Strong Mg ii and Fe ii Absorbers at 2.2 $<~{}z~{}<$ 6.0

We selected 31 quasars with signal-to-noise ratios (S/N) greater than 10. The S/N is a mean S/N per resel measured from the ‘clean’ continuum region of the spectra without strong OH skylines or water vapor absorption features. The mean S/N values of all spectra are presented in Table 1. Given the low resolution (R$\sim$ 800) of GNIRS spectra, we did not use the lower-S/N spectra. We first fitted each quasar spectrum with a continuum. The continuum was selected interactively with knots in the absorption free wavelength region. The region between two knots was fitted with a spline curve. Then the spectrum was normalized with this continuum. We then used our algorithm to automatically search and identify metal absorbers in the normalized spectra. The absorption feature was identified with a Gaussian kernel filter, which has a rest-frame velocity FWHM between 376 km s^-1 and 600 km s^-1 (six pixels, empirically selected). If $W_{r}$ of this Gaussian kernel is greater than 0.8 Å, which is our detection limit (e.g. observe equivalent width $\sim$ 3 Å around wavelength 10,000 Å) for Mg ii line, then it was considered as an absorption feature. The $W_{r}$ was measured from the flux summation over $\Delta\lambda$ where the Gaussian kernel is within 3% of the continuum. For Mg ii doublet, the two kernels of the doublet are separated by $\sim$ 770 km s^-1 and cross-correlated with the spectrum simultaneously. The selection criteria of Mg ii candidates relate to $W_{r},\sigma(W_{r})$ and S/N in the continuum. We calculated the $\sigma(W_{r}$) by using a method by Vollmann & Eversberg (2006). For a normalized spectrum, the $W_{r}$ of an absorption line is defined as:

where $\Delta\lambda_{r}$ = $(\lambda_{2}-\lambda_{1})/(1+z)$ is the rest-frame absorption line width. $\overline{F}$ is the mean normalized flux density of the absorption line. Equation 2 can be expanded in a Taylor series:

According to Equation 2, there is $\frac{\partial W_{r}}{\partial\overline{F}}=-\Delta\lambda$. Together with $\sigma(F)$ = $\frac{\overline{F}}{\textrm{S/N}_{c}}$, we have

(S/N)_c is the average S/N per resel of $\pm$ 10 pixels adjacent to $\Delta\lambda$. The specific Mg ii candicates selection criteria are in the following:

1) $W_{r}(\lambda$2796) / $\sigma~{}(W_{r})~{}>$ 3.

2) $W_{r}(\lambda$2796) $>$ 0.8 Å and $W_{r}(\lambda$2803) $>$ 0.4 Å,

3) S/N $>$ 3 per resel in three or more contiguous pixels beyond the $\Delta\lambda$ region.

We searched for Mg ii systems in all 31 quasar spectra using the above criteria and obtained 110 candidates. Afterward, at least two Fe ii lines (at 1608, 1611, 2344, 2374, 2586, or 2600 Å) were visually inspected at the same redshift to further confirm the identified Mg ii doublet. In the end, we confirmed 32 Mg ii and Fe ii absorbers at 2.2 $<z<$ 6.0. The spectra of the absorbers are presented in Figure 1. We found that all these Mg ii absorbers have $W_{r}(\lambda$2796) $>$ 1.0 Å, and 13 of them are very strong with $W_{r}(\lambda$2796) $>$ 2.0 Å. The median $W_{r}$ is 1.86 Å. The redshift distribution (with a median $z$ = 3.743) of these absorbers is shown in Figure 2.

We also measured $W_{r}$ of an absorption candidate from a Voigt profile fit. The line is fitted using the VoigtFit package (Krogager, 2018). During the visual inspection process, we noticed that our detection algorithm detected a few absorbers as candidates but they are strongly blended, e.g. Mg ii ($\lambda$2803) lines at $z$ = 3.059 (J0002+2550), $z$ = 5.595 (J0840+5624), $z$ = 4.201 (J1250+3130) and $z$ = 4.530 (J1335+3533). Due to this blending, the $W_{r}$ would be overestimated from the flux boxcar summation. The Doppler parameter $b$ values of the fits are between 20–60 km s^-1. Because that the relatively low resolution would introduce large uncertainties on the $b$ and column density measurements, we only use Voigt fits to calculate the $W_{r}$, which is independent of spectral resolution. We compared the measurements from the fits and the flux summation. Except for systems with obvious blending, the differences between the two measurements have a median of 0.13 Å and a maximum of 0.5 Å.

