effect is of order $10^{-16}$ day$^{-1}$ for M87* so that the observation by the Event Horizon Telescope set a constraint on either the maximum accretion rate or on a time variation of $|\dot{G}/G| < 10^{-3}$ yr$^{-1}$.

This is indeed not competitive with existing bounds and one also needs to be aware that this computation assumes comoving observer, lens, and sources, and lens with static mass distribution, which is an ideal situation. Effects of transverse proper motions, mass accretion, growth of density perturbations can modify a lens system over time and shall be estimated. Besides, the use of the standard lensing equation implicitly assumes that the geodesic equation is not modified when $G$ becomes dynamical. This is the case in scalar-tensor theories since the Maxwell Lagrangian is conformally invariant in 4 dimensions. But one shall keep in mind these theoretical limitations.

### 6.4.4 New proposals: gravitational waves

Among the recent developments in physics and astrophysics, the detection of gravitational waves (Abbott et al. 2016) has opened a whole new field of investigation. Some applications for fundamental constants are starting to be explored.

In General relativity, structurally the speed of gravitational waves is equal to the speed of light in vacuum. Gravitational waves detection offers the possibility to measure GW speed $c_T$. Setting $c_T^2 = (1 + \alpha_T)c^2$ the lack of gravi-Cerenkov effect from cosmic rays sets $\alpha_T > -10^{-15}$ (Moore and Nelson 2001). Then, a upper bound from the travel time of GW detected by LIGO imply $\alpha_T < 0.42$ (Blas et al. 2016; Cornish et al. 2017). Thanks to the electromagnetic counterpart Baker et al. (2017) concluded $|\alpha_T| \lesssim 10^{-15}$ and Mastrogiovanni et al. (2020) $|\alpha_T| \lesssim 10^{-17}$.

Then GW observations allow for the measurement of the luminosity distance of standard sirens, that can be compared to the one derived from electromagnetic measurement, such as SNIa. This has opened a new and widely studied field to constrain theories of gravitation beyond General Relativity. An application can be done thanks to the determination of the gravitational luminosity distance (Abbott et al. 2017a) and the electromagnetic luminosity distance (Coughlin et al. 2020) for the system GW170817 (Abbott et al. 2017b); see also Colaço et al. (2024).

Calmet (2020) pointed out that LISA could probe the cosmological evolution of the Higgs vev $\nu$ through the GW produced during the electroweak phase transition. Indeed that would require the Higgs mass to be below 72 GeV at temperature above 100 GeV, much below the actual mass of 125 GeV. Nevertheless any sign of GW at frequencies merger than $10^{-5}$ Hz could be a sign of a smaller Higgs self-coupling constant in the past.

Then, Vijaykumar et al. (2021) argues that observations of gravitational waves emitted during the merging of binary neutron stars can be a window on the time variation of $G$. This relies on the determination of the masses of the neutron stars and on them being consistent with the theoretically allowed range since a varying-$G$ would induce a shift of all the masses by $(1 + \Delta G/G)^{-3/2}$. From the data of
GW170817, they concluded that $-1 \lesssim \Delta G/G \lesssim 8$. The idea can also be applied to black hole-neutron star mergers.

## 7 Models with varying constants

The models that can be constructed are numerous and cannot all be reviewed. Several frameworks have however become widely used to interpret experimental and observational data. We shall thus discuss (1) the string dilaton model in Sect. 7.1, (2) Damour–Donoghue model in Sect. 7.2 and (3) the Bekenstein framework in Sect. 7.3. To finish, we shall reconsider in Sect. 7.4 the case of a scalar field coupled to electromagnetism, already introduced in Sect. 3.1.2 in the discussion on dark energy, and even though it is formally a subcase of the three other models. These studies are of importance to investigate how well new data can improve constraints on a set of parameters. They also allow one to evaluate to which extent some conclusions are dependent on the model. They also allow one to connect different constraints and compare the efficiency of each data set constraints.

### 7.1 String dilaton and runaway dilaton models

#### Model definition

Damour and Polyakov (1994a, 1994b) considered the effective action for the massless modes of string theory taking into account the full string loop expansion. They argued it should be of the form

$$ S = \int d^4\mathbf{x} \sqrt{-\hat{g}} \left[ m_s^2 \left\{ B_g(\Phi)\hat{R} + 4B_\phi(\Phi) \left[ \hat{\Box}\Phi - (\nabla\hat{\Phi})^2 \right] \right\} - B_F(\Phi) \frac{k}{4} \hat{F}^2 - B_\psi(\Phi) \bar{\psi} \mathcal{D}_\nu \hat{\psi} + \dots \right] \quad (335) $$

in the string frame, $m_s$ being the string mass scale. The functions $B_i$ are not known but can be expanded in the limit $\Phi \to -\infty$ as

$$ B_i(\Phi) = e^{-2\Phi} + c_0^{(i)} + c_1^{(i)} e^{2\Phi} + c_2^{(i)} e^{4\Phi} + \dots \quad (336) $$

where the first term is the tree level term. It follows that these functions can exhibit a local maximum. After a conformal transformation ($g_{\mu\nu} = CB_g\hat{g}_{\mu\nu}$, $\psi = (CB_g)^{-3/4}B_\psi^{1/2}\hat{\psi}$), the action in the Einstein frame takes the form

$$ S = \int \frac{d^4 \mathbf{x}}{16\pi G} \sqrt{-g} \left[ R - 2(\nabla\varphi)^2 - \frac{k}{4} B_F(\varphi) F^2 - \bar{\psi} \mathcal{D}_\nu \psi + \dots \right] \quad (337) $$

where the field $\varphi$ is defined as
$$ \varphi \equiv \int \left[ \frac{3}{4} \left( \frac{B_g'}{B_g} \right)^2 + 2 \frac{B_\Phi'}{B_\Phi} + 2 \frac{B_\Phi''}{B_g} \right] d\Phi. $$

It follows that the Yang–Mills couplings all behave as $g_{YM}^{-2} = k B_F(\varphi)$. This implies that the QCD mass scale is given by

$$ \Lambda_{\text{QCD}} \sim m_s (CB_g)^{-1/2} e^{-8\pi^2 k B_F / b} \quad (338) $$

where b depends on the matter content. Hence, the mass of any hadron, proportional to $\Lambda_{\text{QCD}}$ in first approximation, depends on the dilaton, $m_A(B_g, B_F, \dots)$.

## Phenomenology

If, as allowed by the ansatz (336), $m_A(\varphi)$ has a minimum $\varphi_m$ then the scalar field will be driven toward this minimum during the cosmological evolution. However, if the various coupling functions have different minima then the minima of $m_A(\varphi)$ will depend on the particle A. To avoid violation of the equivalence principle at an unacceptable level, it is necessary to assume that all the minima coincide in $\varphi = \varphi_m$, which can be implemented by setting $B_i = B$. This was shown to be realized when assuming that $\varphi_m$ is a special point in field space, for instance it could be associated to the fixed point of a $Z_2$ symmetry of the T- or S-duality (Damour and Lilley 2008).

Expanding $\ln B$ around its maximum $\varphi_m$ as $\ln B \propto -\kappa(\varphi - \varphi_m)^2/2$, Damour and Polyakov (1994a, 1994b) constrained the set of parameters $(\kappa, \Delta\varphi_0)$ with

$$ \Delta\varphi_0 \equiv \varphi_0 - \varphi_m $$

using the different observational bounds. $\kappa$ is the only model parameter while $\Delta\varphi_0$ is an environmental parameter, the value of which depends on the cosmological evolution. The model can be extended to include a mass term, $V = m_\varphi(\varphi - \varphi_m)^2/2$, with same minimum has the coupling function B to ensure the attraction mechanism is not spoiled.

This model allows one to address the unsolved problem of the dilaton stabilization, to study all the experimental bounds together and to relate them in a quantitative manner (e.g., by deriving a link between equivalence-principle violations and time-variation of $\alpha_{EM}$). In particular, the Klein–Gordon equation (61) together with the Friedmann equations (453) allows one to determine $\varphi(z)$ and then $\alpha_{EM}(z)$, $\mu(z)$, etc. so that local constraints can be consistently interpreted together with astrophysical and cosmological data. The Klein–Gordon equation involves the effective potential (63) with $\beta_i$ defined in Eq. (59), hence offering a screening mechanisms as discussed in Sect. 3.2.

## Lowest order composition-independent effects

An important feature of this model lies in the fact that at lowest order the masses of all nuclei are proportional to $\Lambda_{\text{QCD}}$ so that at this level of approximation, the coupling is universal and the theory reduces to a scalar-tensor theory, as described
in Sect. 3.1.1. Hence, there will be no violation UFF and the deviation from General Relativity are characterized by the PPN parameters

$$ \gamma^{\text{PPN}} - 1 \approx -2\alpha_A^2 = -2\beta_s^2\kappa^2\Delta\phi_0^2, \qquad \beta^{\text{PPN}} - 1 \approx \frac{1}{2}\alpha_A^2 \frac{d\alpha_A}{d\phi} = \frac{1}{2}\beta_s^3\kappa^3\Delta\phi_0^2 $$

with

$$ \alpha_A = \frac{\partial \ln \Lambda_{\text{QCD}}(\phi)}{\partial \phi} = - \left[ \ln \frac{m_s}{m_A} + \frac{1}{2} \right] \frac{d \ln B}{d \phi} \equiv -\beta_s \frac{d \ln B}{d \phi} = \beta_s \kappa \Delta \phi_0 \quad (339) $$

with $\beta_s \sim 40$ (Damour and Polyakov 1994b). The variation of the gravitational constant is, from the general expression (55), simply

$$ \frac{\dot{G}}{G} = 2\alpha_A \dot{\phi}_0 = -2 \left[ \ln \frac{m_s}{m_A} + \frac{1}{2} \right] \frac{d \ln B}{d \phi} \dot{\phi}_0. $$

The value of $\dot{\phi}_0 = H_0\phi'_0$ is obtained from the Klein-Gordon equation (56) and is typically given by $\phi'_0 = -Z\beta_s\kappa H_0\Delta\phi_0$ where Z is a number that depends on the equation of state of the fluid dominating the matter content of the universe in the last e-fold and the cosmological parameters so that

$$ \left. \frac{\dot{G}}{G} \right|_{0} = 2\alpha_{A} \dot{\phi}_{0} = -2ZH_{0}\beta_{s}^{2}\kappa^{2}\Delta\phi_{0}^{2}. \qquad (340) $$

The factor Z is model-dependent. Another way to estimate $\dot{\phi}_0$ is to use the Friedmann equations, which imply that $\dot{\phi}_0 = H_0\sqrt{1+q_0-\frac{3}{3}\Omega_{m0}}$ where q is the deceleration parameter defined in Eq. (452).

### *Composition-dependent effects*

When one considers the quark masses and binding energies, various composition-dependent effects appear. First, the fine-structure constant scales as $B^{-1}$ so that

$$ \left. \frac{\dot{\alpha}}{\alpha} \right|_{0} = \kappa \Delta \phi_{0} \dot{\phi}_{0} = -ZH_{0} \beta_{s} \kappa^{2} \Delta \phi_{0}^{2}. \qquad (341) $$

Then, as already discussed in Sect. 3, one expects a violation of UFF since

$$ m_A(\phi) = N \Lambda_{\text{QCD}}(\phi) \left[ 1 + \sum_q \epsilon_A^q \frac{m_q}{\Lambda_{\text{QCD}}} + \epsilon_A^{\text{EM}} \alpha_{\text{EM}} \right]. \qquad (342) $$

Using an expansion of the form (30), it was concluded that

$$ \eta_{AB} = \kappa^2 \Delta \phi_0^2 \left[ C_B \Delta \left( \frac{B}{M} \right) + C_D \Delta \left( \frac{D}{M} \right) + C_E \Delta \left( \frac{E}{M} \right) \right] \qquad (343) $$

with $B = N + Z$, $D = N - Z$ and $E = Z(Z - 1)/(N + Z)^{1/3}$ and where the value of the parameters $C_i$ are model-dependent.
## Generic conclusions

It follows from this model that:

* The PPN parameters, the time variation of $\alpha$ and $G$ today and the violation of the university of free-fall all scale as $\Delta\phi_0^2$.

* The field is driven toward $\phi_m$ during the cosmological evolution, a point at which the scalar field decouples from the matter field. The mechanism is usually called the least coupling principle; see Sect. 3.2 for the general discussion on screening mechanisms.

* Once the dynamics for the scalar field is solved, $\Delta\phi_0$ can be related to $\Delta\phi_i$ at the end of inflation. Interestingly, this quantity can be expressed in terms of amplitude of the density contrast at the end of inflation, that is to the energy scale of inflation.

* The numerical estimations (Damour and Polyakov 1994b) indicate that $\eta_{U,H} \sim -5.4 \times 10^{-5}(\gamma^{\text{PPN}} - 1)$ showing that in such a class of models, the constraint on $\eta$, once reevaluated to take into account the MICROSCOPE bound (11), $\eta \sim 10^{-15}$ implies $1 - \gamma^{\text{PPN}} \sim 2 \times 10^{-11}$, hence $\alpha_A^2 \lesssim 10^{-11}$, which is a better constraint that the one obtained directly.

* The model was compared to Oklo, QSO and CMB data by Landau et al. (2006a) to conclude that $|\Delta\phi_0| < 3.4\kappa 10^{-6}$.

## Runaway dilaton model

Following Gasperini et al. (2002), the light dilaton model was extended (Damour et al. 2002b) to the case where the coupling functions have a smooth finite limit for infinite value of the bare string coupling, so that

$$B_i = C_i + \mathcal{O}(e^{-\phi}).$$

Hence, the dilaton runs away toward its attractor at infinity during a stage of inflation and then during the matter dominated era. The late time dynamics of the scalar field is similar as in quintessence models, so that it can also explain the late time acceleration of the cosmic expansion. The amplitude of residual dilaton interaction is related to the amplitude of the primordial density fluctuations and it induces a variation of the fundamental constants, provided it couples to dark matter or dark energy. It is concluded that, in this framework, the largest allowed variation of $\alpha_{\text{EM}}$ is of order $2 \times 10^{-6}$, which is reached for a violation of the universality of free fall of order $10^{-12}$ and it was established that

$$\left. \frac{\dot{\alpha}_{\text{EM}}}{\alpha_{\text{EM}}} \right|_0 \sim \pm 10^{-16} \sqrt{1 + q_0 - \frac{3}{2} \Omega_{\text{m0}} \sqrt{10^{12} \eta}} \text{yr}^{-1}, \quad (344)$$

where the first square-root arises from the computation of $\dot{\phi}_0$. The formalism was later used to discuss the time variation of $\alpha_{\text{EM}}$ and $\mu$ (Chiba et al. 2007). Hence, the Klein-Gordon equation links $\dot{\phi}_0$ to the deceleration parameter (if the field is slow-rolling) and hence to the dark energy equation of state and the matter content of the universe. It follows that the model can be constrained on the one side locally (i.e.,
using the Solar system and laboratory constraints on the fundamental constants) and on the other hand by low-redshift astrophysical data and cosmological data than constrain the cosmological parameters.

As examples, Martins et al. (2015b) and then Martins and Vacher (2019) have considered the simplified case in which $\varphi$ has either (1) a constant coupling $\alpha_V$ to the dark matter or (2) $\alpha_A$, as defined in Eq. (339) to the hadronic cosmological matter, or (3) self-coupling $\alpha_\varphi$ to dark energy to set constraint from low-redshift QSO data and local clock and WEP constraints. Assuming that the dilaton behaves as $\varphi \sim \varphi_0 + \varphi'_0 \ln a$ at low-redshift, they concluded, taking into account the constraint on the dark matter equation of state (Tutusaus et al. 2016), that $|\varphi'_0| < 0.1$, $|\alpha_A| < 1.5 \times 10^{-5}$ and $|\alpha_V| < 0.09$. Vacher et al. (2023) followed the same line using local (clocks + MICROSCOPE) data, Oklo and astrophysical (QSO) data to constrain the redshift evolution of $\alpha_{EM}$ and the latest BBN+CMB+BAO+SNIa to constrain the background cosmology. The model assumes a coupling to hadron $\alpha_A = \alpha_{A0}e^{-(\varphi-\varphi_0)}$ and to dark matter $\alpha_m = \alpha_{m0}e^{-(\varphi-\varphi_0)}$ and either (I) a massless field and cosmological constant or (2) a model with a potential $V \propto e^{c(\varphi-\varphi_0)}$ to conclude that $\alpha_{A0} \lesssim 3 \times 10^{-6}$, $\alpha_{m0} \lesssim 6 \times 10^{-2}$.

## Conclusions

In these two string-inspired scenarios, the amplitude of the variations of the constants are related to the one of the density fluctuations during inflation and the cosmological evolution. Both the amplitude of the violation of UFF and the variation of all constants are related to the same scalar field. The models then required an attraction mechanism to avoid the Solar system constraints, either at a finite $\varphi_m$ for the light dilaton or at infinity for the runaway dilaton. The scalar field can also act as a quintessence field so that the time variations of the fundamental constants are also related to the equation of state of the dark energy component and thus to the cosmological expansion.

## 7.2 Damour–Donoghue interaction light dilaton model

### Model definition

The Damour–Donoghue model (Damour and Donoghue 2010a, b) considered the standard action of General Relativity and the standard matter field plus a scalar field

$$S = \int \left\{ \frac{1}{16\pi G} \left[ R - 2g^{\mu\nu} \partial_{\mu} \varphi \partial_{\nu} \varphi - V(\varphi) \right] + \mathcal{L}_{\text{SM}}(\Psi_i, g_{\mu\nu}) \right\} \sqrt{-g} d^4 x$$

to which one adds an interaction term to the baryonic sector that takes into account a direct coupling to the Faraday tensor of electromagnetism $F_{\mu\nu}$ the gluon tensor $F^{A}_{\mu\nu}$ and the fermion spinors $\psi_i$ as

$$S_{\text{int}} = \int \varphi \left[ \frac{d_e}{4e^2} F^2 - \frac{\beta_3 d_g}{2g_3} F^{A}_{\mu\nu} F^{\mu\nu}_A - \sum_{i=e,u,d} (d_{m_i} + \gamma_{m_i} d_g) m_i \bar{\psi}_i \psi_i \right] \sqrt{-g} d^4 x. \quad (345)$$

$g_3$ stands for the QCD gauge coupling and $\beta_3$ for the $\beta$-function of the running of $g_3$.
<table><caption>**Table 20** Summary of the main constraints on a secular change of G</caption><thead><tr><th>System</th><th>ΔG/G</th><th>˙G/G (yr<sup>-1</sup>)</th><th>Ref</th></tr></thead><tbody><tr><td>Earth-Moon</td><td>x</td><td>(-0.7 ± 3.8) × 10<sup>-13</sup></td><td>Hofmann et al. (2010)</td></tr><tr><td>Earth-Moon</td><td>x</td><td>(-5.0 ± 9.6) × 10<sup>-15</sup></td><td>Biskupek et al. (2021)</td></tr><tr><td>Planet ranging</td><td>x</td><td>(-0.01 ± 0.91) × 10<sup>-13</sup></td><td>Pinto et al. (2011)</td></tr><tr><td>Planet ranging</td><td>x</td><td>-2.9 × 10<sup>-14</sup> &lt; ˙G/G &lt; 4.6 × 10<sup>-14</sup></td><td>Pitjeva and Pitjev (2012)</td></tr><tr><td>Large Magellanic Cloud</td><td>-0.07<sup>+0.05</sup><sub>-0.04</sub></td><td>x</td><td>Desmond et al. (2021)</td></tr><tr><td>PSR B1913+16</td><td>x</td><td>(4 ± 5) × 10<sup>-12</sup></td><td>Kaspi et al. (1994)</td></tr><tr><td>PSR B1855+09</td><td>x</td><td>(-9 ± 18) × 10<sup>-12</sup></td><td>Kaspi et al. (1994)</td></tr><tr><td>PSR J1738+0333</td><td>x</td><td>(-0.7 ± 7.6) × 10<sup>-13</sup></td><td>Freire et al. (2012)</td></tr><tr><td>PSR J1713+0747</td><td>x</td><td>(-0.6 ± 1.1) × 10<sup>-13</sup></td><td>Zhu et al. (2015)</td></tr><tr><td>PSR J0437-4715</td><td>x</td><td>&lt;2.3 × 10<sup>-11</sup></td><td>Verbiest et al. (2008)</td></tr><tr><td>Globular clusters</td><td>x</td><td>(-1.4 ± 2.1) × 10<sup>-11</sup></td><td>Del'Innocenti (1996)</td></tr><tr><td>Helioseismology</td><td>x</td><td>&lt;1.6 × 10<sup>-12</sup></td><td>Guenther et al. (1998)</td></tr><tr><td>Asteroseismology</td><td>x</td><td>1.2 ± 2.6) × 10<sup>-12</sup></td><td>Bellinger and Christensen-Dalsgaard (2019)</td></tr><tr><td>NGC 6791</td><td>x</td><td>(-0.9 ± 0.9) × 10<sup>-12</sup></td><td>Garcia-Berro et al. (2011)</td></tr><tr><td>G117-B15A</td><td>x</td><td>(-0.9 ± 0.9) × 10<sup>-10</sup></td><td>Corsico et al. (2001)</td></tr><tr><td>R548</td><td>x</td><td>(-0.65 ± 0.65) × 10<sup>-10</sup></td><td>Corsico et al. (2001)</td></tr><tr><td>G117-B15A</td><td>x</td><td>&lt;4.1 × 10<sup>-11</sup></td><td>Benvenuto et al. (2004), Biesiada and Malec (2004)</td></tr><tr><td>SNIa</td><td>x</td><td>-3 × 10<sup>-11</sup> &lt; ˙G/G &lt; 7.3 × 10<sup>-11</sup></td><td>Mould and Uddin (2014)</td></tr><tr><td>SNII</td><td>x</td><td>-0.6 ± 4.2) × 10<sup>-12</sup></td><td>Thorsett (1996)</td></tr><tr><td>BBN</td><td>-0.01<sup>+0.06</sup><sub>-0.05</sub></td><td>x</td><td>Alvey et al. (2020)</td></tr></tbody></table>

All details in the text

One considers 3 fermions (electrons, up and down quarks) of mass $m_i$. Their anomalous dimensions $\gamma_{m_i}$ give the energy running of the quark masses. The interaction is thus characterized by 5 dimensionless couplings ($d_e, d_g, d_{m_i}$) that characterize the strength of the coupling of $\phi$ to the matter sectors: $d_e$ and $d_g$ parametrise the coupling with the electromagnetic and gluonic fields while $d_{m_e}, d_{m_u}$ and $d_{m_d}$ are the couplings to the electron, u and d quarks mass terms. It is usual to redefine the latter two as

$$ \hat{m} = \frac{1}{2}(m_u + m_d), \quad \delta m = m_d - m_u. \qquad (346) $$

While Eq. (345) is linear in $\phi$, a similar interaction term with higher powers of $\phi$ have been considered (Hees et al. 2018). In such a case the previous couplings are labelled $d^{(1)}$ while the couplings to $\phi^2/2$ are ($d_e^{(2)}, d_g^{(2)}, d_{m_i}^{(2)}$) respectively. Stadnik and Flambaum (2016b) considered a coupling of the form $d_j(\phi - \phi_j)^2$ which then
boils down to a combination of the parameters $d^{(1)}$ and $d^{(2)}$. The model can be easily generalized to any coupling functions by replacing in the action (345) $1 + d_i\varphi$ by $B_i(\varphi)$ and then $d_i$ by $\alpha_i = B'_i/B_i$.

This model has been constrained from the explicit violation of the UFF it generates as well as all the others tests on the fundamental constants. It has become a prototypical example for several ultra-light dark matter scenarios. All the constraints depend on the choice of the potential, with the two cases of a massless ($V=0$) and light ($V=2m_{\varphi}^2\varphi^2$).

### Field dependencies of the constants

It follows that the fine structure constant, the quark masses and the QCD mass scale behave as

$$ 
\begin{aligned}
\alpha_{\text{EM}}(\varphi) &= (1 + d_e\varphi)\alpha_{\text{EM}}, \\
m_i(\varphi) &= (1 + d_{m_i}\varphi)m_i, \\
\Lambda_3(\varphi) &= (1 + d_g\varphi)\Lambda_{\text{QCD}}
\end{aligned}
\qquad (347) $$

and Eq. (346) implies that

$$ \hat{m}(\varphi) = (1 + d_{\hat{m}}\varphi)\hat{m}, \qquad \delta m(\varphi) = (1 + d_{\delta m}\varphi)\delta m \qquad (348) $$

with

$$ d_{\hat{m};\delta m} = \frac{m_d d_{m_d} \pm m_u d_{m_u}}{m_d \pm m_u} $$

Hence, the sensitivities (446) of the constants to the scalar field are simply $s_i(\varphi) = d_i$. In the case of a quadratic coupling (Hees et al. 2018), the expressions (347) have to be modified by relabelling the linear coefficients $d_i$ as $d_i^{(1)}$ and then defining the second order coefficients according to the replacement $d_i\varphi \to d_i^{(2)}\varphi^2/2$ so that $s_i^{(2)}(\varphi) = d_i^{(2)}\varphi$.

