New Limit on Space-Time Variations in the Proton-to-Electron Mass Ratio from Analysis of Quasar J110325-264515 Spectra

Abstract

Astrophysical tests of current values for dimensionless constants known on Earth, such as the fine-structure constant, $\alpha$, and proton-to-electron mass ratio, $\mu =m_p/m_e$, are communicated using data from high-resolution quasar spectra in different regions or epochs of the universe. The symmetry wavelengths of [Fe II] lines from redshifted quasar spectra of J110325-264515 and their corresponding values in the laboratory were combined to find a new limit on space-time variations in the proton-to-electron mass ratio, $∆\mu /\mu =\left(0.096\pm 0.182\right)\times {10}^{-7}$. The results show how the indicated astrophysical observations can further improve the accuracy and space-time variations of physics constants.

1. Introduction

A motivating challenge is related to varying dimensionless physics constants for example, the fine-structure constant, $\alpha =\frac{e^2}{4{\pi \epsilon }_0\hslash c}$ (where $e$ is the elementary charge, $\hslash$ is the reduced Planck constant, ${\epsilon }_0$ is the electric constant, $c$ is the speed of light in vacuum), and the ratio of proton-to-electron mass ratio, $\mu =m_p/m_e$ (where $m_p$ is the rest mass of the proton and $m_e$ is the rest mass of the electron). This continues to be an interesting problem in both experimental and theoretical physics phenomena for the validation of the Standard Model (SM) [1,2,3,4,5] because values and dependence on parameters over the cosmological timescales of $\alpha$ and $\mu$ values have not been explained or predicted by the SM. Therefore, variations of these constants over cosmological time or space would violate the Einstein Equivalence Principle (EEP), which is the basic assumption of General Relativity (GR). It is presently the most well-tested gravity theory and is of great interest in the context of cosmologically relevant scalar fields such as quintessence. However, the GR is not consistent with quantum mechanics and it is limited by some energy threshold. On the basis of the tests passed with flying colors, the deviations of GR would only be predicted at some lengths or energy scales [6,7,8]. Thus, using new sources at current and future detectors allows us to indicate non-GR effects that help us better understand the important relation. So far, this has been a motivator for astrophysical and cosmological studies beyond GR without including dark energy or dark matter, as it is used for explaining the present accelerating and expanding universe and other interesting problems in galaxies such as missing mass. It is well known that there is a highly significant indication that the temporal or spatial variations of physics constants reported have orbital effects [9,10]. Evidence for these effects motivates us to interpret other data in terms of spatial variation. At the same time, these studies provide an important tool for searching for any deviations from GR and to constrain them in the future [11,12,13,14,15]. Thus, it is possible to identify mystery phenomena beyond the SM [16,17,18]. On the other hand, measurements of space-time variations of physics constants are essential in accepting the Grand Unification Theory. Based on high-resolution spectroscopic observations from the large redshifts (0.2 < z < 4.2) of quasars from Keck Observatory and Very Large Telescope, variations of $\alpha$ and $\mu$ have been predicted with high precision using the Many-Multiplet method [19], such as temporal [20] and spatial [21] variations. Moreover, these variations would be the result of orders of magnitude compared with previous methods, referring to a new method that was recently established and can be applied to wide-spanning redshifts for both emission and absorption spectra [22]. On the other hand, sensitive variation in $\mu$ over cosmological timescales can be found from a comparison of the wavelengths of molecular transitions detected in quasar spectra with their corresponding laboratory values, while others used the ratio of the chromodynamics-to-electroweak scale [23] and molecular absorption [24,25]. By using ammonia [26,27,28] and methanol [29,30] to test the observed spectra with redshifts $z <1$, the limit of $∆\mu /\mu$ was estimated at the level of $-{10}^{-7}$ at 1σ. With higher redshifts, $z >2$, observations in the universe have shown almost abundant molecular lines like hydrogen in which the Lyman series fall into in the optical band, whereas usually, one associates Lyman lines with UV. These were used to investigate the μ-variation [31,32,33,34,35,36,37,38,39,40,41,42,43,44]. Applying nine ${\mathrm{H}}_2$ absorption systems with a redshift of $2.05 \le z \le 4.22$, the current results find $\left|∆\mu /\mu \right|\le {10}^{-6}$ at 3 σ [45]. Moreover, the benefit of the ${\mathrm{H}}_2$ method is that it allows us to find how μ depends on the transition wavelength shifts for each line via identification of sensitive coefficients of this molecule [46,47]. On the other hand, both α and μ were sensitive, with some molecules found in the observations of the previous study using the combination of H I 21-cm lines with UV metal absorption lines to identify the combination of ${\alpha }^2{g}_p\mu$, with $g_p$, and the dimensionless proton $g$-factor varied with time [48]. These combined variations would also be found via OH microwave line observation [49]. The second abundant molecule candidate, carbon monoxide, can be used for the study of $\mu$-variations [50,51]. These studies used six absorption systems combining high-quality spectra of both ${\mathrm{H}}_2$ and CO to find $\mu$-variation.

