# Time Variation of Fine-Structure Constant Constrained by [O III] Emission-
Lines at $1.1<z<3.7$

Ge Li1, Luming Sun2, Xiangjun Chen1, and Hongyan Zhou3,1
1Department of Modern Physics,University of Science and Technology of China,
Hefei, Anhui, 230026, China.
2Department of Physics, Anhui Normal University, Jiuhuanan Road 189, Wuhu
241002, China
3Polar Research Institute of China, 451 Jinqiao Road, Shanghai, China E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

[O III]$\lambda\lambda$4960,5008 doublet are often the strongest narrow
emission lines in starburst galaxies and quasi-stellar objects (QSOs), and
thus are a promising probe to possible variation of the fine-structure
constant $\alpha$ over cosmic time. Previous such studies using QSOs optical
spectra were limited to $z<1$. In this work, we constructed a sample of 40
spectra of Ly$\alpha$ emitting galaxies (LAEs) and a sample of 46 spectra of
QSOs at $1.09<z<3.73$ using the VLT/X-Shooter near-infrared spectra publicly
available. We measured the wavelength ratios of the two components of the
spin-orbit doublet and accordingly calculated $\alpha(z)$ using two methods.
Analysis on all of the 86 spectra yielded $\Delta\alpha/\alpha=(-3\pm 6)\times
10^{-5}$ with respect to the laboratory $\alpha$ measurements, consistent with
no variation over the explored time interval. If assuming a uniform variation
rate, we obtained $\alpha^{-1}{\rm d}\alpha/{\rm d}t=(-3\pm 6)\times 10^{-15}$
yr-1 within the last 12 Gyrs. Extensive tests indicate that $\alpha$ variation
could be better constrained using starburst galaxies’ spectra than using QSO
spectra in future studies.

††pubyear: ††pagerange: Time Variation of Fine-Structure Constant Constrained
by [O III] Emission-Lines at $1.1<z<3.7$–LABEL:lastpage

##  1 Introduction

Dirac (1937) proposed that fundamental physical constants may change with the
evolution of the universe. The fine structure constant $\alpha$ is defined as:

$\rm\alpha=\frac{e^{2}}{4\pi\varepsilon_{0}\hbar c},$ (1)

where $e$ is the electron charge, $\varepsilon_{0}$ is the permittivity of
free space, $\hbar$ is the Planck constant, and $c$ is the speed of light. It
is a fundamental physical constant charactering the strength of the
electromagnetic interaction between elementary charged particles. It also
quantifies the gap in the fine structure of the spectral lines of atom. This
gap is proportional to the energy of main level by a factor of $\alpha^{2}$
(e.g., Dzuba et al., 1999; Uzan, 2003). Thus variation of $\alpha$ over time
could be directly constrained by comparing the wavelengths of fine-structure
splitting of atomic lines from another epoch with that from today. Possible
variation of $\alpha$ over time has been studied with the help of measurements
from laboratories, geology and astronomy (see review by Uzan, 2003, 2011).
Among these, the studies involving astronomical spectra can provide possible
variation of $\alpha$ with the longest lookback time and the greatest spatial
spans, in which case trivial variation of $\alpha$ with spacetime might
accumulate to a measurable level.

Observational studies using astronomical spectroscopy began in the 1950s
(e.g., Savedoff, 1956). From the 1960s, doublet lines of [O III]
$\lambda\lambda$4960, 5008 in the spectra of galaxies and QSOs have been used
to study the variation of $\alpha$ over time. Bahcall & Schmidt (1967)
analyzed the [O III] doublet in five QSOs at $0.17<z<0.26$ and found no
variation in $\alpha$ as they measured
$\Delta\alpha/\alpha=\frac{\alpha(z)}{\alpha(0)}-1=(1\pm 2)\times 10^{-3}$,
where $\alpha(z)$ and $\alpha(0)$ are the $\alpha$ values at redshift $z$ and
in laboratory. The [O III] doublet method was sidelined for decades and
returned in the 2000s when the SDSS project produced a large sample of QSO
spectra with resolution $R\approx 2000$. The final precision can be greatly
improved by averaging the results of the measurements from a large number of
QSOs. Bahcall et al. (2004) measured $\Delta\alpha/\alpha=(0.7\pm 1.4)\times
10^{-4}$ using [O III] doublet in spectra of 42 QSOs at $0.16<z<0.80$ from the
SDSS Early Data Release. Gutiérrez & López-Corredoira (2010) measured
$\Delta\alpha/\alpha=(2.4\pm 2.5)\times 10^{-5}$ using 1568 QSOs at $z<0.8$
from SDSS Data Release 6. Rahmani et al. (2014) measured
$\Delta\alpha/\alpha=(-2.1\pm 1.6)\times 10^{-5}$ using 2347 QSOs at
$0.02<z<0.74$ from SDSS Data Release 7. Albareti et al. (2015) measured
$\Delta\alpha/\alpha=(0.9\pm 1.8)\times 10^{-5}$ using 13,175 QSOs at $z<1$
from SDSS Data Release 12. In addition to the [O III] doublet, other doublet
emission lines had been attempted, such as [Ne III] $\lambda\lambda$3870,3969,
to study the $\alpha$ variation at $1<z<1.5$. However, Albareti et al. (2015)
demonstrated that the accuracy is limited to be worse than $10^{-3}$ due to
systematic errors caused by the contamination from H$\epsilon$ to [Ne III]
$\lambda$3969.

At $z>1$, when the [O III] doublet is redshifted beyond the wavelength ranges
of optical spectrometers, the studies on the variation of $\alpha$ using
astronomical spectroscopy can be carried out using alkali doublet absorption
lines in the ultraviolet spectra of QSOs (e.g. Bahcall et al., 1967; Murphy et
al., 2001). With a principle similar to the [O III] doublet method,
measurements of $\Delta\alpha/\alpha$ involving this alkali-doublet method
have reached a precision of $\sim 10^{-5}$, slightly better than the [O III]
doublet method and did not detect variation in $\alpha$ either.

The most precise astronomical limits on $\alpha$ variation till now were
achieved using many-multiplet method (e.g., Dzuba et al., 1999; Webb et al.,
2001). In this method, the wave number of a spectra line, in producing which
the fine-structure splitting of an energy level is involved, can be expressed
as $\omega_{z}=\omega_{0}+q_{1}x+q_{2}y$ (e.g., Dzuba et al., 1999), where
$\omega_{z}$ and $\omega_{0}$ are the wave numbers in vacuum at redshift $z$
and at today, respectively, and $x=(\frac{\alpha_{z}}{\alpha_{0}})^{2}-1$,
$y=(\frac{\alpha_{z}}{\alpha_{0}})^{4}-1$. The parameters $q_{1}$ and $q_{2}$
can be computed using relativistic many-body perturbation and experimental
data, while their values for light elements and heavy elements can differ by a
factor of tens to thousands. This great diversity of the values of $q_{1}$ and
$q_{2}$ can amplify possible variation of $\alpha$ and then reveal it in the
differences between the wave numbers of the spectral lines of different atoms
and ions. Multiple absorption lines in the damped Ly$\alpha$ absorption
systems in QSOs’ spectra, including Mg II, Fe II, and others, were used. In
practice, the absorption lines generally have complex profiles, and therefore,
a fitting technique involving many Voigt profiles is used. All absorption
lines in an absorption system are fitted simultaneously, in which every line
is decomposed into multiple Voigt components, and the number of free
parameters is minimized by linking the physically related absorption
components of different lines. Some early works claimed to have found evidence
of significant differences between the past and present values of $\alpha$
(e.g., Webb et al., 2001; Murphy et al., 2003). However, subsequent works
(e.g., Murphy et al., 2007; Griest et al., 2010; Whitmore & Murphy, 2015)
indicated that the early results were affected by wavelength distortion. After
taking the wavelength distortion into account, later works, including those
using QSO spectra with resolutions of $40,000\sim 60,000$ aiming at several to
hundreds of absorbers at $0.2<z<4$ (e.g., King et al., 2012; Evans et al.,
2014; Songaila & Cowie, 2014; Murphy & Cooksey, 2017) and those using spectra
of bright QSO HE 0515-4414 with resolutions up to 145,000 (e.g., Kotuš et al.,
2017; Milaković et al., 2021; Murphy et al., 2022a), measured
$\Delta\alpha/\alpha$ with precisions of $(0.8\sim 4)\times 10^{-6}$ and did
not detect variation in $\alpha$ over time. Recently, Murphy et al. (2022b)
applied the many-multiple method to the absorption spectra of neighbouring
stars within 50 pc, finding $\Delta\alpha/\alpha$ consistent with 0 with a
precision of $1.2\times 10^{-8}$.

Though having archived great precision, possible problems lurking in works
using the many-multiplet method have been pointed out (see a review by Webb et
al., 2022), which may emerge from the techniques for correcting wavelength
distortion (e.g., Dumont & Webb, 2017), the physically assumed linking that
the Voigt fitting technique relies on (e.g., Levshakov, 2004), the technical
details of the fitting process (e.g., Bainbridge & Webb, 2017; Lee et al.,
2023), the unconsidered systematic errors (e.g., Lee et al., 2021), and other.
These problems can lead to biases in the best-fitting $\Delta\alpha/\alpha$
values or to underestimations of their errors, and some of the problems lack
clear solutions despite much effort. Anyhow, the results of the many-multiplet
method need to be tested with entirely different methods. The [O III] doublet
method relies on fewer assumptions and suffers from fewer systematic errors.
No assumptions on chemical composition, ionization state, and distribution of
energy levels are required, because the [O III] doublet lines originate in the
downward transitions from the same upper level of the same ion. Also, there is
no need to decompose one [O III] emission line into multiple components by
assuming that it originates from multiple clouds, as was usually the routine
in the study of absorption lines, for the [O III] doublet lines must have the
same profile. The [O III] doublet method is more tolerant of the wavelength
distortion because of the small wavelength range used in the measurement
(doublet line interval $47.9(1+z)$ Å). At present, although the precision of
the $\Delta\alpha/\alpha$ measured by the [O III] doublet method is much lower
than that measured by the many-multiplet method, improvement in the future can
be achieved by using more spectra of QSOs and starburst galaxies with better
quality. In addition, current measurements based on absorption lines were
limited to $z<7.1$ (Wilczynska et al., 2020). Studying the $\alpha$ variation
using absorption lines at higher redshifts is challenging because QSOs are
extremely rare in the early universe, and the normal galaxies’ continua are
too weak. Fortunately, many starburst galaxies have been discovered in the
early universe, and their [O III] doublet or other doublet emission lines can
be used.

In studies of $\alpha$ variation using the [O III] doublet method, works with
large sample sizes ($N>20$) have used only spectra of QSOs, not starburst
galaxies. This may be because QSOs have higher [O III] luminosities, or it may
be related to the observational strategy of the SDSS project. In reality,
starburst galaxies far outnumber QSOs in the universe. They have narrower [O
III] emission lines than QSOs, which is advantageous for improving the
accuracy of $\Delta\alpha/\alpha$ measurement and may compensate for their
disadvantage of lower [O III] luminosity. If this can be proved, the number of
available spectra could be significantly increased by including starburst
galaxies in studies of $\alpha$ variation, and the precision of the final
results improved. In addition, previous works only used optical spectra
(wavelength $<1$ $\mu$m), and hence the $\Delta\alpha/\alpha$ measurements
were limited to $z<1$. Applying this method to $z>1$ demands infrared
spectroscopy.

The XShooter(Vernet et al., 2011) is an intermediate-resolution echelle
spectrometer mounted on UT2 of the Very Large Telescope (VLT) since 2009.
Entering celestial radiation is split into three arms optimized for the
UVB,VIS and NIR wavelength ranges by dichroic mirrors. Each arm has an echelle
and a detector, and the wavelength ranges of the UVB, VIS and NIR arms are
299–556, 534–1020 and 994–2478 nm, respectively. By splicing the spectra from
the three arms, a continuous wavelength coverage from 300 to 2470 nm can be
achieved. The XShooter can work under the long-slit or the integral field unit
modes. For the long-slit mode, slits with widths of 0.4 to 1.5$\arcsec$ are
available in the NIR arm. The slits with widths of 0.6, 0.9 and 1.2 $\arcsec$
are often used, producing NIR spectra in resolutions of 8030, 5570 and 4290,
respectively. After an observation finishes on the Xshooter, the raw data will
be processed automatically by the pipeline (Modigliani et al., 2010), giving
the extracted spectra of the targets. Overall, the XShooter has a unique
wavelength coverage of up to 2.47$\mu$m with a moderate spectra resolution,
and it is highly sensitive due to the large aperture of the VLT and the high
efficiency of the spectrometer, not to mention the pipeline makes accurate and
reliable wavelength calibration. Thus, spectra taken by the XShooter are
suitable for studies of the variation of $\alpha$ via the [O III] doublet
method at $z>1$.

In this work, we measured the $\alpha$ variation by the [O III] doublet method
using XShooter spectra of a sample of Lyman-$\alpha$ emitters (LAEs) and QSOs.
LAEs are distant galaxies selected by their strong Lyman-$\alpha$ emission
lines, among which most are starburst galaxies, and few are obscured active
galactic nuclei. We achieved a measurement at $1.1<z<3.7$ for the first time
with this method. In addition, we found that $\alpha$ variation can be better
constrained by using LAE spectra than QSO spectra. Throughout the paper, we
assume a $\Lambda$CDM cosmology and use the cosmological parameters obtained
by Bennett et al. (2014), which are $H_{0}=69.6$ km s-1 Mpc-1,
$\Omega_{M}=0.286$, and $\Omega_{\Lambda}=0.714$.

##  2 Data Reduction and Sample Selection

###  2.1 Data reduction

We collected information on the normal, the large, the small, the ToO and the
DDT XShooter programs in period 84 $-$ period 104 between 2009 and 2020.
Projects related to LAEs and QSOs at $1<z<4$ and observed in the long-slit
mode were selected. The redshifts of the included targets were then determined
roughly by inspecting their optical/NIR spectra visually. Although these rough
redshifts were only used for selecting targets, their accuracy was guaranteed
later by the more precise redshifts obtained in further analysis by fitting
the spectra. The uncertainties of the rough redshifts are on the 0.01 level,
which is sufficient for selecting reliable targets at this stage. Targets with
redshifts between 1.07 and 3.77 were selected, ensuring the NIR spectra cover
the wavelength range of 4800–5200 Å in the rest frame.

