#  Cosmological-Scale Lyman-alpha Forest Absorption Around Galaxies and AGN
Probed with the HETDEX and SDSS Spectroscopic Data

Dongsheng Sun Institute for Cosmic Ray Research, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Department of Astronomy,
Graduate School of Science, the University of Tokyo, 7-3-1 Hongo, Bunkyo,
Tokyo 113-0033, Japan  Ken Mawatari National Astronomical Observatory of
Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Institute for Cosmic Ray
Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582,
Japan  Masami Ouchi National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  Institute for Cosmic Ray Research, The
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Kavli
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
Yoshiaki Ono Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5
Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Hidenobu Yajima Center for
Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba,
Ibaraki 305-8577, Japan  Yechi Zhang Institute for Cosmic Ray Research, The
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
Department of Astronomy, Graduate School of Science, the University of Tokyo,
7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Kavli 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  Makito Abe  Center for
Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba,
Ibaraki 305-8577, Japan  William P. Bowman Department of Astronomy, Yale
University, New Haven, CT 06520  Erin Mentuch Cooper Department of Astronomy,
The University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712,
USA  McDonald Observatory, The University of Texas at Austin, 2515 Speedway
Boulevard, Austin, TX 78712, USA  Dustin Davis Department of Astronomy, The
University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA
Daniel J. Farrow University Observatory, Fakultät für Physik, Ludwig-
Maximilians University Munich, Scheinerstrasse 1, 81679 Munich, Germany  Max-
Planck Institut für extraterrestrische Physik, Giessenbachstrasse 1, 85748
Garching, Germany  Karl Gebhardt Department of Astronomy, The University of
Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA  Gary J. Hill
McDonald Observatory, The University of Texas at Austin, 2515 Speedway
Boulevard, Austin, TX 78712, USA  Department of Astronomy, The University of
Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA  Chenxu Liu
South-Western Institute for Astronomy Research, Yunnan University, Kunming,
Yunnan, 650500, People’s Republic of China  Department of Astronomy, The
University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA
Donald P. Schneider Department of Astronomy & Astrophysics, The Pennsylvania
State University, University Park, PA 16802, USA  Institute for Gravitation
and the Cosmos, The Pennsylvania State University, University Park, PA 16802,
USA

###### Abstract

We present cosmological-scale 3-dimensional (3D) neutral hydrogen (Hi)
tomographic maps at $z=2-3$ over a total of 837 deg2 in two blank fields that
are developed with Ly$\alpha$ forest absorptions of 14,736 background Sloan
Digital Sky Survey (SDSS) quasars at $z$=2.08-3.67. Using the tomographic
maps, we investigate the large-scale ($\gtrsim 10$ $h^{-1}$cMpc) average Hi
radial profiles and two-direction profiles of the line-of-sight (LoS) and
transverse (Trans) directions around galaxies and AGN at $z=2-3$ identified by
the Hobby-Eberly Telescope Dark Energy eXperiment (HETDEX) and SDSS surveys,
respectively. The peak of the Hi radial profile around galaxies is lower than
the one around AGN, suggesting that the dark-matter halos of galaxies are less
massive on average than those of AGN. The LoS profile of AGN is narrower than
the Trans profile, indicating the Kaiser effect. There exist weak absorption
outskirts at $\gtrsim 30$ $h^{-1}$cMpc beyond Hi structures of galaxies and
AGN found in the LoS profiles that can be explained by the Hi gas at $\gtrsim
30$ $h^{-1}$cMpc falls toward the source positions. Our findings indicate that
the Hi radial profile of AGN has transitions from proximity zones ($\lesssim$
a few $h^{-1}$cMpc) to the Hi structures ($\sim 1-30$ $h^{-1}$cMpc) and the
weak absorption outskirts ($\gtrsim 30$ $h^{-1}$cMpc). Although there is no
significant dependence of AGN types (type-1 vs. type-2) on the Hi profiles,
the peaks of the radial profiles anti-correlate with AGN luminosities,
suggesting that AGN’s ionization effects are stronger than the gas mass
differences.

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

##  1 Introduction

Galaxy formation in the Universe is closely related to the neutral hydrogen
(Hi) gas in the intergalactic medium (IGM). Within the modern paradigm of
galaxy formation, galaxies form and evolve in the filament structure of Hi gas
(e.g., Meiksin 2009; Mo et al. 2010). Cosmological hydrodynamics simulations
suggest that the picture of galaxy formation and evolution is associated with
large-scale baryonic gas exchange between the galaxy and the IGM (fox 2017;
van de Voort 2017).

The circulation of gas is one of the keys to understanding galaxy formation
and evolution. The interplay of gravitational and feedback-driven processes
can have surprisingly large effects on the large scale behavior of the IGM.
Some of the radiation produced by massive stars and black hole accretion disks
can escape from the dense gaseous environments and propagate out of galaxies
and photoionize the Hi gas in the circumgalactic medium (CGM) and even in the
IGM (Mukae et al., 2020; National Academies of Sciences, Engineering, 2021).

Great progress has been achieved in exploring the Ly$\alpha$ forest absorption
around galaxies and active galactic nuclei (AGN). The cross-correlation of the
Hi in the IGM and galaxies has been detected by Ly$\alpha$ absorption features
in the spectra of background quasars (e.g., Rauch 1998; Faucher-Giguère et al.
2008a; Prochaska et al. 2013) and bright star-forming galaxies (Steidel et
al., 2010; Mawatari et al., 2016; Thomas et al., 2017). The Keck Baryon
Structure Survey (KBSS: Rudie et al., 2012; Rakic et al., 2012; Turner et al.,
2014), the Very Large Telescope LBG Redshift Survey (VLRS: Crighton et al.,
2011; Tummuangpak et al., 2014), and other spectroscopic programs (e.g.,
Adelberger et al., 2003, 2005) have investigated the detailed properties of
the Ly$\alpha$ forest absorption around galaxies. These observations target Hi
gas around galaxies on the scale of the circumgalactic medium (CGM). Recently,
3-dimensional (3D) Hi tomography mapping, a powerful technique to reconstruct
the large scale structure of Hi gas, has been developed by Lee et al. (2014,
2016, 2018). Hi tomography mapping is originally proposed by Pichon et al.
(2001) and Caucci et al. (2008) with the aim of reconstructing the 3D matter
distribution from the Ly$\alpha$ transmission fluctuation of multiple
sightlines. By this technique, the COSMOS Ly$\alpha$ Mapping and Tomography
Observations (CLAMATO) survey (Lee et al., 2014, 2018) has revealed Hi large
scale structures with spatial resolutions of 2.5 $h^{-1}$ comoving Megaparsec
(cMpc). This survey demonstrates the power of 3D Hi tomography mapping in a
number of applications, including the study of a protocluster at $z=2.44$ (Lee
et al., 2016) and the identification of cosmic voids (Krolewski et al., 2018).
Due to an interpolation algorithm (Section 4.3) used in the reconstruction of
the 3D Hi tomography map, we are able to estimate the Ly$\alpha$ forest
absorption along lines-of-sight where there are no available background
sources. Based on the 3D Hi tomography map of the CLAMATO survey, Momose et
al. (2021) have reported measurements the IGM Hi–galaxy cross-correlation
function (CCF) for several galaxy populations. Due to the limited volume of
the CLAMATO 3D IGM tomography data, Momose et al. (2021) cannot construct the
CCFs at scales over 24 $h^{-1}$cMpc in the direction of transverse to the
line-of-sight. Mukae et al. (2020) have investigated a larger field than the
one of Momose et al. (2021) using 3D Hi tomography mapping and report that a
huge ionized structure of Hi gas associated with an extreme QSO overdensity
region in the EGS field. Mukae et al. (2020) interpret the large ionized
structure as the overlap of multiple proximity zones which are photoionized
regions created by the enhanced ultraviolet background (UVB) of quasars.
However, Mukae et al. (2020) found only one example of a huge ionized bubble,
and no others have been reported in the literature.

Dispite the great effort made by previous studies, the limited volume of
previous work prevents us from understanding how ubiquitous or rare these
large ionized structures are. In order to answer this question, we must
investigate the statistical Ly$\alpha$ forest absorptions around galaxies and
AGN at much larger spatial scales ($\gtrsim 10~{}h^{-1}$cMpc). Although Momose
et al. (2021) derived CCFs for different populations: Ly$\alpha$ emitters
(LAEs), H$\alpha$ emitters (HAEs), [Oiii] emitters (O3Es), active galactic
nuclei (AGN), and submillimeter galaxies (SMGs), on a scale of more than $20$
$h^{-1}$cMpc, the limited sample size results in large uncertainties in the
CCF at large scales and prevents definitive conclusions to be made regarding
the statistical Ly$\alpha$ forest absorptions around galaxies and AGN.

Another open question is the luminosity and AGN type dependence of the large
scale Ly$\alpha$ forest absorption around AGN. Font-Ribera et al. (2013) have
estimated the Ly$\alpha$ forest absorption around AGN using the Sloan Digital
Sky Survey (SDSS; York et al. 2000) data release 9 quasar catalog (DR9Q;
(Pâris et al., 2011)) and find no dependence of the Ly$\alpha$ forest
absorption on AGN luminosity. In this study, we investigate the luminosity
dependence using the SDSS data release 14 quasar (DR14Q; Pâris et al. 2018)
catalog, which includes sources $\sim 2$ magnitude fainter than those used by
Font-Ribera et al. (2013). In the AGN unification model (Antonucci & Miller
1985; see also Spinoglio & Fernández-Ontiveros 2021), which provides a
physical picture that a hot accretion disk of super-massive blackhole is
obscured by a dusty torus, the type-1 and type-2 classes are produced by
different accretion disk viewing angles. In this picture, the type-1 (type-2)
AGN is biased to AGN with a wide (narrow) opening angle. In the case of type-1
AGN, one can directly observe the accretion disks and the broad line region,
while for type-2 AGN, only the narrow line region is observable. Previous
studies have identified the proximity effect that the IGM of type-1 AGN is
statistically more ionized due to the local enhancement of the UV background
on the line-of-sight passing near the AGN (Faucher-Giguère et al., 2008b).
Based on the unification model, the type-2 AGN obscured on the line of sight
statistically radiates in transverse direction. The investigation of the AGN
type dependence on the surrounding Hi can reveal the large scale Ly$\alpha$
forest absorption influenced by the direction of radiation from the AGN.

To investigate the Ly$\alpha$ forest absorptions around galaxies and AGN on
large scales, over tens of $h^{-1}$cMpc, we need conduct a new study in a
field with length of any side larger than 100 $h^{-1}$cMpc. We reconstruct a
3D Hi tomography maps of Ly$\alpha$ forest absorption at $z\sim$ $2-3$ in a
total area of $837$ deg2. We use $\gtrsim 15,000$ background sightlines from
SDSS quasars (Pâris et al., 2018; Lyke et al., 2020) for the Hi tomography map
reconstruction and have a large number of unbiased galaxies and AGN from the
Hobby Eberly Telescope Dark Energy eXperiment (HETDEX; Gebhardt et al. 2021)
and SDSS surveys for the investigations of the large scale Ly$\alpha$ forest
absorptions around galaxies and AGN.