As in previous studies, a new method of searching for space-time variations of fundamental coupling constants over cosmological timescales was developed [22] and refs. therein. This method is sensitive to these constants at a level below 10⁻⁶ with wide-spanning redshifts, making it orders of magnitude better than previous methods such as the many-multiplet method, ${\mathrm{H}}_2$ method, and line-by-line method [19,29,30,31,32,33,34]. Most of the published results using these methods are conflicted because they used different methods, and these analyses of $∆\alpha /\alpha$ and $∆\mu /\mu$ were certainly not explicitly used as the fitting parameter. They just used the ${\chi }^2$ versus $∆\alpha /\alpha$ and $∆\mu /\mu$ curve to acquire the best-fitting values of $∆\alpha /\alpha$ and $∆\mu /\mu$. This inconsistency is due to a systematic effect in the used laboratory wavelengths. It is a very important problem because this effect is dependent on transition wavelengths, just like a wavelength shift due to $∆\alpha /\alpha$ and $∆\mu /\mu$. Moreover, the sensitivity of this technique estimates and controls systematic effects with high precision. The most important of these are the accuracy of the wavelength calibrations, assumptions of uniform velocity structure, and uniform spatial abundance patterns in [Fe II] lines. In this way, the advantage of this procedure and the fitting procedure is that they can determine $∆\alpha /\alpha$ with higher precision than the previous results. On the basis of the updated spectra with uncertainty around parts of million and laboratory wavelengths with uncertainties around parts of 10⁻⁷, this would provide a good way to estimate the changing values of $∆\alpha /\alpha$ below 10⁻⁶ and $∆\mu /\mu$ below 10⁻⁸ better than results using the same data. The method is based on the line ratios for the identification of $\mu$ independently. In this case, the sizes of statistical and systematic errors were determined by the exact wavelength splitting of the line pairs with small separations [52,53,54,55,56,57].

Combined wavelengths of [Fe II] ($\lambda 1608$–$\lambda 2600)$ from quasar J110325-264515 at a redshift of $z_{abs}=1.8389$ and the laboratory wavelengths to constrain past variations in $\mu$ over cosmological timescales were used [52,53,54,55,56,57]. Based on existing astrophysical observations of [Fe II] in absorption systems, we identified the variation of the proton-to-electron mass ratio in our galaxy, and provide significant improvement for future astrophysical measurements of higher precision to more stringently test whether the $∆\alpha /\alpha$ and $∆\mu /\mu$ vary.

