number of photoelectron at the continuum level and *M* is the number of pixel covering the line profile.

The developments of high resolution and ultras-stable spectrographs, such as EXPRESSO (Echelle Spectrograph for PREcision Super Stable Observation) (Cristiani et al. 2007; Pepe et al. 2013, 2021; Schmidt et al. 2021) on 4 VLT units are key developments with a resolution *R* ~ 145 000. Projects such as CODEX (COsmic Dynamics EXplorer) on E-ELT (Molaro et al. 2006; Molaro 2009; Uzan et al. 2008a), ELET/HIRES (Maiolino et al. 2013) or ANDES (Martins et al. 2024) consider fundamental constants as both science and design drivers. ANDES shall be able to compare the values of fundamental constants up to 12 Gyr ago and 15 Gpc and provide a photon-limited uncertainty of about 3 × 10⁻⁷, corresponding to 6 m/s relative line shifts, on their measurements.; see Fig. 3 of Martins et al. (2024) for synthetic spectra with α<sub>EM</sub> variation. Besides, it will provide measurement of molecular H₂ allowing it to combine α<sub>EM</sub> and μ measurements. At lower redshifts, ALMA may provide additional information (Fish et al. 2013; Tilanus et al. 2014).

As a concrete example, EXPRESSO was specifically designed to suppress wavelength calibration errors and allows for cm/s (photon-limited) calibration precision; see Murphy et al. (2022b, 2022c) for a description of the progresses it led to. As an example, the analysis of the single system HE 0515-4414 (Murphy et al. 2022b) reached an total uncertainty of 1.4 ppm, i.e., similar to the ensemble precision of the previous large samples of absorbers from HIRES and UVES that indicated variations at about the 5 ppm level (Webb et al. 2011). Hence one can safely conclude that it arose from long-range distortions in the wavelength scale, as discussed above. Recently, Schmidt and Bouchy (2024) improved the calibration accuracy to provide fully consistent measurements with a scatter of the order of a m/s ensuring that the instrument-related systematics can be nearly eliminated over most of the spectral range.

Thanks to the progresses to reach an increasing precision of the laboratory measurements of the rest frame frequencies of molecular transitions, and an increasing sensitivity and spectral resolution of astronomical observations, one may hope to reach constraints at the level of 10⁻⁸ for α<sub>EM</sub> and μ.

## 5.6 Quasar absorption spectra: constraints on combination of α<sub>EM</sub>, μ and g<sub>p</sub>

### 5.6.1 H I-21 cm vs. UV: $x = g_p \alpha_{EM}^2 \mu$

The comparison of UV heavy element transitions with the hyperfine H I transition allows to extract (Tzanavaris et al. 2005) a measurement of

$$x \equiv g_p \alpha_{EM}^2 \mu,$$

since the hyperfine transition is proportional to α<sub>EM</sub><sup>2</sup>g<sub>p</sub>μ⁻¹R<sub>∞</sub> while optical transitions are simply proportional to R<sub>∞</sub>. Hence constraints on the time variation of *x* can be obtained from high resolution 21 cm spectra compared to UV lines, e.g., of Si II, Fe II and/or Mg II, as first performed in Wolfe et al. (1976) in z ~ 0.524 absorber.
<table><caption>**Table 14** Constraints on the combination of $g_p$, $\alpha_{EM}$ and $\mu = m_p/m_e = 1/\bar{\mu}$</caption><thead><tr><th></th><th>Object</th><th>z<sub>abs</sub></th><th>constraint</th><th>Reference</th></tr></thead><tbody><tr><td rowspan="10">x</td><td>3 sources</td><td>0.23--2.35</td><td>(-0.63 ± 0.99) × 10<sup>-5</sup></td><td>Tsanavaris et al. (2007)</td></tr><tr><td>5 sources</td><td>0.23--0.53</td><td>(1.02 ± 1.68) × 10<sup>-5</sup></td><td>Dent et al. (2008)</td></tr><tr><td>4 sources</td><td>1.7--2.35</td><td>(0.58 ± 1.94) × 10<sup>-5</sup></td><td>Dent et al. (2008)</td></tr><tr><td>J1337+3152</td><td>3.174</td><td>(1.7 ± 1.7) × 10<sup>-6</sup></td><td>Srianand et al. (2010)</td></tr><tr><td>7 sources</td><td>0.24--2.04</td><td>(-1.2 ± 1.4) × 10<sup>-6</sup></td><td>Darling (2012)</td></tr><tr><td>J2358-1020</td><td>1.173</td><td>(1.8 ± 2.7) × 10<sup>-6</sup></td><td>Rahmani et al. (2012)</td></tr><tr><td>J1623+0718</td><td>1.336</td><td>(-3.7 ± 3.4) × 10<sup>-6</sup></td><td>Rahmani et al. (2012)</td></tr><tr><td>J2340-0053</td><td>1.361</td><td>(-1.3 ± 2.0) × 10<sup>-6</sup></td><td>Rahmani et al. (2012)</td></tr><tr><td>J0501-0159</td><td>1.5605</td><td>(3.0 ± 3.1) × 10<sup>-6</sup></td><td>Rahmani et al. (2012)</td></tr><tr><td>PKS1413+135</td><td>0.247</td><td>(-4 ± 6) × 10<sup>-5</sup></td><td>Wiklind and Combes (1997)</td></tr><tr><td></td><td></td><td>(-0.2 ± 0.44) × 10<sup>-5</sup></td><td>Murphy et al. (2001b)</td></tr><tr><td></td><td>B0218+357</td><td>0.6847</td><td>(-0.16 ± 0.54) × 10<sup>-5</sup></td><td>Murphy et al. (2001b)</td></tr><tr><td rowspan="4">F</td><td>Q0248+430</td><td>0.0519</td><td>(5.2 ± 4.5) × 10<sup>-6</sup></td><td>Gupta et al. (2018)</td></tr><tr><td>PKS1413+135</td><td>0.247</td><td>(-0.51 ± 1.26) × 10<sup>-5</sup></td><td>Darling (2004)</td></tr><tr><td>B0218+357</td><td>0.685</td><td>(0.35 ± 0.4) × 10<sup>-5</sup></td><td>Chengalur and Kanekar (2003)</td></tr><tr><td>J0134-0931</td><td>0.765</td><td>(-0.44 ± 0.36 ± 1<sub>syst</sub>) × 10<sup>-5</sup></td><td>Kanekar et al. (2005)</td></tr><tr><td></td><td></td><td>(-5.2 ± 4.3) × 10<sup>-6</sup></td><td>Kanekar et al. (2012)</td></tr><tr><td rowspan="14">F'</td><td>J1024+4709</td><td>2.285</td><td>(100 ± 40) × 10<sup>-6</sup></td><td>Curran et al. (2011)</td></tr><tr><td>J2135-0102</td><td>2.326</td><td>(100 ± 100) × 10<sup>-6</sup></td><td>Curran et al. (2011)</td></tr><tr><td>J636+6612</td><td>2.517</td><td>(-100 ± 120) × 10<sup>-6</sup></td><td>Curran et al. (2011)</td></tr><tr><td>H1413+117</td><td>2.558</td><td>(-40 ± 80) × 10<sup>-6</sup></td><td>Curran et al. (2011)</td></tr><tr><td>J1401+0252</td><td>2.565</td><td>(-140 ± 80) × 10<sup>-6</sup></td><td>Curran et al. (2011)</td></tr><tr><td>J0911+0551</td><td>2.796</td><td>(6.9 ± 3.7) × 10<sup>-6</sup></td><td>Weiss et al. (2012)</td></tr><tr><td>APM0828+5255</td><td>3.913</td><td>(-360 ± 90) × 10<sup>-6</sup></td><td>Curran et al. (2011)</td></tr><tr><td>MM1842+5938</td><td>3.930</td><td>(-180 ± 40) × 10<sup>-6</sup></td><td>Curran et al. (2011)</td></tr><tr><td>PSS2322+1944</td><td>4.112</td><td>(170 ± 130) × 10<sup>-6</sup></td><td>Curran et al. (2011)</td></tr><tr><td>BR1202-0725</td><td>4.695</td><td>(1.4 ± 1.5) × 10<sup>-5</sup></td><td>Levshakov et al. (2008b)</td></tr><tr><td></td><td></td><td>(-5 ± 15) × 10<sup>-5</sup></td><td>Lentati et al. (2013)</td></tr><tr><td>J0918+5142</td><td>5.243</td><td>&lt;2 × 10<sup>-5</sup></td><td>Levshakov et al. (2012)</td></tr><tr><td>J2310+1855</td><td>6.003</td><td>(2.3 ± 3.4) × 10<sup>-5</sup></td><td>Levshakov et al. (2020)</td></tr><tr><td>J0439+1634</td><td>6.519</td><td>(0.0 ± 2.7) × 10<sup>-5</sup></td><td>Levshakov et al. (2020)</td></tr><tr><td>J1148+5251</td><td>6.420</td><td>(1 ± 10) × 10<sup>-5</sup></td><td>Levshakov et al. (2008b)</td></tr><tr><td></td><td></td><td>(-33 ± 23) × 10<sup>-5</sup></td><td>Lentati et al. (2013)</td></tr><tr><td rowspan="3">G</td><td>PKS1413+135</td><td>0.247</td><td>(-1.18 ± 0.46) × 10<sup>-5</sup></td><td>Kanekar et al. (2010a)</td></tr><tr><td></td><td></td><td>(0.97 ± 1.52) × 10<sup>-6</sup></td><td>Kanekar et al. (2018)</td></tr><tr><td></td><td></td><td>(-1.0 ± 1.3) × 10<sup>-6</sup></td><td>combined</td></tr></tbody></table>

The object determines the line of sight and the redshift refers to the absorption system. We recall that $x \equiv \alpha_{EM}^2 g_p / \mu$, $y \equiv g_p \alpha_{EM}^2$, $F \equiv g_p (\alpha_{EM}^2 \mu)^{1.57}$, $F' \equiv \alpha_{EM}^2 \mu$ and $G = g_p (\alpha_{EM}^2 \mu)^{1.85}$. The observations are ordered by increasing redshift and then by the date of the constraint for each system
Using 9 absorption systems, Tsanavaris et al. (2007) showed no evidence for any variation of $x$, $\Delta x/x = (-0.63 \pm 0.99) \times 10^{-5}$ for $0.23 < z < 2.35$. This constraint was criticized by Kanekar et al. (2006) on the basis that the systems have multiple components and that it is not necessary that the strongest absorption arises in the same component in both types of lines. However, the error analysis of Tsanavaris et al. (2007) tries to estimate the effect of this assumption. Dent et al. (2008) noticed that the systems lie in two widely-separated ranges and that the two samples have completely different scatter. Therefore it can be split into two samples of respectively 5 and 4 systems to get

$$ \Delta x/x = (1.02 \pm 1.68) \times 10^{-5}, \quad 0.23 < z < 0.53, \qquad (196) $$

$$ \Delta x/x = (0.58 \pm 1.94) \times 10^{-5}, \quad 1.7 < z < 2.35. \qquad (197) $$

In such an approach two main difficulties arise: (1) the radio and optical source must coincide (in the optical, QSO can be considered pointlike and it must be checked that this is also the case for the radio source), (2) the clouds responsible for the 21 cm and UV absorptions must be localized in the same place. Therefore, the systems must be selected with care and today the number of such systems is small and are still actively looked for (Petitjean et al. 2009a).

The recent detection of 21 cm and molecular hydrogen absorption lines in the same damped Lyman-$\alpha$ system at $z_{abs} = 3.174$ towards SDSS J1337+3152 constrains (Srianand et al. 2010) the variation $x$ to $\Delta x/x = -(1.7 \pm 1.7) \times 10^{-6}$ at $z = 3.174$. This system is unique since it allows for 21 cm, H$_2$ and UV observation so that in principle one can measure $\alpha_{EM}$, $x$ and $\mu$ independently. However, as the H$_2$ column density was low, only Werner band absorption lines are seen so that the range of sensitivity coefficients is too narrow to provide a stringent constraint, $\Delta\mu/\mu < 4 \times 10^{-4}$. It was also shown that the H$_2$ and 21 cm are shifted because of the inhomogeneity of the gas, hence emphasizing this limitation. Petitjean et al. (2009a) also mentioned that 4 systems at $z = 1.3$ sets $\Delta x/x = (0.0 \pm 1.5) \times 10^{-6}$ and that another system at $z = 3.1$ gives $\Delta x/x = (0.2 \pm 0.5) \times 10^{-6}$. Note also that the comparison (Kanekar et al. 2010b) with C I at $z \sim 1.4 - 1.6$ towards Q0458-020 and Q2337-011, yields $\Delta x/x = (6.8 \pm 1.0 \pm 6.7_{syst}) \times 10^{-6}$ over the band o redshift $0 < \langle z \rangle \le 1.46$, but this analysis ignores an important wavelength calibration estimated to be of the order of $6.7 \times 10^{-6}$. It was argued that, using the existing constraints on $\Delta\mu/\mu$, this measurement is inconsistent with claims of a smaller value of $\alpha_{EM}$ from the many-multiplet method, unless fractional changes in $g_p$ are larger than those in $\alpha_{EM}$ and $\mu$.

From the Green Bank Telescope digital data, Darling (2012) selected 10 objects with H I absorption lines with redshifts spanning from 0.09 to 0.69 with the primary goal to compare literature analog spectra to contemporary digital spectra and then measure the time drift of redshift, $\dot{z}$. Besides, by comparing UV metal absorption lines (from the data by Tsanavaris et al. 2007) to the 21 cm line from GBT allowed them to set constraints on a variation of $x$. Their seven data points are gathered in
their Table 1 and summarized in Table 3 of Martins (2017). This leads to the average constraint

$$ \Delta x/x = (-1.2 \pm 1.4) \times 10^{-6} \qquad (198) $$

in the redshift range 0.24–2.04.

Rahmani et al. (2012) analyzed high resolution optical spectra from VLT/UVES and 21-cm absorption spectra from Giant Metrewave Radio Telescope and the Green Bank Telescope for 5 quasars with redshifts ranging from 1.17 to 1.56 and for which 21 cm has already been detected. The data of 4 out of 5 systems (towards J0501-0159, J1623+0718, J2340-0053 and J2358-1020) are used to constrain $x$ and were confirmed for 2 systems thanks to Keck/HIRES spectra. They constrain the weighted and the simple means

$$ \Delta x/x = (-0.1 \pm 1.3) \times 10^{-6}, \quad \text{and} \quad \Delta x/x = (0.0 \pm 1.5) \times 10^{-6} \qquad (199) $$

for $z \in [1.17, 1.56]$ with mean redshift $\langle z \rangle = 1.36$. We summarized the measurements for these 4 systems in Table 14.

### 5.6.2 H I vs. molecular transitions: $y \equiv g_p \alpha_{EM}^2$

The H I 21 cm hyperfine transition frequency is proportional to $g_p \mu^{-1} \alpha_{EM}^2 R_\infty$ (see Sect. 5.1.1). On the other hand, the rotational transition frequencies of diatomic are inversely proportional to their reduced mass $M$. As on the example of Eq. (151) where we compared an electronic transition to a vibro-rotational transition, the comparison of the hyperfine and rotational frequencies is proportional to

$$ \frac{v_{hf}}{v_{rot}} \propto g_p \alpha_{EM}^2 \frac{M}{m_p} \approx g_p \alpha_{EM}^2 \equiv y, $$

where the variation of $M/m_p$ is usually suppressed by a large factor of the order of the ratio between the proton mass and nucleon binding energy in nuclei, so that we can safely neglect it. The constraint on the variation of $y$ is directly determined by comparing the redshift as determined from H I and molecular absorption lines,

$$ \frac{\Delta y}{y} = \frac{z_{mol} - z_H}{1 + z_{mol}}. $$

This method was first applied (Varshalovich and Potekhin 1996) to the CO molecular absorption lines (Wiklind and Combes 1997) towards PKS 1413+135 to get $\Delta y/y = (-4 \pm 6) \times 10^{-5}$ at $z = 0.247$. The most recent constraint (Murphy et al. 2001b) relies on the comparison of the published redshifts of two absorption systems determined both from H I and molecular absorption. The first is a system at $z = 0.6847$ in the direction of B 0218+357 for which the spectra of CO(1-2), $^{13}$CO(1-2), $^{18}$O(1-2), CO(2-3), HCO+(1-2) and HCN(1-2) are available. They concluded that
$$ \Delta y/y = (-0.16 \pm 0.54) \times 10^{-5} \qquad z = 0.6847. \qquad (200) $$

The second system lies in direction of PKS 1413+135 for which the molecular lines of CO(1-2), HCO+(1-2) and HCO+(2-3) have been detected. Its analysis led to

$$ \Delta y/y = (-0.2 \pm 0.44) \times 10^{-5}, \qquad z = 0.247. \qquad (201) $$

Carilli et al. (2000) obtained the constraints $|\Delta y/y| < 3.4 \times 10^{-5}$ at $z \sim 0.25$ and $z \sim 0.685$.

The radio domain has the advantage of heterodyne techniques, with a spectral resolution of $10^6$ or more, and dealing with cold gas and narrow lines. The main systematics is the kinematical bias, i.e., that the different lines do not come exactly from the same material along the line of sight, with the same velocity. To improve this method one needs to find more sources, which may be possible with the radio telescope ALMA.<fnref n="9" />

### 5.6.3 OH - 18 cm: $F = g_p(\alpha_{EM}^2\mu)^{1.57}$

Using transitions originating from a single species, as with SIDAM, allows to reduce the systematic effects. The 18 cm lines of the OH radical offers such a possibility (Chengalur and Kanekar 2003; Kanekar and Chengalur 2004).

The ground state, $^2\Pi_{3/2}J = 3/2$, of OH is split into two levels by A-doubling and each of these doubled level is further split into two hyperfine-structure states. Thus, it has two “main” lines ($\Delta F = 0$) and two “satellite” lines ($\Delta F = 1$). Since these four lines arise from two different physical processes (A-doubling and hyperfine splitting), they enjoy different dependencies in $g_p$ and $\alpha_{EM}$. By comparing the four transitions to the H I hyperfine line, one has access to

$$ F \equiv g_p(\alpha_{EM}^2\mu)^{1.57} \qquad (202) $$

and it was also proposed to combine them with HCO$^+$ transitions to lift the degeneracy.

Using the four 18 cm OH lines from the gravitational lens at $z \sim 0.765$ toward J0134-0931 and comparing the H I 21 cm and OH absorption redshifts of the different components allowed Kanekar et al. (2005) to set $\Delta F/F = (-0.86 \pm 0.86) \times 10^{-5}$ at $z_{abs} = 0.765$. Combining with the constraint by Chengalur and Kanekar (2003) in the lens toward B0218+357,

$$ \Delta F/F = (0.35 \pm 0.40) \times 10^{-5}, \qquad z = 0.685, \qquad (203) $$

led to $\Delta F/F = (-0.44 \pm 0.36 \pm 1.0_{syst}) \times 10^{-5}$ (Kanekar et al. 2005) at $z_{abs} = 0.765$ where the second error is due to velocity offset between OH and H I velocity dispersion of 3 km/s. Thanks to observations with deep Green Bank Telescope spectroscopy, Kanekar et al. (2012) concluded that

<footnote n="9">⁹ http://www.eso.org/sci/facilities/alma/</footnote>
<table><caption>**Table 15** Available measurements of μ from molecular absorption spectra</caption><thead><tr><th>Object</th><th>z</th><th>Δμ/μ</th><th>Molecules</th><th>Reference</th></tr></thead><tbody><tr><td>B0218+357</td><td>0.685</td><td>(0.74 ± 0.47 ± 0.76<sub>syst</sub>) × 10<sup>-6</sup><br>(-0.35 ± .012) × 10<sup>-6</sup></td><td>NH<sub>3</sub>, HCN,<br>HCO<sup>+</sup><br>NH<sub>3</sub>, CS, H<sub>2</sub>CO</td><td>Murphy et al. (2008a)<br>Kanekar (2011)</td></tr><tr><td>PKS1830-211</td><td>0.886</td><td>(0 ± 1.4) × 10<sup>-6</sup><br>(-2.04 ± 0.74) × 10<sup>-6</sup><br>(-1.95 ± 0.47)) × 10<sup>-6</sup><br>(0.8 ± 2.1) × 10<sup>-7</sup><br>(0.0 ± 1.0) × 10<sup>-7</sup><br>(1.5 ± 1.5) × 10<sup>-7</sup><br>(0 ± 4) × 10<sup>-7</sup><br>(-1.8 ± 1.2) × 10<sup>-7</sup><br>&lt;9 × 10<sup>-6</sup></td><td>NH<sub>3</sub><br>NH<sub>3</sub><br>CH<sub>3</sub>OH<br>CH<sub>3</sub>OH<br>CH<sub>3</sub>OH<br>CH<sub>3</sub>OH<br>CH<sub>3</sub>OH<br>CH<sub>3</sub>NH<sub>2</sub></td><td>Henkel et al. (2009)<br>Muller et al. (2011)<br>Muller et al. (2011)<br>Ellingsen et al. (2012)<br>Bagdonaitė et al. (2013b)<br>Bagdonaitė et al. (2013a)<br>Kanekar et al. (2015)<br>Muller et al. (2021)<br>Ilyushin et al. (2012)</td></tr><tr><td>J2123-0050</td><td>2.059</td><td>(5.6 ± 5.5 ± 2.9<sub>syst</sub>) × 10<sup>-6</sup><br>(8.5 ± 3.6 ± 2.2<sub>syst</sub>) × 10<sup>-6</sup><br>(7.6 ± 3.5) × 10<sup>-6</sup> [*]</td><td>H<sub>2</sub>, HD<br>H<sub>2</sub>, HD (HIRES)<br>H<sub>2</sub>, HD (UVES)</td><td>Malec et al. (2009)<br>van Weerdenburg et al. (2011)<br>Ubachs et al. (2016a)</td></tr><tr><td>Q1232+082</td><td>2.34</td><td>(19 ± 9 ± 5<sub>syst</sub>) × 10<sup>-6</sup></td><td>H<sub>2</sub>, HD</td><td>Daprà et al. (2017b)</td></tr><tr><td>HE0027-1836</td><td>2.402</td><td>(-7.6 ± 8.1 ± 6.3<sub>syst</sub>) × 10<sup>-6</sup></td><td>H<sub>2</sub></td><td>Rahmani et al. (2013)</td></tr><tr><td>Q2348-011</td><td>2.426</td><td>(-6.8 ± 27.8) × 10<sup>-6</sup></td><td>H<sub>2</sub></td><td>Bagdonaitė et al. (2012)</td></tr><tr><td>J0000+0048</td><td>2.52</td><td>(1.8 ± 2.2 ± 0.4<sub>syst</sub>) × 10<sup>-5</sup></td><td>CO, H<sub>2</sub></td><td>Daprà et al. (2017c)</td></tr><tr><td>Q00405-443</td><td>2.595</td><td>(27.8 ± 8.8) × 10<sup>-6</sup><br>(5.2 ± 7.4) × 10<sup>-6</sup><br>(0.6 ± 10) × 10<sup>-6</sup><br>(7.5 ± 5.3) × 10<sup>-6</sup> [*]</td><td>H<sub>2</sub><br>H<sub>2</sub><br>H<sub>2</sub><br>H<sub>2</sub></td><td>Reinhold et al. (2006)<br>King et al. (2008)<br>Thompson et al. (2009)<br>Ubachs et al. (2016a)</td></tr><tr><td>J0643-5041</td><td>2.659</td><td>(7.4 ± 4.3 ± 5.1<sub>syst</sub>) × 10<sup>-6</sup></td><td>H<sub>2</sub></td><td>Albornoz Vásquez et al. (2014)</td></tr><tr><td>B0642-5038</td><td></td><td>(12.7 ± 4.5 ± 4.2<sub>syst</sub>) × 10<sup>-6</sup><br>Δμ/μ = (10.3 ± 4.6) × 10<sup>-6</sup> [*]</td><td>H<sub>2</sub><br>H<sub>2</sub></td><td>Bagdonaitė et al. (2014b)<br>Ubachs et al. (2016a)</td></tr><tr><td>J1237+0647</td><td>2.69</td><td>(-5.4 ± 6.3 ± 4.0<sub>syst</sub>) × 10<sup>-6</sup><br>(0.7 ± 1.6 ± 0.5<sub>syst</sub>) × 10<sup>-5</sup><br>(5.6 ± 5.6 ± 3.1<sub>syst</sub>) × 10<sup>-6</sup></td><td>H<sub>2</sub>, HD<br>CO<br>CO,H<sub>2</sub></td><td>Daprà et al. (2015)<br>Daprà et al. (2016)<br>Daprà et al. (2016)</td></tr><tr><td>Q0528-250</td><td>2.811</td><td>(-1.4 ± 3.9) × 10<sup>-6</sup><br>(0.3 ± 3.2 ± 1.9<sub>syst</sub>) × 10<sup>-6</sup><br>(-0.5 ± 2.7) × 10<sup>-6</sup> [*]</td><td>H<sub>2</sub><br>H<sub>2</sub>, HD<br>H<sub>2</sub></td><td>King et al. (2008)<br>King et al. (2011)<br>Ubachs et al. (2016a)</td></tr><tr><td>Q0347-383</td><td>3.025</td><td>(20.6 ± 7.9) × 10<sup>-6</sup><br>(5.2 ± 7.4) × 10<sup>-6</sup><br>(-28 ± 16) × 10<sup>-6</sup><br>(15 ± 9 ± 6<sub>syst</sub>) × 10<sup>-6</sup><br>(4.3 ± 7.2) × 10<sup>-6</sup></td><td>H<sub>2</sub><br>H<sub>2</sub><br>H<sub>2</sub><br>H<sub>2</sub><br>H<sub>2</sub></td><td>Reinhold et al. (2006)<br>King et al. (2008)<br>Thompson (2009)<br>Wendt and Molaro (2011)<br>Wendt and Molaro (2012)</td></tr></tbody></table>
<table><caption>**Table 15** continued</caption><thead><tr><th>Object</th><th>z</th><th>Δμ/μ<br>(5.1 ± 4.5) × 10<sup>-6</sup> [*]</th><th>Molecules<br>H<sub>2</sub></th><th>Reference</th></tr></thead><tbody><tr><td>J1443+2724</td><td>4.224</td><td>(-9.5 ± 5.4 ± 5.3<sub>syst</sub>) × 10<sup>-6</sup></td><td>H<sub>2</sub></td><td>Ubachs et al. (2016a)<br>Bagdonaitė et al. (2015)</td></tr></tbody></table>

Listed are, respectively, the object along each line of sight, the redshift of the measurement, the measurement itself, the molecule(s) used, and the original reference. Low-redshift measurements were obtained with various facilities in the radio/mm band, while high-redshift ones were obtained in the UV/optical with the UVES spectrograph. [*] indicates that this value is the mean obtained from other values and thus is not an independent measurement; see text

$$ \Delta F/F = (-5.2 \pm 4.3) \times 10^{-6}, \quad z = 0.765. \qquad (204) $$

A similar analysis (Darling 2004) in a system in the direction of PKS1413+135 gave

$$ \Delta F/F = (0.51 \pm 1.26) \times 10^{-5}, \quad z = 0.2467. \qquad (205) $$

From the first detection of OH in a system in a star-forming galaxy at $z_{\text{abs}} = 0.0519$ toward QO248+430 and its combination with H1 21 cm, Gupta et al. (2018) derived

$$ \Delta F/F = (5.2 \pm 4.5) \times 10^{-6}, \quad z = 0.0519. \qquad (206) $$

### 5.6.4 Far infrared fine-structure lines: $F' = \alpha_{\text{EM}}^2\mu$

Another combination (Kozlov et al. 2008) of constants is obtained from the comparison of far infrared fine-structure spectra with rotational transitions, which respectively behaves as $R_{\infty}\alpha_{\text{EM}}^2$ and $R_{\infty}/\mu$ so that they give access to

$$ F' = \alpha_{\text{EM}}^2\mu. $$

A good candidate for the rotational lines is CO since it is the second most abundant molecule in the Universe after H$_2$.