### Note on normalisation

Again, we assume that $\varphi$ is dimensionless so that the action is consistent with the normalisation of e.g., Damour and Nordtvedt (1993a, 1993b), Stadnik and Flambaum (2015a), Hees et al. (2018) while other works such as e.g., Arvanitaki et al. (2015), Stadnik and Flambaum (2015b, 2015c), Arvanitaki et al. (2016), Bergé et al. (2018a) consider the coupling to $\phi = \varphi/\sqrt{4\pi G}$. In that case, energy scales $\Lambda_i$, are introduced in the relations (347), i.e., the terms $(1 + d_i\varphi)$ are expressed as $(1 + \phi/\Lambda_i)$ so that $d_i = \sqrt{4\pi G}/\Lambda_i$ for the linear couplings. The comparison of normalisations of the quadratic coupling models (Stadnik and Flambaum 2015a, b, c, 2016a, b; Kalaydzhyan and Yu 2017; Flambaum and Stadnik 2023) are detailed in Appendix B of Hees et al. (2018).
<table><caption>**Table 21** Dilatonic charges for the materials used to test the UFF</caption><thead><tr><th>Material</th><th>-Q'<sub>ṁ</sub> [×10<sup>-3</sup>]</th><th>Q'<sub>e</sub> [×10<sup>-3</sup>]</th><th>-Q'<sub>m<sub>e</sub></sub> [×10<sup>-5</sup>]</th><th>Q'<sub>δm</sub> [×10<sup>-4</sup>]</th></tr></thead><tbody><tr><td>H/He [70:30]</td><td>45.51</td><td>0.36</td><td>-18.9</td><td>-11.7</td></tr><tr><td>Fe</td><td>9.94</td><td>2.32</td><td>1.89</td><td>1.17</td></tr><tr><td>Be</td><td>17.64</td><td>0.45</td><td>3.05</td><td>1.91</td></tr><tr><td>Al</td><td>12.30</td><td>1.47</td><td>1.00</td><td>0.62</td></tr><tr><td>Ti</td><td>10.42</td><td>2.01</td><td>2.24</td><td>1.38</td></tr><tr><td>U-238</td><td>7.63</td><td>4.28</td><td>6.24</td><td>3.86</td></tr><tr><td>Cu</td><td>6.93</td><td>2.46</td><td>2.18</td><td>1.35</td></tr><tr><td>Pb</td><td>7.73</td><td>3.92</td><td>5.30</td><td>3.28</td></tr><tr><td>Pt/Rh [90:10]</td><td>7.83</td><td>3.92</td><td>5.30</td><td>3.28</td></tr><tr><td>Ti/Al/V [90:6:4]</td><td>10.52</td><td>1.98</td><td>2.17</td><td>1.34</td></tr></tbody></table>

Adapted from Hees et al. (2018), Bergé (2023)

## Sensitivities of the atomic masses

As explained in Sect. 2.2.2 and illustrated in Sect. 3.1.2, this implies that all the atomic masses, as well as the masses of the proton and neutron, depend on $\phi$. Starting from the general expression of the atomic masses (27-30), the sensitivity $\beta_A$ defined in Eqs. (447) have been obtained to be (Damour and Donoghue 2010a, b)

$$ \beta_A = d_g + (d_{\hat{m}} - d_g)Q_{\hat{m}} + (d_{\delta m} - d_g)Q_{\delta m} + (d_{m_e} - d_g)Q_{m_e} + d_e Q_e \quad (349) $$

in terms of the four charges $Q_i$ that depend on the atomic and mass numbers (Z, A) as

$$ Q_{\hat{m}} = F_A \left[ 93 - \frac{36}{A^{1/3}} - 20 \left( \frac{A-2Z}{A} \right)^2 - 0.14 \frac{Z(Z-1)}{A^{4/3}} \right] \times 10^{-3} \quad (350) $$

$$ Q_{\delta m} = F_A \left[ 17 \frac{A-2Z}{A} \right] \times 10^{-4} \quad (351) $$

$$ Q_{m_e} = F_A \left[ 5.5 \frac{Z}{A} \right] \times 10^{-4} \quad (352) $$

$$ Q_e = F_A \left[ -1.4 + 8.2 \frac{Z}{A} + 7.7 \frac{Z(Z-1)}{A^{4/3}} \right] \times 10^{-4} \quad (353) $$

with

$$ F_A = A \frac{931 \text{ MeV}}{m_A}. \quad (354) $$
Universality of free fall

This allows one to translate the experimental tests of the UFF to constraints on the model parameters.

First, it has been shown (Damour and Donoghue 2010a, b) (see also Hees et al. 2018; Bergé 2023 for technical details) that the expression (349) for the coupling can be approximated as

$$ \alpha_A \approx d_g^* + [(d_{\bar{m}} - d_g)Q'_{\bar{m}} + d_eQ'_e + (d_{m_e} - d_g)Q'_{m_e} + (d_{\delta m} - d_g)Q'_{\delta m}]_A, $$

in which the first term $d_g^* = d_g + 0.093(d_{\bar{m}} - d_g) + 2.7 \times 10^{-4}d_e$ represents the composition independent part and where
$Q'_{\bar{m}} = -0.036A^{1/3} - 1.4 \times 10^{-4}Z(Z-1)/A^{4/3} - 0.02(A-2Z)^2/A^2$,
$Q'_e = [7.7Z(Z-1)/A^{4/3} - 4.1(A-2Z)/Z] \times 10^{-4}$,
$Q'_{m_e} = -2.75 \times 10^{-4}(A - 2Z)/A$ and $Q'_{\delta m} = 1.7 \times 10^{-3}(A - 2Z)/A$. Their values for different materials used experimentally are summarized in Table 21. Note that the term in $A - 2Z$ in these expressions can be neglected for heavy elements for which $Z/A \sim 1/2$ so that, given the experimental precision, one can rely on the value

$$ \alpha_A \approx d_g^* + [(d_{\bar{m}} - d_g)Q'_{\bar{m}} + d_eQ'_e + (d_{m_e} - d_g)Q'_{m_e}]_A. $$

For a massless scalar field ($V=0$) or any field for which the Compton wavelength, $\lambda \propto m_\phi^{-1}$ is much larger than any other spatial scales, the Eötvös parameter (20) reduces to $\eta_{ij} = (\alpha_i - \alpha_j)\alpha_{\text{Earth}}$ for two test bodies falling in the gravitational field of the Earth. Hence,

$$ \eta_{ij}(m_\phi = 0) = D_{\bar{m}}([Q'_{\bar{m}}]_i - [Q'_{\bar{m}}]_j) + D_e([Q'_e]_i - [Q'_e]_j), \quad (355) $$

with the coefficients

$$ D_{\bar{m}} = d_g^*(d_{\bar{m}} - d_g), \quad \text{and} \quad D_e = d_g^*d_e. $$

The constraints are summarized on Fig. 25 [left] combining the Eötvös constraints from Braginskii and Panov (1971), Wagner et al. (2012b) and Touboul et al. (2019).

For a massive scalar field, the range of the composition-dependent fifth force is finite so that the expression for the Eötvös parameter is modified to

$$ \eta(m_\phi) = \eta(m_\phi = 0) \times \Phi\left(\frac{R_E}{\lambda_{\phi}}\right) \left(1 + \frac{r}{\lambda_{\phi}}\right) e^{-m_\phi r}; \quad (356) $$

Atomic clocks

Using the decomposition of the frequencies of atomic clocks, it is clear, given the Lagrangian (345) that the best set of parameters to be used is $(\alpha_{\text{EM}}, X_q, X_e)$ and thus the expression (114) from which one easily deduces that the sensitivity of the frequency $v_A$ to the scalar field $\phi$ defined by
<figure><img src="image_32.png" /><figcaption>Fig. 25 Constraints on the couplings of a massless dilaton ($D_{\tilde{m}}$, $D_e$). The region allowed by the MICROSCOPE (Touboul et al. 2019) measurement (black band) is compared to earlier constraints by torsion pendulum experiments by Braginskii and Panov (1971) (green) and Wagner et al. (2012b) (yellow, cyan). The difference of slopes arises from the difference of material used in these 3 experiments. MICROSCOPE allows one to shrink the allowed region by one order of magnitude. Images reproduced with permission from Bergé et al. (2018a), copyright by APS</figcaption></figure>

$$d \ln v_A = \kappa_A d\varphi$$

is given by

$$\kappa = K_{\alpha}d_e + K_e(d_{m_e} - d_g) + (K_q - 0.048)(d_{\tilde{m}} - d_g) \quad (357)$$

with the coefficients $K$ given in Table 6 for the experiments described in Sect. 5.1 and summarized in Table 9.

The most stringent constraint on the time variation of $\alpha_{EM}$ alone (142) translates to

$$d_e \dot{\varphi}_0 = (1.8 \pm 2.5) \times 10^{-19} \text{yr}^{-1} \iff d_e \frac{d\varphi}{d\ln a} \Big|_{a=a_0} = (1.76 \pm 2.45) \times 10^{-9} \quad (358)$$

using the value (456) of the Hubble constant.

These bounds are valid for linear couplings. They set strong constraints on the decoupling of the field in the Solar system and hence on all the cosmological dynamics.

## Constraints

The model has been extensively used to interpret local and astrophysical data to the point that it can be considered as a standard lore to that goal, in particular in connection with the search for a high scalar dark matter component (Arvanitaki et al. 2015, 2016; Stadnik and Flambaum 2015a, c; Van Tilburg et al. 2015; Derevianko 2016) as described in Sect. 2.3.4.

The first MICROSCOPE constraints on linear case where published in (Bergé et al. 2018a), see Fig. 25. From the modified post-Newtonian equations of motion of an N-body system and the light time travel for a massless dilaton with quadratic coupling, Bernus et al. (2022) predicted the ephemeris of the main planets of the Solar system. The comparison with the planetary ephemeris INPOP19a, Fienga
et al. (2015) constrained the coupling at the level of $O(10^{-5})$; see also Fienga and Minazzoli (2024) for a review on the tests of gravity with ephemeris.

A series of works analyzed clock experiments in this framework. Hees et al. (2016) searched for oscillations in the Cs-Rb clock data by Guena et al. (2012) induced by a massive oscillating scalar field with linear coupling. While no sign of a scalar dark matter was found, it constrained the combination $d_e + k_q(d_m - d_g/k_\alpha)$ of the non universal coupling to standard matter. Stadnik and Flambaum (2016b) constrained the linear coupling of the light dilaton to the Higgs boson thanks to atomic clock spectroscopy. Stadnik and Flambaum (2015c) constrained the linear model through variation of fundamental constants with laser and maser interferometry. Hees et al. (2018) performed an extensive comparison to UFF and clocks and extended their analysis to quadratic couplings. Regarding the linear case, the scalar field being the sum of an oscillating contribution and a Yukawa contribution, the oscillating contribution can be identified as dark matter while the Yukawa interaction leads to a “standard” fifth force. Interestingly clocks are more sensitive to the first while UFF are better designed to constrain the Yukawa component. It was concluded that “natural coupling” of order unity are excluded for mass smaller than $10^{-5}$ eV ($d_e$ only). They pointed out that the quadratic coupling has a richer phenomenology since no Yukawa interaction is generated and that, Instead, the scalar field exhibits an oscillatory behavior that is perturbed or enhanced by the presence of a massive body. The scalar field oscillations can be amplified (negative $d^{(2)}$) or screened (positive) with the consequence that experiments in space are more interesting to detect or constrain DM with a quadratic coupling - the field is mediating long-distance forces despite being massive. Kobayashi et al. (2022) searched for the oscillating scalar DM fields with frequency comparison between an Yb-171 optical lattice clock and a Cs-133 fountain clock that span 298 days. Brzeminski et al. (2022) investigated the prospect of increasing the sensitivities of atomic and nuclear clocks by placing them in an eccentric orbit around the Earth. Sherrill et al. (2023) restricted to two couplings, one to $F^2$ and the second to the electron mass, and the considered scalar field undergoes damped harmonic oscillations with amplitude $\phi_0 \exp(-\Gamma t/2)$, pulsation $\omega_d$. The model was constrained thanks to atomic clock through the induced variations of $\alpha_{EM}$ and $\bar{\mu}$. It was extended by Banerjee et al. (2023b) to include both a CP odd pseudo-scalar, and a CP-even scalar.

## 7.3 Bekenstein and related models

Bekenstein (1982, 2002) introduced a theoretical framework in which only the electromagnetic sector was modified by the introduction of a dimensionless scalar field $\epsilon$ so that all electric charges vary in unison $e_i = e_{0i}\epsilon(x^\alpha)$ so that only $\alpha_{EM}$ is assumed to possibly vary.
## Early formulation

To avoid the arbitrariness in the definition of $\epsilon$, which can be rescaled by a constant factor while $e_{0i}$ is inversely rescaled, it was postulated that the dynamics of $\epsilon$ be invariant under global rescaling so that its action should be of the form

$$S_{\epsilon} = -\frac{\hbar c}{2l^2} \int \frac{g^{\mu\nu} \partial_{\mu} \epsilon \partial_{\nu} \epsilon}{\epsilon^2} \sqrt{-g} d^4 x, \qquad (359)$$

$l$ being a constant length scale. Then, $\epsilon$ is assumed to enter all electromagnetic interaction via $e_i A_{\mu} \rightarrow e_{0i} \epsilon A_{\mu}$ where $A_{\mu}$ is the usual electromagnetic potential and the gauge invariance is then preserved only if $\epsilon A_{\mu} \rightarrow \epsilon A_{\mu} + \lambda_{,\mu}$ for any scalar function $\lambda$. It follows that the action for the electromagnetic sector is the standard Maxwell action

$$S_{\epsilon} = -\frac{1}{16\pi} \int F^{\mu\nu} F_{\mu\nu} \sqrt{-g} d^4 x, \qquad (360)$$

for the generalized Faraday tensor

$$F_{\mu\nu} = \frac{1}{\epsilon} [(\epsilon A_{\nu})_{,\mu} - (\epsilon A_{\mu})_{,\nu}]. \qquad (361)$$

To finish the gravitational sector is assumed to be described by the standard Einstein-Hilbert action. Finally, the matter action for point particles of mass $m$ takes the form $S_m = \int [-mc^2 + (e/c)u^{\mu}A_{\mu}]\gamma^{-1}\delta^3(x^i - x^i(\tau))d^4x$ where $\gamma$ is the Lorentz factor and $\tau$ the proper time. Note that the Maxwell equation becomes

$$\nabla_{\mu} (\epsilon^{-1} F^{\mu\nu}) = 4\pi j^{\nu}, \qquad (362)$$

which is the same as for electromagnetism in a material medium with dielectric constant $\epsilon^{-2}$ and permeability $\epsilon^2$ (this was the original description proposed by Fierz 1956 and Lichnerowicz 1955; see also Dicke 1964).

As discussed previously, this class of models predict a violation of the universality of free fall and, from Eq. (25), it is expected that the anomalous acceleration is given by $\delta a = -M^{-1}(\partial E_{EM}/\partial \epsilon)\nabla \epsilon$. From the confrontation of the local and cosmological constraints on the variation of $\epsilon$ Bekenstein (1982) concluded, given his assumptions on the couplings, that $\alpha_{EM}$ “is a parameter, not a dynamical variable” (see, however, Bekenstein 2002 and then Kraiselburd and Vucetich 2011) and concluded the model not to be viable.

## Extensions

It was proposed (Sandvik et al. 2002) to rewrite this theory by introducing the two fields $a_{\mu} \equiv \epsilon A_{\mu}$ and $\psi \equiv \ln \epsilon$ so that the theory takes the form

$$S = \frac{c^3}{16\pi g} \int R \sqrt{-g} d^4 x - \frac{1}{16\pi} \int e^{-2\psi} f^{\mu\nu} f_{\mu\nu} \sqrt{-g} d^4 x - \frac{1}{8\pi\kappa^2} \int (\partial_{\mu} \psi)^2 \sqrt{-g} d^4 x \qquad (363)$$

with $\kappa = l/(4\pi\hbar c)$ and $f_{\mu\nu}$ the Faraday tensor associated with $a_{\mu}$. The model was
further extended to include a potential for $\psi$ (Barrow and Li 2008) and to include the electroweak theory (Shaw and Barrow 2005).

Olive and Pospelov (2002) generalized the model to allow for additional couplings of a scalar field $\epsilon^{-2} = B_F(\phi)$ to non-baryonic dark matter (as first proposed in Damour et al. 1990) and cosmological constant. They argue that in supersymmetric dark matter, it is natural to expect that $\phi$ would couple more strongly to dark matter than to baryon. For instance, supersymmetrizing Bekenstein model, $\phi$ will get a coupling to the kinetic term of the gaugino of the form $M_*^{-1}\phi\bar{\chi}\partial\chi$. Assuming that the gaugino is a large fraction of the stable lightest supersymmetric particle, the coupling to dark matter would then be of order $10^3 - 10^4$ times larger. Such a factor could almost reconcile the constraint arising from the test of the universality of free fall with the order of magnitude of the cosmological variation. This generalisation of the Bekenstein model is defined as

$$ S = \frac{1}{2} M_*^4 \int R \sqrt{-g} d^4 \mathbf{x} - \int \left[ \frac{1}{2} M_*^2 \partial_\mu \varphi \partial^\mu \varphi + \frac{1}{4} B_F(\varphi) F_{\mu\nu} F^{\mu\nu} \right] \sqrt{-g} d^4 \mathbf{x} \\ - \int \left\{ \sum \bar{N}_i [i \not D - m_i B_{N_i}(\varphi)] N_i + \frac{1}{2} \bar{\chi} \partial \chi \right. \\ \left. + M_*^2 B_A(\varphi) \Lambda + \frac{1}{2} M_\chi B_\chi(\varphi) \chi^T \chi \right\} \sqrt{-g} d^4 \mathbf{x} \quad (364) $$

where the sum is over proton [$\not D = \gamma^\mu (\partial_\mu - ie_0 A_\mu)$] and neutron [$\not D = \gamma^\mu \partial_\mu$]. The functions B can be expanded (since one focuses on small variations of the fine-structure constant and thus of $\varphi$) as $B_X = 1 + \zeta_X \varphi + \zeta_X \varphi^2/2$. It follows that $\alpha_{EM}(\varphi) = e_0^2/4\pi B_F(\varphi)$ so that $\Delta\alpha_{EM}/\alpha_{EM} = \zeta_F \varphi + (\zeta_F - 2\zeta_F^2)\varphi^2/2$. Under this form, the effective theory is very similar to the one detailed in Sect. 3.2.

This framework extends the analysis of Bekenstein (1982) to a 4-dimensional parameter space ($M_*, \zeta_F, \zeta_m, \zeta_A$). It contains

* the original Bekenstein model ($\zeta_F = -2$, $\zeta_A = 0$, $\zeta_m \sim 10^{-4} \xi_F$),

* a Jordan–Brans–Dicke model ($\zeta_F = 0$, $\zeta_A = -2\sqrt{2/2\omega + 3}$, $\zeta_m = -1/\sqrt{4\omega + 6}$),

* a string-like model ($\zeta_F = -\sqrt{2}$, $\zeta_A = \sqrt{2}$, $\zeta_m = \sqrt{2}/2$) so that $\Delta\alpha_{EM}/\alpha_{EM} = 3$) and

* a supersymmetrized Bekenstein model ($\zeta_F = -2$, $\zeta_\chi = -2$, $\zeta_m = \zeta_\chi$ so that $\Delta\alpha_{EM}/\alpha_{EM} \sim 5/\omega$).

In all these models, the universality of free fall sets a strong constraint on $\zeta_F/\sqrt{\omega}$ (with $\omega \equiv M_*/2M_*^2$) and the authors showed that a small set of models may be compatible with a variation of $\alpha_{EM}$ from quasar data while being compatible with the equivalence principle tests. A similar analysis (Marra and Rosati 2005) concluded that such models can reproduce the variation of $\alpha_{EM}$ from quasars while being compatible with Oklo and meteorite data. Onegin (2014) argued that the $|\zeta_m(l/l_p)| < 6 \times 10^{-7}$ to fullfil Oklo bounds so that the characteristic length of the model shall be smaller than the Planck scale if $\zeta_m \sim 10^{-4}$.
## Constraints

A series of works (Vielzeuf and Martins 2014; Martins 2017; Martins et al. 2022b; Vacher et al. 2022; Tohfa et al. 2024) have considered the constraints that arise from cosmological data, the latest astrophysical data on $\alpha_{EM}$ and local constraints. They all point to the conclusion that typically $\zeta < 10^{-7}$ and that the local data dominate the constraints. In particular, the connection between $\eta$ and $\zeta$ (Sandvik et al. 2002), $\eta = 3 \times 10^9 \zeta$, implies that MICROSCOPE sets immediately $\eta < 10^{-5}$. Leite and Martins (2016) estimated that astrophysical data ae about 20 times more constraining for $\zeta$ and that the coming E-ELT observations could reach a factor 50 compared to the universality of free fall. Li et al. (2011) developed N-body simulations including the resolution of the scalar field equation. While it does not cluster significantly, because of its weak coupling to matter, it modifies the structure formation. This was complemented by the study of gravitational collapse (Chakrabarti 2023) in which dark matter is modeled by a Bekenstein scalar field. Liu et al. (2021) constrained the model by use of the SZ effect (see Sect. 5.12) using two galaxy cluster samples, the 61 clusters provided by the Planck collaboration and the 58 clusters detected by the South Pole Telescope, to conclude that $\zeta > O(10^{-2})$.

On smaller scales, Danielsson et al. (2017) investigated the phenomenology of a heavy Bekenstein scalar for accelerator physics, in particular at LHC since it would be produced through photon-photon fusion, leading to diphoton final state, and by quark-antiquark fusion. This was later generalized to include varying gauge coupling in the electroweak sector (Danielsson et al. 2019) with either one or two scalars and the signatures were discussed. This allowed them to set constraints on the masses of these scalars and the energy scale of the theory from the LHC $\gamma\gamma, \gamma Z, ZZ, WW$ and $jj$ resonance search results. For instance $\gamma\gamma$ implies that the energy scale for new physics has to be larger than 20 TeV for a scalar of mass $\sim 1$ TeV.

## Further studies

This theory was also used (Bekenstein and Schiffer 2009) to study the spacetime structure around charged black-hole, which corresponds to an extension of dilatonic charged black hole. It was concluded that a cosmological growth of $\alpha_{EM}$ would decrease the black-hole entropy but with half the rate expected from the earlier analysis (Davies et al. 2002; MacGibbon 2007).

Leszczynska et al. (2019) considered an extension of the Bekenstein theory in which the scalar field enjoys an extra-coupling to the baryon current, i.e., a term $\propto \lambda J^{\mu}_{\text{baryon}} \partial_{\mu} \varphi$ that spontaneously breaks the baryon symmetry (Dimopoulos and Susskind 1978), hence relating baryogenesis to varying constants. It enjoys the interesting feature that the current baryon entropy $\eta \sim 8.6 \times 10^{-11}$ can then be obtained for a large range of parameters.
## 7.4 Light scalar quintessence

### Model

A series of works have investigated the connection between quintessence and the variation of $\alpha_{EM}$ with an action of the form (57) assuming $\varphi$ is a quintessence field, i.e., that it enjoys a potential leading to a late time cosmic acceleration. Even though, they are a subcase of the light-dilaton models described in Sect. 7.2 with only $d_e$ non-vanishing, this class of models has been extensively studied since it provides a simple example to study the connection between the dark energy equation of state and the time variation of the fine structure constant. As we have seen in Sect. 3.1.2, even if there is no direct coupling to the matter field but the Faraday tensor, the masses of protons, neutrons and all nuclei will be time varying due to their electromagnetic binding energy so that such a coupling leads to a strong violation of the universality of free fall.

We recall that the action is given by

$$S = \frac{1}{16\pi G} \int \left[ R - 2(\hat{\alpha}_{\mu}\varphi)^2 - V(\varphi) - \frac{1}{4}B_F(\varphi)F_{\mu\nu}^2 \right] \sqrt{-g} d^4 x + S_{\text{matter}}[\psi; g_{\mu\nu}].$$

This model was initially considered by Bekenstein (1982) with the coupling

$$B_F = 1 - \zeta(\varphi - \varphi_0). \qquad (365)$$

It follows that

$$\frac{\Delta\alpha_{EM}}{\alpha_{EM}} = \frac{1 - B_F(\varphi)}{B_F(\varphi)} \approx \zeta(\varphi - \varphi_0) \qquad (366)$$

As discussed in Sect. 2.3.3, light field models are generically thought to lead to a coupling to $F^2$ so that linear coupling were first investigated. This was then generalized to quintessence models with a couplings of the form $Z(\varphi)F^{\mu\nu}F_{\mu\nu}$ (Wetterich 2003a; Anchordoqui and Goldberg 2003; Parkinson et al. 2004; Copeland et al. 2004; Doran 2005; Lee et al. 2005; Marra and Rosati 2005; Lee 2007) and then to the runaway dilaton (Damour et al. 2002a, b); Sect. 7.1. The evolution of the scalar field drives both the acceleration of the universe at late time and the variation of the constants.

The dynamics of $\varphi$ depends on its potential. It has two main regimes, either as dark energy if $V$ allows for slow-roll or as dark matter if $\varphi$ oscillates; see Sect. 2.3.4. In both case there is a strong connection between either $\dot{\phi}_0$ or $\varphi_0$ and the cosmological parameters hence offering a local/global issue. In particular, whatever the model, it has to satisfy the strong bound (358).

### Connection with the universality of free fall

As detailed in Sect. 3.1.2 such a coupling induces a strong violation of the universality of free fall, which was one of the motivation of the original work by Bekenstein (1982).

This was revisited in Chiba and Khori (2002), Dvali and Zaldarriaga (2002), Wetterich (2003b) in which the dependence of $\alpha_{EM}$ on the scalar field responsible
for its variation is expanded as $\alpha_{EM} = \alpha_{EM}(0) + \lambda\phi + O(\phi^2)$. The QSO data on $\alpha_{EM}$ were then pointing to $\lambda\Delta\phi \sim 10^{-7}$ at best during the last Hubble time.