2. Data Analysis

High-resolution observation quasar spectra offer a key tool to test the space-time possibility of variations of fundamental constants like α and μ over cosmological timescales. Observations including almost all of the atomic and molecular absorption lines can provide us with an important tool for testing these variations with a wide range of redshifts. The small variation can be inferred from a previous study with high precision, as follows [58]:

$$\frac{∆\alpha }{\alpha }\approx \frac{1}{2}(\frac{\frac{1}{2}\left(\frac{{\lambda }_2\left(t\right)}{{\lambda }_1\left(t\right)}\right)-1}{\frac{1}{2}\left(\frac{{\lambda }_2\left(0\right)}{{\lambda }_1\left(0\right)}\right)-1}-1)$$

(1)

Using a comparison of wavelengths, ${\lambda }_1\left(t\right)$ and ${\lambda }_2\left(t\right),$ found from quasar spectra with the redshift parameter $z= {\lambda }_{observ}/{\lambda }_{lab}-1$ and their corresponding laboratory wavelengths ${\lambda }_1\left(0\right)$ and ${\lambda }_2\left(0\right)$ with a redshift of z = 0, the α-variation can be detected with high sensitivity and precision. The advantage of this method is that it can be applied to both emission and absorption systems with wide-spanning redshifts to search for space-time variations in α with high accuracy. Theoretical and experimental astrophysics and cosmology predict that fundamental constants such as α and μ change with time and space during the evolution of the universe, which is an interesting problem for laboratory and observation work. The variations could be tested by using extra-galactic measurements and in a broad class of unification scenarios that are linked by the following relation [59,60]:

$$\frac{∆\mu }{\mu }=\left[0.8R-0.3\left(1+S\right)\right]\frac{∆\alpha }{\alpha }$$

(2)

Dimensionless fundamental couplings $R$ and $S$ can be found from the Quantum Chromodynamics and Electro-Weak sector. The characterized values of $R$ and $S$ are different when using different models. These values have the same or opposite signs based on the dependent model that is applied to observation systems for testing the variations of α and μ with high accuracy. Based on the particular models, the unification parameters $R$ and $S$ have values of $R=273\pm 86$ and $S=630\pm 230$ relating to the unification scenarios [60,61]. In the present analysis, a combination of the Ritz wavelengths (the Ritz wavelengths are the wavelengths derived from the lower and upper levels of the transitions. They are available only if both levels of the transition are known. If they are available, they are usually more accurate than the observed wavelengths, especially in the vacuum ultraviolet spectral region.) and quasar spectra to search for a variation of α-μ over cosmic timescales using previous studies [52,53,54,55,56,57]. Statistical and systematic errors including wavelength calibration were discussed in detail in [52,53,54,55,56,57]. We applied the previous analysis method with non-linear least-squares and several $m\mathsf{\AA }$ to determine the laboratory uncertainty errors of [Fe II] lines [58]. The obtained results are shown in

Table 1. $∆\mu /\mu =\left(0.096\pm 0.182\right)\times {10}^{-7}$ was identified at a redshift of $z_{abs}=1.8389$ from the combined [Fe II] symmetry wavelengths from quasar spectra of J110325-264515 with corresponding laboratory and weighted average data of all lines.

z	$∆\mathit{\mu }/\mathit{\mu }$$\left({10}^{-7}\right)$	${\mathit{\sigma }}_{∆\mathit{\mu }/\mathit{\mu }}$$\left({10}^{-7}\right)$
1.83704	−0.08415	0.0660
1.83743	−0.17439	0.1164
1.83752	−0.29722	0.1032
1.83766	−0.12876	0.0504
1.83772	0.14416	0.3240
1.83772	−0.09147	0.0852
1.83816	−0.06879	0.4632
1.83819	−0.13683	0.0840
1.83825	−0.15553	0.0720
1.83832	0.12753	0.0948
1.83836	−0.22846	0.0704
1.83847	−0.14485	0.2280
1.83854	0.37092	0.2016
1.83855	−0.49429	0.1692
1.83857	−0.50071	0.1203
1.83857	−0.16694	0.0732
1.83865	−0.11433	0.1116
1.83870	0.08293	0.1926
1.83874	−0.25422	0.1344
1.83886	0.25605	0.1572
1.83888	−0.11963	0.1560
1.83888	−0.06850	0.0806
1.83891	−0.09760	0.1536
1.83895	−0.20794	0.0624
1.83905	0.37487	0.0900
1.83704	−0.08415	0.0660
1.83743	−0.17439	0.1164

, and the plot is illustrated in Figure 1 for $∆\mu /\mu$ over cosmological space-time at a 10⁻⁷ level with a high-accuracy estimation of systematic error.

