Catch Me if You Can: Biased Distribution of Ly$\alpha$-emitting Galaxies according to the Viewing Direction

We report that Ly$\alpha$-emitting galaxies (LAEs) may not faithfully trace the cosmic web of neutral hydrogen (HI), but their distribution is likely biased depending on the viewing direction. We calculate the cross-correlation (CCF) between galaxies and Ly$\alpha$ forest transmission fluctuations on the near and far sides of the galaxies separately, for three galaxy samples at $z\sim2$: LAEs, [OIII] emitters (O3Es), and continuum-selected galaxies. We find that only LAEs have anisotropic CCFs, with the near side one showing lower signals up to $r=3-4~h^{-1}$ comoving Mpc. This means that the average HI density on the near side of LAEs is lower than that on the far-side by a factor of $2.1$ under the Fluctuating Gunn-Peterson Approximation. Mock LAEs created by assigning Ly$\alpha$ equivalent width ($EW_\text{Ly$\alpha$}^\text{obs}$) values to O3Es with an empirical relation also show similar, anisotropic CCFs if we use only objects with higher $EW_\text{Ly$\alpha$}^\text{obs}$ than a certain threshold. These results indicate that galaxies on the far side of a dense region are more difficult to be detected ("hidden") in Ly$\alpha$ because Ly$\alpha$ emission toward us is absorbed by dense neutral hydrogen. If the same region is viewed from a different direction, a different set of LAEs will be selected as if galaxies are playing hide-and-seek using HI gas. Care is needed when using LAEs to search for overdensities.


INTRODUCTION
Matter in the universe is not uniformly distributed in space but forms large-scale filamentary structure or the cosmic web. Many galaxy redshift surveys have depicted the cosmic web of the present-day universe (e.g., de Lapparent et al. 1986;Tegmark et al. 2004;Guzzo et al. 2014;Libeskind et al. 2018). For the young universe, the most successfully used galaxies to map the web are those with hydrogen Lyα emission at 1216Å (Lyα-emitting galaxies, LAEs). LAEs are one of the major galaxy populations in the young universe (e.g., Hu et al. 1998;Mal-hotra & Rhoads 2004;Shibuya et al. 2018). LAEs in the post reionization epoch (z ≤ 6) can be easily detected by narrow-band imaging whose wavelength is tuned to Lyα emission at the target redshift. Many LAE surveys have revealed filamentary structures and proto-clusters in the young universe (e.g., Shimasaku et al. 2003Shimasaku et al. , 2004Cai et al. 2017 ).
However, several studies have recently pointed out that the density peak of LAEs' distribution does not always match that traced by other galaxy populations. For instance, Shimakawa et al. (2017) have found a few Mpc deviation of LAEs' density peak in a z = 2 protocluster from that of Hα-emitting galaxies, which are normal star-forming galaxies commonly seen at high redshifts. A similar offset from continuum-selected galaxies has also been reported (Overzier et al. 2008;Toshikawa et al. 2016;Shi et al. 2019). More importantly, Momose et al. (2020) have suggested that the peak of LAEs' distribution does not match that of intergalactic neutral hydrogen (IGM Hi). With its density increasing with matter density, IGM Hi is a faithful tracer of the cosmic web that has recently come into use, although the areas mapped by IGM Hi are still limited because the observations are very time-consuming. These reported discrepancies might imply that LAEs intrinsically avoid dense parts of the cosmic web for some reason (e.g., Shimakawa et al. 2017;Oteo et al. 2018;Shi et al. 2019;Momose et al. 2020).
Alternatively, the discrepancies may be a bias caused by the absorption of Lyα photons by IGM Hi in dense regions (e.g., Dijkstra et al. 2007;Zheng et al. 2011;Laursen et al. 2011;Gurung-López et al. 2020;Hayes et al. 2021). In this case, a biased distribution of LAEs along the line-of-sight is expected. LAEs located on the far side of a dense region are more difficult to detect because their Lyα emission toward us is heavily absorbed by IGM Hi during passing through the region (Zheng et al. 2011;Gurung-López et al. 2020). As a result, fewer LAEs will be detected on the far side even if the original number is identical. So far, however, no observation has reported such an anisotropic distribution.
Aiming at finding such anisotropic features, we investigate the connection between LAEs and IGM Hi with a publicly available 3D IGM tomography map of the Lyα forest transmission fluctuation (δ F ), called the COSMOS Lyα Mapping And Tomography Observations (CLAMATO, Lee et al. 2016Lee et al. , 2018, as an extension of Momose et al. (2020). This paper is organized as follows. We introduce the data and methodology used in this study in Section 2 and present the results in Section 3. An examination of CCFs using mock LAEs is presented in Section 4. We summarize our study together with discussions in Section 5. We use a cosmological parameter set of (Ω m , Ω Λ , h) = (0.31, 0.69, 0.7) throughout this paper (Lee et al. 2016(Lee et al. , 2018. All distances are described in a comoving unit unless otherwise stated. We indicate "cosmic web" and "IGM" as those traced by Hi unless otherwise specified.