This paper is organized as follows. Section 2 describes the details of the
HETDEX survey and our spectroscopic data. Our foreground and background
samples of galaxies and AGN are presented in Section 3. The technique of
creating the Hi tomography mapping and the reconstructed Hi tomography map are
described in Section 4, and the observational results of Ly$\alpha$ forest
absorptions around galaxies and AGN are given in Section 5. In this section,
we also interpret our results in the context of previous studies, and
investigate the dependence of out tomography maps on AGN type and luminosity.
We adopt a cosmological parameter set of ($\Omega_{m}$, $\Omega_{\rm\Lambda}$,
$h$) = (0.29, 0.71, 0.7) in this study.

##  2 Data

###  2.1 HETDEX Spectra

HETDEX provides an un-targeted, wide-area, integral field spectroscopic
survey, and aims to determine the evolution of dark energy in the redshift
range $1.88-3.52$ using $\sim 1$ million Lyman-$\alpha$ emitters (LAEs) over
540 deg2 in the northern and equatorial fields that are referred to as
“Spring” and “Fall” fields, respectively. The total survey volume is $\sim
10.9$ comoving Gpc3.

The HETDEX spectroscopic data are gathered using the 10 m Hobby-Eberly
Telescope (HET; Ramsey et al., 1994; Hill et al., 2021) to collect light for
the Visible Integral-field Replicable Unit Spectrograph (VIRUS; Hill et al.,
2018, 2021) with 78 integral field unit (IFUs; Kelz et al., 2014) fiber
arrays. VIRUS covers a wavelength, with resolving power ranging from
$750-950$. Each IFU has 448 fibers with a $1^{\prime\prime}.5$ diameter. The
$78$ IFUs are spread over the $22$ arcmin field of view, with a $1/4.6$ fill
factor. Here we make use of the data release 2 of the HETDEX (HDR2; Cooper et
al., 2023) over the Fall and Spring fields. In this study, we investigate the
fields where HETDEX survey data are taken between 2017 January and 2020 June.
The effective area is 11542 arcmin2. The estimated depth of an emission line
at S/N$=5$ reaches $3-4\times 10^{-17}$ erg cm-2 s-1.

###  2.2 Subaru HSC Imaging

The HETDEX-HSC imaging survey was carried out in a total time allocation of 3
nights in $2015-2018$ (semesters S15A, S17A, and S18A; PI: A. Schulze) and
$2019-2020$ (semester S19B; PI: S. Mukae) over a $\sim$250 deg2 area in the
Spring field, accomplishing a 5$\sigma$ limiting magnitude of $r=25.1$ mag.
The SSP-HSC program has obtained deep multi-color imaging data on the 300 deg2
sky, half of which overlaps with the HETDEX footprints. In this study, we use
the $r$-band imaging data from the public data release 2 (PDR2) of SSP-HSC.
The 5$\sigma$ depth of the SSP-HSC PDR2 $r$-band imaging data is typically
$27.7$ mag for the $3^{\prime\prime}.0$ diameter aperture. The data reduction
of HETDEX-HSC survey and SSP-HSC program are processed with HSC pipeline
software, hscPipe (Bosch et al., 2018) version $6.7$.

Because the spectral coverage width of the HETDEX survey is narrow, only 2000
Å, most sources appear as single-line emitters. Furthermore, since the Oii
doublet is not resolved, we rely on the equivalent width (EW) to distinguish
Ly$\alpha$ from Oii. The high-$z$ Ly$\alpha$ emission is typically stronger
than low-$z$ [Oii] lines, due to the intrinsic line strengths and the
cosmological effects. The continuum estimate from the HETDEX spectra reach
about g$=25.5$ (Davis et al., 2021; Cooper et al., 2023) and we improve on
this using the deep HSC imaging. We estimate EW using continua measured from
two sets of images taken by HSC r-band imaging survey for HETDEX (HETDEX-HSC
survey) and the Subaru Strategic Program (SSP-HSC; Aihara et al., 2018). Davis
et al. and Cooper et al. find that our contamination of Oii emitters in the
LAE sample to be below 2%.

###  2.3 SDSS-IV eBOSS Spectra

We use quasar data from eBOSS (Dawson et al., 2016), which is publically
available in the SDSS Data Release 14 and 16 quasar catalog (DR14Q, DR16Q;
Pâris et al., 2018; Lyke et al., 2020). The cosmology survey, eBOSS, is part
of SDSS-IV. The eBOSS quasar targets are selected by the XDQSOz method (Bovy
et al., 2012) and the color cut

$m_{opt}-m_{WISE}\geq(g-i)+3,$ (1)

where $m_{opt}$ is a weighted stacked magnitude in the $g,r$ and $i$ bands and
$m_{WISE}$ is a weighted stacked magnitude in the W1 and W2 bands of the Wide-
Field Infrared Survey (WISE; Wright et al. 2010). The aim of the eBOSS is to
accomplish precision angular-diameter distance measurements and the Hubble
parameter determination at $z\sim 0.6-3.5$ using different tracers of the
underlying density fields over 7500 deg2. Its final goal is to obtain spectra
of $\sim 2.5$ million luminous red galaxies, $\sim 1.95$ million emission line
galaxies, $\sim$ 450,000 QSOs at $0.9\leq z\leq 2.2$, and the Lyman-$\alpha$
forest of 60,000 QSOs at $z>2$ over four years of operation.

The eBOSS program is conducted with twin SDSS spectrographs (Smee et al.,
2013), which are fed by 1,000 fibers connected from the focal plane of the
2.5m Sloan telescope (Gunn et al., 2006) at Apache Point Observatory. SDSS
spectrographs have a fixed spectral bandpass of $3600-10000$ Å over the 7 deg2
field of view. The spectral resolution varies from 1300 at the blue end to
2600 at the red end, where one pixel corresponds to $1.8-5.2$ Å.

##  3 Samples

Our study aims to map the statistical distribution of Hi gas on a cosmological
scale around foreground galaxies and AGN by the 3D Hi tomography mapping
technique with background sources at $z=2-3$. We use the foreground galaxies,
foreground AGN, and background sources presented in Sections 3.1, 3.2, and
3.3, respectively.

Two of the goals of this study are to explore the dependence of luminosity and
AGN type on the Ly$\alpha$ forest absorption. To examine statistical results,
we need a large number of bright AGN and type-2 AGN. Compared to moderately
bright AGN and type-1 AGN, bright AGN and type-2 AGN are relatively rare. To
obtain a sufficiently large samples of bright AGN and type-2 AGN, we expand
the Spring and Fall fields of the HETDEX survey, from which we are able to
investigate the statistical luminosity and AGN type dependence of the HI
distribution around AGN (Section 3.2). The northern extended Spring field
flanking the HETDEX survey fields, referred to as the “ExSpring field”, covers
over 738 deg2, while the equatorial extended Fall field flanking the HETDEX
survey fields, here after “ExFall field”, covers 99 deg2. The total area of
our 3D Hi tomography mapping field is 837 deg2 in the ExSpring and ExFall
fields that is referred to as “our study field”. Our analysis is conducted in
our study field where the foreground galaxies+AGN and the background sources
overlap on the sky. As an example, we present the foreground galaxies+AGN in
the ExFall field at ${\rm z}=2.0-2.2$ in Figure 1. We also present the sky
distribution of the background sources within the ExFall field in Figure 2.
The rest of the foreground and background sources are shown in the Appendix.

Figure 1:  Sky distribution of the foreground AGN and galaxies at ${\rm z}=2.0-2.2$ in the ExFall field. The squares present the positions of All-AGN sample sources. Pink (magenta) squares represent the sources of the T1-AGN (T2-AGN) sample. The cyan and blue dots show the positions of the Galaxy and T1-AGN(H) sample sources, respectively. The black dashed line indicates the border of the Hi tomography map in the Exfall field. Figure 2: Sky distribution of background AGN in the ExFall field. The gray crosses indicate background AGN that are used to reconstruct our Hi tomography map. The back dashed line has the same meaning as that in Figure 1. Table 1: Sample size of foreground samples at $z=2-3$ Name of sample | ExFall | ExSpring | Total | Survey | Criteria
---|---|---|---|---|---
Galaxy | 3431 | 11436 | 14867 | HETDEX | EW${}_{0}>20$ Å, FWHMLyα $<1000$ km/s, Muv$>-22$ mag
T1-AGN(H) | 438 | 1349 | 1787 | HETDEX | EW${}_{0}>20$ Å, FWHMLyα $>1000$ km/s
T1-AGN | 2393 | 12300 | 14693 | SDSS | FWHMLyα $>1000$ km/s
T2-AGN | 436 | 1633 | 2069 | SDSS | FWHMLyα $<1000$ km/s
Table 2: Sample size of background sample at $z=2.08-3.67$ Name of sample | ExFall | ExSpring | Total | Survey | Criteria
---|---|---|---|---|---
background AGN | 2181 | 12555 | 14736 | SDSS | $\langle S/N\rangle_{\rm Ly\alpha forest}>1.4$

###  3.1 Foreground Galaxy Sample

We make a sample of foreground galaxies from the data of the HETDEX spectra
(Section 2.1) and the Subaru HSC images (Section 2.2). With these data, Zhang
et al. (2021) have build a catalog of LAEs that have the rest-frame equivalent
widths (${\rm EW}_{0}$) of ${\rm EW}_{0}>20$ Å and the HETDEXs Emission Line
eXplorer (ELiXer) probabilities (Davis et al., 2021, 2023) larger than 1. This
${\rm EW}_{0}$ cut is similar to previous LAE studies (e.g., Gronwall et al.
2007; Konno et al. 2016). This catalog of LAEs is composed of 15959 objects.
Because the LAE catalog of Zhang et al. (2021) consists of galaxies, type-1
AGN, and type-2 AGN, we isolate galaxies from the sources of the LAE catalog
with the limited observational quantities, Ly$\alpha$ and UV magnitude
($M_{\rm UV}$), that can be obtained from the HETDEX and Subaru/HSC data.
Because type-1 AGN have broad-line Ly$\alpha$ emission, we remove sources with
broad-line Ly$\alpha$ whose full width half maximum (FWHM) of the Ly$\alpha$
emission lines are greater than 1000 km s-1. To remove clear type-2 AGN from
the LAE catalog, we apply a UV magnitude cut of $M_{\rm UV}>-22$ mag that is
the bright end of the UV luminosity function dominated by star-forming
galaxies (Zhang et al., 2021). We then select sources in our study field, and
apply the redshift cut of $z=2.0-3.0$ (as measured by the principle component
analysis of multiple lines; Pâris et al. 2018) to match the redshift range
over which we construct Hi tomography map. These redshifts are measured with
Ly$\alpha$ emission (Zhang et al., 2021), because Ly$\alpha$ is the only
emission available for all of the sources.