Using the C II fine-structure and CO rotational emission lines from the quasars J1148+5251 and BR 1202-0725, Levshakov et al. (2008b) concluded that

$$ \Delta F'/F' = (0.1 \pm 1.0) \times 10^{-4}, \quad z = 6.42, \qquad (207) $$

$$ \Delta F'/F' = (1.4 \pm 1.5) \times 10^{-5}, \quad z = 4.69, \qquad (208) $$

which represent the constraints at highest redshift. As usual, when comparing the frequencies of two different species, one must account for random Doppler shifts caused by non-identical spatial distributions of the two species. Several other candidates for microwave and FIR lines with good sensitivities are discussed in Kozlov et al. (2009). Lentati et al. (2013) used two sensitive observations toward these two systems. First, combining [Cu] fine structure and CO(2-1) rotational
transitions towards J1148+5251 taken with the Plateau de Bure Interferometer (PdBI) and Jansky Very Large Array (JVLA) respectively, led to

$$ \frac{\Delta F'}{F'} = (-33 \pm 23) \times 10^{-5}, \quad z = 6.42. \qquad (209) $$

Then, combining [CII] fine structure and CO(2-1) transitions from BR1202-0725 and its sub-millimeter companion galaxy at $z = 4.695$ taken with ALMA and the PdBI, they derived

$$ \frac{\Delta F'}{F'} = (-5 \pm 15) \times 10^{-5}, \quad z = 4.69. \qquad (210) $$

Curran et al. (2011) fit the observed [CII] and CO profiles of the redshifted systems known at the time in order to derive constraints on $F'$. They exhibit an anti-correlation between $\Delta F'$ and the quality of the carbon detection and claim that “current instruments are incapable of the sensitivities required to measure changes in the constants through the comparison of CO and carbon lines” before studying the case for ALMA. We shall not describe their analysis further and refer to their Table 1. Their results for their 8 systems with absorption redshift between 2.285 and 4.11 are indeed included in our summary table 15.

The quasar host galaxy RXJ0911.4+0551 at $z = 2.796$ is considered very attractive for $F'$-measurements since it emits very strong and narrow CO(7-6) and [C I] lines. Weiss et al. (2012) derived

$$ \frac{\Delta F'}{F'} = (6.9 \pm 3.7) \times 10^{-6}, \quad z = 2.796. \qquad (211) $$

Levshakov et al. (2012) used unique observations of the CO(7-6) rotational line and the [C I] lensed galaxy HLSJ091828.6+514,223 at redshift $z = 5.243$ to derive the upper bound

$$ |\frac{\Delta F'}{F'}| < 2 \times 10^{-5}. \qquad (212) $$

From IRAM/NOEMA spectra of J0439+1634 at $z = 6.519$ that exhibit four rotational transitions of CO(6-5), CO(9-8) and CO(10-9), Levshakov et al. (2020) used only the two first since the third is blended, together with [CII] and [CII] lines to get

$$ \frac{\Delta F'}{F'} = (0.0 \pm 2.7) \times 10^{-5}. \qquad (213) $$

Similarly from new ALMA observations of the quasar J2310+1855 at $z = 6.003$ let them conclude that

$$ \frac{\Delta F'}{F'} = (2.3 \pm 3.4) \times 10^{-5}. \qquad (214) $$

### 5.6.5 “Conjugate” satellite OH lines: $G = g_p(\alpha_{EM}^2\mu)^{1.85}$

The satellite OH 18 cm lines are conjugate so that the two lines have the same shape, but with one line in emission and the other in absorption. This arises due to an inversion of the level of populations within the ground state of the OH molecule. This behavior has recently been discovered at cosmological distances and it was
shown (Chengalur and Kanekar 2003) that a comparison between the sum and difference of satellite line redshifts probe

$$G = g_p(\alpha_{EM}^2\mu)^{1.85}.$$

From the analysis of the two conjugate satellite OH systems at $z \sim 0.247$ towards PKS 1413+135 and at $z \sim 0.765$ towards PMN J0134-0931, Chengalur and Kanekar (2003) concluded that

$$|\Delta G/G| < 7.6 \times 10^{-5} \qquad z \sim 0.247. \qquad (215)$$

It was also applied to a nearby system, Centaurus A, to give $|\Delta G/G| < 1.6 \times 10^{-5}$ at $z \sim 0.0018$. A more recent analysis (Kanekar et al. 2010a) claims for a tentative evidence (with $2.6\sigma$ significance, or at 99.1% confidence) for a smaller value of G

$$\Delta G/G = (-1.18 \pm 0.46) \times 10^{-5} \qquad z \sim 0.247$$

for the system towards PKS 1413+135. Further observation of the same system at the Arecibo telescope, in emission (1720 MHz) and absorption (1612 MHz) by Kanekar et al. (2018) led however to the constraint

$$\Delta G/G = (0.97 \pm 1.52) \times 10^{-6}, \qquad z = 0.2467. \qquad (216)$$

Combined with their former data from the Arecibo Telescope and the Westerbork Synthesis Radio Telescope (Kanekar et al. 2010a), they concluded $\Delta G/G = (-1.0 \pm 1.3) \times 10^{-6}$.

One strength of this method is that it guarantees that the satellite lines arise from the same gas, preventing from velocity offset between the lines. Also, the shape of the two lines must agree if they arise from the same gas.

### 5.6.6 Summary of the constraints on x,y,F,F' and G

The constraints discussed in Sect. 5.6.1 to 5.6.5 are summarized in Table 14 and Fig. 16 for each of the combination of $g_p$, $\alpha_{EM}$ and $\mu$ as a function of the redshift of the absorption system.

## 5.7 Molecular absorption spectra and the electron-to-proton mass ratio

### 5.7.1 Introduction

As was pointed out in Sect. 5.1, molecular lines provide a test of the variation<fnref n="10" /> (Thompson 1975) of $\mu$ since rotational and vibrational transitions are respectively inversely proportional to their reduce mass and its square-root [see Eq. (151)].

<footnote n="10">¹⁰ Again, $\mu$ is used either from $m_e/m_p$ or $m_p/m_e$ in the literature. In this review, it has been chosen to use $\mu = m_p/m_e$ and $\bar{\mu} = m_e/m_p$.</footnote>
<figure><img src="image_23.png" /><figcaption>**Fig. 16** Constraints on the variation of—from the top to bottom panels—$x \equiv \alpha_{\text{EM}}^2 g_p / \mu$, $y \equiv g_p \alpha_{\text{EM}}^2$, $F \equiv g_p (\alpha_{\text{EM}}^2 \mu)^{1.57}$, $F' \equiv \alpha_{\text{EM}}^2 \mu$ and $G \equiv g_p (\alpha_{\text{EM}}^2 \mu)^{1.85}$ as function of the redshift of the absorption system, as summarized in Table 14</figcaption></figure>

While many molecules are observed in our local environment, only 19 diatomic molecules have been observed extragalactically and about 125 polyatomic molecules have been detected in the interstellar medium (McGuire 2018). There are two main issues to select the astrophysical molecules to be used: (1) their sensitivity to a variation of $\mu$ and (2) how abundant and “easy” to observe them and (3) the intensity of the transitions. This depends on both the intrinsic strength of the transition (i.e., its Einstein A coefficient) and the population of the lower energy states, i.e., of the temperature of the system. Unfortunately intergalactic absorption systems are usually low temperature. Moreover molecules with different sensitivities, with different signs, allow for a better control of the systematic effects and (4) the availability of telescope to measure those molecular spectral lines (while observing in the optical and radio band is quite easy from Earth, IR and far-IR band are important to study nearby systems and require space observations).

For almost 3 decades, vibrorotational lines of H₂ were the sole method to access $\mu$ but H₂ absorbers were difficult to detect even though it is the most abundant molecule in the universe. Today about 10 absorption systems have been extensively studied with both H₂ and HD transitions (Ubachs 2018). However so far one has been unable to observe H₂ directly in systems below redshift 2. It is now complemented by constraints arising from amonia (2 systems), methanol (Milky
Way+1 system), carbon monoxyde (2 systems) and preliminary results from oxygenated water and methylamine.

Intensive works have been performed to compute the sensitivities of the required transitions and to compile data on molecular spectra with the highest accuracy. We refer e.g., to Ubachs et al. (2016a), Ubachs (2018), Ubachs et al. (2019) for H₂ and HD and to Kozlov and Levshakov (2013) for the computation of the sensitivities of CH, OH, NH⁺, NH₃, ND₃, NH₂D, NHD₂, H₂O₂, H₃O⁺, CH₃OH, and CH₃NH₂, Yurchenko et al. (2013) for CH₄. Syme et al. (2019), Syme and McKemmish (2020) compiled the sensitivities of all diatomic molecules that have been observed astrophysically.

From a theoretical point of view, constraints from molecules are complementary to those from atoms since molecules are usually found in denser environment. Hence comparison of μ constraints from atoms and molecules allow one to question the chameleon mechanism (see Sect. 3.2) and independent constraints on μ and α<sub>EM</sub> allows one to test unification scenarios; see Sect. 4.5.

### 5.7.2 Constraints with H₂

H₂ is the most abundant molecule in the universe and there were many attempts to use it to set constraints on the variation of μ despite the fact that H₂ is very difficult to detect (Noterdaeme et al. 2007). As proposed in Varshalovich and Levshakov (1993), the sensitivity of a vibro-rotational wavelength to a variation of μ can be parameterized as

$$ \lambda_i = \lambda_i^0 (1 + z_{\text{abs}}) \left( 1 + K_i \frac{\Delta\mu}{\mu} \right), $$

where λᵢ⁰ is the laboratory wavelength (in the vacuum) and λᵢ is the wavelength of the transition i in the rest-frame of the cloud, that is at a redshift z<sub>abs</sub> so that the observed wavelength is λᵢ/(1 + z<sub>abs</sub>). Kᵢ is a sensitivity coefficient analogous to the q-coefficient introduced in Eq. (187), but with different normalization since in the parameterisation we would have qᵢ = ωᵢ⁰Kᵢ/2,

$$ K_i \equiv \frac{d \ln \lambda_i}{d \ln \mu} $$

corresponding to the Lyman and Werner bands of molecular hydrogen. From this expression, one deduces that the observed redshift measured from the transition i is simply

$$ z_i = z_{\text{abs}} + bK_i, \quad b \equiv -(1 + z_{\text{abs}}) \frac{\Delta\mu}{\mu}, $$

which implies in particular that z<sub>abs</sub> is not the mean of the zᵢ if Δμ ≠ 0. Indeed zᵢ is measured with some uncertainty of the astronomical measurements λᵢ and by errors of the laboratory measurements λᵢ⁰. But if Δμ ≠ 0 there must exist a correlation between zᵢ and Kᵢ so that a linear regression of zᵢ (measurement) as a function of Kᵢ
(computed) allows to extract ($z_{abs}$, b) and their statistical significance. We refer to Sect. V.C of FVC03 (Uzan 2003b) for earlier studies.

A first analysis of VLT/UVES of the molecular hydrogen of two damped Lyman-α absorption systems at $z = 2.3377$ and $3.0249$ in the direction of two quasars (Q1232+082 and Q0347-383) showed (Ivanchik et al. 2002) a slight indication of a variation, $\Delta\mu/\mu = (5.7 \pm 3.8) \times 10^{-5}$ at $1.5\sigma$ for the combined analysis. The lines were selected so that they are isolated, unsaturated and unblended (using only 12 lines over 50 detected for the first quasar and 18 over 80 for the second) and the two selected spectra had no transition in common. The authors performed their analysis with two laboratory catalogs and got different results, pointing out that the errors on the laboratory wavelengths are comparable to those of the astronomical measurements.

It was further improved with an analysis of two absorption systems at $z = 2.5947$ and $z = 3.0249$ in the directions of Q0405-443 and Q 0347-383 observed with the VLT/UVES spectrograph. The same selection criteria where applied, letting respectively 39 (out of 40) and 37 (out of 42) lines for each spectrum and 7 transitions in common. The combined analysis of the two systems led (Ivanchik et al. 2005) to $\Delta\mu/\mu = (1.65 \pm 0.74) \times 10^{-5}$ or $\Delta\mu/\mu = (3.05 \pm 0.75) \times 10^{-5}$, according to the laboratory measurements that were used. The same data were reanalyzed with new and highly accurate measurements of the Lyman bands of H$_2$, which implied a reevaluation of the sensitivity coefficient $K_i$. It leads to the two constraints (Reinhold et al. 2006)

$$ \Delta\mu/\mu = (2.78 \pm 0.88) \times 10^{-5}, \quad z = 2.59, \quad [Q00405-443] \quad (217) $$

$$ \Delta\mu/\mu = (2.06 \pm 0.79) \times 10^{-5}, \quad z = 3.02, \quad [Q0347-383] \quad (218) $$

leading to a $3.5\sigma$ detection for the weighted mean $\Delta\mu/\mu = (2.4 \pm 0.66) \times 10^{-5}$. Reinhold et al. (2006) did not claim for a detection and are cautious enough to state that systematics dominate the measurements. King et al. (2008) employed a comprehensive fitting method for these 2 quasars to which they added a new system at $z_{abs} = 2.81$ in direction of Q0528-250, to get

$$ \Delta\mu/\mu = (10.1 \pm 6.2) \times 10^{-6}, \quad z = 2.59, \quad [Q00405-443] \quad (219) $$

$$ \Delta\mu/\mu = (-1.4 \pm 3.9) \times 10^{-6}, \quad z = 2.80, \quad [Q0528-250] \quad (220) $$

$$ \Delta\mu/\mu = (5.2 \pm 7.4) \times 10^{-6}, \quad z = 3.02, \quad [Q0347-383] \quad (221) $$

with a weighted mean of $(2.6 \pm 3.0) \times 10^{-6}$ at $z \sim 2.81$. Concerning Q0347-383, Thompson et al. (2009) and Thompson (2009) reanalyzed the data using an advanced line-by-line method to get

$$ \Delta\mu/\mu = (-28 \pm 16) \times 10^{-6}. \quad z = 3.02, \quad [Q0347-383]. \quad (222) $$

New observations led to the bound (Wendt and Reimers 2008) $|\Delta\mu/\mu| < 4.9 \times 10^{-5}$ at a $2\sigma$, thus contradicting (218). Then Wendt and Molaro (2011) derived $\Delta\mu/\mu =$
(15 ± 9 ± 6syst) × 10⁻⁶ at z = 3.02 towards Q0347-383 while Wendt and Molaro (2012) improved to

$$ \Delta\mu/\mu = (4.3 \pm 7.2) \times 10^{-6}. \quad z = 3.02, \quad [Q0347-383], \quad (223) $$

leading to a weighted mean of the four most accurate results of Δμ/μ = (5.1 ± 4.5) × 10⁻⁶ (Ubachs et al. 2016a).

Concerning Q0528-250, renewed VLT observations (King et al. 2011) concluded from the analysis of Using 76 H₂ and 7 HD transitions

$$ \Delta\mu/\mu = (0.3 \pm 3.2 \pm 1.9_{\text{syst}}) \times 10^{-6} \quad z = 2.80, \quad [Q0528-250]. \quad (224) $$

so that an average value is Δμ/μ = (−0.5 ± 2.7) × 10⁻⁶ (Ubachs et al. 2016a).

Concerning Q1232+082, from the analysis of VLT/UVES observations of 106 H₂ and HD lines with a comprehensive fitting technique, Daprà et al. (2017b) concluded that

$$ \Delta\mu/\mu = (19 \pm 9 \pm 5_{\text{syst}}) \times 10^{-6} \quad z = 2.34, \quad [Q1232+082]. \quad (225) $$

A line-by-line analysis of Q00405-443 (Thompson et al. 2009) gave

$$ \Delta\mu/\mu = (0.6 \pm 10) \times 10^{-6}. \quad z = 2.59, \quad [Q00405-443], \quad (226) $$

leading to an average value of Δμ/μ = (7.5 ± 5.3) × 10⁻⁶ (Ubachs et al. 2016a).

The molecular system at z = 2.059 toward J2123-0050 was observed by the HIRES spectrometer at the Keck telescope. It exhibited 86 H₂ transitions and 7 HD transitions. The analysis by (Malec et al. 2009) concluded that

$$ \Delta\mu/\mu = (5.6 \pm 5.5 \pm 2.7_{\text{syst}}) \times 10^{-6}, \quad z = 2.059, \quad [J2123-0050]. \quad (227) $$

An independent spectrum was obtained with the VLT/UVES for 90 H₂ lines and 6 HD lines from which van Weerdenburg et al. (2011) derived

$$ \Delta\mu/\mu = (8.5 \pm 3.6 \pm 2.2_{\text{syst}}) \times 10^{-6}, \quad z = 2.059, \quad [J2123-0050]. \quad (228) $$

Averaging over these two independent results (Ubachs et al. 2016a) lead to Δμ/μ = (7.6 ± 3.5) × 10⁻⁶ for J2123-0050.

Rahmani et al. (2013) observed a system at z<sub>abs</sub> = 2.4018 towards HE0027-1836 with VLT over 3 years to obtain

$$ \Delta\mu/\mu = (-7.6 \pm 10.2) \times 10^{-6}, \quad z = 2.4018, \quad [HE0027-1836]. \quad (229) $$

The system at z<sub>abs</sub> = 2.659 in the direction of J0643-5041 exhibiting a single H₂ absorption feature, was analyzed in a line-by-line analysis by Albornoz Vásquez et al. (2014), Rahmani et al. (2014a) from high-resolution VLT-UVES data from more than 23 h exposure, to get Δμ/μ = (7.4 ± 4.3 ± 5.1<sub>syst</sub>) × 10⁻⁶. Reanalyzed by Bagdonaite et al. (2014b) it gave Δμ/μ = (12.7 ± 4.5 ± 4.2<sub>syst</sub>) × 10⁻⁶ and Ubachs et al. (2016a) adopted the mean value
$$ \Delta\mu/\mu = (10.3 \pm 4.6) \times 10^{-6}, \quad z = 2.659 \quad [J0643-5041], \quad (230) $$

which corresponds to $(7.4 \pm 6.7) \times 10^{-6}$ after combining the uncertainties.

The absorption system at $z = 2.426$ in direction of Q2348-011 has a complex velocity structure featuring at least 7 H$_2$ absorption sub-systems. Analyzed by Bagdonaitė et al. (2012) it led to

$$ \Delta\mu/\mu = (-6.8 \pm 27.8) \times 10^{-6}, \quad z = 2.426, \quad [Q2348-011]. \quad (231) $$

The analysis of an absorption system at $z_{abs} = 2.69$ towards J1237+064 which exhibits 3 clouds with 137 lines of H$_2$ and HD was analyzed by Daprà et al. (2015) to give

$$ \Delta\mu/\mu = (-5.4 \pm 6.3 \pm 4.0_{\text{syst}}) \times 10^{-6}, \quad z = 2.69, \quad [J1237+064] \quad (232) $$

and Ubachs et al. (2016a) adopt $(-5.4 \pm 7.2) \times 10^{-6}$ after combining the uncertainties.

The high redshift system at $z_{abs} = 4.224$ in the direction of J1443+2724 was observed by VLT/UVES. Combining archival data (Ledoux et al. 2006) with 2013-data by Bagdonaitė et al. (2015) gave

$$ \Delta\mu/\mu = (-9.5 \pm 5.4 \pm 5.3_{\text{syst}}) \times 10^{-6}, \quad z = 4.224 \quad [J1443+2724] \quad (233) $$

which corresponds to $(-9.5 \pm 7.5) \times 10^{-6}$ after combining the uncertainties.

Note that the detection of several deuterated molecular hydrogen HD transitions makes it possible to test the variation of $\mu$ in the same way as with H$_2$ but in a completely independent way, even though today it has been detected only in 2 places in the universe. The sensitivity coefficients have been published in Ivanov et al. (2009) and HD was first detected by Noterdaeme et al. (2007). HD was recently detected (Srianand et al. 2008) together with CO and H$_2$ in a DLA cloud at a redshift of 2.418 toward SDSS1439+11 with 5 lines of HD in 3 components together with several H$_2$ lines in 7 components. It allowed to set the $3\sigma$ limit of $|\Delta\mu/\mu|<9 \times 10^{-5}$ (Petitjean et al. 2009b). Even though the small number of lines does not allow to reach the level of accuracy of H$_2$ it is a very promising system in particular to obtain independent measurements.

This method is subject to important systematic errors among which (1) the sensitivity to the laboratory wavelengths—since the use of two different catalogs yields different results (Reinhold et al. 2006)—(2) the molecular lines are located in the Lyman-$\alpha$ forest where they can be strongly blended with intervening H I Lyman-$\alpha$ absorption lines, which requires a careful fitting (King et al. 2008) since it is hard to find lines that are not contaminated. From an observational point of view, very few damped Lyman-$\alpha$ systems have a measurable amount of H$_2$ so that only a dozen systems is actually known even though more systems will hopefully be obtained soon (Petitjean et al. 2009a).

To finish, the sensitivity coefficients are usually low, typically of the order of $10^{-2}$. Some advantages of using H$_2$ arise from the fact there are several hundred available H$_2$ lines so that many lines from the same ground state can be used to
eliminate different kinematics between regions of different excitation temperatures. The overlap between Lyman and Werner bands also allows one to reduce the errors of calibration.

### 5.7.3 Amonia observations

Amonia is one of the most abundant polyatomic molecules observed in the interstellar medium. Unfortunately only two extragalactic sources, B0218+357 and PKS1830-211, are known and both are in low redshift galaxies. Following the idea by van Veldhoven et al. (2004) to use inversion transition in ND₃ to probe a variation of μ, Flambaum and Kozlov (2007a, 2007b) adapted this technique to compute inversion spectrum of NH₃ and show that this allows for a better sensitivity to μ (see Owens et al. 2016 for extensive computation of the sensitivity coefficients). The inversion vibro-rotational mode is described by a double well with the first two levels below the barrier. The tunneling implies that these two levels are split in inversion doublets. It was concluded that the inversion transitions scale as $v_{inv} \sim \bar{\mu}^{4.46}$, compared with a rotational transition, which scales as $v_{rot} \sim \bar{\mu}$. This implies that the redshifts determined by the two types of transitions are modified according to $\delta z_{inv} = 4.46(1 + z_{abs})\Delta\mu/\mu$ and $\delta z_{rot} \sim (1 + z_{abs})\Delta\mu/\mu$ so that

$$ \Delta\mu/\mu = 0.289 \frac{z_{inv} - z_{rot}}{1 + z_{abs}}. $$

For the first quasar absorption system displaying NH₃ at $z = 0.68466$ in the direction of B0218+357, Flambaum and Kozlov (2007a) estimated from the published redshift uncertainties that a precision of $\sim 2 \times 10^{-6}$ on $\Delta\mu/\mu$ can be achieved. A detailed measurement (Murphy et al. 2008a) of the ammonia inversion transitions by comparison to HCN and HCO⁺ rotational transitions concluded that

$$ \Delta\mu/\mu = (0.74 \pm 0.47 \pm 0.76_{\text{syst}}) \times 10^{-6}, \quad z = 0.685, \quad [\text{B0218+357}] \quad (234) $$

which corresponds to the $2\sigma$ bound on the variation of μ, $|\Delta\mu/\mu| < 1.8 \times 10^{-6}$. Combining inversion (NH₃) and rotational (CS, H₂CO) absorption lines detected with the Green Bank Telescope allowed Kanekar (2011) to conclude that

$$ \Delta\mu/\mu = (-0.35 \pm 0.12) \times 10^{-6}, \quad z = 0.685, \quad [\text{B0218+357}]. \quad (235) $$

For the second known system, the analysis of the comparison of NH₃ to HC₃N spectra was performed toward the gravitational lens system PKS 1830-211 ($z \simeq 0.89$), which is a much more suitable system, with 10 detected NH₃ inversion lines and a forest of rotational transitions. Henkel et al. (2009) reached the conclusion that

$$ |\Delta\mu/\mu| < 1.4 \times 10^{-6}, \quad z = 0.89, \quad [\text{PKS 1830-211}] \quad (236) $$

at a $3\sigma$ level. From a comparison of the ammonia inversion lines with the NH₃ rotational transitions, Menten et al. (2008) concluded
$$|\Delta\mu/\mu| < 3.8 \times 10^{-6}, \quad z = 0.89, \quad [\text{PKS 1830-211}] \quad (237)$$

at 95% C.L. Recent works (Owens et al. 2015, 2016) took into account that the rotational energy levels of amonia exhibit a non-negligible centrifugal distortion dependence, which was disregarded. Hence the single sensitivity K = 4.46 shall be replaced with sensitivities ranging from 4.3 to 4.9. From the spectroscopic measurements by Henkel et al. (2008), the reanalysis of this system (Bagdonaitė 2015) led to $\Delta\mu/\mu = (1.1 \times 7.0) \times 10^{-6}$.

One strength of this analysis is to focus on lines arising from only one molecular species but it was mentioned that the frequencies of the inversion lines are about 25 times lower than the rotational ones, which might cause differences in the absorbed background radio continuum. From a survey with the Australia Telescope Compact Array Muller et al. (2011) detected 28 different molecules toward the south-west absorption region, located about 2 kpc from the center, making this system the one with the largest number of detected molecular species of any extragalactic object at the time of the work. The analysis of the inversion lines of amonia led to

$$\Delta\mu/\mu = (-2.04 \pm 0.74) \times 10^{-6}, \quad z = 0.89, \quad [\text{PKS 1830-211}] \quad (238)$$

assuming no time variation between the different observations.