We obtained the two-dimensional (2D) and one-dimensional (1D) NIR spectral
data of the selected targets from the ESO database111The spectral data
products query form, http://archive.eso.org/wdb/wdb/adp/phase3_spectral/form ,
which had been processed by the XShooter pipeline (version is related to the
observation time). These data are referred to as pip-2D and pip-1D hereafter.
The pip-2D data underwent processes including wavelength calibration, CCD
cleaning, target tracking and straightening, combining different exposures,
removing cosmic rays, and merging spectra from different orders. They had also
been interpolated to a wavelength grid with a fixed interval of 0.6 Å. Based
on the pip-2D data, the pip-1D data underwent spectral extraction and flux
calibration. Instead of using the pip-1D data directly, we used the pip-2D
data to extract the final 1D spectrum. In this section, we briefly introduce
the typical extraction approach and display more details and the treatment of
some exceptional cases in Appendix A.

Dither mode was used in most observations, and under this mode, a target will
experience several consecutive exposures. The pipeline combines the data from
the group of multiple exposures of a target, leaving three parallel images of
the target in its pip-2D data, in which one image in the middle has positive
flux, and the other two have negative fluxes. The extraction of such pip-2D
data of a target is as follows. First, we determined the aperture for
extraction by fitting the brightness profiles of the three images at the
spatial direction of the 2D data. Second, we extracted the spectra of the
three images and combined them. Third, we made flux calibration and telluric
correction for the combined spectrum. Finally, we identified the unreliable
pixels and assigned marking masks for them. Pixels seriously affected by sky
emission lines (SELs) or telluric absorption lines (TALs) were masked in this
step, and Figure 1 shows the examples. In addition, if a target has more than
one group of exposures in a project and hence several spectra were extracted,
we would combine these extracted spectra into one.

We extracted 95 LAE and 601 QSO spectra (including obscured QSOs). Note that
if one target was observed in different projects, we extracted a spectrum for
each project.

Figure 1: Left: Examples of masks caused by SELs. We identified strong lines
in the sky emission spectrum and labelled the pixels strongly affected by them
in grey. Right: Examples of masks caused by TALs. We show the normalized
telluric absorption spectrum (1 for unabsorbed, 0 for fully absorbed). Pixels
with a value less than 0.5 (red dotted line) on the normalized absorption
spectrum are masked (grey).

###  2.2 Spectral Analysis

Aimed to build a sample with spectra that have strong and clean [O III]
doublet lines, we fit the [O III] doublet lines and their adjacent regions in
each spectrum for the present. The fitting model includes one component for
the continuum and one for the [O III] doublet. The former component is a
linear function for an LAE and a second-order polynomial for a QSO. We did not
set a separate component for the H$\beta$ broad emission line in a QSO
spectrum because it has been included in the second-order polynomial. For the
latter component, a sum of multiple Gaussian components was used for each [O
III] line. The $\lambda$4960 and $\lambda$5008 lines contain the same number
of Gaussian components, and for each Gaussian component, we set:

$\left\\{\begin{array}[]{lr}{w_{i,5008}=w_{i,4960}\times(1+\eta)}\\\
{\sigma_{i,5008}=\sigma_{i,4960}\times(1+\eta)}\\\
{f_{i,5008}=f_{i,4960}\times A}\end{array}\right.$ (2)

where $w_{i,5008}$, $\sigma_{i,5008}$ and $f_{i,5008}$ are the central
wavelength, standard deviation and flux of the $i$th Gaussian component of the
$\lambda$5008 line, and those with a subscript 4960 are the corresponding
parameters of the $\lambda$4960 line. The definitions of $\eta$ and $A$
followed Bahcall et al. (2004), where $1+\eta$ is the ratio of the wavelengths
of the doublet, and $A$ is the ratio of the fluxes of the doublet. We fixed
$\eta$ as the theoretical value $\eta_{0}$ in this stage to distinguish the
doublet lines when blended or avoid misidentification of the $\lambda$4960
line when weak. We adopted the vacuum wavelengths of 4960.295 and 5008.240 Å
for the doublet, which were taken from the NIST Atomic Spectra
Database222https://www.nist.gov/pml/atomic-spectra-database, and hence
$\eta_{0}=0.00966576$. For spectra with $>10\sigma$ detections of both doublet
lines, fixing $\eta$ to $\eta_{0}$ generally leads to an increase of the
signal-to-noise ratios (SNR) of the $\lambda$4960 by $<$20%, and an increase
of that of the $\lambda$5008 by $<$5%. Here an SNR was calculated as the flux
of a line divided by its error.

The rest-wavelength ranges for fitting are different for LAEs and QSOs. In
most cases, the rest-wavelength range for LAEs is 4900–5100 Å and 4900–5140 Å
for QSOs. When fitting some QSOs’ spectra, the result contains Gaussian
components with FWHMs of several $10^{4}$ km s-1, which lack astronomical
interpretations and are actually caused by a miscalculation of the continuum.
In these cases, we adjusted the red end of the range to 5160, 5180 or 5200 Å
to define the continuum component better. For QSOs with broad and blueshifted
[O III] emission lines, such as the obscured QSOs with strong [O III] outflows
in the sample of Zakamska et al. (2016), we adjusted the blue end of the range
to 4880 Å to decompose the continuum and the $\lambda$4960 line effectively.
For any spectrum, pixels with masks were excluded from the fitting.

We fit the spectra by minimizing $\chi^{2}$. First we tried one Gaussian
component for each line, and then gradually added more Gaussian components.
One component more will lower the $\chi^{2}$ and the degree of freedom by 3 at
the same time. Supposing the fitting were improved after adding a component,
we calculated the confidence probability using the F test:

$F=\frac{\Delta\chi^{2}/\chi^{2}}{\Delta dof/dof},$ (3)

where $\chi^{2}$ and $dof$ are the chi-square and the degree of freedom after
adding the component, $\Delta\chi^{2}$ and $\Delta dof$ are the decreased
amounts, and $\Delta dof$ is 3. We calculated the probability $p(F,dof,\Delta
dof)$ corresponding to this $F$ value. If $p$ is greater than 99.99%, we would
add this component, and vice versa. This threshold of $p$ corresponds to
$\Delta\chi^{2}$ of 20–30 for most spectra ($dof$ is 1000–1500). Figure 2
shows an example of how we gradually added the Gaussian components. All the
doublets in the spectra can be fitted with no more than 4 components, while
when 4 components have been used, adding one more only improves the fitting
slightly but raise difficulty in finding the best solution.

Figure 2:  Example of the multiple Gaussian fitting. From top to bottom
panels, we show the models containing 1–4 Gaussian components for [O III]
doublet. In each panel, we display the observed spectrum (black), the
continuum model (green), the sum of continuum and emission line models (red),
the residual (blue), and the observed errors (cyan). Note that we only show
spectra and model in 4900–5100 Å while the wavelength range of fitting is 4900
to 5140 Å.

Bahcall et al. (2004) indicated that the SDSS pipeline overestimated the
errors for lower flux levels(see their section 4.4). This situation also
happened to the Xshooter pipeline. Therefore we corrected the error following
Bahcall et al. (2004). If the fitting’s reduced chi-square ($\chi^{2}/dof$) is
less than 1 when the original error is used, we multiplied the error of all
pixels by a correction factor (equal to the square root of the reduced chi-
square) and redid the fit. We corrected the error for 43% of the spectra, and
the correction factors in our final sample are 0.6–1.0. This correction
decreases the errors of model parameters and increases the SNRs of [O III]
doublet lines. After correction, the SNRs are more consistent with visual
estimates (e.g., 3–5 for a weak detection or $>7$ for a clear detection).

###  2.3 Sample Selection

Our criteria for sample selection include the following aspects.

First, we required the SNRs of both the $\lambda$4960 and $\lambda$5008 lines
to be above 10.

Second, we required that the [O III] doublet in a spectrum hardly bears any
SELs, TALs, cosmic rays or bad CCD pixels. Although the pixels affected by
these factors had been masked in Section 2.1 and were excluded from spectral
analysis, these factors can still cause problems because automated programs
may fail to mask all affected pixels when the effects are severe. In addition,
for an emission line with half the pixels masked, including those near the
peak, setting $\eta$ as a free parameter (as we will do in Section 3) may lead
to mistaken identification of the peak. We calculated a mask index $I_{\rm
mask}$ for each line to describe these effects quantitatively. As shown in
Figure 1, this index was calculated as the sum of the fluxes of the masked
pixels (grey region) divided by the total flux of the line. After visual
inspections of the effects mentioned above, we required that the sum of
$I_{\rm mask}$ of $\lambda$4960 and $\lambda$5008 lines be below 0.5. The LAE
spectra excluded by this criterion were mainly due to SELs, while the QSO
spectra excluded were mainly due to TALs.

Third, we excluded QSOs with [O III] doublet lines severely blended. If so,
the decomposition of the doublet may be wrong by setting $A$ as free
parameters if that by fixing $A$ at theoretical value treated as correct. A
blending index $I_{\rm blend}$ was calculated to quantify the blending, as is
shown in Figure 3(b). On the multiple Gaussian models of the [O III] doublet
lines, we measured the flux of the highest point near 4960 Å as $f_{1}$ and
the flux of the lowest point in the range between 4960–5008 Å as $f_{2}$.
$I_{\rm blend}$ was set to be the ratio between $f_{2}$ and $f_{1}$. According
to visual inspections, to ensure the doublet lines be properly decomposed, we
required that $I_{\rm blend}$ is below 0.2. The QSO spectra excluded by this
criterion all have broad and blueshifted [O III] emission lines. The
decomposition by fixing $A$ at theoretical value show that the [O III] FWHMs
of these excluded QSOs are 800–4500 km s-1, significantly greater than the
FWHMs of the final sample, which has a median value of 610 km s-1.

Finally, we excluded QSOs with strong Fe II bumps. The Fe II bump between 4900
and 5050 Å may lead to a severe systematic error when fitting the continuum
with a simple second-order polynomial. To measure the intensity of the Fe II
bump, we calculated a Fe II index $I_{\rm FeII}$ as follows. As shown in
Figure 3(a), we measured $I_{4590}$ and $I_{5250}$, which represent the
intensity of the Fe II bumps around 4590 Å (Fe II $\lambda$4590) and around
5100–5400 Å (Fe II $\lambda$5250). We inferred the continuum below the Fe II
bumps using the spectra in the green shade in Figure 3(a): the wavelength
ranges for fitting the continuum are 4420-4460 Å and 4720–4760 Å for Fe II
$\lambda$4590, and 5060–5100 Å and 5400–5440 Å for Fe II $\lambda$5250; the
model used was a power law; the best-fitting model for the continuum is
expressed as $f_{\rm con}(\lambda)$, as is shown by the blue line in Figure
3(a). We fit the continuum-subtracted spectra in 4460–4720 Å and 5100–5400 Å
using cubic spline functions whose nodes have a uniform interval of 20 Å, and
the fitting result is expressed as $f_{\rm FeII}(\lambda)$. Hence we
calculated:

$I_{4590}=\frac{\int_{4460}^{4720}f_{\rm FeII}(\lambda){\rm d}\lambda}{f_{\rm
con}(4590\AA)},\ I_{5250}=\frac{\int_{5100}^{5400}f_{\rm FeII}(\lambda){\rm
d}\lambda}{f_{\rm con}(5250\AA)}$ (4)

Note that for most QSOs, only one of $I_{4590}$ and $I_{5250}$ can be reliably
measured because of the telluric absorption bands. For uniformity, if
$I_{5250}$ can be reliably measured, then we set $I_{\rm FeII}=I_{5250}$. And
if not, we set $I_{\rm FeII}=1.3\times I_{4590}$. The coefficient 1.3 here was
chosen to be the ratio between $I_{5250}$ (16 Å) and $I_{4590}$ (12 Å)
measured on the Xshooter QSO composite spectrum (Selsing et al., 2016). We
excluded QSOs that meets the following criteria: $I_{\rm FeII}>10\AA$ and
$I_{\rm FeII}>0.5\times EW_{\rm[OIII]}$, where $EW_{\rm[OIII]}$ is the
equivalent width of [O III] $\lambda$5008.

Figure 3:  (a): An example of measuring $I_{\rm FeII}$. The green shade shows
the continuum. The continuum model is blue, and the wavelength ranges used for
fitting it are green shades. The red line shows the sum of the continuum and
Fe II models. (b): An example of measuring $I_{\rm blend}$. The black line in
the upper panel is the sum of $\lambda$5008 and $\lambda$4960 models.

The final sample passing these criteria contains 86 spectra, including 40
spectra of 32 LAEs and 46 spectra of 45 QSOs. The observing information of
these 86 spectra is listed in Table 1 (the LAE subsample) and Table 2 (the QSO
subsample). Six of the LAE spectra come from a galaxy at $z=2.37$ named
“Sunburst arc” (SA, Dahle et al., 2016; Rivera-Thorsen et al., 2017), which is
highly magnified by a gravitational lens (magnification $>20$). We refer to
these six spectra as SA1 to SA6. LAE J0332-2746 has three spectra, J0217-0502
has two spectra, and each of the other 29 LAEs has only one spectrum. QSO
J1313-2716 has two spectra, and each of the other 44 QSOs has one spectrum.