We then measured the velocity width from the best-fitted parameters. The intrinsic rest-frame velocity width $\Delta v$ was determined by the instrument broadening and observed velocity width $\Delta v_{\mathrm{obs}}$. The observed $\Delta v_{\mathrm{obs}}$ was measured between the leftmost and rightmost pixels with optical depth $\tau<0.1$. The optical depth $\tau$ equals ln (1/$F$), where $F$ is the normalized flux at this wavelength. This measurement is similar to the standard $\Delta v_{90}$ definition (Prochter et al., 2006). The idea is to include all satellite absorption and to have a good representation of the kinematic extent of the absorption. We measured instrument broadening from lamp/arc lines used for wavelength calibration. The average FWHM of the arc lines FWHM_arc is roughly 376 $\pm$ 31 km s^-1(with 1 $\sigma$ error). The rest-frame intrinsic FWHM was calculated by ${\rm FWHM}=\sqrt{{\rm FWHM_{obs}}^{2}-{\rm FWHM_{arc}}^{2}}$ / ($1+z$), where FWHM_obs is the observed FWHM of the line. Then we assume the ratio of intrinsic FWHM and $\Delta v$ is the same as the ratio of observed FWHM and $\Delta v_{\mathrm{obs}}$, i.e. $\Delta v$ = FWHM$\times(\Delta v_{\mathrm{obs}}/{\rm FWHM_{obs}})$.

To minimize the low resolution impact on our velocity spread measurements, we create 50 mock Mg ii absorption spectra and convolved them into FIRE resolution of R = 6000 (i.e. 50 km s^-1) and GNIRS resolution of R $\sim$ 800 (i.e. 376 km s^-1), respectively. Then we measured the $\Delta v$ from the mock spectra with different resolutions using the same method described above. The measurements are consistent within errors $\sim$ 20 km s^-1 (see Figure 3). We also used a FIRE spectra of Fe ii ($\lambda$ 2374) system at $z=$ 3.495 towards QSO J0148+0702. We degraded the spectral resolution into R = 800. The velocity width measurement difference between the original and degraded spectra is within 30 km s^-1.

The intrinsic and observed velocity widths of all the detected absorbers are shown in Table 1 column (7) and (8), respectively. We found that 15 out of 32 absorbers have $\Delta v>300$ km s^-1. We fit the relation between $\Delta v$ and $W_{r}$ using a polynomial curve fitting technique considering the errors from two variables (see Figure 4).

We compared measurements of five overlapping sightlines (J0203+0012, J0836+0054, J0842+1218, J1148+0702 and J2310+1855) between our sample and the FIRE sample in C17. The details are presented in the following and Table 2.

All Mg ii systems in J0203+0012 and the one at $z$ = 2.299 toward J0836+0054 reported in C17 are below 1 Å, which are beyond our detection limit. The $W_{r}$, $\Delta v$ and $z$ measurements of the system at $z$ = 3.745 toward J0836+0054 are consistent. For J0841+1218, three systems were detected in this work and C17 at $z=5.050,2.540,2.392$. The $W_{r}$, $\Delta v$ and $z$ measurements are consistent with errors. For J1148+0702, we detected two systems at $z$ = 4.369 ($W_{r}(\lambda$2796) = 4.23 $\pm$ 0.52 Å) and $z$ = 3.495 ($W_{r}(\lambda$2796) = 6.50 $\pm$ 1.20 Å). The measurements of the system at $z$ = 4.369 are consistent with that in C17. The system at $z$ = 3.495 has extremely large velocity width ($>$ 800 km s^-1 for Mg ii ($\lambda$2796) line) and the absorptions are strongly blended within the doublet. Thus, the $W_{r}$ and $\Delta v$ measurements of this system inevitably have large uncertainties. We did not use this measurement when calculating the relation in Equation 5. For J2310+1855, the two systems at $z$ = 3.299 and 2.351 detected in C17 have $W_{r}<$ 1.0 Å, which are beyond our detection ability. The one at $z$ = 2.243 is located in a noisy region where the line are not able to be detected in our spectrum. We detected two systems at $z$ = 4.244 and $z$ = 4.013, which are not included in C17. The first one has $W_{r}(\lambda$2796) = 1.86 $\pm$ 0.35 Å and $\Delta v$ = 221 $\pm$ 34 km s^-1. The second one has $W_{r}(\lambda$2796) = 1.19 $\pm$ 0.21 Å and $\Delta v<$ 165 km s^-1 (see the last two panels in Figure 1). The system at $z$ = 4.244 was present in an inspection of the spectrum used in C17, but was rejected by their automated search algorithm because $W_{r}(\lambda$2803) $>$ $W_{r}(\lambda$2796) , likely because of blending in the Mg ii ($\lambda$2803) line from interloping systems at lower redshift. The system at $z$ = 4.013 has severe telluric noise in FIRE spectrum (R. Simcoe, private communication).

