MAMMOTH-Subaru. V. Effects of Cosmic Variance on Lyα Luminosity Functions at z = 2.2–2.3

The fields used for this study are selected using the background quasar spectra from the Baryon Oscillation Spectroscopic Survey (BOSS) of the Sloan Digital Sky Survey (Dawson et al. 2013, 2016). We target a total number of eight overdense fields centered toward MAMMOTH candidates, with two narrowband filters covering four fields each. For the NB387 filter, the fields covered are BOSS J0210+0052 (J0210), BOSS J0222-0224 (J0222), BOSS J0924+1503 (J0924), and BOSS J1419+0500 (J1419). For the NB400 filter, the fields covered are BOSS J0240-0521 (J0240), BOSS J0755+3108 (J0755), BOSS J1133+1005 (J1133), and BOSS J1349+2427 (J1349). The relevant information of the eight fields is summarized in Table 1. See also Liang et al. (2021) and Z. Cai et al. (2023, in preparation) for details of field selections.

We obtained our observations using the gigantic mosaic CCD camera HSC, with a wide FoV of 15 in diameter, and a pixel scale of 0168. Narrowband imagings are carried out using two NB filters, NB387 (λ_c = 3862 Å, FWHM = 56 Å) and NB400 (λ₀= 4003 Å, FWHM = 92Å). The corresponding redshifts of detecting Lyα emissions are z = 2.18 ± 0.02 and z = 2.29 ± 0.04, respectively. We also use the g band to evaluate the continuum of the detected objects. Figure 1 shows the transmission curve of the three filters used, which takes the transmittance accounting in CCD quantum efficiency, dewar window, and the primary focus unit and the reflectivity of the prime mirror into account. The imaging data are reduced with the HSC pipeline, hscPipe (Bosch et al. 2018; Aihara et al. 2019), see Liang et al. (2021) and Zhang et al. (2023) for the details of the data reduction. Due to the poor quality of the J0210 NB387 data, NB387 and g-band data of the J0210 field are reduced with two different versions of hscPipe, which results in a slight difference in the final image depth, as shown in Table 1. The final catalog of J0210 is a combination of the two.

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Object detection and the following photometry are performed using the SExtractor program (Bertin & Arnouts 1996). The g-band and NB387/400 images are PSF matched by convolving a proper Gaussian kernel in each field (Liang et al. 2021; Zhang et al. 2023). The detection threshold is set as 15 continuous pixels over the 1.2σ sky background. We also apply masks to the pixels with low signal-to-noise ratio signals or saturated by bright stars and artifacts during object detection. In addition, the sky background rms map is used as the weighting map in SExtractor, to minimize the influence of the fluctuations in the image depth for each field. The aperture diameters for photometry are 15 pixels (∼25) for the J0210 field and 10 pixels (∼17) for the other seven fields.

We estimate the total magnitude using AUTO-MAG, which uses the automatically determined elliptical aperture for Kron photometry by SExtractor (Bernardeau & Kofman 1995). Both AUTO-MAG and APER-MAG output by SExtractor are used in the selection criteria described in the next section. Furthermore, we also replace the g-band magnitudes with the corresponding 2σ limiting magnitudes, for the objects with a g band fainter than the 2σ limit. The 5σ limiting magnitudes within the 17 (25 for J0210) aperture of the final stacked g-band and NB387/400 images are listed in Table 1. The depths of the NB400 images are about 1 mag deeper than that for the NB387 images in general, and the seeing and the depth of J0210 are relatively poorer than all other fields.

We use the narrowband color excesses in the broad band to select out our LAE samples, this approach has been widely used in previous studies (e.g., Ouchi et al. 2008; Guaita et al. 2010; Konno et al. 2016). Most of these studies use multiwavelength data and thus multiple color excesses to define their selection criteria (e.g., the U band), while we only use one broad band (HSC g band) to estimate the continuum. However, Liang et al. (2021) have proved that it is sufficient enough for the z ∼ 2.2 LAE selection.

We assume the LAE spectrum has a Gaussian-like redshifted Lyα emission with a rest-frame equivalent width (EW) of 20 Å and a continuum that follows a simple power law of f_λ = λ^β . We also take the absorption of the intergalactic medium (Madau 1995; Madau & Dickinson 2014) into account when calculating the observed magnitude (Inoue et al. 2014). In addition, we find that the value of g-NB calculated from APER-MAG is not necessarily the same as that from AUTO-MAG, as shown in Figure 2. (The differences between g-NB values calculated from AUTO-MAG and those from APER-MAG are not zero for every object). As a result, for some objects with APER-MAG values passing the first three criteria, their g-NB values calculated from AUTO-MAG can still be smaller than the corresponding EW₀ = 20 Å values, or even smaller than 0. Since we use AUTO-MAG to estimate the total magnitude, this could be a problem when calculating the Lyα luminosity. To address this issue, we also apply the criteria for assessing the color excess and color error to AUTO-MAG, so that (g-NB)_AUTO is also above the required values.