IGM Hi Data
We use the CLAMATO 1 as a tracer of IGM Hi (Lee et al. 2016(Lee et al. , 2018 . The CLAMATO is a 3D data-cube of δ F over z = 2.05 − 2.55, which is reconstructed from 240 galaxies and quasars spectra taken by the LRIS on 1 The data is from: http://clamato.lbl.gov the Keck I (Oke et al. 1995;Steidel et al. 2004). Here, δ F is defined by where F and F z are the Lyα forest transmission and its cosmic mean from Faucher-Giguère et al. (2008). Hence, δ F is the excess transmission to Lyα photons. Higher IGM Hi densities give lower F , and negative (positive) δ F (x) values indicate higher (lower) Hi densities than the cosmic mean. The effective transverse separation of the CLAMATO is 2.04 comoving h −1 Mpc (hereafter h −1 cMpc). The spatial resolution in the line-of-sight direction is 2.35 h −1 cMpc at z = 2.3. The final 3D data cube of the CLAMATO spans (x, y, z) = (30, 24, 438) h −1 cMpc with 0.5 h −1 cMpc pixel size.

Galaxy Samples
We use three galaxy samples constructed in Momose et al. (2020). One is a sample of 19 LAEs whose redshifts have been determined by nebular lines of Hα and/or [Oiii] 2 (Nakajima et al. 2012(Nakajima et al. , 2013Hashimoto et al. 2013;Shibuya et al. 2014;Konno et al. 2016). The other two used as control samples are 85 [Oiii]λλ4959, 5007 emitting galaxies (O3Es), and 570 continuumselected galaxies with a spectroscopic redshift. O3Es and continuum-selected galaxies are also star-forming galaxies like LAEs, but their selections, based on [Oiii] and far-ultraviolet luminosities, respectively, are not affected by IGM Hi.

Analyses
We conduct two different analyses. The first is the cross-correlation (CCF) between galaxies and the CLAMATO using the same definition of our previous work (Momose et al. 2020): where ξ δF is the cross-correlation at a separation r; δ F,i and δ ran,j are the transmission fluctuation at places i  Figure 1. Schematic diagrams of the CLAMATO data used in each analysis. The CCF and the 1D-CCFs along the line-ofsight and transverse directions are displayed from left to right. Note that the CLAMATO spans comoving dimensions of (x, y, z) = (30, 24, 438) h −1 cMpc. The sub-volume of the CLAMATO used in each analysis is colored in light blue (near side), faint red (far side), and gray (two transverse directions). The exact area used to calculate the CCF at radius r is colored in dark blue and dark red for the near and far sides, respectively. and j separated by r from a galaxy and random point, respectively, in question with σ F,i and σ ran,j being their errors; N (r) and M (r) represent the numbers of pixelgalaxy and pixel-random pairs with separation r, respectively. The σ F and σ ran are evaluated with the CLAM-ATO's 3D noise standard deviation measurements, including pixel noise, finite skewer sampling, and the intrinsic variance of the Lyα forest (Lee et al. 2018). Unlike our previous studies (Momose et al. 2020), we calculate the CCFs on the near and far sides of galaxies separately. The CCF on the near (far) side is derived only using the data in the near (far) hemispheres (regions colored in blue and red of Figure 1-1). For both calculations, the data within a line-of-sight separation of 1 h −1 cMpc from each galaxy is excluded to eliminate the influence of Hi in the circumgalactic medium (CGM). The second analysis is to compute a one dimensional CCF (hereafter "1D-CCF") defined by for the line-of-sight and two transverse directions. For the line-of-sight direction, we use narrow tomography data of the projected 1 × 1 (h −1 cMpc) 2 area centered at individual galaxies ( Figure 1-2). Note that regions colored in light blue and red are referred to as the near and far sides. Similarly, for the transverse directions, we use narrow data of a cross-section of 1 × 1 (h −1 cMpc) 2 along with the North-South (NS) and East-West (EW) directions (Figure 1-3). We measure δ F (∆r) in 2.5 h −1 cMpc steps over −16 ≤ (∆r/h −1 cMpc) ≤ 16 for the line-of-sight direction and −11 ≤ (∆r/h −1 cMpc) ≤ 11 for the transverse directions, excluding −1 ≤ (∆r/h −1 cMpc) ≤ 1.
In both analyses, errors are estimated with Jackknife resampling by removing one object. Therefore, the number of Jackknife samples is the same as that of galaxies in the original sample. Although Jackknife resampling for the cross-correlation is usually evaluated by dividing the survey volume into several small subvolumes, to be conservative, we do not use this resampling in this study because it gives smaller errors than obtained above (Momose et al. 2020).