By these selections, we obtain 14130 star-forming galaxies from the LAE
catalog. These 14130 star-forming galaxies are referred to as the “galaxy”
sample in this study.

###  3.2 Foreground AGN Samples

In this subsection, we describe how we select foreground AGN from two sources,
(a) the combination of the HETDEX spectra and the HSC imaging data and (b) the
SDSS DR14Q catalog. The type-$1$ AGN are identified with the sources of (a)
and (b), while the type-$2$ AGN are drawn from the source of (b).

With the source (a) that is the same as the one stated in Section 3.1, Zhang
et al. (2021) have constructed the LAE catalog. We use the catalog of Zhang et
al. (2021) to select LAEs at $z\sim 2-3$ that fall in our study field.
Applying a Ly$\alpha$ line width criterion of FWHM $>1000$ km s-1 with the
HETDEX spectra, we identify broad-line AGN, i.e. type-1 AGN, from the LAEs. We
thus obtain 1829 type-1 AGN that are referred to as T1-AGN(H).

We use the width of Ly$\alpha$ emission line for the selection of type-1 AGN.
This is because no other emission lines characterising AGN, e.g. Civ, are
available for all of the LAEs due to the limited wavelength coverage and the
sensitivity of HETDEX. Similarly, the redshifts of T1-AGN(H) objects are
measured with Ly$\alpha$ emission whose redshifts may be shifted from the
systemic redshifts by up to a few 100 km s-1 (See Section 3.1). We do not
select type-2 AGN from the source of (a), because we cannot identify type-2
AGN easily with the given data set of source (a).

From the source (b), we obtain the other samples of foreground AGN. We first
choose objects with a classification of QSOs of the SDSS DR14Q, and remove
objects outside the redshift range of $z=2.0-3.0$ in our study field. We
obtain 23721 AGN. For 16762 out of 23721 AGN, Ly$\alpha$ FWHM measurements are
available from Rakshit et al. (2020). The other AGN without FWHM measurement
are removed due to the poor quality of the Ly$\alpha$ line. We thus use these
16762 AGN with good quality of the Ly$\alpha$ line to compose our AGN sample,
referred to as All-AGN sample.

To investigate the type dependence, we classify these 16762 AGN into type-$1$
and type-$2$ AGN. In the same manner as the T1-AGN(H) sample construction, we
use Ly$\alpha$ line width measurements of Rakshit et al. (2020) for the type-1
and type-2 AGN classification. For the 16762 AGN, we apply the criterion of
Ly$\alpha$ FWHM $>1000$ km s-1 (Villarroel & Korn, 2014; Panessa & Bassani,
2002) to select type-1 AGN, and obtain 14693 type-1 AGN. Following Villarroel
& Korn (2014); Panessa & Bassani (2002), we classify type-2 AGN by the
criterion of Ly$\alpha$ FWHM $<1000$ km s-1 and obtain 2069 type-2 AGN (c.f.
Alexandroff et al., 2013; Zakamska et al., 2003). These type-1 and type-2 AGN
are referred to as T1-AGN and T2-AGN, respectively.

Table 1 presents the summary of foreground samples. We obtain 14693 and 1829
type-1 AGN, which referred to as T1-AGN and T1-AGN(H), from the SDSS and
HETDEX surveys, respectively. We select 2069 type-2 AGN that are referred to
as T2-AGN from the SDSS survey.

###  3.3 Background Source Sample

In this subsection, we describe how the background sources are selected. We
select the background sources with the SDSS DR16Q catalog, following the three
steps below.

In the first step, we extract QSOs in our study field from the SDSS DR16Q
catalog. We then select QSOs falling in the range of redshifts from 2.08 to
3.67. The lower and upper limits of the redshift range are determined by the
Ly$\alpha$ forest. Our goal is to probe Hi absorbers at $z=2.0-3.0$ with the
Ly$\alpha$ forest. Because the Ly$\alpha$ forest is observed in the rest-frame
$1040-1185$ Å of the background sources, we obtain the lower and upper limits
of the redshifts, 2.08 and 3.67, by $1216\times(1+2.0)/1185-1=2.08$ and
$1216\times(1+3.0)/1040-1=3.67$, respectively. By this step, we have selected
26899 background source candidates.

In the second step, we choose background source candidates with good quality.
We calculate the average signal to noise ratio, $\langle$S/N$\rangle$, in the
wavelength range of the Ly$\alpha$ forest for the 26899 background source
candidates, and select 15573 candidates with $\langle$S/N$\rangle$ greater
than 1.4. To maximize the special resolution of the tomography map, we set the
threshold, $\langle$S/N$\rangle$ $>1.4$, smaller than the value used by Mukae
et al. (2020). This threshold is more conservative than the value, 1.2, used
in Lee et al. (2018). In the third step, we remove damped Ly$\alpha$ absorbers
(DLAs) and broad absorption lines (BALs) from the Ly$\alpha$ forest of the
15573 candidates, because the DLAs and BALs cause an overestimation of the
absorption of the Ly$\alpha$ forest. We identify and remove DLAs using the
catalog of Chabanier et al. (2022), which is based on the SDSS DR16Q (Lyke et
al., 2020). We mask out the wavelength ranges contaminated by the DLAs of the
Chabanier et al. (2022) catalog (see Section 4.1 for the procedures). We
conduct visual inspection for the 15573 candidates to remove 115 BALs. In this
way, we obtain 15458 ($=15573-115$) sources whose spectra are free from DLAs
and BALs, which we refer to as the background source sample. Table 2 lists the
number of background sources in each field.

##  4 HI Tomography and Mapping

In this section we describe the process to construct Hi tomography maps with
the spectra of the background sources. For Hi tomography, we need to obtain
intrinsic continua of the background sources. Section 4.2 explains masking the
biasing absorption features in the background sources, while Section 4.3
determines the intrinsic continua of the background source spectra. In Section
4.3, we construct Hi tomography maps with the intrinsic continuum spectra.

###  4.1 DLA and Intrinsic Absorption Masking

Because a DLA is an absorption system with a high neutral hydrogen column
density $N_{\mathrm{\sc HI}}>2\times 10^{20}$ cm-2, the intervening DLA
completely absorbs a large portion of the Ly$\alpha$ forest over $\Delta$v
$\sim$ 103 km s-1, which gives bias in the estimates of the intrinsic continua
of the background sources. For the spectra of the background sources, we mask
out the DLAs identified in Section 3.3. We determine the range of wavelengths
for masking with the IDL code of Lee et al. (2012). The wavelength range
corresponds to the equivalent width of each DLA (Draine, 2011):

$W\sim\lambda_{\mathrm{\alpha}}\left[\frac{e^{2}}{m_{\mathrm{e}}c^{2}}N_{\mathrm{HI}}f_{\mathrm{\alpha}}\lambda_{\mathrm{\alpha}}\left(\frac{\gamma_{\mathrm{\alpha}}\lambda_{\mathrm{\alpha}}}{c}\right)\right]^{1/2}.$
(2)

In the formula, $\lambda_{\alpha}$ is the rest-frame wavelength of the
hydrogen Ly$\alpha$ line (i.e. 1216 Å), while $c$, $e$, $m_{e}$, $f_{\alpha}$,
$N_{\sc Hi}$, and $\gamma_{\alpha}$ are the speed of light, the electron
charge, the electron mass, the Ly$\alpha$ oscillator strength, the Hi column
density of the DLA, and the sum of the Einstein A coefficients. We mask out
these wavelength ranges of the background source spectra. In Figure 3, the
masked DLA is indicated by yellow hatches.

We also mask out the intrinsic absorption lines of the metal absorption lines,
which are the other sources of bias. We mask SIv $\lambda$1062, Nii
$\lambda$1084, Ni $\lambda$1134, and Ciii $\lambda$1176 (Lee et al., 2012),
which are shown by the dashed lines in Figure 3. Because the spectral
resolutions of SDSS DR14Q are $\Delta\lambda=1.8-5.2$ Å, we adopt the masking
size of $10$ Å in the observed frame.

###  4.2 Intrinsic Continuum Determination

In order to obtain the intrinsic continuum of the background source (Section
3.3) in the Ly$\alpha$ forest wavelength range (rest-frame $1040-1185$ Å), we
conduct mean-flux regulated principle component analysis (MF-PCA) fitting with
the IDL code (Lee et al., 2012) for the background sources after the masking
(Section 4.1).

There are two steps in the MF-PCA fitting process. The first step is to
predict the shape of the intrinsic continuum of the background sources in the
Ly$\alpha$ forest wavelength range. We conduct least-squares principle
component analysis (PCA) fitting (Suzuki et al., 2005; Lee et al., 2012) to
the background source spectrum in the rest frame $1216-1600$ Å:

$f_{\rm PCA}(\lambda)=\mu(\lambda)+\sum_{j=1}^{8}c_{j}\xi_{j}(\lambda),$ (3)

where $\lambda$ is the rest-frame wavelength. The values of $c_{j}$ are the
free parameters for the weights. The function of $\mu(\lambda)$ is the average
spectrum calculated from the 50 local QSO spectra in Suzuki et al. (2005). The
function of $\xi_{j}(\lambda)$ represents the $j$th principle component (or
‘eigenspectrum’) out of the 8 principle components taken from the PCA template
derived Suzuki et al. (2005).

In the second step, we predict the intrinsic continuum of the background
source in the Ly$\alpha$ forest wavelength range. Because the PCA template is
obtained with the local QSO spectra, the best-fit $f_{\rm PCA}$ in the
Ly$\alpha$ forest does not include cosmic evolution on the average
transmission rate. On average, the best-fit $f_{\rm PCA}$ in the Ly$\alpha$
forest should agree with the cosmic mean-flux evolution (Faucher-Giguère et
al., 2008c):

$\langle F(z)\rangle=\exp[-0.001845(1+z)^{3.924}],$ (4)

where $z$ is the redshift of the absorber. We use $f_{\rm PCA}$ and a
correction function of $a+b\lambda$ to estimate the intrinsic continuum
$f_{\rm intrinsic}(\lambda)$ for large-scale power along the line of sight
with the equation:

$f_{\rm intrinsic}(\lambda)=f_{\rm PCA}(\lambda)\times(a+b\lambda),$ (5)

where $a$ and $b$ are the free parameters. Because the ratio of $f_{\rm
obs}(\lambda)/f_{\rm intrinsic}(\lambda)$ should agree with the cosmic average
$\langle F(z)\rangle$ for $z=(\lambda/1216)-1$ in the wavelength range of the
Ly$\alpha$ forest, we conduct least-squares-fitting to find the values of $a$
and $b$ providing the best fit between the mean ratio and the cosmic average.
The red line shown by the bottom panel of Figure 3 presents a MF-PCA fitted
continuum derived from the spectrum of one of our background sources.