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For NB387 data, the selection criteria are defined as

$$\begin{eqnarray}&&\begin{array}{l}20.5\lt \mathrm{NB}{387}_{\mathrm{APER}}\lesssim {\mathrm{NB}}_{\mathrm{lim},5\sigma }\\ {\left(g-\mathrm{NB}387\right)}_{\mathrm{APER}}\gt 0.3\\ {\left(g-\mathrm{NB}387\right)}_{\mathrm{APER}}\gt 3\sigma (\mathrm{NB}{387}_{\mathrm{APER}})\\ {\left(g-\mathrm{NB}387\right)}_{\mathrm{AUTO}}\gt 0.3\\ {\left(g-\mathrm{NB}387\right)}_{\mathrm{AUTO}}\gt 3\sigma (\mathrm{NB}{387}_{\mathrm{AUTO}}).\end{array}\end{eqnarray} \tag{ 1 }$$

For NB400 data, the selection criteria are defined as

$$\begin{eqnarray}&&\begin{array}{l}18.0\lt \mathrm{NB}{400}_{\mathrm{APER}}\lesssim {\mathrm{NB}}_{\mathrm{lim},5\sigma }\\ {\left(g-\mathrm{NB}400\right)}_{\mathrm{APER}}\gt 0.4\\ {\left(g-\mathrm{NB}400\right)}_{\mathrm{APER}}\gt 3\sigma (\mathrm{NB}{400}_{\mathrm{APER}})\\ {\left(g-\mathrm{NB}400\right)}_{\mathrm{AUTO}}\gt 0.4\\ {\left(g-\mathrm{NB}400\right)}_{\mathrm{AUTO}}\gt 3\sigma (\mathrm{NB}{400}_{\mathrm{AUTO}}).\end{array}\end{eqnarray} \tag{ 2 }$$

Here, The first line defines the detection limits using APER-MAG, where the lower limits are set to avoid saturation, and the upper limits are set as the corresponding 5σ limiting magnitudes to ensure the reliability of the source detection. The color excesses are equivalent to setting EW₀ > 20 Å, which are ∼0.3 for the NB387 filter and ∼0.4 for the NB400 filter. The 3σ (NB) corresponds to the color error, and is defined following Shibuya et al. (2018) as

$$\begin{eqnarray}&&3\sigma (\mathrm{NB})=-2.5\mathrm{log}\left(1-3\times \displaystyle \frac{\sqrt{{f}_{1\sigma ,\mathrm{NB}}^{2}+{f}_{1\sigma ,{\rm{g}}}^{2}}}{{f}_{\mathrm{NB}}}\right),\end{eqnarray} \tag{ 3 }$$

which is designed to reject false selections of faint contaminants passing the criteria due to statistical fluctuations. We plot the LAE selections for each field in Figure 2. Moreover, the NB387 data are re-selected from the original catalog of Liang et al. (2021), where a 2σ color error and APER-MAG-only criteria are used. We make the selection criteria consistent across two filters in this study to investigate the field-to-field variations.

The detection completeness f_det is estimated by Monte Carlo simulations with the GALSIM package (Rowe et al. 2015), following the procedure described in previous studies (e.g., Konno et al. 2016; Itoh et al. 2018; Konno et al. 2018). Mock galaxies are constructed according to 14 specified NB387/400 magnitude bins with centers ranging from 20.25–26.75 with a bin width of Δm = 0.5. Source detection and photometry of the mocks are performed using SExtractor 2.25.0 (Bertin & Arnouts 1996, in the same manner as our sample selection (see Section 2). The detection completeness ${f}_{\det }$ is defined as the ratio between the total number of mocks detected and that of mocks created (i.e., 500 for each bin), as a discrete function of the NB387/400 magnitudes. We repeat the process 10 times for each field with varying random seeds and obtain the final value by taking the average. The results are displayed in Figure 3. Values of f_det are typically above 90% for the bright-end sources and fall rapidly to zero beyond the 5σ limiting magnitudes.

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Possible contaminants that can pass our selection criteria are mainly considered to be [O ii] λ3727 emitters at z ∼ 0.04, C iii] λ1909 emitters at z ∼ 1, and C iv λ1549 emitters at z ∼ 1.5. For the [O ii] emitters, the corresponding survey volume is about three orders of magnitude smaller than that of LAEs at z ∼ 2.2, together with the [O ii] LFs at z ∼ 0.1 investigated by Ciardullo et al. (2013) we conclude that the amount of low-z [O ii] emitters is negligible for our sample. For C iii] and C iv contaminants, the rest-frame EW would be larger than 30 Å, much larger than that of star-forming galaxies. To investigate the portion of these C iii] and C iv contaminants in our sample, we perform a spectroscopic follow-up for our LAEs.

We present an estimate of the overall contamination rate using spectroscopic observations with Magellan/ImacS on 2022 September 29 and 30. The Magellan/IMACS is set in the multi-slit spectroscopy mode with the f/2 camera, with an FoV of 12'' in radius. Slits have a width of 12 and a length of 80. We observe one pointing in each of the J0210 and J0222 fields, and two pointings in the J0240 field. The on-source exposure times are 7500 and 6000 s for NB387 and NB400 LAEs, respectively.

The spectra cover wavelengths between ∼3600 and 5700 Å with a spectral resolution of ≈7 Å. Since the spectral resolution is not small enough to resolve the doublet of low-z [O ii] contaminants, we only use the spectra ≥ 2 emission lines when estimating the contamination rate. After data reduction, we obtained 120 spectra with the detected emission lines at the expected wavelength of Lyα. 22 spectra are confirmed LAEs at z = 2.2 with ≥ 2 emission lines, and two spectra are foreground objects. Therefore, we estimate that the contamination rate is about 2/(2+22) = 8%. The narrowband magnitudes of the 24 objects with ≥ 2 emission lines are 21.3–25.3 mag.

Although we use sources with ≥ 2 emission lines to estimate the contamination rate due to the limitation in the spectral resolution, we note that most LAEs at z ∼ 2.2 should be detected only in a single line on our spectra. Moreover, as mentioned earlier, [O ii] contaminants can only be seen over a very small survey volume in this study. Therefore, 8% is likely an overestimation, and the real contamination rate can be even lower.

Before we move on to the investigation of LFs, we estimate the internal uncertainties associated with the measurements. Since the goal of this paper is to examine the cosmic variance when comparing different fields, an inspection for the field-to-field errors introduced during measurements becomes necessary.