Concentrating only on the electromagnetic binding energy contribution to the proton and of the neutron masses—see Eq. (31)—it was concluded that a test body composed of $n_n$ neutrons and $n_p$ protons will have a sensitivity $\lambda(v_pB_p + v_nB_n)/m_N$ where $v_n$ (resp. $v_p$) is the ratio of neutrons (resp. protons). Assuming<fnref n="17" /> that $v_{n,p}^{\text{Earth}} \sim 1/2$ and using that the compactness of the Moon-Earth system $\partial\ln(m_{\text{Earth}}/m_{\text{Moon}})/\partial\ln\alpha_{EM} \sim 10^{-3}$, one gets $\eta_{12} \sim 10^{-3}\lambda^2$. Dvali and Zaldarriaga (2002) obtained the same result by considering that $\Delta v_{n,p} \sim 6 \times 10^{-2} - 10^{-1}$. This implies that $\lambda < 10^{-5}$, which is compatible with a variation of $\alpha_{EM}$ if $\Delta\phi > 10^{-2}$ during the last Hubble period. From the cosmology one can deduce that $(\Delta\phi)^2 \sim (\rho_\phi + P_\phi)/\rho_{\text{total}}$. If $\phi$ dominates the matter content of the universe, $\rho_{\text{total}}$, then $\Delta\phi \sim 1$ so that $\lambda \sim 10^{-7}$ whereas if it is sub-dominant $\Delta\phi \ll 1$ and $\lambda \gg 10^{-7}$. In conclusion $10^{-7} < \lambda < 10^{-5}$. This makes explicit the tuning of the parameter $\lambda$. This analysis was extended in Dent (2007) who included explicitly the electron and related the violation of the universality of free fall to the variation of $\mu$.

Similarly, Wetterich (2003b) assumed the scalar field $\phi$ to be responsible for both a variation of $\alpha_{EM}$ and the acceleration of the cosmic expansion. Assuming its equation of state is $w_\phi \neq 1$, one can express its time variation as

$$ \dot{\phi} = H\sqrt{3\Omega_{\phi}(1+w_{\phi})}. \qquad (367) $$

Hence, the expected violation of the universality of free fall is related to the time variation of $\alpha_{EM}$ today by

$$ \eta = -1.75 \times 10^{-2} \left( \frac{\partial \ln \alpha_{EM}}{\partial z} \right)^2 \frac{(1 + \tilde{Q}) \Delta \frac{z}{z+N}}{\Omega_{\phi}^{(0)} (1 + w_{\phi}^{(0)})}, $$

where $\tilde{Q}$ is a parameter taking into account the influence of the mass ratios. Again, this shows that in the worse case in which the Oklo bound is saturated (so that $\partial\ln\alpha_{EM}/\partial z \sim 10^{-6}$), one requires $1 + w_\phi^{(0)} \gtrsim 10^{-2}$ for $\eta \lesssim 10^{-13}$, hence providing a string bond between the dark energy equation of state and the violation of the universality of free fall. Dent et al. (2009) extended it to the phenomenological unification model presented in Sect. 4.1. In the case of the string dilaton and runaway dilaton models, one reaches a similar conclusion [see Eq. (344) in Sect. 7.1]. Similarly, Martin et al. (2006) related the equation of state to the post-Newtonian parameters.

In all these models, the link between the local constraints and the cosmological constraints arise from the fact that local experiments constrain the upper value of $\dot{\phi}_0$, which quantify both the deviation of its equation of state from $-1$ and the variation of the constants. It was conjectured that most realistic quintessence models suffer from such a problem (Brax and Martin 2007a).

<footnote n="17">¹⁷ For copper $v_p = 0.456$, for uranium $v_p = 0.385$ and for lead $v_p = 0.397$.</footnote>
<figure><img src="image_33.png" /><figcaption>Fig. 26 Example of the complementarity of the local and astrophysical data. The blue and red regions represent the allowed values of $\alpha_F(\varphi_0)$ and $\dot{\varphi}_0$—see Eqs. (368–369)—for a for a massless scalar field with unique $B_F F^2$ coupling. This shows the complementarity of clock and UFF experiments to constrain $(\varphi_0, \dot{\varphi}_0)$. Note that the UFF constraint is boosted when other couplings are active since the electromagnetic binding energy is only a small part of the total mass of the Earth and test masses. Then, if this field accounts for the acceleration of the cosmic expansion, any constraints on the dark energy equation of state would imply an upper bound on $\dot{\varphi}_0$. Within a theoretical framework, one can make similar plot at different redshifts, hence providing a constraints on the dynamics of the new degree of freedom</figcaption></figure>

### Generic local constraints: UFF vs $\alpha_{EM}$

This simplest model allow one to compare the strength of the different local constraints.

To that goal, we can simply starts from the analysis of Sect. 7.2 in which the only non vanishing coupling is $B_F$ and perform the replacement $1 + d_e \varphi \to B_F(\varphi)$ so that $d_e \to \alpha_F(\varphi) \equiv \partial_\varphi \ln B_F$. In such a case $d_g^*$ just gets the contribution for the electromagnetic binding energy so that $d_g^* \to 2.7 \times 10^{-4} \alpha_F$ and $D_e \to d_g^* \alpha_F$. Hence, Eq. (355) for the Eötvös parameter simplifies to

$$ \eta_{ij}(m_{\varphi} = 0) = 2.7 \times 10^{-4} ([Q'_{e}]_i - [Q'_{e}]_j) \alpha_F^2(\varphi_0). $$

Hence from MICROSCOPE (11), $\eta_{ij}(m_{\varphi} = 0) = 5.2 \times 10^{-7} \alpha_F^2$ and from atomic clocks. (358), one gets can be rewritten thanks to Eq. (366),

$$ \left. \frac{\alpha_{EM}'}{\alpha_{EM}} \right|_0 = \alpha_F(\varphi_0) \dot{\varphi}_0. $$

It follows that the combination atomic clocks and UFF provides a constraint on $(\alpha_F(\varphi_0), \dot{\varphi}_0)$ that is on $(\varphi_0, \dot{\varphi}_0)$,
<figure><img src="image_34.png" /><figcaption>Fig. 27 Constraints on $d_e$ for a massive dilaton coupled linearly only to the electromagnetic sector, compared with constraints (dot-dahed) from atomic spectroscopy (Van Tilburg et al. 2015; Hees et al. 2016) and Eöt-Wash WEP test (dashed) by Schlamminger et al. (2008). Image reproduced with permission from Bergé et al. (2018a), copyright by APS</figcaption></figure>

$$ \alpha_F(\phi_0)\dot{\phi}_0 = (1.8 \pm 2.5) \times 10^{-19} \text{yr}^{-1} \qquad (368) $$

$$ \alpha_F^2(\phi_0) = (-7.8 \pm 11.96 \pm 7.8_{\text{syst}}) \times 10^{-9}, \qquad (369) $$

for a massless scalar field. This sets a strong constraints on $(\phi_0, \dot{\phi}_0)$ whatever $B_F(\phi)$ and $V(\phi)$.

One shall also take the constraints arising from the annual modulation (see Sect. 8.2 below for the discussion on those constraints). Since $k_\alpha = \alpha_\odot(\phi_0)\alpha_F(\phi_0)$ and since in this restricted model with only $B_F$ non-vanishing, $\alpha_\odot = 3.6 \times 10^{-4}\alpha_F(\phi_0)$—see Table 21—we conclude that the bound (404) translates to

$$ \alpha_F^2(\phi_0) = (-6.6 \pm 8.3) \times 10^{-6}, $$

which remains 3 orders of magnitude less constraining than MICROSCOPE. Indeed, in a more realistic model, the two observables are complementary since they exhibit different dependencies on the $\alpha_i$. The MICROSCOPE constraints on the case of a linearly coupled massive scalar field are summarized on Fig. 27.

Then for all models in which this scalar fields also account for dark energy, we shall have

$$ \dot{\phi}^2 = 3H_0^2\Omega_\phi(1+w_\phi) $$

so that the constraints on the dark energy equation of state, $1+w = (-0.03 \pm 0.03)$ (Aghanim et al. 2020), bounds $\dot{\phi}_0$ independently.
## Constraints

This simple extension of scalar field cosmology enjoys 2 free functions, the potential that can turn $\varphi$ as either a dark energy or dark matter component, and the coupling function $B_F$. It offers an easy way to compare observations and connect them to the cosmological dynamics, in particular to evaluate the gain of including non-cosmological data when forecasting future experiments. In particular Eq. (367) implies that

$$ \varphi - \varphi_0 = \int_0^z \sqrt{3\Omega_\varphi(z)[1 + \omega_\varphi(z)]} \frac{dz}{1+z} $$

Besides, the conservation equation, $\dot{\rho}_\varphi + 3H(1+w_\varphi)\rho_\varphi = 0$ when one neglects the coupling to matter integrates to

$$ \rho_{\varphi} = \rho_{\varphi 0} \exp \left( 3 \int_{0}^{z} [1 + w_{\varphi}(z)] \frac{dz}{1+z} \right). $$

Besides Eq. (365) implies that

$$ \frac{\Delta\alpha_{\text{EM}}}{\alpha_{\text{EM}}} = \zeta \int_{0}^{z} \sqrt{3\Omega_{\varphi}(z)(1 + \omega_{\varphi}(z))} \frac{dz}{1+z}. $$

It is clear that the local, astrophysical and cosmological constraints on $\alpha_{EM}(z)$ provide a direct monitoring on the field dynamics $\varphi(z)$ and thus new bounds on the cosmological dynamics.

As a consequence, Avelino et al. (2012) addressed the reconstruction problem of the dark energy equation of state in order to discuss its degeneracies with $\alpha_{EM}$ and the limits of the reconstruction from data. This was applied to quasar absorption spectra and clocks data for $\alpha_{EM}$ and SN Ia data for dark energy (Martins and Pinho 2015; Martins et al. 2015a, 2016) for various quintessence models; see also Leite et al. (2016), Alves et al. (2018) for the discussion of the improvement expected from ESPRESSO and later E-ELT. Zhai et al. (2013) constrained a model with inverse power law or exponential potential from QSO data on $\alpha_{EM}$ to get $\zeta \simeq 10^{-5}$ but they did not check whether this fit is viable with respect to UFF constraints.

This is complemented by more phenomenological studied parametrizing the dark energy equation of states or the scalar field redshift evolution. Martins et al. (2022a) discussed the possibility to constrain a series expansion of $\Delta\alpha_{EM}/\alpha_{EM}$ in powers of $z/(1+z)$ in a similar way as what is done for the dark energy equation of state. Leite et al. (2014) described the quintessence equation of state by a step function in redshift space to get a parametric form of $\varphi(z)$ and thus of $\alpha_{EM}(z)$ in terms of $N$ parameters to investigate the power of astrophysical data on $\alpha_{EM}$ to unveil the equation of state of dark energy. Assuming $\varphi - \varphi_0 \propto \lambda \ln a$, so that $\Delta\alpha_{EM}/\alpha_{EM} = -\zeta\lambda \ln(1+z)$, which implies that the quintessence equation of state is $w = -1 + \lambda^2/3(1 - \Omega_m)$. Da Fonseca et al. (2022) showed that the constraints $\zeta \lesssim 1.4 \times 10^{-4}$ (CMB + QSO data including the latest ESPRESSO data) increases to $\zeta \lesssim 1.1 \times 10^{-4}$ when atomic clocks are added and to $\zeta \lesssim 1.6 \times 10^{-6}$ with MICROSCOPE. The constraints on $\lambda$ do not change much and remain at the level
of $(2-3) \times 10^{-2}$. More speculatively, Schöneberg et al. (2023) showed, using CMB, SNIa, BAO, O'clocks, QSO, Oklo for $\alpha_{EM}$, that a linear coupling to electromagnetism sets a new constraint on the swampland criteria since data favor $\zeta < 1.5 \times 10^{-1} \ll 1-10$ as expected from naturalness Martinelli et al. (2021) investigated how well the ESO/Euclid data can improve the bounds on $\zeta$ thanks to its sensitivity the dark energy equation of state through weak lensing and large-scale structure growth. It is expected that the Euclid satellite mission will shrink the allowed values for $\zeta$.

The latest developments are related to the new constraint on $\alpha_{EM}/\alpha_{EM}$ by Filzinger et al. (2023) which implies the constraint (358). First, it allowed Filzinger et al. (2023) to improve the bound on $d_e$ by more than one order of magnitude in the mass range $(10^{-24} - 10^{-17})$ eV. Then, it allowed Vacher and Schöneberg (2024) to show that models of single field coupled to $F^2$ with a potential $V$ have either a too large fine-structure constant at recombination while remaining compatible with Filzinger et al. (2023) today or require an extreme fine tuning of the potential or its initial conditions in order to alleviate the Hubble tension. Again, this is a nice example of the importance of the complementarity between local and cosmological data.

## 7.5 Other ideas

Since it is impossible to give an exhaustive description of all the models, let us mention without details other theoretical frameworks proposed to accommodate varying constants.

* Models involving a late time phase transition in the electromagnetic sector (Chacko et al. 2003; Anchordoqui et al. 2008);

* Braneworld models (Lorén-Aguilar et al. 2003; Palma et al. 2003; Byrne and Kolda 2004; Li and Chu 2006b; Amarilla and Vucetich 2010) or extra-dimensions (Steinhardt and Wesley 2010);

* Models with pseudo-scalar couplings (Flambaum et al. 2009b);

* Growing neutrino models (Wetterich 2007; Amendola et al. 2008) in which the neutrino masses are function of a scalar field, that is also responsible for the late time acceleration of the universe. In these models the neutrinos freeze the evolution of the scalar field when they become non-relativistic while its evolution is similar as in quintessence when the neutrinos are ultra-relativistic;

* Models based on discrete quantum gravity (Gambini and Pullin 2003) or on loop quantum gravity in which the Barbero-Immirzi parameter controls the minimum eigenvalue of the area operator and could be promoted to a field, leading to a classical coupling of Einstein's gravity with a scalar-field stress-energy tensor (Taveras and Yunes 2008; Mercuri and Taveras 2009)

* "Varying speed of light" models for which we refer to the review by Magueijo (2003) and (Ellis and Uzan 2005) for a critical view;

* Quintessence models with a non-minimal coupling of the quintessence field (Chiba and Khori 2002; Anchordoqui and Goldberg 2003; Wetterich 2003a; Copeland et al. 2004; Parkinson et al. 2004; Doran 2005; Lee et al. 2005; Marra
and Rosati 2005; Avelino et al. 2006; Lee 2007; Fujii 2009; Nunes et al. 2009)
[see discussion Sect. 2.3];

* It has be shown (von Harling and Servant 2017) that, within the Randall–Sundrum mechanism, a Goldberger–Wise scalar field couples to the fermions, leading to a model of varying Yukawa couplings;

* Holographic dark energy models with non-minimal couplings. (Granda and Escobar 2009);

* van de Bruck et al. (2015) extended the light field model coupled solely to $F^2$ to a situation in which electromagnetism is disformally coupled to the scalar field, i.e., the electromagnetism Lagrangian is of the form $h(\varphi)\tilde{g}^{\mu\nu}\tilde{g}^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta}$ with $\tilde{g}_{\mu\nu} = C(\varphi)g_{\mu\nu} + D(\varphi)\partial_{\mu}\varphi\partial_{\nu}\varphi$. Similarly matter is coupled to the same metric but with possibly different functions C and D;

* Barros and da Fonseca (2023) generalized this same model to include a coupling to the scalar field kinetic term, i.e., an interaction Lagrangian of the form $h[\phi, X]F^2$ with $X = -g^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi/2$. Assuming $h = h(X) \propto X^{-\eta}$ and that $\varphi$ accounts for quintessence on cosmological scales. CMB, BBN, QSO, atomic clocks and MICROSCOPE imply $\zeta \lesssim 10^{-7}$;

* Brans–Dicke theories are generically highly constrained by Solar system tests unless—see Sect. 3.1.1—, their couplings to ordinary matter are much suppressed relative to gravitational strength. As discussed, this is a major obstacle to construct realistic models of light dilatons. On the other hand, these scalars appear naturally in UV complete theories like string theory, where they arise as dilatons for the various accidental scaling symmetries that are generic in higher-dimensional supergravities. This led Burgess and Quevedo (2022) to remark that if matter also couples to a light axion, this would qualitatively change how the fields respond to a gravitating source. As consequence, matter-dilaton couplings that would be excluded in the absence of an axion can become acceptable given an additional small axion-matter coupling. Fifth forces on matter test bodies are then controlled by the much weaker axion-matter couplings and can easily be small enough to escape detection. This axion homeopathy mechanism is further explored by Burgess et al. (2022), Brax et al. (2023a).

* Brax et al. (2023b) studied the dynamics of a light scalar field responsible for the $\mu$ term of the Higgs potential and coupled to matter via the Higgs-portal mechanism. These models could be subject to a screening akin to the symmetron mechanism so that local gravitational tests are evaded thanks to the weakness of the quadratic coupling in the dark matter halo.

# 8 Spatial variations

The constraints on the variation of the fundamental constants that we have described so far are mainly related to their cosmological evolution so that, given the Copernican principle, they reduce to constrain on their eventual time variation. Indeed, spatial variations shall also occur, at least in two regimes:
* On cosmological scales, one shall consider two situations since (1) the fields dictating the variation of the constants have fluctuations that can let their imprint in some cosmological observables, in particular on large scales if they are light and (2) one can check if the distributions are compatible with the cosmological principle, in particular by constraining their possible dipolar, quadrupolar, etc. deviations from homogeneity.

* On local scales (e.g., our Solar system or the Milky Way) the fields at the origin of the variation of the constants are sourced by the local matter distribution so that one expect that the constants are not homogeneous on these scales. The motion of the Earth and the laboratory in which experiments are performed can then reveal this spatial dependence through modulations of the experimental results.

Besides, these studies can set constraints on the variation of the constant with either the gravitational potential, as expected in many scalar field models, or the local density of matter, as in chameleon mechanism; see Sect. 3.2. They are thus highly complementary to those in the laboratory and in intergalactic absorption systems.

## 8.1 Generalities

In order to determine the profile of the constants in the Solar system, assume that their values derive from a scalar field. As in Sect. 3.1, we consider that at lowest order the profile of this scalar field is described by the scalar-tensor theory of the type (60) in which fall most of the models studied in Sect. 7, taking into account that all masses scale as $\Lambda_{QCD}(\phi_*)$ where $\phi_*$ is the value of the field in the Einstein frame.

## 8.2 Local scales

### 8.2.1 Solar system scales

In order to infer the field distribution within the Solar system, we restrict to the weakly self-gravitating ($\Phi_*/c^2 \ll 1$) and slow moving ($T^{01} \ll T^{00}$) localized material systems and follow Damour and Esposito-Farèse (1992).

#### Scalar field profile

In the Einstein frame and for generic scalar field theories of the form (47) or (60), the Einstein equation (48) reduces to

$$ \Box_* \Phi_* = -4\pi G_* \sigma_* \qquad (370) $$

where $\Box_*$ is the d'Alembertian in Minkowski spacetime and $\sigma_*$ is given by

$$ \sigma_* = T_*^{00} + T_*^{ii} = \rho_* + 3P_* $$

The Klein-Gordon equation (49) reduces to Eq. (61) with the generic effective potential (62). Assuming the scalar-field energy density is negligible compared to
the matter energy density and restricting to non-relativistic matter ($\rho \gg P/3$), we end up with the system

$$ \Box_* \Phi_* = -4\pi G_* \rho_*, \qquad \Box_* \varphi_* = \frac{dV}{d\varphi_*} + 4\pi G \alpha(\varphi_*) \rho_* \quad (371) $$

where have gathered the sum on $\alpha_i \rho_i$ of Eq. (62) in a single term. Restricting to the static case with a single massive point source, the only non-vanishing source terms are $\sigma_*(\mathbf{r}) = M_* \delta^3(\mathbf{r}_*)$ so that the set of equations reduces to two Poisson equations

$$ \Delta_* \Phi_* = -4\pi G_* M_* \delta^3(\mathbf{r}_*) \qquad \Delta_* \varphi_* = \frac{dV}{d\varphi_*} + 4\pi G_* M_* \alpha(\varphi_*) \delta^3(\mathbf{r}_*) \quad (372) $$

The equation for $\varphi_*$ can be solved iteratively by expanding the field as $\varphi_* = \varphi_{0*} + \varphi_{1*}(r_*)$ so that at lowest order $\varphi_{1*}$ is solution of

$$ (\Delta_* - m_{\varphi}^2) \varphi_{1*} = 4\pi G_* M_* \alpha_0 \delta^3(\mathbf{r}_*) $$

with $\alpha_0 \equiv \alpha(\varphi_{0*})$ and $m_{\varphi}^2 = V''(\varphi_0)$. It follows that we end up with

$$ \Delta_* \Phi_* = -4\pi G_* M_* \delta^3(\mathbf{r}_*) \qquad (\Delta_* - m_{\varphi}^2) \varphi_{1*} = 4\pi G_* M_* \alpha_0 \delta^3(\mathbf{r}_*) \quad (373) $$

This set of equations are solved by means of the Green function

$$ G_{m_{\varphi}}(\mathbf{x} - \mathbf{x}') = -\frac{1}{4\pi} \frac{e^{-m_{\varphi}|\mathbf{x} - \mathbf{x}'|}}{|\mathbf{x} - \mathbf{x}'|} $$

to get

$$ \Phi_*(r_*) = \frac{G_* M_*}{r_*}, \qquad \varphi_{1*}(r) = -\frac{G_* M_* \alpha_0}{r_*} e^{-m_{\varphi} r_*} $$

This can be used to determine the metric and scalar field in the Jordan frame. It follows that at lowest order, they are given by

$$ \Phi_N = \frac{GM}{r}, \qquad \varphi_*(r) = \varphi_{*0} - \alpha_0 \Phi_N(r) e^{-m_{\varphi}r}, \quad (374) $$

where we have neglected the corrections $-\alpha(\varphi)(\varphi - \varphi_0)$ for the gravitational potential, which, given the Solar system constraints on $\alpha_0$, is a good approximation.

### Seasonal modulations of the fundamental constant

Now, let us consider any constant $c_i$ function of $\varphi$. Thus, its profile is given by $c_i(r) = c_i[\varphi(r)] = c_i(\varphi_0) - \alpha_0(d c_i / d\varphi)_{\varphi_0} \Phi_N(r) e^{-m_{\varphi}r}$ so that

$$ \frac{\Delta c_i}{c_i}(r) = -s_i(\varphi_0)\alpha_0 \Phi_N(r) e^{-m_{\varphi}r} \quad (375) $$

where $s_i(\varphi_0)$ is the sensitivity of the constant $\alpha_i$ to a variation of the scalar field as defined in Eq. (446), i.e., $s_i \equiv d \ln \alpha_i / d\varphi$.

For a laboratory in orbit on an elliptic trajectory,
$$r = \frac{a(1 - e^2)}{1 + e \cos \psi}, \quad \cos \psi = \frac{\cos E - e}{1 - e \cos E}, \quad t = \sqrt{\frac{a^3}{GM}(E - e \sin E)}$$

where a is the semi-major axis, e the eccentricity and ψ the true anomaly, which holds if the mass of the scalar field is small enough on the scale of the orbit, $m_φa \ll 1$, which we assume in the following. Hence, expending in e,

$$\frac{\Delta c_i}{c_i}(a, \psi) = -s_0 \alpha_0 \frac{GM}{a} - s_0 \alpha_0 \frac{GM}{a} e \cos \psi + \mathcal{O}(e^2).$$

The first term represents the variation of the mean value of the constant on the orbit compared to its cosmological value. This shows that local terrestrial and Solar system experiments do measure the effects of the cosmological variation of the constants (Damour and Esposito-Farèse 1992; Shaw and Barrow 2005, 2006; Shaw 2007). The second term is a seasonal modulation arising from the motion of the Earth within the static profile of φ.

## Parameterisation of the seasonal modulation

It is usually parameterized (Flambaum and Shuryak 2008)—coming back to our original notion $\alpha_i$ for a fundamental constant now that no confusion with $\alpha_0$ is possible—as

$$\left. \frac{\Delta \alpha_i}{\alpha_i} \right|_{\text{seasonal}} = k_i \frac{\Delta \Phi_N}{c^2}, \qquad (376)$$

defining the parameters $k_i$.

For a clock relying on the transition A, one can define the gravitational redshift—see Eq. (9)—as

$$\frac{\Delta v_A}{v_A} = (1 + \beta_A) \frac{\Delta \Phi_N}{c^2}. \qquad (377)$$

In General Relativity, as a consequence of LPI and as discussed in Eq. 8, $\beta_A = 0$. In particular when the WEP holds, the redshift does not depend on the clock. Note that the definition (377) involves $\Phi_N$ which may differ from U used in the definition (9). If LPI is violated, then the relative frequency drift between two clocks,

$$\Delta \ln y_{AB} = \beta_{AB} \frac{\Delta \Phi}{c^2} \qquad \beta_{AB} = \beta_A - \beta_B, \qquad (378)$$

may depend on the clock so that redshifts are no more achromatic.

From the definition of the sensitivity coefficients (116, 117), $\Delta \ln y_{AB} = \sum_i K_{AB}^i \Delta \ln \alpha_i$, which reduces thanks to (376) to $\sum_i K_{AB}^i k_i \Delta \Phi_N$. Hence,

$$\beta_{AB} = \sum_{i=\alpha_{EM}, \mu, q} K_{AB}^i k_i \Leftrightarrow \beta_A = \sum_{i=\alpha_{EM}, \mu, q} K_A^i k_i. \qquad (379)$$

In this expression the relative redshift, $\beta_{AB}$, is split into a linear combination of the clocks properties encoded in the sensitivity coefficients $K_A^i$ and on the parameter $k_i$ that describes the physics beyond the standard model, e.g., how the light field
couples to the standard model fields. It follows that one can extract constraints on the $k_i$ from different experiments.

