Measuring the Time Variation of the Fine-structure Constant with Quasars Detected by LAMOST

Fundamental physical constants are assumed to be universal and constant under the current Standard Model of particle physics. However, some modern theories beyond the Standard Model predict that the fundamental constants of nature may vary across space in time (Martins 2017). In the last few decades, many efforts have been devoted to search for possible variations of these constants, either through laboratory experiments or astrophysical observations (see Uzan 2003, 2011 for a review).

One of the particularly interesting fundamental constants is the fine-structure constant $\alpha$, defined by $\alpha\equiv e^{2}/(4\pi\epsilon_{0}\hbar c)$, where $e$, $\epsilon_{0}$, $\hbar$, and $c$ are the electron charge, permittivity of free space, reduced Planck constant, and speed of light in vacuum, respectively. It is a dimensionless physical constant that characterizes the strength of the electromagnetic force between electrically charged particles. Any possible variations in $\alpha$ could indicate that the laws of physics are not the same everywhere in the Universe or have changed over time. This challenges the assumption of the constancy of physical laws, which is a cornerstone of modern physics. Moreover, variations in $\alpha$ could imply the existence of new physics beyond the Standard Model, such as extra dimensions or varying scalar fields. This would have profound implications for our understanding of the Universe. By studying potential variations in $\alpha$, physicists and astronomers can examine the robustness of the fundamental laws of nature, test alternative cosmological models that account for varying $\alpha$ (Martins & Pinho 2015; Martins et al. 2015), and potentially uncover new aspects of the Universe’s underlying structure (Mota & Barrow 2004).

The fundamental constant $\alpha$ also quantifies the separation in the fine structure of atomic spectral lines (e.g., Dzuba et al. 1999a; Uzan 2003). Any relative $\alpha$ variation (i.e., $\Delta\alpha/\alpha$) over time can therefore be measured directly by comparing the wavelengths of fine-structure splitting of atomic lines at two different epochs. Astrophysical spectra invloving long look-back times have been widely used to investigate the possible variation of $\alpha$. The first measurements on the $\alpha$ variation from astronomical spectroscopy reached an accuracy of $\Delta\alpha/\alpha\approx 10^{-2}-10^{-3}$ (Savedoff 1956; Bahcall & Salpeter 1965; Bahcall & Schmidt 1967; Bahcall et al. 1967). Since then, the methodologies for analyzing spectra and our understanding of systematic errors have improved significantly. At present, there are two main methods to measure the relative separation of absorption lines in the spectra of quasars, i.e., the alkali-doublet (AD) method (Bahcall et al. 1967) and the many-multiplet (MM) method (Dzuba et al. 1999b; Webb et al. 1999).

In the AD method, the adopted quasar absorption lines were mainly fine-structure doublet lines, such as C iv, N v, Mg ii, and Si iv (e.g., Potekhin & Varshalovich 1994; Cowie & Songaila 1995; Murphy et al. 2001b; Chand et al. 2005). The current best constraints obtained using the AD method are those based on the analysis of Si iv absorption lines, yielding $\Delta\alpha/\alpha=(1.5\pm 4.3)\times 10^{-6}$ over a redshift range of $1.59\leq z\leq 2.92$ (Chand et al. 2005). The MM method simultaneously analyzes all (or most) doublets of many atomic species, thereby achieving a higher precision compared to the AD method. Using the MM method, some early works claimed to have found tentative evidence for variation of $\alpha$ at a level of $\Delta\alpha/\alpha\sim(1-10)\times 10^{-6}$ (e.g., Murphy et al. 2001a, 2003; Webb et al. 2001, 2011). However, subsequent works indicated that such variations were likely to be caused by wavelength distortions and other systematic effects (e.g, Evans et al. 2014; Whitmore & Murphy 2015). Recently, Murphy et al. (2022) applied the MM method to the absorption spectra of nearby star twins within 50 pc, and found no variations in $\alpha$ with a precision of $5.0\times 10^{-8}$. Although more precise, the MM method still suffers from a number of uncertainties (see Webb et al. 2022 for a recent review), which may arise from the techniques for correcting wavelength distortion (Dumont & Webb 2017), the assumptions underlying the Voigt fitting technique (Levshakov 2004), the technical details of profile fitting (Bainbridge & Webb 2017; Lee et al. 2023), the unconsidered systematic errors (Lee et al. 2021), and other. These uncertainties may induce biases on the values of $\Delta\alpha/\alpha$ and underestimations of their errors. Thus, different methods are encouraged to cross-check the results of the MM method.