By the MF-PCA fitting, we have obtained the estimates of $f_{\rm
intrinsic}(\lambda)$ for 14736 out of the 15458 background sources. We find
the other background sources show poor fitting results found by visual
inspection. We do not use these background sources in the following analyses.
Figure 4 shows an example of poor fitting result due to the unknown
absorption. We adopt continuum fitting errors of $\sim 7\%,\sim 6\%$, and
$\sim 4\%$ for Ly$\alpha$ forests with mean S/N values of $<4$, $4-10$, and
$>10$, respectively (Lee et al., 2012).

Figure 3: Example of a background source spectrum that was used for the
reconstruction of the Hi tomography map. Bottom panel: Estimation of intrinsic
continuum. The thin black line is the spectrum of a background source taken
from the SDSS survey. The red and magenta lines are the results of MF-PCA and
PCA fitting, respectively. The vertical dashed lines present the central
wavelengths of the metal absorptions. The gray hatches represent the
wavelength ranges that are not used for the Hi tomography map reconstructions.
The yellow hatch indicates the wavelength ranges of DLA. Top panel: Spectrum
of $\delta_{\mathrm{F}}$ extracted from the bottom panel in the Ly$\alpha$
forest wavelength range. The vertical yellow and gray hatches are the same as
those in the bottom panel. The black and pink lines show the spectrum of
$\delta_{\mathrm{F}}$ and the error of $\delta_{\mathrm{F}}$ at the
corresponding wavelength extracted from the bottom panel. The horizontal line
indicates the cosmic average of Ly$\alpha$ forest transmission. Figure 4: Same
as the bottom panel of Figure 3, but for the background spectrum with a poor
fitting result. The red and magenta lines are the results of MF-PCA and PCA
continuum fitting, respectively. The yellow hatch indicates the wavelength
range of unknown absorption.

###  4.3 HI Tomography Map Reconstruction

We reconstruct our Hi tomography maps by a procedure similar to Lee et al.
(2018). We define Ly$\alpha$ forest fluctuations $\delta_{F}$ at each pixel on
the spectrum by

$\delta_{\rm F}=\frac{f_{\rm obs}/f_{\rm intrinsic}}{\langle F(z)\rangle}-1,$
(6)

where $f_{obs}$ and $f_{intrinsic}$ are the observed spectrum and estimated
intrinsic continuum, respectively. ${\langle F(z)\rangle}$ is the cosmic
average transmission. We calculate $\delta_{\rm F}$ with our background source
spectra. The top panel of Figure 3 shows the ‘spectrum’ of $\delta_{\rm F}$
derived from the $f_{obs}$ and $f_{intrinsic}$ in the bottom panel. For the
pixels in the wavelength ranges of masking (Section 4.1), we do not use
$\delta_{\rm F}$ in our further analyses. We thus obtain $\delta_{\rm F}$ in
876,560 pixels.

For the the HI tomography map of the Extended Fall field, we define the cells
of the Hi tomography map in the three-dimensional comoving space. We choose a
volume of $30^{\circ}\times 3.3^{\circ}$ in the longitudinal and latitudinal
dimensions, respectively, in the redshift range of $2.0<z<3.0$. The comoving
size of our Hi tomography map is 2257 $h^{-1}{\rm cMpc}$ $\times$ 233
$h^{-1}{\rm cMpc}$ $\times$ 811 $h^{-1}{\rm cMpc}$ in the right ascension
(R.A.), declination (Dec), and $z$ directions, respectively in the same manner
as Mukae et al. (2020). Our Hi tomography map has $451\times 46\times 162$
cells, and one cell is a cubic with a size of 5.0 $h^{-1}{\rm cMpc}$ on a
side, where the line-of-sight distance is estimated under the assumption of
the Hubble flow.

We conduct a Wiener filtering scheme for reconstructing the sightlines that do
not have background sources. We use the calculation code developed by Stark et
al. (2015). The solution for each cell of the reconstructed sightline is
obtained by

$\delta_{\rm F}^{\rm
rec}=\mathrm{C}_{\mathrm{MD}}\cdot(\mathrm{C}_{\mathrm{DD}}+\mathrm{N})^{-1}\cdot\delta_{\rm
F},$ (7)

where $\mathrm{C}_{\mathrm{MD}}$, $\mathrm{C}_{\mathrm{DD}}$, and $\mathrm{N}$
are the map-data, data-data, and noise covariances, respectively. We assume
Gaussian covariances between two points $\mathrm{r}_{1}$ and $\mathrm{r}_{2}$:

$\mathrm{C}_{\mathrm{MD}}=\mathrm{C}_{\mathrm{DD}}=\mathrm{C}(\mathrm{r}_{\mathrm{1}},\mathrm{r}_{\mathrm{2}}),$
(8)
$\mathrm{C}(\mathrm{r}_{\mathrm{1}},\mathrm{r}_{\mathrm{2}})=\sigma^{2}_{F}\exp\left[-\frac{(\Delta
r_{\|})^{2}}{2L^{2}_{\|}}\right]\exp\left[-\frac{(\Delta
r_{\bot})^{2}}{2L^{2}_{\bot}}\right],$ (9)

where $\Delta r_{\|}$ and $\Delta r_{\bot}$ are the distances between
$\mathrm{r}_{\mathrm{1}}$ and $\mathrm{r}_{\mathrm{2}}$ in the directions of
parallel and transverse to the line of sight, respectively. The values of
$L_{\bot}$ and $L_{\|}$ are the correlation lengths for vertical and parallel
to the line-of-sight (LoS) direction, respectively, and defined with
$L_{\bot}$ = $L_{\|}$ = 15 $h^{-1}{\rm cMpc}$. The value of $\sigma^{2}_{F}$
is the normalization factor that is $\sigma^{2}_{F}=0.05$. Stark et al. (2015)
develop this Gaussian form to obtain a reasonable estimate of the true
correlation function of the Ly$\alpha$ forest. We perform the Wiener filtering
reconstruction with the values of $\delta_{\rm F}$ at the 898390 pixels, using
the aforementioned parameters of the Stark et al. (2015) algorithm with a
stopping tolerance of $10^{-3}$ for the pre-conditioned conjugation gradient
solver. As noted by Lee et al. (2016), the boundary effect that leads to an
additional error on $\delta_{\rm F}$ occurs at the positions that are near the
boundaries of an Hi tomography map. The boundary effect is caused by the
background sightlines not covering the region that contribute to the
calculation of the $\delta_{F}$ values for cells near the Hi tomography map
boundaries. To avoid the boundary effect, we extend a distance of 40 h-1cMpc
for each side of the Hi tomography map of the ExFall field. The resulting map
is shown in Figure 5.

For the HI tomography map reconstruction of the Extended Spring field
(hereafter ExSpring field), we perform almost the same procedure as the one of
the ExFall field. The area of the ExSpring field is more than 6 times larger
than that of the ExFall field. We separate the ExSpring field into $8\times
3=24$ footprints to save calculation time. Each footprint covers an area of
$10^{\circ}\times 5^{\circ}$ in the R.A. and Dec directions, respectively. We
reconstruct the Hi tomography map one by one for the footprints of the
ExSpring field.

To weaken the boundary effect, we extend a distance of 40 h-1cMpc for each
side of the footprints. The extensions mean that every two adjacent footprints
has an overlapping region of 80 h-1cMpc width. The width of the overlapping
regions is a conservative value to weaken the boundary effect since it is much
larger than the resolution, 15 h-1cMpc, of our Hi tomography maps. By the 40
h-1cMpc extension, we reduce the uncertainty in the $\delta_{F}$ value for the
edge of each footprint caused by boundary effect to $\pm$0.01. This value
corresponds to the $1/10$ of the typical error for each cell of the Hi
tomography map (Mukae et al., 2020) The remaining additional error caused by
boundary effect is negligible compared to the statistical uncertainties in the
HI distributions obtained in Section 5. Then we follow the reconstruction
procedure for the ExFall field to reconstruct HI tomography maps of the
footprints and cut off all the cells within 40 h-1cMpc to the borders that are
affected by the boundary effect. Finally we obtain the Hi tomography map of
the ExSpring field with a special volume of 3475 $h^{-1}{\rm cMpc}$ $\times$
1058 $h^{-1}{\rm cMpc}$ $\times$ 811 $h^{-1}{\rm cMpc}$ in the R.A., Dec, and
$z$ directions, respectively (Figure 6).

Figure 5: 3D Hi tomography map of the ExFall field. The color contours
represent the values of $\delta_{\rm F}$ from negative (red) to positive
(blue). The spatial volume of the Hi tomography map is $2257\times 233\times
811$ $h^{\rm-3}$cMpc3. The redshift range is $z=2.0-3.0$. Figure 6: Same as
Figure 5, but for the ExSpring field. The spatial volume of the Hi tomography
map is $3475\times 1058\times 811$ $h^{\rm-3}$cMpc3.

##  5 Results and Discussions

###  5.1 Average HI Profiles around AGN: Validations of our AGN Samples

In this section we present the Hi profile, $\delta_{\mathrm{F}}$ as a function
of distance, with the All-AGN sample sources, using the reconstructed Hi
tomography maps. We compare the Hi profile of the All-AGN sample to the one of
the previous study (Font-Ribera et al., 2013). We also present the comparison
of the Hi profiles between T1-AGN(H) and T1-AGN samples that are made with the
HETDEX and SDSS data. In this study, we only discuss the structures having
size $\gtrsim 15$ $h^{-1}$cMpc corresponding to the resolution of our 3D Hi
tomography maps.

For the Hi profiles with the All-AGN sample, we extract $\delta_{\mathrm{F}}$
values around the 16978 All-AGN sample sources in the Hi tomography map. We
cut the Hi tomography map centered at the positions of the All-AGN sample
sources, and stack the $\delta_{\mathrm{F}}$ values to make a two dimensional
(2D) map of the average $\delta_{\mathrm{F}}$ distribution around the sources
that is referred to as a 2D Hi profile of the All-AGN sample sources. The two
dimensions of the 2D Hi profile correspond to the transverse distance
$D_{\mathrm{Trans}}$ and the LoS Hubble distance. The velocity corresponding
to the LoS Hubble distance is referred to as the LoS velocity.

Figure 7 shows the 2D Hi profile with values of $\delta_{\rm F}$ for All-AGN
sample. The solid black lines denote the contours of $\delta_{\rm F}$. In each
cell of the 2D Hi profile, we define the $1\sigma$ error with the standard
deviation of $\delta_{\rm F}$ values of the 100 mock 2D Hi profiles. Each mock
2D Hi profile is obtained in the same manner as the real 2D Hi profile, but
with random positions of sources whose number is the same as the one of All-
AGN sample sources. In Figure 7, the dotted black lines indicate the contours
of the 6$\sigma$, 9$\sigma$ and $12\sigma$ confidence levels, respectively. We
find the $19.5\sigma$ level detection of $\delta_{\rm F}$ at the source
position (0,0). The $\delta_{\rm F}$ value at the source position indicates
the averaging value over the ranges of ($-7.5$ $h^{-1}\mathrm{cMpc}$, $+7.5$
$h^{-1}\mathrm{cMpc}$) in both the LoS and transverse directions. The
$19.5\sigma$ level detection at the source position is suggestive that obvious
Ly$\alpha$ forest absorption exists near the All-AGN sources on average The 2D
Hi profile is more extended in the transverse direction than along the line of
sight. We discuss this difference in Section 5.2.