Similarly, for the universality of free fall, the definition (376) implies that the general expression (26) for the Eötvös parameter reduces

$$ \eta_{AB} = \sum_i |\lambda_{Ai} - \lambda_{Bi}|k_i, \qquad (380) $$

since $|\nabla|\alpha_i = \alpha_i k_i g_N$ and making use of the relation (445) between the sensitivity coefficients $f_{Ai}$ and $\lambda_{Ai}$.

### 8.2.2 Atomic clock constraints

The parameters $k_i$ can be constrained from laboratory measurements on Earth and in particular atomic clocks experiments described in Sect. 5.1.2. by searching seasonal modulations instead of a linear drift.

#### Methodology

To that goal, one needs to recall that the Earth has an eccentricity $e \simeq 0.0167$ so that the Sun gravitational potential on the Terrestrial orbit should have a peak-to-peak amplitude of $2 GMe/ac^2 \sim 3.3 \times 10^{-10}$ on a period of 1 year. This shows that the order of magnitude of the constraints will be roughly of $10^{-16}/10^{-10} \sim 10^{-6}$ for clocks with an accuracy of $10^{-16}$. The expected magnitude of the constraints on $\beta$ for other systems are summarized in Table 22.

More precisely, the gravitational potential of the Sun on the Earth orbit can be approximated by

$$ \Phi(t) = \Phi_p + \Delta\Phi \cos\left[2\pi \frac{(t - t_p)}{T_0}\right], \qquad \Delta\Phi/c^2 \sim 1.65 \times 10^{-10} $$

with $\Phi_p$ and $t_p$ the potential and time at perihelion. $\Omega_0 = 2\pi/T_0$ is the pulsation of the motion so that $T_0$ can be chosen as annual, diurnal, etc. modulation timescale. Then, from the data, one can fit a profile of the form

<table><caption>**Table 22** The typical values of the potential and accuracy of frequency measurements for different systems allow one to estimate the expected magnitude of the constraints they can set on the parameters $\beta$</caption><thead><tr><th>Physical system</th><th>ΔΦ<sub>N</sub>/c<sup>2</sup></th><th>Typical clock accuracy</th><th>Expected β constraint</th></tr></thead><tbody><tr><td>Earth orbit</td><td>3.3 × 10<sup>-10</sup></td><td>10<sup>-16</sup> – 10<sup>-18</sup></td><td>10<sup>-6</sup> – 10<sup>-8</sup></td></tr><tr><td>Solar stars</td><td>2 × 10<sup>-7</sup></td><td>10<sup>-7</sup> – 10<sup>-8</sup></td><td>10<sup>-1</sup></td></tr><tr><td>White dwarfs</td><td>3 × 10<sup>-4</sup></td><td>10<sup>-7</sup> – 10<sup>-8</sup></td><td>10<sup>-3</sup></td></tr><tr><td>Neutron stars</td><td>0.5</td><td>10<sup>-2</sup></td><td>10<sup>-1</sup></td></tr></tbody></table>
<table><caption>**Table 23** Expressions of the $\beta_{12} = \beta_1 - \beta_2$ coefficients derived from Table (7) and Eq. (379) for the clock experiments summarized in Table 24</caption><thead><tr><th>Clock 1</th><th>Clock 2</th><th>(z<sub>EM</sub>, μ̄, X<sub>q</sub>)</th><th>(z<sub>EM</sub>, X<sub>e</sub>, X<sub>q</sub>)</th></tr></thead><tbody><tr><td>H</td><td>Cs</td><td>-0.83k<sub>z</sub> - 0.102k<sub>q</sub></td><td>id</td></tr><tr><td>H</td><td>Rb</td><td>-0.34k<sub>z</sub> - 0.081k<sub>q</sub></td><td>id</td></tr><tr><td>Rb</td><td>Cs</td><td>-0.49k<sub>z</sub> - 0.021k<sub>q</sub></td><td>id</td></tr><tr><td>Sr</td><td>Cs</td><td>-2.77k<sub>z</sub> - k<sub>μ</sub> - 0.002k<sub>q</sub></td><td>-2.77k<sub>z</sub> - k<sub>e</sub> + 0.046k<sub>q</sub></td></tr><tr><td>Hg</td><td>Cs</td><td>-5.77k<sub>z</sub> - k<sub>μ</sub> - 0.002k<sub>q</sub></td><td>-5.77k<sub>z</sub> - k<sub>e</sub> + 0.046k<sub>q</sub></td></tr><tr><td>Yb</td><td>Cs</td><td>-2.52k<sub>z</sub> - k<sub>μ</sub> - 0.002k<sub>q</sub></td><td>-2.52k<sub>z</sub> - k<sub>e</sub> + 0.046k<sub>q</sub></td></tr><tr><td>Yb-E3</td><td>Cs</td><td>-8.78k<sub>z</sub> - k<sub>μ</sub> - 0.002k<sub>q</sub></td><td>-8.78k<sub>z</sub> - k<sub>e</sub> + 0.046k<sub>q</sub></td></tr><tr><td>CSO</td><td>H</td><td>3k<sub>z</sub> - k<sub>μ</sub> + 0.1k<sub>q</sub></td><td>3k<sub>z</sub> - k<sub>e</sub> + 0.148k<sub>q</sub></td></tr><tr><td>Al<sup>+</sup></td><td>Hg<sup>+</sup></td><td>-2.948k<sub>z</sub></td><td>id</td></tr><tr><td>Dy-162</td><td>Dy-164</td><td>k<sub>z</sub></td><td>id</td></tr><tr><td>Yb-E3</td><td>Yb-E2</td><td>6.95k<sub>z</sub></td><td>id</td></tr></tbody></table>

$$y_{AB}(t) = y_0 + A_{AB} \cos \left[ 2\pi \frac{(t - t_p)}{T_0} \right].$$

The constraint on the amplitude $A_{AB}$ of the modulation of the relative frequencies with periodicity $T_0$ translates in a constraint on the redshift parameter

$$\beta_{AB} = A_{AB} (\Delta\Phi/c^2)^{-1}. \qquad (381)$$

$A_{AB}$ is an experimental constraint, $\Delta\Phi/c^2$ is known from the trajectory. Thanks to the $K_{AB}^i$ listed in Table 6, this provides a constraint on $k_i$.

### Atomic clock constraints

All the experiments used here are described in Sect. 5.1.2 and the summary of all the experiments are gathered in Table 9 and Fig. 28.

* Hg/Cs The data of Fortier et al. (2007) gives the constraint

$$\beta_{Hg,Cs} = (2 \pm 3.5) \times 10^{-6}. \qquad (382)$$

* H/Cs and H/Rb Bauch and Weyers (2002) compared a caesium atomic fountain frequency standard with a hydrogen maser for about 1 year to conclude that the frequency ratio did not change by more than $7 \times 10^{-15}$ over 6 months, hence

$$|\beta_{H,Cs}| < 2.1 \times 10^{-5}. \qquad (383)$$

This was improved thanks to the 7-yr comparison of caesium and hydrogen atomic clocks by Ashby et al. (2007) to
$$ \beta_{H,Cs} = (0.1 \pm 1.4) \times 10^{-6}. \qquad (384) $$

Peil et al. (2013) compared 3 atomic clocks based on the hyperfine transitions of a set of 70 commercial Cs-133 clocks, a Rb-87 fountain and a set of H-masers in a continuous operation over 1.5 years. They concluded

$$ \beta_{H,Cs} = (-0.7 \pm 1.1) \times 10^{-6} \qquad (385) $$

$$ \beta_{H,Rb} = (2.7 \pm 4.9) \times 10^{-7} \qquad (386) $$

as well as a constraint on $\beta_{Rb,Cs}$; see below. The comparison of H-maser to 3 separate Cs fountain clocks and one Rb fountain clock over more than 8 years (Tobar et al. 2013) concluded that

$$ \beta_{H,Rb} = (6.3 \pm 10) \times 10^{-6} \qquad (387) $$

$$ \beta_{H,Cs} = (3.6 \pm 4.8) \times 10^{-6}. $$

The latter was improved by the comparison between the long-term fractional frequency variation of 4 H-masers with caesium clocks (Ashby et al. 2018)

$$ \beta_{H,Cs} = (2.2 \pm 2.5) \times 10^{-7}. \qquad (388) $$

* Rb/Cs The data by Guena et al. (2012) translated to

<figure><img src="image_35.png" /><figcaption>Fig. 28 Atomic clock constraints on $\beta_{AB}$. The data are given in Table 24 and are split in 3 families according to the way they depend on $(k_\alpha, k_\mu, k_q)$; see Table 23</figcaption></figure>
$$ \beta_{Rb,cs} = (0.11 \pm 1.04) \times 10^{-6}. \qquad (389) $$

Peil et al. (2013) concluded from the experiment described above that

$$ \beta_{Rb,Cs} = (-1.6 \pm 1.3) \times 10^{-6}. \qquad (390) $$

* Dy-162/164 The atomic dyprosium experiment by Cingöz et al. (2008), was shown to imply (Ferrel et al. 2007)

$$ \beta_{Dy-162,Dy-164} = (-8.7 \pm 6.6) \times 10^{-6}. \qquad (391) $$

Similarly, the local dyprosium-clock constraints (141) by Leefer et al. (2013) can be translated to

$$ \beta_{Dy-162,Dy-164} = (-5.5 \pm 5.2) \times 10^{-7}. \qquad (392) $$

* Sr/Cs The caesium-strontium experiment by Blatt et al. (2008) led to

$$ \beta_{Sr,Cs} = (-5.7 \pm 9.1) \times 10^{-6}. \qquad (393) $$

Schwarz et al. (2020) later used 42 measurements of the transition frequency of ¹S₀-³P₀ in ⁸⁷Sr over 3 years from 2017 to 2019 compared to caesium to set

$$ \beta_{Sr,Cs} = (-1.1 \pm 5.2) \times 10^{-7}. \qquad (394) $$

* CSO/H-maser Tobar et al. (2010) compared a cryogenic sapphire oscillator (CSO) with various H-masers from 2001 to 2008. They constrained both the annual and diurnal modulations to set the bounds $\beta_{CSO,H-maser} = (-2.7 \pm 1.4) \times 10^{-4}$ and $\beta_{CSO,H-maser} = (-6.9 \pm 4.0) \times 10^{-4}$ respectively with a weighted mean of

$$ \beta_{CSO,H} = (-3.2 \pm 1.3) \times 10^{-4}. \qquad (395) $$

The sensitivity coefficients $K_{CSO,H}^i$ are gathered in Table 23.

* Al/Hg Dzuba and Flambaum (2017) reanalyzed the constraint by Rosenband et al. (2008) on Al⁺-Hg⁺ to conclude that

$$ \beta_{Al,Hg} = (0.16 \pm 0.3) \times 10^{-6}. \qquad (396) $$
* Yb/Cs McGrew et al. (2019a) compared their 8 month data of the measured Yb/Cs frequency ratio to get an amplitude $A = (0.1 \pm 1.5) \times 10^{-16}$ that translates, using Eq. (381), to

$$ \beta_{Yb,Cs} = (0.3 \pm 4.5) \times 10^{-7} \qquad (397) $$

* Yb E2-E3 The most recent comparisons of these Yb$^+$ E2 and E3 transitions and Cs clocks allowed Lange et al. (2021) to conclude that

$$ \beta_{Yb-E3,Cs} = (-4 \pm 44) \times 10^{-8}, \qquad (398) $$

$$ \beta_{Yb-E3,Yb-E2} = (-9.7 \pm 7.7) \times 10^{-8}. \qquad (399) $$

The second constraints was improved by Filzinger et al. (2023) to

$$ \beta_{Yb-E3,Yb-E2} = (-2.4 \pm 3.0) \times 10^{-9}. \qquad (400) $$

## Constraints on the parameters $k_i$

The previous experimental constraints can be combined to get bounds on ($k_\alpha, k_\mu, k_q, k_e$). They have evolved with the accuracy of the experiments and some of the sensitivity coefficients were reevaluated.

The first bounds (382) and (383) were translated by Flambaum and Shuryak (2008) into $k_\alpha + 0.17k_e = (-3.5 \pm 6) \times 10^{-7}$ and $|k_\alpha + 0.13k_q| < 2.5 \times 10^{-5}$, neglecting the contribution in $k_q$ for Hg/Cs. Then, it was combined with the dyprosium result (391) by Ferrel et al. (2007) to conclude that $k_e = (4.9 \pm 3.9) \times 10^{-5}$ and $k_q = (6.6 \pm 5.2) \times 10^{-5}$ while the Ashby et al. (2007) data on H/Cs improved to $k_\alpha + 0.13k_q = (-1 \pm 17) \times 10^{-7}$. When combined with H-maser (Ashby et al. 2007), the Sr/Cs bound (393) allowed Blatt et al. (2008) to get

$$ k_{\alpha} = (2.5 \pm 3.1) \times 10^{-6}, \quad k_{\mu} = (-1.3 \pm 1.7) \times 10^{-5}, \quad k_{q} = (-1.9 \pm 2.7) \times 10^{-5}. \qquad (401) $$

Barrow and Shaw (2008); Shaw and Barrow (2009) reanalyzed the data by Peik et al. (2008) on Yb$^+$-E2/Cs to conclude that $k_\alpha + 0.51k_\mu = (7.1 \pm 3.4) \times 10^{-6}$. Combined with (393), they got

$$ k_{\mu} = (3.9 \pm 3.1) \times 10^{-6}, \quad k_{q} = (0.1 \pm 1.4) \times 10^{-5}. \qquad (402) $$

Barrow and Shaw (2008) also used the data by Rosenband et al. (2008) on Al/Hg clocks to get $k_\alpha = (-5.4 \pm 5.1) \times 10^{-8}$. Combining with (397) the former constraints on $k_\alpha$ and $k_q$, McGrew et al. (2019a) concluded $k_\mu = (0.7 \pm 1.4) \times 10^{-6}$. Similarly from their result (394) combined with those by Rosenband et al. (2008), Dzuba and Flambaum (2017), Ashby et al. (2018), Schwarz et al. (2020) they reached $k_\mu = (3.5 \pm 5.9) \times 10^{-7}$. Then, to tighten the constraints, McGrew et al. (2019b) employed a multi-species analysis that can be used to link absolute
frequency measurements of different types of atomic clocks with optical ratios that have been measured with sufficient precision to get $k_μ = (-0.19 \pm 0.94) \times 10^{-6}$. Peil et al. (2013) performed a global fit including their data on H/Cs, H/Rb and Rb/Cs together with those by Ashby et al. (2007), Fortier et al. (2007), Blatt et al. (2008), Guena et al. (2012) to conclude

$$k_α = (1.7 \pm 37.5) \times 10^{-7} \qquad k_μ = (-2.5 \pm 5.4) \times 10^{-6} \qquad k_q = (3.8 \pm 4.9) \times 10^{-6}. \qquad (403)$$

Then, a series of constraints on $k_α$ independently of $(k_μ, k_q)$ were obtained (1) from the Al/Hg (396) as $k_α = (5.3 \pm 10) \times 10^{-8}$ which, combined with (388) implies $k_q = (-2.6 \pm 2.6) \times 10^{-6}$; (2) from Dy-clocks (Leefer et al. 2013) as $(-5.5 \pm 5.2) \times 10^{-7}$ and (3) from Yb:E3-E2, first by Lange et al. (2021) to get $k_α = (1.4 \pm 1.1) \times 10^{-8}$, which combined with their YE3-Cs data implied $k_μ = (7 \pm 45) \times 10^{-8}$, and finally improved by Filzinger et al. (2023) to

$$k_α = (-2.4 \pm 3.0) \times 10^{-9}. \qquad (404)$$

## Conclusion

To conclude this analysis, we perform a combined fit of the latest data; see Fig. 29. First, the best constraint on $k_α$ alone is given by Eq. (404).

The atomic clock constraints on seasonal modulations are summarized in Table 24. It allows us to get the up-to-date constraints on the parameters $k$.

<figure><img src="image_36.png" /><figcaption>Fig. 29 Atomic clock constraints on $(k_α, k_μ, k_q)$. The data are given in Table 24 and dependence on $(k_α, k_μ, k_q)$ in Table 23. The left plot gathers the experiments that depend on $(k_α, k_μ + 0.002k_q)$, Red: Hg-Cs (Fortier et al. 2007); Blue: Sr-Cs (Schwarz et al. 2020); Green: Yb-Cs (McGrew et al. 2019b); and Purple: Yb(E3)-Cs (Lange et al. 2021). The right plot gathers the experiments that depend on $(k_α, k_q)$, Red: H-Cs (Ashby et al. 2018); Blue: H-Rb (Peil et al. 2013) and Green: Rb-Cs (Guena et al. 2012); On both plots the vertical line stands for the $1σ$ contour on $k_α$ from Filzinger et al. (2023); see Eq. (404) while the red and black dots are the best fits of all the data and all the data but the constraints on $k_α$ respectively</figcaption></figure>
<table><caption>**Table 24** Summary of the constraints on the $\beta$ discussed in the text</caption><thead><tr><th>Clock 1</th><th>Clock 2</th><th>β<sub>12</sub></th><th>Note</th><th>Ref</th></tr></thead><tbody><tr><td>H</td><td>Cs</td><td>&lt;2.1 × 10<sup>-5</sup></td><td>k<sub>z</sub>, k<sub>q</sub></td><td>Bauch and Weyers (2002)</td></tr><tr><td>H</td><td>Cs</td><td>(0.1 ± 1.4) × 10<sup>-6</sup></td><td></td><td>Ashby et al. (2007)</td></tr><tr><td>H</td><td>Cs</td><td>(3.6 ± 4.8) × 10<sup>-6</sup></td><td></td><td>Tobar et al. (2013)</td></tr><tr><td>H</td><td>Cs</td><td>(-0.7 ± 1.1) × 10<sup>-6</sup></td><td></td><td>Peil et al. (2013)</td></tr><tr><td>H</td><td>Cs</td><td>(0.22 ± 0.25) × 10<sup>-6</sup></td><td></td><td>Ashby et al. (2018)</td></tr><tr><td>H</td><td>Rb</td><td>(0.27 ± 0.49) × 10<sup>-6</sup></td><td></td><td>Peil et al. (2013)</td></tr><tr><td>H</td><td>Rb</td><td>(6.3 ± 10) × 10<sup>-6</sup></td><td></td><td>Tobar et al. (2013)</td></tr><tr><td>Rb</td><td>Cs</td><td>(0.11 ± 1.04) × 10<sup>-6</sup></td><td></td><td>Guena et al. (2012)</td></tr><tr><td>Rb</td><td>Cs</td><td>(-1.6 ± 1.3) × 10<sup>-6</sup></td><td></td><td>Peil et al. (2013)</td></tr><tr><td>Hg</td><td>Cs</td><td>(2 ± 3.5) × 10<sup>-6</sup></td><td>k<sub>z</sub>, k<sub>μ</sub> + 0.002k<sub>q</sub></td><td>Fortier et al. (2007)</td></tr><tr><td>Sr</td><td>Cs</td><td>(-5.7 ± 9.1) × 10<sup>-6</sup></td><td></td><td>Blatt et al. (2008)</td></tr><tr><td>Sr</td><td>Cs</td><td>(-0.11 ± 0.52) × 10<sup>-6</sup></td><td></td><td>Schwarz et al. (2020)</td></tr><tr><td>Yb</td><td>Cs</td><td>(0.3 ± 4.5) × 10<sup>-6</sup></td><td></td><td>McGrew et al. (2019b)</td></tr><tr><td>Yb-E3</td><td>Cs</td><td>(-4 ± 44) × 10<sup>-8</sup></td><td></td><td>Lange et al. (2021)</td></tr><tr><td>Dy-162</td><td>Dy-164</td><td>(-0.55 ± 0.52) × 10<sup>-6</sup></td><td>k<sub>z</sub></td><td>Leefer et al. (2013)</td></tr><tr><td>Al<sup>+</sup></td><td>Hg<sup>+</sup></td><td>(0.16 ± 0.3) × 10<sup>-6</sup></td><td></td><td>Rosenband et al. (2008)</td></tr><tr><td>Yb-E3</td><td>Yb-E2</td><td>(-9.7 ± 7.7) × 10<sup>-8</sup></td><td></td><td>Lange et al. (2021)</td></tr><tr><td>Yb-E3</td><td>Yb-E2</td><td>(-2.4 ± 3.0) × 10<sup>-9</sup></td><td></td><td>Filzinger et al. (2023)</td></tr><tr><td>CSO</td><td>H</td><td>(-2.7 ± 1.4) × 10<sup>-4</sup></td><td>[annual] k<sub>z</sub>, k<sub>μ</sub>, k<sub>q</sub></td><td>Tobar et al. (2010)</td></tr><tr><td>CSO</td><td>H</td><td>(-6.9 ± 4.0) × 10<sup>-4</sup></td><td>[diurnal]</td><td>Tobar et al. (2010)</td></tr><tr><td>CSO</td><td>H</td><td>(-3.2 ± 1.3) × 10<sup>-4</sup></td><td>[mean]</td><td>Tobar et al. (2010)</td></tr></tbody></table>

$$k_{\alpha} = (-2.4 \pm 3.0) \times 10^{-9} \qquad (405)$$

$$k_{\mu} = (0.28 \pm 4.2) \times 10^{-7} \qquad (406)$$

$$k_q = (-2.31 \pm 2.27) \times 10^{-6}, \qquad (407)$$

from which one extracts, thanks to Eq. (96),

$$k_e = (1.2 \pm 4.3) \times 10^{-7}. \qquad (408)$$

All were derived using the sensitivity coefficients computed in Angstmann et al. (2004), Flambaum and Tedesco (2006) and gathered in Tables 23.

Still, we shall mention Jenkins et al. (2009) who claimed for an unexplained seasonal variation that demonstrated the difficulty to interpret this phenomena. We also note that Roberts et al. (2020) used data from a European network of fiber-linked optical atomic clocks to constrain transient variation of the fine structure constant.
All these bounds can and will be improved by comparing clocks on Earth and onboard of satellites (Salomon et al. 2001; Maleki and Prestage 2004; Flambaum and Shuryak 2008) while the observation of atomic spectra near the Sun can lead to an accuracy of order unity (Flambaum and Shuryak 2008). A space mission with atomic clocks onboard and sent to the Sun (Maleki and Prestage 2004), on highly eccentric terrestrial orbit (Schiller et al. 2009) or on a Solar system escape trajectory (Wolf et al. 2009) could reach an accuracy of $10^{-8}$; see Sect. 5.1.4.

### 8.2.3 Milky Way

All the observational methods that have been described in Sect. 5.5 can be applied locally within our Galaxy, the Milky Way, provided the detection of local interstellar absorption system.

#### Amonia inversion method

The first attempt (Levshakov et al. 2008a) to study $\mu$ in the Milky Way was based on 207 individual measurements in the Perseus molecular cloud (PC), the Pipe Nebula (PN), and the infrared dark clouds (IRDCs) and the analysis relied on measurements of the relative radial velocity offsets between NH$_3$ inversion lines and other molecular transitions. It concluded that $\Delta\mu/\mu = (3.5 \pm 1.4) \times 10^{-8}$, which was at that time the most accurate estimate of this quantity based on the spectral analysis of astronomical objects. The Perseus molecular cloud is located towards the galactic anti-center, whereas the Pipe Nebula and the IRDCs in the direction of the galactic center. Modeling the gravitational potential as

$$ \Phi = \Phi_{\text{MW}} + \Phi_{\text{cloud}} $$

where $\Phi_{\text{MW}}$ can be calculated from analytic models of the Milky Way and $\Phi_{\text{cloud}}$ is the molecular cloud gravitational potential, one can translate this bound to $k_{\mu} \sim 1$. This early result was in contradiction with the local constraint (401) if $\Delta\mu/\mu$ follows the gradient of the local gravitational potential. Indeed a logical possibility would be that a chameleon field is at work—see Sect. 3.2—which motivates the effort to obtain measurements in different environments. Molaro et al. (2009) applied the ammonia method to an atlas of 193 dense protostellar and prestellar cores of low masses in the Perseus molecular cloud as well as the dark cloud L183 with observations of NH$_3$(1,1) and (2,2) inversion lines. They concluded that $\Delta\mu/\mu < 10^{-7}$. Two similar analysis were performed first by Levshakov et al. (2010b) using high resolution spectral observations of molecular core in lines of NH$_3$, HC$_3$N and N$_2$H$^+$ with 3 radio-telescopes to get $|\Delta\mu/\mu| < 3 \times 10^{-8}$ between the cloud and the local laboratory environments. However, an offset was measured that could be interpreted as a variation of $\mu$ of amplitude $\Delta\bar{\mu}/\bar{\mu} = (2.2 \pm 0.4_{\text{stat}} \pm 0.3_{\text{sys}}) \times 10^{-8}$. Then, a second analysis (Levshakov et al. 2010a) mapped four molecular cores L1498, L1512, L1517, and L1400K selected from the previous sample in order to estimate systematic effects due to possible velocity gradients. The measured velocity offset, once expressed in terms of $\Delta\bar{\mu}$, gives $\Delta\bar{\mu} = (26 \pm 1_{\text{stat}} \pm 3_{\text{sys}}) \times 10^{-9}$. The same technique was followed by Levshakov et al. (2013, 2014) with observation from the
Medicina 32-m (towards the cores L1512 and L1498 are among the narrowest molecular lines known in the interstellar medium) and the Effelsberg 100-m radiotelescopes (to observe the two molecular cores L1512 and L1498). They revealed a systematic error in the radial velocities of an amplitude of about 0.02 km s⁻¹ that was conservatively assign to the possible systematic error of the Medicina dataset, hence concluding

$$ \Delta\bar{\mu}/\bar{\mu} = (0.1 \pm 2.2) \times 10^{-8} \text{ [L1521]} \qquad \Delta\bar{\mu}/\bar{\mu} = (-0.1 \pm 2.3) \times 10^{-8} \text{ [L1498]} \tag{409} $$

leading to $\Delta\mu/\mu < 2 \times 10^{-8}$ at $3\sigma$.