Table 1: Information of the 40 LAE spectra ProgID | ObjID | Ra | Dec | z | $t_{\rm exp}$ | $R$ |  log $L_{5008}$ | FWHM |  SNR4960 |  SNR5008
---|---|---|---|---|---|---|---|---|---|---
084.A-0303(C) | J1000+0201 | 10:00:49.17 | +02:01:19.4 | 2.246 | 3600 | 4287 | 42.48 | 95 | 10.89 | 30.23
088.A-0672(A) | J0332-2759 | 03:32:49.44 | -27:59:51.6 | 2.205 | 9600 | 5573 | 42.40 | 112 | 10.02 | 37.63
| J0332-2800 | 03:32:32.41 | -28:00:51.8 | 2.172 | 9600 | 5573 | 43.05 | 182 | 33.03 | 65.07
091.A-0413(A) | J2346+1247 | 23:46:09.17 | +12:47:54.5 | 2.327 | 4640 | 8014 | 42.24 | 176 | 10.20 | 16.68
| J2346+1249 | 23:46:29.41 | +12:49:42.9 | 2.173 | 9280 | 8014 | 42.41 | 138 | 23.12 | 46.13
093.A-0882(A) | J0217-0511 | 02:17:47.27 | -05:11:44.1 | 2.243 | 7200 | 4287 | 42.40 | 131 | 23.44 | 50.26
| J0217-0512 | 02:17:43.41 | -05:12:45.5 | 2.183 | 3600 | 5573 | 42.76 | 115 | 18.38 | 46.60
| J0332-2745 | 03:32:24.53 | -27:45:34.9 | 2.311 | 7200 | 4287 | 42.88 | 131 | 41.96 | 87.17
| J0332-2746 | 03:32:36.85 | -27:46:18.3 | 2.226 | 3600 | 5573 | 42.68 | 164 | 19.74 | 43.15
| J2358-1014 | 23:58:21.33 | -10:14:30.8 | 2.261 | 10800 | 4287 | 42.68 | 133 | 30.13 | 37.91
096.A-0348(A) | J0217-0514 | 02:17:03.97 | -05:14:16.2 | 2.185 | 7200 | 4287 | 42.69 | 132 | 11.82 | 41.86
| J0217-0515 | 02:17:43.60 | -05:15:27.1 | 2.065 | 3600 | 4287 | 42.56 | 143 | 11.10 | 25.66
| J0332-2741 | 03:32:38.91 | -27:41:37.5 | 2.312 | 7200 | 4287 | 42.74 | 207 | 22.31 | 53.97
097.A-0153(A) | J0145-0946 | 01:45:16.78 | -09:46:04.5 | 2.357 | 7200 | 5573 | 42.93 | 221 | 42.37 | 103.25
| J0209-0005A | 02:09:49.27 | -00:05:33.0 | 2.167 | 3600 | 5573 | 42.65 | 133 | 22.69 | 32.17
| J0209-0005B | 02:09:43.19 | -00:05:51.9 | 2.187 | 7200 | 5573 | 42.79 | 225 | 23.87 | 37.23
099.A-0254(A) | J0217-0502 | 02:17:46.02 | -05:02:58.6 | 2.209 | 8960 | 4287 | 42.45 | 140 | 11.31 | 19.81
099.A-0758(A) | J0332-2746 | 03:32:35.39 | -27:46:15.1 | 2.171 | 13200 | 5573 | 42.53 | 119 | 18.64 | 32.16
| J0332-2749 | 03:32:17.95 | -27:49:42.7 | 2.033 | 10800 | 5573 | 42.60 | 148 | 20.27 | 68.82
0101.B-0262(A) | J0014-3024 | 00:14:20.71 | -30:24:30.8 | 2.016 | 4800 | 5573 | 42.34 | 109 | 10.31 | 23.39
| J0120-5143 | 01:20:42.35 | -51:43:52.0 | 1.295 | 4800 | 5573 | 42.10 | 103 | 9.99 | 22.83
| J0131-1335 | 01:31:51.96 | -13:35:38.1 | 2.467 | 4800 | 5573 | 42.63 | 150 | 18.91 | 42.20
| J2129+0006 | 21:29:39.55 | +00:06:48.9 | 2.405 | 4800 | 5573 | 42.38 | 163 | 16.26 | 31.20
| J2129-0741 | 21:29:21.92 | -07:41:02.1 | 2.274 | 4800 | 5573 | 42.71 | 188 | 19.86 | 59.34
| J2140-2339 | 21:40:19.08 | -23:39:30.9 | 2.486 | 4800 | 5573 | 42.95 | 148 | 28.84 | 44.96
| J2248-4431 | 22:48:37.00 | -44:31:18.5 | 2.065 | 4800 | 5573 | 42.49 | 123 | 11.99 | 30.04
0101.B-0779(A) | J1000+0236 | 10:00:37.34 | +02:36:45.5 | 3.423 | 7200 | 5573 | 43.17 | 164 | 20.45 | 40.89
0102.A-0391(A) | J0332-2745 | 03:32:03.18 | -27:45:17.7 | 3.211 | 14400 | 5573 | 43.01 | 129 | 20.34 | 51.12
0102.A-0652(A) | J0217-0502 | 02:17:46.19 | -05:02:55.6 | 2.209 | 8800 | 5573 | 42.71 | 118 | 25.07 | 45.14
0103.A-0688(C) | SA1 | 15:50:04.55 | -78:10:57.1 | 2.369 | 2700 | 5573 | 44.37 | 124 | 163.07 | 266.60
| SA2 | 15:50:04.55 | -78:10:57.1 | 2.369 | 2700 | 5573 | 44.29 | 114 | 187.94 | 467.47
| SA3 | 15:50:00.02 | -78:11:12.9 | 2.369 | 2700 | 8031 | 44.12 | 114 | 113.17 | 167.20
| SA4 | 15:50:00.02 | -78:11:12.9 | 2.369 | 2700 | 8031 | 43.86 | 136 | 94.52 | 259.79
| SA5 | 15:49:59.05 | -78:11:12.4 | 2.369 | 1800 | 8031 | 43.96 | 116 | 114.40 | 209.85
| SA6 | 15:49:59.05 | -78:11:12.4 | 2.369 | 1800 | 8031 | 43.68 | 151 | 71.30 | 184.56
0103.B-0446(A) | J0224-0449 | 02:24:18.99 | -04:49:12.3 | 2.550 | 14400 | 5573 | 42.68 | 123 | 32.15 | 64.78
| J0332-2752 | 03:32:12.98 | -27:52:34.7 | 2.566 | 1800 | 5573 | 42.99 | 365 | 21.42 | 43.28
| J1001+0206 | 10:00:08.86 | +02:06:39.7 | 2.435 | 1800 | 5573 | 42.73 | 489 | 10.62 | 19.49
0104.A-0236(A) | J0332-2746 | 03:32:35.31 | -27:46:18.1 | 2.171 | 21600 | 5573 | 42.25 | 127 | 19.13 | 57.12
0104.A-0236(B) | J0332-2746 | 03:32:35.32 | -27:46:17.9 | 2.171 | 12600 | 5573 | 42.29 | 132 | 14.89 | 48.34
Table 2: Information of the 46 QSO spectra ProgID | ObjID | Ra | Dec | z | $t_{\rm exp}$ | $R$ |  log $L_{5008}$ | FWHM |  SNR4960 |  SNR5008
---|---|---|---|---|---|---|---|---|---|---
086.B-0320(A) | J1242+0232 | 12:42:20.04 | +02:32:55.6 | 2.221 | 3000 | 4287 | 43.71 | 842 | 11.62 | 18.53
087.A-0610(A) | J0914+0109 | 09:14:32.07 | +01:09:10.0 | 2.139 | 2400 | 8031 | 43.27 | 792 | 14.79 | 34.64
087.B-0229(A) | J1512+1119 | 15:12:49.39 | +11:19:27.3 | 2.100 | 8700 | 5573 | 44.46 | 560 | 122.60 | 142.96
088.B-1034(A) | J0148+0003 | 01:48:12.79 | +00:03:21.6 | 1.481 | 2880 | 4287 | 43.42 | 550 | 10.29 | 25.29
| J0240-0758 | 02:40:28.97 | -07:58:45.8 | 1.533 | 5760 | 4287 | 43.40 | 335 | 16.65 | 24.15
| J0323-0029 | 03:23:49.63 | -00:29:51.3 | 1.625 | 2880 | 4287 | 43.71 | 483 | 16.71 | 26.16
| J0842+0151 | 08:42:40.68 | +01:51:31.4 | 1.496 | 1440 | 4287 | 43.51 | 972 | 13.54 | 46.10
| J1005+0245 | 10:05:13.74 | +02:45:08.0 | 1.484 | 2880 | 4287 | 43.76 | 577 | 21.83 | 30.65
089.B-0275(A) | J1226-0006 | 12:26:07.94 | -00:06:03.1 | 1.126 | 2400 | 5567 | 43.01 | 610 | 25.91 | 47.80
089.B-0936(A) | J0904-2552 | 09:04:52.43 | -25:52:51.0 | 1.644 | 1260 | 5567 | 43.29 | 385 | 12.59 | 20.73
| J1107-4449 | 11:07:08.47 | -44:49:07.8 | 1.598 | 960 | 5567 | 43.52 | 485 | 14.00 | 41.30
089.B-0951(A) | J0958+0251 | 09:58:52.18 | +02:51:52.8 | 1.408 | 3780 | 5567 | 43.18 | 395 | 22.65 | 38.30
| J1313-2716 | 13:13:47.30 | -27:16:46.1 | 2.198 | 760 | 5567 | 44.16 | 822 | 23.80 | 35.45
090.A-0830(A) | J1002+0137 | 10:02:11.41 | +01:37:08.0 | 1.591 | 3600 | 5567 | 42.86 | 833 | 11.99 | 31.60
| J1003+0209 | 10:03:08.99 | +02:09:03.2 | 1.472 | 3600 | 5567 | 42.99 | 1219 | 10.88 | 34.57
090.B-0424(B) | J0811+1720 | 08:11:14.58 | +17:20:55.2 | 2.332 | 14400 | 5573 | 44.10 | 643 | 72.58 | 133.05
091.B-0900(A) | J1313-2716 | 13:13:47.17 | -27:16:50.1 | 2.200 | 1030 | 5567 | 44.10 | 766 | 22.14 | 39.14
| J1404-0130 | 14:04:45.89 | -01:30:24.6 | 2.515 | 1860 | 5567 | 43.71 | 373 | 46.12 | 113.68
| J1544+0407 | 15:44:59.42 | +04:07:43.9 | 2.182 | 1860 | 5567 | 43.99 | 695 | 57.54 | 140.20
092.A-0391(A) | J0209-0438 | 02:09:30.68 | -04:38:30.0 | 1.132 | 2400 | 5573 | 43.94 | 441 | 80.07 | 115.24
092.B-0860(A) | J1154-0215 | 11:54:32.66 | -02:15:40.6 | 2.181 | 7200 | 5573 | 43.21 | 392 | 23.04 | 59.62
093.B-0553(A) | J0940+0034 | 09:40:25.62 | +00:33:58.8 | 2.332 | 1920 | 5573 | 43.44 | 511 | 13.41 | 25.51
| J1019+0345 | 10:19:25.94 | +03:45:33.9 | 2.212 | 2400 | 5573 | 43.25 | 702 | 12.93 | 31.01
| J1323+0022 | 13:23:07.77 | +00:22:59.6 | 2.258 | 1920 | 5573 | 43.62 | 438 | 31.80 | 65.65
| J2330-0216 | 23:30:16.90 | -02:16:45.9 | 2.496 | 1920 | 5573 | 43.46 | 493 | 14.83 | 25.71
094.B-0111(A) | J0108+0134 | 01:08:38.66 | +01:34:57.6 | 2.099 | 1800 | 5567 | 43.65 | 560 | 28.85 | 57.16
095.B-0507(A) | J0114-0812 | 01:14:20.47 | -08:12:45.5 | 2.104 | 2400 | 4287 | 43.45 | 717 | 21.29 | 42.48
| J2136-1631 | 21:36:55.62 | -16:31:39.8 | 1.659 | 2400 | 4287 | 43.82 | 397 | 25.70 | 30.77
098.B-0556(A) | J0111-3502 | 01:11:43.50 | -35:02:58.3 | 2.405 | 4800 | 5573 | 44.63 | 473 | 240.67 | 399.55
| J0331-3824 | 03:31:06.20 | -38:24:04.9 | 2.435 | 4800 | 5573 | 44.11 | 673 | 99.02 | 211.71
099.A-0018(A) | J1338+0010 | 13:38:31.46 | +00:10:54.4 | 2.300 | 4800 | 5567 | 43.02 | 486 | 15.63 | 29.17
099.B-0118(A) | J0020-1540 | 00:20:25.22 | +15:40:52.5 | 2.020 | 1200 | 5567 | 43.93 | 721 | 42.93 | 76.68
0101.A-0528(A) | J2124-4948 | 21:24:28.77 | -49:48:07.4 | 2.480 | 1800 | 4287 | 43.57 | 271 | 24.67 | 44.02
0101.A-0528(B) | J0607-6031 | 06:07:55.01 | -60:31:54.9 | 1.097 | 1800 | 4287 | 43.40 | 481 | 19.57 | 56.62
0101.B-0739(A) | J1352+1302 | 13:52:18.38 | +13:02:39.7 | 1.600 | 800 | 4287 | 43.76 | 859 | 12.33 | 51.04
| J1616+0931 | 16:16:39.73 | +09:31:15.7 | 1.468 | 1200 | 4287 | 43.66 | 420 | 25.73 | 72.43
| J2239+1222 | 22:39:32.01 | +12:22:22.3 | 1.484 | 400 | 4287 | 44.13 | 997 | 20.61 | 57.26
0102.A-0335(A) | J0259+1635 | 02:59:42.96 | -16:35:41.6 | 2.160 | 4800 | 4287 | 44.64 | 525 | 460.18 | 641.76
| J0630-1201 | 06:30:09.16 | -12:01:18.7 | 3.344 | 4800 | 4287 | 43.96 | 763 | 11.69 | 22.70
| J1042+1641 | 10:42:22.25 | +16:41:16.7 | 2.519 | 4800 | 4287 | 44.84 | 963 | 287.70 | 300.13
0103.A-0253(A) | J2245-2945 | 22:45:31.61 | -29:45:54.3 | 2.379 | 3200 | 4287 | 43.46 | 341 | 15.60 | 39.16
189.A-0424(A) | J1018+0548 | 10:18:18.58 | +05:48:20.8 | 3.512 | 3600 | 5573 | 44.22 | 616 | 17.57 | 37.83
| J1042+1957 | 10:42:34.03 | +19:57:16.3 | 3.628 | 3600 | 5573 | 44.21 | 1267 | 12.76 | 29.44
| J1117+1311 | 11:17:01.97 | +13:11:13.1 | 3.623 | 3600 | 5573 | 44.09 | 832 | 12.48 | 28.53
| J1312+0841 | 13:12:42.94 | +08:41:02.9 | 3.735 | 3600 | 5573 | 44.36 | 859 | 13.90 | 27.94
| J1621-0042 | 16:21:17.05 | -00:42:52.9 | 3.703 | 3600 | 5573 | 44.44 | 914 | 16.72 | 42.25

For each spectrum, we measured the peak wavelength of the [O III]
$\lambda$5008 line and calculated a [O III] redshift as the approximation of
the systematic redshift. For typical QSOs, this would cause an error of
$<$0.001 ($<$300 km s-1) except for a small fraction of QSOs called “blue
outliers” (Marziani et al., 2016), but those blue outliers generally have
blended [O III] doublet and would not meet our criteria and hence would not be
included in our sample. We also measured the luminosity of [O III]
$\lambda$5008 in each spectrum. The statistical errors of the luminosities are
$<$6% , corresponding to $<$0.03 dex in logarithm. Changes in the slit width,
the seeing condition and the atmospheric transmittance can cause systematic
errors in the luminosities, and we estimated them to be 0.1–0.2 dex based on
the multiple observations of some targets. These redshifts and [O III]
luminosities of the spectra are listed in Table 1 and Table 2 and are
displayed in Figure 4. We found that all the targets are distributed in three
redshift intervals, $1.09<z<1.66$, $2.01<z<2.57$, and $3.21<z<3.74$,
corresponding to the observed wavelengths of the J, H and K bands
respectively. Most of the LAEs are in the redshift range corresponding to the
H-band, and this may be caused by the methods with which the LAEs were
selected. The [O III] luminosities of the QSOs are in the range between
$10^{42.8}$ and $10^{44.8}$ erg s-1, and have a median value of $10^{43.7}$
erg s-1. The [O III] luminosities of the SA spectra are in the range of
$10^{43.6-44.4}$ erg s-1, and those of other LAEs are in the range between
$10^{42.1}$ and $10^{43.2}$ erg s-1, and have a median value of $10^{42.7}$
erg s-1, which is one order of magnitude lower than that of QSOs.