We then define a 3D distance, $D$, under the assumption of the Hubble flow in
the LoS direction. We derive $\delta_{\rm F}$ as a function of $D$ that is
referred to as ”Hi radial profile”, averaging $\delta_{\rm F}$ values of the
2D Hi profile over the 3D distance. Figure 8 shows the Hi radial profile of
the All-AGN sample. We find that the $\delta_{\rm F}$ values increase towards
a large distance. This trend is consistent with the one found by Ravoux et al.
(2020) with the SDSS quasars.

Figure 7:  2D Hi profile of the All-AGN sample sources. The color map
indicates the $\delta_{\mathrm{F}}$ values of each cell of the 2D Hi profile.
The solid lines denote constant $\delta_{\mathrm{F}}$ values in steps of
$-0.01$ starting at $-0.01$. The dotted lines correspond to multiples of
$3\sigma$ starting at $6\sigma$.

Ravoux et al. (2020) have obtained the average Ly$\alpha$ transmission
fluctuation distribution around the AGN taken from the SDSS data release 16
quasar (SDSS DR16Q) catalog in the field of Strip 82. The criteria of the
target selection for the SDSS DR16Q and SDSS DR14Q sources are the same. The
luminosity distribution of AGN for Ravoux et al. (2020) is almost the same as
that of our All-AGN sample sources that are taken from the SDSS DR14Q catalog.
We derive the average radial Hi profile of the Ravoux et al. (2020) AGN
sources by the same method as for our All-AGN sample, using the 3D Hi
tomography map reconstructed by Ravoux et al. (2020). We compare the radial Hi
profile of the All-AGN sample with the one derived from the 3D Hi tomography
map of Ravoux et al. (2020). The comparison is shown in Figure 8. Our result
agrees with that of Ravoux et al. (2020) within the error range at scale
$D>10$ $h^{-1}$ cMpc. The peak values of $\delta_{\rm F}$ showing the
strongest Ly$\alpha$ absorption are comparable, $\delta_{\rm F}\simeq-0.02$.
The slight difference between the peak values of our and Ravoux et al.’s
results can be explained by the different approaches of the estimation for the
intrinsic continuum adopted by Ravoux et al. and us. Ravoux et al. conduct
power law fitting, which is different from the MF-PCA fitting that we used,
for the intrinsic continuum in the wavelength range of the Ly$\alpha$ forest.
Given the low ($\sim 15$ $h^{-1}$) spatial resolution of both our Hi
tomography map and that of Ravoux et al. (2020), neither studies are able to
search for the proximity effect making a photoionization region around AGN
(D’Odorico et al., 2008). From the comparison shown by Figure 8, we conclude
that the Ly$\alpha$ forest absorption derived from our Hi tomography map is
reliable.

Figure 8:  Hi radial profile of the All-AGN and Ravoux et al. (2020) AGN
samples. The black and gray data points and error bars show the Hi radial
profiles of our All-AGN sample sources and the AGN of Ravoux et al. 2020,
respectively. The horizontal dashed line shows the cosmic average Ly$\alpha$
transmission fluctuation, $\delta_{\mathrm{F}}=0$.

To check the reliability of the HETDEX survey results, we use the reliable
result of the SDSS AGN to compare with the result derived by the HETDEX AGN.

We select type-1 AGN from the HETDEX’s T1-AGN(H) and SDSS’s T1-AGN samples to
make sub-samples of T1-AGN(H) and T1-AGN with matching rest-frame 1350 Å
luminosity ($L_{\mathrm{1350}}$). For T1-AGN, the measurements directly from
the SDSS spectra ($L^{\mathrm{spec}}_{\mathrm{1350}}$) are available (Rakshit
et al., 2020). For T1-AGN(H), we do not have
$L^{\mathrm{spec}}_{\mathrm{1350}}$ measurements from the HETDEX spectra, we
estimate it using HSC r-band imaging. Since the central wavelength of the
r-band imaging is rest-frame $\sim 1700{\rm\AA}$, we calibrate the conversion
between r-band luminosity, $L^{\mathrm{phot}}_{\mathrm{UV}}$, and
$L^{\mathrm{spec}}_{\mathrm{1350}}$. We examine the 283 type-1 AGN sources
that appear in both the SDSS and HETDEX surveys (and, thus, have both
$L^{\mathrm{spec}}_{\mathrm{1350}}$ measurements from SDSS and r-band
luminosities from HSC) to calibrate the relationship. The results are
displayed in Figure 9. The $L^{\mathrm{phot}}_{\mathrm{UV}}$ are always
smaller than those of $L^{\mathrm{spec}}_{\mathrm{1350}}$ (Rakshit et al.,
2020). Due to the blue UV slope of the spectra for the AGN both categorized in
the T1-AGN(H) and T1-AGN samples, the luminosity of the rest-frame 1350 Å
always shows a larger value than the one of rest-frame 1700 Å. We conduct
linear fitting to the data points of Figure 9, and obtain the best-fit linear
function. With the best-fit linear function, we estimate
$L^{\mathrm{spec}}_{\mathrm{1350}}$ values for the HETDEX’s T1-AGN(H) sample
sources.

We show the $L^{\mathrm{spec}}_{\mathrm{1350}}$ distributions of all the
T1-AGN(H) and T1-AGN sample sources in the upper panel of Figure 10. The
$L^{\mathrm{spec}}_{\mathrm{1350}}$ distribution of T1-AGN(H) covers a wider
luminosity range than the one of T1-AGN. To make sure the comparison between
the SDSS and HETDEX AGN is fair, we make the sub-samples of T1-AGN and
T1-AGN(H) that consist of the sources with matching
$L^{\mathrm{spec}}_{\mathrm{1350}}$ distributions. We present the
$L^{\mathrm{spec}}_{\mathrm{1350}}$ distributions of the T1-AGN and T1-AGN(H)
sub-samples in the bottom panel of Figure 10. We obtain 540 and 4338 type-1
AGN for the sub-samples of T1-AGN(H) and T1-AGN, respectively, whose
$L^{\mathrm{spec}}_{\mathrm{1350}}$ distributions are shown in the bottom
panel of Figure 9.

We derive the Hi radial profiles for the sub-samples of T1-AGN(H) and T1-AGN
sample sources, as shown in Figure 11. The Hi radial profiles of T1-AGN(H) and
T1-AGN sub-sample sources are in good agreement.

Figure 9: Relations of $L^{\mathrm{phot}}_{\mathrm{UV}}$ against
$L^{\mathrm{spec}}_{\mathrm{1350}}$ for the sources both categorized in the
T1-AGN(H) and T1-AGN samples. The $L^{\mathrm{phot}}_{\mathrm{UV}}$ and
$L^{\mathrm{spec}}_{\mathrm{1350}}$ are measured from the HSC r-band imaging
and SDSS spectra (Rakshit et al., 2020), respectively. The gray points show
the distribution of $L^{\mathrm{spec}}_{\mathrm{1350}}$ $-$
$L^{\mathrm{phot}}_{\mathrm{UV}}$ relations for the sources both categorized
in the T1-AGN(H) and T1-AGN samples. The black dashed line indicates the
relation where $L^{\mathrm{spec}}_{\mathrm{1350}}$ $=$
$L^{\mathrm{phot}}_{\mathrm{UV}}$. The red dashed line represents the linear
best fit of the blue points. Figure 10: Top panel:
L${}^{\mathrm{spec}}_{\mathrm{1350}}$ distributions of the T1-AGN and
T1-AGN(H) samples with blue and red histograms, respectively. Bottom panel:
Same as the top panel, but for the T1-AGN and T1-AGN(H) sub-sample sources.
Figure 11: Hi radial profiles of the T1-AGN and T1-AGN(H) sub-samples. The
blue and red triangles show the values of $\delta_{\rm F}$ as a function of
distance, $D$, for the T1-AGN and T1-AGN(H) sample sources, respectively. The
horizontal dashed line shows the cosmic average Ly$\alpha$ transmission
fluctuation, $\delta_{\rm F}=0$.

###  5.2 AGN Average Line-of-Sight and Transverse Hi Profiles

Based on the 2D Hi profile of the All-AGN sample (Figure 7), we find that the
Ly$\alpha$ forest absorptions of the All-AGN sample sources are more extended
in the transverse direction. In this section, we present the Hi radial
profiles of All-AGN sample in the LoS and transverse directions and compare
these two Hi radial profiles.

To derive the Hi radial profile of the All-AGN sample with the absolute LoS
distance, which is referred to as the LoS Hi radial profile (Figure 13), we
average $\delta_{\rm F}$ values of the 2D Hi profiles of All-AGN over
$D_{\mathrm{Trans}}$ $<7.5$ $h^{-1}\mathrm{cMpc}$ (from $-7.5$
$h^{-1}\mathrm{cMpc}$ to $+7.5$ $h^{-1}\mathrm{cMpc}$ in the transverse
direction) that corresponds to the spatial resolution of the 2D Hi profile
map, $15$ $h^{-1}\mathrm{cMpc}$. Among the 16,978 All-AGN sample sources,
10,884 sources are used as both background and foreground sources. In this
case, the Ly$\alpha$ transmission fluctuation ($\delta_{\rm F}$) of these
10,884 sources at the LoS velocity $\lesssim-5250$ km s-1 is estimated mainly
from their own spectrum. As the discussion in Youles et al. (2022), the
redshift uncertainty of the SDSS AGN causes the overestimation of intrinsic
continuum and the underestimation of $\delta_{\rm F}$ around the metal
emission lines such as Ciii $\lambda$1176\. This leads to a systemic error
toward negative $\delta_{\rm F}$ in the Hi radial profile of LoS velocity (LoS
distance) at the LoS velocity $\lesssim 5250$ km s-1 (Figure 12). The Hi
radial profile of LoS velocity (LoS distance) is derived by averaging
$\delta_{\rm F}$ values over $D_{\mathrm{Trans}}$ $<7.5$ $h^{-1}\mathrm{cMpc}$
as a function of the negative and positive LoS velocity (LoS distance). In
this study, we only use the values of $\delta_{\rm F}$ at the LoS distance
$>-52.5h^{-1}\mathrm{cMpc}$ (LoS velocity $>-5250$ km s-1) to derive the LoS
Hi radial profile of the All-AGN sample (Figure 13). The scale, LoS distance
$>-52.5h^{-1}\mathrm{cMpc}$ (LoS velocity $>-5250$ km s-1), is determined by
the maximum wavelength of the Ly$\alpha$ forest we used, the smoothing scale
of the Wiener filtering scheme, and the AGN redshift uncertainty, assumed by
Youles et al. (2022). After removing the $\delta_{\rm F}$ values affected the
systemics in the 2D Hi profile, we present the LoS Hi radial profile of the
All-AGN sample in Figure 13.