## CO vs C₁ and C₁₁

Following the same method as in Sect. 5.6.4, Levshakov et al. (2010c) measured the offsets between the radial velocities of the rotational transitions of carbon-13 monoxide and the fine structure transitions of neutral and singly ionized carbon that probe the variation of $F = \alpha_{EM}^2\mu$ over the galaxy. It concluded that

$$ |\Delta F'/F'| < 3.7 \times 10^{-7} \tag{410} $$

between high (terrestrial) and low (interstellar) densities of baryonic matter. Combined with the previous constraint on $\mu$, it implies that $|\Delta\alpha_{EM}/\alpha_{EM}| < 2 \times 10^{-7}$. This was updated (Levshakov et al. 2009) to $|\Delta F'/F'| < 2.3 \times 10^{-7}$ so that $|\Delta\alpha_{EM}/\alpha_{EM}| < 1.1 \times 10^{-7}$. A later study (Levshakov et al. 2017) with $^{12}$CO and $^{13}$CO in the dark cloud L1599B concluded that $|\Delta F'/F'| < 3 \times 10^{-7}$.

An analysis of [CII] lines (Herschell observatory) and CO(2,1) (IRAM) from the regions BCLMP691 and BCLMP303 close to the dynamical center of the Triangulum galaxy (M33) led Levshakov et al. (2017) to set

$$ \Delta F'/F' = (-0.3 \pm 4.6) \times 10^{-7} \quad \text{[M33].} \tag{411} $$

Levshakov et al. (2019) applied this method to the Magellanic Clouds dwarf galaxies that are known to be dark matter dominated. Thanks to the Herschel Space Observatory they concluded that $\Delta F'/F' < 2 \times 10^{-7}$ after averaging over 9 positions in the LMC while one position observed with a higher spectral resolution gave

$$ \Delta F'/F' = (-1.7 \pm 0.7) \times 10^{-7}. \tag{412} $$

This offset is still not fully understood and further investigations are required to test whether it can be related to chemical segregation in the emitting gas or merely due to Doppler noise. Nevertheless this opens a window to track a specific coupling to dark matter.

## CH

Truppe et al. (2013) developed a spectroscopic method for pulsed beams of cold molecules, and use it to measure the frequencies of microwave transitions in CH with accuracy down to 3 Hz to compare this transition to the ground state A-doublet transition of OH. From 5 absorption systems toward 5 stellar sources, each system
having a density estimated 10¹⁹ times smaller than on Earth, they conclude that $\Delta F'/F' = (-0.7 \pm 2.2) \times 10^{-7}$ after averaging on their data.

### CH₃OH

The detection of methanol in the Galaxy led Levshakov et al. (2011) to conclude that it can reach constraint of $\Delta\mu/\mu < 29 \times 10^{-9}$ at $1\sigma$ so that this technique deserves attention. Daprà et al. (2017a) applied it to the dense dark cloud core L1498 in the Taurus-Auriga complex. They detected 5 methanol transitions towards two different positions (IRAM) leading to the two independent constraints $\Delta\mu/\mu(1) = (-3.2 \pm 2.0) \times 10^{-8}$ and $\Delta\mu/\mu(2) = (-3.8 \pm 6.6) \times 10^{-8}$ with weighted mean

$$ \Delta\mu/\mu = (-3.3 \pm 1.9) \times 10^{-8} \quad [\text{L1498}]. \qquad (413) $$

Combined with (409) for L1498, one obtains $\Delta\mu/\mu = (-3.2 \pm 1.5) \times 10^{-8}$ and, if averaged with L1521, $\Delta\mu/\mu = (-1.2 \pm 0.9) \times 10^{-8}$ at $1\sigma$ in the Milky Way.

Levshakov et al. (2022) measured $\Delta\mu/\mu$ in the Milky Way disk thanks to methanol masers from the northern Galactic hemisphere distributed over the galactocentric distance range of 4–12 kpc. Only 7 maser sources out of 229 could be selected from the MSX Source (RMS) catalogue and 11 objects from 144 sources from the Bolocam Galactic Plane Survey. The analysis allowed them to set a constraint on the spatial variation of $\mu$ in the Galaxy and its absolute variation in the range 4–12 kpc,

$$ |\Delta\mu/\mu_{4-12\,\text{kpc}}| < 2 \times 10^{-8}, \quad k_{\mu} < 2 \times 10^{-9}. \qquad (414) $$

### 8.2.4 Towards strong field environments

Measuring fundamental constants in stronger field regions opens a window to models with environmental dependencies such as the chameleon theory and better tests on their dependencies on the gravitational potential. It was proposed by Berengut et al. (2013b) to measure $\alpha_{EM}$ in strong field environment using metal lines in the absorption spectra of white dwarfs since their local gravitational potential is higher than in the Solar system.

#### White Dwarf atmosphere

White dwarfs are interesting systems to test for the variation of fundamental constants. First, they have a strong surface gravity,

$$ \log g \equiv \log\left(\frac{GM}{r^2}/1\,\text{cm s}^{-2}\right), \qquad (415) $$

reaching 7–8. Second, DA white dwarfs have a pure hydrogen photosphere but thanks to levitation and accretion from the interstellar medium, metallic elements can also be found in their atmosphere. Third, they are bright in UV, require pace observation and comprise about 85% of all white dwarfs.
This idea was first applied by Berengut et al. (2013b) to the hot hydrogen rich white dwarf G191-B2B located at about 45 pc from the Earth observed by HST Imaging Spectrograph with absorption lines from Fe v and Ni v in its atmosphere. With a radius of $0.22R_⊙$ and a mass of $0.51M_⊙$, one estimates that $g \equiv GM/R^2 \sim 10.5g_⊙ \sim 280g_{Earth}$ so that $\Phi_N/c^2 \sim 4.9 \times 10^{-5}$ much larger than $3 \times 10^{-10}$ on the Earth orbit. Comparing with laboratory spectrum, the 106 Fe v transitions led to, using Eq. (376),

$$ \Delta\alpha_{EM}/\alpha_{EM} = (4.2 \pm 1.6) \times 10^{-5} \quad (\text{Fe v, G191-B2B}) \quad (416) $$

showing a $2.6\sigma$ deviation from zero, that can be rephrased as $k_{\alpha_{EM}} = 0.7 \pm 0.3$. The analysis of 32 Ni V transitions gave

$$ \Delta\alpha_{EM}/\alpha_{EM} = (-6.1 \pm 5.8) \times 10^{-5} \quad (\text{Ni v, G191-B2B}) \quad (417) $$

showing a $1.05\sigma$ deviation from zero. These two measurements on the same system seem inconsistent since they indicate variation with different signs. Berengut et al. (2013b) suggested that this inconsistency is due to a systematic effect in the laboratory wavelengths. This first analysis has the advantage to use typically hundreds of lines compare to tens for QSO observation. Bainbridge et al. (2017) presented preliminary results on a similar analysis by considering new laboratory data and by including nine more hot, bright white dwarfs and sub-dwarfs and three bright white dwarfs known to have photospheric Fe and Ni absorption lines (see their Fig. 1 depicting $\Delta\alpha_{EM}/\alpha_{EM}$ for various surface gravity (log g ranging from 3.9 to 7.9).

Hu et al. (2019) estimated the effect of the stellar magnetic field that will alter absorption line profiles and introduce additional uncertainties in measurements of $\alpha_{EM}$. In the case of G191-B2B they obtained an upper limit on its magnetic field of $B<2300$ G ($3\sigma$) and concluded that the impact of quadratic Zeeman shifts on measurements of $\alpha_{EM}$ is 4 orders of magnitude below laboratory wavelength uncertainties. Hu et al. (2020) proposed a new analysis with two new and independent laboratory samples for Fe v, the re-reduction of HST data, new atomic calculation of the sensitivities for Fe v, and a better analysis of the blending of the absorption lines. This led to

$$ \begin{aligned} \Delta\alpha_{EM}/\alpha_{EM} &= [6.36 \pm (0.33_{\text{stat}} + 1.94_{\text{syst}})] \times 10^{-5} \\ \Delta\alpha_{EM}/\alpha_{EM} &= [4.21 \pm (0.47_{\text{stat}} + 2.35_{\text{syst}})] \times 10^{-5} \end{aligned} \quad (418) $$

for each of the laboratory data, showing that the systematic uncertainties are dominated by laboratory wavelength precision, see e.g., Hu (2019), Lee et al. (2022) for an analysis of the systematic effects. Lee et al. (2025) considered the effects of continuum placement error that was concluded to impact significantly the $\alpha_{EM}$ measurements. From the analysis of Ni v, they concluded for an inconsistency with the Fe v measurements (418)—the least negative of their measurements being $\Delta\alpha_{EM}/\alpha_{EM} = (-1.462 \pm 1.121) \times 10^{-5}$—suspecting that this $3.2\sigma$ difference arises from unknown laboratory wavelength systematics.

Bagdonaitė et al. (2014a), Salumbides et al. (2015) followed the same strategy while focusing on the Lyman transitions molecular H$_2$ in the atmosphere of the
white dwarfs GD133 with a surface potential $\Phi_N/c^2 \sim 1.2 \times 10^{-4}$ and G29-38 (with $\Phi_N/c^2 \sim 1.9 \times 10^{-4}$) observed with the HST Cosmic Origin Spectrograph. They concluded

$$ \begin{aligned} \Delta\mu/\mu &= (-2.3 \pm 4.7) \times 10^{-5} & \text{GD133} \\ \Delta\mu/\mu &= (-5.8 \pm 3.7) \times 10^{-5} & \text{GD29-38.} \end{aligned} \qquad (419) $$

## Solar twins

The method can be applied to any stellar atmosphere. The *Solar twin method* was proposed by Berke et al. (2022a) who analysed the $\alpha_{EM}$-dependence of the absorption spectra of 17 stars, selected to have almost identical atmospheric properties as the Sun in order to reduce systematic effects, observed by ESO/VLT HARPS spectrograph. A weighted mean of the individual stellar constraints (Murphy et al. 2022a) (see Berke et al. 2022b for discussion of systematic errors) gave

$$ \Delta\alpha_{EM}/\alpha_{EM} = [7 \pm 5_{\text{stat}} \pm 11_{\text{syst}}) \times 10^{-6} \qquad (420) $$

on a local neighborhood of 50 pc around the Sun.

## Compact objects

Stronger field environments exist, e.g., around black holes. Hees et al. (2020) analyzed the spectra of 5 late-type evolved giant stars from the S-star cluster

<table><caption>**Table 25** Summary of the constraints on the variations of $\mu$ and $\alpha_{EM}$ in strong field environments</caption><thead><tr><th>Object</th><th>Φ<sub>N</sub>/c<sup>2</sup></th><th>constraint</th><th>constant</th><th>method</th><th>Ref</th></tr></thead><tbody><tr><td rowspan="4">G191-<br>B2B</td><td rowspan="4">4.9 × 10<sup>-5</sup></td><td>(4.2 ± 1.6) × 10<sup>-5</sup></td><td>α<sub>EM</sub></td><td>Fe v</td><td>Berengut et al.<br>(2013b)</td></tr><tr><td>(-6.1 ± 5.8) × 10<sup>-5</sup></td><td>α<sub>EM</sub></td><td>Ni v</td><td>Berengut et al.<br>(2013b)</td></tr><tr><td>[4.21 ± (0.47<sub>stat</sub> + 2.35<sub>syst</sub>)]</td><td>α<sub>EM</sub></td><td>Fe v</td><td>Hu et al. (2020)</td></tr><tr><td>[6.36 ± (0.33<sub>stat</sub> + 1.94<sub>syst</sub>) × 10<sup>-5</sup>]</td><td>α<sub>EM</sub></td><td>Fe v</td><td>Hu et al. (2020)</td></tr><tr><td>GD133</td><td>1.2 × 10<sup>-4</sup></td><td>(-2.3 ± 4.7) × 10<sup>-5</sup></td><td>μ</td><td>H<sub>2</sub></td><td>Bagdonaitė et al.<br>(2014a)</td></tr><tr><td>GD29-<br>38</td><td>1.9 × 10<sup>-4</sup></td><td>(-5.8 ± 3.7) × 10<sup>-5</sup></td><td>μ</td><td>H<sub>2</sub></td><td>Bagdonaitė et al.<br>(2014a)</td></tr><tr><td>S0-6</td><td>2.4 × 10<sup>-6</sup></td><td>(1.0 ± 1.2) × 10<sup>-4</sup></td><td>α<sub>EM</sub></td><td>MM</td><td>Hees et al. (2020)</td></tr><tr><td>S0-12</td><td>1.6 × 10<sup>-6</sup></td><td>(-0.3 ± 1.4) × 10<sup>-4</sup></td><td>α<sub>EM</sub></td><td>MM</td><td>Hees et al. (2020)</td></tr><tr><td>S0-13</td><td>9.4 × 10<sup>-7</sup></td><td>(0.03 ± 1.3.5) × 10<sup>-4</sup></td><td>α<sub>EM</sub></td><td>MM</td><td>Hees et al. (2020)</td></tr><tr><td>S1-5</td><td>6.5 × 10<sup>-7</sup></td><td>(-0.7 ± 2.4) × 10<sup>-4</sup>,</td><td>α<sub>EM</sub></td><td>MM</td><td>Hees et al. (2020)</td></tr><tr><td>S1-23</td><td>4.6 × 10<sup>-7</sup></td><td>(0.9 ± 5.8) × 10<sup>-4</sup></td><td>α<sub>EM</sub></td><td>MM</td><td>Hees et al. (2020)</td></tr></tbody></table>

We recall that $\Phi_N/c^2 \sim 9.8 \times 10^{-9}$ and $g \sim 9.8 \text{ m s}^{-2}$ at the Earth surface. The two results by Hu et al. (2020) rely on the same data but on different laboratory wavelengths
orbiting the central supermassive black hole SgrA* in a similar way as for the many-multiplet method. For each star, one got ($\Delta\alpha_{EM}/\alpha_{EM}$, $\Phi_N/c^2$,) (individual constraints are gathered in Table 25) and the combined analysis imposes

$$ \Delta\alpha_{EM}/\alpha_{EM} = (1.0 \pm 5.8) \times 10^{-9} \qquad (421) $$

between the Galactic center and the Earth. Indeed, those stars are relatively far from the black hole so that $\Phi_N$ is of the same order as for white dwarf stars but the eccentricity of their trajectory is an asset to control systematics.

### Further ideas

To reach higher fields, it was proposed (Bambi 2014) that the analysis of fluorescent emission lines in the reflection spectra of black holes can potentially test variations of fundamental constants in gravitational fields up to $\Phi_N/c^2 \sim 1$ however the construction of a theoretical model to analyze a full reflection spectrum in the presence of variations of fundamental constants remains a challenge (see the example of Davis et al. (2016) who considered a model in which the masses of elementary particles depend on a scalar field around a black hole).

From the theory side, let us mention that the effects of varying fundamental constants on the white dwarf mass-radius relation were recently investigated by Magano et al. (2017) for polytropic models. It indicates that independent measurements of the mass and radius, together with direct spectroscopic measurements of $\alpha_{EM}$ in white dwarf atmospheres such as those discussed above, could constrain unification scenarios which interestingly are almost orthogonal to those coming from atomic clocks (see Sect. 4.5).

## 8.3 Cosmological scales

### 8.3.1 Primordial fluctuations

During inflation, any light scalar field develop super-Hubble fluctuations of quantum origin, with an almost scale invariant power spectrum (see Chapt. 8 of Peter and Uzan 2009). It follows that if the fundamental constants depend on such a field, their value must fluctuate and have a non-vanishing correlation function. More important these fluctuations can be correlated with the metric perturbations. More speculative is the effect that such fluctuations can have during preheating after inflation since the decay rate of the inflaton in particles may fluctuate on large scales (Kofman 2003; Kofman et al. 2004).

In such a case, the fine-structure constant will behave as $\alpha_{EM} = \alpha_{EM}(t) + \delta\alpha_{EM}(x, t)$, the fluctuations being a stochastic variable. As seen earlier, $\alpha_{EM}$ enters the dynamics of recombination, which would then become patchy. This has several consequences for the CMB anisotropies. In a very general way, assume a constant, X say, depends on the local value of a dynamical scalar field $\phi$. The value of X at the observation point is compared to its value here and today,
$$ \Delta X/X_0 \equiv X(\phi)/X(\phi_0) - 1. $$

Decomposing the scalar field as $\phi = \phi_0 + \Delta\phi$, one gets that $\Delta X/X_0 = s_X(\phi)\Delta\phi$, with $s_X$ defined in Eq. (446). Now the scalar field can be decomposed into a background and perturbations as $\phi = \bar{\phi}(t) + \delta\phi(\mathbf{x}, t)$ where the background value depends only on $t$ because of the Copernican hypothesis. It follows that

$$ \begin{aligned} \frac{\Delta X(\mathbf{x}, t)}{X_0} &= s_X(\bar{\phi})[\bar{\phi}(t) - \phi_0] + \{s_X(\bar{\phi}) + s'_X(\bar{\phi})[\bar{\phi}(t) - \phi_0]\}\delta\phi(\mathbf{x}, t) \quad (422) \\ &\equiv s_X(\bar{\phi})\Delta\bar{\phi} + S_X(\bar{\phi})\delta\phi(\mathbf{x}, t). \end{aligned} $$

The first term of the r.h.s. depends only on time while the second is spacetime dependent. It is also expected that the second term in the curly brackets is negligible with respect to the first, i.e., $S_X(\bar{\phi}) \sim s_X(\bar{\phi})$. It follows that one needs $\delta\phi(\mathbf{x}, t)$ not to be small compared to the background evolution term $\Delta\bar{\phi}$ for the spatial dependence to dominate over the large scale time dependence. This can be achieved for instance if $\phi$ is a seed field whose mean value is frozen. Because of statistical isotropy, and in the same way as for CMB anisotropies (see, e.g., Peter and Uzan 2009), one can express the equal-time angular power spectrum of $\Delta X/X_0$ for two events on our past lightcone as

$$ \left\langle \frac{\Delta X(\mathbf{n}_1, r, t)}{X_0} \frac{\Delta X(\mathbf{n}_2, r, t)}{X_0} \right\rangle = \sum_{\ell} \frac{2\ell + 1}{4\pi} C_{\ell}^{(XX)}(z) P_{\ell}(\mathbf{n}_1 \cdot \mathbf{n}_2). \quad (423) $$

If $\delta\phi$ is a stochastic field characterized by its power spectrum, $\langle\delta\phi(\mathbf{k}_1, t)\delta\phi(\mathbf{k}_2, t)\rangle = P_\phi(k, t)\delta(\mathbf{k}_1 + \mathbf{k}_2)$ in Fourier space, then

$$ C_{\ell}^{(XX)}(z) = \frac{2}{\pi} S_X^2[\bar{\phi}(z)] \int P_{\phi}(k, z) j_{\ell}[k(\eta_0 - \eta)] k^2 dk, \quad (424) $$

$j_\ell$ being a spherical Bessel function. For instance, if $P_\phi \propto k^{n_s-1}$ where $n_s$ is a spectral index, $n_s = 1$ corresponding to scale invariance, one gets that $\ell(\ell+1)C_{\ell}^{(XX)} \propto \ell^{n_s-1}$ on large angular scales.

Besides, similarly to weak gravitational lensing, these fluctuations of $\alpha_{\text{EM}}$ modify the mean power spectra (this is a negligible effect) and induce a curl component (B mode) to the polarization (Sigurdson et al. 2003). Such spatial fluctuations also induce non-Gaussian temperature and polarization correlations in the CMB (Sigurdson et al. 2003; Pitrou et al. 2008). Such correlations have not allowed to set observational constraints yet but they need to be included for consistency, see e.g., the example of CMB computation in scalar-tensor theories (Riazuelo and Uzan 2002). The effect on large-scale structure was also studied in Mota and Barrow (2004), Barrow (2005a) and the Keck/HIRES QSO absorption spectra showed (Murphy et al. 2001d) that the correlation function of the fine-structure constant is consistent on scales ranging between 0.2 and 13 Gpc.

From the 4-point correlation of the CMB temperature anisotropies of the Planck data, O'Bryan et al. (2015) constrained the fractional rms of the fluctuations of a
stochastic $\alpha_{EM}$ to $(1.34 \pm 5.82) \times 10^{-2}$ at 95% C.L. on scales larger than 10 deg. Smith et al. (2019) put a 95% C.L. limit on the amplitude of a scale-invariant angular power spectrum of $\alpha_{EM}$ fluctuation, $C_{\ell}(\alpha_{EM}) = A\ell(\ell + 1)$, of $A < 1.6 \times 10^{-5}$ form Planck measurements of the temperature and polarization power spectrum.

### 8.3.2 Dipolar modulation

The astronomical measurements of the fundamental constants described in Sect. 5.5 have been used to question the homogeneity of $\alpha_{EM}$ and $\mu$ by constraining a dipole modulation. Such analysis may reveal a spatial variation that could have consequences for the validity of the Copernican principle. Indeed if the putative scalar field $\phi$ causing the variation had a dipole, it would reflect in the distribution of the fundamental constants. Such an assumption enlarges the space of free parameters by including the dipole direction $d$ in the data analysis.

* QSO data. In the analysis of the spectroscopic data from QSO, it has been claimed (Berengut et al. 2011; Webb et al. 2011; King et al. 2012b) that $\alpha_{EM}$ may have a dipolar variation that would explain consistently the data from the Southern and Northern hemispheres (see Sect. 5.5.3). Webb et al. (2011) decomposed the value of the fine structure of each absorption system as

$$ \frac{\Delta\alpha_{EM}}{\alpha_{EM}}(z, \mathbf{n}) = A(z) + B(z) \cos \theta \quad (425) $$

where $\theta$ is the angle between the dipole direction and the line of sight $\mathbf{n}$, i.e., $\cos \theta = \mathbf{n} \cdot \mathbf{d}$. Three main models where considered for $B(z)$:

* (dip1) $B = B_0$,

* (dip2): $B = B_0r(z)$ with $r(z)/c$ the look-back time,

* (dip3): $B = B_0z^\beta$.

Webb et al. (2011) concluded that in scenario (dip1) the dipole has an amplitude

$$ B = (1.02 \pm 0.21) \times 10^{-5} \quad (426) $$

and direction (RA,DEC) = $(17.4 \pm 0.9 \text{ hour}, -58^\circ \pm 9^\circ)$. In the scenario (dip2)

$$ B = (1.01 \pm 0.25) \times 10^{-6} \text{Gly}^{-1} \quad (427) $$

with almost the same direction. The analysis by Berengut et al. (2012) concluded that $A = -0.19(8) \times 10^{-5}$ and $B = 0.106(22) \times 10^{-5} \text{Gly}^{-1}$ for the parameterisation (dip2). Mariano and Perivolaropoulos (2012) pointed out that the $\alpha_{EM}$ dipole is anomalously aligned with the dark energy dipole obtained through the Union2 sample at $2\sigma$ level (Antoniou and Perivolaropoulos 2010) and suggested a model (“extended topological quintessence”) to explain the correlation between the two dipoles—see also Bronnikov et al. (2013) for a model with dark energy and $\alpha_{EM}$ variations arising from extra-dimensions. We refer to Aluri et al. (2023) for a
<table><caption>**Table 26** Summary of the constraints on a dipolar modulation of $\alpha_{EM}$ for the two ansatz (dip1) and (dip2) of the parameterisation (425)</caption><thead><tr><th>Method</th><th>Model</th><th>A</th><th>B</th><th>Ref</th></tr></thead><tbody><tr><td>QSO</td><td>dip1</td><td>–</td><td>(1.02 ± 0.21) × 10<sup>-5</sup></td><td>Webb et al. (2011)</td></tr><tr><td>QSO</td><td>dip2</td><td>–</td><td>(1.01 ± 0.25) × 10<sup>-6</sup> Gly<sup>-1</sup></td><td>Webb et al. (2011)</td></tr><tr><td>QSO</td><td>dip2</td><td>-0.19(8) × 10<sup>-5</sup></td><td>0.106(22) × 10<sup>-5</sup> Gly<sup>-1</sup></td><td>Berengut et al. (2012)</td></tr><tr><td>QSO (Archival)</td><td>dip1</td><td>–</td><td>(9.4 ± 2.2) × 10<sup>-6</sup></td><td>Aluri et al. (2023)</td></tr><tr><td>QSO (Dedicated)</td><td>dip1</td><td>–</td><td>2.9 × 10<sup>-6</sup></td><td>Aluri et al. (2023)</td></tr><tr><td>SNIa (Union-2.1)</td><td>dip1</td><td>–</td><td>(-3.11 ± 1.16) × 10<sup>-2</sup></td><td>Negrelli et al. (2018)</td></tr><tr><td>SNIa (JLA)</td><td>dip1</td><td>–</td><td>(0.22 ± 0.47) × 10<sup>-2</sup></td><td>Negrelli et al. (2018)</td></tr><tr><td>Clusters</td><td>dip2*</td><td>–</td><td>(-5.5 ± 7.9) × 10<sup>-3</sup> Gly<sup>-1</sup></td><td>Galli (2013)</td></tr><tr><td>Clusters</td><td>dip2*</td><td>–</td><td>(-3 ± 3) × 10<sup>-3</sup> Gly<sup>-1</sup></td><td>Bora and Desai (2021)</td></tr><tr><td>Clusters</td><td>dip1*</td><td>(6 ± 4) × 10<sup>-3</sup></td><td>(8 ± 9) × 10<sup>-3</sup></td><td>De Martino et al. (2016b)</td></tr><tr><td>Clusters</td><td>dip2*</td><td>(3 ± 3) × 10<sup>-3</sup></td><td>(6 ± 5) × 10<sup>-3</sup> Gly<sup>-1</sup></td><td>De Martino et al. (2016b)</td></tr><tr><td>Clusters</td><td>dip1</td><td>(2.1 ± 2.9) × 10<sup>-2</sup></td><td>(-3 ± 1.4) × 10<sup>-2</sup></td><td>De Martino et al. (2016b)</td></tr><tr><td>Clusters</td><td>dip2</td><td>(1.9 ± 1.1) × 10<sup>-2</sup></td><td>(-2.7 ± 5.1) × 10<sup>-3</sup> Gly<sup>-1</sup></td><td>De Martino et al. (2016b)</td></tr><tr><td>CMB</td><td>dip1</td><td>–</td><td>(-2.4 ± 3.7) × 10<sup>-2</sup></td><td>Ade et al. (2015)</td></tr></tbody></table>

A star indicates that the direction of the dipole is fixed to the one of (426) in the analysis

review on the different dipoles that appear in cosmology and to Mariano and Perivolaropoulos (2013) for their comparison. As discussed in Sect. 5.5, the QSO results can be performed with the archival or dedicated sample, which do not agree. Concerning a pure dipole (dip1), Aluri et al. (2023) got (see Pinho and Martins 2016 for an earlier analysis)

$$B = (9.4 \pm 2.2) \times 10^{-6} \quad \text{(archival)} \qquad |B| \le 2.9 \times 10^{-6} \quad \text{(dedicated)}. \quad (428)$$

Hence the Archival data have a statistical preference, at just over four standard deviations, for a dipole with an amplitude of about 9 ppm, while the Dedicated data show no preference for a dipole, and the amplitude is constrained to be less than about 3 ppm at the 95.4% C.L.. To finish, from their analysis of VLT/X-SHOOTER, Wilczynska et al. (2020) (see Sect. 5.5.3) constrained the amplitude of a constant dipole to $A = (-0.70 \pm 0.16) \times 10^{-5}$.