Different seeings of the field images can lead to a field-to-field variation. To address this issue, we have smoothed our images before carrying out photometry. Table 1 shows the seeing sizes measured in narrowband and g-band images in each field before smoothing. As we mentioned in Section 2.3, when performing photometry, two algorithms APER-MAG and AUTO-MAG can produce different values. This is because not all of our LAEs are point sources. However, this difference does not affect our analysis as we only use AUTO-MAG (suitable for both extended and point sources) to calculate Lyα luminosity. The APER-MAG measured with a fixed aperture is only used for accurate source detection and g-NB color selection. Besides, we have matched the seeing of each field image before performing photometry. As a result, this photometric methodology issue does not contribute to the field-to-field variations.

Due to the limited depths of each field image, the number of faint galaxies measured is underestimated. We carry out a completeness correction as described in Section 3.1 to reduce the effect of this issue. However, one may still be concerned about the contribution to the field-to-field errors if these underestimations differ substantially between fields. Here we argue that, since our mock galaxies for estimating the completeness are generated directly on the field images, and they go through the same detection and selection process as real sources, the field-to-field variations caused by photometric random noise are also taken into account by the completeness correction. Furthermore, we have checked the photometric errors versus luminosity for each field, and the correlations are consistent across fields.

We have estimated the field-to-field variation raised when calibrating the photometric zero-points for each field and identified an upper limit of a variation of 0.1 mag. We have performed a simulation by propagating this photometric error into the LFs (see the next section for the calculation of LFs) and found that the resultant error is no larger than 60% of the statistical (i.e., Poisson) error (see Figure 4). We include this internal photometric error in the LF error bars in the next section and also propagate it into the cosmic variance investigation in Section 4.

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We derive the Lyα LFs in the same manner as many previous studies (e.g., Ouchi et al. 2008; Itoh et al. 2018). We calculate Lyα luminosities using the total magnitude of NB387/400 and the g band, estimated by AUTO-MAG. We calculate the Lyα line flux (f_line) and the rest-frame UV continuum flux (f_c) from the NB387/400 and g-band magnitudes (m_NB and m_BB) using the following equation:

$$\begin{eqnarray}&&{m}_{\mathrm{NB},\mathrm{BB}}+48.6=-2.5\mathrm{log}\displaystyle \frac{{\int }_{0}^{\infty }({f}_{{\rm{c}}}+{f}_{\mathrm{line}}){T}_{\mathrm{NB},\mathrm{BB}}d\nu /\nu }{{\int }_{0}^{\infty }{T}_{\mathrm{NB},\mathrm{BB}}d\nu /\nu }.\end{eqnarray} \tag{ 4 }$$

Here, T_NB and T_BB are the transmittance curves in the frequency space of the NB and BB filters, respectively. We take the same assumption as Itoh et al. (2018) that f_line is a δ function and f_c is a constant.

The volume number density $\phi (\mathrm{log}\,L)$ in a Lyα luminosity bin of $[\mathrm{log}\,L-d\mathrm{log}\,L/2,\mathrm{log}\,L+d\mathrm{log}\,L/2]$ is defined as follows:

$$\begin{eqnarray}&&\phi (\mathrm{log}\,L)d\mathrm{log}\,L=\displaystyle \sum _{i}\displaystyle \frac{1}{{f}_{{\rm{i}},\ \det }{V}_{\mathrm{eff}}},\end{eqnarray} \tag{ 5 }$$

where the sum is taken over all LAEs in the specified bin. ${f}_{{\rm{i}},\det }$ is the detection completeness for the corresponding NB387/NB400 magnitude of the ith LAE, and V_eff is the effective survey volume assuming a top-hat filter transmittance. We set the bin width to $d\mathrm{log}\,L=0.1\mathrm{dex}$, which is the same as that of Konno et al. (2016). The actual NB387 and NB400 filter transmission curves are not perfect top hats. Sobral et al. (2017) investigated the incompleteness due to this effect and found that the correction factor ranges from 0.97 in the faintest bin to 1.3 in the brightest bin. This study mainly focuses on the faint end and excludes the bright end as described below; therefore, we take the assumption that the transmittance shapes of our filters are top hats. Any systematic errors introduced due to this assumption will not affect the cosmic variance investigation since it is a differential analysis.

The left two panels of Figure 5 show LFs derived from eight fields, as well as the overall LFs of NB387 and NB400. The error bars include the Poisson uncertainties calculated from Gehrels (1986), the internal photometric uncertainties described in Section 3.3, as well as the field-to-field variance estimated from our measurement in Section 4. The values of these field-to-field variances are 0.052 for NB387 and 0.025 for NB400 (σ_g ; see Section 4.2 for details). We include the field-to-field variance in this section. The low luminosity point has been excluded for low completeness. Note that we choose the value of log L > 42.3 for all fields to keep the fitting process consistent. Using the minimum χ² fitting, the LFs are fitted to a Schechter function (Schechter 1976), defined as

$$\begin{eqnarray}&&\phi \left(\mathrm{log}\,L\right)d\mathrm{log}\,L=\mathrm{log}{\,}_{e}(10){\phi }^{* }{\left(\displaystyle \frac{L}{{L}^{* }}\right)}^{\alpha +1}{e}^{-\tfrac{L}{{L}^{* }}}d\mathrm{log}\,L.\end{eqnarray} \tag{ 6 }$$

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Here, Schechter parameters α, L^*, and ϕ^* characterize the faint-end slope, luminosity, and density, respectively (Schechter 1976). Konno et al. (2016) surveyed the Lyα LFs for z ∼2.2 LAEs using the NB387 data from Subaru/Suprime-Cam (Miyazaki et al. 2014) in five fields and fitted three Schechter parameters simultaneously as free parameters. They reported a faint-end slope α of $-{1.75}_{-0.09}^{+0.10}$, and since this value is reasonably well constrained (Konno et al. 2016), we fix our α at this value. The goal of this study is to investigate the field-to-field variation, and thus having α fixed at the same value has the benefit of reducing differences in the expected galaxy abundance for all fittings so that the statistical fluctuations become significant. We also calculate the integral of each best-fit Schechter function n:

$$\begin{eqnarray}&&n={\int }_{\mathrm{log}{L}_{{\rm{lim}}}}^{\infty }\phi (\mathrm{log}L)d\mathrm{log}L.\end{eqnarray} \tag{ 7 }$$