We estimate the Hi radial profiles of $D_{\mathrm{Trans}}$, which is referred
to as the Transverse Hi radial profile,by averaging the $\delta_{\rm F}$
values over the LoS velocity of $(-750,+750)$ km s-1 whose velocity width
corresponds to $15$ $h^{-1}$ cMpc in the Hubble-flow distance. The Hi radial
profile of $D_{\mathrm{Trans}}$ is also shown in Figure 13.

We compare the LoS and Transverse Hi radial profile. The $\delta_{\rm F}$
value increase toward large-scale more rapidly in the LoS direction than those
in the Transverse direction (Figure 13). This difference may be explained by
an effect similar to the Kaiser effect (Kaiser, 1987), doppler shifts in AGN
redshifts caused by the large-scale coherent motions of the gas towards the
AGN. The LoS Hi radial profile is positive, $\delta_{\rm F}\sim 0.002\pm
0.0008$, at the large scale, $\gtrsim 30$ $h^{-1}$cMpc. In Section 5.5, we
discuss the positive $\delta_{\rm F}$ values of LoS Hi radial profiles at
large scales and compare our observational result to the models of a previous
study, Font-Ribera et al. (2013).

Figure 12: Hi radial profiles of LoS velocity (LoS distance) for the All-AGN
sample. The black solid line shows the $\delta_{\rm F}$ values as a function
of LoS velocity (LoS distance) for the All-AGN sample. The vertical dashed
line presents the position of LoS velosity $=0$ km s-1 (LoS distance $=0$
$h^{-1}$cMpc). The horizontal dashed indicates the cosmic average Ly$\alpha$
transmission fluctuation ,$\delta_{\rm F}=0$. The gray shaded area shows the
range of the $\delta_{\rm F}$ not used to derive LoS Hi radial profile. Figure
13: LoS and Transverse Hi radial profiles the All-AGN sample. The black and
gray lines show the $\delta_{\mathrm{F}}$ values as a function of LoS distance
and $D_{\rm Trans}$, respectively. The horizontal dashed line indicates
$\delta_{\mathrm{F}}$=0.

###  5.3 Source Dependences of the AGN Average HI Profiles

In this section, we present 2D and Hi radial profiles of the AGN sub-samples
to investigate how the average Hi density depends on luminosity and AGN type.

####  5.3.1 AGN Luminosity Dependence

We study the AGN-luminosity dependence of the average Hi profiles. Figure 14
presents the $L_{1350}^{\rm spec}$ distribution of All-AGN. We make 3 sub-
samples of All-AGN that are All-AGN-L3, All-AGN-L2 and All-AGN-L1. The
luminosity ranges of the sub-samples are $43.70<\log(L_{1350}^{\rm
spec}/{\rm[erg\ s^{-1}]})<45.41$, $45.41<\log(L_{1350}^{\rm spec}/{\rm[erg\
s^{-1}]})<45.75$, and $45.75<\log(L_{1350}^{\rm spec}/{\rm[erg\
s^{-1}]})<47.35$, respectively. The luminosity ranges of the 3 sub-samples are
defined in a way that the numbers of the AGN are same 5695 in each subsamples.
We derive the 2D Hi profiles of the sub-samples in the same manner as Section
5.1, and present the profiles in Figures 15. In these 2D Hi profiles, The
brightest sub-sample of All-AGN-L1 (the faintest sub-sample of All-AGN-L3)
shows the weakest (the strongest) Ly$\alpha$ transmission fluctuations around
the source position, $D=0$.

We then extract the Hi radial profiles from the 2D Hi profiles of the All-AGN
sub-samples, and present the Hi radial profiles in Figure 16. In this figure,
we find that the peak values of $\delta_{\rm F}$ for the All-AGN sub-samples
is anti-correlates with AGN luminosities. The peak $\delta_{\rm F}$ values
near the source position drops from the faintest All-AGN-L3 subsample to the
brightest All-AGN-L1 subsample. The gas densities around bright AGN are higher
than (or comparable to) those around faint AGN, this result would suggest that
the ionization fraction of the hydrogen gas around bright AGN is higher than
the one around faint AGN on average.

We also present the LoS and Transverse Hi radial profiles of the All-AGN sub-
samples derived by the same method as that for the All-AGN sample in Figure
17. Similar to what we found in the comparison of the Hi radial profiles for
the All-AGN sub-samples, the peak values of the LoS and Transverse Hi profiles
also decrease from the faintest sub-sample, All-AGN L3, to the brightest sub-
sample, All-AGN L1. For the LoS (Transverse) Hi radial profiles at the scales
beyond 25 $h^{-1}$ cMpc, we do not find any significant differences in the
comparison of the LoS (Transverse) Hi radial profiles for the All-AGN sub-
samples.

Figure 14: log$L^{\mathrm{spec}}_{\mathrm{1350}}$ distribution of the bright
and All-AGN sample sources. The vertical dashed lines indicate the boarders of
$L^{\mathrm{spec}}_{\mathrm{1350}}$ where
log$(L^{\mathrm{spec}}_{\mathrm{1350}}/[{\rm erg\ s^{-1}}])$ $=45.41$ and
$45.75$, respectively. These three borders separate the All-AGN sample into 3
sub-samples of All-AGN-L3, All-AGN-L2, and All-AGN-L1, respectively.

Figure 15: Same as Figure 7, but for the All-AGN-L3 (top), All-AGN-L2 (middle)
and All-AGN-L1 (bottom) samples. Figure 16: Same as Figure 11, but for the
All-AGN-L3 (red), All-AGN-L2 (gray) and All-AGN-L1 (black) samples.

Figure 17: LoS and Transverse Hi radial profiles of the All-AGN-L3, All-
AGN-L2, and All-AGN-L1 sub-samples. The top figure (bottom figure) presents
the LoS (Transverse) Hi radial profiles of the All-AGN-L3, All-AGN-L2, and
All-AGN-L1 sub-samples, shown by the red, gray, and black lines, respectively.
The meaning of the horizontal dashed lines both in the top and bottom figures
are the same as the one in Figure 8.

####  5.3.2 AGN Type Dependence

We investigate the dependence of Hi profiles on type-1 and type-2 AGN. To
remove the effects of the AGN luminosity dependence (Section 5.3.1), we make
sub-samples of T1-AGN and T2-AGN with the same
$L^{\mathrm{spec}}_{\mathrm{1350}}$ distribution by the same manner as the one
we conduct for the selection of T1-AGN and T1-AGN(H) sub-samples in Section
5.1. The top panel of Figure 18 presents the
$L^{\mathrm{spec}}_{\mathrm{1350}}$ distributions of T1-AGN and T2-AGN
samples, while the bottom panel of Figure 18 shows those of the T1-AGN and
T2-AGN sub-samples. The sub-samples of T1-AGN and T2-AGN are composed of 10329
type-1 AGN and 1462 type-2 AGN, respectively. We derive the 2D Hi profiles
from the T1-AGN and T2-AGN sub-samples. The profiles are presented in Figure
19. We find $17.7$ and $7.9$ $\sigma$ detections at the source center position
(0,0) of the T1-AGN and T2-AGN sub-samples, respectively. We calculate the Hi
radial profiles from the 2D Hi profiles of the T1-AGN and T2-AGN sub-samples.
In Figure 20, we compare the Hi radial profiles of the T1-AGN and T2-AGN sub-
samples. No notable difference is found within 1$\sigma$ error. The peak value
of $\delta_{\rm F}=0$ of the T2-AGN subsample is within $1\sigma$ error of the
peak value of the T1-AGN subsample near the source position.

To compare the Ly$\alpha$ forest absorptions of type-1 and type-2 AGN in the
LoS and transverse directions, we derive the LoS and Transverse Hi radial
profiles of the T1-AGN and T2-AGN sub-samples and present the profiles in
Figure 21. Similar to the trend of the Hi radial profiles, the peak values of
the LoS and Transverse Hi radial profiles for T1-AGN and T2-AGN sub-samples
are not significantly different. The comparable peak values of the LoS and
Transverse Hi radial profiles suggest that the selectively different
orientation and opening angles of the dusty tori of the type-1 and type-2 AGN
do not significantly affect the Ly$\alpha$ forest absorption at the scale
$\lesssim 15$ $h^{-1}\mathrm{cMpc}$ or our measurement dose not have enough
sensitivity to detect the difference of Ly$\alpha$ forest absorption between
type-1 and type-2 AGN.

For the Hi radial profiles at the scale $>15$ $h^{-1}\mathrm{cMpc}$, we find
that the $\delta_{\rm F}$ value for the LoS Hi radial profile of the T1-AGN
sub-sample is smaller than those of the T2-AGN sub-sample over the 1$\sigma$
error bar at the scale around $25$ $h^{-1}\mathrm{cMpc}$. This result may hint
that the type-2 AGN have a stronger power of ionization at $25$
$h^{-1}\mathrm{cMpc}$ than the type-1 AGN. The interpretation of ionization at
large-scales is in Section 5.5.

Figure 18: Same as Figure 10, but for the T1-AGN (blue) and T2-AGN (red)
samples.

Figure 19: Same as Figure 7, but for the T1-AGN (top figure) and T2-AGN
(bottom figure) sub-samples. Figure 20: Same as Figure 11, but for the T1-AGN
(blue) and T2-AGN (red) sub-samples and the galaxy (gray) sample.

Figure 21: Same as Figure 17, but for the T1-AGN and T2-AGN sub-samples.

###  5.4 Average HI Profiles around Galaxy

We derive the 2D Hi profile at the positions of the galaxy sample sources in
the same manner as the one of the All-AGN sample sources. Figure 22 presents
the 2D Hi profile of the galaxy sample sources. There is a clear $10.5\sigma$
detection at the source position of (0,0). Similarly, we calculate the Hi
radial profile from the 2D Hi profile of the galaxy sample (Figure 23). The Hi
radial profile of the galaxy sample shows a trend similar to those of the All-
AGN sample. Both for the galaxy and All-AGN samples, the Hi radial profile
decreases towards the large scales, reaching cosmic average.

In Figure 22, we find that the Ly$\alpha$ forest absorptions in the LoS and
transverse directions are different. A similar difference between the values
of $A_{\rm F}$ in LoS and transverse directions of 2D Hi profiles is claimed
by Mukae et al. (2020). To investigate the difference between the Ly$\alpha$
forest absorptions in LoS and transverse directions for the galaxy sample, we
present the LoS and Transverse Hi radial profiles of the galaxy sample in
Figure 25. We find that the LoS and Transverse Hi radial profiles of the
galaxy sample show different gradient of the increasing $\delta_{\rm F}$ at
the scale D $\sim 3.75-50$ $h^{-1}$cMpc. This difference can be explained by
the gas version of the Kaiser effect that we discussed in Section 5.2. In the
LoS Hi radial profile of the galaxy sample, we find that the $\delta_{\rm F}$
values are positive on the scale of D $=25-70$ $h^{-1}$cMpc, which is similar
to the positive $\delta_{\rm F}$ values we found on the large scale of the LoS
Hi radial profile for the All-AGN sample. We discuss these positive
$\delta_{\rm F}$ values on the LoS Hi radial profile of the galaxy sample in
Section 5.5.