RESULTS
The results of the two analyses are shown in Figure 2. Figure 2 (a) shows that the near side CCF of LAEs is consistently higher than the far side one up to r = 3 − 4 h −1 cMpc, possibly up to r ∼ 20 h −1 cMpc, meaning that the average density of IGM Hi gas around LAEs is systematically lower on the near side. The IGM Hi density at 3 h −1 cMpc between the two sides differs by a factor of 2.1 under the Fluctuating Gunn-Peterson Approximation (Appendix A). In contrast, no such systematic difference is seen in the other two galaxy populations (Figure 2

(b) and (c)).
A similar trend is found in the 1D-CCF along the lineof-sight in Figure 2 (d)-(f). Although the difference is marginal, the 1D-CCF of LAEs is higher on the near side up to ∼ 8 h −1 cMpc. In contrast, the 1D-CCFs of O3Es and continuum-selected galaxies are nearly symmetric around ∆r = 0, indicating an isotropic Hi density distribution along the line-of-sight with respect to the position of those galaxies. Unlike the 1D-CCF along the lineof-sight, those along transverse directions (Figure 2 (g)-(i)) show nearly symmetric distributions within the 1σ errors with a negative peak at r = 0, indicating an isotropic Hi density distribution.  ) and 1D-CCFs along the line-of-sight (panels (d)-(f)) and two transverse directions (panels (g)-(i)) for LAEs, O3Es, and continuum-selected galaxies. Panels (a)-(c). Blue and red lines indicate, respectively, the CCFs on the near and far sides of galaxies. Gray shades represent the CCFs calculated using all directions (Momose et al. 2020). A vertical dotted line indicates the possible largest radius for 3D cross-correlation calculations. Panels (d)-(f ). δF along the line-of-sight direction in 2.5 h −1 cMpc steps over 1 ≤ (∆r/h −1 cMpc) ≤ 16, plotted against the absolute distance from galaxies ∆rLOS, with blue (red) color indicating the near (far) side of them. A gray zone indicates the distance range excluded from the analysis. Panels (g)-(i). δF along two transverse directions in 2.5 h −1 cMpc steps over −11 ≤ (∆r/h −1 cMpc) ≤ 11, except the range in −1 ≤ (∆r/h −1 cMpc) ≤ 1, where ∆rTRANS indicates the distance from galaxies shown in Figure 1-3. Filled and open triangles denote δF of the NS-and EW-directions, respectively, while filled circles indicate the average of the two directions.
We calculate the probability of an anisotropic CCF and 1D-CCF along the line-of-sight like those seen in LAEs being caused by chance due to a small sample, by conducting the same analyses to randomly selected 19 continuum-selected galaxies and O3Es. The probability is found to be only 2% even if we loosen the criteria for anisotropy. 3 Therefore, the anisotropy of the Hi density found for LAEs is unlikely to be by chance.
One may be concerned that the limited spatial distribution of our LAEs along the transverse directions ((∆x, ∆y) = (30, 24) h −1 cMpc) and anlog the line- 3 We set two criteria. The first is that the near side CCF is higher than the far side one beyond the error bars at any radius up to r = 4 h −1 cMpc. The second criterion is the same as the first one except for the innermost radius by considering the case in mock LAEs presented in Section 4.2. The probability of satisfying each criterion is 0 and 2% (1 and 2%) for continuum-selected galaxies (O3Es).  of-sight (z = 2.14 − 2.22 4 or 69.7 h −1 cMpc) causes an anisotropic Hi density distribution along the lineof-sight. We also conduct the same analyses to 17 O3Es and 108 continuum-selected galaxies in this redshift range and find anisotropy in the CCF and 1D-CCF along the line-of-sight around O3Es, but with an opposite sign to that of LAEs: the Hi density is lower on the far side than on the near side. Besides, the probability that randomly selected galaxies reproduce an anisotropic CCF and 1D-CCF along the line-of-sight like those found for real LAEs is less than 1%. These results indicate that the anisotropy seen around LAEs is unlikely to be due to a small sample in a limited redshift range.