Here, log ${L}_{\mathrm{lim}}$ corresponds to the integration limit for Lyα luminosities. We adopt the value from Konno et al. (2016) of $\mathrm{log}\,{L}_{\mathrm{lim}}=41.41$ erg s⁻¹ for all fittings (see Section 4.1 to obtain integrals of the Schechter functions). The right panels of Figure 5 show the 68% and 90% confidence levels, as well as the best-fit values of the parameter L^* and the integral of the corresponding Schechter function n. The best-fitting results are shown in Table 2.

Table 2. Lyα LF Fitting Information of Individual Fields and the Overall Data of NB387, NB400, and All Eight Fields

Field Name	L* (1042 erg s−1)	ϕ* (10−4 Mpc−3)	n (10−3 Mpc−3)	χ2	NLAE_fit
(1)	(2)	(3)	(4)	(5)	(6)
J0210	${4.03}_{-1.15}^{+1.74}$	${12.52}_{-5.64}^{+10.28}$	${9.60}_{-2.39}^{+3.21}$	0.464	204
J0222	${10.53}_{-4.98}^{+21.09}$	${3.49}_{-2.42}^{+4.51}$	${6.40}_{-1.58}^{+2.09}$	0.338	243
J0924	${3.92}_{-1.33}^{+2.27}$	${10.61}_{-5.58}^{+12.19}$	${7.92}_{-2.31}^{+3.49}$	0.514	226
J1419	${4.71}_{-1.89}^{+3.88}$	${6.18}_{-3.63}^{+8.73}$	${5.50}_{-1.70}^{+2.68}$	0.510	198
J0240	${8.65}_{-3.09}^{+7.19}$	${2.64}_{-1.33}^{+2.12}$	${4.07}_{-0.79}^{+0.96}$	0.126	333
J0755	${4.55}_{-1.10}^{+1.62}$	${8.19}_{-3.10}^{+4.92}$	${7.05}_{-1.35}^{+1.69}$	0.453	338
J1133	${5.87}_{-1.70}^{+2.90}$	${4.34}_{-1.84}^{+2.96}$	${4.74}_{-0.93}^{1.15}$	0.190	279
J1349	${6.45}_{-1.77}^{+3.02}$	${5.05}_{-2.02}^{+3.03}$	${6.00}_{-1.06}^{+1.26}$	0.297	426
NB387	${5.37}_{-1.24}^{+1.88}$	${6.91}_{-2.53}^{+3.65}$	${6.94}_{-1.25}^{+1.46}$	0.834	871
NB400	${6.18}_{-1.09}^{+1.43}$	${4.67}_{-1.19}^{+1.57}$	${5.34}_{-0.61}^{+0.70}$	0.586	1376
Overall	${5.95}_{-0.96}^{+1.22}$	${5.26}_{-1.27}^{+1.65}$	${5.82}_{-0.65}^{+0.75}$	1.047	2247

Note. Column (1) is the field name. Column (2) is the best-fit L^*. Column (3) is the best-fit ϕ^*. Column (4) is the integral of the best-fit Schechter function n. Column (5) is the reduced χ², and Column (6) is the number of LAEs used in the fitting N_{LAE_fit}. All fittings are performed within a log L range of 42.3–43.1 erg s⁻¹ and the faint-end slopes are fixed at −1.75.

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We combine NB387 and NB400 data to compute the overall LF for all eight fields (see the red points and line in Figure 6). Due to the redshift difference between NB387 and NB400, we argue that the difference in the LF between NB387 and NB400 is due to field-to-field variation rather than redshift evolution.

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An excess in number densities at the bright end with log L ≳ 43.1 is presented in the left panels of Figure 5. One possible explanation is the magnification bias (e.g., Mason et al. 2015), where the luminosity of several LAEs is raised due to gravitational lensing by foreground galaxies. However, past studies (e.g., Konno et al. 2016; Sobral et al. 2017) suggest that the number density excess at the bright end is due to type I AGN rather than magnification bias. Zhang et al. (2021) have also confirmed this conclusion using the HETDEX spectroscopic surveys. Since no AGN LFs reaching ∼43.1 at faint ends is available in the literature, we choose a cutoff at log L = 43.1 to exclude the bright-end hump in our fitting. Konno et al. (2016) argue that, due to the presence of significantly smaller error bars at the faint end, the humps at the bright end do not affect the fitting results significantly. We fit a Schechter function to our overall LF both with (42.3 < log L <44.1) and without the bright end (42.3 < log L < 43.1), and obtain two sets of best-fit parameters of α = −1.75 (fixed), ${L}^{* }={6.23}_{-0.54}^{+0.61}$ ×10⁴² erg s⁻¹, and ${\phi }^{* }={3.81}_{-0.43}^{+0.51}$ × 10⁻⁴ Mpc⁻³, and α = −1.75 (fixed), ${L}_{\mathrm{Ly}\alpha }^{* }={5.95}_{-0.96}^{+1.22}\times {10}^{42}$ erg s⁻¹, and ${\phi }_{\mathrm{Ly}\alpha }^{* }\,={5.26}_{-1.27}^{+1.65}$ ×10⁻⁴ Mpc⁻³. We thus conclude that the exclusion of the bright-end hump does not affect our fitting significantly. We choose to perform curve fitting without the bright-end hump to make the following cosmic variance investigation as accurate as possible. This results in a log L_Lyα coverage range of 42.3–43.1.