In summary, except for the systems have $W_{r}(\lambda$2796) $<$ 1 Å or where the spectra S/N is too low, our absorption redshfit, equivalent width and velocity spread measurements are consistent with that in C17 within errors. Though it is possible that, for systems have $\Delta v_{\textrm{obs}}<$ 400 km s^-1 (close to GNIRS resolution), our velocity widths uncertainties would be large.

For each quasar spectrum, we performed a Monte Carlo simulation by inserting uniformly distributed, virtual Mg ii doublets in the wavelength range between 8500 Å  and 20,000Å, corresponding to the absorber redshift of 2.0 and 6.2, respectively. The inserted $W_{r}(\lambda$2796) varies between 1.0 and 4.5 Å, which is the observed $W_{r}$ range of our detected Mg ii absorbers (except for the strongly blended one toward J1148+0702). For each $W_{r}$, its velocity width follows the relation $\Delta v$ = 103.37 km s^-1 Å${}^{-1}\times W_{r}$ + 399.60 km s^-1, which we measured from the $\Delta v_{\textrm{obs}}$ and $W_{r}$. Two strong water vapor regions (1.35 - 1.42 $\mu$m between $J$ and $H$ and 1.82 - 1.93 $\mu$m between $H$ and $K$ band) are discarded in the statistical analysis of $dN/dz$ and $dN/dX$. Then we use the algorithm introduced in Section 2.1 to detect the inserted virtual absorbers. We measured the uncertainties between inserted and retrieved measurements from 1000 mock inserted Mg ii systems. The measurement errors of $W_{r}$ and $z$ are 0.015 Å and 0.002, respectively (see Figure 5). This bias would be affect the final completeness significantly. The detection result is denoted as a Heaviside function $H(z,W_{r})$:

The redshift-weighted density $g(z,W_{r})$ is a function of $W_{r}$ and $z$ denoted as

where $N$ is the total number of sightlines. The total path $g(z)$ is obtained as the integral of the path density over the whole range that we selected (see Figure 7):

where $W_{0}$ is the $W_{r}$ limit. For each sightline, its completeness is the detection rate of the inserted absorbers. The completeness of pathlength is a function of redshift and $W_{r}$,

We show the pathlength-averaged completeness $\overline{C}(z,W_{r})$ for the selected 31 sightlines with $W_{r}(\lambda$2796) $>$ 0.8 Å, 1 Å and $>$ 3 Å in Figure 6. For $W_{r}(\lambda$2796) $>$ 1 Å  the completeness is around 40% $\sim$ 80% in the $J$ band (1.17–1.37 $\mu$m), and around 20% $\sim$ 60% in the $H$ band (1.49–1.80 $\mu$m). The low completeness in the $H$ band is due to the contamination of strong sky lines.

Even if the S/N of a spectrum is high enough to detect weak lines, visual inspection may miss some weak absorbers. The probability for users to confirm true absorbers is defined as user acceptance. In M13 and C17, the user acceptance rate is defined as a function of S/N and has been considered in their calculations. M13 suggests that when S/N $>$ 10, the user acceptance is close to 1 and the rejection fraction of false-positive candidates is close to 0.