In this work, we employ the method based on the [O iii] emission lines, first proposed by Bahcall & Salpeter (1965), to constrain possible variations in $\alpha$. Since the [O iii] doublet method relies only on a pair of lines, the limits on $\Delta\alpha/\alpha$ are not as stringent as those obtained with the MM method, but has the advantage of being more transparent and less subject to systematics. The [O iii] $\lambda\lambda\ 4960,5008\,$doublet lines originate in the downward transitions from the same upper energy level of the same ion, so no assumptions on ionization state, chemical composition, or distribution of energy levels are required in practice. The [O iii] doublet method therefore represents an excellent alternative for measuring the $\alpha$ variation on a firm basis. The $\Delta\alpha/\alpha$ constraints obtained by recent works based on the [O iii] emission lines are summarized as follows. By analyzing 42 quasar spectra from the Early Data Release of Sloan Digital Sky Survey (SDSS), Bahcall et al. (2004) derived $\Delta\alpha/\alpha=(0.7\pm 1.4)\times 10^{-4}$ over the range $0.16<z<0.8$. Gutiérrez & López-Corredoira (2010) obtained $\Delta\alpha/\alpha=(2.4\pm 2.5)\times 10^{-5}$ using 1568 quasar spectra at $0.0<z<0.8$ from SDSS Data Release 6 (DR6). Rahmani et al. (2014) derived $\Delta\alpha/\alpha=(-2.1\pm 1.6)\times 10^{-5}$ using 2347 quasar spectra at $0.02<z<0.74$ from SDSS DR7. Albareti et al. (2015) obtained $\Delta\alpha/\alpha=(0.9\pm 1.8)\times 10^{-5}$ using 13,175 quasar spectra at $0.04<z<1.0$ from SDSS DR12. Li et al. (2024) analyzed 40 spectra of Ly $\alpha$ emitting galaxies and 46 spectra of quasars at $1.09<z<3.73$ using the VLT/X-Shooter spectra publicly available, from which they yielded $\Delta\alpha/\alpha=(-3\pm 6)\times 10^{-5}$. Jiang et al. (2024a) measured $\Delta\alpha/\alpha=(2\sim 3)\times 10^{-5}$ by utilizing $\sim 110,000$ [O iii] emission-line galaxies at $0<z<0.95$ from the Dark Energy Spectroscopic Instrument. Jiang et al. (2024b) obtained $\Delta\alpha/\alpha=(0.4\pm 0.7)\times 10^{-4}$ using 572 JWST spectra from 522 [O iii] emission-line galaxies at $3<z<10$. In addition to the [O iii] emission lines, other emission doublets, such as [Ne iii] $\lambda\lambda\ 3869,3968\,$and [S ii] $\lambda\lambda\ 6717,6731$, have also been used to explore the $\alpha$ variation (e.g., Gutiérrez & López-Corredoira 2010; Albareti et al. 2015). However, the limits of their accuracy are worse, because all these doublets in quasars are fainter than [O iii] and some of them are affected by systematic errors.

Recently, the Large Sky Area Multi-object Fiber Spectroscopic Telescope (LAMOST) released the results of its 9 yr quasar survey (Jin et al. 2023). Here, we use the latest LAMOST DR9 quasar sample, for the first time, to measure the time variation of $\alpha$ through the [O iii] doublet method. Our analysis can serve to corroborate previous results of SDSS with another independent survey, thereby discarding possible systematic errors in the wavelength calibration of quasar spectra in SDSS. The rest of this paper is organized as follows. In Section 2, we introduce our quasar sample and spectroscopic data. Our resulting constraints on $\Delta\alpha/\alpha$ are then presented in Section 3. Finally, a brief summary and discussions are drawn in Section 4.

In this section, we will first clarify why the [O iii] doublet can provide an ideal testbed for measuring the $\alpha$ variation. We will then describe the LAMOST quasar survey and the refined sample used for our analysis. Finally, we will introduce the measurements of emission-line wavelengths in detail.

We use a total of 209 quasar spectra, drawn from the LAMOST DR9 catalog, after applying the selection criteria (i)-(iii) (see Section 2.2), to measure $\Delta\alpha/\alpha$. With the wavelength measurements of the [O iii] $\lambda\lambda\ 4960,5008\,$doublet lines for each quasar spectra, we calculate $\Delta\alpha/\alpha$ using Equation (1). Our results show that most of the $\Delta\alpha/\alpha$ measurements are consistent with $0$ within $3\sigma$ confidence level, and the accuracies of $\Delta\alpha/\alpha$ are between $10^{-4}$ and $10^{-2}$. Figure 2 shows the rest-frame wavelength measurements of the [O iii] doublet lines of all 209 quasar spectra in our final sample. One can see from this plot that the wavelengths of the two lines are aligned along a line from bottom left to top right, showing no systematic effects.

With a series of measured values $x_{i}=(\Delta\alpha/\alpha)_{i}$, we calculate the weighted average for the final sample through

where $\sigma_{i}$ is the error of $x_{i}$. The corresponding uncertainty on $\overline{x}$ can be obtained from

The weighted average for all the 209 spectra is $\Delta\alpha/\alpha=(0.5\pm 3.7)\times 10^{-4}$. This value is compatible with previous results obtained using other observational samples with the same method (Bahcall et al. 2004; Jiang et al. 2024b).