Further, we also fit a power-law ${\mathrm{log}}_{10}\phi =A\times {\mathrm{log}}_{10}(L)+B$ according to Sobral et al. (2017) to the bright-end LF with log L > 43.1, and obtain the best-fit parameters $A=-{0.99}_{-0.35}^{+0.33}$ and $B={38.82}_{-14.45}^{+15.10}$. We show this using the red-dashed line in Figure 6.

We show the comparison between our Lyα LFs and those of previous studies in Figure 6. Results from blank-field spectroscopic surveys (Blanc et al. 2011; Cassata et al. 2011; Ciardullo et al. 2014) are plotted with black lines with different line styles, while the results from narrowband imaging surveys (Hayes et al. 2010; Ciardullo et al. 2012; Konno et al. 2016; Sobral et al. 2017) are plotted with dashed lines with different colors. We also plot Lyα LF from the HETDEX spectroscopic survey by Zhang et al. (2021) using black triangles. They have confirmed that type I AGNs can fully explain the bright-end hump.

The LAE samples used by Blanc et al. (2011) and Ciardullo et al. (2014) are obtained from spectroscopic observations from the Hobby Eberly Telescope Dark Energy Experiment (HETDEX) Pilot Survey, while Cassata et al. (2011)'s sample comes from the VIMOS survey Fevre et al. (2003). The redshift range covered by these spectroscopic surveys is 1.9 < z < 3.8, 2.0 < z < 3.2, and 1.90 < z < 2.35, respectively, with a Lyα EW₀ greater than 20 Å for most LAEs. Hayes et al. (2010) investigated Lyα LFs as well as the escape fraction of z = 2.2 LAEs in the GOODS-South field, obtained by narrowband imaging using NB388 filter for VLT/FORS1. Lyα LFs from Ciardullo et al. (2012) are derived from the z = 2.1 LAE samples of Guaita et al. (2010), which is obtained from an ultra-deep narrowband MUSYC in the ECDF-S field, with a central wavelength of 3727 Å. We also overplot the data points from Zhang et al. (2021), who confirmed that the bright-end hump of z ∼ 2 LAE LFs are mainly caused by type I AGNs.

Konno et al. (2016) investigated Lyα LFs of z ∼ 2.2 LAEs using the NB387 data of Subaru/Suprime-Cam and constructed their full LAE sample from the SXDS, COSMOS, CDFS, HDFN, and SSA22 fields. Sobral et al. (2017) carried out deep narrowband imaging with a custom-built NB392 filter for the Isaac Newton Telescope's Wide Field Camera (INT/WFC), they derived Lyα LFs from z ∼ 2.23 LAEs detected in the COSMOS and UDS fields. The Lyα EW₀ criteria of color excess in the narrow band in these narrowband imaging surveys are all EW₀ = 20 Å, which are consistent with our selections. The best-fit Schechter parameters of the literature mentioned above and that of our studies are summarized in Table 3.

Table 3. Best-fit Schechter Parameters for the Lyα LFs of Previous z ∼ 2.2 LAEs, and Their Corresponding Total Survey Area and Survey Volumes

References	α	L* (1042 erg s−1)	ϕ* (10−4 Mpc−3)	Atot (deg2)	Vtot (106 Mpc3)
This study	−1.75 (fixed)	${5.95}_{-0.96}^{+1.22}$	${5.26}_{-1.27}^{+1.65}$	11.62	8.71
Blanc et al. (2011)	−1.7 (fixed)	${16.3}_{-10.8}^{+94.6}$	${1.0}_{-0.9}^{+5.4}$	0.047	1
Cassata et al. 2011	${-1.6}_{-0.12}^{+0.12}$	5.0 (fixed)	${7.1}_{-1.8}^{+2.4}$	0.78	2.3
Ciardullo et al. (2014)	−1.6 (fixed)	${39.8}_{-16.4}^{+98.2}$	0.36	0.044	1.03
Hayes et al. 2010	${-1.49}_{-0.27}^{+0.27}$	${14.5}_{-7.54}^{+15.7}$	${2.34}_{-1.64}^{+5.42}$	0.016	0.005
Ciardullo et al. (2012)	−1.65 (fixed)	${2.14}_{-0.52}^{+0.68}$	${13.8}_{-1.5}^{+1.7}$	0.36	0.17
Konno et al. (2016)	-${-1.75}_{-0.09}^{+0.10}$	${5.29}_{-1.13}^{+1.67}$	${6.32}_{-2.31}^{+3.08}$	1.43	1.32
Sobral et al. (2017)	-${-1.75}_{-0.25}^{+0.25}$	${3.89}_{-0.65}^{+1.73}$	${9.13}_{-4.41}^{+3.09}$	1.43	0.73

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In Figure 6, our results show a small offset with Konno et al. (2016). Apart from the result of Hayes et al. (2010), which is subject to a huge uncertainty owing to small statistics, all Lyα LFs from the narrowband imaging surveys yield similar of α and L^*, but small deviations in ϕ^*.