### 5.7.4 Methanol observations

Levshakov et al. (2011) identified methanol as a tracer of $\mu$. They pointed out that its spectrum includes purely rotational transitions as well as transitions with contributions of the internal degrees of freedom associated with the hindered rotation of OH and that the latter is highly sensitive to $\mu$. Sensitivity coefficients are calculated and range between -88 and 330. Note that there are two types of methanol: in E-type, one of the protons in the hydrogen atoms of the CH$_3$ group has an antiparallel nuclear spin with respect to the others while in A-type methanol, the nuclear spins of the three protons in the CH$_3$ methyl group are parallel. The two methanol types have different transition frequencies and may arise in different physical environments and have both been observed in molecular clouds. Recently, Agafonova et al. (2024) presented the simultaneous observations of Class I methanol masers at 25, 36, and 44 GHz towards 22 Galactic targets thanks to the Effelsberg 100-m telescope.

This led Jansen et al. (2011) to conclude that the 6.7 and 12.2 GHz masers transitions in methanol, corresponding to the transitions $5_1 \rightarrow 6_0A^+$ and $2_0 \rightarrow 3_{-1}$, are among the brightest radio objects in the sky and enjoy a sensitivity of $K_\mu = -42$ and $K_\mu = -33$ respectively, making methanol an excellent candidate to constrain the variation of fundamental constants. The first detection of methanol at cosmological distance (Muller et al. 2011) in the molecular system located in the disk of a spiral galaxy at $z_{\text{abs}} = 0.8859$ in direction of the quasar PKS1830-211. They concluded that
$$ \Delta\mu/\mu = (-1.95 \pm 0.47) \times 10^{-6} \quad z = 0.8859 \quad [\text{PKS1830-211}]. \quad (239) $$

While a 4σ detection in agreement with their NH3 results, Muller et al. (2011) considered their results as “tentative”. Taking into account the velocity dispersion of a large number of molecular species, they concluded $|\Delta\mu/\mu| < 4 \times 10^{-6}$. The first constraint based only on methanol was obtained by Ellingsen et al. (2012) from the observation of the former transitions in the same system to get

$$ \Delta\mu/\mu = (0.8 \pm 2.1) \times 10^{-7} \quad z = 0.8859 \quad [\text{PKS1830-211}]. \quad (240) $$

which corresponds to a 3σ upper bound of $6.3 \times 10^{-7}$. Bagdonaitė et al. (2013b) combined the 12.2 GHz line with nine other transitions and performed a detailed analysis to the error budget to conclude that

$$ \Delta\mu/\mu = (0.0 \pm 1.0) \times 10^{-7} \quad z = 0.8859 \quad [\text{PKS1830-211}]. \quad (241) $$

After a study of the systematic effects of chemical segregation, excitation temperature, frequency dependence and time variability of the background source are quantified, Bagdonaitė et al. (2013a) concluded that

$$ \Delta\mu/\mu = (1.5 \pm 1.5) \times 10^{-7} \quad z = 0.8859 \quad [\text{PKS1830-211}]. \quad (242) $$

These studies did not test whether the different lines arise from the same region of the gas. Kanekar et al. (2015) actually concluded from their VLA spectrum that the 12.2 GHz line has different properties than the 3 other transitions they consider ($0_0 \rightarrow 1_0$E at 48.382, $0_0 \rightarrow 1_0$A$^+$ at 48.377 and $2_{-1} \rightarrow 1_0$E at 60.531 GHz) and shall not be included in the analysis since it introduces a bias. From these 3 lines they deduced that

$$ |\Delta\mu/\mu| < 4 \times 10^{-7} \quad (243) $$

at 2σ and that including the 12.2 GHz line would bias the result to $\Delta\mu/\mu = (-2.9 \pm 5.7) \times 10^{-8}$. Indeed, for gravitational lensed systems, additional systematics have to be considered since, even though lensing is achromatic, the structure of the background quasar being lensed does vary both with frequency and time. Thanks to observations up to a few milliarcseconds Marshall et al. (2017) resolved the northeastern and southwestern as well as the Einstein ring (The quasar jet has a steep spectral index and in PKS B1830-211 it is this which causes an Einstein ring to be present only at lower frequencies). Muller et al. (2021) confirmed that methanol is detected in only one, the southwest, of the two lines of sight. The observation of 14 transitions of methanol, five of the A-form and nine of the E-form with ALMA allowed them to set the 1σ constraint

$$ \Delta\mu/\mu = (-1.8 \pm 1.2) \times 10^{-7} \quad z = 0.8859 \quad [\text{PKS1830-211}], \quad (244) $$

corresponding to at $|\Delta\mu/\mu| < 3.6 \times 10^{-7}$ at 3σ.

In conclusion, methanol offers the advantage of a test based on a single species, avoiding the location bias of multi-species techniques. But for now it has been
observed in a single absorption system that has been shown to be subject to many disturbing phenomena such as the time variability of the background blazar and the existence of chromatic substructures.

### 5.7.5 CO observations

CO is the second most abundant molecule in gas form in the universe. From the study of transitions from electronic A-X, Salumbides et al. (2012) suggested to use the electronic $A^1Π - X^1Σ^+$ transition in CO to supplement $H_2$ even though their sensitivities are low (Daprà et al. 2016) and thus expected to set less stringent constraints.

A combined analysis of CO and $H_2$ in a system at $z_{abs} = 2.69$ towards J1237+0647 observed with the VLT/UVES. Thirteen CO vibrational bands in this absorber were detected set

$$ \Delta\mu/\mu = (0.7 \pm 1.6 \pm 0.5_{\text{syst}}) \times 10^{-5} \quad [\text{J1237+0647}] \quad (245) $$

from the analysis of CO only while it reaches

$$ \Delta\mu/\mu = (-5.6 \pm 5.6 \pm 3.1_{\text{syst}}) \times 10^{-6} \quad [\text{J1237+0647}] \quad (246) $$

once combined with $H_2$ in the same system

From the observations of 8 vibrational singlet-singlet bands and 1 singlet-triplet band of carbon monoxide in the spectrum of a damped Lyman-$\alpha$ system at $z = 2.52$ towards quasar SDSS J000015.16+004833.2 with VLT/UVES and correction for long-range wavelength scale distortions using the supercalibration technique, Daprà et al. (2017c) concluded that

$$ \Delta\mu/\mu = (1.8 \pm 2.2 \pm 0.4_{\text{syst}}) \times 10^{-5} \quad [\text{J0000+0048}] \quad (247) $$

So far, only two systems have been studied while optical absorption bands of CO have been detected in 6 absorption systems at redshifts larger than 1, SDSS J16045+2203 (Noterdaeme et al. 2008), SDSS J0857+1855, SDSS J1047+2057, SDSS J1705+3543 (Noterdaeme et al. 2011), SDSS J1439+1117 (Srianand et al. 2008) and SDSS J1237+0647 (Noterdaeme et al. 2010).

### Other molecules

Ilyushin et al. (2012) identified enhancement of the sensitivity to $\mu$ in torsion-wagging-rotation transitions in the ground state of methylamine ($CH_3NH_2$) due to energy cancellations between internal rotational, overall rotational and inversion energy splitting. This molecule is present in our Galaxy and in a system at $z_{abs} = 0.8859$ in direction of the quasar PKS1830-211 (Muller et al. 2011). Using 3 transitions at 78.135, 79.008 and 89.956 GHz with $k_{\mu} = -0.87$ for the first two and $-1.4$ for the third, Ilyushin et al. (2012) concluded that $|\Delta\mu/\mu| < 9 \times 10^{-6}$. Combining with the radial velocity of the methanol line at 60.531 GHz in methanol ($K_{\mu} = -7.4$, Jansen et al. 2011), they conclude that $|\Delta\mu/\mu| < 10^{-6}$, with the caveat
that the two molecules may not be distributed in the same region of the absorption system.

The detection of H₂O₂ in interstellar clouds have been reported by Bergman et al. (2011). It has been identified as a good candidate for testing variation of μ (Kozlov 2011; Polyansky et al. 2013) since the estimations of the sensitivity coefficients of its microwave transitions lead to the largest coefficient for 14.8 GHz transition: K_μ = 37.

## Summary and further possibilities

All these constraints are summarized in Table 15, showing the tremendous progresses of the observation of the molecular absorption spectra. Molecular H₂ observations allowing one to reach 10⁻⁸ are expected from the E-ELT/ANDES spectrograph (Martins et al. 2024). We refer to Chen et al. (2019) for the prospects on methanol observation with the FAST telescope. Let us also mention that FIR lines are expected to be observed by a new generation of telescopes such as HERSCHEL<fnref n="11" /> and that surveys in radio are being carried out so that the number of known redshift OH, HI and HCO⁺ absorption systems will increase. For instance the future Square Kilometer Array (SKA) will be able to detect relative changes of the order of 10⁻⁷ in α<sub>EM</sub>.

## 5.8 Emission spectra

Similar analysis to constrain the time variation of the fundamental constants were also performed with emission spectra. Very few such estimates have been performed, since it is less sensitive and harder to extend to sources with high redshift. In particular, emission lines are usually broad as compared to absorption lines and the larger individual errors need to be beaten by large statistics.

The O III analysis considered the forbidden line doublet of O III at λ = 4960.295 Å and λ = 5008.239 Å. These two particular spectral lines are produced by magnetic dipole interactions with a small contribution of electric quadrupole radiation. It can be shown (Bahcall et al. 2004) that

$$R \equiv \frac{\lambda_{5008} - \lambda_{4960}}{\lambda_{5008} + \lambda_{4960}} \propto \alpha_{\text{EM}}^2. \qquad (248)$$

The early analysis by Bahcall and Schmidt (1967) concluded that Δα<sub>EM</sub>/α<sub>EM</sub> = (1 ± 2) × 10⁻³ in the redshift band [0.17 – 0.26].

The [O III] doublet method was sidelined for decades thought to be non-competitive compared to absorption spectra. It was revived in the 2000s thanks to the SDSS project that produced a large sample of QSO spectra with resolution of order R ~ 2000. Most the constraints gets an improved precision by averaging on a large number of QSOs so that constraints are obtained for a band of redshifts. As such, the analysis of a sample of 42 quasars from SDSS early data release gave (Bahcall et al. 2004)

<footnote n="11">¹¹ http://sci.esa.int/science-e/www/area/index.cfm?fareaid=16</footnote>
$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (7 \pm 14) \times 10^{-5}, \quad 0.16 < z < 0.8, \qquad (249) $$

while the DR1 release (165 quasars) gave $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (12 \pm 7) \times 10^{-5}$. From the analysis of a sample of 1568 quasar of the SDSS-DR2, Gutiérrez and López-Corredoira (2010) concluded that

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (2.4 \pm 2.5) \times 10^{-5}, \quad 0.0 < z < 0.8. \qquad (250) $$

Using 2347 quasars of the SDSS-DR7, Rahmani et al. (2014b) obtained

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (-2.1 \pm 1.6) \times 10^{-5}, \quad 0.02 < z < 0.74, \qquad (251) $$

consistent with a non-varying $\alpha_{\text{EM}}$ at a level of $2 \times 10^{-5}$. Applying the same method on a much larger data base of 13,175 quasars from SDSS DR12 yielded (Albareti et al. 2015)

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (0.9 \pm 1.8) \times 10^{-5}, \quad 0.04 < z < 1.0. \qquad (252) $$

They also provide the measurement of $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}}$ in 10 redshift bins from 0.390 to 1.00. Laker et al. (2022) analyzed 12 000 galaxies from the SDSS-DR8 to get

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (1.3 \pm 1.26) \times 10^{-5}, \quad 0.0 < z < 0.467. \qquad (253) $$

The method was then extended straightforwardly along the lines of the MM method and applied (Grupe et al. 2005) to the fine-structure transitions in Ne III, Ne v, O III, O I and S II multiplets from a sample of 14 Seyfert 1.5 galaxies to derive the constraint

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (150 \pm 70) \times 10^{-5}, \quad 0.035 < z < 0.281. \qquad (254) $$

Ne III and S II were also analyzed by Gutiérrez and López-Corredoira (2010) to get $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (36 \pm 1) \times 10^{-4}$ while Albareti et al. (2015) got $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (34 \pm 1) \times 10^{-4}$. These doublets are more difficult to analyze since they are fainter than those of O III. These results are significantly different from zero with a clear tendency for a positive variation of $\alpha_{\text{EM}}$, a general tendency incompatible with the O III constraints and that need to be better understood and it was demonstrated (Albareti et al. 2015) that the accuracy of Ne III is below $10^{-3}$ due to contamination by other emission lines.

This method requires to use large samples (typically $N > 20$) and has been applied mostly with QSO probably because they have higher O III luminosities or simply due to the observational strategy of SDSS. Li et al. (2023) noted that (1) starburst galaxies far outnumber QSO and (2) have narrower O III emission lines, which is advantageous for improving the accuracy of the $\alpha_{\text{EM}}$ measurement. They claim that it can compensate their disadvantage of lower O III luminosity. Besides, it will allow one to reach higher redshifts. To that goal, Li et al. (2023) constructed a sample of 40 spectra of Ly$\alpha$ emitting galaxies and of 46 spectra of QSO with
redshift ranging from 1.09 to 3.73 using the VLT/X-Shooter near-IR spectra that are publicly available (Vernet et al. 2011). The analysis of the 86 spectra gave

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (-3 \pm 6) \times 10^{-5}, \quad 1.09 < z < 3.73. \qquad (255) $$

Jiang et al. (2024) used the DESI (Dark Energy Spectroscopic Instrument) observation of about 110 000 galaxies with strong and narrow [O III] emission lines on a redshift band up to 0.95. The sample was split in 10 bins of $\Delta z = 0.1$ and $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}}$ was measured for each subsample. They concluded for an apparent variation, $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (2-3) \times 10^{-5}$ and argued that this was likely to be due to systematics associated with wavelength calibration. Since it allows for the use of other emission lines, in particular [Ne III] and a better calibration, it is claimed that DESI may probe higher redshifts (up to 1.45 in the near future).

Jiang et al. (2025) applied the same method to a galaxy sample consists of 572 spectra with strong and narrow [O III] emission lines from 522 galaxies with 267 spectra at $z > 5$. The sample was split into 5 subsamples, all consistent with zero within $1\sigma$ error of $(1-2) \times 10^{-4}$; see Fig. 5 of Jiang et al. (2025). The whole sample led to the conclusion that

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (4 \pm 7) \times 10^{-5}, \quad 3 < z < 7. \qquad (256) $$

This is the first constraint to reach such high redshifts. From the LAMOST (Large Sky Area Multi-object Fiber Spectroscopic Telescope) Data Release 9 quasar catalog, Wei et al. (2024) exhibited a sample of 209 quasar spectra with strong and narrow [O III] emission lines over a redshift range between 0 and 0.8 to conclude that

$$ \Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} = (0.5 \pm 3.7) \times 10^{-4}, \quad 0 < z < 0.8. \qquad (257) $$

so far not competitive with the previous constraints from the Sloan Digital Sky Survey.

The [O III] emission spectra method has regained interest. In particular because it relies on fewer assumptions and suffers from fewer systematic errors compared the MM method: no assumptions on chemical composition, ionization state, and distribution of energy levels have to be made since the doublet lines originate in the transitions from the same upper level of the same ion. The method is also considered more tolerant of the wavelength distortion because of the small wavelength range used in the measurement (Li et al. 2023). Besides, while it is difficult to detect absorption spectra at high redshifts (see e.g., Wilczynska et al. 2020 for a discussion of this limitation), many starburst galaxies have been discovered in the early universe, and their doublet emission lines can be used. The data discussed in this paragraph are summarized in Table 16 and Fig. 18.
<table><caption>**Table 16** Available measurements of $z_{EM}$ from the emission spectra</caption><thead><tr><th>Redshift band</th><th>Δz<sub>EM</sub>/z<sub>EM</sub></th><th>Data</th><th>Ref</th></tr></thead><tbody><tr><td>[0.17 – 0.26]</td><td>(1 ± 2) × 10<sup>-3</sup></td><td>5 QSO</td><td>(1)</td></tr><tr><td>[0.16 – 0.8]</td><td>(7 ± 14) × 10<sup>-5</sup></td><td>42 QSO (SDSS EDR)</td><td>(2)</td></tr><tr><td>[0.16 – 0.8]</td><td>(12 ± 7) × 10<sup>-5</sup></td><td>165 QSO (SDSS-DR1)</td><td>(3)</td></tr><tr><td>[0 – 0.8]</td><td>(2.4 ± 2.5) × 10<sup>-5</sup></td><td>1568 QSO (SDSS-DR2)</td><td>(4)</td></tr><tr><td>[0.02 – 0.74]</td><td>(–2.1 ± 1.6) × 10<sup>-5</sup></td><td>2347 QSO (SDSS-DR7)</td><td>(5)</td></tr><tr><td>[0.04 – 1.0]</td><td>(0.9 ± 1.8) × 10<sup>-5</sup></td><td>13 175 QSO (SDSS DR12)</td><td>(6)</td></tr><tr><td>[0 – 0.467]</td><td>(1.3 ± 1.26) × 10<sup>-5</sup></td><td>12,000 galaxies (SDSS-DR8)</td><td>(7)</td></tr><tr><td>[0.035 – 0.281]</td><td>(150 ± 70) × 10<sup>-5</sup></td><td>14 Seyfert 1.5 galaxies</td><td>(8)</td></tr><tr><td>[1.09 – 3.73]</td><td>(–3 ± 6) × 10<sup>-5</sup></td><td>40 galaxies+46 QSO (VLT/X-Shooter)</td><td>(9)</td></tr><tr><td>[3 – 7]</td><td>(4 ± 7) × 10<sup>-5</sup></td><td>522 galaxies (JWST)</td><td>(10)</td></tr><tr><td>[0 – 0.8]</td><td>(0.5 ± 3.7) × 10<sup>-4</sup></td><td>209 quasras (LAMOST)</td><td>(11)</td></tr></tbody></table>

These constraints are averaged on a redshift bin but do not require any assumptions on chemical composition, ionization state. Besides they are sensitive to $z_{EM}$ alone

Refs.: (1) Bahcall and Schmidt (1967; 2, 3) Bahcall et al. (2004; 4) Gutiérrez and López-Corredoira (2010); (5) Rahmani et al. (2014b; 6) Albareti et al. (2015; 7) Laker et al. (2022); (8) Grupe et al. (2005; 9) Li et al. (2023); (10) Jiang et al. (2025; 11) Wei et al. (2024)

<figure><img src="image_24.png" /><figcaption>**Fig. 17** Molecular absorption spectra constraints on the variation $\mu = m_p/m_e$ from H$_2$, NH$_3$, CO and CH$_3$OH as summarized in Table 15</figcaption></figure>
## 5.9 Stellar constraints

Stars start to accumulate helium produced by the pp-reaction and the CNO cycle in their core. Furthermore, the products of further nuclear reactions of helium with either helium or hydrogen lead to isotopes with $A = 5$ or $A = 8$, which are highly unstable. In order to produce elements heavier than $A > 7$ by fusion of lighter isotopes, the stars need to reach high temperatures and densities. In these conditions, newly produced $^{12}$C would almost immediately be fused further to form heavier elements so that one expects only a tiny amount of $^{12}$C to be produced, in contradiction with the observed abundances. This led Hoyle (1954) to conclude that a then unknown excited state of the $^{12}$C with an energy close to the $3\alpha$-threshold should exist since such a resonance would increase the probability that $^{8}$Be captures an $\alpha$-particle. It follows that the production of $^{12}$C in stars relies on the three conditions:

* the decay lifetime of $^{8}$Be, of order $10^{-16}$ s, is four orders of magnitude longer than the time for two $\alpha$ particles to scatter, so that a macroscopic amount of beryllium can be produced, which is sufficient to lead to considerable production of carbon,

* an excited state of $^{12}$C lies just above the energy of $^{8}$Be+$\alpha$, which allows for

<figure><img src="image_25.png" /><figcaption>**Fig. 18** Constraints on the variation of $\alpha_{EM}$ as a function of $\log(z)$. The plot gathered the data summarized in Table 16 together with the 5 data points from Table 5 of Jiang et al. (2025). The orange and blue zones correspond respectively to QSO sources and galaxy sources. The extension in redshift accounts for the width of the bin on which the data are averaged</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Fig. 19 Left: Level scheme of nuclei participating to the $^4\text{He}(\alpha\alpha, \gamma)^{12}\text{C}$ reaction. Right: Central abundances at the end of the CHe burning as a function of $\delta_{NN}$ for a $60\,\text{M}_\odot$ star with $Z=0$. From Coc et al. (2009a)</figcaption></figure>

$$ ^4\text{He} + ^4\text{He} \leftrightarrow ^8\text{Be}, \qquad ^8\text{Be} + ^4\text{He} \leftrightarrow ^{12}\text{C}^* \rightarrow ^{12}\text{C} + 7.367\,\text{MeV}, $$

* the energy level of $^{16}\text{O}$ at 7.1197 MeV is non resonant and below the energy of $^{12}\text{C} + \alpha$, of order 7.1616 MeV, which ensures that most of the carbon synthesized is not destroyed by the capture of an $\alpha$-particle. The existence of this resonance, the $J_f^\pi = 0_2^+$-state of $^{12}\text{C}$ was actually discovered (Cook et al. 1957) experimentally later, with an energy of $372 \pm 4$ keV [today, $E_{0_2^+} = 379.47 \pm 0.15$ keV], above the ground state of three $\alpha$-particles (see Fig. 19).

The variation of any constant that would modify the energy of this resonance would also endanger the stellar nucleosynthesis of carbon, so that the possibility for carbon production has often been used in anthropic arguments. Qualitatively, if $E_{0_2^+}$ is increased then the carbon would be rapidly processed to oxygen since the star would need to be hotter for the triple-$\alpha$ process to start. On the other hand, if $E_{0_2^+}$ is decreased, then all $\alpha$-particles would produce carbon so that no oxygen would be synthesized. It was estimated (Livio et al. 1989) that the carbon production in intermediate and massive stars is suppressed if the various of the energy of the resonance is outside the range $-250 \text{ keV} \lesssim \Delta E_{0_2^+} \lesssim 60 \text{ keV}$, which was further improved (Schlattl et al. 2003) to, $-5 \text{ keV} \lesssim \Delta E_{0_2^+} \lesssim 50 \text{ keV}$ in order for the C/O ratio to be larger than the error in the standard yields by more than 50%. Indeed, in such an analysis, the energy of the resonance was changed by hand. However, we expect that if $E_{0_2^+}$ is modified due to the variation of a constant other quantities, such as the resonance of the oxygen, the binding energies and the cross-sections will also be modified in a complex way.
In practice, to draw a constraint on the variation of the fundamental constants from the stellar production of carbon, one needs to go through different steps, each of them involving assumptions,

1. to determine the effective parameters, e.g., cross-sections, which affects the stellar evolution. The simplest choice is to modify only the energy of the resonance but it may not be realistic since all cross-sections and binding energies should also be affected. This requires one to use a stellar evolutionary model;

2. relate these parameters to nuclear parameters. This involves the whole nuclear physics machinery;

3. to relate the nuclear parameters to fundamental constants. As for the Oklo phenomenon, it requires to link QCD to nuclear physics.

A first analysis (Oberhummer et al. 2000, 2001; Schlattl et al. 2003) used a model that treats the carbon nucleus by solving the 12-nucleon Schrödinger equation using a three-cluster wavefunction representing the three-body dynamics of the ¹²C state. The NN interaction was described by the Minnesota model (Korennov and Descouvemont 2004; Thomson et al. 1977) and its strength was modified by multiplying the effective NN-potential by an arbitrary number *p*. This allows to relate the energy of the Hoyle level relative to the triple alpha threshold, $\epsilon \equiv Q_{ααα}$, and the gamma width, $\Gamma_γ$, as a function of the parameter *p*, the latter being almost not affected. The modified 3α-reaction rate was then given by

$$r_{\alpha} = 3^{3/2}N_{\alpha} \left( \frac{2\pi\hbar^2}{M_{\alpha}k_{\text{B}}T} \right)^3 \frac{\Gamma}{\hbar} \exp \left[ -\frac{\epsilon(p)}{k_{\text{B}}T} \right], \quad (258)$$

where $M_α$ and $N_α$ are the mass and number density of the α-particle. The resonance width $\Gamma = \Gamma_α\Gamma_γ/(\Gamma_α + \Gamma_γ) \sim \Gamma_γ$. This was included in a stellar code and ran for red giant stars with 1.3, 5 and 20 $M_⊙$ with Solar metallicity up to thermally pulsating asymptotic giant branch (Oberhummer et al. 2000) and in low, intermediate and high mass (1.3, 5, 15, 25 $M_⊙$) with Solar metallicity also up to TP-AGB (Schlattl et al. 2003) to conclude that outside a window of respectively 0.5% and 4% of the values of the strong and electromagnetic forces, the stellar production of carbon or oxygen will be reduced by a factor 30 to 1000.

In order to compute the resonance energy of the ⁸Be and ¹²C a microscopic cluster model was developed (Korennov and Descouvemont 2004). The Hamiltonian of the system is then of the form $H = \sum_{i}^{A} T(\mathbf{r}_{i} + \sum_{j<i}^{A} V(\mathbf{r}_{ij}))$, where A is the nucleon number, T the kinetic energy and V the NN interaction potential. In order to implement the variation of the strength of the nuclear interaction with respect to the electromagnetic interaction, it was taken as

$$V(\mathbf{r}_{ij}) = V_C(\mathbf{r}_{ij}) + (1 + \delta_{NN})V_N(\mathbf{r}_{ij}),$$

where $\delta_{NN}$ is a dimensionless parameter that describes the change of the nuclear interaction, $V_N$ being described in Thomson et al. (1977). When $A > 4$ no exact solution can be found and approximate solutions in which the wave function of the
$^8$Be and $^{12}$C are described by clusters of respectively 2 and 3 $\alpha$-particle is well adapted.

First, $\delta_{NN}$ can be related to the deuterium binding energy as

$$ \Delta B_D/B_D = 5.7701 \times \delta_{NN}, \qquad (259) $$

which, given the discussion in Sect. 5.14.4, allows one to relate $\delta_{NN}$ to fundamental constants, as, e.g., in Coc et al. (2007). Then, the resonance energy of the $^8$Be and $^{12}$C scale as

$$ \begin{aligned} E_R(^8\text{Be}) &= (0.09208 - 12.208 \times \delta_{NN}) \text{ MeV}, \\ E_R(^{12}\text{C}) &= (0.2877 - 20.412 \times \delta_{NN}) \text{ MeV}, \end{aligned} \qquad (260) $$

so that the energy of the Hoyle level relative to the triple-$\alpha$ threshold is $Q_{\alpha\alpha\alpha} = E_R(^8\text{Be}) + E_R(^{12}\text{C})$.