Figure 4: Redshift versus [O III] luminosity for our final sample.

We measured the FWHMs of the [O III] doublet lines in every spectrum of the
final sample and showed them in Figure 5 with the SNR of [O III] $\lambda$4960
(SNR4960). These two parameters are critical indicators for the measuring
precision of the [O III] wavelengths (see Section 3.3 and Appendix D for
analysis and simulation). In brief, the larger the SNR4960 or the smaller the
FWHM, the more accurate the measurement of the wavelengths. The SNR4960 of the
six SA spectra is 60–200, that of other LAE spectra is 10–50, and that of the
QSO spectra is 10–460. The median values of the SNR4960 of the LAE and the QSO
subsamples are both around 20. The FWHM of [O III] of the LAEs is 90–490 km
s-1 with a median value of 130 km s-1, and that of the QSOs is 270–1260 km s-1
with a median value of 610 km s-1. The [O III] emission lines of the LAEs are
narrower and less luminous than those of the QSOs, while the SNRs of the two
subsamples are similar.

Figure 5: The distributions of SNR4960 and [O III] FWHM for the final sample.

##  3 Measurements of Emission-line Wavelengths

Following Uzan (2003), we calculated the variation in the fine structure
constant as:

$\frac{\Delta\alpha}{\alpha}=\frac{1}{2}\left\\{\frac{[(\lambda_{2}-\lambda_{1})/(\lambda_{2}+\lambda_{1})]_{z}}{[(\lambda_{2}-\lambda_{1})/(\lambda_{2}+\lambda_{1})]_{0}}-1\right\\},$
(5)

where $\lambda_{1}$ and $\lambda_{2}$ are the wavelengths of $\lambda$4960 and
$\lambda$5008 lines, respectively. Those with a subscript of 0 are the
wavelength values at $z=0$ from laboratories, and those with a subscript of
$z$ are the observed values of the doublet from a target’s spectrum at a
redshift of $z$. Equation (5) can be rewritten as:

$\frac{\Delta\alpha}{\alpha}=\frac{\eta-\eta_{0}}{\eta_{0}(2+\eta)},$ (6)

where $\eta\equiv(\lambda_{2}/\lambda_{1}-1)_{z}$,
$\eta_{0}\equiv(\lambda_{2}/\lambda_{1}-1)_{0}=0.00966576$. The Taylor
expansion of equation (6) around $\eta_{0}$ gives:

$\frac{\Delta\alpha}{\alpha}=51.4802(\eta-\eta_{0})-25.6163(\eta-\eta_{0})^{2}+o\left((\eta-\eta_{0})^{2}\right).$
(7)

We will show later that for our sample, $|\eta-\eta_{0}|$ is less than
$3\times 10^{-4}$. In this case, ignoring the quadratic and the high-order
terms only results in a relative error of $\Delta\alpha/\alpha$ no more than
$\sim 10^{-4}$. Thus, we calculated the change in the fine structure constant
and estimated its error with:

$\frac{\Delta\alpha}{\alpha}\approx 51.4802(\eta-\eta_{0}).$ (8)

In this way, the measurement of $\Delta\alpha/\alpha$ can be converted into
the measurement of $\eta$.

###  3.1 Two Methods to Measure $\eta$

We measured $\eta$ using the following two methods. The first is called the
Multiple-Gaussian (MG) method. The $\lambda$4960, $\lambda$5008 lines were
fitted simultaneously with the model expressed as equation (2). The fitting is
basically the same as that described in section 2.2, and the only difference
is that $\eta$ is now a free parameter instead of the fixed value of
$\eta_{0}$. The wavelength ranges and the number of Gaussian functions are
consistent with those used in section 2.2. We fit the spectra using a
Levenberg-Marquardt technique with the MPFIT package. We set the initial
parameters as those obtained in section 2.2. We took the values of $\eta$ and
$A$ yielding the minimum $\chi^{2}$ (expressed as $\chi_{\rm min}^{2}$) as the
measurement result and calculated the statistical errors $\sigma_{\rm
stat}(\eta)$ and $\sigma_{\rm stat}(A)$ according to the range of $\eta$ and
$A$ yielding $\chi^{2}(\eta,A)<\chi_{\rm min}^{2}+1$. An example of measuring
$\eta$ and $A$ using the MG method is shown in the upper row of Figure 6.

The second is called the Profile-Matching (PM) method. This method is based on
the fact that the [O III] doublet originates from the same upper energy level
and should have identical line profiles. The analysis approach is similar to
that adopted by Bahcall et al. (2004). After subtracting the best-fitting
continuum model from the observed spectrum, we obtained the profiles of
$\lambda$5008 and $\lambda$4960 emission lines. We moved the $\lambda$5008
line leftward ($\lambda$ divided by $1+\eta$) and decreased the amplitude
($f_{\lambda}$ divided by $\frac{A}{1+\eta}$). Theoretically, at some specific
values of $\eta$ and $A$, the adjusted $\lambda$5008 line should be the same
with the $\lambda$4960 line, in which case they can be regarded as two
samplings from one profile beneath. So for a pair of values of $\eta$ and $A$,
we fit the adjusted $\lambda$5008 line and the $\lambda$4960 line
simultaneously with a cubic spline function, from which a $\chi^{2}$ value was
obtained. With different values of $\eta$ and $A$ yielding different values of
$\chi^{2}$, we obtained a $\chi^{2}$ surface in the 2D-parameter space. The
$\eta$ and $A$ yielding the minimum $\chi^{2}$ were taken as the result, and
their statistical errors were obtained using $\chi^{2}(\eta,A)<\chi_{\rm
min}^{2}+1$. In the matching, we used the pixels in the following wavelength
ranges: the range for the $\lambda$5008 is where the flux exceeds 20% of the
peak flux (we use the flux obtained from the best-fitting Gaussian models to
avoid random fluctuations), while the range for the $\lambda$4960 is the one
of $\lambda$5008 divided by $1+\eta$. The pixels with masks were excluded.
When $\eta$ changes, the wavelength range of the $\lambda$4960 line changes
accordingly, and the degree of freedom may change when the $\lambda$4960 line
contains masked pixels. If so, we slightly adjusted the wavelength range to
keep the degree of freedom constant. An example of measuring $\eta$ and $A$
using the PM method is shown in the bottom row of Figure 6.

Figure 6:  Examples of measuring $\eta$ and $A$ using the two methods. The MG
method is shown in the upper row, and the PM method is shown in the bottom
row. (a): Observed spectrum (black) and best-fitting multiple-Gaussian model
(red). (b): The profile of $\lambda$4960 line (blue point), and that of
$\lambda$5008 line which has been moved leftward and downsized (red point),
and the best-fitting cubic spline function (black line). (c): The
$\chi^{2}(\eta,A)$ surfaces obtained by the two methods. (d): The contours
that yielding $\chi^{2}=\chi^{2}_{\rm min}+1$, and the resultant 1$\sigma$
confidence intervals of $\eta$ (green shade) and $A$ (red shade). (e): The
minimum $\chi^{2}$ value for different $\eta$ (black line) and the 1$\sigma$
confidence interval of $\eta$ (green shade) corresponding to
$\chi^{2}<\chi^{2}_{\rm min}+1$ (blue shade).

For all the 86 spectra, the $\chi^{2}(\eta,A)$ surfaces obtained by the MG
method look good: they are smooth, and their contours are close to ellipses in
the vicinity of the minimum $\chi^{2}$, as shown in Figure 6(c) and 6(d). This
holds in the range of $\chi^{2}<\chi_{\rm min}^{2}+1$ for all spectra and in
the range of $\chi^{2}<\chi_{\rm min}^{2}+10$ for most spectra. Therefore, the
measurements on $\eta$, $A$ and the corresponding statistical error are
reliable. However, only a small fraction of $\chi^{2}$ surfaces obtained by
the PM method look good, and we showed two bad examples in Appendix B. The PM
method works well only for spectra with narrow [O III] doublet lines that have
high SNR and almost bear no bad pixels. We visually checked the $\chi^{2}$
surfaces and selected 16 LAE spectra and 1 QSO spectrum with which $\eta$ can
be reliably measured.

The consistency between the two methods was checked by the measurements of
$\eta$ from the 17 spectra. The $\eta$ values are showed in Figure 7(a), and
their statistical errors $\sigma_{\rm stat}(\eta)$ in Figure 7(b). The values
of $\sigma_{\rm stat}(\eta)$ obtained from the two methods differ by less than
12%. To check whether the $\eta$ values were consistent, we calculated
deviation $\Delta$ as:

$\Delta=\frac{\eta({\rm MG})-\eta({\rm PM})}{\sqrt{\sigma_{\rm
stat}^{2}(\eta,{\rm MG})+\sigma_{\rm stat}^{2}(\eta,{\rm PM})}}.$ (9)

The distribution of $\Delta$ is shown in Figure 7(c). The average value is
close to 0, and their absolute values are no more than 1.12. These indicate
that the results from the two methods agree with each other. As the MG method
works well for all the spectra in our sample, we adopted the measurements
using it as the final result. The best-fitting Multiple-Gaussian models are
displayed in Appendix C. The number of Gaussian functions, the best estimates
and the statistical errors of $\eta$ and $A$ are listed in Table 3, 4 and 5.
As the measurements of the six spectra of SA enormously contributed to the
final results, we also list the estimates of $\eta$ and $A$ using the PM
method in Table 3 for comparison.

Figure 7:  Comparison of the $\eta$ values measured using the two methods for
the 17 spectra. (a): Comparison of $\eta$ values measured using the two
methods. (b): Comparison of the statistical errors. (c): Distribution of
$\Delta$ parameter, which characterizes the degree of deviation.

###  3.2 Uncertainties from the Wavelength Calibration

The wavelength calibration for the XShooter NIR Arm has a systematic error of
$\Delta\lambda=0.04$Å (recorded in the Xshooter pipeline, Modigliani et al.,
2010). As $\eta\equiv(\lambda_{2}/\lambda_{1}-1)$, assuming that the
wavelengths of $\lambda$4960 and $\lambda$5008 lines in the observer’s frame
both have a systematic error of $\Delta\lambda=0.04$ Å, we obtained:

$\sigma_{\rm
sys}(\eta)=\frac{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}}{\lambda_{1}^{2}}\Delta\lambda=\frac{1.146\times
10^{-5}}{1+z}.$ (10)

The total error $\sigma(\eta)$ can be calculated as:

$\sigma(\eta)=\sqrt{\sigma_{\rm stat}^{2}(\eta)+\sigma_{\rm sys}^{2}(\eta)}$
(11)

The added systematic errors will be on the order of $10^{-6}\sim 10^{-5}$. We
checked whether or not these systematic errors should be considered using 17
spectra yielding $\sigma_{\rm stat}(\eta)<10^{-5}$. If $\eta$ values from the
17 spectra were not to suffer from the systematic errors and their current
statistical errors were enough to be responsible for their scatter around
$\eta_{0}$, the values of $(\eta-\eta_{0})/\sigma_{\rm stat}(\eta)$ would obey
the standard normal distribution with the standard deviation around 1.
However, using the actual measured $\eta$, we obtained the standard deviation
of $(\eta-\eta_{0})/\sigma_{\rm stat}(\eta)$ to be 1.28, indicating that the
probability of the current statistical errors being account for the scatter of
$\eta$ around $\eta_{0}$ to be only 4.7%. Therefore, the systematic errors
caused by the uncertainties of raising from the wavelength calibration should
be considered. After considering the systematic errors, the standard deviation
becomes 1.04, consistent with the prediction of a standard normal
distribution. Therefore, involving the above systematic errors is reasonable,
and we list the systematic errors of $\eta$ for all spectra in Tables 3, 4 and
5.