In previous studies (e.g., Ouchi et al. 2008; Konno et al. 2016), the field-to-field variation is characterized by σ_g , which represents the expected dispersion in galaxy counts owing to the cosmic variance. The expected galaxy counts N can be obtained by integrating LFs over the effective survey volume probed by a field (e.g., Robertson 2010b). Therefore, we make a series of this value N by assigning pointing on each field image, with the coverage smaller than one HSC FoV. For each pointing assigned, a Lyα LF can be derived from a subsample consisting of LAEs within that pointing. By integrating the best-fit Schechter functions, the galaxy abundance N of each pointing can be obtained as

$$\begin{eqnarray}&&N/{V}_{\mathrm{eff}}={\int }_{\mathrm{log}\,{L}_{\mathrm{lim}}}^{\infty }\phi (\mathrm{log}\,L)d\mathrm{log}\,L.\end{eqnarray} \tag{ 8 }$$

As mentioned in Section 3.3, we adopt the value from Konno et al. (2016) ${\text{}}{\rm{log}}\,{L}_{\mathrm{lim}}=41.41$ erg s⁻¹ for all fittings. Moreover, $\mathrm{log}\,{L}_{\mathrm{lim}}$ has also been discussed in Lyα luminosity densities (e.g., Ouchi et al. 2008; Konno et al. 2016), which characterize the mean Lyα luminosity per unit volume of the pointing. Different choices of log ${}_{\mathrm{lim}}$ can result in systematic errors in the integral, thus shifting the position of mean galaxy abundances, but leaving the distribution unaffected.

To keep the calculation process of galaxy abundances N consistent for each pointing, we apply the same fitting procedure for each pointing as follows: fixing the faint-end slope α at −1.75, as mentioned in Section 3.3, and excluding the data points with log L < 42.3, which are beyond the completeness limit, and the data points with log L > 43.1, contaminated by AGNs. The lower limit for integrating LFs is set to be ${\text{}}{\rm{log}}\,{L}_{\mathrm{lim}}=41.41$. In addition, since we aim to investigate the field-to-field variation itself, we only include Poisson uncertainties in the error bars when fitting the LFs for this section.

Furthermore, MAMMOTH fields are selected by targeting overdensities. Overdensities could bias the field-to-field variations, which are supposed to be measured in random fields. To reduce the bias on the cosmic variance measurements caused by these overdensities, we clip out the regions preselected by MAMMOTH on each field image. Six of our fields (J0210, J0222, J0924, J1419, J1133, and J1349) are selected targeting coherently strong Lyα absorption systems (CoSLAs; Cai et al. 2016; Liang et al. 2021), and two (J0240 and J0755) are selected targeting at grouping quasars (see Z. Cai et al. 2023, in preparation). We refer to these CoSLAs/grouping quasars as the MAMMOTH targets. The MAMMOTH preselected regions are defined to cover these MAMMOTH targets, with an aperture size of 15 comoving megaparsec, which is the typical scale of protoclusters at z ∼2.2 (see, e.g., Chiang et al. 2013). These regions are treated as masks and the LAEs within these regions are excluded when sampling the pointings.

The pointings are assigned with circular shapes and fixed volume coverage. We scale up the pointing areas assigned in the NB387 field images by a factor of depth_NB400/depth_NB387 so that the volumes probed by pointings in NB387 images are the same as those in NB400 images. The centers are created uniformly within the FoV for each field. We calculate the galaxy number density n_i for the ith pointing as follows:

$$\begin{eqnarray}&&{n}_{i}={\int }_{\mathrm{log}\,{L}_{\mathrm{lim}}}^{\infty }{\phi }_{i}(\mathrm{log}\,L)d\mathrm{log}L,\end{eqnarray} \tag{ 9 }$$

where ${\phi }_{i}(\mathrm{log}\,L)$ is the best-fit Schechter function for the ith pointing. In such a way, the set {n_i } can be considered to have the same effective survey volume ${\overline{V}}_{\mathrm{eff}}$. Our reason for quantifying galaxy numbers by integrated Schechter functions is to investigate the effect of cosmic variance on Lyα LF, which takes the completeness correction into account.

Log-normal distributions provide an excellent fit for cosmological density distribution function from cold dark matter (CDM) nonlinear dynamics (Bernardeau & Kofman 1995). The skewness behavior toward zero can effectively imitate the dark matter fluctuations at the high-mass end (Coles & Jones 1991). Yang et al. (2010) investigated Lyα nebulae in the CDFS, CDFN, and COSMOS fields, and reported a strong field-to-field variance. They quantified the field-to-field variance by assuming a log-normal distribution for their Lyα blob counts. Here, we adopted a similar model, which assumes the galaxy number density n follows a log-normal distribution $n\sim \mathrm{log}-N(\bar{n},{\sigma }_{\mathrm{LN}}^{2})$, where ${\sigma }_{\mathrm{LN}}^{2}$ is the log-normal variance and is related to the actual variance by ${\sigma }_{v}^{2}=\exp ({\sigma }_{\mathrm{LN}}^{2})-1$. Since the dispersion of N can be estimated as the field-to-field variance plus the Poisson variance (i.e., ${\sigma }_{g}^{2}{\overline{N}}^{2}+\overline{N}$, see, e.g., Robertson 2010b), we then estimate σ_g using the relation

$$\begin{eqnarray}&&{\sigma }_{g}^{2}=\exp ({\sigma }_{\mathrm{LN}}^{2})-1-\displaystyle \frac{1}{\overline{n}{\overline{V}}_{{\rm{eff}}}},\end{eqnarray} \tag{ 10 }$$

where the last term corresponds to the Poisson variance. $\overline{n}$ and ${\overline{V}}_{{\rm{eff}}}$ correspond to the mean values of the galaxy number density and effective survey volume, respectively.

We fit each set of {n_i } to this log-normal distribution,

$$\begin{eqnarray}&&\phi (\mathrm{log}\,n)=\displaystyle \frac{C}{\sqrt{2\pi {\sigma }_{\mathrm{LN}}^{2}}}\exp (-\displaystyle \frac{{\left(\mathrm{log}\,n-\mathrm{log}\overline{n}\right)}^{2}}{2{\sigma }_{\mathrm{LN}}^{2}}),\end{eqnarray} \tag{ 11 }$$

where $\phi (\mathrm{log}n)$ is defined as the number counts of log n within a specified bin divided by the total number, per bin width. We also take the error ranges of n for each pointing into account, and obtain $\phi (\mathrm{log}n)$ weighted by the reciprocal of these error ranges. C is an arbitrary normalization factor. We fit the three parameters C, log $\overline{n}$ and ${\sigma }_{\mathrm{LN}}^{2}$ in Equation (11) simultaneously, and obtain a best-fit Gaussian mean log ${\overline{n}}_{\mathrm{field}}$ for each field.