Zhu & Ménard (2013) identified 40,000 Mg ii absorbers from SDSS at 0.4 $<z<$ 2.3. By requiring the simultaneous detection of Fe ii lines ($\lambda\lambda$ 2344, 2383, 2586, 2600) for each Mg ii absorption line, they recovered close to 100% of strong absorbers in the Pittsburgh catalogs (Quider et al., 2011). In this work, we focus on the strong system with $W_{r}(\lambda$2796) $\geq 1$ Å  for which we also confirm detection of Fe ii candidate lines at the same absorption redshift. Given this and that our database consists of spectra with S/N $>$ 10, we assume that our visual inspection is correct at a rate of ca 95%. In the case that one or two Fe ii candidate lines reside in the water vapor region and are affected significantly by OH lines, the bias would be within this rate given the wide wavelength coverage of Fe ii candidate lines. We calculated the Fe ii lines association with strong Mg ii systems in M13. We found that there are 35 out of 37 (94.5%) strong Mg ii systems ($W_{r}(\lambda$2796) $>$ 1 Å) associated with at least three clear Fe ii lines. Additionally, spurious detection caused e.g. by C iv doublets $W_{r}(\lambda$1548 $>$ 0.5 Å) are not detected in our spectra. Therefore, the false positive detection rate from the weaker lines is close to 0.

We calculate the incompleteness-corrected line-of-sight density of strong Mg ii absorbers at different redshift bins. The results at four redshift bins between 2.2 and 6.0 are shown in Figure 8 and Table 3. The relation between $dN/dz$ and redshift can be expressed as,

where $N_{0}$ is the normalization and $\beta$ is the slope. We apply the Maximum Likelihood Estimation (MLE) method to the relation and find that $N_{0}$ and $\beta$ are 1.882$\pm$3.252 and –0.952$\pm$1.108, respectively.

Previous studies (e.g., M13 and C17) have found that the $dN/dz$ of strong Mg ii absorbers generally decreases with increasing redshift at $2<z<6$. In particular, the $dN/dz$ or $dN/dX$ at $z>4.5$ drops rapidly (see Figure 8). Codoreanu et al. (2017) studied Mg ii systems using four quasars from VLT-Xshooter and found that the $dN/dz$ is relatively flat at $2<z\leq 4$. This is likely due to the larger uncertainties from their small sample, as they have pointed out in the paper. As shown in Figure 8, our results are consistent with the previous results within errors. The trend at $2<z<4$ is not clear due to the large errors, but the density decreases significantly at $z>4.5$.

The evolution of Mg ii incidence has implications for the origin of Mg ii absorbers. One possible scenario is that superwinds give rise to strong Mg ii absorbers in starburst galaxies (Bond et al., 2001; Heckman, 2001; Bouché et al., 2006). Superwinds are gas bubbles generated by starbursts. They escape from gravitational wells and then blow into galaxy halos. Low-ions such as Mg ii and Na i reside in the shells of these superwinds. Another possible scenario is that strong Mg ii systems would reside in a galaxy groups environment. For example, Gauthier (2013) find their ultra-strong Mg ii absorber ($W_{r}(\lambda$2796) = 4.2 Å) at $z$ = 0.5624 is associated with five galaxies within 60 kpc.

To further investigate the possible scenarios for the origin of our strong Mg ii systems, we compare the velocity widths of our Mg ii systems with those at similar and lower redshift. We compare with three samples in the literature: a blindly-searched Mg ii sample from SDSS DR12 (Zhu & Ménard, 2013) at 0.4 $<z<$ 2.3, Mg ii systems associated with a DLA sample from the XQ-100 survey at 2 $<z<$ 4 (Berg et al., 2017), and Mg ii systems traced by a neutral atomic carbon (C i) sample at 1.5 $<z<$ 2.7 (Zou et al., 2018). The comparison is plotted in Figure 9.