Table 3: Measurements of $\eta$ and $A$ for the 6 spectra of SA (z=2.37). ObjID | $N_{G}$ | $(\eta-\eta_{0})\times 10^{6}$ | $A$ | $\Delta\alpha/\alpha\times 10^{4}$
---|---|---|---|---
|  | MG | PM | MG | PM | MG | PM
SA1 | 4 | $-1.5\pm 1.5\pm 3.4$ | $-0.3\pm 1.5\pm 3.4$ | $2.99\pm 0.02$ | $3.00\pm 0.02$ | $-0.8\pm 1.9$ | $-0.2\pm 1.9$
SA2 | 3 | $-0.7\pm 1.4\pm 3.4$ | $-1.4\pm 1.5\pm 3.4$ | $3.07\pm 0.02$ | $3.07\pm 0.02$ | $-0.4\pm 1.9$ | $-0.7\pm 1.9$
SA3 | 4 | $-1.8\pm 1.8\pm 3.4$ | $-4.2\pm 1.9\pm 3.4$ | $3.02\pm 0.03$ | $3.01\pm 0.02$ | $-0.9\pm 2.0$ | $-2.2\pm 2.0$
SA4 | 3 | $6.4\pm 3.3\pm 3.4$ | $4.0\pm 3.8\pm 3.4$ | $3.02\pm 0.03$ | $3.05\pm 0.03$ | $3.3\pm 2.5$ | $2.0\pm 2.6$
SA5 | 4 | $-1.9\pm 2.1\pm 3.4$ | $-3.0\pm 2.1\pm 3.4$ | $2.99\pm 0.03$ | $2.99\pm 0.02$ | $-1.0\pm 2.0$ | $-1.5\pm 2.1$
SA6 | 3 | $-5.1\pm 4.6\pm 3.4$ | $-6.2\pm 5.1\pm 3.4$ | $3.03\pm 0.05$ | $3.03\pm 0.04$ | $-2.6\pm 3.0$ | $-3.2\pm 3.2$
Table 4: Measurements of $\eta$ and $A$ for 34 other LAE spectra ProgID | ObjID | z | $N_{G}$ | $(\eta-\eta_{0})\times 10^{5}$ | $A$ | $\Delta\alpha/\alpha\times 10^{4}$
---|---|---|---|---|---|---
084.A-0303(C) | J1000+0201 | 2.246 | 1 | $1.5\pm 1.7\pm 0.4$ | $3.27\pm 0.55$ | $7\pm 9$
088.A-0672(A) | J0332-2759 | 2.205 | 1 | $-1.2\pm 2.3\pm 0.4$ | $2.99\pm 0.36$ | $-5\pm 11$
| J0332-2800 | 2.172 | 3 | $-2.3\pm 1.0\pm 0.4$ | $3.06\pm 0.12$ | $-11\pm 5$
091.A-0413(A) | J2346+1247 | 2.327 | 2 | $0.1\pm 3.7\pm 0.3$ | $3.06\pm 0.42$ | $0\pm 19$
| J2346+1249 | 2.173 | 2 | $1.6\pm 1.4\pm 0.4$ | $3.28\pm 0.16$ | $8\pm 7$
093.A-0882(A) | J0217-0511 | 2.243 | 1 | $0.4\pm 1.2\pm 0.4$ | $3.01\pm 0.18$ | $1\pm 6$
| J0217-0512 | 2.183 | 2 | $-1.9\pm 1.3\pm 0.4$ | $3.35\pm 0.20$ | $-9\pm 6$
| J0332-2745 | 2.311 | 3 | $0.9\pm 0.7\pm 0.3$ | $3.09\pm 0.09$ | $4\pm 4$
| J0332-2746 | 2.226 | 2 | $-0.5\pm 1.8\pm 0.4$ | $2.92\pm 0.18$ | $-2\pm 9$
| J2358-1014 | 2.261 | 2 | $0.2\pm 0.8\pm 0.4$ | $2.92\pm 0.12$ | $0\pm 4$
096.A-0348(A) | J0217-0514 | 2.185 | 1 | $-2.1\pm 2.4\pm 0.4$ | $3.40\pm 0.35$ | $-10\pm 12$
| J0217-0515 | 2.065 | 2 | $-0.5\pm 2.5\pm 0.4$ | $2.88\pm 0.35$ | $-2\pm 13$
| J0332-2741 | 2.312 | 2 | $0.8\pm 1.9\pm 0.3$ | $2.87\pm 0.19$ | $4\pm 10$
097.A-0153(A) | J0145-0946 | 2.357 | 3 | $1.4\pm 0.8\pm 0.3$ | $3.06\pm 0.08$ | $7\pm 4$
| J0209-0005A | 2.167 | 1 | $0.1\pm 1.4\pm 0.4$ | $3.13\pm 0.17$ | $0\pm 7$
| J0209-0005B | 2.187 | 2 | $2.5\pm 2.6\pm 0.4$ | $3.24\pm 0.23$ | $12\pm 13$
099.A-0254(A) | J0217-0502 | 2.209 | 1 | $1.8\pm 2.8\pm 0.4$ | $2.73\pm 0.45$ | $9\pm 14$
099.A-0758(A) | J0332-2746 | 2.171 | 2 | $0.8\pm 1.1\pm 0.4$ | $3.10\pm 0.21$ | $4\pm 5$
| J0332-2749 | 2.033 | 2 | $2.1\pm 1.8\pm 0.4$ | $2.82\pm 0.16$ | $10\pm 9$
0101.B-0262(A) | J0014-3024 | 2.016 | 1 | $0.3\pm 1.9\pm 0.4$ | $3.51\pm 0.48$ | $1\pm 9$
| J0120-5143 | 1.295 | 2 | $3.6\pm 2.0\pm 0.5$ | $3.34\pm 0.42$ | $18\pm 10$
| J0131-1335 | 2.467 | 1 | $-0.2\pm 1.7\pm 0.3$ | $3.12\pm 0.21$ | $-1\pm 8$
| J2129+0006 | 2.405 | 1 | $0.5\pm 2.4\pm 0.3$ | $2.59\pm 0.20$ | $2\pm 12$
| J2129-0741 | 2.274 | 1 | $-1.9\pm 1.8\pm 0.3$ | $3.09\pm 0.19$ | $-10\pm 9$
| J2140-2339 | 2.486 | 2 | $0.9\pm 0.9\pm 0.3$ | $3.14\pm 0.14$ | $4\pm 5$
| J2248-4431 | 2.065 | 2 | $-1.2\pm 2.0\pm 0.4$ | $3.30\pm 0.34$ | $-6\pm 10$
0101.B-0779(A) | J1000+0236 | 3.423 | 1 | $0.6\pm 1.7\pm 0.3$ | $3.17\pm 0.21$ | $3\pm 8$
0102.A-0391(A) | J0332-2745 | 3.211 | 2 | $-0.0\pm 1.4\pm 0.3$ | $3.37\pm 0.21$ | $0\pm 7$
0102.A-0652(A) | J0217-0502 | 2.209 | 2 | $-1.2\pm 1.1\pm 0.4$ | $3.18\pm 0.28$ | $-6\pm 5$
0103.B-0446(A) | J0224-0449 | 2.550 | 2 | $-1.1\pm 0.8\pm 0.3$ | $3.04\pm 0.13$ | $-5\pm 4$
| J0332-2752 | 2.566 | 1 | $-1.5\pm 3.2\pm 0.3$ | $2.83\pm 0.21$ | $-7\pm 16$
| J1001+0206 | 2.435 | 2 | $8.1\pm 9.5\pm 0.3$ | $2.95\pm 0.41$ | $41\pm 48$
0104.A-0236(A) | J0332-2746 | 2.171 | 1 | $0.3\pm 1.4\pm 0.4$ | $2.86\pm 0.19$ | $1\pm 7$
0104.A-0236(B) | J0332-2746 | 2.171 | 1 | $-1.1\pm 1.8\pm 0.4$ | $3.35\pm 0.25$ | $-5\pm 9$
Table 5: Measurements of $\eta$ and $A$ for the 46 QSO spectra ProgID | ObjID | z | $N_{G}$ | $(\eta-\eta_{0})\times 10^{5}$ | $A$ | $\Delta\alpha/\alpha\times 10^{4}$
---|---|---|---|---|---|---
086.B-0320(A) | J1242+0232 | 2.221 | 3 | $-6.1\pm 7.1\pm 0.4$ | $3.36\pm 0.34$ | $-31\pm 36$
087.A-0610(A) | J0914+0109 | 2.139 | 1 | $6.0\pm 9.4\pm 0.4$ | $3.46\pm 0.25$ | $30\pm 48$
087.B-0229(A) | J1512+1119 | 2.100 | 4 | $-0.5\pm 0.4\pm 0.4$ | $2.96\pm 0.03$ | $-2\pm 2$
088.B-1034(A) | J0148+0003 | 1.481 | 2 | $1.0\pm 10.0\pm 0.5$ | $3.66\pm 0.54$ | $4\pm 51$
| J0240-0758 | 1.533 | 2 | $2.8\pm 4.6\pm 0.5$ | $3.13\pm 0.23$ | $14\pm 23$
| J0323-0029 | 1.625 | 3 | $-1.0\pm 3.0\pm 0.4$ | $3.45\pm 0.33$ | $-5\pm 15$
| J0842+0151 | 1.496 | 2 | $-7.2\pm 6.8\pm 0.5$ | $3.90\pm 0.30$ | $-36\pm 35$
| J1005+0245 | 1.484 | 4 | $5.1\pm 4.8\pm 0.5$ | $3.14\pm 0.26$ | $26\pm 24$
089.B-0275(A) | J1226-0006 | 1.126 | 2 | $-4.4\pm 4.1\pm 0.5$ | $3.11\pm 0.16$ | $-22\pm 21$
089.B-0936(A) | J0904-2552 | 1.644 | 2 | $-1.8\pm 5.1\pm 0.4$ | $3.61\pm 0.34$ | $-9\pm 26$
| J1107-4449 | 1.598 | 2 | $-7.4\pm 7.2\pm 0.4$ | $3.42\pm 0.26$ | $-38\pm 36$
089.B-0951(A) | J0958+0251 | 1.408 | 2 | $0.4\pm 3.2\pm 0.5$ | $3.22\pm 0.17$ | $2\pm 16$
| J1313-2716 | 2.198 | 3 | $-5.3\pm 4.6\pm 0.4$ | $3.36\pm 0.18$ | $-27\pm 23$
090.A-0830(A) | J1002+0137 | 1.591 | 1 | $28.7\pm 12.8\pm 0.4$ | $2.61\pm 0.23$ | $147\pm 65$
| J1003+0209 | 1.472 | 1 | $25.3\pm 19.9\pm 0.5$ | $3.41\pm 0.33$ | $130\pm 102$
090.B-0424(B) | J0811+1720 | 2.332 | 4 | $1.9\pm 1.6\pm 0.3$ | $3.26\pm 0.05$ | $9\pm 8$
091.B-0900(A) | J1313-2716 | 2.200 | 4 | $2.6\pm 3.8\pm 0.4$ | $3.21\pm 0.20$ | $13\pm 19$
| J1404-0130 | 2.515 | 2 | $1.0\pm 1.9\pm 0.3$ | $3.41\pm 0.08$ | $5\pm 10$
| J1544+0407 | 2.182 | 2 | $-3.9\pm 2.4\pm 0.4$ | $3.46\pm 0.07$ | $-19\pm 12$
092.A-0391(A) | J0209-0438 | 1.132 | 4 | $0.6\pm 0.6\pm 0.5$ | $3.14\pm 0.05$ | $3\pm 4$
092.B-0860(A) | J1154-0215 | 2.181 | 2 | $2.2\pm 3.3\pm 0.4$ | $3.37\pm 0.16$ | $11\pm 16$
093.B-0553(A) | J0940+0034 | 2.332 | 2 | $8.1\pm 6.6\pm 0.3$ | $3.36\pm 0.28$ | $41\pm 34$
| J1019+0345 | 2.212 | 1 | $-20.0\pm 10.0\pm 0.4$ | $2.91\pm 0.24$ | $-102\pm 51$
| J1323+0022 | 2.258 | 3 | $1.6\pm 2.5\pm 0.4$ | $3.19\pm 0.11$ | $8\pm 13$
| J2330-0216 | 2.496 | 2 | $-11.8\pm 6.1\pm 0.3$ | $2.67\pm 0.21$ | $-60\pm 31$
094.B-0111(A) | J0108+0134 | 2.099 | 2 | $-2.1\pm 3.3\pm 0.4$ | $3.36\pm 0.13$ | $-10\pm 17$
095.B-0507(A) | J0114-0812 | 2.104 | 2 | $5.5\pm 6.0\pm 0.4$ | $2.23\pm 0.12$ | $28\pm 30$
| J2136-1631 | 1.659 | 4 | $0.2\pm 1.4\pm 0.4$ | $3.00\pm 0.15$ | $0\pm 7$
098.B-0556(A) | J0111-3502 | 2.405 | 3 | $-0.5\pm 0.3\pm 0.3$ | $3.07\pm 0.02$ | $-2\pm 2$
| J0331-3824 | 2.435 | 2 | $0.5\pm 1.4\pm 0.3$ | $3.24\pm 0.04$ | $2\pm 7$
099.A-0018(A) | J1338+0010 | 2.300 | 2 | $-10.9\pm 6.7\pm 0.3$ | $3.09\pm 0.23$ | $-55\pm 34$
099.B-0118(A) | J0020-1540 | 2.020 | 3 | $-1.8\pm 2.5\pm 0.4$ | $2.87\pm 0.08$ | $-9\pm 13$
0101.A-0528(A) | J2124-4948 | 2.480 | 2 | $4.2\pm 2.4\pm 0.3$ | $3.28\pm 0.21$ | $21\pm 12$
0101.A-0528(B) | J0607-6031 | 1.097 | 3 | $8.5\pm 3.8\pm 0.5$ | $3.15\pm 0.17$ | $43\pm 19$
0101.B-0739(A) | J1352+1302 | 1.600 | 1 | $8.8\pm 12.2\pm 0.4$ | $4.11\pm 0.34$ | $45\pm 62$
| J1616+0931 | 1.468 | 2 | $0.8\pm 3.5\pm 0.5$ | $3.24\pm 0.13$ | $4\pm 18$
| J2239+1222 | 1.484 | 3 | $5.8\pm 6.7\pm 0.5$ | $3.41\pm 0.22$ | $29\pm 34$
0102.A-0335(A) | J0259+1635 | 2.160 | 3 | $-0.2\pm 0.2\pm 0.4$ | $3.09\pm 0.01$ | $-1\pm 2$
| J0630-1201 | 3.344 | 2 | $9.9\pm 9.7\pm 0.3$ | $3.48\pm 0.34$ | $50\pm 49$
| J1042+1641 | 2.519 | 4 | $-0.4\pm 0.5\pm 0.3$ | $2.84\pm 0.01$ | $-1\pm 2$
0103.A-0253(A) | J2245-2945 | 2.379 | 2 | $6.9\pm 3.7\pm 0.3$ | $2.96\pm 0.39$ | $35\pm 18$
189.A-0424(A) | J1018+0548 | 3.512 | 2 | $10.8\pm 6.9\pm 0.3$ | $3.21\pm 0.20$ | $55\pm 35$
| J1042+1957 | 3.628 | 1 | $30.8\pm 16.1\pm 0.2$ | $2.81\pm 0.24$ | $158\pm 82$
| J1117+1311 | 3.623 | 1 | $-4.3\pm 11.8\pm 0.2$ | $2.87\pm 0.25$ | $-22\pm 60$
| J1312+0841 | 3.735 | 1 | $15.4\pm 10.9\pm 0.2$ | $2.67\pm 0.22$ | $79\pm 56$
| J1621-0042 | 3.703 | 1 | $-6.9\pm 9.2\pm 0.2$ | $3.77\pm 0.24$ | $-35\pm 47$

###  3.3 Robustness of The Measurements

Results from the final sample were checked statistically. We first checked if
$\eta$ correlated with $A$ or not. The distribution of $\eta$ and $A$ measured
from the 86 spectra is shown in Figure 8. The Pearson correlation coefficients
of the LAE subsample, the QSO subsample and the whole sample are $-0.13$,
$-0.15$ and $-0.11$, respectively, indicating no correlation. This is
consistent with theoretical expectations, as Bahcall et al. (2004) discussed.
Using a bootstrap method described in Section 4.1, we calculated the average
values of $A$ of the LAE and the QSO subsamples, which are $3.03\pm 0.03$ and
$3.05\pm 0.06$, respectively. These values are consistent with $2.99\pm 0.02$
measured by Bahcall et al. (2004) and $2.96\pm 0.02$ measured by Albareti et
al. (2015) within 3$\sigma$, and are also consistent with the theoretical
value of 2.98 (Storey & Zeippen, 2000).