This was implemented in Coc et al. (2009a), Ekström et al. (2010) to population III stars with typical masses, 15 and 60 $M_{\odot}$ with zero metallicity, in order to compute the central abundances at the end of the core He burning. From Fig. 19, one can distinguish 4 regimes (I) the star ends the CHe burning phase with a core composed of a mixture of $^{12}$C and $^{16}$O, as in the standard case; (II) if the $3\alpha$ rate is weaker, $^{12}$C is produced slower, the reaction $^{12}\text{C}(\alpha, \gamma)^{16}\text{O}$ becomes efficient earlier so that the star ends the CHe burning phase with a core composed mostly of $^{16}\text{O}$; (III) for weaker rates, the $^{16}\text{O}$ is further processed to $^{20}\text{Ne}$ and then $^{24}\text{Mg}$ so that the star ends the CHe burning phase with a core composed of $^{24}\text{Mg}$ and (IV) if the $3\alpha$ rate is stronger, the $^{12}\text{C}$ is produced more rapidly and the star ends the CHe burning phase with a core composed mostly of $^{12}\text{C}$. Typically this imposes that

$$ -5 \times 10^{-4} < \delta_{NN} < 1.5 \times 10^{-3}, \quad -3 \times 10^{-4} < \Delta B_D / B_D < 9 \times 10^{-3}, \qquad (261) $$

at a redshift of order $z \sim 15$, to ensure the ratio C/O to be of order unity.

To finish, a recent study (Adams 2008) focuses on the existence of stars themselves, by revisiting the stellar equilibrium when the values of some constants are modified. In some sense, it can be seen as a generalisation of the work by Gamow (1967) to constrain the Dirac model of a varying gravitational constant by estimating its effect on the lifetime of the Sun. In this semi-analytical stellar structure model, the effect of the fundamental constants was reduced phenomenologically to 3 parameters, $G$, which enters mainly on the hydrostatic equilibrium, $\alpha_{EM}$, which enters in the Coulomb barrier penetration through the Gamow energy, and a composite parameter $C$, which describes globally the modification of the nuclear reaction rates. The underlying idea is to assume that the power generated per unit volume, $\varepsilon(r)$, and which determines the luminosity of the star, is proportional to the fudge factor $C$, which would arise from a modification of the nuclear fusion factor, or equivalently of the cross-section. Thus, it assumes that all cross-sections are affected in a similar way. The parameter space for which stars can form and for which stable nuclear configurations exist was determined, showing that no fine-tuning seems to be required.
This system is very promising and will provide new information on the fundamental constants at redshifts smaller than $z \sim 15$ where no constraints exist at the moment, even though drawing a robust constraint seems to be difficult at the moment. In particular, an underlying limitation arises from the fact that the composition of the interstellar media is a mixture of ejecta from stars with different masses and it is not clear which type of stars contribute the most the carbon and oxygen production. Besides, one would need to include rotation and mass loss (Ekström et al. 2008). As for the Oklo phenomenon, another limitation arises from the complexity of nuclear physics.

## 5.10 Cosmic microwave background

The CMB radiation is composed of photons emitted at the time of the recombination of hydrogen and helium when the universe was about 300,000 years old [see, e.g., Peter and Uzan (2009) for details on the physics of the CMB]. This radiation is observed to be a black-body with a temperature $T_0 = 2.725$ K with small anisotropies of order of the $\mu$K. The temperature fluctuation in a direction $(\vartheta, \varphi)$ is usually decomposed on a basis of spherical harmonics as

$$ \frac{\delta T}{T}(\vartheta, \varphi) = \sum_{\ell} \sum_{m=-\ell}^{m=+\ell} a_{\ell m} Y_{\ell m}(\vartheta, \varphi). \qquad (262) $$

The angular power spectrum multipole $C_\ell = \langle |a_{lm}|^2 \rangle$ is the coefficient of the decomposition of the angular correlation function on Legendre polynomials. Given a model of structure formation and a set of cosmological parameters, this angular power spectrum can be computed and compared to observational data in order to constrain this set of parameters.

### Influence of the fundamental constants

The CMB temperature anisotropies mainly depend on $G$, $\alpha_{EM}$ and $m_e$.

The gravitational constant enters in the Friedmann equation and in the evolution of the cosmological perturbations. It has mainly three effects (Riazuelo and Uzan 2002) that are detailed in Sect. 6.4.1. $\alpha_{EM}$, $m_e$ affect the dynamics of the recombination. Their influence is complex and must be computed numerically. However, we can trace their main effects since they mainly modify the CMB spectrum through the change in the differential optical depth of photons due to the Thomson scattering

$$ \dot{t} = x_e n_e c \sigma_T, \qquad (263) $$

which enters in the collision term of the Boltzmann equation describing the evolution of the photon distribution function and where $x_e$ is the ionization fraction (i.e., the number density of free electrons with respect to their total number density $n_e$).

The first dependence arises from the Thomson scattering cross-section given by
$$ \sigma_T = \frac{8\pi}{3} \frac{\hbar^2}{m_e^2 c^2} \alpha_{\text{EM}}^2 \qquad (264) $$

and the scattering by free protons can be neglected since $m_e/m_p \sim 5 \times 10^{-4}$.

The second, and more subtle dependence, comes from the ionization fraction. Recombination proceeds via 2-photon emission from the 2s level or via the Ly-α photons, which are redshifted out of the resonance line (Peebles 1968) because recombination to the ground state can be neglected since it leads to immediate re-ionization of another hydrogen atom by the emission of a Ly-α photons. Following Peebles (1968) and Ma and Bertschinger (1995) and taking into account, for the sake of simplicity, only the recombination of hydrogen, the equation of evolution of the ionization fraction takes the form

$$ \frac{dx_e}{dt} = C \left[ \beta(1 - x_e) \exp\left(-\frac{B_1 - B_2}{k_B T_M}\right) - \mathcal{R} n_p x_e^2 \right], \qquad (265) $$

where $T_M$ is the temperature. At high redshift, $T_M$ is identical to the one of the photons $T_\gamma = T_0(1+z)$ but evolves according to

$$ \frac{dT_M}{dt} = -\frac{8\sigma_T a_R}{3m_e} T_R^4 \frac{x_e}{1+x_e} (T_M - T_\gamma) - 2HT_M \qquad (266) $$

where the radiation constant $a_R = 4\sigma_{SB}/c$ with $\sigma_{SB} = k_B^4 \pi^2/(60\pi c^2 \hbar^3)$ the Stefan-Boltzmann constant. In Eq. (265), $B_n = -E_l/n^2$ is the energy of the n$^{th}$ hydrogen atomic level, $\beta$ is the ionization coefficient, $\mathcal{R}$ the recombination coefficient, $C$ the correction constant due to the redshift of Ly-α photons and to 2-photon decay and $n_p = n_e$ is the number density of protons. $\beta$ is related to $\mathcal{R}$ by the principle of detailed balance so that

$$ \beta = \mathcal{R} \left( \frac{2\pi m_e k_B T_M}{\hbar^2} \right)^{3/2} \exp \left( -\frac{B_2}{k_B T_M} \right). \qquad (267) $$

The recombination rate to all other excited levels is

$$ \mathcal{R} = \frac{8\pi}{c^2} \left( \frac{k_B T}{2\pi m_e} \right)^{3/2} \sum_{n,l}^{*} (2l+1) e^{B_n/k_B T} \int_{B_n/k_B T}^{\infty} \sigma_{nl} \frac{y^2 dy}{e^y - 1} $$

where $\sigma_{nl}$ is the ionization cross-section for the $(n, l)$ excited level of hydrogen. The star indicates that the sum needs to be regularized and the $\alpha_{EM}^-$, $m_e$-dependence of the ionization cross-section is complicated to extract. However, it can be shown to behave as $\sigma_{nl} \propto \alpha_{EM}^{-1} m_e^{-2} f(hv/B_1)$. Finally, the factor $C$ is given by

$$ C = \frac{1 + K \Lambda_{2s} (1 - x_e)}{1 + K (\beta + \Lambda_{2s}) (1 - x_e)} \qquad (268) $$

where $\Lambda_{2s}$ is the rate of decay of the 2s excited level to the ground state via 2 photons; it scales as $m_e \alpha_{EM}^8$. The constant $K$ is given in terms of the Ly-α photon $\lambda_\alpha = 16\pi\hbar/(3m_e\alpha_{EM}^2c)$ by $K = n_p\lambda_\alpha^3/(8\pi H)$ and scales as $m_e^{-3}\alpha_{EM}^{-6}$.
<figure><img src="image_27.png" /><figcaption>**Fig. 20** *Left:* relative difference between a CMB TT power spectrum calculated using (i) a value of $\alpha/\alpha_0$ different from 1 in one, few or all the terms where it appears, and a power spectrum calculated using (ii) a standard value of $\alpha/\alpha_0 = 1$. We thus plot $\Delta C_\ell / C_\ell = (C_\ell - C_{\ell,\text{st}})/C_{\ell,\text{st}}$[%]. The cases considered are: $\alpha$ varying only in the hydrogen binding energy (solid light blue), only in the Ly$\alpha$ energy (solid yellow), in both the previous two terms (solid purple), in both the previous terms and in the Thomson scattering cross-section (dashed dark blue), in the previous three terms and in the 2-photon decay rate (dashed red), in all terms (solid green). In each case, we assume that $\alpha$ varies of +5% ($\alpha/\alpha_0 = 1.05$) only in the terms considered, while it is $\alpha/\alpha_0 = 1$ in all the others. *Right:* same cases as left, but for a variation of $m_e$ of 10.025% ($m_e/m_{e0} = 1.1025$). Image reproduced with permission from Ade et al. (2015), copyright by ESO</figcaption></figure>

In summary, both the temperature of the decoupling and the residual ionization after recombination are modified by a variation of $\alpha_{EM}$ or $m_e$. This was first discussed in Battye et al. (2001) and Kaplinghat et al. (1999). The last scattering surface can roughly be determined by the maximum of the visibility function $g = \dot{\tau} \exp(-\tau)$, which measures the differential probability for a photon to be scattered at a given redshift. Increasing $\alpha_{EM}$ shifts $g$ to a higher redshift at which the expansion rate is faster so that the temperature and $x_e$ decrease more rapidly, resulting in a narrower $g$. This induces a shift of the $C_\ell$ spectrum to higher multipoles and an increase of the values of the $C_\ell$. The first effect can be understood by the fact that pushing the last scattering surface to a higher redshift leads to a smaller sound horizon at decoupling. The second effect results from a smaller Silk damping.

## Degeneracy

Since the atomic energy levels scale as $B \propto \alpha_{EM}^2 m_e$, this ensures that the effect of a 5% change in $\alpha_{EM}$ or 10.025% in $m_e$ in the hydrogen binding energy and in the Ly$\alpha$ energy level produces identical effects on the angular power spectra. This degeneracy is in Fig. 20: the blue line (relative to the change of hydrogen binding energy only), the yellow line (relative to a change in the Ly$\alpha$ energy level only) and the purple line (sum of the previous two effects are identical for $\alpha_{EM}$ and for $m_e$.

It is evident from the figures that the major contribution to the change in the angular power spectrum induced by a variation of $\alpha_{EM}$ or $m_e$ comes from the change in the hydrogen binding energy and Ly$\alpha$ energy which leads to a degeneracy at small angular scales. A difference in the effects of $\alpha_{EM}$ or $m_e$ is however introduced
<figure><img src="image_28.png" /><figcaption>**Fig. 21** Evolution of the free electron fraction with redshift for various values of $\alpha_{EM}$ (left) or $m_e$ (right) of -5% (blue), 0% (black) and +5% (red). The decrease in the plateaus at $z \sim 6\,000$ and $2\,000$ correspond to the first and second recombination of helium while the larger one at $z \sim 1300$ is due to the recombination of hydrogen. Image reproduced with permission from Ade et al. (2015), copyright by ESO</figcaption></figure>

when one considers the impact on the Thomson scattering cross-section $\sigma_T$. Adding the effect of the constants on the Thomson cross-section, shown in the dark-blue dashed lines in Fig. 20, increases the amplitude of the peaks for a larger value of $\alpha$, while it decreases it for a larger value of $m_e$. Alternatively, this is the reason why $\alpha_{EM}$ and $m_e$ have different effects on the amplitude of the peaks.

<figure><img src="image_29.png" /><figcaption>**Fig. 22** CMB TT, TE and EE angular power spectrum from left to right for various values of $\alpha_{EM}$ (left) or $m_e$ (right) of -5% (blue), 0% (black) and +5% (red). Image reproduced with permission from Ade et al. (2015), copyright by ESO</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>**Fig. 23** Summary of the main results on $m_e$ and their consequences for the Hubble constant; see Table 17. The vertical gray zone indicates the Cepheid-calibrated supernovae determination of the Hubble constant, $H_0 = (73.04 \pm 1.04)$ km/s/Mpc by the SH0ES experiment (Riess et al. 2022)</figcaption></figure>

### CMB code modifications

Most studies have introduced those modifications in the RECFAST code (Seager et al. 1999) including similar equations for the recombination of helium. Our previous analysis shows that the dependencies in the fundamental constants have various origins, since the binding energies $B_i$ scale has $m_e\alpha_{EM}^2$, $\sigma_T$ as $\alpha_{EM}^2 m_e^{-2}$, $K$ as $m_e^{-3}\alpha_{EM}^{-6}$, the ionisation coefficients $\beta$ as $\alpha_{EM}^3$, the transition frequencies as $m_e\alpha_{EM}^2$, the Einstein's coefficients as $m_e\alpha_{EM}^5$, the decay rates $\Lambda$ as $m_e\alpha_{EM}^8$ and $\mathcal{R}$ has complicated dependence, which roughly reduces to $\alpha_{EM}^{-1}m_e^{-2}$. In earlier works (Hannestad 1999; Kaplinghat et al. 1999) it was approximated by the scaling $\mathcal{R} \propto \alpha_{EM}^{2(1+\xi)}$ with $\xi \sim 0.7$. Note that a change in the fine-structure constant and in the mass of the electron are degenerate according to $\Delta\alpha_{EM} \approx 0.39\Delta m_e$ but this degeneracy is broken for multipoles higher than 1500 (Battye et al. 2001). This is illustrated in the analysis using the first Planck data release (Ade et al. 2015) that implemented a variation of ($\alpha_{EM}$, $m_e$) in RECFAST. Fig. 21 illustrates the evolution of the free electron fraction $x_e(z)$ under a relative variation of either $\alpha_{EM}$ or $m_e$. One can clearly witness the shift of the recombination epoch to earlier time/higher redshift when the constants increase, and the effect on the TT, TE and EE angular power spectra as depicted in Fig. 22. An earlier recombination has 3 main effects: (1) a smaller sound horizon et recombination and hence a larger diameter distance resulting in a shift of the Doppler peaks to higher multipoles; (2) a decrease of the Silk damping since at lowest order the Thomson cross-section scales as $\sigma_T \propto \alpha_{EM}^2/m_e^2$; (3) an increase of
the spectra at large scales due to the shorter time interval between matter-radiation equality and recombination (see Ade et al. 2015 for a detailed discussion).

### Observational constraints

The first studies (Hannestad 1999; Kaplinghat et al. 1999) focused on the sensitivity that can be reached by WMAP<fnref n="12" /> and Planck.<fnref n="13" /> They concluded that they should provide a constraint on $\alpha_{EM}$ at recombination, i.e., at a redshift of about $z \sim 1$, with a typical precision $|\Delta\alpha_{EM}/\alpha_{EM}| \sim 10^{-2} - 10^{-3}$.

The first attempt (Avelino et al. 2000) to actually set a constraint was performed on the first release of the data by BOOMERanG and MAXIMA. It concluded that a value of $\alpha_{EM}$ smaller by a few percents in the past was favored but no definite bound was obtained, mainly due to the degeneracies with other cosmological parameters. It was later improved (Avelino et al. 2001) by a joint analysis of BBN and CMB data that assumes that only $\alpha_{EM}$ varies and that included 4 cosmological parameters ($\Omega_{\text{mat}}$, $\Omega_b$, $h$, $n_s$) assuming a universe with Euclidean spatial section, leading to $-0.09 < \Delta\alpha_{EM} < 0.02$ at 68% confidence level. A similar analysis (Landau et al. 2001), describing the dependence of a variation of the fine-structure constant as an effect on recombination the redshift of which was modeled to scale as $z_* = 1080[1 + 2\Delta\alpha_{EM}/\alpha_{EM}]$, set the constraint $-0.14 < \Delta\alpha_{EM} < 0.02$, at a $2\sigma$ level, assuming a spatially flat cosmological models with adiabatic primordial fluctuations that. The effect of re-ionisation was discussed in Martins et al. (2004). These works assume that only $\alpha_{EM}$ is varying but, as can been seen from Eqs. (262–268), assuming the electron mass constant.

With the WMAP first year data, the bound on the variation of $\alpha_{EM}$ was sharpened (Rocha et al. 2004) to $-0.05 < \Delta\alpha_{EM}/\alpha_{EM} < 0.02$, after marginalizing over the remaining cosmological parameters ($\Omega_{\text{mat}}h^2$, $\Omega_bh^2$, $\Omega h^2$, $n_s$, $\alpha_s$, $\tau$) assuming a universe with Euclidean spatial sections. Restricting to a model with a vanishing running of the spectral index ($\alpha_s \equiv dn_s/d\ln k = 0$), it gives $-0.06 < \Delta\alpha_{EM}/\alpha_{EM} < 0.01$, at a 95% confidence level. In particular it shows that a lower value of $\alpha_{EM}$ makes $\alpha_s = 0$ more compatible with the data. These bounds were obtained without using other cosmological data sets. This constraint was confirmed by the analysis of Ichikawa et al. (2006), which got $-0.097 < \Delta\alpha_{EM}/\alpha_{EM} < 0.034$, with the WMAP-1yr data alone and $-0.042 < \Delta\alpha_{EM}/\alpha_{EM} < 0.026$, at a 95% confidence level, when combined with constraints on the Hubble parameter from the HST Hubble Key project.

The analysis of the WMAP-3yr data allows Stefanescu (2007) to improve this bound to $-0.039 < \Delta\alpha_{EM}/\alpha_{EM} < 0.010$, at a 95% confidence level, assuming ($\Omega_{\text{mat}}$, $\Omega_b$, $h$, $n_s$, $z_{\text{re}}$, $A_s$) for the cosmological parameters ($\Omega_{\Lambda}$ being derived from the assumption $\Omega_K = 0$, as well as $\tau$ from the re-ionisation redshift, $z_{\text{re}}$) and using both temperature and polarization data (TT, TE, EE).

The WMAP 5-year data were analyzed, in combination with the 2dF galaxy redshift survey, assuming that both $\alpha_{EM}$ and $m_e$ can vary and that the universe was spatially Euclidean. Letting 6 cosmological parameters [($\Omega_{\text{mat}}h^2$, $\Omega_bh^2$, $\Theta$, $\tau$, $n_s$, $A_s$),

<footnote n="12">¹² http://map.gsfc.nasa.gov/</footnote>

<footnote n="13">¹³ http://astro.estec.esa.nl/SA-general/Projects/Planck/</footnote>
$\Theta$ being the ratio between the sound horizon and the angular distance at decoupling] and 2 constants vary, Scoccola et al. (2008, 2009) concluded $-0.012 < \Delta\alpha_{EM}/\alpha_{EM} < 0.018$ and $-0.068 < \Delta m_e/m_e < 0.044$, the bounds fluctuating slightly with the choice of the recombination scenario. A similar analysis (Nakashima et al. 2008) not including $m_e$ gave $-0.050 < \Delta\alpha_{EM}/\alpha_{EM} < 0.042$, which can be reduced by taking into account some further prior from the HST data. Including polarisation data from ACBAR, QUAD and BICEP, it was also obtained (Menegoni et al. 2009) $-0.043 < \Delta\alpha_{EM}/\alpha_{EM} < 0.038$ at 95% C.L. and $-0.013 < \Delta\alpha_{EM}/\alpha_{EM} < 0.015$ including HST data, also at 95% C.L. Let us also mention the work by Martins et al. (2010) trying to include the variation of the Newton constant by assuming that $\Delta\alpha_{EM}/\alpha_{EM} = Q\Delta G/G$, Q being a constant and the investigation of Nakashima et al. (2010) taking into account $\alpha_{EM}$, $m_e$ and $\mu$, G being kept fixed. Considering $(\Omega_{mat}, \Omega_b, h, n_s, \tau)$ for the cosmological parameters they concluded from WMAP-5 data (TT, TE, EE) that $-8.28 \times 10^{-3} < \Delta\alpha_{EM}/\alpha_{EM} < 1.81 \times 10^{-3}$ and $-0.52 < \Delta\mu/\mu < 0.17$

The analysis of Scoccola et al. (2008, 2009) was updated by Landau and Scoccola (2010) to the WMAP-7yr data (Komatsu et al. 2011), including polarisation and SDSS data. It leads to $-0.025 < \Delta\alpha_{EM}/\alpha_{EM} < -0.003$ and $0.009 < \Delta m_e/m_e < 0.079$ at a $1\sigma$ level. Menegoni et al. (2012) focused on the small angular scale up to $\ell_{max} = 3000$ by adding data from ACT (Dunkley et al. 2011), ACBAR (Reichardt et al. 2009) and SPT (Keisler et al. 2011) in order to study the relation between $\alpha_{EM}$ and the number of relativistic degrees of freedom $N_{eff}$ that is reported to be higher than its nominal value $N_{eff} = 3.046$—the deviation from 3 is related to the non-spontaneous neutrino decoupling, see e.g., Pitrou et al. (2018). Assuming a 6 parameters cosmological model $(\Omega_{mat}h^2, \Omega_b h^2, H_0, \tau, n_s, A_s)$ they consider 3 models with (1) $\alpha_{EM}$ free ($N_{eff}, Y_p$ fixed to (3.046,0.24), (2) $(\alpha_{EM}, N_{eff})$ free and $Y_p$ fixed and (3) $(\alpha_{EM}, N_{eff}, Y_p)$ free to conclude $\alpha_{EM}/\alpha_{EM}(0) = 0.984 \pm 0.005$ (case 1), $\alpha_{EM}/\alpha_{EM}(0) = 0.99 \pm 0.006$ (case 2), $\alpha_{EM}/\alpha_{EM}(0) = 0.987 \pm 0.014$ (case 3). The freedom of a free $\alpha_{EM}$ increases the error bar on $N_{eff}$ by about 30%. (Ade et al. 2015) analyzed the first Planck data release combined with several other data as described in Ade et al. (2014) and fit a model with 6 cosmological parameters $(\Omega_{mat}h^2, \Omega_b h^2, H_0, \tau, n_s, A_s)$. Assuming $m_e$ constant, they get for the Planck data + WMAP polarization data + high $\ell$ data

$$ \Delta\alpha_{EM}/\alpha_{EM} = (3.6 \pm 3.7) \times 10^{-3}. \qquad (269) $$

From CMB alone, Planck data the constraint on the variation of $\alpha_{EM}$ by a factor of about 5 compared to WMAP-9 and also better that WMAP combined with small scale experiment (Sievers et al. 2013), mainly because Planck data is able to break the strong $\alpha_{EM} - H_0$ degeneracy from the observation of the damping tail. The study of the correlation shows that the constraints on the cosmological parameters change very little, exceptions being $n_s$ and $H_0$ and the degeneracy with $Y_p$ is confirmed. Finally, they also conclude that $\alpha_{EM}$ is weakly degenerate with foreground, beam and calibration parameters. Assuming $\alpha_{EM}$ constant, Ade et al. (2014) notes that Planck cannot break the strong degeneracy between $H_0$ and $m_e$ since the latter does not affect much the damping tail. But adding external data such as BAO decreases the uncertainty by a factor of order 5 to get
$$ \Delta m_e/m_e = (4 \pm 11) \times 10^{-3}. \qquad (270) $$

When considering the simultaneous variation of $\alpha_{EM}$ and $m_e$ it is shown that thanks to small scale data, the degeneracy is lifted and that the constraints are not substantially changed.

Hart and Chluba (2018) revisited this result with the recombination codes CosmoRec (Chluba and Thomas 2011) and Recfast++. From Planck-2015 data combined with BAO data, they concluded that $\Delta\alpha_{EM}/\alpha_{EM} = (0.7 \pm 2.5) \times 10^{-3}$ and $\Delta m_e/m_e = (3.9 \pm 7.4) \times 10^{-3}$ for independent variations, while for a joint variation they get

$$ \Delta\alpha_{EM}/\alpha_{EM} = (1.1 \pm 2.6) \times 10^{-3} \qquad \Delta m_e/m_e = (5.6 \pm 8) \times 10^{-3} \qquad (271) $$

including BAO data. Hart and Chluba (2020) included Planck-2018 polarisation to get, with either $m_e$ or $\alpha_{EM}$ fixed,

$$ \Delta\alpha_{EM}/\alpha_{EM} = (0.05 \pm 0.24) \times 10^{-2} \qquad \Delta m_e/m_e = (-11.2 \pm 5.9) \times 10^{-2} \qquad (272) $$

from CMB alone and

$$ \Delta\alpha_{EM}/\alpha_{EM} = (0.19 \pm 0.22) \times 10^{-2} \qquad \Delta m_e/m_e = (0.78 \pm 0.67) \times 10^{-2} \qquad (273) $$

once BAO are included, while the joint analysis on $(\alpha_{EM}, m_e)$ gives

$$ \Delta\alpha_{EM}/\alpha_{EM} = (0.10 \pm 0.24) \times 10^{-2} \qquad \Delta m_e/m_e = (0.54 \pm 0.80) \times 10^{-2}. \qquad (274) $$

while adding SNIa data pushes the electron mass to $\Delta\alpha_{EM}/\alpha_{EM} = (1.9 \pm 0.55) \times 10^{-2}$. These two works also more model-dependent constraints that assume a redshift dependence of the form $(1+z)^p$. To go beyond the hypothesis of a time-independent changes of $(\alpha_{EM}, m_e)$, Hart and Chluba (2022) proposed a model-independent principal component analysis using an eigenmode decomposition of the varying constant during recombination. From Planck-2018 data, they show that for each constant, three independent modes can be constrained at present. The analysis was extended by Hart and Chluba (2023) to include early dark energy. This results highlight the connection between $m_e$ and $H_0$ and the dependence of the constraints on the time-variation around the decoupling as well as the importance to extend them up to reionisation. An attempt to deal with the whole ionization history (i.e., recombination + reionization) was addressed in Wang and Chen (2022) by adding SFR density from UV and IR measurements to the Planck-2018+BAO+SNIa data. They concluded that at 68% C.L. $(\Delta\alpha_{EM}/\alpha_{EM})_{CMB} = (1.49 \pm 2) \times 10^{-3}$ and $(\Delta\alpha_{EM}/\alpha_{EM})_{Rei} = -1.46_{-0.27}^{+0.31} \times 10^{-1}$. These constraints suffer a 4.64$\sigma$ discrepancy so that the work mostly illustrate the importance not to restrict to CMB.