A TOY MODEL WITH MOCK LYα-EMITTING GALAXIES
We show in Section 3 that the IGM Hi density distribution around LAEs is anisotropic, with the density on the near side being lower than on the far side. Considering that the Hi density averaged over all directions decreases with the distance from LAEs, this result leads to the picture that LAEs for a given dense region are preferentially distributed on its near side; at any position on the near side, the Hi gas in front of that position should be more transparent to Lyα than on the opposite side.
The most natural explanation of this anisotropy is a selection effect. LAEs on the far side of dense regions are more difficult to detect because Lyα photons emitted from them toward us have to pass through the dense regions of low Lyα transmission. To test this hypothesis, we construct a toy model with mock LAEs and examine if galaxies, whose intrinsic distribution is normal, show an anisotropic Hi density distribution like actual LAEs if the sample is limited to those with strong Lyα emission in the observed-frame owing to their location relative to a dense region.

Methodology
Our toy model is based on a mock LAE sample created from the O3Es. We carry out the examination as follows. First, we assign each object of the O3E sample an "intrinsic" equivalent width, EW int Lyα (hence an "intrinsic" Lyα luminosity) according to an empirical EW Lyαstellar mass (M ) relation by Cullen et al. (2020): We use M estimates in the photometric redshift catalog of Straatman et al. (2016). Note that we adopt not the best-fit relation of Cullen et al. (2020) but its upper 1σ envelope because the best-fit relation assigns a large fraction of the sample negative EW int Lyα , 5 which cannot be used for selecting mock LAEs. Nonetheless, this prescription is consistent with the fact that the Lyα luminosities of LAEs are boosted compared with other star-forming galaxies.
Next, we attenuate the Lyα luminosity of each object using the actual line-of-sight δ F at the position of this object to obtain an "observed" equivalent width (EW obs Lyα ) by taking the attenuation by the IGM into account: where F LOS near is the transmission in the line-of-sight direction averaged over −5.0 ≤ (∆r/h −1 cMpc) ≤ −1.7 on the near side, in a projected square of 1×1 (h −1 cMpc) 2 area center at the galaxy. Equation 6 comes from the following consideration. First, we assume a double-peaked   Lyα profile where the red peak is located 200 km s −1 redward of λ(Lyα) while the blue peak, contributing 24% of the total Lyα emission, is centered at −200 km s −1 with a 200 km s −1 width. This profile is based on a stacked Lyα spectrum of z = 2.2 LAEs presented in Matthee et al. (2021) but also roughly consistent with other observations of individual LAEs (e.g., Rakic et al. 2011;Hashimoto et al. 2013Hashimoto et al. , 2015. We then assume that only the blue peak is subject to the IGM attenuation, neglecting the effect of the peculiar motion of galaxies relative to the IGM. Because Lyα photons in the blue peak, which have relative velocities of −300 to −100 km s −1 , are redshifted to λ(Lyα) after traveling over 1.7 ≤ ∆r/h −1 cMpc ≤ 5.1, we use the line-of-sight transmission averaged over this distance range, F LOS near . We calculate F LOS near from the excess transmission averaged over the same distance range, δ F LOS near , by: We assume F LOS near = 1 for four O3Es whose F LOS near is larger than unity so that their EW obs Lyα is equal to EW int Lyα . We should also note that an observed Lyα luminosity is determined not only by the line-of-sight IGM Hi absorption but to some degree by the absorption in all other directions, as pointed out by Zheng et al. (2011). However, since it is extremely complicated to evaluate the impact of Lyα suppression in all directions, we consider the line-of-sight direction alone.
The resultant EW obs Lyα estimates, presented by black circles in Figure 3, are lower than those of actual LAEs plotted as red stars. It is probably because LAEs generally have higher Lyα ISM/CGM escape fractions than continuum and nebular emission line selected galaxies, owing to their lower dust extinction (e.g., Stark et al. 2010;Wardlow et al. 2014;Kusakabe et al. 2015), lower Hi column density of the ISM (e.g., Hashimoto et al. 2015), and/or higher ionization parameter (e.g., Nakajima & Ouchi 2014; Sobral et al. 2018). However, our goal here is not to accurately reproduce observed LAEs but to create mock galaxies whose EW obs Lyα are sufficiently widely scattered around the selection threshold, to examine if the selection effect can cause anisotropy in the IGM density similar to the observed one.
Finally, by dropping objects whose EW obs Lyα is below some threshold, we obtain a sample of mock LAEs. We try two threshold EW obs Lyα values. One is 30Å that is actually adopted to select the z = 2.2 LAEs. The other is 40Å; we try this slightly higher threshold to evaluate how sensitive the results are to the threshold. We select 35 (12) O3Es with EW obs Lyα ≥ 30 (40)Å as mock LAEs. Using those mock LAEs, we conduct the same analyses (CCF and 1D-CCF) as for the real galaxies. Figure 4 presents the results of the toy model. A similar trend to that of real LAEs is seen in the EW obs Lyα ≥ 40 A sample, whose CCF is lower on the far side up to r = 5 h −1 cMpc, implying a factor 1.5 density difference within r = 3 h −1 cMpc with the FGPA. The 1D-CCF along the line-of-sight of this sample is also systemati-cally lower on the far side over |∆r TRANS | < 6 h −1 cMpc (Figure 4 (d)) in contrast to those along transverse directions, which are nearly symmetric over a similar distance (Figure 4 (f)). On the other hand, the EW obs Lyα ≥ 30Å sample does not show significant line-of-sight anisotropy. We note anisotropy of the 1D-CCFs along transverse directions seen beyond |∆r TRANS | ≥ 5 h −1 cMpc in Figures 4 (e) and (f). This could be due to the small sample size of mock LAEs because the anisotropic feature becomes weaker with increasing sample size, probably by mitigating the large-scale variation in the IGM Hi distribution around them.

