. 

 Our LAEs are estimated to have the total SFR=5.7 M_⊙ yr^-1 on average from SED fitting <cit.>. With the fiducial ξ_ion value of ∼1.3×10^25 Hz erg^-1 <cit.>, this SFR is converted into N( ion)∼5.3×10^53 photon s^-1, which is three times lower than the minimum required value obtained above.  In order for N( ion) to reach ∼1.8×10^54 photon s^-1, ξ_ion must be as high as ∼4×10^25 Hz erg^-1. 
 The minimum value of the required ξ_ion is higher than the estimated ξ_ion (∼2×10^25 Hz erg^-1 ) for the large-EW LAEs (EW_0,ps( Lyα)∼70Å ) from <cit.>'s relation (see section <ref>). Note that the required ξ_ion is consistent with a high ξ_ion estimated for LAEs at z∼3 in <cit.>. We can also estimate the minimum value of the escape fraction of ionizing photons from the ISM as ∼10% from N( ion)_ LAH/(N( ion)_ LAH, corr+N( ion)_Hα, corr). This is larger than 2% for LAEs at z∼3.5 in <cit.>. 

The fluorescence scenario requires a high ξ_ion and a high escape fraction of ionizing photons for the LAEs in subregion of SXDS field with Hα observation even without including ionizing photons escaping out to the IGM. However, we can not exclude the fluorescence scenario completely because of the lack of direct observations of ξ_ion and a high escape fraction of ionizing photons. Further observational and theoretical studies are required to examine the fluorescence scenario for star forming galaxies. 


apj



natexlab#1#1

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