## Links with the Hubble tension

The possibility to solve or alleviate the Hubble tension thanks to a variation of the constant has first been stressed by Hart and Chluba (2020) while it was noticed by Ade et al. (2014) that “for WMAP there is a strong degeneracy between $H_0$ and
<table><caption>**Table 17** Summary of the main results on a constant shift on $m_e$ and their consequences for the Hubble constant</caption><thead><tr><th>Δm<sub>e</sub>/m<sub>e</sub> (×10<sup>2</sup>)</th><th>H<sub>0</sub> [km/s/Mpc]</th><th>Hypothesis</th><th>CMB</th><th>BAO</th><th>SNIa</th><th>Ref</th></tr></thead><tbody><tr><td>1.1<sup>+7,7</sup><sub>-5,7</sub></td><td>72 ± 10</td><td></td><td>W9</td><td></td><td></td><td>Ade et al. (2014)</td></tr><tr><td>4 ± 9.1</td><td>73 ± 10</td><td>α<sub>EM</sub></td><td>W9</td><td></td><td></td><td>Ade et al. (2014)</td></tr><tr><td>2.7 ± 1.2</td><td>73.5 ± 2.4</td><td></td><td>P13</td><td>SDSS+BOSS</td><td>HST</td><td>Ade et al. (2014)</td></tr><tr><td>0.39 ± 0.74</td><td>68.1 ± 1.3</td><td></td><td>P15</td><td>SDSS</td><td></td><td>Hart and Chluba (2018)</td></tr><tr><td>0.56 ± 0.80</td><td>68.1 ± 1.3</td><td>α<sub>EM</sub></td><td>P15</td><td>SDSS</td><td></td><td>Hart and Chluba (2018)</td></tr><tr><td>0.47 ± 0.66</td><td>68.46 ± 1.26</td><td></td><td>P18</td><td>SDSS</td><td></td><td>Hart and Chluba (2020)</td></tr><tr><td>1.5 ± 1.8</td><td>69.29 ± 2.11</td><td>Ω<sub>K</sub></td><td>P18</td><td>SDSS</td><td></td><td>Hart and Chluba (2020)</td></tr><tr><td>0.3 ± 0.6</td><td>68.0 ± 1.1</td><td></td><td>P18+SPT</td><td>SDSS</td><td>Pan</td><td>Khalife et al. (2024)</td></tr><tr><td>0.35 ± 1.64</td><td>68.2 ± 1.6</td><td>Ω<sub>K</sub></td><td>P18+SPT</td><td>SDSS</td><td>Pan</td><td>Khalife et al. (2024)</td></tr><tr><td>3 ± 3</td><td>69.8<sup>+1,8</sup><sub>-2,9</sub></td><td>Ω<sub>K</sub>, m<sub>v</sub></td><td>P18+SPT</td><td>SDSS</td><td>Pan</td><td>Khalife et al. (2024)</td></tr><tr><td>0.92 ± 0.55</td><td>69.44 ± 0.84</td><td></td><td>P18</td><td>SDSS+DESI</td><td>Pan</td><td>Seto and Toda (2024)</td></tr><tr><td>1.3 ± 1.4</td><td>69.7 ± 1.4</td><td>Ω<sub>K</sub></td><td>P18</td><td>SDSS+DESI</td><td>Pan</td><td>Seto and Toda (2024)</td></tr><tr><td>-1.00<sup>+1.09</sup><sub>-1.04</sub></td><td>65.1<sup>+2,2</sup><sub>-2,0</sub></td><td></td><td>P18</td><td></td><td>Pan+</td><td>Baryakhtar et al. (2024)</td></tr><tr><td>1.21 ± 0.63</td><td>70.03 ± 1.06</td><td></td><td>P18</td><td>DESI</td><td></td><td>Schöneberg and Vacher (2025)</td></tr><tr><td>1.88 ± 0.52</td><td>71.61 ± 1.00</td><td>Ω<sub>K</sub></td><td>P18</td><td>DESI</td><td>Pan+</td><td>Schöneberg and Vacher (2025)</td></tr><tr><td>0.72 ± 0.84</td><td>69.38 ± 2.17</td><td>ω</td><td>P18</td><td>DESI</td><td>Pan+</td><td>Schöneberg and Vacher (2025)</td></tr><tr><td>0.61 ± 0.65</td><td>68.8 ± 1.1</td><td></td><td>P18</td><td>SDSS</td><td>Pan</td><td>Toda et al. (2024)</td></tr><tr><td>1.9 ± 1.8</td><td>69.7 ± 1.7</td><td>Ω<sub>K</sub></td><td>P18</td><td>SDSS</td><td>Pan</td><td>Toda et al. (2024)</td></tr></tbody></table>

The data use and the extra parameters than have been varied are also indicated; see text for definitions

$m_e$, which is why the uncertainty on $m_e/m_{e0}$ is much larger than for Planck". This hints that a combined modification of recombination and reionisation physics could be at work. This has motivated a renewed attention to constrain a *constant shift* of the electron mass between CMB and today with many studies (Hart and Chluba 2018, 2020; Schöneberg et al. 2022; Khalife et al. 2024; Baryakhtar et al. 2024;
Seto and Toda 2024; Toda et al. 2024; Schöneberg and Vacher 2025; Toda et al. 2024) considering different data sets. Most of them consider the Planck-2015 (Ade et al. 2014) or Planck-2018 (Aghanim et al. 2020) for the CMB, BAO data mostly from the SDSS-DR12 LRG (Alam et al. 2017), DR14-QSO/Ly$\alpha$ (Blomqvist et al. 2019) and DR7 MGS (Ross et al. 2015) or DESI (Adame et al. 2025) SNIa data from either Pantheon (Scolnic et al. 2018) or PantheonPlus (Scolnic et al. 2022). Then, from a theoretical point of view they may or may not consider a non-vanishing curvature, $\Omega_K$, massive neutrinos, $m_\nu$, a varying $\alpha_{EM}$, an equation of state for the dark energy component or a sign shift of the cosmological constant in Toda et al. (2024). The results of their analysis are summarized in Table 17. We refer to Chluba and Hart (2024) and Schöneberg and Vacher (2025) for summaries of the current status of this question as well as to Sect. 2.3.6.

## Summary

The main limitation of these analyses lies in the fact that the CMB angular power spectrum depends on the evolution of both the background spacetime and the cosmological perturbations. It follows that it depends on the whole set of cosmological parameters as well as on initial conditions, that is on the shape of the initial power spectrum. Hence the results will always be conditional to the model of structure formation. The constraints on $\alpha_{EM}$ or $m_e$ can then be seen mostly as constraints on a delayed recombination. A strong constraint on the variation of $\alpha_{EM}$ can be obtained from the CMB only if the cosmological parameters are independently known. Rocha et al. (2004) forecasts that CMB alone can determine $\alpha_{EM}$ to a maximum accuracy of 0.1% and Martinelli et al. (2012) showed that including lensing data from Euclid experiment will also to reach an accuracy of $8 \times 10^{-4}$. These tests push our understanding of the recombination and turn to be important in view of the Hubble tension.

## 5.11 21 cm

After recombination, the CMB photons are redshifted and their temperature drops as $(1+z)$. However, the baryons are prevented from cooling adiabatically since the residual amount of free electrons, that can couple the gas to the radiation through Compton scattering, is too small. It follows that the matter decouples thermally from the radiation at a redshift of order $z \sim 200$.

The intergalactic hydrogen atoms after recombination are in their ground state, which hyperfine-structure splits into a singlet and a triple states ($1s_{1/2}$ with $F=0$ and $F=1$ respectively, see Sect. III.B.1 of FCV [Uzan 2003b]). Khatri and Wandelt (2007) proposed that the observation of the 21-cm emission can provide a test on the fundamental constants. We refer to Furlanetto et al. (2006) for a detailed review on 21 cm.

The fraction of atoms in the excited (triplet) state versus the ground (singlet) state is conventionally related by the spin temperature $T_s$ defined by the relation
<table><caption>**Table 18** Summary of the constraints on the variation of $\alpha_{EM}$ obtained from the analysis of CMB data</caption><thead><tr><th>Constraint (Δα<sub>EM</sub>/α<sub>EM</sub> × 10<sup>2</sup>)</th><th>Data</th><th>Comment</th><th>Ref</th></tr></thead><tbody><tr><td>(-3.5 × 5.5)</td><td>pW, BBN</td><td>α<sub>EM</sub> only</td><td>Avelino et al. (2001)</td></tr><tr><td>(-6 ± 8)</td><td>pW</td><td>α<sub>EM</sub> only</td><td>Landau et al. (2001)</td></tr><tr><td>(-1.5 ± 3.5)</td><td>W1</td><td>include α<sub>s</sub></td><td>Rocha et al. (2004)</td></tr><tr><td>(-2.5 ± 3.56)</td><td>W1</td><td>α<sub>s</sub> = 0</td><td>Rocha et al. (2004)</td></tr><tr><td>(-3.15 ± 6.55)</td><td>W1</td><td>m<sub>e</sub> free</td><td>Ichikawa et al. (2006)</td></tr><tr><td>(10.9 ± 15.1)</td><td>W1, H<sub>0</sub></td><td>same</td><td>Ichikawa et al. (2006)</td></tr><tr><td>(-1.45 ± 2.45)</td><td>W3 (T+Pol), H<sub>0</sub></td><td></td><td>Stefanescu (2007)</td></tr><tr><td>(0.3 ± 1.5)</td><td>W5, 2df</td><td>m<sub>e</sub> free</td><td>Scoccola et al. (2008)</td></tr><tr><td>(-0.4 ± 4.6)</td><td>W5, H<sub>0</sub></td><td></td><td>Nakashima et al. (2008)</td></tr><tr><td>(-0.25 ± 4.05)</td><td>W5 + Pol</td><td></td><td>Menegoni et al. (2009)</td></tr><tr><td>(0.1 ± 1.4)</td><td>W5 + Pol, H<sub>0</sub></td><td></td><td>Menegoni et al. (2009)</td></tr><tr><td>(-0.324 ± 0.505)</td><td>W5 (T+Pol)</td><td>m<sub>e</sub> free</td><td>Nakashima et al. (2010)</td></tr><tr><td>(-1.4 ± 1.1)</td><td>W7 (T+Pol), SDSS</td><td>m<sub>e</sub> free</td><td>Landau and Scoccola (2010)</td></tr><tr><td>(1.6 ± 0.5)</td><td>W7+Pol</td><td></td><td>Menegoni et al. (2012)</td></tr><tr><td>(-1 ± 0.6)</td><td>W7+Pol</td><td>Y<sub>p</sub> fixed</td><td>Menegoni et al. (2012)</td></tr><tr><td>(1.3 ± 1.4)</td><td>W7+Pol</td><td>(N<sub>eff</sub>, Y<sub>p</sub>) free</td><td>Menegoni et al. (2012)</td></tr><tr><td>(0.36 ± 0.37)</td><td>P15 + W7(Pol)</td><td>m<sub>e</sub> free</td><td>Ade et al. (2015)</td></tr><tr><td>(0.07 ± 0.25)</td><td>P15 + BAO</td><td></td><td>Hart and Chluba (2018)</td></tr><tr><td>(0.11 ± 0.26)</td><td>P15 + BAO</td><td>m<sub>e</sub> free</td><td>Hart and Chluba (2018)</td></tr><tr><td>(0.05 ± 0.24)</td><td>P18 (T+Pol)</td><td></td><td>Hart and Chluba (2020)</td></tr><tr><td>(0.19 ± 0.22)</td><td>P18 (T+Pol), BAO</td><td></td><td>Hart and Chluba (2020)</td></tr><tr><td>(0.10 ± 0.24)</td><td>P18 (T+Pol), BAO</td><td>m<sub>e</sub> free</td><td>Hart and Chluba (2020)</td></tr><tr><td>(0.149 ± 0.2)</td><td>P18, BAO, SNI</td><td></td><td>Wang and Chen (2022)</td></tr></tbody></table>

All, unless specified, assume $\Omega_K = 0$. Concerning the data W-(1,3,5,7) refers to the WMAP 1,...,7 years data, P15/P18 to the Planck 2015 and 2018 releases and pW to pre-WMAP data such as BOOMERanG, MAXIMA or COBE—see text—specifying the use of temperature only (T) or temperature +polarisation (T+Pol). BBN refers to the use of BBN data. The cosmological parameters involved on the statistical analysis are specified in the text

$$ \frac{n_t}{n_s} = 3 \exp \left( - \frac{T_*}{T_s} \right) \qquad (275) $$

where $T_* \equiv hc/(\lambda_{21}k_B) = 68.2$ mK is the temperature corresponding to the 21 cm transition and the factor 3 accounts for the degeneracy of the triplet state (note that this is a very simplified description since the assumption of a unique spin temperature is probably not correct (Furlanetto et al. 2006)). The population of the two states is determined by two processes, the radiative interaction with CMB photons with a wavelength of $\lambda_{21} = 21.1$ cm (i.e., $\nu_{21} = 1420$ MHz) and spin-changing atomic collision. Thus, the evolution of the spin temperature is dictated by (Furlanetto et al. 2006).
<table><caption>**Table 19** Summary of the main dependencies of the 21 cm relevant parameters as listed by Lopez-Honorez et al. (2020) assuming $m_p$ and $g_p$ constant</caption><thead><tr><th>Constraint</th><th></th><th>dependence</th></tr></thead><tbody><tr><td>hyperfine transition</td><td>v<sub>21</sub></td><td>α<sub>EM</sub><sup>4</sup>m<sub>e</sub><sup>2</sup></td></tr><tr><td>Spontaneous emission coefficient of the 21 cm transition</td><td>A<sub>10</sub></td><td>α<sub>EM</sub><sup>13</sup>m<sub>e</sub></td></tr><tr><td>Brightness temperature factor</td><td>τ<sub>v21</sub>T<sub>S</sub></td><td>α<sub>EM</sub><sup>5</sup></td></tr><tr><td>Lyman-α frequency</td><td>v<sub>α</sub></td><td>α<sub>EM</sub><sup>2</sup>m<sub>e</sub><sup>2</sup></td></tr><tr><td>Proper Lyman-α intensity</td><td>J̃<sub>α★</sub></td><td>α<sub>EM</sub><sup>-2</sup>m<sub>e</sub><sup>-1</sup></td></tr><tr><td>Spontaneous emission coefficient of the Lyman-α transition</td><td>A<sub>α</sub></td><td>α<sub>EM</sub><sup>5</sup>m<sub>e</sub></td></tr><tr><td>Gunn-Peterson optical depth</td><td>τ<sub>GP</sub></td><td>α<sub>EM</sub><sup>-1</sup>m<sub>e</sub><sup>-2</sup></td></tr><tr><td>Lyman-α coupling</td><td>x<sub>α</sub></td><td>S<sub>α</sub>α<sub>EM</sub><sup>-10</sup>m<sub>e</sub><sup>-4</sup></td></tr><tr><td>Recombination case B-coefficient</td><td>α<sub>B</sub></td><td>α<sub>EM</sub><sup>3</sup>m<sub>e</sub><sup>-3/2</sup></td></tr><tr><td>Ground state energy of species i</td><td>hν<sub>i</sub></td><td>α<sub>EM</sub><sup>2</sup>m<sub>e</sub></td></tr><tr><td>Ionization cross-section of species i</td><td>σ<sub>i</sub></td><td>G<sub>i</sub>α<sub>EM</sub><sup>-1</sup>m<sub>e</sub><sup>-2</sup></td></tr></tbody></table>

The expression of $S_α$ if given in Eq. (2.17) of Lopez-Honorez et al. (2020), as obtained from Furlanetto and Pritchard (2006) and $G_i$ a function detailed in Madau and Fragos (2017)

$$ \frac{dT_s}{dt} = 4C_{10} \left( \frac{1}{T_s} - \frac{1}{T_g} \right) T_s^2 + (1+z)HA_{10} \left( \frac{1}{T_s} - \frac{1}{T_\gamma} \right) \frac{T_\gamma}{T_*} \quad (276) $$

The first term corresponds to the collision de-excitation rate from triplet to singlet and the coefficient $C_{10}$ is decomposed as

$$ C_{10} = \kappa_{10}^{HH} n_p + \kappa_{10}^{eH} x_e n_p $$

with the respective contribution of H-H and e-H collisions. The second term corresponds to spontaneous transition and $A_{10}$ is the Einstein coefficient. The equation of evolution for the gas temperature $T_g$ is given by Eq. (266) with $T_M = T_g$ (we recall that we have neglected the contribution of helium) and the electronic density satisfies Eq. (265).

It follows (Khatri and Wandelt 2007, 2009) that the change in the brightness temperature of the CMB at the corresponding wavelength scales as $T_b \propto A_{12}/v_{21}^2$, where the Einstein coefficient $A_{12}$ is defined below. Observationally, we can deduce the brightness temperature from the brightness $I_v$, that is the energy received in a given direction per unit area, solid angle and time, defined as the temperature of the black-body radiation with spectrum $I_v$. Thus, $k_B T_b \approx I_v c^2 / 2v^2$. It has a mean value, $\bar{T}_b(z_{obs})$ at various redshifts where $1+z_{obs} = v_{21}^{\text{today}}/v_{obs}$. Besides, as for the CMB, there will also be fluctuation in $T_b$ due to imprints of the cosmological perturbations on $n_p$ and $T_g$. It follows that we also have access to an angular power spectrum $C_l(z_{obs})$ at various redshifts (see Lewis and Challinor 2007 for details on this computation).
Both quantities depend on the value of the fundamental constants. Beside the same dependencies of the CMB that arise from the Thomson scattering cross-section, we have to consider those arising from the collision terms. In natural units, the Einstein coefficient scaling is given by $A_{12} = \frac{2}{3}\pi\alpha_{\text{EM}}v_{21}^3m_e^{-2} \sim 2.869 \times 10^{-15} \text{ s}^{-1}$. It follows that it scales as $A_{10} \propto g_p^3\mu^3\alpha_{\text{EM}}^{13}m_e$. The brightness temperature depends on the fundamental constant as $T_b \propto g_p\mu\alpha_{\text{EM}}^5/m_e$. Note that the signal can also be affected by a time variation of the gravitational constant through the expansion history of the universe. Khatri and Wandelt (2007) (see also Furlanetto et al. 2006 for further discussions), focusing only on $\alpha_{\text{EM}}$, showed that this was the dominant effect on a variation of the fundamental constant (the effect on $C_{10}$ is much complicated to determine but was argued to be much smaller). It was estimated that a single station telescope like LWA<sup>14</sup> or LOFAR<sup>15</sup> can lead to a constraint of the order of $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} \sim 0.85\%$, improving to 0.3% for the full LWA. The fundamental challenge for such a measurement is the subtraction of the foreground.

The 21 cm absorption signal in a available on a band of redshift typically ranging from $z \lesssim 1000$ to $z \sim 20$, which is between the CMB observation and the formation of the first stars, offering a unique window on the “dark age”. Thus, it offers an interesting possibility to trace the constraints on the evolution of the fundamental constants between the CMB epoch and the quasar absorption spectra.

As for CMB, the knowledge of the cosmological parameters is a limitation since a change of 1% in the baryon density or the Hubble parameter implies a 2% (3% respectively) on the mean bolometric temperature. The effects on the angular power spectrum have been estimated but still require an in depth analysis along the lines of, e.g., Lewis and Challinor (2007). It is motivating since $C_\ell(z_{\text{obs}})$ is expected to depend on the correlators of the fundamental constants, e.g., $\langle\alpha_{\text{EM}}(\mathbf{x}, z_{\text{obs}})\alpha_{\text{EM}}(\mathbf{x}', z_{\text{obs}})\rangle$ and thus in principle allows to study their fluctuation, even though it will also depend on the initial condition, e.g., power spectrum, of the cosmological perturbations.

Using the public tool 21vmFASTv2 (Mesinger et al. 2011) and the Rec-fast++ code for reionisation (in the version by Hart and Chluba (2018) implementing varying constant to get the initial conditions on $x_e$ and the temperature of the gas at $z=30$, Lopez-Honorez et al. (2020) discussed the dependence of the 21-cm signal on ($\alpha_{\text{EM}}, m_e$) and their degeneracies with astrophysical parameters (see Table 19) to conclude that their variations can be hope to be constrained at a level of $\mathcal{O}(10^{-3})$ with future SKA data (Weltman et al. 2020).

Thanks to its high resolution in radio spectral lines, SKA1-Low has good prospects to use e.g., lines from HI and the OH radical) to constrain $\alpha_{\text{EM}}$ (Weltman et al. 2020). The covered redshifts for SKA1-Low will be, e.g., $z<13$ for the HI-21 cm absorption and $z<16$ for the ground-state OH-18 cm absorption (Curran et al. 2004). Khatri and Wandelt (2007) proposed another method to measure $\alpha_{\text{EM}}$

<footnote n="14"><sup>14</sup> http://lwa.unm.edu</footnote>

<footnote n="15"><sup>15</sup> http://www.lofar.org</footnote>
through the 21 cm absorption of CMB photons. They estimated that the change of 1% of its value shall modify the mean brightness temperature decrement of the CMB due to 21 cm absorption by 5% over the redshift range $z \in [30, 50]$.

In conclusion, the 21 cm observation will open a observational window on the fundamental at redshifts ranging typically from 30 to 100, but full in-depth analysis is still required (see Flambaum and Porsev 2010; Khatri and Wandelt 2010 for a critical discussion of this probe).

## 5.12 Galaxy clusters

Galli (2013) proposed a new method based on the idea that hot clusters radiate in the X-ray mainly due to bremsstrahlung, while they leave an imprint on the CMB frequency spectrum through the Sunyaev-Zel'dovich effect. The SZ Compton parameter $y$ depends on the Thomson cross-section as

$$y = \sigma_T \int n_e \frac{k_B T}{m_e c^2} dl = \frac{\sigma_T}{m_e c^2} \int P dl \quad (277)$$

with $P = n_e k_B T$ the pressure of the intracluster medium. The integrated Compton parameter, $Y_{SZ}$, is defined as

$$Y_{SZ} = \int y(\mathbf{n}) d^2\Omega = \frac{\sigma_T}{m_e c^2} \int P d\Omega = \frac{\sigma_T}{m_e c^2} \int D_A^{-2} P d^3V \quad (278)$$

with $D_A$ the angular distance. Using Eq. (264) for $\sigma_T$ and since the integral on the line of sight peaks at the redshift on the cluster one gets

$$Y_{SZ} D_A^2(z) = \frac{8\pi}{3} \frac{\hbar^2}{m_e^3 c^4} \alpha_{EM}^2 \int P d^3V. \quad (279)$$

Concerning the X-ray emission, one observes the surface brightness derived from the thermal bremsstrahlung emissivity (Galli 2013). It is common to define the parameter $Y_X = MT_X$ in term of the mass and temperature of the cluster. The former is shown to scale (Sasaki 1996) as

$$M \sim \alpha_{EM}^{-3/2} m_e^{3/4} m_H D_L D_A^{3/2},$$

with $m_H$ the mass of hydrogen, that we identify to $m_p$ for simplicity here. The ratio $Y_{SZ} D_A^2 / Y_X$ is then expected to be constant from numerical simulations and current observations, and it scales as $\alpha_{EM}^{7/2}$. More precisely we get that

$$\frac{Y_{SZ} D_A^2}{Y_X} = \left( \frac{\sigma_T}{m_e m_p \mu_e c^2} \right) \left( \frac{\alpha_{EM}}{\alpha_{EM}(0)} \right)^{3/2} \left( \frac{m_e}{m_e(0)} \right)^{-3/4} \left( \frac{m_H}{m_H(0)} \right)^{-1} \\ \times \eta(z) \frac{\int n_e T d^3V}{T_X \int n_e d^3V} \quad (280)$$

with $\eta(z) \equiv D_L/(1+z)^2 D_A$ a parameter that is equal to 1 as long as distance duality holds (Uzan et al. 2004). Hence
$$ \left( \frac{Y_{SZ} D_A^2}{Y_X} \right)_i = \left( \frac{\alpha_{EM}}{\alpha_{EM}(0)} \right)^{7/2} \left( \frac{m_e}{m_e(0)} \right)^{-15/4} \left( \frac{m_p}{m_p(0)} \right)^{-2} \eta(z) \left( \frac{Y_{SZ} D_A^2}{Y_X} \right)_0 . \quad (281) $$

Using a catalog of 61 selected clusters with $z \in [0.2, 1.5]$ with SZ detected by SPT and X-ray measurement from XMM-Newton, and assuming no variation in $m_e$ and $m_p$ and that $\eta(z) = 1$, Galli (2013) concluded that

$$ |\Delta\alpha_{EM}/\alpha_{EM}| < 8 \times 10^{-3} \qquad (282) $$

at 66% C.L., not including uncertainties on the cosmological parameters used to determine the angular diameter distance. As pointed out by the author, the data are neither a complete nor a representative sample of clusters and used to provide a first estimate of the accuracy that can be reached by this new method. This constraint was later confirmed by Bora and Desai (2021) with a catalog of 58 SPT clusters with $Y_X$ determined by XMM-Newton. Assuming a redshift evolution of the form $\Delta\alpha_{EM}/\alpha_{EM} = -\gamma \ln(1+z)$ they concluded that $\gamma = 0.95^{+0.12}_{-0.11}$ hence showing no evolution with $z$. Colaço et al. (2019) got $\gamma = -0.15 \pm 0.17$ and De Martino et al. (2016b) used the Planck 2013 data to measure the thermal SZ effect at the location of 618 X-ray selected clusters but focused on a dipole variation (see Sect. 8.3.2). Albuquerque et al. (2025) applied this test by combining the catalog of 44 X-ray data by Mantz et al. (2021) and SNIa data to infer that $\Delta\bar{\mu} < 1\%$ on a redshift band [0.018, 1.160]. See also Gonçalves et al. (2020) for the influence of $\eta$ on this test.

This method is promising given the growing activity of SZ surveys (such as SPTpol, SPT-3 G, ACTpol) that shall detect an order of magnitude more clusters up to $z \sim 2$ and X-ray surveys (such as eROSITA that shall discover approx. $10^5$ clusters). It allows one for a test of $\alpha_{EM}$ variation and a possible dipolar modulations. It still assumes the validity of the distance duality that is intertwined with a variation of $\alpha_{EM}$ and $m_e$ (see e.g., Holanda et al. 2016, 2019) which is today bounded to $|\Delta\eta| < 10^{-4}$ (Ellis et al. 2013) between decoupling and today.