We found that the velocity widths of our Mg ii absorbers are larger than those associated with DLAs with similar equivalent widths at 2$<z<4$, this feature is also seen in C17 strong Mg ii systems. In the C17 sample of 287 absorbers, 104 of which have $W_{r}(\lambda$2796) $>$ 1 Å, and 58 out of 104 have $\Delta v>$ 300 km s^-1. Note that the $\Delta v$ given in C17 is defined as the total velocity interval under the continuum. Even if only 90% of their intervals are considered, half of the strong absorbers still have $\Delta v>$ 300 km s^-1. In the DLA-tracing Mg ii sample, 18 out of 29 Mg ii absorbers have $W_{r}>$ 1 Å  but only one has velocity width greater than 300 km s^-1. Moreover, large velocity widths for Mg ii absorbers are also seen in the C i-tracing Mg ii absorbers at 1.5 $<z<$ 2.7. The velocity widths were measured by the same method decribed in Setion 2.2. In the 17 systems of C i-tracing Mg ii absorbers, 15 of which have $W_{r}>$ 1 Å  and 13 out of 15 (87%) have $\Delta v>$ 300 km s^-1. C i has been shown to effectively trace molecular and cold gas at $z\sim 2$, and thus star formation activities. As discussed in Zou et al. (2018), the C i systems can be highly disturbed by superwinds or the interactions between several galaxies. Therefore, large velocity widths of our Mg ii absorber suggest that our systems are potentially strongly affected by the galactic superwinds and/or the interaction within galaxy groups.

The two dashed lines in panel (b) are for $W_{r}(\lambda$2803) /$W_{r}(\lambda$2796) = 1 and 0.5 respectively. The ratio greater than one implies that Mg ii doublets are strongly saturated. In our sample, about 42% of the absorbers have this line ratio greater than 0.8. This fraction is $\sim 55$% in the 0.4 $<z<$ 2.3 SDSS sample. Our Mg ii systems are slightly less saturated than the absorbers at $z<$ 2.3.

Another piece of possible supportive evidence is the equivalent width ratio of Fe ii and Mg ii lines ($W_{r}(\lambda$2600) /$W_{r}(\lambda$2796) ). We compare our sample with the DLA-tracing Mg ii sample at 2$<z<4$ and a sample from Rodríguez Hidalgo et al. (2012) at low redshift. Rodríguez Hidalgo et al. (2012) analyzed 87 Mg ii system with $W_{r}(\lambda$2796) $>$ 0.3 Å at 0.2 $<z<$ 2.5. They found that strong systems ($W_{r}(\lambda$2796) $>$ 1 Å) do not have small $W_{r}(\lambda$2600) /$W_{r}(\lambda$2796) ratios in their sample. In panel (c) of Figure 9, our sample covers a wide range of the $W_{r}(\lambda$2600) /$W_{r}(\lambda$2796) ratios. In particular, four systems among the strongest Mg ii absorbers have smaller ratios ($W_{r}$ ($\lambda$2600)/$W_{r}$ ($\lambda$2796) $<$ 0.5) than most of the other systems in the sample. The small $W_{r}(\lambda$2600) /$W_{r}(\lambda$2796) values can be due to many reasons, e.g. kinematics evolution, dust depletion and intrinsic [Mg/Fe] abundance in the gas phase. We here propose that the kinematic evolution of the profiles of the very strong absorbers is a possible reason. Which means, at high redshift, the number of unresolved sub-components associated with strong Mg ii absorbers may grow.

We have a small sample of HST snapshot images observed in the WFC3/IR F105W band (Program ID: 12184, PI: X. Fan). The images cover 7 of our quasars with strong absorbers with $W_{r}(\lambda$2796) $>$ 1.5 Å or $\Delta v>$ 300 km s ^-1, namely J0002+2550, J0050+3445, J0842+1218, J1207+0630, J1335+3533, J2310+1855. The median redshift of the absorbers is 3.982. For each quasar, there is at least one candidate galaxy with $\geq$7$\sigma$ detection ($\leq$ 25.5 mag) within a 5$\arcsec$ circular radius in the HST image (see Table 4 and Figures 10, 11). The photometry is performed using SExtractor (Bertin & Arnouts, 1996). All these galaxies are fainter than 24 mag in F105W, which is fainter than $L^{*}$ galaxies at $z\sim$ 4 ($m^{*}$ = 23.31 $\pm$ 0.08) (Bouwens et al., 2015). The median magnitude in F105W for our sample is 24.78.

The rest-frame band of F105W at the median redshift of Mg ii ($z$ = 3.743) is $U$ band. The LBGs UV-continuum slope $\gamma$ ($f_{\lambda}$ = $\lambda^{\gamma}$) measured by Bouwens et al. (2012) at $z\sim 4$ is around $-$2, therefore we have $u-b\sim$ 0. If the galaxy is a passive galaxy, it is unable to be detected with present images. We then obtained $L_{B}/L_{B}^{*}=$ 0.25, where $L_{B}$ and $L_{B}^{*}$ are the $B$ band luminosity of our galaxy candidates and $L^{*}$ galaxies, respectively. The result is consistent with the estimates of the Mg ii associated galaxy luminosity in M13.

