# Fundamental constants: from measurement to the universe, a window on gravitation and cosmology

Jean-Philippe Uzan

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## To cite this version:

Jean-Philippe Uzan. Fundamental constants: from measurement to the universe, a window on gravitation and cosmology. Living Reviews in Relativity, 2025, 28 (1), pp.6. 10.1007/s41114-025-00059-y. hal-04751688

HAL Id: hal-04751688

https://hal.science/hal-04751688v1

Submitted on 31 Oct 2025

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# Fundamental constants: from measurement to the universe, a window on gravitation and cosmology

Jean-Philippe Uzan¹,² ID

Received: 9 October 2024 / Accepted: 26 May 2025

© The Author(s) 2025

## Abstract

Fundamental constants are a cornerstone of our physical laws. Any constant varying in space and/or time would signal a violation of local position invariance and be associated with a violation of the universality of free fall, and hence of the weak equivalence principle at the heart of the geometrisation of gravity. It will also reflect the existence of new degrees of freedom that couple to standard matter fields. Thus, testing for the stability of fundamental constants is of utmost importance for our understanding of gravity and for characterizing the domain of validity of general relativity. Besides, it opens an independent window on the dark matter and dark energy components. As a consequence, thanks to the active developments of experiments, fundamental constants have become a key player in our search for physics beyond the standard model of particle physics and general relativity. This review details the various roles of the fundamental constants in the laws of physics and in the construction of the international system of units, which now depends strongly on them. This requires to distinguish the concepts of *fundamental units* and *fundamental parameters*. Then, the relations between constants, the tests of the local position invariance and of the universality of free fall are presented, as well as the construction of field theories that account for “varying constants” and the motivations arising from high-energy physics and string theory. From a theoretical perspective any varying fundamental parameter is related to a dynamical field, the dynamics of which is dictated from the whole theory so that it remains fully consistent: no arbitrary law of variation has to be postulated. Then, the main experi-

This article is a revised version of https://doi.org/10.12942/lrr-2011-2.

**Change summary** Major revision, updated and expanded.

**Change details** Updated and restructured the article, added new subsections. Previous Sect. 5 on “Theories with varying constants” has been split into new Sects. 3, 4, and 7. due to the increase of results and also to simplify the discussion of the experimental results. Appendices C and D have been added. The number of references has increased from 554 to 1198.

Extended author information available on the last page of the article
mental and observational constraints that have been obtained from atomic clocks, the Oklo phenomenon, solar system observations, meteorite dating, quasar absorption spectra, stellar physics, pulsar timing, the cosmic microwave background and Big Bang nucleosynthesis are described. It details the basics of each system, its dependence with respect to the primary parameters the variation of which can be constrained from observations, the known systematic effects and the most recent constraints. It also describes how these primary parameters can be related to the fundamental constants and the model-dependencies that is involved. Both time and space variations are considered. To finish, it contains a short discussion on the more speculative possibility of understanding the numerical values of the fundamental parameters in view of the apparent fine-tuning that they confront us with, by invoking anthropic arguments. Given the huge increase of data and constraints and the difficulty to standardize them, a general scheme to present experimental and observational results and to construct a collaborative data base that will be more efficient for the community and allow us for better traceability, is proposed.

**Keywords** General theory of gravitation · Fundamental physical constants · Theoretical cosmology · Metrology · System of units