To explore the possible time variation of $\Delta\alpha/\alpha$, we divide the final sample into eight subsamples with redshifts from low to high, with each subsample containing approximately the same number ($N\simeq 26$) of spectra. For each subsample, we also compute the average value of $\Delta\alpha/\alpha$ and its uncertainty through Equations (4) and (5). The averages of $\Delta\alpha/\alpha$ as a function of redshift (or look-back time) are shown in Figure 3 and Table 1. We do not find any variation of $\Delta\alpha/\alpha$ over cosmic time, because none of the $\Delta\alpha/\alpha$ averages deviate from $0$ by more than $1.5\sigma$ confidence level

The redshift range of quasars in our final sample is between 0.033 and 0.8, corresponding to a look-back time ($t_{\rm LB}$) of 0.5–7.0 Gyr, or the age of the Universe of 6.8–13.3 Gyr. We assume that $\alpha$ shows linear change with time:

where $\kappa$ and $\omega$ are two free parameters. A linear fit of $\Delta\alpha/\alpha$ with respect to $t_{\rm LB}$ for all 209 spectra gives a slope of $\kappa=(-3.4\pm 2.4)\times 10^{-13}$ $\mathrm{yr^{-1}}$ and an intercept of $\omega=(1.3\pm 1.0)\times 10^{-3}$. Here the slope $\kappa=\mathrm{d}(\Delta\alpha/\alpha)/\mathrm{d}t_{\rm LB}$ refers to the mean rate of change in $\Delta\alpha/\alpha$ (Bahcall et al. 2004).

We emphasize that all our results are based on the [O iii] emission lines. In principle, by comparing the multiple absorption lines in the damped Ly$\alpha$ systems of quasar spectra, the MM method can achieve higher accuracy than the [O iii] doublet method. As the MM method simultaneously analyzes all absorption lines, it requires accurate measurements of both the line wavelengths in the observed spectra and the laboratory wavelengths for all involved atomic transitions. Due to the low resolution of LAMOST, the absorption lines of different atoms are difficult to extract accurately from the observed quasar spectra. Therefore, it is hard to apply the MM method to the quasar sample used here.

In this work, we have used the LAMOST quasar survey, for the first time, to constrain the possible time variation of the fine-structure constant $\alpha$ through the [O iii] doublet method. A great advantage of [O iii] is that its doublet lines have a wide wavelength separation, which makes it very sensitive to the measurement of the relative $\alpha$ variation $\Delta\alpha/\alpha$. The other advantage is that the [O iii] emission lines are much stronger than any other doublets in many quasars, which is crucial for the $\Delta\alpha/\alpha$ measurement.

From 56,175 objects identified as quasars in the LAMOST DR9 quasar catalog, we have extracted a sample of 209 quasars with strong [O iii] emission lines up to redshift 0.8. With this refined sample, we estimated a weighted average value of $\Delta\alpha/\alpha=(0.5\pm 3.7)\times 10^{-4}$ during the last 7.0 Gyr. Due to the smaller number of quasars and the lower resolution of LAMOST with respect to SDSS, our measuring precision of $\Delta\alpha/\alpha$ is worse than previous results obtained using different SDSS quasar samples with the same method by one order of magnitude (Gutiérrez & López-Corredoira 2010; Rahmani et al. 2014; Albareti et al. 2015). While our LAMOST-based constraint is not competitive, there is merit to the result. Our analysis serves to confirm the results of SDSS with another independent survey, so we can exclude possible systematic errors in the wavelength calibration of spectra in SDSS.

To analyze the value of $\Delta\alpha/\alpha$ as a function of redshift, we divided the sample into eight redshift bins, with each bin containing approximately the same number of quasars. We found that the averages of $\Delta\alpha/\alpha$ in all redshift bins are consistent with 0 within $1.5\sigma$ confidence level. This indicates that there is no evidence of changes in $\Delta\alpha/\alpha$ with redshift. We limited the mean rate of change in $\Delta\alpha/\alpha$ to be $(-3.4\pm 2.4)\times 10^{-13}$ $\mathrm{yr^{-1}}$ within the last 7.0 Gyr.

To achieve better constraints on $\Delta\alpha/\alpha$ ($<10^{-6}$) using the emission-line method, high-resolution spectroscopy ($R\sim 100,000$) is required. The measurement of $\Delta\alpha/\alpha$ using the LAMOST low resolution spectra is doomed to be unable to reach the best precision from previous quasar observations. Nonetheless, the LAMOST ongoing survey is still collecting useful data, and much more valuable quasars are expected to be identified in the future. The LAMOST quasar survey not only provides an independent measurement of $\Delta\alpha/\alpha$, but also helps to cross-check the results of other surveys. This work only focused on the [O iii] doublet, but we will consider more emission lines (e.g., [S ii] ) in future research. In addition, we plan to discuss collaborative efforts with other observatories to combine datasets, thereby enhancing statistical power for constraining variations in $\alpha$. For example, there are a total of 56,175 identified quasars in the LAMOST DR9 quasar catalog, of which 24,127 are newly discovered and not reported by SDSS (Jin et al. 2023). We plan to combine the SDSS quasar survey with those new ones discovered by LAMOST to achieve a more robust constraint on $\Delta\alpha/\alpha$.