Figure 8:  Plots of $\eta$ and $A$ of the final sample measured by the MG
method. LAE results are in red, and QSO results are in blue. The size of the
dots represents the error of $\eta$ (larger for greater error). We did not
show the error of $A$, but those with greater error of $\eta$ generally have
greater error of $A$.

We then checked whether the measured $\sigma_{\rm stat}(\eta)$ is related to
the SNR and FWHM of the [O III] lines as expected. As can be seen in Figure 9,
$\sigma_{\rm stat}(\eta)$ is inversely related to SNR4960 for similar FWHM and
is positively related to FWHM for similar SNR4960. We verified this finding
through numerical simulations, as detailed in Appendix D. Numerical
simulations give quantitative correlations, as expressed in formulas D1 and
D2. In brief, $\sigma_{\rm stat}(\eta)$ is roughly inversely proportional to
SNR4960 and positively proportional to FWHM. In Figure 9, we show the relation
with FWHMs of 130 and 610 km s-1, which are the median FWHM values of LAE and
QSO subsamples, respectively. For both the subsamples, the measurements are in
good agreement with the theoretical expectations with scatters less than 0.4
dex.

Figure 9:  Relation between $\sigma_{\rm stat}(\eta)$ and SNR4960. The
pentagrams represent LAEs, and the dots represent QSOs. The colour represents
the [O III] FWHM (see the colour bar below). The two black dashed lines show
theoretical expectations for FWHMs of 130 and 610 km s-1, respectively.

We finally checked whether $\chi=(\eta-\eta_{0})/\sigma(\eta)$ obeys the
standard normal distribution or not, which should be the case if the true
value of all $\eta$ were $\eta_{0}$ and $\sigma(\eta)$ could be account for
the uncertainty in the measurement of $\eta$. Figure 10 shows the distribution
of $\chi$. The mean value and standard deviation of $\chi$ and the mean value
of $\chi^{2}$ are 0.11, 1.02 and 1.04, respectively. These values are
consistent with the theoretical values with a sample size 86, which are $0\pm
0.11$, $1\pm 0.08$ and $1\pm 0.15$ (68.3% confidence level), respectively. We
made a Kolmogorov-Smirnov (KS) test. We calculated a KS statistic value of
0.11, corresponding to a probability of 0.21, $>$0.05. Thus, we concluded that
the distribution of $\chi$ obeys the standard normal distribution.

Figure 10:  The distribution of $\chi$ and a comparison with the standard
normal distribution (red line).

##  4 Results of Variation in the Fine Structure Constant

We converted all the measurements of $\eta$ and their errors $\sigma(\eta)$
into values and errors of $\Delta\alpha/\alpha$ using equation (8). The
results are listed in Tables 3, 4, and 5, and shown in Figure 11. None of the
measurements with all 86 spectra shows a deviation of $\Delta\alpha/\alpha$
from 0 by more than 3$\sigma$, and accuracies are between $2\times 10^{-4}$
and $10^{-2}$.

According to equation (8), $\Delta\alpha/\alpha$ is proportional to
$\eta-\eta_{0}$. Hence $\chi=(\eta-\eta_{0})/\sigma(\eta)$ defined in section
3.3 equals $\frac{\Delta\alpha/\alpha}{\sigma(\Delta\alpha/\alpha)}$, and also
represents the deviation of $\Delta\alpha/\alpha$ from 0. In section 3.3, we
have shown that $\chi$ for the 86 spectra obeys the standard normal
distribution. This is consistent with the assumption that the truth value of
$\Delta\alpha/\alpha$ is 0, and the errors were estimated reasonably.

Figure 11:  The upper panel shows the measurements of $\Delta\alpha/\alpha$
using all 86 spectra. The lower panel shows those using 37 spectra with errors
$<10^{-3}$.

###  4.1 Averages

We calculated average values for the final sample and different subsamples
using two methods. The first is the weighted average method (WM). For values
$x_{i}=(\Delta\alpha/\alpha)_{i}$ and errors $\sigma_{i}$, we calculated the
weight as:

$w_{i}=\frac{1}{\sigma_{i}^{2}}.$ (12)

Thus, the weighted average is:

$\bar{x}_{\rm WM}=\frac{\sum_{i}w_{i}x_{i}}{\sum_{i}w_{i}},$ (13)

and its error is:

$\sigma(\bar{x})_{\rm
WM}=\frac{\sqrt{\sum_{i}(w_{i}\sigma_{i})^{2}}}{\sum_{i}w_{i}}.$ (14)

The second is the bootstrap method (BS, e.g., Bahcall et al., 2004). For a
sample containing $N$ elements, we randomly generated $10^{5}$ fake bootstrap
samples, each containing $N$ elements by put-back sampling. We calculated the
weighted averages of these bootstrap samples with equations (12) to (14).
These $10^{5}$ weighted averages typically obey a Gaussian distribution. We
adopted the mean value and standard deviation of this Gaussian distribution
$\bar{x}_{\rm BS}$ and $\sigma(\bar{x})_{\rm BS}$ as the bootstrap average and
its error. Both WM and BS methods have their advantages and disadvantages. On
the one hand, when the errors $\sigma_{i}$ are estimated inaccurately, the
error of average value by the BS method is more accurate. For example, suppose
all the errors are overestimated or underestimated by a constant factor; the
error by the BS method will not change, but the error by the WM method will be
overestimated or underestimated correspondingly. On the other hand, when the
sample size $N$ is small or the errors $\sigma_{i}$ vary significantly from
one another, the weighted averages of bootstrap samples in the BS approach may
not obey Gaussian distribution. In this case, the results of the WM method are
more reliable. For each sample or subsample, we calculated the average value
of $\Delta\alpha/\alpha$ and its error using both methods, and the results are
listed in Table 6.

The average of $\Delta\alpha/\alpha$ of all the 86 spectra is $(-3\pm 6)\times
10^{-5}$ and $(-3\pm 5)\times 10^{-5}$ using WM and BS methods, respectively.
The two values are consistent and indicate no deviation between
$\Delta\alpha/\alpha$ and 0 in the $6\times 10^{-5}$ error level. We
calculated the averages of the LAE and QSO sub-samples and further divided the
former into SA and other LAE sub-samples and calculated the averages. For each
subsample, the averages by both two methods agree with 0.

We found that for the SA subsample, the error of average by the BS and WM
methods have a great difference, while for other subsamples, the difference is
slight. This may be because the SA subsample has a small size $N$. Because we
have shown that the errors $\sigma(\Delta\alpha/\alpha)$ are estimated
reasonably, we accept the results by the WM method as the final results.

The above results are based on $\eta$ measured by the MG method. As described
in section 3.1, there are 17 spectra with which $\eta$ can be measured by the
PM method, including 6 spectra of SA. The averages of $\Delta\alpha/\alpha$
measured by the PM method are shown in Table 6, which are also consistent with
0. This indicates that the coincidence of $\Delta\alpha/\alpha$ and 0 does not
depend on how we measured $\eta$.

Table 6: Average $\Delta\alpha/\alpha$ for different samples subsample | N | $\Delta\alpha/\alpha\times 10^{5}$
---|---|---
|  | WM | BS
All | 86 | $-3.1\pm 6.0$ | $-3.1\pm 5.0$
All QSO | 46 | $-7.6\pm 11.1$ | $-7.3\pm 10.6$
All LAE | 40 | $-1.2\pm 7.2$ | $-1.3\pm 6.2$
SA | 6 | $-4.1\pm 8.7$ | $-4.3\pm 5.6$
Other LAE | 34 | $4.9\pm 12.7$ | $5.1\pm 13.4$
SA PM | 6 | $-9.1\pm 8.9$ | $-9.2\pm 5.3$
All good PM | 17 | $-2.6\pm 7.9$ | $-2.6\pm 6.8$
$1.09<z<1.66$ | 18 | $48\pm 31$ | $47\pm 31$
$2.01<z<2.57$ | 60 | $-5.7\pm 6.2$ | $-5.8\pm 5.0$
$3.21<z<3.74$ | 8 | $39\pm 55$ | $41\pm 119$
z2 without SA | 54 | $-7.3\pm 8.8$ | $-7.3\pm 8.3$

Due to telluric absorption bands, the whole sample can be naturally divided
into three subsamples in three redshift ranges. The averages of
$\Delta\alpha/\alpha$ of the three subsamples, listed in Table 6, are all
consistent with 0. The errors of these averages are $3\times 10^{-4}$,
$6\times 10^{-5}$, and $5\times 10^{-4}$, respectively. Because the results
obtained from the spectra of SA ($z=2.37$) have high precision, one may worry
that the average value in the redshift range of $2.01<z<2.57$ is greatly
affected by SA and cannot be treated as the representative value of this
redshift range. Thus, we calculated the average of $\Delta\alpha/\alpha$
obtained from other 54 spectra in this redshift range, and the result agrees
with 0 with an error of $9\times 10^{-5}$.

###  4.2 Variation over Time

We do not find any variance of $\Delta\alpha/\alpha$ over time because it
agrees with 0 in all the redshift ranges. We limited the rate at which
$\alpha$ may vary, which can be used to constrain cosmological models further.

The redshifts of LAEs and QSOs in our sample are in the range of 1.097–3.735,
corresponding to look-back time ($t_{\rm LB}$) of 8.2–12.0 billion years, or
the age of the universe of 1.7–5.5 billion years. We assumed that $\alpha$
varies uniformly over time from then to now:

$\frac{\Delta\alpha}{\alpha}=kt_{\rm LB}.$ (15)

The $k$ here means the same as $\alpha^{-1}{\rm d}\alpha/{\rm d}t$ in Bahcall
et al. (2004). We fit the $\Delta\alpha/\alpha$ measurements of all the 86
spectra, and obtained $k=(-3\pm 6)\times 10^{-15}$ yr-1.

We then included $\Delta\alpha/\alpha$ measured by Albareti et al. (2015)
using SDSS QSO spectra (see their Table 3) into the rate limitation. Note that
we did not use the data in the redshift bin 0.580–0.625 in their results
because it may be seriously affected by the SELs. If only using measurements
at $z<1$, the result is $k=(2\pm 5)\times 10^{-15}$ yr-1. And by combining the
measurements at $z<1$ and our measurements at $1.0<z<3.8$, we obtained
$k=(0\pm 4)\times 10^{-15}$ yr-1. This is by far the most precise limitation
of the rate using the [O III] doublet method. Moreover, our results extend the
limitation to an era when the universe is only 2 to 5 billion years old.

Figure 12:  Tests of whether $\alpha$ changes over time. The upper panel shows
the result using only our sample, and the bottom panel shows the result using
both our sample and the measurements using SDSS QSO spectra by Albareti et al.
(2015). The best-fitting model assuming that $\alpha$ varies at a uniform rate
over time is shown in red, and the 1$\sigma$ confidence interval is shown in
green shade.

##  5 Discussions

###  5.1 Considerations on Selecting LAE and QSO for $\alpha$ Measurement

We discuss the reliability and necessity of our final sample selection
criteria.

First, we only selected spectra where SNRs of both the [O III] doublet lines
are greater than 10. An SNR of $>$10 guarantees that the contours of the
$\chi^{2}(\eta,A)$ surface near the minimum value are close to ellipses, and
hence, the probability density distributions of $\eta$ and $A$ are close to
Gaussian, which is convenient for subsequent analysis. In addition, the $\eta$
measured from spectra with low [O III] SNR have large errors and hence have
small weights when calculating average and rate. Therefore, discarding them
hardly affects the precision of the final results. Some previous work studying
$\alpha$ variation with optical QSO spectra used the SNR of $\lambda$5008 as a
criterion for sample selection. We suggest the SNR of $\lambda$4960 line to be
included in the criteria when using NIR spectra, because it is not strictly
correlated with the SNR of $\lambda$5008 line (Figure D1) due to the influence
of telluric lines.

Second, we added masks to pixels affected by SELs, TALs, cosmic rays and bad
CCD pixels, and excluded these pixels when measuring line wavelengths. In
addition, we excluded spectra where many pixels near [O III] doublet are
affected, using a mask index $I_{\rm mask}$ for quantification. Note that this
approach differs from previous works using optical spectra, where pixels
affected by SELs and TALs were included in the measurements but were given
lower weights due to larger errors. In analysis concerning NIR spectra,
systematic errors caused by corrections for SELs and TALs must be addressed,
and therefore, the errors of the pixels where strong SELs and TALs are located
may be significantly underestimated. Fortunately, the VLT/XShooter spectra
have high spectral resolutions, so the impacting ranges of single SEL or TAL
are narrow. For a typical XShooter spectrum, the proportion of pixels strongly
affected by SELs or TALs is generally between 10% and 20%, except for
wavelength ranges where telluric absorption lines are dense. Most spectra can
pass the criterion that the sum of the $I_{\rm mask}$ of the doublet lines is
less than 0.5. Even if the affected pixels are directly masked for these
spectra, the remaining pixels are sufficient to achieve reliable wavelength
measurements.