3. Results and Discussion

Astrophysical tests of $∆\mu /\mu$ using the ${\mathrm{H}}_2$ absorption systems found $∆\mu /\mu \le {10}^{-5}$ at $2 \le z \le 3$ [60,61,62,63,64,65,66,67,68]. As for a single system toward Q 0528-250, results reported $\mu /\mu =\left(0.3\pm 3.7\right)\times {10}^{-6}$, while the other study found $\mu /\mu =-\left(3.5\pm 1.2\right)\times {10}^{-7}$ and $∆\mu /\mu =\left(1.0\pm 4.7\right)\times {10}^{-7}$ at $z \le 1.0$ using ammonia and methanol transitions of ${\mathrm{NH}}_3$ and other rotational molecular transitions [69]. The analysis of ammonia from quasar PKS-1830-211 at $z=0.89$ provided a better determination of variation: $∆\mu /\mu =\left(0.0\pm 1.0\right)\times {10}^{-7}$ [70]. However, these studies indicated that this technique could only be applied to low-redshift systems, such as $z \le 1.0$ for ${\mathrm{NH}}_3$ and ${\mathrm{CH}}_3\mathrm{OH}$ [71]. With higher redshifts, such as $z \sim 3.17$ towards J1337 + 3152, the best results were $∆\mu /\mu =\left(-1.7\pm 1.7\right)\times {10}^{-6}$ and $∆\mu /\mu =\left(0.0\pm 1.5\right)\times {10}^{-6}$ at $z_{abs}\sim 1.3,$ relating to the best choice of four 21 cm absorption systems [72]. Almost all of these various studies identified that the changed values of $∆\mu /\mu$ were at the level of 10⁻⁷. Thus, to improve the limit variation of $∆\mu /\mu$ on cosmological timescales, a combination of Ritz wavelengths with those seen in the quasar spectra of [Fe II] lines was used. In the present study, $1\sigma$ error was assigned for a statistical process to outline the required changing value of $∆\mu /\mu$ with the computed expression: $∆{\chi }^2= {\chi }^2-{\chi }_{min}^2=1$. Then, the maximal change in $∆\mu /\mu$ was indicated by $∆{\chi }^2= 1$ for the error estimation of $∆\mu /\mu$. I obtained ${\chi }^2$-minimal for each one of the whole fits, and these were plotted as a general function of $∆\mu /\mu$. The study concludes that the changing value of $∆\mu /\mu$ was below 10⁻⁸ with a higher precision based on the standard statistical and systematic error deviation: ${\sigma }_{tot}^2= {\sigma }_{∆\mu /\mu }^2+{\sigma }_{sys}^2$ [22,58,60]. In this case, the present study provides an estimation of $∆\mu /\mu ,$ an order magnitude better than previous studies using the same data [60,61,62,63,64,65,66,67,68,69,70,71,72]. The expected result provides a strong limit to $∆\mu /\mu$ in comparison with previous results, which are indicated in Table 1, and the plot is illustrated in Figure 1, including the estimation of wavelength calibration, data reduction, and statistical and systematic errors using our previous method with high accuracy [59,60,61,62,63,64,65,66,67,68,69,70,71,72].

4. Conclusions

In this work, the combined wavelengths of the [Fe II] quasar J110325-264515 and Ritz wavelengths were used to test the possible proton-to-electron mass ratio μ over cosmological timescales. The obtained results from this analysis provide a stronger limit on μ-variation than previous studies [59,60,61,62,63,64,65,66,67,68,69,70,71,72]. With the improvement of the laboratory and observation wavelengths, in the near future, the best candidate selections of such abundant interstellar molecules, ${\mathrm{CH}}_3\mathrm{OH}$, OH, and CH, will provide us with important samples for checking how the proton-to-electron mass ratio varies with cosmic space-time by using more stringent limits than the best present estimations [73,74,75,76,77,78].

Studies at wide-spanning redshifts, based on the variations of fundamental dimensionless constants, yield a constraint of checking parameters beyond the Standard Model.