Results
Interestingly, the degree of anisotropy is sensitive to the threshold EW obs Lyα value. The absence of anisotropy in the EW obs Lyα ≥ 30Å sample may indicate that this EW threshold is not high enough to produce detectable anisotropy in our mock LAEs. We also find that the mock LAEs with EW obs Lyα ≥ 40Å show stronger anisotropy in the 1D-CCF along the line-of-sight than the real LAEs. It may be because O3Es are, on average, located in higher IGM density environments that give higher near-to-far density contrasts. Another intriguing fact is a smaller offset of the 1D-CCF along the line-ofsight between the near and far sides seen in Figure 4

Possible impacts of Lyα profile shape and peculiar motion on LAE selection
Here we briefly discuss the possible impacts of Lyα profile shape and peculiar motion on the selection of mock LAEs. Changing the blue peak's width (currently 200 km s −1 ) and position (−200 km s −1 ) will select a different set of mock LAEs because of a change in the distance range over which Lyα photons are attenuated. The IGM attenuation is sensitive to the fraction of the blue (< λ(Lyα)) part (24% is assumed). Galaxies with an expanding Hi shell (e.g., Verhamme et al. 2008) will have low blue fractions because of systematic redshifting of the entire Lyα profile. As an extreme case, if galaxies have no blue part as found for some LAEs (e.g., Rakic et al. 2011;Hashimoto et al. 2013Hashimoto et al. , 2015, no IGM attenuation, and hence no selection effect, will occur. Galaxies with higher receding peculiar velocities than the IGM will suffer weaker selection effects because of decreased fractions of the Lyα emission subject to the IGM attenuation. If the actual selection effect by the IGM attenuation is much weaker than assumed in our toy model, we have to invoke an alternative mechanism as the origin of the observed anisotropy.