Several complementary tests have been proposed. First, one can consider the analogous high-resolution spectroscopic constraints on $\mu$ (see Table 15). They tend to indicate that a pure dipole in $\mu$ is constrained to $|B_{\mu}| < 2 \times 10^{-6}$ (Aluri et al. 2023). In any model where $\alpha_{EM}$ and $\mu$ variations are correlated one would expect the
dipole for the two constants to be aligned. While Berengut et al. (2011) found a weak indication for a μ-dipole aligned with the α<sub>EM</sub>-dipole with B<sub>μ</sub> = (2.6 ± 1.3) × 10<sup>-6</sup>Gly<sup>-1</sup> using parameterisation (dip2), this was not confirmed by the analysis of Aluri et al. (2023). Hence, there is currently no robust evidence of a dipole in α<sub>EM</sub> and μ at a level of a few 10<sup>-6</sup>.

* SNIa data. Negrelli et al. (2018) proposed to use the effect of the variation of the constant on SNIa and hence on the Hubble diagram to conclude from the SNIA compilations Union 2.1 that B = (-3.11 ± 1.16) × 10<sup>-2</sup> and JLA B = (0.22 ± 0.47) × 10<sup>-2</sup>. Indeed such an analysis deriving from the luminosity distance entangles any dipolar variations that can arise from the measurement or the cosmology itself (Chang et al. 2012) so that they are not robust constraints. See also Karpikov et al. (2015) for an analysis spacetime variations of the Ni-46 decay rate from SNIa light curves and a potential window on G<sub>F</sub> or v.

* Galaxy clusters. Galli (2013) extended her method based of galaxy clusters to investigated the signature of the spatial variation of α<sub>EM</sub> using multi-frequency measurements of the thermal Sunyaev-Zeldovich (see Sect. 5.12) to conclude that B = (-5.5 ± 7.9) × 10<sup>-3</sup> Gly<sup>-1</sup> for parameterisation (dip2) assuming the direction of dipole is fixed to the QSO dipole (426). A similar analysis by Bora and Desai (2021) got B = (-3 ± 3) × 10<sup>-3</sup> Gly<sup>-1</sup>. De Martino et al. (2016a, 2016b) applied the same method to a catalog of 618 X-ray clusters including Planck data to measure the thermal SZ effect at the location of the clusters. They concluded that A = (6 ± 4) × 10<sup>-3</sup> and B = (8 ± 9) × 10<sup>-3</sup> for parameterisation (dip1) and A = (3 ± 3) × 10<sup>-3</sup> and B = (6 ± 5) × 10<sup>-3</sup> Gly<sup>-1</sup> for parameterisation (dip2) assuming the direction fixed to the QSO dipole (426) while they got A = (2.1 ± 2.9) × 10<sup>-2</sup> and B = (-3 ± 1.4) × 10<sup>-2</sup> for parameterisation (dip1) and A = (1.9 ± 1.1) × 10<sup>-2</sup> and B = (-2.7 ± 5.1) × 10<sup>-3</sup> Gly<sup>-1</sup> for parameterisation (dip2) if the direction of the dipole is not fixed.

* O III emission spectra. Jiang et al. (2025) tested the spatial variation in α<sub>EM</sub> using the [O III] emission of a JWST galaxy sample with 3 < z < 7 in four directions on the sky. The Δα<sub>EM</sub>/α<sub>EM</sub> values in the different directions are also consistent with zero within 1σ error of ~10<sup>-4</sup>, suggesting no spatial variation up to z ~ 9.

* CMB. To finish with the CMB, it was demonstrated that a spatial variation of α<sub>EM</sub> would induce a dipolar modulation of CMB anisotropies (Moss et al. 2011; Aiola et al. 2015), but at a level incompatible with existing constraints (Prunet et al. 2005). The study by Ade et al. (2015) concluded that the relative constant amplitude of a dipolar modulation of α<sub>EM</sub> is constrained to

$$B = (-2.4 \pm 3.7) \times 10^{-2} \qquad (429)$$

from the Planck data (Ade et al. 2014).

### 8.3.3 Wall of constants

When the cosmological history of the evolution of fundamental constants is interpreted in terms of a time variation which, as we have seen so far, it has to
face the problem of strong local constraints. This has motivated Olive et al. (2011) to interpret the observations as a spatial variation on cosmological scales thanks to the hypothesis of the existence of a low energy domain wall produced in the spontaneous symmetry breaking involving a dilaton-like scalar field coupled to electromagnetism. Domains on either side of the wall exhibit slight differences in their respective values of $\alpha_{EM}$; see Fig. 30. If such a domain wall is present within our Hubble volume, absorption spectra at large redshifts may or may not provide a variation in $\alpha_{EM}$ relative to the terrestrial value, depending on our relative position with respect to the wall and the direction observation.

The first model (Olive et al. 2011) relies on the generalisation of Bekenstein (1982),

$$S = \int \left[ \frac{R}{16\pi G} - \frac{1}{2}(\partial_{\mu}\phi)^2 + V(\phi) + \frac{1}{4}B_F(\phi)F^2 - \sum_j \bar{\psi}_j(i\not D - B_j(\phi)m_j)\psi_j \right] \sqrt{-g}d^4x \quad (430)$$

with $\phi$ is assumed to have the quartic potential

$$V(\phi) = \frac{1}{4}\lambda(\phi^2 - \eta^2)^2 \quad (431)$$

and a coupling to the Faraday tensor of electromagnetism as well as to the fermions $\psi$ with the coupling functions $B_i$ of the form

<figure><img src="image_37.png" /><figcaption>Fig. 30 A domain wall is assumed to cross our Hubble volume, hence intersecting our past light-cone on a 2-dimensional spatial hypersurface characterized by the redshift of the wall in a direction $\mathbf{n}$. $z_*$ is the lowest redshift at which the wall can be observed (left). On a constant time hypersurface, this leads to a spatial distribution of domains in which $\alpha_{EM}$ is mostly constant (right). Image reproduced with permission from Olive et al. (2011), copyright by APS</figcaption></figure>
$$B_i(\phi) = \exp\left(\xi_i \frac{\phi}{M_*}\right) \approx 1 + \xi_i \frac{\phi}{M_*}. \qquad (432)$$

The coefficients $\xi_i$ are constant and $M_*$ is a mass scale. It was assumed for simplicity that only $\xi_F$ is non-vanishing at tree level. Indeed, $\phi$ inevitably couples to nucleons radiatively. To reproduce a change in $\alpha_{EM}$ through the domain wall matching the claimed spatial variation by Webb et al. (2011), one needs

$$\frac{\Delta\alpha_{EM}}{\alpha_{EM}} \approx 2\xi_F \frac{\eta}{M_*} \sim \text{few} \times 10^{-6}. \qquad (433)$$

The contribution of the wall to the energy budget of the universe is

$$\Omega_{\text{wall}} \approx \left( \frac{\eta}{100 \, \text{MeV}} \right)^3. \qquad (434)$$

his led to the idea that $\eta = \mathcal{O}(\text{MeV})$ to construct a fully viable model.

Our local Hubble bubble can then be thought as a patchwork of domains each enjoying constants with different values as, what could be named, a “microland-scape of constants”. Indeed, no variation of any constant will be detectable within each domain but they can be revealed on the past light cone of any observer. Such a model illustrates the complementarity between local and astrophysical constraints and give a counter example to claims that local constraints on the variation of $\alpha_{EM}$ from atomic clocks are sharper than those obtained from QSO absorption spectra. Olive et al. (2012) showed that a single wall configuration is statically comparable to a dipole fit (but is a big improvement over a weighted mean) and that a two-wall solution is a far better fit (despite adding 3 parameters over the single wall solution). Nielsen and Froggatt (2023) actually argued that such domain walls can form a network with typical distance between walls of 100 millions light-year s,hence defining a large number of regions with different low-energy physics within our observable universe.

The model was extended to a runaway potential (Chiba and Yamaguchi 2011) in order to construct an toy model that could explain simultaneously time (Webb et al. 1999; Murphy et al. 2003) and space (Webb et al. 2011) variations that were thought to be indicated by QSO at the time. It was then reformulated as a $F(R)$ theory by Bamba et al. (2012) and in the framework of the symmetron (Silva et al. 2014). Avelino and Sousa (2015) concluded that in order for such a wall to be responsible for a spatial variation of $\alpha_{EM}$ compatible with (Webb et al. 2011; King et al. 2012b) the domain wall network shall contribute to $\Omega_{wall} \in [10^{-10}, 10^{-5}]$ to the matter budget of the universe. Stadnik (2020) showed that for domain walls within the symmetron model with $B_F \propto \phi^2$, the current constraints imply $\Omega_{wall} \ll 10^{-10}$; see Sect. 2.3.5 for a discussion of the detection of topological defects in the laboratory.
### 8.3.4 Non-Copernican field configuration

Another possibility to explain a dipolar modulation of some fundamental constant would be that the Copernican principle is not fully satisfied, such as in various void models. Then the background value of $\phi$ would depend, e.g., on $r$ and $t$ for a spherically symmetric spacetime (such as the Lemaître–Tolman–Bondi spacetime). This would give rise to a dipolar modulation of the constant if the observer is not located at the center of the universe. Note, however, that such a cosmological dipole would also reflect itself, e.g., on CMB anisotropies; see Cusin et al. (2017). Similar possibilities are also offered within the chameleon mechanism where the value of the scalar field depends on the local matter density; see Sect. 3.2.

## 8.4 Universality of free fall

The universality of free fall can also be used to constrain large scale spatial variation of the fundamental constant.

Indeed, the Solar system barycenter moves, on laboratory timescales, almost linearly with a constant speed, at about 250 km/s around the Milky Way with a period of about 240 million years and at about 370 km/s with respect to the comoving rest frame defined by the observed cosmic microwave background (Aghanim et al. 2020). Any spatial variation of the constants on these scales will translate as a time variation in the Earth restframe. Indeed the galactic modulation cannot be tested directly but would have to be taken into account to interpret Oklo data for which the timescales are identical to the Sun orbital period. A cosmological gradient, as e.g., discussed in Sect. 8.3.2, will result in a linear temporal drift of the constants, $\Delta\alpha_i(t)/\alpha_i = v_\odot \cdot \nabla \ln \alpha_i$, hence linking a local measurement to a cosmological variation.

Estimating from Sect. 8.3.2 that $\cos(v_\odot, \nabla \ln \alpha) \sim 0.1$, the constraint by Stadnik (2021) can be updated to take into account (142) on $\alpha_{EM}$ by Filzinger et al. (2023) to deduce that

$$|\nabla \ln \alpha_{EM}| \lesssim 3.5 \times 10^{-6} \text{Gyr}^{-1},$$

which is competitive with the bounds listed in Table 25, highlighting the strong connection between local and cosmological constraints.

Damour and Donoghue (2011) investigated the perturbing influence of a differential force with a fixed direction acting on a gravitationally bound two-body system, such as the Earth–Moon system. This “gravitational Stark effect” (Damour and Schaefer 1991) is an example of singular perturbation where a small perturbing force can have a large effect. In particular, elliptic orbits undergo a complicated secular evolution that can be constrained. It was estimated that LLR experiment can probe a gradient of

$$|\nabla \ln X_q| \lesssim 2.6 \times 10^{-6} \text{Gyr}^{-1},$$

again comparable to the bounds of Table 25.
# 9 Why are the constants just so?

The numerical values of the constants are not determined by the laws of nature in which they appear. One of the most outstanding questions pertains to the values of the fundamental parameters of the Standard Model so that one may wonder why they have the values we observe. In particular, as pointed out by many authors (see below), the constants of nature seem to be fine tuned (Leslie 1989). Many physicists take this fine-tuning to be an explanation that cries for an explanation, hence following Hoyle (1965) who wrote that “one must at least have a modicum of curiosity about the strange dimensionless numbers that appear in physics.” The answer may lie in an as yet undefined future theory, such as a complete string theory, in which case there is hope of a deeper understanding. It is also possible that our Universe is part of a larger structure or a Multiverse, but we have no means to know. In this case, the observed values may be environmental and reflect an observational bias through the anthropic principle, which absolves us, the humans on Earth, from the duty of explaining the values of the governing constants, at least for the time being, until data at higher energies are available and eventually change our description of physics.

We refer to Sloan et al. (2020) for an extensive review of these questions, to Donoghue (2016) for a review on the motivations for a multiverse and a discussion of the fine-tunings in particle physics, to Barnes (2012, 2018, 2017), Adams (2019), Barnes (2022), Sandora et al. (2022) for detailed discussions of the fine-tunings and discussions of the evaluation of the level of fine-tuning in Bayesian approach.

## 9.1 Universe and multiverse approaches

Two possible lines of explanation are usually envisioned: a *design or consistency hypothesis* and an *ensemble hypothesis*, that are indeed not incompatible together. The first hypothesis includes the possibility that all the dimensionless parameters in the “final” physical theory will be fixed by a condition of consistency or an external cause. In the ensemble hypothesis, the universe we observe is only a small part of the totality of physical existence, usually called the multiverse. This structure needs not be fine-tuned and shall be sufficiently large and variegated so that it can contain as a proper part a universe like the one we observe the fine-tuning of which is then explained by an *observation selection effect* (Bostrom 2002).

These two possibilities send us back to the large number hypothesis by Dirac (1937) that has been used as an early motivation to investigate theories with varying constants. The main concern was the existence of some large ratios between some combinations of constants. As we have seen in Sect. 4.1, the running of coupling constants with energy, dimensional transmutation or relations such as Eq. (85) have opened a way to a rational explanation of very small (or very large) dimensional numbers. This follows the ideas developed by Eddington (1936, 1948) aiming at deriving the values of the constants from consistency relations, e.g., he proposed to link the fine-structure constant to some algebraic structure of spacetime. Dicke (1961) pointed out another possible explanation to the origin of Dirac large
numbers: the density of the universe is determined by its age, this age being related to the time needed to form galaxies, stars, heavy nuclei.... This led Carter (1974) to argue that these numerical coincidences should not be a surprise and that conventional physics and cosmology could have been used to predict them, at the expense of using the anthropic principle.

The idea of such a structure called the *multiverse* has attracted a lot of attention in the past years and we refer to Carr (2007) for a more exhaustive account of this debate. While many versions of what such a multiverse could be, one of them finds its root in string theory. In 2000, it was realized (Bousso and Polchinski 2000) that vast numbers of discrete choices, called flux vacua, can be obtained in compactifying superstring theory. The number of possibilities is estimated to range between $10^{100}$ and $10^{500}$, or maybe more. No principle is yet known to fix which of these vacua is chosen. Eternal inflation offers a possibility to populate these vacua and to generate an infinite number of regions in which the parameters, initial conditions but also the laws of nature or the number of spacetime dimensions can vary from one universe to another, hence being completely contingent. It was later suggested by Susskind (2003) that the anthropic principle may actually constrain our possible locations in this vast string landscape. This is a shift from the early hopes (Kane et al. 2002) that M-theory may conceivably predict all the fundamental constants uniquely.

Indeed such a possibility radically changes the way we approach the question of the relation of these parameters to the underlying fundamental theory since we now expect them to be distributed randomly in some range. Among this range of parameters lies a subset, that we shall call the *anthropic range*, which allow for universe to support the existence of observers. This range can be determined by asking ourselves how the world would change if the values of the constants were changed, hence doing *counterfactual cosmology*. However, this is very restrictive since the mathematical form of the law of physics managed as well and we are restricting to a local analysis in the neighborhood of our observed universe. The determination of the anthropic region is not a prediction but just a characterization of the sensitivity of “our” universe to a change of the fundamental constants *ceteris paribus*. Once this range is determined, one can ask the general question of quantifying the probability that we observe a universe as ours, hence providing a probabilistic prediction. This involves the use of the anthropic principle, which expresses the fact that we observe are not just observations but observations made by us, and requires us to state what an observer actually is (Neal 2006).

## 9.2 Fine-tunings and determination of the anthropic range

As we have discussed in the previous sections, the outcome of many physical processes are strongly dependent on the value of the fundamental constants. One can always ask the scientific question of what would change in the world around us if the values of some constants were changed, hence doing some counterfactual cosmology in order to determine the range within which the universe would have developed complex physics and chemistry, what is usually thought to be a
prerequisite for the emergence of complexity and life (we emphasize the difficulty of this exercise when it goes beyond small and local deviations from our observed universe and physics, see, e.g., Harnik et al. (2006) for a possibly life supporting universe without weak interaction). In doing so, one should consider the fundamental parameters entering our physical theory but also the cosmological parameters. We refer to Adams (2019) for an extensive and detailed discussion of the degree of fine-tuning of our description of the world.

## Naturalness and the macroscopic world

Our description of the universe is plagued with incomprehensible instance of anti-naturalness. As we have discussed in Sect. 2.3.1, the first is related to the value of the cosmological constant, the observed value of which is some 120 orders of magnitude smaller than the Planck scale $M_P^4$, believed to be the only fundamental scale; see Bianchi and Rovelli (2010), Gegelia and Meißner (2019) for opposite views on this belief. The second lies in the fact that the electroweak scale is 17 orders of magnitude lower than the Planck scale. The facy that the QCD vacuum angle $\theta$ was 10 orders of magnitude lower than its expected value $\theta \sim 1$ has been addressed by Lee et al. (2020). These two anti-naturalness problems are related to the fact that “we live in a big universe that contains big things.”

Indeed, the suppression of the cosmological constant is vital for the existence of a large and old universe. Even if it were a few orders of magnitude larger, the Universe would have entered an inflationary phase before the onset of galaxy formation. Indeed, the typical radius and typical age of the universe are given by $H_0^{-1} \sim \sqrt{3/\Lambda} \sim 10^{60} \ell_p$. In other word the fact that we leave in a old and big universe is a consequence of the cosmological constant non-naturalness.

Then, concerning matter and the macroscopic structures we observe to exist, the smallness of the u and d quark masses compared to $\Lambda_{QCD}$ and the fact that $m_u < m_d$ are crucial for the genesis of heavier elements in stars. Then, following Carr and Rees (1979)—see also Adams (2008)—we can estimate the typical mass of macroscopic objects such as stars and planets by roughly looking at the balance between the intensities of gravitation and the pressure of the force opposing the gravitational collapse for a given mass $M$. The minimum mass necessary to sustain nuclear fusion is determined by the criterium that the star’s central temperature is high enough leading to (Carr and Rees 1979)

$$M_{\text{min,star}} \sim \frac{1}{2} \mu^{3/4} \alpha_{\text{EM}}^{3/2} M_*$$

with $M_*$ the characteristic stellar mass, $M_* = \alpha_G^{-3/2} m_p$. Similarly the typical mass of a rocky planet is determined by the requirement that the electromagnetic interaction counterbalance gravity, leading to

$$M_{\text{planet}} \sim \left( \frac{\alpha_{\text{EM}}}{\alpha_G} \right)^{3/2} m_p = \alpha_{\text{EM}}^{3/2} M_*$$

while the critical mass scale of life form (Page 2009) is
$$M_{\text{life}} \sim \epsilon_c^{3/4} \left(\frac{\alpha_{\text{EM}}}{\alpha_{\text{G}}}\right)^{3/2} m_p = (\epsilon_c \alpha_{\text{EM}} \alpha_{\text{G}})^{3/4} M_*$$

where $\epsilon_c \sim 10^{-3}$ is the chemical conversion factor. As a consequence, there exist macroscopic objects because gravity is weak compared to the weak scale. For planets, it allows to pack $(\alpha_{\text{EM}}/\alpha_{\text{G}})^{3/2}$ particles without turning to a black hole. Such an object will have a typical size of $(\alpha_{\text{EM}}/\alpha_{\text{G}})^{1/2}a_0 \sim 10^4$ km. The separation of scales then implies that there exist classes of objects with typically the same sizes and masses, hence fixing their dimension in units of the Bohr radius $a_0$ independently of its value. For instance, the minimal mass of a neutron star and its typical size are given by $\alpha_S M_*$ where the fine structure constant has been replaced by the strong structure constant.

As a conclusion the anti-naturalness problems concerning the cosmological constant and the weak scale ensure that we live in a big universe containing big objects. This is the first step toward the anthropic principle.

## Fine tuning on the particle physics parameters

Then, there are several constraints that the fundamental parameters listed in Table 1 have to satisfy in order for the universe to allow for complex physics and chemistry. Let us stress, in a non-limiting way, some examples.

* It has been noted that the stability of the proton requires $m_d - m_u \gtrsim \alpha_{\text{EM}}^{3/2} m_p$. The anthropic bounds on $m_d$, $m_u$ and $m_e$ (or on the Higgs vev) arising from the existence of nuclei, the di-neutron and the di-proton cannot form a bound state, the deuterium is stable have been investigated in many works (Agrawal et al. 1998a, b; Hogan 2000; Dent and Fairbairn 2003; Donoghue et al. 2006; Hogan 2007; Damour and Donoghue 2008; Donoghue et al. 2010), even allowing for nuclei made of more than 2 baryon species (Jaffe et al. 2009). Typically, the existence of nuclei imposes that $m_d + m_u$ and $v$ cannot vary by more than 60% from their observed values.

* If the difference of the neutron and proton masses where less than about 1 MeV, the neutron would become stable and hydrogen would be unstable (Rozental 1988; Hogan 2006) so that helium would have been the most abundant at the end of BBN. Hence, the whole history of the formation and burning of stars would have been different. It can be deduced that (Hogan 2000) one needs $m_d - m_u - m_e \gtrsim 1.2$ MeV so that the universe does not become all neutrons; $m_d - m_u + m_e \lesssim 3.4$ MeV for the pp reaction to be exothermic and $m_e > 0$ leading to a finite domain. Note that among the fine-tuning, one often considers the diproton arguing that if it were bound, stars would burn $10^{18}$ times brighter and faster via strong interactions, resulting in a universe that would fail to support life. Barnes (2015), Barnes and Lewis (2017), Adams et al. (2021) demonstrated that stable, “strong-burning” stars adjust their central densities and temperatures to have familiar surface temperatures, luminosities and lifetimes so that there is no “diproton disaster”. Similarly Coc et al. (2012a) showed that a stable Be-8 nuclei will not affect BBN.
* A coincidence emerges from the existence of stars with convective and radiative envelopes, since it requires (Carr and Rees 1979) that $\alpha_G \sim \alpha_{EM}^{20}$. It arises from the fact that the typical mass that separates these two behavior is roughly $\alpha_G^{-2} \alpha_{EM}^{10} m_p$ while the masses of star span a few decades around $\alpha_G^{-3} m_p$. Both stars seem to be needed since only radiative stars can lead to supernovae, required to disseminate heavy elements, while only convective stars may generate winds in their early phase, which may be associated with formation of rocky planets. This relation while being satisfied numerically in our universe cannot be explained from fundamental principles.

* Similarly, it seems that for neutrinos to eject the envelope of a star in a supernovae explosion, one requires (Carr and Rees 1979) $\alpha_G \sim \alpha_W^4$.

* As discussed in Sect. 5.9, the production of carbon seems to imply that the relative strength of the nuclear to electromagnetic interaction must be tuned typically at the 0.1% level.

* Sandora et al. (2022) investigated the dependence of primordial abundances on physical constants and the implications for the distribution of complex life for various proposed habitability criteria. Interestingly they exhibit a series of predictions (carbon-rich or carbon-poor planets are uninhabitable; slightly magnesium-rich planets are habitable; life does not depend on nitrogen abundance too sensitively; metal-rich planets and phosphorus-poor planets are habitable) that can be checked by probing regions of our universe that closely resemble normal environments in other universes. Then, if any of these predictions are found to be wrong, the multiverse scenario would predict that the majority of observers are born in universes differing substantially from ours, changing our assumption about our typicality.

## Fine tuning on the cosmological parameters

Other coincidences involve also the physical properties, not only of the physical theories, but also of our universe, i.e., the cosmological parameters summarized in Table 30. Indeed those parameters are not on the same footing as fundamental constants since they may be though to vary on large scale if the Copernican principle does not hold on super-Hubble scales even if the physics remains unchanged. Let us remind some examples.