## 5.13 Fast radio bursts

Fast radio bursts (FRB) are millisecond transient events in radio frequency; see Platts et al. (2019), Petroff et al. (2022) for reviews. While many models have been proposed to explain the burst's origin, the emission mechanism remains unknown (Platts et al. 2019). One of the key parameter is the FRB's dispersion measure (DM) related to the density of free electrons along the line of sight. Starting from the time for a pulse to reach Earth from a source at distance $d$ with frequency $\omega$ and group velocity $v_g$,

$$ t_p = \int_0^d \frac{ds}{v_g} \approx d + \int_0^d \frac{\omega_p^2}{2\omega^2} ds $$

where $\omega_p = 4\pi e^2 n_e / m_e$ is the plasma frequency, one defines
$$DM(d) = \int_{0}^{d} n_{e} ds \quad \text{so that} \quad \frac{dt_{p}}{d\omega} = -\frac{4\pi e^{2}}{m_{e}\omega^{3}} DM(d). \quad (283)$$

This expression neglects the expansion of the universe and assumes the constancy of the fundamental constants. In a cosmological framework, this rewrites as (Lemos et al. 2023)

$$\frac{dt_p}{d\omega_0} = -\frac{4\pi\alpha_{EM}(0)}{m_e(0)\omega_0^3} DM(z)$$

with

$$DM = \int_{0}^{z} \frac{n_{e}(z')}{(1+z')^{2}H(z')} \frac{\alpha_{EM}(z')}{\alpha_{EM}(0)} \frac{m_{e}(0)}{m_{e}(z')} dz', \quad (284)$$

where $\omega_0$ is the observed frequency. The main difficulty is to evaluate the contribution of the interstellar medium, Milky Way halo, host galaxy and intergalactic medium. This requires a difficult modelization to extract the contribution of the intergalactic medium. Lemos et al. (2025) applied the method to a catalog of 17 FRB upto a redshift of 0.47 assuming that $\Delta\alpha_{EM}/\alpha_{EM} = -\gamma\ln(1+z)$ and $\mu$ constant to conclude that $\gamma<1\%$. While it still has to face many systematics and difficult modelizations, the growth of FRB catalogues still makes it an idea to be developed further.

# 5.14 Big Bang nucleosynthesis

## 5.14.1 Overview

The amount of $^4$He produced during the Big Bang nucleosynthesis (BBN) is mainly determined by the neutron to proton ratio at the freeze-out of the weak interactions that interconvert neutrons and protons. The results thus depend on $G$, $\alpha_W$, $\alpha_{EM}$ and $\alpha_S$ respectively through the expansion rate, the neutron to proton ratio, the neutron-proton mass difference and the nuclear reaction rates, besides the standard parameters such as, e.g., the number of neutrino families.

The standard BBN scenario (Cyburt et al. 2005; Peter and Uzan 2009; Pitrou et al. 2018) proceeds in three main steps:

1. for $T > 1$ MeV, ($t<1$ s) a first stage during which the neutrons, protons, electrons, positrons and neutrinos are kept in statistical equilibrium by the (rapid) weak interaction

$$n \leftrightarrow p + e^{-} + \bar{v}_{e}, \quad n + v_{e} \leftrightarrow p + e^{-}, \quad n + e^{+} \leftrightarrow p + \bar{v}_{e}. \quad (285)$$

As long as statistical equilibrium holds, the neutron to proton ratio is

$$(n/p) = e^{-Q_{np}/k_B T} \quad (286)$$

where $Q_{np} \equiv (m_n - m_p)c^2 = 1.29$ MeV. The abundance of the other light elements is given by (Peter and Uzan 2009)
$$Y_A = g_A \left(\frac{\zeta(3)}{\sqrt{\pi}}\right)^{A-1} 2^{(3A-5)/2} A^{5/2} \left[\frac{k_B T}{m_N c^2}\right]^{3(A-1)/2} \eta^{A-1} Y_p^Z Y_n^{A-Z} e^{B_A/k_B T}, \quad (287)$$

where $g_A$ is the number of degrees of freedom of the nucleus $Z$X, $m_N$ is the nucleon mass, $\eta$ the baryon-photon ratio and $B_A \equiv (Zm_p + (A-Z)m_n - m_A)c^2$ the binding energy.

2. Around $T \sim 0.8$ MeV ($t \sim 2$ s), the weak interactions freeze out at a temperature $T_f$ determined by the competition between the weak interaction rates and the expansion rate of the universe and thus roughly determined by $\Gamma_w(T_f) \sim H(T_f)$ that is

$$G_F^2(k_B T_f)^5 \sim \sqrt{GN_*}(k_B T_f)^2 \quad (288)$$

where $G_F$ is the Fermi constant and $N_*$ the number of relativistic degrees of freedom at $T_f$. Below $T_f$, the number of neutrons and protons change only from the neutron $\beta$-decay between $T_f$ to $T_N \sim 0.1$ MeV when $p+n$ reactions proceed faster than their inverse dissociation.

3. For $0.05$ MeV $< T < 0.6$ MeV ($3s<t<6$ min), the synthesis of light elements occurs only by two-body reactions. This requires the deuteron to be synthesized ($p+n \to D$) and the photon density must be low enough for the photo-dissociation to be negligible. This happens roughly when

$$\frac{n_d}{n_\gamma} \sim \eta^2 \exp(-B_D/T_N) \sim 1 \quad (289)$$

with $\eta \sim 3 \times 10^{-10}$. The abundance of $^4$He by mass, $Y_p$, is then well estimated by

$$Y_p \approx 2 \frac{(n/p)_N}{1 + (n/p)_N} \quad (290)$$

with

$$(n/p)_N = (n/p)_f \exp(-t_N/\tau_n) \quad (291)$$

with $t_N \propto G^{-1/2}T_N^{-2}$ and $\tau_n^{-1} = 1.636 G_F^2(1 + 3g_A^2)m_e^5/(2\pi^3)$, with $g_A \simeq 1.26$ being the axial/vector coupling of the nucleon. Assuming that $B_D \propto \alpha_S^2$, this gives a dependence $t_N/\tau_p \propto G^{-1/2}\alpha_S^2 G_F^2$.

4. The light element abundances, $Y_i$, are then obtained by solving a series of nuclear reactions

$$\dot{Y}_i = J_i - \Gamma_{ij} Y_j,$$

where $J_i$ and $\Gamma_{ij}$ are time-dependent source and sink terms.
<figure><img src="image_31.png" /><figcaption>Fig. 24 (Left): variation of the light element abundances in function of η compared to the spectroscopic abundances. The vertical line depicts the constraint obtained on η from the study of the cosmic microwave background. The lithium-7 problem lies in the fact that η_spectro < η_WMAP. Reprinted with permission from Pitrou et al. (2018), copyright by Elsevier. (Right): dependence of the light elements abundances on the independent variation of the BBN parameters, assuming η = η_WMAP. Adapted from Coc et al. (2007)</figcaption></figure>

### 5.14.2 Status of standard BBN

Many public codes implement the standard BBN model, PArthENoPE (Consiglio et al. 2018), PRIMAT (Pitrou et al. 2018), PRyMordial (Burns et al. 2024). All the reaction rates required for the BBN predictions are measured in accelerators so that, assuming the Copernican principle, BBN has only two free parameters, namely $\Omega_b h_0^2$ and $N_{\text{eff}}$. The predictions of the light elements abundances can be computed as a function of η and compared to their observed abundance (see Fig. 24 that summarizes the observational constraints obtained on helium-4, helium-3, deuterium and lithium-7). η is related to the baryon density, a parameter measured by other cosmological probes such as the CMB by the relation (Pitrou et al. 2018; Fields et al. 2020; Yeh et al. 2021)

$$ \frac{\Omega_b h_0^2}{0.0224} = \left( \frac{\eta}{6.13197 \times 10^{-10}} \right) \left( \frac{T_{\text{CMB}}}{2.7255 \text{ K}} \right) \left( \frac{1 - 1.759 \times 10^{-3} \frac{Y_{\text{p}}}{0.2471}}{1 - 1.759 \times 10^{-3}} \right) \quad (292) $$

Prior to WMAP, these parameters were adjustable but are now determined with high accuracy from the CMB analysis. WMAP data (Komatsu et al. 2009) have led to the
conclusion that $\eta = \eta_{WMAP} = (6.19 \pm 0.15) \times 10^{-10}$. This value has been reevaluated by Planck-2018 (Aghanim et al. 2020) to

$$ \begin{aligned} \Omega_b h_0^2 &= 0.02237 \pm 0.00015 \quad (\text{CMB}) \\ \Omega_b h_0^2 &= 0.02242 \pm 0.00014 \quad (\text{CMB +BAO}). \end{aligned} \qquad (293) $$

Then, the effective number of neutrinos has been reevaluated to $N_{\text{eff}} = 3.044$ for 3 families, taking into account their non-instantaneous decoupling (Froustey et al. 2020) and the neutron decay constant is now measured to $\tau_n = 879.4 \pm 0.6$ s (Zyla et al. 2020).

Concerning the observations, the spectroscopic abundances. *helium-3* is not very constraining because it is both produced and destroyed in stars so that the evolution of its abundance in time is not very precise. *Lithium-7* exhibits a factor of order 3 discrepancy, which is usually discarded quietly, the consensus being that it cannot arise from the nuclear sector. Finally, the most most recent recommended observed value for *Deuterium* (Cooke et al. 2018) is D/H= (2.527 ± 0.030) × 10⁻⁵ at a redshift $z \sim 2.5 - 3.1$, making it the most constraining BBN observable because both its observational measurement and its theoretical prediction reach 1% accuracy. It has recently been pointed out that differences between BBN codes are attributed to different choices made when modelling the nuclear cross-sections, and not on weak rates. It is important to keep in mind that one would need to control their accuracy at least at the percent level and to take into account the latest data; see Fields et al. (2020), Pisanti et al. (2021), Pitrou et al. (2021a, 2021b).

There exists a long-time discrepancy between the predicted abundance of lithium-7 based on the CMB results (Coc et al. 2004; Coc and Vangioni 2010) for $\eta$, $^{7}\text{Li/H} = (5.14 \pm 0.50) \times 10^{-10}$ and its values measured in metal-poor halo stars in our galaxy (Bonifacio et al. 2007), $^{7}\text{Li/H} = (1.26 \pm 0.26) \times 10^{-10}$, which is a factor of three lower, at least (Cyburt et al. 2008) (see also Spite and Spite 2009), than the predicted value. No solution to this *lithium-7* problem is known. A back of the envelope estimate shows that we can mimic a lower $\eta$ parameter, just by modifying the deuterium binding energy, letting $T_N$ unchanged, since from Eq. (289), one just need $\Delta B_D/T_N \sim -\ln 9$ so that the effective $\eta$ parameter, assuming no variation of constant, is three times smaller than $\eta_{WMAP}$. This rough rule of thumb explains that the solution of the lithium-7 problem may lie in a possible variation of the fundamental constants.

### 5.14.3 Constants everywhere

In complete generality, the effect of varying constants on BBN is difficult to model because of the intricate structure of QCD and its role in low energy nuclear reactions. Thus, a solution is to proceed in *two steps*, first by determining the dependencies of the light element abundances on the BBN parameters and then by relating those parameters to the fundamental constants.

The analysis of the previous Sect. 5.14.1, that was restricted to the helium-4 case, clearly shows that the abundances will depend on: (1) $\alpha_G$, which will affect the
Hubble expansion rate at the time of nucleosynthesis in the same way as extra-relativistic degrees of freedom do, so that it modifies the freeze-out time $T_f$. This is the only gravitational sector parameter. (2) $\tau_n$, the neutron lifetime dictates the free neutron decay and appears in the normalization of the proton-neutron reaction rates. It is the only weak interaction parameter and it is related to the Fermi constant $G_F$, or equivalently the Higgs vev. (3) $\alpha_{EM}$, the fine-structure constant. It enters in the Coulomb barriers of the reaction rates through the Gamow factor, in all the binding energies. (4) $Q_{np}$, the neutron-proton mass difference enters in the neutron-proton ratio and we also have a dependence in (5) $m_N$ and $m_e$ and (6) the binding energies.

Clearly all these parameters are not independent but their relation is often model-dependent. If we focus on helium-4, its abundance mainly depends on $Q_{np}$, $T_f$ and $T_N$ (and hence mainly on the neutron lifetime, $\tau_n$). Early studies (see Sect. III.C.2 of FVC03 [Uzan 2003b]) generally focused on one of these parameters. For instance, Kolb et al. (1986) calculated the dependence of primordial $^4$He on $G$, $G_F$ and $Q_{np}$ to deduce that the helium-4 abundance was mostly sensitive in the change in $Q_{np}$ and that other abundances were less sensitive to the value of $Q_{np}$, mainly because $^4$He has a larger binding energy; its abundances is less sensitive to the weak reaction rate and more to the parameters fixing the value of $(n/p)$. To extract the constraint on the fine-structure constant, they decomposed $Q_{np}$ as $Q_{np} = \alpha_{EM}Q_\alpha + \beta Q_\beta$ where the first term represents the electromagnetic contribution and the second part corresponds to all non-electromagnetic contributions. Assuming that $Q_\alpha$ and $Q_\beta$ are constant and that the electromagnetic contribution is the dominant part of $Q$, they deduced that $|\Delta\alpha_{EM}/\alpha_{EM}| < 10^{-2}$. Campbell and Olive (1995) kept track of the changes in $T_f$ and $Q_{np}$ separately and deduced that $\frac{\Delta Y_p}{Y_p} \simeq \frac{\Delta T_f}{T_f} - \frac{\Delta Q_{np}}{Q_{np}}$ while more recently the analysis (Landau et al. 2008) focused on $\alpha_{EM}$ and $\nu$.

Let us now see how these parameters are now accounted for in BBN codes.

Bergström et al. (1999) started to focus on the $\alpha_{EM}$-dependence of the thermonuclear rates (see also Ichikawa and Kawasaki 2005). In the non-relativistic limit, it is obtained as the thermal average of the product of the cross, the relative velocity and the number densities. Charged particles must tunnel through a Coulomb barrier to react. Changing $\alpha_{EM}$ modifies these barriers and thus the reaction rates. Separating the Coulomb part, the low-energy cross-section can be written as

$$ \sigma(E) = \frac{S(E)}{E} e^{-2\pi\eta(E)} \qquad (294) $$

where $\eta(E)$ arises from the Coulomb barrier and is given in terms of the charges and the reduced mass $M_r$ of the two interacting particles as

$$ \eta(E) = \alpha_{EM} Z_1 Z_2 \sqrt{\frac{M_r c^2}{2E}}. \qquad (295) $$

The form factor $S(E)$ has to be extrapolated from experimental nuclear data but its $\alpha_{EM}$-dependence as well as the one of the reduced mass were neglected. Keeping all other constants fixed, assuming no exotic effects and taking a lifetime of 886.7 s for
the neutron, it was deduced that $|\Delta\alpha_{EM}/\alpha_{EM}|<2 \times 10^{-2}$. This analysis was then extended (Nollet and Lopez 2002) to take into account the $\alpha_{EM}$-dependence of the form factor to conclude that

$$\sigma(E) = \frac{2\pi\eta(E)}{\exp^{2\pi\eta(E)} - 1} \approx 2\pi\alpha_{EM}Z_1Z_2\sqrt{\frac{M_r c^2}{c^2}} \exp^{-2\pi\eta(E)}.$$

Nollet and Lopez (2002) also took into account (1) the effect that when two charged particles are produced they must escape the Coulomb barrier. This effect is generally weak because the $Q_i$-values (energy release) of the different reactions are generally larger than the Coulomb barrier at the exception of two cases, $^3$He(n, p)$^3$H and $^7$Be(n, p)$^7$Li. The rate of these reactions must be multiplied by a factor $(1 + a_i\Delta\alpha_{EM}/\alpha_{EM})$. (2) The radiative capture (photon emitting processes) are proportional to $\alpha_{EM}$ since it is the strength of the coupling of the photon and nuclear currents. All these rates need to be multiplied by $(1 + \Delta\alpha_{EM}/\alpha_{EM})$. (3) The electromagnetic contribution to all masses was taken into account, which modify the $Q_i$-values as $Q_i \rightarrow Q_i + q_i\Delta\alpha_{EM}/\alpha_{EM}$. For helium-4 abundance these effects are negligible since the main $\alpha_{EM}$-dependence arises from $Q_{np}$. Equipped with these modifications, it was concluded that $\Delta\alpha_{EM}/\alpha_{EM} = -0.007_{-0.017}^{+0.010}$ using only deuterium and helium-4 since the lithium-7 problem was still present. This was updated by Meißner et al. (2023) that study the $\alpha_{EM}$-dependence on $Q_{np}$, $\beta$-decays and nuclear cross-sections to conclude that $|\Delta\alpha_{EM}/\alpha_{EM}|<0.2$ assuming all other constants fixed.

Then the focus fell on the deuterium binding energy, $B_D$. Flambaum and Shuryak (2002, 2003), Dmitriev et al. (2004), Dmitriev and Flambaum (2003) illustrated the sensitivity of the light element abundances on $B_D$. Its value mainly sets the beginning of the nucleosynthesis, that is of $T_N$ since the temperature must low-enough in order for the photo-dissociation of the deuterium to be negligible (this is at the origin of the deuterium bottleneck). The importance of $B_D$ is easily understood by the fact that the equilibrium abundance of deuterium and the reaction rate $p(n, \gamma)D$ depends exponentially on $B_D$ and on the fact that the deuterium is in a shallow bound state. Focusing on the $T_N$-dependence, it was concluded (Flambaum and Shuryak 2002) that $\Delta B_D/B_D < 0.075$.

This shows that the situation is more complex and that one cannot reduce the analysis to a single varying parameter. Many studies then tried to determine the sensitivity to the variation of many independent parameters.

The sensitivity of the helium-4 abundance to the variation of 7 parameters was first investigated by Müller et al. (2004) considering the dependence on the parameters $\{X_i\} = \{G, \alpha_{EM}, \nu, m_e, \tau_n, Q_{np}, B_D\}$ independently,

$$\Delta \ln Y_{\text{He}} = \sum_i c_i^{(X)} \Delta \ln X_i$$

and assuming $\Lambda_{QCD}$ fixed (so that the seven parameters are in fact dimensionless quantities). The $c_i^{(X)}$ are the sensitivities to the BBN parameters, assuming the six others are fixed. It was concluded that
$$Y_{\text{He}} \propto \alpha_{\text{EM}}^{-0.043} v^{2.4} m_e^{0.024} \tau_n^{0.24} Q_{\text{np}}^{-1.8} B_D^{0.53} G^{0.405}$$

for independent variations. They further related ($\tau_n$, $Q_{np}$, $B_D$) to ($\alpha_{EM}$, $v$, $m_e$, $m_N$, $m_d$ - $m_u$), as we shall discuss in the next Sect. 5.14.4.

This was generalized by Landau et al. (2006b) up to lithium-7 considering the parameters {$\alpha_{EM}$, $G_F$, $\Lambda_{QCD}$, $\Omega_b h_0^2$}, assuming G constant where the variation of $\tau_n$ and the variation of the masses where tied to these parameters but the effect on the binding energies were not considered.

Coc et al. (2007) considered the effect of ($Q_{np}$, $B_D$, $\tau_n$, $m_e$) on the abundances of the light elements up to lithium-7 neglecting the effect of $\alpha_{EM}$ on cross-sections. Their dependencies on the independent variation of each of them are depicted on Fig. 24. It confirmed the result of Flambaum and Shuryak (2002), Olive (2009a) that the deuterium binding energy is the most sensitive parameter. From the helium-4 data alone, the bounds

$$-8.2 \times 10^{-2} \lesssim \frac{\Delta\tau_n}{\tau_n} \lesssim 6 \times 10^{-2}, \quad -4 \times 10^{-2} \lesssim \frac{\Delta Q_{np}}{Q_{np}} \lesssim 2.7 \times 10^{-2}, \quad (296)$$

and

$$-7.5 \times 10^{-2} \lesssim \frac{\Delta B_D}{B_D} \lesssim 6.5 \times 10^{-2}, \quad (297)$$

at a $2\sigma$ level, were set (assuming $\eta_{WMAP}$). The deuterium data set the tighter constraint $-4 \times 10^{-2} \lesssim \Delta \ln B_D \lesssim 3 \times 10^{-2}$. Note also on Fig. 24 that the lithium-7 abundance can be brought in concordance with the spectroscopic observations provided that $B_D$ was smaller during BBN

$$-7.5 \times 10^{-2} \lesssim \frac{\Delta B_D}{B_D} \lesssim -4 \times 10^{-2},$$

so that $B_D$ may be the most important parameter to resolve the lithium-7 problem. The effect of the quark mass on the binding energies was described in Berengut et al. (2010b). They concluded that a variation of $\Delta m_q/m_q = 0.013 \pm 0.002$ could reconcile the abundance of lithium-7 and the value of $\eta$ deduced from WMAP.

This analysis was extended (Dent et al. 2007) to incorporate the effect of 13 independent BBN parameters including those considered before plus the binding energies of deuterium, tritium, helium-3, helium-4, lithium-6, lithium-7 and beryllium-7. The sensitivity of the light element abundances to the independent variation of these parameters is summarized in Table I of Dent et al. (2007). These BBN parameters were then related to the same 6 “fundamental” parameters used in Müller et al. (2004).

All these analyses demonstrate that the effects of the BBN parameters on the light element abundances are now under control. They have been implemented in BBN codes and most results agree, as well as with semi-analytical estimates. As long as these parameters are assumed to vary independently, no constraints sharper than $10^{-2}$ can be set. One should also not forget to take into account standard
parameters of the BBN computation such as $\eta$ and the effective number of relativistic particles.

### 5.14.4 From BBN parameters to fundamental constants

To reduce the number parameters, we need to relate the BBN parameters to more fundamental ones, keeping in mind that this can usually be done only in a model-dependent way. We shall describe some of the relations that have been used in many studies. They mainly concern $Q_{np}$, $\tau_n$ and $B_D$.

At lowest order, all dimensional parameters of QCD, e.g., masses, nuclear energies etc., are to a good approximation simply proportional to some powers of $\Lambda_{QCD}$. One needs to go beyond such a description and takes the effects of the masses of the quarks into account.

* Proton-neutron mass difference. $Q_{np}$ can be expressed in terms of the mass on the quarks u and d and the fine-structure constant as

$$Q_{np} = a\alpha_{EM}\Lambda_{QCD} + (m_d - m_u),$$

where the electromagnetic contribution today is $(a\alpha_{EM}\Lambda_{QCD})_0 = -0.76$ MeV and therefore the quark mass contribution today is $(m_d - m_u) = 2.05$ (Gasser and Leutwyler 1982) so that

$$\frac{\Delta Q_{np}}{Q_{np}} = -0.59 \frac{\Delta \alpha_{EM}}{\alpha_{EM}} + 1.59 \frac{\Delta(m_d - m_u)}{(m_d - m_u)}. \qquad (298)$$

All the analyses cited above agree on this dependence.

* Neutron lifetime. It can be well approximated by

$$\tau_n^{-1} = \frac{1 + 3g_A^2}{120\pi^3} G_F^2 m_e^5 \left[ \sqrt{q^2 - 1}(2q^4 - 9q^2 - 8) + 15 \ln(q + \sqrt{q^2 - 1}) \right],$$

with $q \equiv Q_{np}/m_e$ and $G_F = 1/\sqrt{2}v^2$. Using the former expression for $Q_{np}$ we can express $\tau_n$ in terms of $\alpha_{EM}$, $v$ and the u, d and electron masses. It follows

$$\frac{\Delta \tau_n}{\tau_n} = 3.86 \frac{\Delta \alpha_{EM}}{\alpha_{EM}} + 4 \frac{\Delta v}{v} + 1.52 \frac{\Delta m_e}{m_e} - 10.4 \frac{\Delta(m_d - m_u)}{(m_d - m_u)}. \qquad (299)$$

Again, all the analyses cited above agree on this dependence.

* Binding energies. Let us focus on $B_D$ that, as we have seen, is a crucial parameter. This is one the better known quantities in the nuclear domain and it is experimentally measured to a precision better than $10^{-6}$ (Audi 2006). Two approaches have been followed.
* Pion mass. A first route is to use the dependence of the binding energy on the pion mass (Epelbaum et al. 2003; Beane and Savage 2003), which is related to the u and d quark masses by

$$m_{\pi}^{2} = m_{q} \langle \bar{u}u + \bar{d}d \rangle f_{\pi}^{-2} \approx \hat{m} \Lambda_{\text{QCD}},$$

where $m_q \equiv \frac{1}{2}(m_u + m_d)$ and assuming that the leading order of $\langle \bar{u}u + \bar{d}d \rangle f_{\pi}^{-2}$ depends only on $\Lambda_{QCD}$, $f_{\pi}$ being the pion decay constant. This dependence was parameterized (Yoo and Scherrer 2003) as

$$\frac{\Delta B_D}{B_D} = -r \frac{\Delta m_{\pi}}{m_{\pi}},$$

where $r$ is a fitting parameter found to be between 6 (Epelbaum et al. 2003) and 10 (Beane and Savage 2003). Prior to this result, the analysis of Flambaum and Shuryak (2002) provided two computations, which respectively lead to $r = -3$ and $r = 18$ while, following the same lines, Chamoun et al. (2007) got $r = 0.082$. Müller et al. (2004), following Pudliner et al. (1997), added an electromagnetic contribution to get

$$\frac{\Delta B_D}{B_D} = -\frac{r}{2} \frac{\Delta m_q}{m_q} - 0.0081 \frac{\Delta \alpha_{\text{EM}}}{\alpha_{\text{EM}}}, \qquad (300)$$

which has not been included in other work.

* Sigma model. In the framework of the Walecka model, where the potential for the nuclear forces keeps only the $\sigma$ and $\omega$ meson exchanges,

$$V = -\frac{g_s^2}{4\pi r} \exp(-m_\sigma r) + \frac{g_v^2}{4\pi r} \exp(-m_\omega r),$$

with $g_s$ and $g_v$ two coupling constants. Describing $\sigma$ as a SU(3) singlet, its mass was related to the one of the strange quark. In this way, one can hope to take into account the effect of the strange quark, both on the nucleon mass and the binding energy. In a second step $B_D$ is related to the meson and nucleon masses by

$$\frac{\Delta B_D}{B_D} = -48 \frac{\Delta m_\sigma}{m_\sigma} + 50 \frac{\Delta m_\omega}{m_\omega} + 6 \frac{\Delta m_N}{m_N}$$

so that $\Delta B_D/B_D \simeq -17\Delta m_s/m_s$ (Flambaum and Shuryak 2003). Unfortunately, a complete treatment of all the nuclear quantities on $m_s$ has not been performed yet.