Figure 22: 2D Hi profile of the galaxy sample sources. The color map indicates
the $\delta_{\mathrm{F}}$ values of each cell of the 2D Hi profile. The dotted
lines show confidence level contours of $3\sigma$ and $6\sigma$. The solid
line presents the contour where $\delta_{\mathrm{F}}$ $=-0.01$.

####  5.4.1 Galaxy-AGN Dependence

We derive 2D Hi profiles for the T1-AGN(H) sample constructed from the HETDEX
data. Figure 22 and 23 show the 2D Hi profiles of the galaxy and T1-AGN(H)
samples. We find $7.6$$\sigma$ detection around the source position for the
T1-AGN(H) sample. Figure 24 presents the Hi radial profiles of the galaxy and
T1-AGN(H) samples derived from the 2D Hi profiles. We also compare the Hi
radial profiles of the galaxy sample with those of T1-AGN and T2-AGN in Figure
20. In the Hi radial profiles of the galaxy and T1-AGN(H) samples, the
$\delta_{\rm F}$ values decrease toward the source position $D=0$. In Figure
24 (20), we find that the $\delta_{\rm F}$ values of T1-AGN(H) (T1-AGN and
T2-AGN) are smaller than those of the galaxies at $\lesssim 20$ $h^{-1}$ cMpc.
These $\delta_{\rm F}$ excesses of the AGN may be explained by the hosting
dark matter halos of the AGN being more massive than those of the galaxies.
Momose et al. (2021) also investigate the Hi radial profile around AGN, and
find Ly$\alpha$ forest absorption decrement at the source center ($\lesssim 5$
$h^{-1}$Mpc). They argure that this trend can be explained by the proximity
effect. On the other hand, their result is different from ours that the
$A_{\rm F}$ values monotonically increase with decreasing distance. This
difference between our and Momose et al.’s results is produced by the fact
that our results for $\lesssim 10$ $h^{-1}$ cMpc are largely affected by the
Ly$\alpha$ transmission fluctuation at $\sim 10$ $h^{-1}$ cMpc due to the
coarse resolution of our Hi tomography map, 15 $h^{-1}$ cMpc, in contrast with
$2.5$ $h^{-1}$ cMpc for the resolution of Momose et al. (2021).

We then derive the LoS and Transverse radial Hi profile of the T1-AGN(H)
sample. The results of the profiles are shown in Figure 25. Similar to the LoS
and Transverse Hi radial profiles of the All-AGN and galaxy samples, the gas
version of the Kaiser effect and the positive $\delta_{\rm F}$ in the LoS
direction on the scale beyond $D=25$ h-1cMpc are also found in those of the
T1-AGN(H) sample.

Figure 23: Same as Figure 15, but for the T1-AGN(H) sample.  Figure 24: Same
as Figure 16, but for galaxy (gray) and T1-AGN(H) (black) samples. Figure 25:
Same as Figure 13, but for the galaxy and T1-AGN(H) samples.

###  5.5 Comparison with Theoretical Models

There are theoretical models of Hi radial profiles around AGN that are made by
Font-Ribera et al. (2013). Font-Ribera et al. (2013) present their Hi radial
profiles with the LoS distance in the form of cross-correlation function
(CCF).

We first calculate theoretical CCFs of All-AGN, following the definition of
the CCF presented in Font-Ribera et al. (2012, 2013). Font-Ribera et al.
(2013) assume the linear cross-power spectrum of the QSOs and Ly$\alpha$
forest,

$P_{\rm qF}(\mathbf{k},z)=b_{\rm q}(z)[1+\beta_{\rm q}(z)\mu^{2}_{\rm
k}]b_{\rm F}(z)[1+\beta_{\rm F}(z)\mu^{2}_{\rm k}]P_{\rm L}(k,z),$ (10)

where $P_{\rm L}(k,z)$ is the linear matter power spectrum. Here $\mu_{\rm k}$
is the cosine of the angle between the Fourier mode and the LoS (Kaiser,
1987). The values of $b_{\rm q}$ and $b_{\rm F}$ ($\beta_{\rm q}$ and
$\beta_{\rm F}$) are the bias factors (redshift space distortion parameters)
of the QSO and Ly$\alpha$ density, respectively.

The redshift distortion parameter of QSO obeys the relation $\beta_{\rm
q}=f(\Omega)/b_{\rm q}$, where $f(\Omega)$ is the logarithmic derivative of
the linear growth factor (Kaiser, 1987), $b_{\rm q}=3.8\pm 0.3$ (White et al.,
2012). We use the condition of Ly$\alpha$ forest, $b_{\rm F}(1+\beta_{\rm
F})=-0.336$ for $b_{F}\propto(1+z)^{2.9}$, that is determined by observations
of Ly$\alpha$ forest at $z\simeq 2.25$ (Slosar et al., 2011). Font-Ribera et
al. (2013) estimate the CCF of QSOs by the Fourier transform of $P_{\rm qF}$
(Hamilton, 1992):

$\xi({\bf r})=\xi_{0}(r)P_{0}(\mu)+\xi_{2}(r)P_{2}(\mu)+\xi_{4}(r)P_{4}(\mu),$
(11)

where $\mu$ is the cosine of angle between the position ${\bf r}$ and the LoS
in the redshift space. The values of $P_{0}$, $P_{2}$, and $P_{4}$ are the
Legendre polynomials, $P_{0}=1$, $P_{2}=(3\mu^{2}-1)$, and
$P_{4}=(35\mu^{4}-30\mu^{2}+3)/8$, respectively. The functions of $\xi_{0}$,
$\xi_{2}$, and $\xi_{4}$ are:

$\xi_{0}(r)=b_{\rm q}b_{\rm F}[1+(\beta_{\rm q}+\beta_{\rm F})/3+\beta_{\rm
q}\beta_{\rm F}/5]\zeta(r),$ (12) $\xi_{2}(r)=b_{\rm q}b_{\rm
F}[2/3(\beta_{\rm q}+\beta_{\rm F})+4/7\beta_{\rm q}\beta_{\rm
F}][\zeta(r)-\bar{\zeta}(r)],$ (13) $\xi_{4}(r)=8/35b_{\rm q}b_{\rm
F}\beta_{\rm q}\beta_{\rm
F}[\zeta(r)-5/2\bar{\zeta}(r)-7/2\bar{\bar{\zeta}}(r)].$ (14)

The function $\zeta(r)$ is the standard CDM linear correlation function in
real space (Bardeen et al., 1986; Hamilton et al., 1991). The functions
$\bar{\zeta}(r)$ and $\bar{\bar{\zeta}}(r)$ are given by:

$\bar{\zeta}(r)\equiv 3r^{-3}\int^{r}_{0}\zeta(s)s^{2}ds,$ (15)
$\bar{\bar{\zeta}}(r)\equiv 5r^{-5}\int^{r}_{0}\zeta(s)s^{4}ds.$ (16)

In Figure 26, we present $D\xi$ as a function of the LoS distance for the
model of Font-Ribera et al. (2013) that is calculated under the assumption of
the mean overdensity of the $15$ $h^{-1}$cMpc corresponding to the spatial
resolution of our observational results.

To compare our observational measurements with the model CCF of Font-Ribera et
al. (2013), we calculate the value of $\xi$ for our All-AGN sample. The value
of $\xi$ in each cell $\xi_{\rm cell}$ is calculated by

$\xi_{\rm cell}=\frac{\sum_{i\in{\rm cell}}\omega_{i}\delta_{{\rm
F}i}}{\sum_{i\in{\rm cell}}\omega_{i}},$ (17)

where $\omega_{i}$ is the weight determined by the observational errors and
the intrinsic variance of the Ly$\alpha$ forest. Noted that the $\delta_{\rm
Fi}$ used in Font-Ribera et al. (2012, 2013) is the raw $\delta_{\rm Fi}$,
which is not undergoing the Wiener filtering scheme. The value of $\omega_{i}$
is obtained by

$\omega_{i}=\left[\sigma^{2}_{\rm F}(z_{i})+\frac{1}{\langle
S/N\rangle^{2}\times\langle F(z_{i})\rangle^{2}}\right]^{-1},$ (18)

where $\sigma_{\rm F}(z_{i})$ is the intrinsic variance of the Ly$\alpha$
forest. The value of $\langle F(z_{i})\rangle$ is the cosmic average
Ly$\alpha$ transmission (Eq.4). We adopt $\langle S/N\rangle=1.4$ that is the
criterion of the background source selection (Section 3.3). The intrinsic
variance, $\sigma_{\rm F}(z_{i})$, of the Ly$\alpha$ forest taken from Font-
Ribera et al. (2013) is:

$\sigma^{2}_{\rm F}(z_{i})=0.065[(1+z_{i})/3.25]^{3.8}.$ (19)

We calculate $\xi$ with our All-AGN sample via the Equations 17, 18, and 19,
using the binning sizes same as those in Font-Ribera et al. (2013). We present
$\xi$ multiplied by $D$ with the black squares in Figure 26. (explanation of
Momose+21) For reference, we also derive the $\xi$ for our galaxy sample shown
by the blue triangles.