Finally, we had two additional requirements for QSO spectra compared with
previous works. One is that there should be no strong Fe II bump, and the
other is that the doublet lines should not blend severely. For these two
requirements, we measured Fe II index $I_{\rm FeII}$ and blending index
$I_{\rm blend}$ as the criteria. We tested whether the two criteria are
reasonable by examining the spectra they excluded. For a continuum-subtracted
spectrum, we obtained the profiles of the doublet emission lines in wavelength
ranges corresponding to a velocity range of $-600$ to $600$ km s-1. We made
linear interpolation for the profile of $\lambda$5008 to align it with the
profile of $\lambda$4960 in velocity space, and then calculated the Pearson
correlation coefficient between them. This coefficient represents the
consistency of the shapes of the doublet lines. In Figure 13, we show the
coefficients of QSO spectra excluded by the Fe II or the blending criterion,
those of QSO spectra in the final sample, and the $A$ values measured using
these spectra. We only show the results of spectra with SNR${}_{5008}>50$ to
ensure the accuracy of the parameters. We found that the doublet profiles have
a good correlation in the final sample as the coefficients have a median value
of 0.95 and are all greater than 0.87, and the measurements of $A$ are all
around the theoretical value of 3. However, the correlation between the
doublet profiles is not as good in QSO spectra excluded by the Fe II
criterion, as the coefficients have a median value 0.83. Also, a large $A$
value of $>$4 is measured for more than half of these excluded spectra. The
likely reason is that the Fe II bump under the [O III] doublet causes the
continuum to be fit improperly. The correlation can be poor in QSO spectra
excluded by the blending criterion as more than half of coefficients are
$<$0.5, and the measured values of $A$ spread over a wide range. The likely
reason is the mistaken decomposition of the very broad [O III] emission
components. The anomalies in the correlation coefficients and measurements of
$A$ indicate that these spectra are unsuitable for the measurements of doublet
wavelengths. Hence, our criteria are necessary and practical.

Figure 13:  We present the correlation coefficients of the doublet line
profiles and the measurements of $A$ for QSO spectra in the final sample
(black pentagram), excluded by the Fe II criterion (red box) and excluded by
the blending criterion (blue origin).

According to the initial redshift limit of $1.07<z<3.77$, 95 LAE spectra and
601 QSO spectra were selected. However, the final sample only contains 40 LAE
and 46 QSO spectra. The passing rate of LAE and QSO subsamples are 42% and 8%,
respectively. About 20% of the LAE spectra are excluded because of SELs: in
some cases, the mask index is too large, and in other cases, the SELs result
in insufficient SNR of [O III] . This fraction is similar to the fraction of
pixels strongly affected by SELs, which is 10%–20% as we previously estimated.

QSO spectra have a lower pass rate. A few spectra are excluded due to
observational factors, such as short exposure time causing insufficient [O
III] SNR or inappropriate redshift causing [O III] to fall into the telluric
absorption bands. These factors are similar for QSOs and LAEs. However, more
than half of the spectra are excluded because of the QSOs’ intrinsic
properties, such as the weak [O III] relative to the continuum, the blending
[O III] doublet, or a strong Fe II bump. The composite of the 46 QSO spectra
in the final sample, displayed in Figure 14, differs from that of general
QSOs. To demonstrate this difference, we also present other QSO composite
spectra for comparison, including that of Selsing et al. (2016) using XShooter
data of $1<z<2$ high-luminosity QSOs, and that of Vanden Berk et al. (2001)
using SDSS data (of $z<1$ QSOs for the wavelength range around [O III] ). The
QSOs in our sample have much stronger [O III] lines, and the EW of [O III]
lines in their composite spectrum can reach 31.9 Å, much higher than those in
the other two composite spectra (Table 7). Also, they have weaker Fe II bumps
as the Fe II indexes are smaller. Thus, QSOs suitable for $\alpha$ measurement
may belong to a particular type, accounting for a small proportion of the
total number of QSOs. This may be the main reason for the low pass rate of QSO
spectra. Assuming that the proportions of LAEs and QSOs excluded due to
observational factors are similar, with a pass rate of $\sim 40$%, we
estimated that only 20% of the QSO spectra are suitable for $\alpha$
measurement.

Figure 14:  The composite spectrum of QSOs in our final sample (black). We show XShooter (blue) and SDSS (red) QSO composite spectra for comparison. All three spectra are normalized at 5100 Å. The insert panel shows the partial enlargement near the [O III] doublet.  Table 7: Comparison between our QSO sample and QSO composite | our | XshQ | SDSSQ
---|---|---|---
z | 1.09–3.73 | 1–2.1 | $<1$
$\nu L_{\nu}(5100\AA)$ (erg s-1) | 8e45 |  $\sim$1e47 |  $\sim$3e44
$\alpha_{\lambda}$ (4450–5500 Å) |  $-$1.7 |  $-$2.7 |  $-$1.1
$[$OIII] $\lambda$5007 EW ($\AA$) | 31.9 | 10.0 | 13.4
$[$OIII] FWHM (km s-1) | 510 | 970 | 460
$I_{\rm 4590}$ ($\AA$) | 7.6 | 12.0 | 18.1
$I_{\rm 5250}$ ($\AA$) | 9.7 | 16.3 | 16.3

###  5.2 Implications for Future Works

We suggest that LAE spectra have more advantages than QSO spectra studying of
$\alpha$ variation using the [O III] doublet method. The reasons are as
follows.

First, the [O III] doublet lines are narrower in LAE spectra than those in
QSOs’ spectra, as the median values of the [O III] FWHM in the LAE and QSO
subsamples are 130 and 610 km s-1, respectively, with a difference of 4.7
times. The value of the FWHM affects the statistical error of $\eta$, as
narrated in Appendix D, where $\sigma_{\rm stat}(\eta)\propto{\rm FWHM^{3/2}}$
for the same flux. Therefore, a difference of 4.7 times in the values of FWHM
would lead to a 10 times difference in $\sigma_{\rm stat}(\eta)$ if other
conditions were the same. In our sample, the resultant precision of
$\Delta\alpha/\alpha$ from the LAE and QSO subsamples are on the same level,
for the higher luminosities of the [O III] lines in the QSOs’ spectra
compensate for their disadvantage in the FWHM. The median [O III] luminosity
of QSOs ($10^{43.7}$ erg s-1) in our sample is 10 times that of LAEs
($10^{42.7}$ erg s-1), as can be seen in Figure 4.

Second, the volume density of LAEs suitable for studying $\alpha$ variation is
much higher than that of QSOs at $z>1$. We calculated the volume density of
LAEs with Ly$\alpha$ luminosity greater than the mean value of our LAEs, and
that of QSOs with continuum luminosity greater than the mean value of our
QSOs, both at $z\sim 2$. We extracted the spectra on the UVB arm of $z\sim 2$
LAEs in our sample and measured the Ly$\alpha$ luminosities, which have a
median value of $10^{43.1}$ erg s-1. Using the Ly$\alpha$ luminosity function
of $z=2.2$ LAEs from Konno et al. (2016), we estimated the volume density of
LAEs with $L_{\rm Ly\alpha}>10^{43.1}$ erg s-1 is $\sim 5\times 10^{-5}$
Mpc-3. We measured the G-band absolute magnitudes of the QSOs in our sample,
with a median value of $-26.2$. Using the continuum luminosity function of
$2.2<z<2.6$ QSOs from Croom et al. (2009), we estimated that the number
density of QSOs with $g<-26.2$ is only $\sim 5\times 10^{-7}$ Mpc-3. In
addition, only a small fraction of QSOs have strong and narrow [O III] lines
and weak Fe II bumps. We estimated this fraction at 20% in section 5.1. As a
result, the number density of LAEs suitable for $\alpha$ measurement is 2–3
orders of magnitude higher than QSOs.

Finally, systematic errors may be larger and more challenging to analyze when
measuring $\alpha$ with QSO spectra. In this work, we considered two possible
sources of systematic error for QSOs, the blending of [O III] doublet and Fe
II bump. We selected QSOs with [O III] not severely blended and with weak Fe
II. The resultant $\eta/\sigma(\eta)$ obeys the standard normal distribution,
implying that the possible systematic errors from these two sources hardly
affect the results of our work. However, if the method is applied to future
$\alpha$ measurements with higher precision, these systematic errors may
become significant as other errors decrease. In addition, theoretically, there
are some other sources of systematic errors, such as the uncertainty of the
shape of H$\beta$ broad emission lines, weak QSO emission lines, and others.
These errors may also affect future works using QSO spectra for accurate
$\alpha$ measurement. In LAE spectra, the continuum is weak and mainly comes
from starlight. Thus, the modelling of the LAE continuum is more reliable than
the modelling of the QSO continuum.

We discuss the possible precision of future $\alpha$ measurements using
spectroscopy of LAE and other starburst galaxies at $z<2$. This work obtained
an error of $\Delta\alpha/\alpha$ of $7\times 10^{-5}$ using 40 XShooter
spectra of LAEs. The error is $9\times 10^{-5}$ if only using 6 spectra of SA
and is $1.3\times 10^{-4}$ if only using the other 34 spectra. To obtain an
error of $\Delta\alpha/\alpha$ below $10^{-6}$ to test the measurements using
the many-multiplet method, $N>50,000$ spectra similar to the SA’s or
$N>600,000$ spectra similar to other LAEs’ are required, assuming that the
error is inversely proportional to $\sqrt{N}$. To achieve the required sample
size, observing multiple spectra of starburst galaxies simultaneously using
multi-object spectrometers is necessary. The ongoing project Dark Energy
Spectroscopic Instrument (DESI, DESI Collaboration et al., 2016) may meet the
sample size demand. The DESI project is expected to observe about a dozen
million emission line galaxies, of which $\sim$7 million at $z<0.95$, suitable
for the study of $\alpha$ variation using [O III] doublet as the wavelength
range is 3600–9800 Å. Spectra with sufficiently high [O III] SNRs may be on
the order of $10^{5}\sim 10^{6}$. Because the sensitivity, resolution and
sample size of DESI are all better than SDSS, the final accuracy of $\alpha$
measurement must be better than the existing results using SDSS spectra ($\sim
2\times 10^{-5}$), and might be comparable with the results using the many-
multiplet method at present.

So far, the highest redshift at which $\alpha$ variation has been measured is
7.1 (Wilczynska et al., 2020). Their work used the absorption lines in the
VLT/XShooter spectra of QSO J1120+0641 at $z=7.085$. Measuring $\alpha$
variation at higher redshift using QSO absorption lines is challenging because
QSOs at $z>7$ are extremely rare. Fortunately, a number of $z>7$ LAEs have
been spectroscopically identified (e.g., Vanzella et al., 2011; Stark et al.,
2017; Jung et al., 2020; Endsley et al., 2021). Their spectral energy
distributions suggest they have strong [O III] emission lines. These LAEs can
be used to constrain the variation of $\alpha$ at $z>7$ as long as deep mid-
infrared spectroscopic observations are obtained because the [O III]
wavelengths are redshifted to $>4$ $\mu$m.

The Near InfraRed Spectrograph (NIRSpec) mounted on James Webb Space Telescope
(JWST, Gardner et al., 2006) can take spectrum in a wavelength range of
0.6–5.3 $\mu$m, and hence can observe [O III] emission lines of LAEs at
$z<9.4$. Depending on the observational mode, the spectral resolution $R$ can
be 100, 1000 or 2700. Assuming that the LAEs have an intrinsic [O III] FWHM of
130 km s-1 and are observed with JWST/NIRSpec with $R$ of 1000 or 2700, we
estimated the observed [O III] FWHM to be 300 or 170 km s-1 using equation D3
in Appendix D. Assuming an SNR of $\lambda$4960 line of 10, we further
estimated that the precision of $\Delta\alpha/\alpha$ measurement could reach
$(2\sim 3)\times 10^{-3}$. Higher precision could be obtained if the SNR is
higher or more than one LAE is observed. This measurement will, for the first
time, provide a constraint on the variation of $\alpha$ at $7.1<z<9.4$,
corresponding to an age of the universe of only 500–700 million years.

##  6 Conclusions

We constructed a sample of 86 public NIR spectra observed by VLT/XShooter of
LAE (40) and QSOs (46) at $1.09<z<3.73$. The selection criteria of the sample
concern the SNRs of the [O III] doublets and three indexes: the first
describes to what extent the data is affected by SELs, TALs, cosmic rays and
bad CCD pixels, the second describes the strength of Fe II bump in the QSO
spectrum, and the third describes the blending of the [O III] doublet. We
measured the parameter $\eta$ describing the ratio of the wavelengths of the
[O III] doublet. We further inferred $\Delta\alpha/\alpha$, the difference
between $\alpha$ in the distant universe and that in the laboratory. When
measuring $\eta$, we tried a Multiple-Gaussian method and a Profile-Matching
method, in which the former applies to all spectra (86) while the latter only
applies to a part (17), and we adopted the results using the former method.
Reassuringly, the two methods yield consistent results for spectra that are
applied to both. Our main results are as follows.

1\. The $\Delta\alpha/\alpha$ measured using the 86 spectra are all consistent
with 0 by $<3\sigma$. The ratio of $\Delta\alpha/\alpha$ to its error obeys
the standard normal distribution, which supports the assumptions that the
truth value is 0 and that our estimates of errors are reliable.

2\. The weighted average value of $\Delta\alpha/\alpha$ agrees with 0, whether
the whole sample or different subsamples is used. If using all the 86 spectra,
the average is $(-3\pm 6)\times 10^{-5}$. The average values using the LAE and
QSO subsamples have similar errors of $7\times 10^{-5}$ and $1.1\times
10^{-4}$, respectively.

3\. The average $\Delta\alpha/\alpha$ at three redshifted ranges of
$1.09<z<1.66$, $2.01<z<2.57$ and $3.21<z<3.73$ are $(5\pm 3)\times 10^{-4}$,
$(-6\pm 6)\times 10^{-5}$ and $(4\pm 5)\times 10^{-4}$, respectively, all in
accordance with 0. We found no variation of $\alpha$ over time. We limited the
rate at which $\alpha$ varies to be $k=(-3\pm 6)\times 10^{-15}$ yr-1 since
$z=3.73$ using our measurements. By combining our results with the
measurements at $z<1$ using SDSS QSO spectra, we limited the rate to be
$k=(0\pm 4)\times 10^{-15}$ yr-1.

In addition, our results suggest that starburst galaxies’ spectra have a
better application prospect than QSO spectra in future studies of $\alpha$
variation.

## Acknowledgements

This work is based on observations obtained with the Very Large Telescope,
programs 084.A-0303, 086.B-0320, 087.A-0610, 087.B-0229, 088.A-0672,
088.B-1034, 089.B-0275, 089.B-0936, 089.B-0951, 0909.A-0830, 090.B-0424,
091.A-0413, 091.B-0900, 092.A-0391, 092.B-0860, 093.A-0882, 093.B-0553,
094.B-0111, 095.B-0507, 096.A-0348, 097.A-0153, 098.B-0556, 099.A-0018,
099.A-0254, 099.A-0758, 099.B-0118, 101.A-0528, 101.B-0262, 101.B-0739,
101.B-0779, 102.A-0335, 102.A-0391, 102.A-0652, 103.A-0253, 103.A-0688,
103.B-0446, 104.A-0236, 189.A-0424. Based on data products from observations
made with ESO Telescopes at the Paranal Observatory under ESO programme ID
179.A-2005.