To eliminate the diversity in the Gaussian mean between different fields, we take the best-fit ${\overline{n}}_{\mathrm{field}}$ from fittings for individual fields, and compute a quantity log ${\left(\tfrac{n}{\bar{n}}\right)}_{\mathrm{field}}$ for each field. This quantity should have the same mean value of 0 for all fields; we thus fit the following equation to the overall distributions of log $(\tfrac{n}{\bar{n}})$ combining multiple fields:

$$\begin{eqnarray}&&\phi (\mathrm{log}\displaystyle \frac{n}{\bar{n}})=\displaystyle \frac{C}{\sqrt{2\pi {\sigma }_{\mathrm{LN}}^{2}}}\ \exp \left(-\displaystyle \frac{{\left(\mathrm{log}\tfrac{n}{\bar{n}}\right)}^{2}}{2{\sigma }_{\mathrm{LN}}^{2}}\right),\end{eqnarray} \tag{ 12 }$$

where ${\sigma }_{\mathrm{LN}}^{2}$ and the arbitrary normalizing factor C are treated as free parameters for fitting. The lower and upper limits for the errors of log $(\tfrac{n}{\bar{n}})$ are calculated as log $\tfrac{{n}_{\mathrm{lower}}}{{\overline{n}}_{\mathrm{upper}}}$ and log $\tfrac{{n}_{\mathrm{upper}}}{{\overline{n}}_{\mathrm{lower}}}$, respectively. Furthermore, we scale back each ϕ( log $\tfrac{n}{\bar{n}}{)}_{\mathrm{field}}$ according to the normalizing factor C_field obtained in Equation (11) when combining $\{\mathrm{log}{\left(\tfrac{n}{\bar{n}}\right)}_{i}\}{}_{\mathrm{field}}$ for different fields.

In the sections above, we described our approach to obtain σ_g by randomly assigning pointings with fixed volume coverages for eight fields, and we investigated the effects on σ_g by changing the effective volume of pointings in this section.

To obtain the relation between σ_g and V_eff, we need to keep V_eff similar for every pointing during each investigation. In the sections above, we randomly created our pointing centers across the whole circular HSC FoV and clipped out the MAMMOTH-specified regions. However, this results in a relatively large scatter in the effective survey volumes V_eff since pointings at the edge lack full coverage of the field image, and pointings overlapping with masked and clipped regions have smaller V_eff than expected. To deal with this effect, we first generate pointing centers within a confined circle with a radius defined as the radius of the HSC FoV minus the aperture size set for the pointings. In this way, we ensure that every pointing fully covers the field image. Next, we repeat the process described in Section 4.1 with various sizes of pointings and obtain multiple pairs of (V_eff, log $(\tfrac{n}{\bar{n}})$), with each pair corresponding to a pointing assigned. We then make bins according to the V_eff distribution. For each V_eff bin, we calculate the mean value of the effective volumes ${\overline{V}}_{\mathrm{eff}}$ and σ_g by fitting the Gaussian to log $(\tfrac{n}{\bar{n}})$ of the pointings whose V_eff lies inside the bin.

We plot σ_g against ${\overline{V}}_{\mathrm{eff}}$ and show the results in Figure 7. The range of ${\overline{V}}_{\mathrm{eff}}$ is limited by the survey depths and effective survey areas of the field images. A too small ${\overline{V}}_{\mathrm{eff}}$ will cause the pointings to pick up a small number of LAEs or even zero LAEs, which leads to a poor Schechter fit of LFs. On the other hand, a large ${\overline{V}}_{\mathrm{eff}}$ value that covers a whole field image will create a delta function for its distribution, which is meaningless for the σ_g investigation. The smallest size assigned for the pointings is 0.2 deg² for NB400 and 0.34 deg² for NB387 (the depths of the two filters are different and the areas are changed accordingly to keep the volume constant), and the corresponding mean number of LAEs per pointing is ∼70. The largest size assigned for the pointings is 0.5 deg² for NB400 and 0.84 deg² for NB387, the corresponding mean number of LAEs per pointing is ∼170. This results in a volume coverage between 1.86 × 10⁵and 4.61 × 10⁵ Mpc⁻³ for the pointings. (Although, due to the overlapping between pointings and masked or clipped regions, the actual values of ${\overline{V}}_{\mathrm{eff}}$ are smaller than expected in general.) Detailed information on each V_eff bin is shown in Table 4.

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Table 4. Information on Each V_eff Bin

${\overline{V}}_{\mathrm{eff}}$ (105 Mpc3)	Npointings	σg	χ2
1.57	1133	${0.664}_{-0.056}^{+0.063}$	0.498
1.97	1313	${0.470}_{-0.034}^{+0.040}$	0.993
2.42	1302	${0.381}_{-0.028}^{+0.033}$	1.563
2.87	1324	${0.293}_{-0.021}^{+0.024}$	1.597
3.32	1201	${0.227}_{-0.018}^{+0.020}$	1.718
3.74	1133	${0.195}_{-0.012}^{+0.015}$	1.062
4.07	209	${0.178}_{-0.028}^{+0.035}$	0.836

Note. From left to right, the columns are the mean value of V_eff inside the bin, the number of pointings within the V_eff bin, best-fit σ_g , and the reduced χ², respectively. Note that the number of pointings in the last bin is relatively smaller than those in other bins, and this results in a larger error of the σ_g calculated, which can also be seen in Figure 7.