In Figure 26, we find that the $D\xi$ profile of our All-AGN sample show a
trend similar to the one of the model predicted by Font-Ribera et al. (2013).
The observational $D\xi$ profile of our All-AGN sample shows a good agreement
with the model $D\xi$ profile of Font-Ribera et al. (2013) at the scale of
$D>30$ $h^{-1}$cMpc. Equation 10 of the linear theory model already includes
the parameter of redshift distortion, $\beta$, which is due to the coherent
motions of the Hi gas around the quasars. The model is a prediction on the
impact of the clustering effect that a quasar statistically gathers Hi gas
from large scales, even $\gtrsim 30$ $h^{-1}$cMpc. The positive $D\xi$
structure $\gtrsim 30$ $h^{-1}$cMpc can be explained by ‘cosmic voids’ like
structure whose Hi gas column density is slightly smaller than the cosmic
average. Though the general trend of the positive $D\xi$ structure of our
results at $\gtrsim 30$ $h^{-1}$cMpc are the same as the model $D\xi$ profile,
the model $D\xi$ profile is slightly lower than the $D\xi$ profiles of the
observations at $\gtrsim 60$ $h^{-1}$cMpc. We can not rule out the possibility
that the ionization of AGN make an extra decreasing on the Ly$\alpha$
absorption at $\gtrsim 60$ $h^{-1}$cMpc. Font-Ribera et al. (2013) also
present the model of ionization. In the model of ionization, Font-Ribera et
al. (2013) assume the spectrum of the AGN at $D=0$ with
$L_{\nu}\propto\nu^{-\alpha}$, where $\alpha=1.5$ (1.0) for the frequency
$\nu$ over (below) the Lyman limit. The luminosity of $\lambda=1420$ Å is
normalized as $L_{\nu}=3.1\times 10^{30}$ erg/s/Hz, which is taken from the
mean luminosity of the SDSS data release 9 quasars. No assumptions of AGN type
have been made in the models of Font-Ribera+13. Based on the model of
ionization, Font-Ribera et al. (2013) calculate $\xi$ for the homogeneous gas
radiated by AGN, and obtain the function

$\xi=0.0065(20\;h^{-1}{\rm cMpc}/D)^{2}.$ (20)

With the $\xi$ function, we calculate $D\xi$ that is presented with the cyan
dashed curve in Figure 26. The cyan dashed curve shows the plateau at $D\geq
40$ $h^{-1}{\rm cMpc}$ with positive $D\xi$ values. To distinguish the large-
scale positive $D\xi$ values, which are referred to as the ‘weak absorption
outskirts’, from the proximity zone created by the proximity effect, we plot
the observational CCF of AGN obtained by Momose et al. (2021) in Figure 26.
The AGN CCF obtained by Momose et al. shows a decreasing Ly$\alpha$ forest
absorption toward source position ($D=0$ $h^{-1}$cMpc) caused by the proximity
effect. If the weak absorption outskirts are created by the combination of the
clustering effect and ionization, our findings indicate that the Hi radial
profile of AGN may has transitions from proximity zones ($\lesssim$ a few
$h^{-1}$cMpc) to the Hi structures ($\sim 1-30$ $h^{-1}$cMpc) and the ionized
outskirts ($\gtrsim 30$ $h^{-1}$cMpc). The hard radiation may pass through the
Hi structure due to the small cross-section and ionizes the Hi gas in the
regions of ionized outskirts. Because of the low recombination rate, the Hi
gas remains ionized in the weak absorption outskirts.

Figure 26 shows that the $D\xi$ values in the range of Hi structure around AGN
and galaxy are also similar. Interestingly, the $D\xi$ profile of our galaxy
sample also shows positive $D\xi$ values towards $\gtrsim 30$ $h^{-1}$cMpc
which is similar to those of the AGN model and our All-AGN sample. This result
may suggest that the Hi gas at large scale ($\gtrsim 20$ $h^{-1}$cMpc) around
galaxies also fall toward the source position ($D=0$). Regions around galaxies
are special as galaxies are clustered together. Galaxies in this work are
bright with $M_{\rm UV}<-22$ mag. The galaxies can be hosted by massive
haloes, and are likely to distribute at overdensity regions. The overdensity
region suggests that each galaxy can be surrounded by several galaxies.
Although it is difficult for a galaxy to trigger the clustering effect for the
Hi gas on a scale of $\gtrsim 20$ $h^{-1}$cMpc, a group of galaxies may have
enough gravitational power to aggregate the Hi on this scale.

Figure 26: Comparison between our All-AGN and galaxy results and the models of
Font-Ribera et al. (2013) in the LoS CCF $\xi$ multiplied by distance ($D$).
The black and blue points are the results derived from the All-AGN and galaxy
samples sources, respectively. The orange curve is the LoS CCF of QSOs with
the Ly$\alpha$ forest derived by Font-Ribera et al. (2013). The cyan dashed
curve shows the ionization of radiation effect taken from Font-Ribera et al.
(2013). The pink line presents the CCF of AGN obtained by Momose et al.
(2021). The gray shade presents the range of the Hi structure. Two white areas
show the regions of proximity zone and weak absorption outskirts. The
horizontal gray line indicates the cosmic average where $D\xi=0$.

##  6 Summary

We reconstruct two 3D Hi tomography maps based on the Ly$\alpha$ forests in
the spectra of 14763 background QSOs from the SDSS survey with no signatures
of damped Ly$\alpha$ system or broad absorption lines. The maps cover the
extended Fall and Spring fields defined by the HETDEX survey. The spatial
volume of the reconstructed 3D Hi tomography maps are $2257\times 233\times
811$ $h^{\rm-3}$cMpc3 and $3475\times 1058\times 811$ $h^{\rm-3}$cMpc3. We
investigate Ly$\alpha$ forest absorption around galaxies and AGN with samples
made from HETDEX and SDSS survey results in our study field. Our results are
summarized below.

  * •

We derive the 2D Hi and Hi radial profiles of the All-AGN sample consisted of
SDSS AGN. We find that the 2D Hi profile is more extended in the transverse
direction than along the line of sight. In the Hi radial profile All-AGN
sample, the values of Ly$\alpha$ transmission fluctuation, $\delta_{\rm F}$,
increase toward the large scale, touching to $\delta_{\rm F}\sim 0$.

  * •

We compare the Hi radial profiles derived from the T1-AGN and T1-AGN(H) sub-
samples, whose $L^{\rm spec}_{\rm 1350}$ distributions are the same. We find
that the Hi radial profile of the T1-AGN sub-sample agrees with that of the
T1-AGN(H) sub-sample. This agreement suggests that the systematic uncertainty
between the SDSS and the HETDEX survey results is negligible.

  * •

We examine the dependence of the Hi profile on AGN luminosity by deriving the
2D Hi, Hi radial, LoS Hi radial, and Transverse Hi radial profiles of the All-
AGN-L3 (the faintest), All-AGN-L2, and All-AGN-L1 (the brightest) sub-samples.
We find that the Ly$\alpha$ forest absorption is the greatest in the lowest-
luminosity AGN sub-sample, and that the Ly$\alpha$ forest absorption becomes
weaker with increasing AGN luminosity This result suggests that, on average,
if the density of Hi gas around the bright AGN is greater than (or comparable
to) those of the faint AGN, the ionization fraction of Hi gas around bright
AGN is higher than that around faint AGN.

  * •

We investigate the AGN type dependence of Ly$\alpha$ forest absorption around
type-1 and type-2 AGN by the 2D Hi, Hi radial, LoS Hi radial, and Transverse
Hi radial profiles extracted from the T1-AGN and T2-AGN sub-samples with the
same $L^{\rm spec}_{\rm 1350}$ distributions. The comparison between the Hi
radial profiles of T1-AGN and T2-AGN sub-samples indicates that the Ly$\alpha$
transmission fluctuation around the T2-AGN sub-sample is comparable to the one
of the T1-AGN sub-sample on average. This trend suggests that, the selectively
different opening angle and orientation of the dusty torus for type-1 and
type-2 AGN do not have a significant impact on the Mpc-scale Ly$\alpha$ forest
absorption or the sensitivity of our result is not enough to detect the
difference.

  * •

We compare the Ly$\alpha$ forest absorptions around galaxies and type-1 AGN
with the 2D Hi, Hi radial, LoS Hi radial, and Transverse Hi radial profiles
derived from the galaxy and T1-AGN(H) sample sources. The Ly$\alpha$
transmission fluctuation values, $\delta_{\rm F}$, around the T1-AGN(H) sample
are larger than those of the Galaxy sample on average. This result may be
caused by the dark matter halos of type-1 AGN having a larger mass than the
one of galaxies on average.

  * •

We find that the Hi radial profiles of the LoS distance for the galaxy and
All-AGN samples show positive $\delta_{\rm F}$ values, which means weak
Ly$\alpha$ forest absorption, at the scale over $\sim 30$ $h^{-1}$cMpc. We
extract the $D\xi$ profile of our galaxy and All-AGN samples to compare with
the model CCF of AGN from Font-Ribera et al. (2013). The general trend of the
positive $D\xi$ at $\gtrsim 30$ $h^{-1}$cMpc is the same as the model CCF.
This results suggest that the Hi radial profile of AGN has transitions from
proximity zones ($\lesssim$ a few $h^{-1}$cMpc) to the Hi rich structures
($\sim 1-30$ $h^{-1}$cMpc) and the weak absorption outskirts ($\gtrsim 30$
$h^{-1}$cMpc).

## Acknowledgements

We thank Nobunari Kashikawa, Khee-Gan Lee, Akio Inoue, Rikako Ishimoto,
Shengli Tang, Yongming Liang, Rieko Momose, and Koki Kakiichi for giving us
helpful comments.

HETDEX is led by the University of Texas at Austin McDonald Observatory and
Department of Astronomy with participation from the Ludwig-Maximilians-
Universität München, Max-Planck-Institut für Extraterrestrische Physik (MPE),
Leibniz-Institut für Astrophysik Potsdam (AIP), Texas A&M University,
Pennsylvania State University, Institut für Astrophysik Göttingen, The
University of Oxford, Max-Planck-Institut für Astrophysik (MPA), The
University of Tokyo and Missouri University of Science and Technology. In
addition to Institutional support, HETDEX is funded by the National Science
Foundation (grant AST-0926815), the State of Texas, the US Air Force (AFRL
FA9451-04-2- 0355), and generous support from private individuals and
foundations. The observations were obtained with the Hobby-Eberly Telescope
(HET), which is a joint project of the University of Texas at Austin, the
Pennsylvania State University, Ludwig-Maximilians-Universität München, and
Georg-August-Universität Göttingen. The HET is named in honor of its principal
benefactors, William P. Hobby and Robert E. Eberly. The authors acknowledge
the Texas Advanced Computing Center (TACC) at The University of Texas at
Austin for providing high performance computing, visualization, and storage
resources that have contributed to the research results reported within this
paper. URL: http://www.tacc.utexas.edu

VIRUS is a joint project of the University of Texas at Austin, Leibniz-
Institut für Astrophysik Potsdam (AIP), Texas A&M University (TAMU), Max-
Planck-Institut für Extraterrestrische Physik (MPE), Ludwig-Maximilians-
Universität Muenchen, Pennsylvania State University, Institut fur Astrophysik
Göttingen, University of Oxford, and the Max-Planck-Institut für Astrophysik
(MPA). In addition to Institutional support, VIRUS was partially funded by the
National Science Foundation, the State of Texas, and generous support from
private individuals and foundations.

This work is supported in part by MEXT/JSPS KAKENHI Grant Number 21H04489
(HY), JST FOREST Program, Grant Number JP-MJFR202Z (HY).

K. M. acknowledges financial support from the Japan Society for the Promotion
of Science (JSPS) through KAKENHI grant No. 20K14516.

This paper is supported by World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan, the joint research program of the
Institute of Cosmic Ray Research (ICRR), the University of Tokyo, and KAKENHI
(19H00697, 20H00180, and 21H04467) Grant-in-Aid for Scientific Research (A)
through the Japan Society for the Promotion of Science.

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Figure 27: Continued from Figure 1. The different panels denote the coverages
over different redshift ranges shown at the top left of each panel.

Figure 28:  Same as Figure 1, but for the foreground sources in the ExSpring
field.

Figure 29: Continued from Figure 28. Figure 30:  Same as Figure 2, but for the
background sources in the ExSpring field.