Because we only have a single photometric band measurement, we do not know their redshifts or whether they are associated with the detected absorption systems. We search the archival images of the Dark Energy Camera Legacy Survey (DECaLS) (Dey et al., 2019) and find that four quasar fields are covered by DECaLS (J0002+2550, J0842+1218, J1207+0630, J1250+3130). None of the above galaxies are detected in the $grz$ bands. The 2$\sigma$ detection limit in the $g$ band (the deepest band) is roughly 25.5 mag (see also Table 4). The red $g$–F105W color implies that these galaxies are likely at high redshift. We further estimate the surface density of $z>$ 2.5 galaxies brighter than 25.5 mag in a few HST  fields (Bouwens et al., 2015) and find that the expected number of random galaxies in a 5$\arcsec$ circular area is about 0.1. This is significantly lower than our density of $>1$, suggesting that the galaxies detected in F105W above are likely associated with the strong absorbers. We can see that within 50 kpc of each absorber, we detected maximum two galaxy candidates. If multiple galaxies are associated with strong Mg ii systems at high redshift, we need higher resolution spectra to disentangle the absorbing gas kinematic structure and deeper images to search for galaxy candidates nearby.

We have limited our search of galaxy candidates within a 5$\arcsec$ (42.36 kpc at $z$ = 2.2) radius, and we detected at least one candidate for each absorber within an impact parameter $D=50$ kpc. This median distance of 23.31 kpc is smaller than the median distance, $<D>$ = 48.7 kpc found in the local Universe (Schulze et al., 2012; Nielsen et al., 2013a, b, 2018), suggesting that strong Mg ii absorbers likely have smaller impact parameters at higher redshift. We can also calculate the absorbing halo gas size from the measured $dN/dX$ and $L_{B}/L_{B}^{*}$ using the relation from Kacprzak et al. (2008). The comoving line density $(dN/dz)$ can be expressed as the product of the absorber physical cross-section $\sigma$ and volume co-moving number density $n(z)$,

where c/${H_{0}}$ is the constant of proportionality and

The gas cross-section is expressed as $\sigma$ = $\pi R_{x}^{2}$, where $R_{x}$ is the absorbing gas halo size. The volume number density $n(z)$ can be expressed as a function of associated galaxy luminosity:

where $\Phi^{*}$ is the number density of $L^{*}$ galaxies in the galaxy luminosity function. $\Gamma(x,y)$ is an incomplete Gamma function with $x=2\beta-\alpha+1$, where $\alpha$ is the faint-end slope of Schechter function and $\beta$ is the factor of a relation between $R_{x}$ and associated galaxy luminosity (Steidel et al., 1995): $R_{x}=R_{*}\times(L/L^{*})^{\beta}$. $y$ is the ratio of the detected galaxy minimum luminosity to $L^{*}$. Therefore $R_{x}$ is:

We take $\alpha=-1.64\pm 0.04$ from Bouwens et al. (2015), $\beta=0.35$, our measured $y=L_{B}/L_{B}^{*}=$ 0.25 and $dN/dz$ at $<z>$ = 3.5, $\Phi^{*}_{B}$ = 1.97 ${}^{+0.34}_{-0.28}\times 10^{-3}$ Mpc ^-3 (Bouwens et al., 2015) to calcualte the $R_{x}$. The $\beta$ value is from Chen et al. (2010), which analysed 47 Mg ii associated galaxies at $z<$ 0.5 and with 0.1 $<$$W_{r}(\lambda$2796) $<$ 2.34 Å. We assume the slope does not change at higher redshift. The $R_{x}$ is then estimated as follows:

