# Updated Constraints on the Variations of the Fine-Structure Constant from an Analysis of White-Dwarf Spectra

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## Abstract

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## 1. Introduction

_{abs}= 2.8 [5]. Additionally, Dzuba et al. and Webb et al. used the MM approach and obtained $\Delta \alpha /\alpha =\left(-1.9\pm 0.4\right)\times {10}^{-5}$ for z > 1 and $\Delta \alpha /\alpha =\left(-0.2\pm 0.4\right)\times {10}^{-5}$ for z < 1 [5,6,7]. Furthermore, Webb et al. used the MM technique on 128 systems with a redshift of 0.2 < z < 3.7, and obtained $\Delta \alpha /\alpha =\left(-0.57\pm 0.10\right)\times {10}^{-5}$ [8]. Using large sets of quasar echelle spectra with a high resolving power, Quast et al. found $\Delta \alpha /\alpha =\left(-0.4\pm 1.9\right)\times {10}^{-6}$ [9,10] and Chand et al. reported $\Delta \alpha /\alpha =\left(-0.5\pm 2.4\right)\times {10}^{-6}$ [11]. However, the drawback of the MM method is the estimation errors in $\Delta \alpha /\alpha $, as the errors in the calibration wavelengths are typically not taken into account. The best estimates of $\Delta \alpha /\alpha $, giving cosmological variations at the level of 10

^{−6}, comes from the work of Levshakov et al., who analyzed [Fe II] absorption lines at z

_{abs}= 1.15 and obtained $\Delta \alpha /\alpha =\left(-0.07\pm 0.84\right)\times {10}^{-6}$ [12,13,14], and from the work of Porsev et al., who found $\Delta \alpha /\alpha =\left(-0.12\pm 1.79\right)\times {10}^{-6}$ and $\Delta \alpha /\alpha =\left(-5.4\pm 2.5\right)\times {10}^{-6}$ at z = 1.84 by analyzing absorption lines of [Fe II] [15]. The differences between these results may stem from the fact that some poorly understood systematic effects and/or correlations between spectral doublets were not properly included in the final analysis. In particular, difficulties with the wavelength calibration of the complex [Fe II] absorption spectra and the different sensitivity of various transitions included in the analysis, coming from different atoms and molecules, affect the resulting error [16,17,18,19,20,21,22,23]. Another problem may arise from possible variety in the asymmetric isotopic abundance of the sources of the emitted radiation, which is hard to quantify.

## 2. Data Analysis

## 3. Results and Discussions

^{−5}. Berengut et al. demonstrated that this limitation is caused by the systematic effect in the laboratory-measured wavelengths used rather than by a gravitational dependence of $\Delta \alpha /\alpha $. Therefore, I have improved the study of Berengut et al. by refining their analytical methodology by incorporating robust techniques from previous asymmetric studies using the wavelength shifts of 32 quadruply ionized iron [Ni V] in the work of Berengut et al. [40]. The precision in the relative change in $\alpha $ achieved in my analyses is three orders of magnitude greater than that achieved by Berengut et al. The results are given as $\Delta \alpha /\alpha =\left(-0.003\pm 0.072\right)\times {10}^{-6}$, and the statistical and systematic errors are estimated with high accuracy compared with those of previous studies [10,11,12,13,14,17,18,19,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]. Recently, significant effort has been invested into improving the laboratory-measured wavelengths of transitions in [Ni V] multiplet lines, which can be utilized in my analyses. This will offer the most robust and constraining test of whether the fine-structure constant varies in gravitational fields, and will also allow further investigation of a large number of systems to reduce the final uncertainties [21,25,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]. This will improve the selection process of the satisfactory spectra of white-dwarf stars. Additionally, the high relative precision of this work reduces the potential for systematic effects, improves the dedication of $\Delta \alpha /\alpha $, and reduces the number of spectra of white-dwarf stars in a sensitive way.

^{−6}. However, these estimations of $\Delta \alpha /\alpha $ are not explicitly utilized as fitting parameters. The most effective studies used the ${\chi}^{2}$ versus $\Delta \alpha /\alpha $ curve to obtain the best-fitting estimate of $\Delta \alpha /\alpha $. The reasons for these disparities are not yet completely understood. However, similar to difficulties in wavelength calibration, problems with the technique used are probably the main reason for these disparities. In order to improve these disadvantages, my analysis was based completely on the main effect on the error budget range consisting of $\alpha $-independent line ratios, which allow us to perceive the real size of statistical and systematic errors and to determine the actual wavelength splitting of the line pairs. This method is most suitable for lines with small separations.

^{−1}that was required for the analysis. Utilizing these spectra allows us to increase the precision in the measurement of $\alpha $ with uncertainties on the order of 10

^{−6}for the observed wavelengths and 10

^{−7}for laboratory-measured wavelengths.

## 4. Conclusions

## Funding

## Conflicts of Interest

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Le, T.D.
Updated Constraints on the Variations of the Fine-Structure Constant from an Analysis of White-Dwarf Spectra. *Symmetry* **2019**, *11*, 936.
https://doi.org/10.3390/sym11070936

**AMA Style**

Le TD.
Updated Constraints on the Variations of the Fine-Structure Constant from an Analysis of White-Dwarf Spectra. *Symmetry*. 2019; 11(7):936.
https://doi.org/10.3390/sym11070936

**Chicago/Turabian Style**

Le, T. D.
2019. "Updated Constraints on the Variations of the Fine-Structure Constant from an Analysis of White-Dwarf Spectra" *Symmetry* 11, no. 7: 936.
https://doi.org/10.3390/sym11070936