* The total density parameter $\Omega$ must lie within an order of magnitude of unity. If it were much larger the universe will have re-collapsed rapidly, on a time scale much shorter than the main-sequence star lifetime. If it were to small, density fluctuations would have frozen before galaxies could form. Typically one expects $0.1 < \Omega_0 < 10$. Indeed, most inflationary scenarios lead to $\Omega_0 \sim 1$ so that this may not be anthropically determined but in that case inflation should last sufficiently long so that this could lead to a fine tuning on the parameters of the inflationary potential.

* The cosmological constant was probably the first one to be questioned in an anthropical way (Weinberg 1987). Weinberg noted that if $\Lambda$ is too large, the universe will start accelerating before structures had time to form. Assuming that
it does not dominate the matter content of the universe before the redshift $z_*$ at which earliest are formed, one concludes that $\rho_V = \Lambda/8\pi G < (1 + z_*)\rho_{\text{mat0}}$. Weinberg (1987) estimated $z_* \sim 4.5$ and concluded that “if it is the anthropic principle that accounts for the smallness of the cosmological constant, then we would expect the vacuum energy density $\rho_v \sim (10 - 100)\rho_{\text{mat0}}$ because there is no anthropic reason for it to be smaller”. Indeed, the observations indicate $\rho_v \sim 2\rho_{\text{mat0}}$

* Tegmark and Rees (1998) pointed out that the amplitude of the initial density perturbation, $Q$ enters into the calculation and determined the anthropic region in the plane $(\Lambda, Q)$. This demonstrates the importance of determining the parameters to include in the analysis.

* Different time scales of different origin seem to be comparable: the radiative cooling, galactic halo virialization, time of cosmological constant dominance, the age of the universe today. These coincidences were interpreted as an anthropic sign (Bousso et al. 2009).

These are just a series of examples. For a multi-parameter study of the anthropic bound, we refer, e.g., to Tegmark et al. (2006) and to Hall and Nomura (2008) for a general anthropic investigation of the standard model parameters. Note that Landsman (2016) suggested that fine-tuning requires no special explanation at all, as an example of the difficulty to put the multiverse on solid theoretical grounds.

## 9.3 Anthropic predictions

The determination of the anthropic region for a set of parameters is in no way a prediction but simply a characterization of our understanding of a physical phenomenon $P$ that we think is important for the emergence of observers. It reflects that, the condition $C$ stating that the constants are in some interval, $C \Rightarrow P$, is equivalent to $!P \Rightarrow !C$.

The anthropic principle (Carter 1974) states that “what we can expect to observe must be restricted by the conditions necessary for our presence as observers”. It has received many interpretations among which the weak anthropic principle stating that “we must be prepared to take account of the fact that our location in the universe in necessarily privileged to the extent of being compatible with our existence as observers”, which is a restriction of the Copernican principle often used in cosmology, and the strong anthropic principle according to which “the universe (and hence the fundamental parameters on which it depends) must be such as to admit the creation of observers within it at some stage.” (see Barrow and Tipler 1986 for further discussions and a large bibliography on the subject).

One can then try to determine the probability that an observer measures the value $x$ of the constant $X$ (that is a random variable fluctuating in the multiverse and the density of observers depend on the local value of $X$). According to Bayes theorem,

$$P(X = x|\text{obs}) \propto P(\text{obs}|X = x)P(X = x).$$

$P(X = x)$ is the prior distribution, which is related to the volume of those parts of
the universe in which $X = x$ at $dx$. $P(\text{obs}|X = x)$ is proportional to the density of observers that are going to evolve when $X = x$. $P(X = x|\text{obs})$ then gives the probability that a randomly selected observer is located in a region where $X = x \pm dx$. It is usually rewritten as (Vilenkin 1995)

$$P(x)dx = n_{\text{obs}}(x)P_{\text{prior}}dx.$$

This highlights the difficulty in making a prediction. First, one has no idea of how to compute $n_{\text{obs}}(x)$. When restricting to the cosmological constant, one can argue (Vilenkin 1995) that $\Lambda$ does not affect microphysics and chemistry and then estimate $n_{\text{obs}}(x)$ by the fraction of matter clustered in giant galaxies and that can be computed from a model of structure formation. This may not be a good approximation when other constants are allowed to vary and it needs to be defined properly. Second, $P_{\text{prior}}$ requires an explicit model of multiverse that would generate subuniverses with different values $x_i$ (continuous or discrete) for $x$. A general argument (Weinberg 1989) states that if the range over which $X$ varies in the multiverse is large compared to the anthropic region $X \in [X_{\text{min}}, X_{\text{max}}]$ one can postulate that $P_{\text{prior}}$ is flat on $[X_{\text{min}}, X_{\text{max}}]$. Indeed, such a statement requires a measure in the space of the constants (or of the theories) that are allowed to vary. This is a strong hypothesis, which is difficult to control. In particular if $P_{\text{prior}}$ peaks outside of the anthropic domain, it would predict that the constants should lie on the boundary of the anthropic domain (Rubakov and Shaposhnikov 1989). It also requires that there are sufficiently enough values of $x_i$ in the anthropic domain, i.e., $\delta x_i \ll X_{\text{max}} - X_{\text{min}}$. Garriga and Vilenkin (2000) stressed that the hypothesis of a flat $P_{\text{prior}}$ for the cosmological constant may not hold in various Higgs models, and that the weight can lower the mean viable value. To finish, one wants to consider $P(x)$ as the probability that a random observer measures the value $x$. This relies on the fact that we are a typical observer and we are implicitly making a self sampling hypothesis. It requires to state in which class of observers we are supposed to be typical—the final result may depend on this choice (Neal 2006)—and this hypothesis leads to conclusions such as the doomsday argument that have been debated actively (Bostrom 2002; Neal 2006).

This approach to the understanding of the observed values of the fundamental constants (but also of the initial conditions of our universe) by resorting to the actual existence of a multiverse populated by a different “low-energy” theory of some “mother” microscopic theory allows us to explain the observed fine-tuning by an observational selection effect. It also sets a limit to the Copernican principle stating that we do not live in a particular position in space since we have to live in a region of the multiverse where the constants are inside the anthropic bound. Such an approach is indeed not widely accepted and has been criticized in many ways (Ellis et al. 2004; Stoeger et al. 2004; Stoeger 2006; Starkman and Trotta 2006; Page 2007; Aguirre 2007; Vaas 2010). Several criticisms have emerged concerning probabilistic measurements of life-permitting intervals and differing opinions inasmuch as the constants of nature that must be considered. When talking about life, consensus is illusive largely because there is not even a consistent definition of what life is, e.g., Adams (2019) did not restrict to carbon-based life, Sandora et al.
(2022) focused on complex intelligent life and Collins (2009) considered “about embodied moral agents”. Díaz-Pachón et al. (2021) adopted a Bayesian approach to assign an upper bound for the probability of tuning, which is invariant with respect to change of physical constants.

Among the issues to be answered before such an approach becomes more rigorous, let us note: (1) what is the shape of the string landscape; (2) what constants should we scan. It is indeed important to distinguish the parameters that are actually fine-tuned in order to determine those that we should hope to explain in this way (Wilczek 2005, 2007). Here theoretical physics is indeed important since it should determine which of the numerical coincidences are coincidences and which are expected for some unification or symmetry reasons; (3) how is the landscape populated; (4) what is the measure to be used in order and what is the correct way to compute anthropically conditioned probabilities.

While considered as not following the standard scientific approach, this is the only existing window on some understanding of the values of the fundamental constants.

# 10 Conclusions

Over the past three decades, the fundamental constants have attracted a larger interest. They have become key players in testing General Relativity in the Solar system but also on the whole cosmological history, in constraining its extensions and in investigating dark matter and dark energy scenarios, providing informations that cannot be accessed only through the study of the large-scale structure of our universe. This complementarity with the other tests of General Relativity on astrophysical scales and the complementarity of local and cosmological constraints have been emphasized for many years (Uzan 2003a, 2007, 2010c; Jain et al. 2013b) and shall now be full part of the cosmological lore.

The study of fundamental constants has witnessed tremendous progresses. Within a decade, the constraints on their possible space and time variations have flourished in many ways. They have reached higher precision and new systems, involving different combinations of constants and located at different redshifts, have been proposed. This has improved our knowledge on the validity of the equivalence principle and allowed us to test it on astrophysical and cosmological scales, as reviewed them in Sects. 5 and 6. Many experimental observational progresses are expected in the coming years thanks to quantum laboratory technologies, or the developments of the E-ELT, radio observations, atomic clocks in space, or gravitational waves. We have provided new combined analysis of the constraints on the variation of fundamental constants.

Fundamental constants have an important role to play in cosmology, in view of its many anomalies (Peebles 2022) and the puzzle of the dark sector. As we have seen, they may be involved in the Hubble tension (Sect. 2.3.6) and in the lithium problem (Sect. 5.14.5). Besides, they are a powerful tool to constrain and eventually detect ultralight dark matter—Sect. 2.3.4 and topological defects (Sect. 2.3.5). Each of these effects has a different signature; see Fig. 5. More important the local
constraints from atomic clock set strong bounds—see Eq. (358) that needs to be satisfied by any viable scalar field model—on any light scalar field that couple to the standard matter fields and hence on any dark energy model (Sect. 2.3.3). They also allow us to question the isotropy of space (Sect. 8.3.2) and hence the Copernican principle. Fundamental constants bridge our local physics to cosmological scale in a unique way and they have become to study any new degree of freedom and constrain its coupling to the standard model fields (see Fig. 10).

From a theoretical point of view, we had to spend time defining the concept of constant (Sect. 2.1.1). Structurally, the fundamental constants are now the heart of the definition of the International System of Units (Sect. 2.1.2) which we have also detailed. We pointed out the conceptual difference between fundamental units and fundamental parameters and the crucial fact that fundamental constants cannot be considered without a theoretical framework that needs to be specified. The first strong connection with theory arises from General Relativity and the test of the equivalence principle on which it rests (Sect. 2.2). Then, we have described in Sect. 3 the high-energy models that predict such variations, as well as the link with the origin of the acceleration of the universe. In all these cases, a spacetime varying fundamental constant reflects the existence of an almost massless field that couples to matter. This will be at the origin of a violation of the universality of free fall and thus of utmost importance for our understanding of gravity and of the domain of validity of General Relativity. Huge progresses have been made in the understanding of the coupled variation of different constants (Sect. 4). While more model-dependent, this allows one to set stronger constraints and eventually to open an observational window on unification mechanisms. Again, it is important to stress that such theories are necessary to interpret the constraints consistently. Section 7 has detailed the main models used in the literature.

Section 9 discussed the ideas that try to understand the value of the fundamental constants. While considered as borderline with respect to the standard physical approach, it reveals the necessity of considering a universe larger than our own, and called the multiverse. It will also give us a hint on our location in this structure in the sense that the anthropic principle limits the Copernican principle at the basis of most cosmological models. We have stressed the limitations of this approach and the ongoing debate on the possibility to make it predictive.

Given the growth of the number of studies, both experimental and theoretical, it becomes almost impossible to keep track of all the evolution. This cannot be the goal of a review, that would necessarily miss some results and eventually misinterpret some others. This shall be a collaborative work for the community. In particular, we shall find an easy way for anyone to include the constraints arising from the fundamental constants in her/his analysis, and more particularly the local constraints that set strong constraints on cosmological models. Epilogue 11 proposes a scheme that could allow us to keep track of the evolutions in almost real time. It proposes a form, that can be used to summarize experimental conclusions in articles or be used to create an open database. Indeed it needs to be thought and designed collectively. The goal would be to compile these data in a standardized way. This would offer traceability, avoid mistakes (since complied by
its authors), be validated by the community, always up-to-date and eventually be accessible online.

To conclude, the puzzle about the large numbers pointed out by Dirac has led to a better understanding of the fundamental constants and of their roles in the laws of physics. They are now part of the general tests of General Relativity, as well as a breadcrumbs to understand the origin of the acceleration of the universe and to more speculative structures, such as a multiverse structure, and possibly a window on string theory.

# 11 Epilogue: Towards the collective creation of a community data base

In order to construct a comprehensive and useful data basis, it is proposed that every article stating a constraint on some variation of fundamental constant tries to provide the following element at least in an appendix. This will allow traceability and possibility to compare the analysis as well as avoid cut/paste mistakes and misinterpretation

## Proposition of a general typology

1. **Physical system.** This can be any of the systems we have been discussed or eventually a new physical system such as

* Equivalence principle

* Clock

* Oklo

* Meteorite dating

* Absorption spectra

* Emission spectra

* Stellar physics

* 21 cm

* Galaxy cluster and SZ

* Lensing

* CMB

* BBN

* Pulsar timing

* Gravitational waves

It can have different level of entry such as clocks/Rb-Cs or Absorption spectra/0 H-18 cm

2. **Redshift of the measurement.** Or more generally spacetime position and environment.

3. **Data set.** What are the data used in taure (absorption spectra/reference, CMB maps/release etc...)
4. **Hypothesis on the physical system.** This shall include all the external parameters that are either fitted or fixed for the analysis. This could be the list of free cosmological parameters, nuisance parameters, parameters describing the environment (temperature, density,...)

5. **Hypothesis on the fundamental constants.** This shall include all the fundamental constants that are either fitted or fixed for the analysis.

6. **Hypothesis on the physical theory.** Specify hypothesis on the link, if any, on the variations of the constants or if one has assumed an independent variation or a single constant.

7. **Raw constraints.** If possible one shall give the constraints on the observable, e.g., relative frequency drift.

8. **Constraints on the constant.** This shall include $1\sigma$ error bar and systematics error bar and the list of hypothesis/models used to link the observable to the constants.

9. **List of possible systematic effects.** Description of the most important systematics of the system.

10. **Other articles that have studied the same dataset.** If any. It could be a reanalysis (in what sens), an extension, a cross-study with another dataset etc...

11. **Contact authors.**

# Appendix A. System of units and the international system of units (SI)

## A.1. System of units

Originally, the sizes of the human body were mostly used to measure the length of objects (e.g., the foot and the thumb gave *feet* and *inches*) and some of these units can seem surprising to us nowadays (e.g., the *span* was the measure of hand with fingers fully splayed, from the tip of the thumb to the tip of the little finger). Similarly weights were related to what could be carried in the hand: the pound, the ounce, the dram.... Needless to say, this system had a few disadvantages since each country, region has its own system (for instance in France there was more than 800 different units in use in 1789). The need to define a system of units based on natural standard led to several propositions to define a standard of length (e.g., the *mille* by Gabriel Mouton in 1670 defined as the length of one angular minute of a great circle on the Earth or the length of the pendulum that oscillates once a second by Jean Picard and Christiaan Huygens). The real change happened during the French Revolution during which the idea of a universal and non anthropocentric system of units arose. In particular, the Assemblée Nationale Constituante adopted the principle of a uniform system of weights and measures on 8 May 1790 and, in March 1791 a decree (these texts are reprinted in Uzan and Lehoucq 2005) was voted, stating that a quarter of the terrestrial meridian would be the basis of the definition of the *meter* (from the Greek metron, as proposed by Borda): a meter would henceforth be one ten millionth part of a quarter of the terrestrial meridian. Similarly the *gram* was defined as the mass of one cubic centimeter of distilled
water (at a precise temperature and pressure) and the second was defined from the
property that a mean Solar day must last 24 h.

To make a long story short, this led to the creation of the metric system and then
of the signature of *La convention du mètre* in 1875. Since then, the definition of the
units has evolved significantly. First, the definition of the meter was related to more
immutable systems than our planet, which, as pointed out by Maxwell in 1870, was
an arbitrary and inconstant reference. He then suggested that atoms may be such a
universal reference.<fnref n="18" /> In 1960, the International Bureau of Weights and Measures
(BIPM) established a new definition of the meter as the length equal to 1,650,763
wavelengths, in a vacuum, of the transition line between the levels 2p₁₀ and 5d₅ of
krypton-86. Similarly the rotation of the Earth was not stable enough for a reference.
It was proposed in 1927 by André Danjon to use the tropical year as a reference, as
adopted in 1952. In 1967, the second was then related to an atomic transition,
defined as the duration of 9 162 631 770 periods of the transition between the two
hyperfine levels of the ground state of caesium-133. To finish, it was decided in
1983, that the meter shall be defined by fixing the value of the speed of light in
vacuum to c = 299 792 458 m s⁻¹ and we refer to BIPM (2006) for an up to date
description of the SI system. Note that this version of the SI definition assumes that
the permittivity of vacuum μ₀ is also fixed so that the permeability of vacuum ε₀ is
also fixed.

This summary illustrates that the system of units is a human product and all SI definitions are historically based on non-relativistic classical physics and slowly drifted to quantum defined units since quantum technologies allow for a much lower uncertainty and a better control. The changes in the definitions were driven by the will to use more stable and more fundamental quantities so that they closely follow the progress of physics. This system has been created for legal use and indeed the choice of units is not restricted to SI.

## A.2. SI systems and the number of basic units

The International System (SI) of Units defines seven basic units (see BIPM 2006 for details): the meter (m), second (s) and kilogram (kg), the ampere (A), kelvin (K), mole (mol) and candela (cd), from which one defines secondary units. While needed for pragmatic reasons, this system of units is unnecessarily complicated from the point of view of theoretical physics. In particular, the kelvin, mole and candela are derived from the four other units since temperature is actually a measure of energy, the candela is expressed in terms of energy flux so that both can be expressed in

<footnote n="18">¹⁸ In his 1870 address, J.C. Maxwell states "If we wish to obtain standards of length, time and mass which shall be absolutely permanent, we must seek them not in the dimensions, or motion or the mass of our planet, but in the wavelength, the period of vibration, and absolute mass of these imperishable and unalterable and perfectly similar molecules." This is the reason why he proposed to use the properties of what he thought were the most fundamental objects of nature at his time, namely atoms (which he then called molecules). Indeed, and as he explained, if a structure is truly fundamental, it cannot change in time. It is imperishable and unalterable. It follows that these references are to be found, as already discussed, in the most fundamental theories and in particular in its constants.</footnote>
mechanical units of length [L], mass [M] and time [T]. The mole is merely a unit denoting numbers of particles and has no dimension associated.

The status of the Ampere is interesting in itself. The discovery of the electric charge [Q] led to the introduction of a new units, the Coulomb (C). The Coulomb law describes the force between two charges as being proportional to the product of the two charges and to the inverse of the distance squared. The dimension of the force being known as [MLT⁻²], this requires the introduction of a new constant ε₀ (which is only a conversion factor), with dimensions [Q²M⁻¹L⁻³T²] in the Coulomb law, and that needs to be measured. Indeed, another route is possible, and was actually adopted by Maxwell following Gauss, since the Coulomb law tells us that no new constant is actually needed if one uses [M¹/²L³/²T⁻¹] as the dimension of the charge. In such a system of units, known as Gaussian units, the numerical value of ε₀ is 1. Hence the coulomb can be expressed in terms of the mechanical units [L], [M] and [T], and so will the ampere. This reduces the number of conversion factors, that need to be experimentally determined, but note that both choices of units assume the validity of the Coulomb law, which has its own domain of validity.

### A.3. Natural units

The previous discussion tends to show that all units can be expressed in terms of the three mechanical units. It follows, as realized by Johnstone Stoney in 1874,<fnref n="19" /> that these three basic units can be defined in terms of 3 independent constants. He proposed (Johnstone-Stoney 1881; Barrow 1983) to use three constants: the Newton constant G, the velocity of light c and the basic units of electricity, i.e., the electron charge e, in order to define, from dimensional analysis a “natural series of physical units” as

$$t_S = \sqrt{\frac{Ge^2}{4\pi\varepsilon_0 c^6}} \sim 4.59 \times 10^{-45} \text{s},$$

$$\ell_S = \sqrt{\frac{Ge^2}{4\pi\varepsilon_0 c^4}} \sim 1.37 \times 10^{-36} \text{m},$$

$$m_S = \sqrt{\frac{e^2}{4\pi\varepsilon_0 G}} \sim 1.85 \times 10^{-9} \text{kg},$$

where the ε₀ factor has been included because we are using the SI definition of the electric charge. In such a system of units, by construction, the numerical value of G, e and c is 1, i.e., c = 1 × ℓₕ · tₛ⁻¹ etc.

<footnote n="19">¹⁹ After studying electrolysis in 1874, Johnstone Stoney suggested the existence of a “single definite quantity of electricity”. He was able to estimate the value of this elementary charge by means of Faraday’s laws of electrolysis. He introduced the term “electron” in 1894 and it was identified as a particle in 1897 by Thomson.</footnote>
A similar approach to the definition of the units was independently proposed by Planck (1900) on the basis of the two constants a and b entering the Wien law and G, which he reformulated later (Planck 1914) in terms of c, G and ℏ as

$$t_P = \sqrt{\frac{G\hbar}{c^3}} \sim 5.39056 \times 10^{-44} \text{ s},$$

$$\ell_P = \sqrt{\frac{G\hbar}{c^3}} \sim 1.61605 \times 10^{-35} \text{ m},$$

$$m_P = \sqrt{\frac{\hbar c}{G}} \sim 2.17671 \times 10^{-8} \text{ kg.}$$

The two systems are clearly related by the fine-structure constant since $e^2/4\pi\epsilon_0 = \alpha_{EM}hc$.

Indeed, we can construct many such systems since the choice of the 3 constants is arbitrary. For instance, we can build a system based on (e, m<sub>e</sub>, h), that we would call the *Bohr units*, well-suited to the study of the atom. The choice may be dictated by the system being studied (e.g., it is indeed far fetched to introduce G in the construction of the units when studying atomic physics) so that the system is well adjusted in the sense that the numerical values of the computations are expected to be of order unity in these units.

Such constructions are very useful for theoretical computations but not adapted to measurement so that one needs to switch back to SI units. More important, this shows that, from a theoretical point of view, one can define the system of units from the laws of nature, which are supposed to be universal and immutable.

### A.4. The new (2018) SI system

The 1983 redefinition of the meter was a first step toward natural units. Indeed the requirement of continuity, which is central to any redefinition of the SI system, forbids to set the numerical value of c to unity. Even though the “natural units” are not convenient and practical enough for high precision metrology, the idea to define units from constants was taken very seriously, mostly because they appear, using the words by J.C. Maxwell (see footnote 18), as the *most fundamental objects in our theories* and hence maybe the most stable objects on which one can build a stable system of units driving to move from physical artefacts to atomic properties to the physical constants; see Table 27 for the effects of such a change on the accuracy of the meter.

Let us first remind that the 1983 definition of the SI system was calling for improvements for many reasons.

* The first is related to the kilogram. Since 1889, it was defined on the basis of a material artefact, the International Prototype of the Kilogram (IPK). Since then, it was compared almost every 40 years to official copies. This shows a relative drift of some tenth of microgram since 1889, which limits the relative accuracy of the kilogram to 50 ppm. Also, the PIK was kept in Paris making comparison difficult. Then, the fact that the kilogram is a material artefact implies that one
can reach a relative accuracy of some $10^{-6}$ of masses of some kg but that such an accuracy was difficult to assess for very small masses, e.g., for particles, or very large masses, e.g., in astrophysics. This led for instance to the use of atomic units in particle physics with which one had a better accuracy for the masses of particles than in kg. To finish, the definitions other base units such as the ampere, candela, mole depend on the one of the kilogram.

* Concerning electronic units, the definition of the ampere through the mechanical force between two infinitely thin and long conductors was illusory. During some decades, this definition had been swapped in practice to definitions based on quantum effects so that electric standards were based on definitions outside the SI to reach higher precisions. This was in particular the case of the quantum Hall effect, related to the von Klitzning constant $R_K = h/e^2$ and the Josephson effect related to the constant $K_J = 2e/h$. It follows that quantum techniques spread largely in one branch of metrology.

* The Kelvin was defined as a fraction of the thermodynamic temperature of the triple point of water, hence depending on the properties of water and of its composition (purity, isotopic composition, etc.) Besides, the temperature scale was not adapted to extreme temperatures, typically above 2 000 K and below 20 K.

* The definition of the mole mixes the concept of quantity of matter and mass, which was conceptually misleading.

* To finish, from a structural point of view, the speed of light was fixed since 1983 to define the meter and the vacuum permeability $\mu_0$ was fixed in the definition of the ampere. Assuming Maxwell theory of electromagnetism, this implies that the vacuum permittivity $\epsilon_0$ was also fixed. Hence constants started to play a key role in the definitions.

These considerations led to the idea to unify the SI system to solve these problems of consistency, stability and precision that became crucial with the development of quantum technologies in particular. The idea to redefine the kilogram in terms of a fixed value of the Planck constant (Karshenboim 2005b; Blaum et al. 2019) through the development of e.g., Watt balance paved the way to a fully quantum-based

<table><caption>**Table 27** Evolution absolute and relative uncertainties of the realisation of the meter with its definition</caption><thead><tr><td>Date</td><td>Reference of the definition</td><td>Absolute uncertainty</td><td>Relative uncertainty</td></tr></thead><tbody><tr><td>1798</td><td>Earth meridian</td><td>0.1 mm</td><td>10<sup>-4</sup></td></tr><tr><td>1799</td><td>Platine prototype</td><td>0.01 mm</td><td>10<sup>-5</sup></td></tr><tr><td>1889</td><td>Platine prototype</td><td>0.1 µm</td><td>10<sup>-7</sup></td></tr><tr><td>1960</td><td>Kr-86 wavelength</td><td>0.01 µm</td><td>10<sup>-8</sup></td></tr><tr><td>1983</td><td>Speed of light in vacuum</td><td>0.1 nm</td><td>10<sup>-10</sup></td></tr><tr><td>2018</td><td>Speed of light in vacuum</td><td>0.1 nm</td><td>10<sup>-10</sup></td></tr></tbody></table>

The shift to atomic units occurred in 1960 and to a fundamental constant in 1983
(coherent and consistent) system of units based on fundamental constants; see e.g.,
Peik (2010).