With our measured $L_{B}/L_{B}^{*}=$ 0.25, $R_{x}$ is smaller than the possible $D$. This is based on the assumption that the covering fraction $f_{c}$ of the gas is unity. Covering fraction of the absorbing gas is defined as the ratio of absorbers associated galaxies and all galaxies at the same redshift bin. Lan (2020) found that covering fraction of strong Mg ii systems evolves with redshift at 0.4 $<z<$ 1.3, similarly to the evolution of star-formation rate of galaxies. In the study of Chen et al. (2010), the Mg ii-absorbing halo gas covering fraction is 70% for $W_{r}(\lambda$2796) $>$ 0.3 Å. Nielsen et al. (2018) studied 74 galaxies at 0.113 $<z<$ 0.888 with $\langle$$W_{r}(\lambda$2796) $\rangle$ = 0.65 Å, and found $f_{c}$ = 0.68 for isolated galaxies. Since we have 1-2 galaxy candidate within 5$\arcsec$ of the absorber, we adopt the $f_{c}$ = 0.68. The covering fraction corrected size $R_{*}$ is 9.23 kpc ($R_{x}^{2}$ = $f_{c}R_{*}^{2}$), which is still smaller than the $<D>$ = 23.31 kpc.

In summary, we searched for associated galaxies around our strong Mg ii absorbers at $z$ = 3–5.1 within 50 kpc and found 1–2 candidates for each absorber. The galaxy candidates are brighter than 25.5 mag and have a median magnitude of 24.78 in F105W band. Theses candidates have a median impact paramter $<D>$ = 23.31 kpc, which is smaller than that at $z<$ 1. If we assume that the $R_{x}$–$L$ slope and $f_{c}$ for strong Mg ii absorbers at $z$ = 3–5.1 are similar to that at $z<$ 1, with a fixed associated galaxy luminosity $L=0.25L*$ and a covering fraction of $f_{c}$ = 0.68, the $f_{c}$-corrected absorbing halo gas $R_{*}$ is smaller than the $<D>$. In another word, within 50 kpc, high redshift strong Mg ii absorbers tend to have a more disturbed environment but smaller halo size than that at $z<1$.

In this subsection, we present a few individual systems with peculiar absorption features or having images in other bands.

J0050+3445. We detected a Mg ii absorber at $z$ = 3.435 with $W_{r}(\lambda$2796) =3.44 $\pm$ 0.88Å. There are two galaxy candidates within 5$\arcsec$ from the quasar, labeled as 1 and 2 in Figure 10. We measure the $g$ and $r$ band magnitudes of the two objects using archived Canada France Hawaii Telescope (CFHT) MegaPrime images. The magnitudes of galaxy 1 are $g$ = 25.44 $\pm$ 0.10 and $i$ =25.64 $\pm$ 0.11. The magnitudes of galaxy 2 are $g$ = 25.43 $\pm$ 0.10 and $i$ =24.89 $\pm$ 0.13. The Ly$\alpha$ emission line at $z$ = 3.435 is redshifted to 5391 Å ($g$ band), so galaxy 2 with $g-i$ = 0.54 is more likely to be the absorber.

J2310+1855. We detected strong Mg ii and Fe ii at $z$= 4.244. In Figure 10 there are two galaxies within 5$\arcsec$ (25 kpc) from the quasar. The magnitudes of galaxy 1 are $r$ = 25.55 $\pm$ 0.14, $z$ =24.34 $\pm$ 0.25, and $F105W$ = 24.26 $\pm$ 0.05. The magnitudes of galaxy 2 are $r$ = 26.18 $\pm$ 0.25 and $F105W$ = 25.10 $\pm$ 0.11. It is not detected in $z$. Based on their colors, galaxy 1 is more likely to be the absorber host at $z=4.244$. In addition, there is a bright object next to galaxy 1. But its $m_{z}$ = 22.15 is much brighter than $L^{*}$ at $z\sim$ 4, so we did not consider it.

J1148+0702. We detected Mg ii and Fe ii at $z$= 4.369 and $z$ = 3.495, the latter one has extremely large velocity spread for a Mg ii doublet. This one at $z$ = 4.369 has a very strong Mg ii ($W_{r}(\lambda$2796) $>$ 4 Å ) absorber with the strongest Fe ii ($W_{r}(\lambda 2600)$ = 4.45 $\pm$ 0.82 Å ) absorption in our sample. Both Mg ii and Fe ii lines are strongly saturated. As discussed in Joshi et al. (2017), strong Mg ii and Fe ii in the same system may indicate star formation nearby. We do not have HST images for this quasar. The Mg ii and Fe ii absorption profiles are presented in Figure 12.