The binding energies of the other elements have been less studied. Dent et al. (2007) follows a route similar than for $B_D$ and relates them to pion mass and assumes that

$$\frac{\partial B_i}{\partial m_{\pi}} = f_i(A_i - 1) \frac{B_D}{m_{\pi}} r \approx -0.13 f_i(A_i - 1),$$

where $f_i$ are unknown coefficients assumed to be of order unity and $A_i$ is the number of nucleons. No other estimates has been performed. Other nuclear potentials (such as Reid 93 potential, Nijmegen potential, Argonne v18 potential and Bonn potential)
have been used in Civitarese et al. (2010) to determine the dependence of $B_D$ on $v$ and agree with previous studies. Note also that the effect of a possible change of the nucleon-nucleon interaction on the reaction rates involving $A = 5$ and $A = 8$ unstable nuclei was investigated in Coc et al. (2012a, 2012b) to conclude that no significant effect on BBN appear seven if beryllium-8 were stable.

These analyses allow one to reduce all the BBN parameters to the physical constants ($\alpha_{EM}$, $v$, $m_e$, $m_d - m_u$, $m_q$) and $G$ that is not affected by this discussion. This set can be further reduced, since all the masses can be expressed in terms of $v$ as $m_i = h_i v$, where $h_i$ are Yukawa couplings. We also refer to the extensive study (Birrell et al. 2014; Rafelski et al. 2024) for a discussion of the effects of fundamental constants of the standard model of particle physics on the neutrino freeze-out process showing that it mostly depends on the Weinberg angle $\theta_W$ and the combination $m_e^3 G_F^2 / G_N^{1/2}$.

Focusing on the resonant nuclear process $^3$He($d, p$)$^4$He dependence on $m_q$, Cheoun et al. (2011) concluded that $-0.5 \times 10^{-2} < \Delta m_q / m_q < 2.5 \times 10^{-2}$ from D/H alone and $\Delta m_q / m_q < 10^{-2}$ from helium-4 alone assuming that $\eta$ is fixed to its WMAP-7 value $\eta = 6.23 \times 10^{-10}$, hence concluding that $-0.5 \times 10^{-2} < \Delta m_q / m_q < 10^{-2}$. Bedaque et al. (2011) considered several effective field theories as well as lattice QCD results, including a nuclear pionless effective theory, to compute the sensitivities of the binding energies on $m_q$. They concluded that BBN implies $-10^{-2} < \Delta m_q / m_q < 7 \times 10^{-3}$. Berengut et al. (2013a) presented a derivation of the sensitivities of the light elements of the binding energies ans nuclear scattering lengths in the chiral perturbation theory in combination with non-perturbative methods. From deuterium and helium-4 data, they concluded $\Delta m_q / m_q = (2 \pm 4) \times 10^{-2}$, a constraints that drop to $|\Delta m_q / m_q| < 9 \times 10^{-3}$ if the variation of $m_q$ is assumed to be related to a variation of $v$ that translated to a variation of $m_e$ that impacts the neutron lifetime. Following Flambaum and Wiringa (2007) for the sensitivities of the binding energies of nucleus masses, Mori and Kusakabe (2019) investigated the sensitivity of the $^7$Be($n, p$)$^7$Li reaction to a change of $m_q$. They concluded that if the variation of the excitation energies of the compound nucleus $^8$Be* is the same as that of the ground state, the shift of the resonance at $E_r = 0.33$ MeV can decrease the $^7$Be abundance significantly for $\Delta m_q / m_q < -5 \times 10^{-3}$.

To go further, one needs more assumptions, such as grand unification, or by relating the Yukawa coupling of the top to $v$ by assuming that weak scale is determined by dimensional transmutation (Coc et al. 2007), or that the variation of the constant is induced by a string dilaton (Campbell and Olive 1995). At each step, one gets more stringent constraints, which can reach the $10^{-4}$ (Dent et al. 2007) to $10^{-5}$ (Coc et al. 2007) level but indeed more model-dependent!

### 5.14.5 Constants and the lithium problem

The success of the BBN model is limited by the long-standing lithium problem, i.e., the fact that the theoretically expected abundance of lithium-7 (given our present
knowledge of astrophysics, nuclear and particle physics) exceeds the observed one by a factor of about 3.5.

As seen on Fig 24, Coc and Vangioni (2010) pointed out that a lower value of the deuterium binding energy at BBN can solve this problem. This was later marginally ruled out by the improvement of the D/H measurements by Cooke et al. (2018). This fact nevertheless rose attention of the importance on correlated variations of constants. Bedaque et al. (2011) stressed that since the values for lithium-7 binding energies are not reliable, the nuclear pionless effective theory may change this and allow us to interpret the lithium problem as a signal of quark mass variation. Then, Mori and Kusakabe (2019) proposed a solution with a finite value of $\Delta m_q/m_q = (4-8) \times 10^{-3}$ using the sensitivities of the binding energies and nucleus masses from Flambaum and Wiringa (2007) while assuming that the resonance energies of excited states do not vary. Mosquera and Civitarese (2017) concluded that the lithium abundance is reduced for $3\sigma$ level variation of ($\alpha_{EM}, \nu$) with introduction of dark energy. Martins (2021), Clara and Martins (2020) reconsidered the problem within the (R, S)-parameterisation (see 4.5) that allowed them to express the variation of all the primordial abundances as a linear combinations of R and S, $\Delta Y_i/Y_i = (x_i + y_i S + z_i R)\Delta\alpha_{EM}/\alpha_{EM}$ to conclude that constants remain an alternative for a solution to the lithium problem. Deal and Martins (2021) extended this to include $\Delta Y_i/Y_i = (x_i + y_i S + z_i R)\Delta\alpha_{EM}/\alpha_{EM} + t_i \Delta\tau_n/\tau_n + v_i \Delta N_{eff}/N_{eff} + w_i \Delta\eta_{10}/\eta_{10}$ to conclude that the lithium problem most likely has an astrophysical solution while the deuterium discrepancy provides a possible hint for $\Delta\alpha_{EM}/\alpha_{EM} > 0$. Similarly, Franchino-Viñas and Mosquera (2021) claimed that a $(G, N_{eff}, \alpha_{EM}, \nu)$ variation can ease both the lithium and Hubble tensions and, to finish, Seto et al. (2023) stressed a variation of $\alpha_{EM}$ to explain the helium-4 data by Matsumoto et al. (2022).

### 5.14.6 Conclusion

Primordial nucleosynthesis offers a possibility to test almost all fundamental constants of physics at a redshift of $z \sim 10^8$. This makes it very rich but indeed the effect of each constant is more difficult to disentangle. The effect of the BBN parameters has been quantified with precision and they can be constrained typically at a $10^{-2}$ level, and in particular it seems that the most sensitive parameter is the deuterium binding energy.

The link with more fundamental parameters is better understood but the dependence of the deuterium binding energy still left some uncertainties and a good description of the effect of the strange quark mass is missing.

We have not considered the variation of $G$ in this section. Its effect is disconnected from the other parameters. Let us just stress that assuming the BBN sensitivity on $G$ by just modifying its value may be misleading. In particular $G$ can vary a lot during the electron-positron annihilation so that the BBN constraints can in general not be described by an effective speed-up factor (Damour and Pichon 1999; Coc et al. 2006).
## 6 The gravitational sector and G

The gravitational constant was the first constant whose constancy was questioned (Dirac 1937). From a theoretical point of view, theories with a varying gravitational constant can be designed to satisfy the equivalence principle in its weak form but not in its strong form (Will 1993) (see Sect. 3.1.1 for details). Most theories of gravity that violate the strong equivalence principle predict that the locally measured gravitational constant may vary with time.

The value of the gravitational constant is

$$G = 6.67430(15) \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}$$

so that its relative standard uncertainty fixed by the CODATA<fnref n="16" /> is 0.01% (Tiesinga et al. 2021); see Wu et al. (2019) for a review on the latest measurement of G. Interestingly, the disparity between different experiments led, in 1998, to a temporary increase of this uncertainty to 0.15% (Gundlach and Merkowitz 2000), which demonstrates the difficulty in measuring the value of this constant. This explains partly why the constraints on the time variation are less stringent than for the other constants.

A variation of the gravitational constant, being a pure gravitational phenomenon, does not affect the local physics, such as, e.g., the atomic transitions or the nuclear physics. In particular, it is equivalent to stating that the masses of all particles are varying in the same way so that their ratios remain constant. Similarly all absorption lines will be shifted in the same way. It follows that most constraints are obtained from systems in which gravity is non-negligible, such as the motion of the bodies of the Solar system, astrophysical and cosmological systems. They are mostly related in the comparison of a gravitational time scale, e.g., period of orbits, to a non-gravitational time scale. It follows that in general the constraints assume that the values of the other constants are fixed. Taking their variation into account would add degeneracies and make the constraints cited below less stringent.

Indeed since G has dimensions, any constraint cited here has to assume that some mass m is kept constant so that it refer to a constraint on $Gm^2/\hbar c$, see Eq. (2). Most studies assume implicitly universal scalar-tensor theories so that any mass can be chosen since all mass ratios, $m_i/m_j$, remain constant. Otherwise most cases considered the proton mass $m_p$ to be constant. Hence, constraints on the variation of G are meant to be constraints on the dimensionless parameter $\alpha_G$; see Eq. (2). The main constraints are summarized in Table 20.

We refer to Sect. IV of FVC03 (Uzan 2003b) for earlier constraints based, e.g., on the determination of the Earth surface temperature, which roughly scales as $G^{2.25} M_{\odot}^{1.75}$ and gives a constraint of the order of $|\Delta G/G|<0.1$ (Gamow 1967), or on the estimation of the Earth radius at different geological epochs.

Note an important difference between G and other fundamental constants. A variation of G would affect all observables achromatically, hence the effect on cosmic spectra would be undistinguishable from a global redshift effect that can

<footnote n="16">¹⁶ The CODATA is the Committee on Data for Science and Technology, see http://www.codata.org/.</footnote>
e.g., arise from the cosmic expansion. If rewritten in terms of varying masses then all mass ratios remain unchanged. To finish, note that while one can measure *GM* with high accuracy, its is more difficult to measure *G* and *M* independently. This explains why the constraints on the time variation of *G* are several order of magnitude worse than those on non-gravitational constants.

## 6.1 Solar system constraints

Monitoring the orbits of various bodies of the Solar system offers a possibility to constrain deviations from General Relativity, and in particular the time variation of *G*. This accounts for comparing a gravitational time scale (related to the orbital motion) and an atomic time scale and it is thus assumed that the variation of atomic constants is negligible over the time of the experiment.

### Earth-Moon system

A time variation of *G* can be related to a variation of the mean motion ($n = 2\pi/P$) of the orbit of the Moon around the Earth. A decrease in *G* would induce both the Lunar mean distance and period to increase. As long as the gravitational binding energy is negligible, one has

$$ \frac{\dot{P}}{P} = -2 \frac{\dot{G}}{G}. $$

Earlier constraints rely on paleontological data and ancient eclipses observations (see Sect. IV.B.1 of FVC03 [Uzan 2003b]) and none of them are very reliable. A main difficulty arises from tidal dissipation that also causes the mean distance and orbital period to increase (for tidal changes $2n_i/n + 3a_i/a = 0$), but not as in the same ratio as for $\dot{G}$.

The Lunar Laser Ranging (LLR) experiment has measured the Earth-Moon distance with an accuracy of the order of 1 cm over 4 decades. An early analysis of this data (Williams et al. 1976) assuming a Brans-Dicke theory concluded that $|\dot{G}/G| \le 3 \times 10^{-11} \text{ yr}^{-1}$, later improved (Müller et al. 1991), by using 20 years of observation, to $|\dot{G}/G| \le 1.04 \times 10^{-11} \text{ yr}^{-1}$, the main uncertainty arising from Lunar tidal acceleration. With, 24 years of data, one reached (Williams et al. 1996) $|\dot{G}/G| \le 6 \times 10^{-12} \text{ yr}^{-1}$ and then (Williams et al. 2004) reached $|\dot{G}/G| \le (4 \pm 9) \times 10^{-13} \text{ yr}^{-1}$. This was improved (Hofmann et al. 2010) to

$$ \left. \frac{\dot{G}}{G} \right|_0 = (-0.7 \pm 3.8) \times 10^{-13} \text{ yr}^{-1} \qquad (301) $$

by using data from 1970 to 2009 and including the effect of a fluid Lunar core in the model. A better analysis of the Lunar orbit, a better distribution of measurements over the Lunar retro-reflectors, as well as higher accuracy of the data, allowed Biskupek et al. (2021) to claim that LLR constraints are improved to
$$ \left. \frac{\dot{G}}{G} \right|_{0} = (-5.0 \pm 9.6) \times 10^{-15} \text{yr}^{-1}. \qquad (302) $$

## Planet ranging

Similarly, Shapiro et al. (1971) compared radar-echo time delays between Earth, Venus and Mercury with a caesium atomic clock between 1964 and 1969. The data were fitted to the theoretical equation of motion for the bodies in a Schwarzschild spacetime, taking into account the perturbations from the Moon and other planets. They concluded that $|\dot{G}/G| < 4 \times 10^{-10} \text{yr}^{-1}$. The data concerning Venus cannot be used due to imprecision in the determination of the portion of the planet reflecting the radar. This was improved to $|\dot{G}/G| < 1.5 \times 10^{-10} \text{yr}^{-1}$ by including Mariner 9 and Mars orbiter data (Reasenberg and Shapiro 1978). The analysis was further extended (Shapiro 1990) to give $\dot{G}/G = (-2 \pm 10) \times 10^{-12} \text{yr}^{-1}$. The combination of Mariner 10 an Mercury and Venus ranging data gives (Anderson et al. 1992) gave $\dot{G}/G = (0.0 \pm 2.0) \times 10^{-12} \text{yr}^{-1}$. Reasenberg et al. (1979) considered the 14 months data obtained from the ranging of the Viking spacecraft and deduced, assuming a Brans-Dicke theory, $|\dot{G}/G| < 10^{-12} \text{yr}^{-1}$. Hellings et al. (1983) using all available astrometric data and in particular the ranging data from Viking landers on Mars deduced that $|\dot{G}/G| = (2 \pm 4) \times 10^{-12} \text{yr}^{-1}$. The major contribution to the uncertainty is due to the modeling of the dynamics of the asteroids on the Earth-Mars range. Hellings et al. (1983) also tried to attribute their results to a variation of the atomic constants. Using the same data but a different modeling of the asteroids, Reasenberg (1983) got $|\dot{G}/G| < 3 \times 10^{-11} \text{yr}^{-1}$, later improved by Chandler et al. (1993) to $|\dot{G}/G| < 10^{-11} \text{yr}^{-1}$.

From the planetary ephemerides (INPOP13c) Fienga et al. (2015) deduced bounds on the variation of $GM_\odot$. Assuming that $M_\odot(t) = (t - J2000) \times M_\odot(J2000)$ and $G(t) = (t - J2000) \times G(J2000)$ and fixing the Sun total mass loss (including radiation and Solar winds) to $\dot{M}_\odot/M_\odot = (-0.55 \pm 0.15) \times 10^{-13}$ (Pinto et al. 2011), they reached

$$ \left. \frac{\dot{G}}{G} \right|_{0} = (-0.01 \pm 0.91) \times 10^{-13} \text{yr}^{-1}. \qquad (303) $$

Pitjeva et al. (2021) estimated the effect of a secular change of $GM_\odot$ on positional observations of planets and spacecrafts using the ephemeris EPM2019. From the determination of $\dot{M}_\odot$ from outgoing Solar radiation and wind and infalling material, they concluded that

$$ -2.9 \times 10^{-14} \text{yr}^{-1} < \dot{G}/G < 4.6 \times 10^{-14} \text{yr}^{-1} \qquad (304) $$

at $3\sigma$, improving their former result (Pitjeva and Pitjev 2012), $-4.2 \times 10^{-14} \text{yr}^{-1} < \dot{G}/G < 7.5 \times 10^{-14} \text{yr}^{-1}$.
## Extension beyond the Solar system

Desmond et al. (2021) proposed to extend these tests to the Large Magellanic Cloud, (LMC) that contains 6 well-studied Cepheid variable stars in detached eclipsing binaries. Radial velocity and photometric observations enable a complete orbital solution, and precise measurements of the Cepheids’ periods permit a detailed stellar modelling. Both are sensitive to G, the former via Kepler’s third law and the latter through the gravitational free-fall time. They concluded that the gravitational constant in the LMC is

$$G_{\text{LMC}}/G = 0.93_{-0.04}^{+0.05}. \qquad (305)$$

## 6.2 Pulsar timing

Contrary to the Solar system, the dependence of the gravitational binding energy cannot be neglected while computing the time variation of the period of denser objects such as pulsars. Here, two approaches can be followed. Either one sticks to a model (e.g., scalar-tensor gravity) to compute all the effects in this model or one relies on a more phenomenological approach to set model-independent bounds.

### Binary pulsars

Eardley (1975) followed the first route to discuss the effects of a time variation of the gravitational constant on binary pulsar in the framework of the Brans-Dicke theory. In that case, both a dipole gravitational radiation and the variation of G induce a periodic variation in the pulse period. Nordtvedt (1990) showed that the orbital period changes as

$$\frac{\dot{P}}{P} = - \left[ 2 + \frac{2(m_1 c_1 + m_2 c_2) + 3(m_1 c_2 + m_2 c_1)}{m_1 + m_2} \right] \frac{\dot{G}}{G} \qquad (306)$$

where $c_i \equiv \delta \ln m_i / \delta \ln G$. He concluded that for the pulsar PSR 1913+16 ($m_1 \simeq m_2$ and $c_1 \simeq c_2$) one gets

$$\frac{\dot{P}}{P} = -[2 + 5c] \frac{\dot{G}}{G}, \qquad (307)$$

the coefficient c being model dependent. As another application, he estimated that $c_{\text{Earth}} \sim -5 \times 10^{-10}$, $c_{\text{Moon}} \sim -10^{-8}$ and $c_{\text{Sun}} \sim -4 \times 10^{-6}$ justifying the formula used in the Solar system.

Damour et al. (1988) used the timing data of the binary pulsar PSR 1913+16. They implemented the effect of the time variation of G by considering the effect on $\dot{P}/P$. They defined, in a phenomenological way, that $\dot{G}/G = -0.5\delta\dot{P}/P$, where $\delta\dot{P}$ is the part of the orbital period derivative that is not explained otherwise (by gravitational waves radiation damping). This theory-independent definition has to be contrasted with the theory-dependent result (307) by Nordtvedt (1990). They got $\dot{G}/G = (1.0 \pm 2.3) \times 10^{-11} \text{ yr}^{-1}$. Damour and Taylor (1991) reexamined the data
of PSR 1913+16 to establish the bound $\dot{G}/G < (1.10 \pm 1.07) \times 10^{-11}$ yr$^{-1}$. Using data from PSR B1913+16 and PSR B1855+09, Kaspi et al. (1994) got respectively

$$ \dot{G}/G = (4 \pm 5) \times 10^{-12} \text{yr}^{-1} \quad \text{and} \quad \dot{G}/G = (-9 \pm 18) \times 10^{-12} \text{yr}^{-1}, \qquad (308) $$

the latter case being more “secure” since the orbiting companion is not a neutron star. A 10-year timing campaign of PSR J1738+0333, a 5.85-ms pulsar in a low-eccentricity 8.5-hour orbit with a low-mass white dwarf companion, allowed Freire et al. (2012) to get

$$ |\dot{G}/G| = (-0.7 \pm 7.6) \times 10^{-13} \text{yr}^{-1}. \qquad (309) $$

From 21-year timing of observation of the 1.3 M$_{\odot}$ millisecond pulsar PSR J1713+0747, one of the most precise pulsars known, orbiting a 2.9 M$_{\odot}$ white dwarf, Zhu et al. (2015) set the constraint

$$ |\dot{G}/G| = (-0.6 \pm 1.1) \times 10^{-13} \text{yr}^{-1}. \qquad (310) $$

### Single pulsars

All the previous results concern binary pulsars but isolated ones can also be used. Heintzmann and Hillebrandt (1975) related the spin-down of the pulsar JP1953 to a time variation of G. The spin-down is a combined effect of electromagnetic losses, emission of gravitational waves, possible spin-up due to matter accretion. Assuming that the angular momentum is conserved so that $I/P = \text{constant}$, one deduces that

$$ \left. \frac{\dot{P}}{P} \right|_G = \left( \frac{d \ln I}{d \ln G} \right) \frac{\dot{G}}{G}. \qquad (311) $$

The observational spin-down can be decomposed as

$$ \left. \frac{\dot{P}}{P} \right|_{\text{obs}} = \left. \frac{\dot{P}}{P} \right|_{\text{mag}} + \left. \frac{\dot{P}}{P} \right|_{\text{GW}} + \left. \frac{\dot{P}}{P} \right|_{G}. \qquad (312) $$

Since $\dot{P}/P_{\text{mag}}$ and $\dot{P}/P_{\text{GW}}$ are positive definite, it follows that $\dot{P}/P_{\text{obs}} \ge \dot{P}/P_G$ so that a bound on $\dot{G}$ can be inferred if the main pulse period is the period of rotation. Heintzmann and Hillebrandt (1975) then modeled the pulsar by a polytropic ($P \propto \rho^n$) white dwarf and deduced that $d \ln I/d \ln G = 2 - 3n/2$ so that $|\dot{G}/G| < 10^{-10}$ yr$^{-1}$. Mansfield (1976) assumed a relativistic degenerate, zero temperature polytropic star and got that, when $\dot{G} < 0$, $0 \le -\dot{G}/G < 6.8 \times 10^{-11}$ yr$^{-1}$ at a $2\sigma$ level. He also noted that a positive $\dot{G}$ induces a spin-up counteracting the electromagnetic spin-down, which can provide another bound if an independent estimate of the pulsar magnetic field can be obtained. Goldman (1990), following Eardley (1975), used the scaling relations $N \propto G^{-3/2}$ and $M \propto G^{-5/2}$ to deduce that $2d \ln I/d \ln G = -5 + 3d \ln I/d \ln N$. He used the data from the pulsar PSR 0655+64 to deduce that
$$0 \leq -\dot{G}/G < 5.5 \times 10^{-11} \text{ yr}^{-1}. \qquad (313)$$

The analysis (Verbiest et al. 2008) of 10 years high precision timing data on the millisecond pulsar PSR J0437-4715 reached

$$|\dot{G}/G| < 2.3 \times 10^{-11} \text{ yr}^{-1}. \qquad (314)$$

It was also argued (Jofré et al. 2006; Reisenegger et al. 2009) that a variation of G would induce a departure of the neutron star matter from β-equilibrium, due to the changing hydrostatic equilibrium. This would force non-equilibrium β-processes to occur, which releases energy that is invested partly in neutrino emission and partly in heating the stellar interior. Eventually, the star arrives at a stationary state in which the temperature remains nearly constant, as the forcing through the change of G is balanced by the ongoing reactions. Comparing the surface temperature of the nearest millisecond pulsar, PSR J0437-4715, inferred from ultraviolet observations, two upper limits for variation were obtained, $|\dot{G}/G| < 2 \times 10^{-10} \text{ yr}^{-1}$, direct Urca reactions operating in the neutron star core are allowed, and $|\dot{G}/G| < 4 \times 10^{-12} \text{ yr}^{-1}$, considering only modified Urca reactions. This was extended in Krastev and Li (2007) in order to take into account the correlation between the surface temperatures and the radii of some old neutron stars to get $|\dot{G}/G| < 2.1 \times 10^{-11} \text{ yr}^{-1}$.

## 6.3 Stellar constraints

Early works, reviewed in Sect. IV.C of FVC03 (Uzan 2003b), studied the Solar evolution in presence of a time varying gravitational constant, concluding that under the Dirac hypothesis, the original nuclear resources of the Sun would have been burned by now. This results from the fact that an increase of the gravitational constant is equivalent to an increase of the star density (because of the Poisson equation).

The idea of using stellar evolution to constrain the possible value of G was originally proposed by Teller (1948). The luminosity of a main sequence star can be expressed as a function of Newton's gravitational constant and its mass by using homology relations (Teller 1948; Gamow 1967). In the particular case in which the opacity is dominated by free-free transitions, Gamow (1967) found that the luminosity of the star approximately scales as $L \propto G^{7.8}M^{5.5}$. In the case of the Sun, this would mean that for higher values of G, the burning of hydrogen will be more efficient and the star evolves more rapidly, therefore we need to increase the initial content of hydrogen to obtain the present observed Sun. In a numerical test of the previous expression, Del'Innocenti (1996) found that low-mass stars evolving from the Zero Age Main Sequence to the red giant branch satisfy $L \propto G^{5.6}M^{4.7}$, which agrees to within 10% of the numerical results, following the idea that Thomson scattering contributes significantly to the opacity inside such stars. Indeed, in the case of the opacity being dominated by pure Thomson scattering, the luminosity of the star is given by $L \propto G^4 M^3$. This shows how the evolution of the main sequence stars is sensitive to G.
The driving idea behind the stellar constraints is that a secular variation of $G$ would affect the hydrostatic equilibrium of the star and in particular its pressure profile. In the case of non-degenerate stars, the temperature, being the only control parameter, will adjust to compensate the modification of the intensity of the gravity. It will then affect the nuclear reaction rates, which are very sensitive to the temperature, and thus the nuclear time scales associated to the various processes. It follows that the main stage of the stellar evolution, and in particular the lifetimes of the various stars, will be modified. As we shall see, basically two types of methods have been used, the first in which on relates the variation of $G$ to some physical characteristics of a star (luminosity, effective temperature, radius), and a second in which only a statistical measurement of the change of $G$ can be inferred. Indeed, the first class of methods are more reliable and robust but is usually restricted to nearby stars. Note also that they usually require to have a precise distance determination of the star, which may depend on $G$.

Let us mention that Sahni and Shtanov (2014) suggested a time variation of $G$ as a possible explanation of the faint young sun problem (Gough 1981), that is to the fact that numerical models of the Sun could indicate its luminosity was 75% of its present value 4 Gyr ago which is not supported by geophysical and climatological data. One usually assumes an early greenhouse effect, but one can estimate that an increase $\Delta G/G \sim 0.02$ on a timescale of 4 Gyr offers a solution compatible with current constraints on $\dot{G}/G$.

### 6.3.1 Ages of globular clusters

These ideas were first applied with globular clusters. Their ages, determined for instance from the luminosity of the main-sequence turn-off, have to be compatible with the estimation of the age of the galaxy. This gave the constraint (Del'Innocenti 1996)

$$ \dot{G}/G = (-1.4 \pm 2.1) \times 10^{-11} \text{ yr}^{-1}. \qquad (315) $$

The effect of a possible time dependence of $G$ on luminosity has been studied in the case of globular cluster H-R diagrams but has not yielded any stronger constraints than those relying on celestial mechanics

### 6.3.2 Solar and stellar seismology

A side effect of the change of luminosity is a change in the depth of the convection zone. This induces a modification of the vibration modes of the star and particularly to the acoustic waves, i.e., p-modes (Demarque et al. 1994).