Structurally, the new SI relies on a single physical artefact (as in the previous
system, a hyperfine transition of Cs-133, together with the (exact) value of six
constants ($c, h, e, k, N_A, K_{cd}$); see Table 28 (the definitions below specify the exact
numerical value of each constant when expressed in the corresponding SI unit). This
set of defining constants has been chosen to provide a fundamental, stable and
universal reference that simultaneously allows for practical realizations with the
smallest uncertainties.

For theorists, this definition is particularly satisfying for at least 2 reasons: (1) it
shed some light of the constants, the nature of which is source of many debates and
(2) everything rely on a clock. This is indeed the most universal way to define units
in General Relativity since all units derive from the second and that the second, as
defined, is the unit of proper time in the sense of the General Theory of Relativity.
This revision of the SI, probably the most significant since its establishment, was
adopted by the 26th CGPM (2018) and is documented<fnref n="20" /> in the 9th edition of the SI
Brochure (BIPM 2024). It has entered into force on May 20th 2019.

**Resolution 1 of the 26th CGPM (2018) –**

**On the revision of the International System of Units (SI)**

The General Conference on Weights and Measures (CGPM), at its 26th
meeting,

considering

[...]

decides that, effective from 20 May 2019, the International System of Units,
the SI, is the system of units in which:

* the unperturbed ground state hyperfine transition frequency of the caesium 133 atom $\Delta v_{Cs}$ is 9 192 631 770 Hz,

* the speed of light in vacuum c is 299 792 458 m/s,

* the Planck constant $h$ is 6.626 070 15 × 10⁻³⁴ J s,

* the elementary charge $e$ is 1.602 176 634 × 10⁻¹⁹ C,

* the Boltzmann constant $k$ is 1.380 649 × 10⁻²³ J/K,

* the Avogadro constant $N_A$ is 6.022 140 76 × 10²³ mol⁻¹,

* the luminous efficacy of monochromatic radiation of frequency 540 × 10¹² Hz, $K_{cd}$, is 683 lm/W,

where the hertz, joule, coulomb, lumen, and watt, with unit symbols Hz, J, C,
lm, and W, respectively, are related to the units second, metre, kilogram,
ampere, kelvin, mole, and candela, with unit symbols s, m, kg, A, K, mol, and
cd, respectively, according to $Hz = s^{-1}$, $J = kg \cdot m^2 \cdot s^{-2}$, $C = A \cdot s$,
$lm = cd \cdot m^2 \cdot s^{-2} = cd \cdot sr$, and $W = kg \cdot m^2 \cdot s^{-3}$

[...]

The integral text of the resolution can be accessed at https://www.bipm.org/en/
committees/cg/cgpm/26-2018/resolution-1

<footnote n="20">²⁰ We also refer to the following online resources for an in depth description of the new SI https://www.bipm.org/en/measurement-units.</footnote>
<table><caption>**Table 28** The seven defining constants of the SI and the seven units they define</caption><thead><tr><th>Defining constant</th><th>Symbol</th><th>Numerical value</th><th>Unit</th></tr></thead><tbody><tr><td>Hyperfine transition frequency of Cs</td><td>Δν<sub>Cs</sub></td><td>9 192 631 770</td><td>Hz</td></tr><tr><td>Speed of light in vacuum</td><td>c</td><td>299 792 458</td><td>m s<sup>-1</sup></td></tr><tr><td>Planck constant</td><td>h</td><td>6.626 070 15 ×10<sup>-34</sup></td><td>J s</td></tr><tr><td>elementary charge</td><td>e</td><td>1.602 176 634 ×10<sup>-19</sup></td><td>C</td></tr><tr><td>Boltzmann constant</td><td>k</td><td>1.380 649 ×10<sup>-23</sup></td><td>J K<sup>-1</sup></td></tr><tr><td>Avogadro number</td><td>N<sub>A</sub></td><td>6.022 140 76 ×10<sup>23</sup></td><td>mol<sup>-1</sup></td></tr><tr><td>luminous efficacy</td><td>K<sub>cd</sub></td><td>683</td><td>lm W<sup>-1</sup></td></tr></tbody></table>

The definition of the new SI has many consequences that shall be mentioned.

* It is a consistent and homogeneous system which ensures a gain in accuracy and precision with a better reliability and stability of the mass unit. It brings back electrical units within the SI and ensures continuity with the previous system while approaching the ideal of theoreticians. It relies on the best-tested equations of physics to connect the value of the constants to measurable quantities in the best tested theories at hand so that the SI system reflects our knowledge of physics at his best.

* It contains only one physical artefact, an atomic clock, to define the second. This implies that any progress in time-frequency metrology will propagate to the other units. The actual reflection thus concerns the definition of the second, see e.g., Bize (2019) and Dimarcq et al. (2024) for the actual roadmap.

* Besides, the new system makes no reference to any particular technology. A variety of experimental methods, referred to as “mises en pratique”, described by the CIPM Consultative Committees may be used to realize the definitions; see BIPM (2024). Realizations may be revised whenever new experiments with higher precision are developed; for this reason advice on realizing the definitions is not included in the definition but is available on the BIPM website. The use of a constant to define a unit disconnects definition from realization. This offers the possibility that completely different or new and superior practical realizations can be developed, as technologies evolve, without the need to change the definition.

* Concerning the nature of the defining constants, it ranges from fundamental constants to technical constants, see Table 28. The seven constants are chosen in such a way that any unit of the SI can be written either through a defining constant itself or through products or quotients of defining constants. According to our former discussions, both the Planck constant *h* and the speed of light in vacuum *c* are properly described as fundamental (class C). The elementary charge *e* corresponds to a coupling strength of the electromagnetic force via the fine-structure constant α<sub>EM</sub> (class B). As discussed in this review, variable α<sub>EM</sub> theories do exist but the constraints discussed in Sect. 5 is a strong argument to
think that any effect of a variation of $\alpha_{EM}$ on foreseeable practical measurements can be excluded. Anyhow, such a variation will not impact the new SI since it will then be interpreted as a variation of $\mu_0$ and $\epsilon_0$ (remember that $\alpha_{EM}$ is dimensionless). Note also that this choice is not unique and was at the core of many debates, both theoretical and experimental. For instance, shall one choose $e$ or $\epsilon_0$ or the vacuum impedance $Z_0 = \mu_0 c$ to define the ampere? The choice emphasizes on the physical interpretation of the fine structure constant, seen either as a property of the vacuum or of matter<fnref n="21" />; see Kovalesky (2006), Jeckelmann and Piquemal (2019). This highlights the importance on our reading of the physical laws. $\Delta v_{Cs}$ the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom is an atomic parameter (class A), which may be affected by the environment, such as electromagnetic fields. However, the underlying transition is well understood, stable and a good choice as a reference transition under practical considerations. It specifies the reference for all other units. The Boltzmann constant $k$ and the Avogadro constant $N_A$ are proportionality constants, the first between temperature (with unit kelvin) and energy (with unit joule), whereby the numerical value is obtained from historical specifications of the temperature scale and the second between the quantity amount of substance (with unit mole) and the quantity for counting entities (with unit one, symbol 1). The luminous efficacy of monochromatic radiation of frequency $540 \times 10^{12}$ Hz, $K_{cd}$, is a technical constant (class A) that gives an exact numerical relationship between the purely physical characteristics of the radiant power stimulating the human eye (W) and its photobiological response defined by the luminous flux due to the spectral responsivity of a standard observer (lm) at that frequency.

* The definitions of the units can be understood as follows; see BIPM (2024). The exact relation $\Delta v_{Cs} = 9192631770$ Hz implies that the unit second is expressed in terms of the defining constant as

$$1 \text{ s} = \frac{9192631770}{\Delta v_{Cs}}. \qquad (435)$$

The definition of the meter derived from the exact relation $c = 299792458 \text{ m.s}^{-1}$, i.e.,

$$1 \text{ m} = \frac{c}{299792458} \text{ s} \approx 30.663319 \frac{c}{\Delta v_{Cs}}. \qquad (436)$$

The definition of the kilogram derives from the fact that the exact value of the Planck constant is set to $6.62607015 \times 10^{-34}$ J s, that can be inverted to give

<footnote n="21">²¹ Assuming $\alpha_{EM}$ is a property of vacuum originates in string theories in which the dilaton determine the string coupling and can source a variation of $\alpha_{EM}$ and other constants. Hence one would then favor the choice of $e$ to define the ampere. If $\alpha_{EM}$ can also be seen as property of the electron since QED in flat spacetime relates the g-factor of the electron to a series in $\alpha_{EM}/2\pi$. It would then favor the choice of the Planck charge $q_P = \sqrt{2\epsilon_0hc} = \sqrt{2h/\bar{Z}_0} = e/\sqrt{\alpha_{EM}}$ which would correspond to keep $Z_0$ constant. To conclude, it is for practical and educational reasons that the elementary charge was preferred.</footnote>
$$1 \text{ kg} = \frac{h}{6.626\,070\,15 \times 10^{-34} \text{m}^{-2}\text{s}} \approx 1.475\,5214 \times 10^{40} \frac{h\Delta v_{\text{Cs}}}{c^2}. \quad (437)$$

The ampere derives from the exact value of e so that

$$1 \text{ A} = \frac{e}{1.602\,176634 \times 10^{-19} \text{s}^{-1}} \approx 6.789\,6868 \times 10^{8} \Delta v_{\text{Cs}}e. \quad (438)$$

Since only e is fixed, $\mu_0$ shall now be determined experimentally, as well as $\varepsilon_0$— see Table 28—but the relation $\mu_0\varepsilon_0 = 1/c^2$ remains exact. The Kelvin derives from the Boltzmann constant as

$$1 \text{ K} = \frac{1.380\,649 \times 10^{-23}}{k} \text{kg m}^{-2}\text{s}^{-2} \approx 2.266\,6653 \times 10^{8} \frac{\Delta v_{\text{Cs}}h}{k}. \quad (439)$$

The mole is not related to a dimension since it is a pure number. The definition in terms of the Avogadro number has the advantage to disentangle the amount of matter from the notion of mass. As a consequence, the molar mass of carbon-12, previously exactly fixed to 0.012 kg/mol, is no longer known exactly and must be determined experimentally and has a relative standard uncertainty of $4.5 \times 10^{-10}$. To finish, the candela is

$$1 \text{ cd} = K_{\text{cd}}683 \text{ kg m}^2\text{s}^{-3}\text{sr}^{-1} \approx 2.614\,8305 \times 10^{10} \Delta v_{\text{Cs}}^2 h K_{\text{cd}}. \quad (440)$$

* Note that with such definitions, primary realizations of the kilogram and the Kelvin can be established, in principle, at any point in the mass scale or the temperature scale.

* There is a big shift in the way to think about constants and units. For instance, the previous definition of the kilogram fixed the value of the mass of the IPK to be equal to one kilogram exactly and the value of the Planck constant h had to be determined with a given uncertainty by experiment. The present definition fixes the numerical value of h exactly and the mass of the IPK has now to be determined by experiment and is thus known with an uncertainty. Thus, uncertainties are shifted from the constants to the physical realization of the units, which then propagates to all measurements (see Table 28). It follows that the temperature of the triple point of water, $\mu_0$, $\varepsilon_0$ and the molar mass of carbon-12 are no more exact and their values and uncertainties shall be determined experimentally. Hence, one can see the exact values of the defining constant as a thermostat which fixes the realization of the units. Indeed, none of the values of the dimensionless parameters are affected by the change of definition.

* All conversion factors between energy units (Joule, eV, kg.c², ...) become exact since all conversions involve constants the values of which are now exact.

* The whole system relies on atomic and quantum physics. Besides, the *quantum metrology Triangle* experiments (see Scherer and Camarota 2012 for a review) combine three quantum electrical effects (the Josephson effect, the quantum Hall effect and the single-electron transport effect) used in metrology and allow for important fundamental consistency tests on the validity of commonly assumed relations between fundamental constants and the quantum electrical effects.
This new SI is probably the best system one can have constructed since it builds on our best understanding of the laws of nature and on the parameters which are thought to be the most fundamental so far, the fundamental constants.

### A.5. Do we actually need 3 natural units?

This is an issue debated at length. For instance, Duff et al. (2002) respectively argue for none, three and two (see also Wignall 2000; Matsas et al. 2007; Duff 2015; Matsas et al. 2024). Arguing for no fundamental constant leads to consider them simply as conversion parameters. Some of them are, like the Boltzmann constant, but some others play a deeper role in the sense that when a physical quantity becomes of the same order as this constant, new phenomena appear; this is the case, e.g., of $\hbar$ and $c$, which are associated respectively to quantum and relativistic effects. Okun (1991) considered that only three fundamental constants are necessary, as indicated by the International System of units. In the framework of quantum field theory + General Relativity, it seems that this set of three constants has to be considered and it allows one to classify the physical theories (with the famous cube of physical theories). However, Veneziano (1986) argued that in the framework of string theory one requires only two dimensionful fundamental constants, $c$ and the string length $\lambda_s$. The use of $\hbar$ seems unnecessary since it combines with the string tension to give $\lambda_s$. In the case of the Nambu–Goto action $S/\hbar = (T/\hbar) \int d(Area) \equiv \lambda_s^{-2} \int d(Area)$ and the Planck constant is just given by $\lambda_s^{-2}$. In this view, $\hbar$ has not disappeared but has been promoted to the role of a UV cut-off that removes both the infinities of quantum field theory and singularities of General Relativity. This situation is analogous to pure quantum gravity (Novikov and Zel'dovich 1983) where $\hbar$ and $G$ never appear separately but only in the combination $\ell_{Pl} = \sqrt{G\hbar/c^3}$ so that only $c$ and $\ell_{Pl}$ are needed. Volovik (2002) made an analogy with quantum liquids to clarify this. There, an observer knows both the effective and microscopic physics so that he can judge whether the fundamental constants of the effective theory remain fundamental constants of the microscopic theory. The status of a constant depends on the considered theory (effective or microscopic) and, more interestingly, on the observer measuring them, i.e., on whether this observer belongs to the world of low-energy quasi-particles or to the microscopic world.

## Appendix B. Notations

### B.1 Constants

The notations and numerical values of the constants used in this review are summarized in Tables 1 and 2.

### B.2 Sensitivity coefficients

The text introduces several sensitivity coefficients. We recall their definitions here.
<table><caption>**Table 29** Uncertainties of the quantities entering the new definition of the SI compared to the one of the former definition of the SI</caption><thead><tr><th>Quantity</th><th>Uncertainty (CODATA-2017) ×10<sup>9</sup></th><th>Uncertainty (new SI) ×10<sup>9</sup></th></tr></thead><tbody><tr><td>Speed of light</td><td>0</td><td>0</td></tr><tr><td>Planck constant</td><td>10</td><td>0</td></tr><tr><td>Electron charge</td><td>5.2</td><td>0</td></tr><tr><td>Boltzmann constant</td><td>370</td><td>0</td></tr><tr><td>Avogadro number</td><td>10</td><td>0</td></tr><tr><td>IPK</td><td>0</td><td>10</td></tr><tr><td>Vacuum permeability</td><td>0</td><td>0.23</td></tr><tr><td>Vacuum permittivity</td><td>0</td><td>0.23</td></tr><tr><td>Water triple point</td><td>0</td><td>370</td></tr><tr><td>C-12 molar mass</td><td>0</td><td>0.45</td></tr></tbody></table>

<table><caption>**Table 30** Main cosmological parameters in the standard Λ-CDM model</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>Reduced Hubble constant</td><td>h<sub>0</sub></td><td>0.6766 ± 0.0042</td></tr><tr><td>Photon density</td><td>Ω<sub>ν</sub>h<sub>0</sub><sup>2</sup></td><td>2.471 × 10<sup>-5</sup></td></tr><tr><td>Neutrino density</td><td>Ω<sub>ν</sub>h<sub>0</sub><sup>2</sup></td><td>(0.0005–0.023)</td></tr><tr><td>Dark matter density</td><td>Ω<sub>CDM</sub>h<sub>0</sub><sup>2</sup></td><td>0.11933 ± 0.00091</td></tr><tr><td>Baryon density</td><td>Ω<sub>b</sub>h<sub>0</sub><sup>2</sup></td><td>0.02242 ± 0.00014</td></tr><tr><td>Matter density</td><td>Ω<sub>m</sub>h<sub>0</sub><sup>2</sup></td><td>0.14240 ± 0.00087</td></tr><tr><td>Matter density</td><td>Ω<sub>m</sub></td><td>0.3111 ± 0.0056</td></tr><tr><td>Cosmological constant</td><td>Ω<sub>Λ</sub></td><td>0.6889 ± 0.0056</td></tr><tr><td>Spatial curvature</td><td>Ω<sub>K</sub></td><td>0.011(12)</td></tr><tr><td>Baryon-to-photon ratio</td><td>η = n<sub>b</sub>/n<sub>γ</sub></td><td>6.12(19) × 10<sup>-10</sup></td></tr><tr><td>Scalar modes amplitude</td><td>Q</td><td>(2.0 ± 0.2) × 10<sup>-5</sup></td></tr><tr><td>Scalar spectral index</td><td>n<sub>S</sub></td><td>0.958(16)</td></tr><tr><td>Dark energy equation of state</td><td>w</td><td>-0.97(7)</td></tr><tr><td>Scalar running spectral index</td><td>α<sub>S</sub></td><td>-0.05 ± 0.03</td></tr><tr><td>Tensor-to-scalar ratio</td><td>T/S</td><td>&lt; 0.36</td></tr><tr><td>Tensor spectral index</td><td>n<sub>T</sub></td><td>&lt; 0.001</td></tr><tr><td>Tensor running spectral index</td><td>α<sub>T</sub></td><td>?</td></tr></tbody></table>

The first part of the table gives the parameters of the background minimal standard cosmological parameters, that is the reduced Hubble parameters together with 7 density parameters—only 6 of them being independent since $\sum \Omega_i = 1$—and the baryon-to-photon ratio. The second part gives the parameters related to the density perturbations for the large-scale structures of the universe. Then, the last part gives several extensions such as a dark energy component and gravity waves. Note that often the spatial curvature is set to $\Omega_K = 0$. (See Aghanim et al. 2020)
* Given an observable O, the value of which depends on a set of primary parameters $G_k$, the sensitivity of the measured value of O to these parameters is

$$ \frac{d \ln O}{d \ln G_k} = c_k. \qquad (441) $$

The value of the quantities $c_k$ requires a physical description of the system.

* The parameters $G_k$ can be related to a set of fundamental constants $\alpha_i$ and we define

$$ \frac{d \ln G_k}{d \ln \alpha_i} = d_{ki}. \qquad (442) $$

The computation of the coefficients $d_{ki}$ requires one to specify the theoretical framework and depends heavily on our knowledge of nuclear physics and the assumptions on unification.

* A particular sets of parameters $d_{ki}$ has been singled out for the sensitivity of the mass of a body A to a variation of the fundamental constants

$$ \frac{d \ln m_A}{d \alpha_i} = f_{Ai}. \qquad (443) $$

One also introduced

$$ \frac{d \ln m_A}{d \ln \alpha_i} = \lambda_{Ai} \qquad (444) $$

so that

$$ \lambda_{Ai} = \alpha_i f_{Ai}. \qquad (445) $$

* In models where the variation of the fundamental constants are induced by the variation of a scalar field we defined

$$ \frac{d \ln \alpha_i}{d \varphi} = s_i(\varphi), \qquad (446) $$

and

$$ \frac{d \ln m_A}{d \varphi} = \alpha_A(\varphi), \qquad (447) $$

which can be expressed in terms of the previous sensitivities as

$$ \alpha_A(\varphi) = \sum_i f_{Ai} \alpha_i s_i(\varphi). $$

We also recall that the dimensionless scalar field $\varphi$ and the one with dimension of mass $\phi$ are related by

$$ \phi = M_* \varphi \qquad (448) $$

where $M_*$ is a mass scale, often chosen as
$$M_* = \sqrt{8\pi G_*}.$$

* In the class of models the variation of the constants can be related to the gravitational potential by

$$\frac{d \ln \alpha_i}{d \Phi_N} = k_i. \qquad (449)$$

## Appendix C. Background cosmological spacetime

We consider that the spacetime is described by a manifold $\mathcal{M}$ with metric $g_{\mu\nu}$ with signature $(-, +, +, +)$. In the case of a Minkowski spacetime $g_{\mu\nu} = \eta_{\mu\nu}$.

In the cosmological context, the universe is described by a Friedmann–Lemaître spacetime with metric

$$ds^2 = -dt^2 + a^2(t) \gamma_{ij} dx^i dx^j \qquad (450)$$

where $t$ is the cosmic time, $a$ the scale factor and $\gamma_{ij}$ the metric on the constant time hypersurfaces. The Hubble function is defined as

$$H \equiv \dot{a}/a \qquad (451)$$

with a dot referring to a derivative with respect to the cosmic time $t$. We also define the redshift by the relation $1+z = a_0/a$, with $a_0$ the scale factor evaluated today. It is usual to introduce the conformal time $\eta$ by $dt = a\eta$ and use a prime for derivative with respect to $\eta$. The deceleration parameter $a$ is defined as

$$q = - \frac{\ddot{a}}{aH^2} = \frac{\dot{H}}{H^2} - 1 \qquad (452)$$

that the Universe accelerates if $q<0$ and decelerates otherwise.

The evolution of the scale factor is dictated by the Friedmann equation

$$H^2 = \frac{8\pi G}{3} \rho - \frac{K}{a^2} + \frac{\Lambda}{3}, \qquad (453)$$

where $\rho = \sum_i \rho_i$ is the total energy density of the matter components in the universe. Assuming the species $i$ has a constant equation of state $w_i = P_i/\rho_i$, each component evolves as $\rho_i = \rho_{i0}(1+z)^{2(1+w_i)}$. The Friedmann equation can then be rewritten as

$$\frac{H^2}{H_0^2} = \sum_i \Omega_i (1+z)^{3(1+w_i)} + \Omega_K (1+z)^2 + \Omega_\Lambda, \qquad (454)$$

with the density parameters defined by
$$ \Omega_i \equiv \frac{8\pi G\rho_{i0}}{3H_0^2}, \qquad \Omega_i \equiv -\frac{K}{3H_0^2}, \qquad \Omega_A \equiv \frac{\Lambda}{3H_0^2}. \qquad (455) $$

They clearly satisfy $\sum \Omega_i + \Omega_K + \Omega_A = 1$.

The value of the Hubble parameter today is

$$ H_0 = 100 h_0 \text{ km s}^{-1} \text{ Mpc}^{-1} = \frac{h_0}{9.78 \times 10^9 \text{ yr}} = \frac{h_0}{3.09 \times 10^{17} \text{ s}}, \qquad (456) $$

where $h_0$ is a dimensionless number, usually denoted by $h$ in the literature but this notation avoids confusion with the Planck constant. For the sake of numerical computations, we shall use the values of the cosmological parameters as estimated by the latest Planck satellite data analysis (Aghanim et al. 2020).

Concerning the properties of the cosmological spacetime, we follow the notations and results of Peter and Uzan (2009).

## Appendix D. Equation of motion for a charged particle

In a theory in which the matter fields couple to the metric $A^2(\varphi)g_{\mu\nu}$, the equation of a point particle of mass $m$ and charge $q$ is

$$ S_{pp} = -c^2 \int mA(\varphi) \sqrt{-g_{\mu\nu}u^\mu u^\nu} d\tau + q \int A_\mu u^\mu d\tau. \qquad (457) $$

The equation of motion follows as

$$ mc^2 \gamma^\mu = \frac{q}{A(\varphi)} F^\mu_\nu u^\nu - mc^2 \frac{\partial \ln A}{\partial \varphi} \perp^{\mu\nu} \nabla_\nu \varphi \qquad (458) $$

with $\perp_{\mu\nu} = g_{\mu\nu} + u_\mu u_\nu / c^2$ the projector on the 3-space normal to $u^\mu$, which indeed ensures that $u^\mu u_\mu = -c^2$; see e.g., Uzan (2011). It follows that the fifth force,

$$ F^\mu = -mc^2\alpha(\varphi) \perp^{\mu\nu} \nabla_\nu \varphi, \qquad (459) $$

remains perpendicular to the 4-velocity, $u_\mu F^\mu = 0$.

**Acknowledgements** I would like to thank all my collaborators on this topic, Alain Coc, Pierre Descouvemont, Sylvia Ekström, George Ellis, Georges Meynet, Nelson Nunes, Keith Olive, Cyril Pitrou and Elisabeth Vangioni as well as Bénédicte Leclercq, Roland Lehoucq, and Ludivine Bantigny. I also thank many colleagues for sharing their thoughts on the subject with me, first at the Institut d'Astrophysique de Paris, Luc Blanchet, Michel Cassé, Gilles Esposito-Farèse, Bernard Fort, Guillaume Faye, Jean-Pierre Lasota, Yannick Mellier, Patrick Petitjean; at BIPM, Richard Davies, Céline Fellag-Ariouet, Martin Milton, Terry Quinn in France, Jibril Ben Achour, Joel Bergé, Francis Bernardeau, Sébastien Bize, Philippe Brax, Françoise Combes, Thibault Damour, Nathalie Deruelle, Hugo Lévy, Martin Pernot-Borrás, Christophe Salomon, Peter Wolf; and to finish worldwide, John Barrow, Thomas Dent, Victor Flambaum, Bala Iyer, Lev Kofman, Carlos Martins, Paolo Molaro, David Mota, Michael Murphy, Jeff Murugan, Anan Srianand, Gabriele Veneziano, John Webb, Amanda Weltman, Christof Wetterich.

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