DISCUSSION AND SUMMARY
We have investigated the IGM Hi density traced by Lyα forest absorption around three galaxy populations at z ∼ 2, particularly paying attention to LAEs. We have found that LAEs show an anisotropic IGM Hi density distribution along the line-of-sight, while continuum-selected galaxies and O3Es do not. Such anisotropy is obtained if LAEs on the far side of dense regions tend not to be selected because of heavier Lyα attenuation by the IGM Hi. An examination with mock LAEs supports this idea, finding an anisotropic Hi density distribution as observed.
Other mechanisms could also create an anisotropic IGM Hi density distribution along the line-of-sight. One candidate is feedbacks from galaxies. If some feedback energy of a galaxy is emitted toward us, Hi gas in front of the galaxy may be swept out or ionized, thus decreasing the IGM Hi density on the near side. Quasars and Active Galactic Nuclei (AGN) jets can drive gas and/or energy in the host galaxies to the surrounding IGM up to distances of a few Mpc (e.g., Perucho et al. 2014;Dabhade et al. 2017Dabhade et al. , 2020Momose et al. 2020). Moreover, Hopkins et al. (2020) have indicated that cosmic ray (CR)-driven outflows extend beyond ∼ 1 Mpc. However, none of our LAEs has an AGN signature (Nakajima et al. 2012;Konno et al. 2016;Momose et al. 2020). CRdriven outflows are also unlikely because their effects on the surrounding IGM become negligible at z ≥ 1 − 2 (Hopkins et al. 2020). Thus, feedbacks are unlikely to be the major cause of the anisotropic density distribution around our LAEs.
Taking all these results into account, we conclude that the observed anisotropy of the IGM Hi distribution around LAEs is likely due to a selection effect of IGM absorption ( Figure 5). If a dense region is viewed from a different direction, some are missed from the original LAE sample, while some are newly selected as if galaxies are playing hide-and-seek using Hi gas. We note that anisotropy may also occur along with transverse directions because of complicated Lyα radiative transfer. Nevertheless, its amplitude seems to be negligible, as we have presented in Sections 3 and 4.2.
Our results also indicate that LAEs may not faithfully trace matter distribution, unlike O3Es and continuumselected galaxies. Some observational studies have found a 3 − 15 h −1 cMpc offset between the overdensity peaks of LAEs and other galaxy populations (Toshikawa et al. 2016;Shimakawa et al. 2017;Shi et al. 2019). Given the anisotropy of Hi density distribution over r = 3 − 5 h −1 cMpc found in this study, the discordance reported by those previous studies can be, at least partly, explained by the selection bias. Hence, we should

Nearside
Density Peak Farside ~ 20 h −1 Mpc Figure 5. Schematic picture of the selection bias against LAEs in a dense region. The IGM Hi density is represented by the grayscale, with darker colors meaning higher densities. Circles indicate galaxies, with larger sizes meaning higher intrinsic Lyα luminosities. The galaxy sample as a whole follows the matter/Hi density distribution. It means that the number of galaxies is proportional to the matter/Hi density where they are located; the number of galaxies in the darkest area is highest. The Lyα emission from each galaxy travels toward us from right to left in the IGM. Galaxies that are detectable in Lyα are colored in dark pink, while those not are in faint pink. Blue arrows indicate the direction and propagation of Lyα photons in the IGM. The Lyα photons emitted from a galaxy cannot reach us if the IGM Hi in 3 h −1 cMpc on the near side of the galaxy exceeds a certain density (translucent arrows).
be careful when using LAEs to search for overdensities such as proto-clusters. As has been found in Sections 3 and 4.2, an anisotropic Hi density distribution seems to be detectable even when the Hi density contrast of a given region is as small as a factor of ∼ 2. Since such a small difference can be easily realized between the densest part and the outskirt of a cosmic filament, we expect an anisotropic distribution across ∼ 10 − 20 h −1 cMpc can be observed if the sightline passes through a filament.
Our finding of an anisotropic Hi distribution around LAEs is based on a small sample (N = 19). To increase the statistical reliability, a larger sample from a widefield spectroscopic survey, such as one planned with the Subaru Prime Focus Spectrograph (PFS), is necessary. Followup spectroscopy of Lyα emission of the sample galaxies is also useful. Theoretically, there seems to be a debate on the impact of Lyα radiative transfer effects on galaxies in current simulations (e.g., Zheng et al. 2011;Hough et al. 2020). Theoretical studies focusing on lineof-sight dependence of the IGM Hi-galaxy connection over a wide range of cosmological environment are also needed. ming language and its packages of numpy, matplotlib, scipy, and astropy (Astropy Collaboration et al. 2013).

A. THE DENSITY DIFFERENCE BETWEEN THE NEAR AND FAR SIDES
The density difference between the near and far sides is evaluated with the Fluctuating Gunn-Peterson Approximation (FGPA, e.g., Rauch et al. 1997;Croft et al. 1998;Weinberg 1999;Becker et al. 2015). In the FGPA, optical depth, τ FGPA , is described by a power-law of normalized gas density, ρ/ρ, with Then, the density difference is expressed as the ratio: The τ FGPA is calculated from δ F with: In this study, we use β = 1.6 (e.g., Croft et al. 1998;Weinberg 1999) and calculate the Hi density difference within r = 3 h −1 cMpc. We estimate τ far FGPA and τ near FGPA for the 19 real LAEs, and obtain ρ far /ρ near = 2.1. Similarly, the mock LAEs satisfying the EW obs Lyα = 30 and 35Å thresholds have ρ far /ρ near = 1.3 and 2.1, respectively.