## Data Availability

The data underlying this article are available in the ESO data archive at
http://archive.eso.org.

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##  Appendix A Details of Data Reduction

We briefly introduce the pip-2D data generated by the XShooter pipeline. The
data is contained in fits files with three header data units (HDU). As shown
in Figure A1(a), the first HDU stores the 2D spectrum of a target (in the unit
of ADU); the second stores the flux error of spectrum; the third stores the
bad CCD map, marking those pixels affected seriously by bad CCD pixels, cosmic
rays, and other factors. The 2D spectrum had been straightened and aligned,
with the wavelength direction going horizontally at a fixed interval of
0.6Åand the spatial direction going vertically. This kind of 2D spectrum was
produced by combining the 4 frames taken in 4 consecutive exposures under the
“ABBA” dither mode. Using an A$-$B$-$B$+$A algorithm, the pipeline eliminated
the sky emission and left 3 images of the target on the combined frame, among
which the bright image originated from the two exposures labelled as A and
have positive fluxes and the other two dark images originated from the two
exposures labelled as B and have negative fluxes.

We extracted 1-D spectrum based on the pip-2D data as follows.

First, the aperture for extracting the spectra from the three images was
determined, as shown in Figure A1(b). Almost all the targets are point-like
sources, or close to point-like sources, and their brightness profiles along
the spatial direction, cutting at the non-emission wavelengths for QSOs and at
the [O III] $\lambda$5008 emission lines for LAEs, were well fitted by
Gaussian functions. The best-fitting Gaussian models give positions of the
three images and the FWHMs of their brightness profiles, in which the latter
were used to determine the width of the aperture for extracting the 1-D
spectra. The only exceptional target whose brightness distribution cannot be
approximated by a Gaussian function is SA, and we will describe its extraction
in another paragraph.

Second, spectra were extracted from the three images and merged into one
spectrum, as shown in Figure A1(c). Using the previously determined aperture
width, we extracted the 1-D spectra from the three images, to which the first
HDU contributed the flux and the second HDU the error. With reference to the
third HDU, we prepared a boolean value named “mask” for each wavelength of the
1-D spectra. The value was set to 0 if the data quality of all the pixels
inside the aperture at this wavelength is 0, and to 1 otherwise. The three
spectra were then rescaled to eliminate the small differences between their
fluxes caused by weather changes, and the negative fluxes were turned
positive. After the rescaling, new contaminated pixels were identified, at
which the flux of one spectrum deviates from those of the other two spectra by
more than 4.5$\sigma$. These pixels were affected by cosmic rays or other
adverse factors but had not been identified by the pipeline, so we changed
their mask values to 1. We calculated the weighted average of the three
spectra using pixels with mask values of 0 and obtained a 1-D spectrum. This
averaged spectrum was also assigned a mask value at each wavelength: it was
set to 1 if the mask values in the three spectra are all 1, and to 0
otherwise. Besides, a few observations were not taken under the dither mode
and each had only one image in its pip-2D data. In these cases, the rescaling
and the averaging were skipped.

Third, for each target, we calibrated the flux of its 1-D spectra and
corrected the telluric absorption, as shown in Figure A1(d). The flux of the
1-D spectra of a target was calibrated using the sensitivity function (the
conversion from ADU to flux) stored in its flux-calibrated pip-1D data. As
telluric absorptions appeared in the targets’ spectra, they must have also
appeared in the spectra of standard stars observed close in time. These
standard stars’ spectra, which belong to project 60.A-9022(C), can be obtained
from the ESO archive. So, to correct the telluric absorption in each target’s
spectrum, we built a telluric absorption model using the spectrum of the
standard star, which was observed closest in time and of whose continuum the
SNR is $>$300 in the wavelength range of the NIR arm. For the spectrum of each
standard star, we searched for a similar stellar spectrum in the XShooter
Spectral Library (Gonneau et al., 2020) and considered it as the template, and
a telluric absorption model can be calculated by dividing the observed
spectrum by the template. Corrected QSO spectra using such telluric absorption
models were examined in the continuum. The correction was successful at most
of the wavelengths. At some wavelengths with severe telluric absorption, the
correction was less successful, which may be caused by the inhomogeneity or
the variability in the properties of the atmosphere.

Finally, for each spectrum pixel, once it meets any of the following
situations, we changed its mask value to 1. The situations are: 1\. The
telluric absorption model of the pixel is below 0.5. 2\. A strong SEL affects
the pixel. We adopted the list of SELs from Oliva et al. (2015), and selected
184 strong lines with flux rates $>1000$. With the information on these lines
and with that revealed by the 1-D error spectrum of each target, we identified
the sky emission lines in each spectrum and calculated their width for
masking, in which an algorithm was involved to give greater widths for
stronger lines. 3\. The pixel lies near the [O III] doublet and is affected by
visually identified cosmic rays that the pipeline had missed. The final mask
values label pixels affected by strong SELs, TALs, cosmic rays and CCD bad
pixels.

We extracted a 1-D spectrum for each pip-2D data. For a target, if there were
more than one group of exposures taken under the dither mode and hence
multiple pip-2D data in the ESO archive, we combined the extracted spectra, as
we had done in combining spectra of the three images in the same pip-2D data.

Figure A1:  Example of a typical approach of extracting 1-D spectrum. (a):
Display of pip-2D data. (b): Example of brightness distribution (black line)
of QSO continuum and determination of apertures for extracting (red shade).
(c): Example of combining the spectra of the three images. (d): Example of
correction for telluric absorption.

SA is a lensed galaxy with multiple clumps, which could be star clusters. The
VLT/XShooter observation of SA has been described in detail by Vanzella et al.
(2020). This observation contained 3 exposures, during which the slit was
placed differently but covered at least two clumps each time. The 2-D spectra
of the three exposures are displayed in Figure A2. We extracted 6 spectra
using 6 apertures shown in the red line in the figure, denoted as SA1 to SA6.

Figure A2: The apertures for extracting the spectra of SA.

##  Appendix B Applicability of Profile-Matching Method

As we have demonstrated in section 3.1, the $\chi^{2}(\eta,A)$ surfaces
obtained by the MG method are smooth, and the contours are close to ellipses,
and the best estimates and statistical errors of $\eta$ and $A$ can be
reliably obtained. However, the results obtained by the PM method do not
guarantee these properties. We found two types of problems. We illustrate each
type with an example (Figure B1) and analyze the possible origin of the
problem.

We show the first type of problem with LAE J0332-2746 as an example. When the
PM method was used, a feature similar to a geological fault occurs on the
$\chi^{2}$ surface. Accordingly, jumps can be seen on the $\chi^{2}(\eta)$
curve which are roughly evenly spaced with an interval of
$\Delta\eta=\frac{0.6\AA}{\lambda_{1}}=\frac{1.2\times 10^{-4}}{1+z}$. Such
jumps in the values of $\chi^{2}$ come from the changes in the pixels of the
$\lambda$4960 line included in the matching. When $\eta$ increases, the
rightmost pixel is excluded, and at the opposite end, a new pixel joins in to
become the new leftmost pixel. The jumps are not obvious in the range of
$\chi^{2}(\eta)<\chi^{2}_{\rm min}+1$ when the $\lambda$4960 line has high SNR
and small FWHM, but they can be very prominent when the $\lambda$4960 line
suffers from great noises, which will make the measurement of $\eta$
unreliable.

We then show the second type with LAE J1001+0206 as an example. When measured
by the PM method, wavy structures appear on the $\chi^{2}$ surface. The origin
of this structure is as follows. Strong noise causes fault signals to be
mistaken for structures on the line profiles. When the fault signals from the
two emission lines happen to be aligned, a local minimum appears on the
$\chi^{2}(\eta)$ curve. The width of the [O III] emission lines’ profile plays
an important role in the emergence of such wavy structures. The broader the [O
III] lines, the more likely the wavy structures will appear.

By visually examining the $\chi^{2}$ surfaces, we selected 17 spectra with
which $\eta$ can be reliably measured by the PM method, including 16 LAE and 1
QSO spectra. The [O III] emission lines in these spectra are narrow, have high
SNR and are little affected by masks. Notably, the FWHMs of these spectra’s [O
III] emission lines span within 16 pixels. The previous work of (e.g., Bahcall
et al., 2004) used QSOs’ SDSS spectra, in which the typical [O III] emission
lines’ FWMHs of 400-1000 km s-1 span for 6–14 pixels. These are informative to
the applicability of the PM method. We suggest that for spectra taken by
Xshooter, SDSS or other spectrometers, as long as the [O III] emission lines’
FWMHs span no more than 14–16 pixels, the application of the PM method can be
considered.

Figure B1:  Two examples of spectra with which $\eta$ can be reliably measured
by the MG method but not by the PM method. The top row shows the spectra, the
middle two rows show the $\chi^{2}(\eta,A)$ surface and $\chi^{2}(\eta)$ curve
obtained by the MG method, and the bottom two rows show those by the PM
method.

##  Appendix C The Multi-Gaussian Fitting Results

We present all 86 spectra and the best-fitting results using the multiple-
Gaussian model in Figures C1, C2, C3 and C4.

Figure C1: 6 SA’s spectra and best-fitting models. Figure C2: 34 other LAEs’
spectra and best-fitting models. Figure C3: 46 QSOs’ spectra and best-fitting
models (part 1). Figure C4: 46 QSO spectra and best-fitting models (part 2).

##  Appendix D Factors Affecting Measurement Precision of $\eta$

Factors that affect the precision of $\eta$ include the width and the SNRs of
the [O III] doublet, and we explored their influence using simulated spectra.
Assuming that both [O III] lines have the same Gaussian profile, we generate
spectra in which the FWHMs and SNRs of the [O III] doublet vary independently.
In a simulated spectrum, the FWMHs of the doublet were set to be equal, and
the SNRs of the doublet, which were adjusted by assigning varied flux errors,
are uncorrelated and can be different. Measurements on $\eta$ using these
simulated spectra revealed a relation between $\sigma_{\rm stat}(\eta)$ and
the FWHM and SNRs of the doublet, which can be expressed as:

$\sigma_{\rm stat}(\eta)\approx 1.9\times 10^{-4}\left(\frac{\rm FWHM}{\rm
100\ km\ s^{-1}}\right)\sqrt{\frac{1}{\rm SNR_{4960}^{2}}+\frac{1}{\rm
SNR_{5008}^{2}}},$ (D1)

This relation holds when the FWHM is 100–1200 km s-1 and SNRs are 10–500.

We plot the observed [O III] SNRs of the final sample in Figure D1. Figure D1
also shows the contour of the values of $\sigma_{\rm stat}(\eta)$ with the
FWHM of the doublet fixed at 300 km s-1. In our sample, the values of ${\rm
SNR}_{5008}$ are greater than those of ${\rm SNR}_{4960}$ for all the spectra,
and their average ratio is 2.06. By assuming that SNR${}_{5008}=2.06{\rm
SNR}_{4960}$, we can simplify Equation D1 as:

$\sigma_{\rm stat}(\eta)\approx 2.1\times 10^{-6}\left(\frac{\rm
SNR_{4960}}{100}\right)^{-1}\left(\frac{\rm FWHM}{\rm 100\ km\
s^{-1}}\right).$ (D2)

In the simulation, when we were adjusting the SNRs of the doublets by
assigning varied flux errors, we found that the values of SNR follow such
relation with the value of FWHMs and the flux errors: ${\rm SNR}\propto{\rm
FWHM}^{-1/2}{\rm Err}$. Hence if considering the influence of the FWHM alone,
$\sigma_{\rm stat}(\eta)$ will be proportional to ${\rm FWHM}^{3/2}$.

Figure D1:  The contours show the simulated relation between $\sigma_{\rm
stat}(\eta)$ and SNRs of the doublet lines when FWHM$=300$ km s-1 is assumed.
The data points show the observed [O III] SNRs in the final sample, red for
LAEs and blue for QSOs. The dashed line shows the relation assuming
SNR${}_{5008}=2.06{\rm SNR}_{4960}$.

The spectral resolution $R$ was generally considered to be one of the factors
affecting the measurement precision. Higher resolution leads to more accurate
wavelength calibration because the calibrating emission lines of arc lamps and
SELs will be narrower and more resolved, offering more information for finding
the wavelength solution. It then helps reduce the systematic error
$\sigma_{\rm sys}(\eta)$ originating from the wavelength calibration. Higher
resolution also helps reduce the statistical error $\sigma_{\rm stat}(\eta)$,
because the SELs and TALs will be narrower and affect fewer pixels, leading to
an increase in the SNRs of [O III] lines and hence reducing $\sigma_{\rm
stat}(\eta)$. In addition, when the resolution is poor, the observed lines
will appear broader. Assuming that the intrinsic profile of a [O III] emission
line is a Gaussian with a FWHM of ${\rm FWHM_{intr}}$, and assuming that the
broadening function of spectrometry is also a Gaussian, the observed FWHM
(${\rm FWHM_{obs}}$) of this line is roughly:

${\rm FWHM_{obs}^{2}=FWHM_{intr}^{2}}+(c/R)^{2},$ (D3)

where c is the speed of light. Higher resolution leads to smaller ${\rm
FWHM_{obs}}$, and then smaller $\sigma_{\rm stat}(\eta)$. Nevertheless, to
what extent can $\sigma_{\rm stat}(\eta)$ be reduced by improving the
resolution is limited by ${\rm FWHM_{intr}}$. For Xshooter, the resolution $R$
is 4200–8100 (Tables 1 and 2), corresponding to $c/R$ of 37–70 km s-1. To a [O
III] emission line with ${\rm FWHM_{intr}}$ of 100 –- 150 km s-1, the
instrumental broadening from the Xshooter will bring an increase in ${\rm
FWHM_{obs}}$ of about 10%–30%. When the [O III] emission line’s ${\rm
FWHM_{intr}}$ is above 150 km s-1, the increase will be no more than 10%. For
our sample, the observed FWHMs of the LAE subsample are affected by $<$30%,
while those of the QSO subsample are almost unaffected. In summary, increasing
resolution benefits on improving the precision of $\eta$ measurement. In
future studies, if the LAE spectra are used for constraining $\alpha$
variation, we suggest that the resolution $R$ should be no less than 4000.
Otherwise, the observed width of the [O III] lines would be significantly
larger than the intrinsic width, and the measurement precision would be
unpromising.