#### Helioseismology

These waves are observed for our star, the Sun, and helioseismology allows one to determine the sound speed in the core of the Sun and, together with an equation of state, the central densities and abundances of helium and hydrogen. Demarque et al. (1994) considered an ansatz in which $G \propto t^{-\beta}$ and showed that $|\beta| < 0.1$ over the
last $4.5 \times 10^9$ years, which corresponds to $|\dot{G}/G| < 2 \times 10^{-11} \text{ yr}^{-1}$. Guenther et al. (1995) also showed that g-modes could provide even much tighter constraints but these modes are up to now very difficult to observe. Nevertheless, they concluded, using the claim of detection by Hill and Gu (1990), that $|\dot{G}/G| < 4.5 \times 10^{-12} \text{ yr}^{-1}$. Guenther et al. (1998) then compared the p-mode spectra predicted by different theories with varying gravitational constant to the observed spectrum obtained by a network of six telescopes and deduced that

$$|\dot{G}/G| < 1.6 \times 10^{-12} \text{ yr}^{-1}. \qquad (316)$$

The standard Solar model depends on few parameters and G plays a important role since stellar evolution is dictated by the balance between gravitation and other interactions. Astronomical observations determine $GM_\odot$ with an accuracy better than $10^{-7}$ and a variation of G with $GM_\odot$ fixed induces a change of the pressure ($P = GM_\odot^2/R_\odot^3$) and density ($\rho = M_\odot/R_\odot^3$). The experimental uncertainties in G between different experiments have important implications for helioseismology. In particular the uncertainties for the standard Solar model lead to a range in the value of the sound speed in the nuclear region that is as much as 0.15% higher than the inverted helioseismic sound speed (Lopes and Silk 2001). While a lower value of G is preferred for the standard model, any definite prediction is masked by the uncertainties in the Solar models available in the literature. Ricci and Villante (2002) studied the effect of a variation of G on the density and pressure profile of the Sun and concluded that present data cannot constrain G better than $10^{-2}\%$. It was also shown (Lopes and Silk 2001) that the information provided by the neutrino experiments is quite significant because it constitutes an independent test of G complementary to helioseismology.

To finish, let us mention a recent claim on a possible evolution of G, that requires confirmation. Using 8640 days of low-l BiSON data, corrected for the Solar cycle variation, Bonanno and Fröhlich (2020) obtained

$$\dot{G}/G = (1.25 \pm 0.3) \times 10^{-13} \text{ yr}^{-1} \qquad (317)$$

including possible systematic effects such as uncertainties on the efficiency of the proton-proton fusion cross-section. This $4\sigma$ effect is argued to significantly outclass models with no secular variation of G.

## Asteroseismology

This method can be extended to other stars. The analysis of the stellar oscillation of the low mass Solar-like star on the main sequence KIC 7,970,740 for which one can determine dozens of oscillation modes (3 years of observation with the Kepler mission) allowed Bellinger and Christensen-Dalsgaard (2019) to conclude that

$$\dot{G}/G = (1.2 \pm 2.6) \times 10^{-12} \text{ yr}^{-1}. \qquad (318)$$
### 6.3.3 White dwarfs pulsation

The observation of the period of non-radial pulsations of white dwarfs allows one to set similar constraints. White dwarfs represent the final stage of the stellar evolution for stars with a mass smaller to about $10 M_\odot$. Their structure is supported against gravitational collapse by the pressure of degenerate electrons. It was discovered that some white dwarfs are variable stars and in fact non-radial pulsator. This opens the way to use seismological techniques to investigate their internal properties. In particular, their non-radial oscillations are mostly determined by the Brunt–Väisälä frequency

$$N^2 = g \frac{d \ln P^{1/\gamma_1} / \rho}{dr}$$

where g is the gravitational acceleration, $\Gamma_1$ the first adiabatic exponent and P and $\rho$ the pressure and density (see, e.g., Ketchum and Adams 2008 for a white dwarf model taking into account a varying G). A variation of G induces a modification of the degree of degeneracy of the white dwarf, hence on the frequency N as well as the cooling rate of the star, even though this is thought to be negligible at the luminosities where white dwarfs are pulsationally unstable (Biesiada and Malec 2004). Althaus et al. (2011) stressed that the impact of a varying G is more notorious in the case of more massive white dwarfs so that the very accurate white dwarf cooling age derived for the old and metal-rich open cluster NGC 6791 makes it an excellent candidate to constrain $\dot{G}/G$.

Using the observation of G117-B15A that has been monitored during 20 years, Benvenuto et al. (2004) was concluded that $-2.5 \times 10^{-10} \text{yr}^{-1} < \dot{G}/G < 4.0 \times 10^{-11} \text{yr}^{-1}$ at a $2\sigma$-level. The same observations were reanalyzed in Biesiada and Malec (2004) to obtain

$$|\dot{G}/G| < 4.1 \times 10^{-11} \text{yr}^{-1}. \qquad (319)$$

Corsico et al. (2001) pointed out that among white dwarfs there is a specific class of stars, known as ZZ-Ceti objects, which have a hydrogen-rich envelope and show periodic variations in their light curves. From the study secularly varying G on the pulsational properties of such variable white dwarfs G117-B15A and R548, they concluded

$$\dot{G}/G = (-0.9 \pm 0.9) \times 10^{-10} \text{yr}^{-1} \qquad (320)$$

$$\dot{G}/G = (-0.65 \pm 0.65) \times 10^{-10} \text{yr}^{-1} \qquad (321)$$

### 6.3.4 Late stages of stellar evolution and supernovae

A variation of G can influence the white dwarf cooling and the light curves of Type Ia supernovae.
## White dwarf cooling

García-Berro et al. (1995) considered the effect of a variation of the gravitational constant on the cooling of white dwarfs and on their luminosity function. As first pointed out by Vila (1976), the energy of white dwarfs, when they are cool enough, is entirely of gravitational and thermal origins so that a variation of G will induce a modification of their energy balance and thus of their luminosity. Restricting to cold white dwarfs with luminosity smaller than ten Solar luminosity, their luminosity can be related to the star binding energy B and gravitational energy, $E_{grav}$, as

$$L = -\frac{dB}{dt} + \frac{\dot{G}}{G}E_{grav}, \qquad (322)$$

which simply results from the hydrostatic equilibrium. Again, the variation of the gravitational constant intervenes via the Poisson equation and the gravitational potential. The cooling process is accelerated if $\dot{G}/G<0$, which then induces a shift in the position of the cut-off in the luminosity function. García-Berro et al. (1995) concluded that $0 \le -\dot{G}/G < (1 \pm 1) \times 10^{-11} \text{ yr}^{-1}$. The result depends on the details of the cooling theory, on whether the C/O white dwarfs are stratified or not and on hypothesis on the age of the galactic disk. For instance, with no stratification of the C/O binary mixture, one would require $\dot{G}/G = -(2.5 \pm 0.5) \times 10^{-11} \text{ yr}^{-1}$ if the Solar neighborhood has a value of 8 Gyr (i.e., one would require a variation of G to explain the data). In the case of the standard hypothesis of an age of 11 Gyr, one obtains $0 \le -\dot{G}/G < 3 \times 10^{-11} \text{ yr}^{-1}$. More recently, from the effect of a secularly varying G on the main sequence ages and on white dwarf cooling ages, García-Berro et al. (2011) concluded

$$\dot{G}/G = (-0.9 \pm 0.9) \times 10^{-12} \text{ yr}^{-1} \qquad (323)$$

using the white dwarf luminosity function and the distance of the open Galactic cluster NGC 6791.

## Supernovae

The late stages of stellar evolution are governed by the Chandrasekhar mass ($\hbar c/G)^{3/2}m_n^{-2}$ mainly determined by the balance between the Fermi pressure of a degenerate electron gas and gravity.

Simple analytical models of the light curves of Type Ia supernovae predict that the peak of luminosity is proportional to the mass of nickel synthesized. In a good approximation, it is a fixed fraction of the Chandrasekhar mass. In models with a varying G, this induces a modification of the luminosity distance-redshift relation (García-Berro et al. 2006; Gaztañaga et al. 2002; Riazuelo and Uzan 2002). Mould and Uddin (2014) concluded from the analysis of the Hubble diagram of SNIa (Suzuki et al. 2012) that

$$-3 \times 10^{-11} \text{ yr}^{-1} < \dot{G}/G < 7.3 \times 10^{-11} \text{ yr}^{-1} \qquad (324)$$

over the past 9 Gyr. Note that this constraint is degenerate with the cosmological parameters. In particular, the Hubble diagram is sensitive to the whole history of
$G(z)$ between the highest redshift observed and today so that one needs to rely on a better defined model, such as, e.g., scalar-tensor theory (Riazuelo and Uzan 2002). The effect of the Fermi constant was also considered in Ferrero and Altschul 2010.

In the case of Type II supernovae, the Chandrasekhar mass also governs the late evolutionary stages of massive stars, including the formation of neutron stars. Assuming that the mean neutron star mass is given by the Chandrasekhar mass, one expects that $\dot{G}/G = -2\dot{M}_{NS}/3M_{NS}$. Thorsett (1996) used the observations of five neutron star binaries for which five Keplerian parameters can be determined (the binary period $P_b$, the projection of the orbital semi-major axis $a_1 \sin i$, the eccentricity $e$, the time and longitude of the periastron $T_0$ and $\omega$) as well as the relativistic advance of the angle of the periastron $\dot{\omega}$. Assuming that the neutron star masses vary slowly as $M_{NS} = M_{NS}^{(0)} - \dot{M}_{NS}t_{NS}$—so that their age is determined by the rate at which $P_b$ is increasing, i.e., $t_{NS} \simeq 2P_b/\dot{P}_b$—and that the mass follows a normal distribution, Thorsett (1996) deduced that, at $2\sigma$,

$$ \dot{G}/G = (-0.6 \pm 4.2) \times 10^{-12} \text{yr}^{-1}. \qquad (325) $$

### 6.3.5 New developments from gravitational-wave astronomy

Zhao et al. (2018) pointed out that the luminosity distance can be independently determined from GW standard siren caused by the coalescence of binary neutron stars, which could leas to a constraint of the order of 1.5% on $\Delta G/G$ for redshifts up to 1.3. Note however that the effect of the variation of $G$ on the standard siren was not taken into account in this analysis. Indeed, a variation of $G$ induces a modification of the binary’s binding energy so that it should affect the gravitational wave luminosity, hence leading to corrections in the chirping frequency (Yunes et al. 2010). It was estimated that a LISA observation of an equal-mass inspiral event with total redshifted mass of $10^5 M_\odot$ for three years should be able to measure $\dot{G}/G$ at the time of merger to better than $10^{-11}/\text{yr}$. This method paves the way to constructing constraints in a large band of redshifts as well as in different directions in the sky, which would be an invaluable constraint for many models.

The development of GW astronomy opens new possibilities. Barbieri et al. (2023) considered the possibility for spaceborne GW detectors at milli-Hz and deci-Hz frequencies, such as LISA or DECIGO, to measure the first and second time derivatives of the GW frequency and how this can set bounds of the time evolution of $G$. They reached the conclusion that the most favorable sources among a simulated population of LISA galactic binaries could yield $\dot{G}/G \sim 10^{-6}/\text{yr}$ and that chirping stellar-mass compact binaries detected by DECIGO-like detectors at Mpc scales can lead to $\dot{G}/G \sim 10^{-11}/\text{yr}$. From the observation of the binary neutron stars GW170817, Vijaykumar et al. (2021) got $-1 \le \Delta G/G \le 8$, corresponding to $-7 \times 10^{-9}/\text{yr}^{-1} \le \dot{G}/G \le 5 \times 10^{-8}/\text{yr}^{-1}$ assuming a monotonic variation. See also Amendola et al. (2018), Belgacem et al. (2019), Sun et al. (2024).

More speculative is the idea (Bambi and Drago 2008) that a variation of $G$ can lead a neutron stars to enter into the region where strange or hybrid stars are the true
ground state. This would be associated with gamma-ray bursts that are claimed to be able to reach the level of $10^{-17}/\text{yr on } \dot{G}$.

Feldman et al. (2016) proposed an experiment in deep space employing the classic gravity train mechanism. With a setup with three bodies (a larger layered solid sphere with a cylindrical hole through its center, a much smaller retroreflector which will undergo harmonic motion within the hole and a host spacecraft with laser ranging capabilities to measure round trip light-times to the retroreflector), measurements of the period of oscillation of the retroreflector in terms of host spacecraft clock time using existing technology could give determinations of $G$ at $6 \times 10^{-8}$, i.e., nearly three orders of magnitude more accurate than current measurements here on Earth.

We refer to Yunes et al. (2025) for a recent review on the possibilities opened by gravitational-wave astronomy.

## 6.4 Cosmological constraints

Cosmological observations are more difficult to use in order to set constraints on the time variation of $G$. In particular, they require to have some ideas about the whole history of $G$ as a function of time but also, as the variation of $G$ reflects an extension of General Relativity, it requires to modify all equations describing the evolutions of the universe and of the large-scale structure in a consistent way. We refer to Uzan (2010a, 2007, 2010c) for an early but detailed discussion of the use of cosmological data to test deviations from General Relativity.

### 6.4.1 Cosmic microwave background

A time-dependent gravitational constant will have mainly three effects on the CMB angular power spectrum (see Riazuelo and Uzan 2002 for discussions in the framework of scalar-tensor gravity in which $G$ is considered as a field):

1. The variation of $G$ modifies the Friedmann equation and therefore the age of the Universe (and, hence, the sound horizon). For instance, if $G$ is larger at earlier time, the age of the Universe is smaller at recombination, so that the peak structure is shifted towards higher angular scales.

2. The amplitude of the Silk damping is modified. At small scales, viscosity and heat conduction in the photon-baryon fluid produce a damping of the photon perturbations. The damping scale is determined by the photon diffusion length at recombination, and therefore depends on the size of the horizon at this epoch, and hence, depends on any variation of the Newton constant throughout the history of the Universe.

3. The thickness of the last scattering surface is modified. In the same vein, the duration of recombination is modified by a variation of the Newton constant as the expansion rate is different. It is well known that CMB anisotropies are affected on small scales because the last scattering “surface” has a finite thickness. The net effect is to introduce an extra, roughly exponential, damping term, with the cutoff length being determined by the thickness of the last
scattering surface. When translating redshift into time (or length), one has to use the Friedmann equations, which are affected by a variation of the Newton constant. The relevant quantity to consider is the visibility function g. In the limit of an infinitely thin last scattering surface, τ goes from ∞ to 0 at recombination epoch. For standard cosmology, it drops from a large value to a much smaller one, and hence, the visibility function still exhibits a peak, but it is much broader.

In full generality, the variation of G on the CMB temperature anisotropies depends on many factors: (1) modification of the background equations and the evolution of the universe, (2) modification of the perturbation equations, (3) whether the scalar field inducing the time variation of G is negligible or not compared to the other matter components, (4) on the time profile of G that has to be determined to be consistent with the other equations of evolution. This explains why it is difficult to state a definitive and model-independent constraint. For instance, in the case of scalar-tensor theories, one has two arbitrary functions that dictate the variation of G. As can be seen, e.g., from Riazuelo and Uzan (2002), Nagata et al. (2004), the profiles and effects on the CMB can be very different and difficult to compare since they are entangled with the cosmological parameters.

In the case of Brans–Dicke theory, one just has a single constant parameter $\omega_{BD}$ characterizing the deviation from General Relativity and the time variation of G; see Eq. (42). Thus, it is easier to compare the different constraints. Chen and Kamionkowski (1999) showed that CMB experiments such as WMAP will be able to constrain these theories for $\omega_{BD} < 100$ if all parameters are to be determined by the same CMB experiment, $\omega_{BD} < 500$ if all parameters are fixed but the CMB normalization and $\omega_{BD} < 800$ if one uses the polarization. For the Planck mission these numbers are respectively, 800, 2500 and 3200. Acquaviva et al. (2005) concluded from the analysis of WMAP, ACBAR, VSA and CBI, and galaxy power spectrum data from 2dF, that $\omega_{BD} > 120$, in agreement with the former analysis of Nagata et al. (2004). Wu and Chen (2010) indicated that The ‘WMAP-5yr data’ and the ‘all CMB data’ both favor a slightly non-zero (positive) $\dot{G}/G$ but with the addition of the SDSS power spectrum data, the best-fit value is back to zero, concluding that $-0.083 < \Delta G/G < 0.095$ between recombination and today, which corresponds to

$$-1.75 \times 10^{-12} \text{yr}^{-1} < \dot{G}/G < 1.05 \times 10^{-12} \text{yr}^{-1}. \qquad (326)$$

Assuming Brans–Dicke theory, Li et al. (2013) concluded from Planck that

$$\dot{G}/G = (-1.315 \pm 2.375) \times 10^{-13} \text{yr}^{-1}. \qquad (327)$$

A more recent analysis based on Planck-2018 CMB temperature, polarization and lensing, with a compilation of BAO measurements from the release DR12 of the BOSS Survey has been performed by Ballardini et al. (2022) in the framework of a Brans–Dicke model.

From a more phenomenological prospect, some works modeled the variation of G with time in a purely ad-hoc way, for instance (Chan and Chu 2007) by assuming
a linear evolution with time or a step function. The latter hypothesis has however gain some visibility since it is a candidate to resolve the $H_0$ tension; see e.g., Ballesteros et al. (2020), Braglia et al. (2020), Marra and Perivolaropoulos (2021), Braglia et al. (2021), Sakr and Sapone (2022), Alestas et al. (2022), Ruchika et al. (2024).

### 6.4.2 Big Bang nucleosynthesis

As detailed in Sect. 5.14.1, the gravitational constant affects the freeze-out temperature $T_f$. A larger value of $G$ corresponds to a higher expansion rate. This rate is determined by the combination $G\rho$ and in the standard case the Friedmann equations imply that $G\rho t^2$ is constant. The density $\rho$ is determined by the number $N_*$ of relativistic particles at the time of nucleosynthesis so that nucleosynthesis allows to put a bound on the number of neutrinos $N_\nu$. Equivalently, assuming the number of neutrinos to be three, leads to the conclusion that $G$ has not varied from more than 20% since nucleosynthesis. But, allowing for a change both in $G$ and $N_\nu$ allows one for a wider range of variation. Contrary to the fine structure constant the role of $G$ is less involved.

It can be described, in its simplest but still useful form, by introducing a speed-up factor, $\xi = H/H_{GR}$, that arises from the modification of the value of the gravitational constant during BBN. Other approaches consider the full dynamics of the problem but restricted themselves to the particular class of Jordan-Fierz-Brans-Dicke theories (Barrow 1978; Yang et al. 1979; Rothman and Matzner 1982; Arai et al. 1987; Accetta et al. 1990; Damour and Gundlach 1991; Casas et al. 1992; Clifton et al. 2005). Casas et al. (1992) concluded from the study of helium and deuterium that $\omega_{BD} > 380$ when $N_\nu = 3$ and $\omega_{BD} > 50$ when $N_\nu = 2$. Then, other analysis considered a massless dilaton with a quadratic coupling (Coc et al. 2006, 2009b; Damour and Pichon 1999; Santiago et al. 1997) or a general massless dilaton (Serna and Alimi 1996). It should be noted that a combined analysis of BBN and CMB data was investigated in Copi et al. (2004), Kneller and Steigman (2003). The former considered $G$ constant during BBN while the latter focused on a non-minimally quadratic coupling and a runaway potential. It was concluded that from the BBN in conjunction with WMAP determination of $\eta$ set that $\Delta G/G$ has to be smaller than 20%. However, we stress that the dynamics of the field can modify CMB results (see previous Sect. 6.4.1) so that one needs to be careful while inferring $\Omega_b$ from WMAP unless the scalar-tensor theory has converged close to General Relativity at the time of decoupling.

In early studies, Barrow (1978) assumed that $G \propto t^{-n}$ and obtained from the helium abundance that $-5.9 \times 10^{-3} < n < 7 \times 10^{-3}$, which implies $|\dot{G}/G| < (2 \pm 9.3) h \times 10^{-12} \text{yr}^{-1}$, assuming a flat universe. This corresponds in terms of the Brans-Dicke parameter to $\omega_{BD} > 25$. Yang et al. (1979) included the deuterium and lithium to improve the constraint to $n < 5 \times 10^{-3}$, which corresponds to $\omega_{BD} > 50$. It was further improved by Rothman and Matzner (1982) to $|n| < 3 \times 10^{-3}$ implying $|\dot{G}/G| < 1.7 \times 10^{-13} \text{yr}^{-1}$. Accetta et al. (1990) studied the dependence of the abundances of D, $^3$He, $^4$He and $^7$Li upon the variation of $G$ and
concluded that $-0.3 < \Delta G/G < 0.4$, which roughly corresponds to $|\dot{G}/G| < 9 \times 10^{-13} \text{yr}^{-1}$. All these investigations assumed that the other constants are kept fixed and that physics is unchanged. Kolb et al. (1986) assumed a correlated variation of $G$, $\alpha_{\text{EM}}$ and $G_F$ and got a bound on the variation of the radius of the extra dimensions.

Although the uncertainty in the helium-4 abundance has been argued to be significantly larger than what was assumed in the past (Olive and Skillman 2004), interesting bounds can still be derived (Cyburt et al. 2005). In particular translating the bound on extra relativistic degress of freedom ($-0.6 < \delta N_v < 0.82$) to a constraint on the speed-up factor ($0.949 < \xi < 1.062$), it was concluded (Cyburt et al. 2005), since $\Delta G/G = \xi^2 - 1 = 7\delta N_v/43$, that

$$-0.10 < \frac{\Delta G}{G} < 0.13. \qquad (328)$$

With similar hypothesis, this was improved to (Yeh et al. 2022)

$$-0.040 < \frac{\Delta G}{G} < 0.006. \qquad (329)$$

Using the PRIMAT code (Pitrou et al. 2018) and assuming only a variation of $G$ (i.e., all cosmological parameters are fixed), Alvey et al. (2020) concluded that

$$\Delta G/G = -0.01^{+0.06}_{-0.05}, \qquad (330)$$

at $2\sigma$. Giri and Scherrer (2024) considered in which the coupling function in Jordan frame behaves as $F^{-1} \propto 1 + A \cos(\omega t + \phi)$ to conclude that if $\rho_{\phi}$ is negligible during the whole cosmic history, BBN imposes that $A\omega < 4.4 \times 10^{-4} \text{s}^{-1}$. Note that $G_{\text{eff}}$ oscillating does not imply that the Newton constant measured in a Cavendish experiment (52) is oscillating. For instance, it is clear that with $A = \cos \phi$, $G_{\text{cav}} = G_*$ while $G_{\text{eff}} = G_* \cos \phi$.

The relation between the speed-up factor, or an extra number of relativistic degrees of freedom, with a variation of $G$ is only approximate since it assumes that the variation of $G$ affects only the Friedmann equation by a renormalization of $G$. This is indeed accurate only when the scalar field is slow-rolling. For instance (Coc et al. 2006), the speed-up factor is given (with the notations of Sect. 3.1.1) by

$$\xi = \frac{A(\varphi_*)}{A_0} \frac{1 + \alpha(\varphi_*) \varphi'_*}{\sqrt{1 - \varphi_*^{2\prime}/3}} \frac{1}{\sqrt{1 + \alpha_0^2}}$$

so that

$$\xi^2 = \frac{G}{G_0} \frac{(1 + \alpha(\varphi_*))\varphi_*^{\prime 2}}{(1 + \alpha^2)(1 - \varphi_*^{2\prime}/3)}, \qquad (331)$$

so that $\Delta G/G_0 = \xi^2 - 1$ only if $\alpha \ll 1$ (small deviation from General Relativity) and $\varphi_*^{\prime} \ll 1$ (slow rolling dilaton). The BBN in scalar-tensor theories was investigated (Coc et al. 2006; Damour and Pichon 1999) in the case of a two-parameter family involving a non-linear scalar field-matter coupling function. Both concluded
that even in the cases where before BBN the scalar-tensor theory was far from General Relativity, BBN enables to set quite tight constraints on the deviations from General Relativity today. In particular, neglecting the cosmological constant, BBN imposes $\alpha_0^2 < 10^{-6.5} \beta^{-1} (\Omega_{\text{mat}}h^2/0.15)^{-3/2}$ when $\beta > 0.5$—with the definitions introduced below Eq. (50).

### 6.4.3 New proposals: strong lensing

Lensing is a generic prediction of General Relativity. In the thin lens approximation, the lensing equation relates the angular positions of the image $\theta$ and of the source $\theta_s$ by

$$ \theta = \theta_s + \frac{4GM}{\theta} \frac{D_{LS}}{D_{OL}D_{OS}} \qquad (332) $$

where $D_i$ are angular distances between the source S, the lens L and the observer O. This equation is used to describe the position of multiple images and times delays in strong lensing; see e.g., Peter and Uzan (2009).

Then, in a Freidmann-Lemaître spacetime, independently of the gravity theory, the kinematics of the cosmic expansion implies that redshifts are evolving with cosmic time (Sandage 1962; Mc Vittie 1962) as

$$ \frac{dz}{dt} = H_0(1+z) - H(z). \qquad (333) $$

The expansion of the universe also induces an aberration of the directions of observations (see Marcori et al. 2018). High precision astrometry now enables us to measure the time drift of astrophysical observables in real time, hence providing new ways to probe different cosmological models.

Piattella and Giani (2017) remarked that a similar effects the positions and time delays of a strong lensing system. Giani and Frion (2020) proposed to extend this very general result to the situation in which $G$ is time dependent. From Eq. (332), they concluded that the positions of the multiple images $\theta$ and time delays $\Delta$ for a lens at redshift $z_L$ evolve as

$$ 2 \frac{\dot{\theta}}{\theta} = H_0 - \frac{H(z_L)}{1+z_L} + \frac{\dot{G}}{G}, \qquad \frac{\Delta}{\Delta} = 2 \left( H_0 - \frac{H(z_L)}{1+z_L} + \frac{\dot{G}}{G} \right) \frac{\theta_E^2}{(\theta_+ - \theta_-)^2} \qquad (334) $$

where $\theta_E$ is the Einstein radius and $\theta_\pm$ the angular positions of the two images. As an example, the quasar such as QSO0957+561 at $z = 1.41$ lensed by a cluster at $z_L = 0.36$ enjoys 2 images separated by $6.1''$ with a time delay of $417 \pm 3$ days. It follows that $\dot{\theta}_E \sim 10^{-10}$ arcsec/yr and $\Delta \sim 10^{-3}$ s/yr. Covone and Sereno (2022) argued that this lensing drifts may be accessible by Gaia or VLBI in the future. On that basis and using the light curves of the lens quasar system DES J0408-5354, Giani and Frion (2020) concluded that a constraints of $10^{-1} - 10^{-2}$yr on $\dot{G}/G$ could be reached. Applying this technique to the apparent drift of the angular size of the shadow of a black hole Frion et al. (2021) argued that the amplitude of this