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In previous studies (e.g., Ouchi et al. 2008; Robertson 2010b; Konno et al. 2016), σ_g is also defined by the following:

$$\begin{eqnarray}&&{\sigma }_{g}={b}_{g}{\sigma }_{\mathrm{DM}}(R,z),\end{eqnarray} \tag{ 13 }$$

where b_g is the bias parameter of galaxies, defined as the ratio between the galaxy and matter correlation functions ξ_gg /ξ_mm (Robertson 2010b), and ${\sigma }_{\mathrm{DM}}^{2}(R,z)$ is the dark matter fluctuation in a sphere with radius R at redshift z. This suggests that σ_g only depends on the survey volume at a given redshift at large scales. σ_DM can be calculated as follows (e.g., Newman & Davis 2002; Moster et al. 2010):

$$\begin{eqnarray}&&{\sigma }_{\mathrm{DM}}^{2}(V,z)=\displaystyle \frac{1}{{V}^{2}}{\int }_{0}^{R}\xi (| \vec{{r}_{1}}-\vec{{r}_{2}}| ){{dV}}_{1}{{dV}}_{2},\end{eqnarray} \tag{ 14 }$$

where ξ is the two-point correlation function of galaxies, which can be treated as a power-law $\xi {(r)=({r}_{0}/r)}^{\gamma }$ in previous studies (e.g., Somerville et al. 2004; Moster et al. Moster 2010). This makes Equation (14) analytically solvable into a closed form of ${\sigma }_{\mathrm{DM}}^{2}=C{({r}_{0}/r)}^{\gamma }$ (Somerville et al. 2004) Since the bias parameter b_g only depends on the redshift, the volume dependency of σ_g comes from σ_DM only. In this study, we assume a simple power-law ${\sigma }_{\mathrm{DM}}^{2}\propto {V}^{\beta }$, and thus fit our σ_g and ${\overline{V}}_{\mathrm{eff},\mathrm{pointing}}$ to a simple relation ${\sigma }_{g}=k\times {\left(\tfrac{{V}_{\mathrm{eff}}}{{10}^{5}{\mathrm{Mpc}}^{3}}\right)}^{\beta }$ with some constant β and k. We find a best-fit value of $-{1.399}_{-0.156}^{+0.160}$ for β and ${1.249}_{-0.193}^{+0.213}$ for k. This is shown as the black-dashed line in Figure 7. Using this relation, we find a σ_g of ${0.052}_{-0.021}^{+0.035}$ for NB387 with a survey volume of 9.72 × 10⁵ Mpc³, and ${0.025}_{-0.011}^{+0.021}$ for NB400 with a survey volume of 16.34 × 10⁵ Mpc³. We use these two values as the field-to-field variance when reporting the LF results in Section 3 (See Section 3.3).

In previous studies investigating Lyα LFs (e.g., Ouchi et al. 2008; Robertson 2010b; Konno et al. 2016), cosmic variance is estimated from predictions of the ΛCDM model and N-body simulations. Using this approach, Moster et al. (2010) derived a model to calculate σ_g for five surveys (UDF, GOODS, GEMS, AEGIS, COSMOS), as a function of mean redshift $\bar{z}$, redshift bin size Δz, and the stellar mass of the galaxy population m_*. We apply their model with $\bar{z}=2.2$, Δz = 0.08 (consistent with our data), and $8.5\lt \mathrm{log}({m}_{* }/{M}_{\odot })\lt 9.0$ (typical stellar mass for LAEs; see, e.g., Ouchi et al. 2020), and obtain the theoretical values of σ_g predicted for five surveys at z = 2.16–2.24. We overplot this σ_g against their corresponding survey volume as blue squares in Figure 8. Since the corresponding survey volume of the COSMOS survey is ∼18.8 × 10⁵ Mpc³, which exceeds the range of our measurements, we exclude this data point. We also overplot the cosmic variance adopted by Konno et al. (2016) as a blue square in Figure 8, which is also estimated using the same method.

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Measurements on σ_g taken by our study and the fitting model described in Section 4.2 are plotted as red dots and lines in Figure 8. It is shown that our measurements of σ_g are significantly larger than the theoretical values (about two to four times). This suggests that the cosmic variance of LAEs might differ from that of general star-forming galaxies. This conclusion is consistent with Ito et al. (2021)'s results, which showed that the bias factor of LAEs might be strongly influenced by the H i distribution, and thus might be different from that of general galaxies.

Our results also suggest that the cosmic variances used by previous Lyα LF studies are likely underestimated. A possible cause is that previous approaches use correlation functions (e.g., Robertson 2010a) to estimate the galaxy bias, which only depends on the coordinates of detected galaxies, and then use the galaxy bias and σ_DM to calculate σ_g . On the other hand, our measurements are carried out by integrating the LFs, which take Lyα luminosity and completeness corrections into account. As a result, introducing Lyα luminosity and completeness correction may cause a higher cosmic variance, which is likely more realistic for LF studies.

The underestimated cosmic variance can also explain the inconsistencies in LF fittings between different narrowband imaging surveys (see, e.g., Konno et al. 2016; Sobral et al. 2017). For instance, Konno et al. (2016) adopted a value of 0.099 ± 0.017 for σ_g using predictions by simulation and ΛCDM, while our model gives a value of ${0.361}_{-0.095}^{+0.126}$ for σ_g under the same survey volume, which is significantly larger. This underestimation can further enlarge the error range of Konno et al. (2016)'s LFs, and thus provide another reason for the offset between Konno et al. (2016)'s LFs and ours presented in Figure 6. Similar arguments can be applied to other LF studies, and thus explain the inconsistencies in the LF fittings by different surveys. However, given that the cosmic variances are only added to the error bars during the calculation of the LF, this underestimation will likely leave the major conclusions unchanged, only enlarging their error ranges.