## Contents

<table><tr><td>1</td><td>Introduction</td><td>5</td></tr><tr><td>2</td><td>Constants and fundamental physics</td><td>11</td></tr><tr><td>2.1</td><td>Toward a definitions of fundamental constants</td><td>11</td></tr><tr><td>2.1.1</td><td>Characterizing the fundamental constants</td><td>12</td></tr><tr><td>2.1.2</td><td>Constants and metrology</td><td>18</td></tr><tr><td>2.1.3</td><td>Definitions: fundamental parameters and fundamental units</td><td>21</td></tr><tr><td>2.2</td><td>The constancy of constants as a test of general relativity</td><td>23</td></tr><tr><td>2.2.1</td><td>General relativity</td><td>23</td></tr><tr><td>2.2.2</td><td>Varying constants and free fall: the link between LPI and UFF</td><td>32</td></tr><tr><td>2.3</td><td>Relations with cosmology</td><td>35</td></tr><tr><td>2.3.1</td><td>ΛCDM standard cosmological model</td><td>36</td></tr><tr><td>2.3.2</td><td>Dark energy and its implications</td><td>36</td></tr><tr><td>2.3.3</td><td>Light fields phenomenology</td><td>37</td></tr><tr><td>2.3.4</td><td>Ultra-light dark matter models</td><td>38</td></tr><tr><td>2.3.5</td><td>Dark matter topological defects</td><td>39</td></tr><tr><td>2.3.6</td><td>Varying constants and cosmology</td><td>40</td></tr><tr><td>2.4</td><td>Conclusion</td><td>44</td></tr><tr><td>3</td><td>Theories with varying constants</td><td>44</td></tr><tr><td>3.1</td><td>Introducing new fields: generalities</td><td>47</td></tr><tr><td>3.1.1</td><td>The example of scalar-tensor theories</td><td>47</td></tr><tr><td>3.1.2</td><td>Making non-gravitational constants dynamical</td><td>53</td></tr><tr><td>3.1.3</td><td>Summary</td><td>55</td></tr><tr><td>3.2</td><td>Hiding light degrees of freedom</td><td>57</td></tr><tr><td>3.2.1</td><td>Decoupling mechanisms</td><td>58</td></tr><tr><td>3.2.2</td><td>Chameleon mechanism</td><td>59</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>3.2.3</td>
      <td>Environment-dependent dilaton model</td>
      <td>62</td>
    </tr>
    <tr>
      <td>3.2.4</td>
      <td>Conclusion</td>
      <td>63</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>Searching for new degrees of freedom</td>
      <td>63</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>High-energy theories and varying constants</td>
      <td>65</td>
    </tr>
    <tr>
      <td>3.4.1</td>
      <td>Higher-dimension and Kaluza-Klein theory</td>
      <td>65</td>
    </tr>
    <tr>
      <td>3.4.2</td>
      <td>String theory</td>
      <td>67</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Relations between constants</td>
      <td>69</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Implication of gauge coupling unification</td>
      <td>70</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Masses and binding energies in terms of the fundamental constants</td>
      <td>75</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Gyromagnetic factors in terms of the fundamental constants</td>
      <td>78</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Expression of the atomic transitions</td>
      <td>80</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Phenomenological (R, S)-parameterisation</td>
      <td>84</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Experimental and observational constraints on non-gravitational constants</td>
      <td>86</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Atomic and molecular clocks</td>
      <td>88</td>
    </tr>
    <tr>
      <td>5.1.1</td>
      <td>Atomic spectra and constants</td>
      <td>88</td>
    </tr>
    <tr>
      <td>5.1.2</td>
      <td>Experimental constraints from atomic clocks</td>
      <td>90</td>
    </tr>
    <tr>
      <td>5.1.3</td>
      <td>Constraints on primary parameters</td>
      <td>96</td>
    </tr>
    <tr>
      <td>5.1.4</td>
      <td>Fiber-linked networks and clocks in space</td>
      <td>98</td>
    </tr>
    <tr>
      <td>5.1.5</td>
      <td>Experimental constraints from molecular transitions</td>
      <td>99</td>
    </tr>
    <tr>
      <td>5.1.6</td>
      <td>Nuclear clocks as a possible new system</td>
      <td>103</td>
    </tr>
    <tr>
      <td>5.1.7</td>
      <td>Future evolutions</td>
      <td>105</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Macroscopic bodies and chemistry</td>
      <td>105</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>The Oklo phenomenon</td>
      <td>106</td>
    </tr>
    <tr>
      <td>5.3.1</td>
      <td>A natural nuclear reactor</td>
      <td>106</td>
    </tr>
    <tr>
      <td>5.3.2</td>
      <td>Constraining the shift of the resonance energy</td>
      <td>108</td>
    </tr>
    <tr>
      <td>5.3.3</td>
      <td>From the resonance energy to fundamental constants</td>
      <td>112</td>
    </tr>
    <tr>
      <td>5.3.4</td>
      <td>Conclusion</td>
      <td>115</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Meteorite dating</td>
      <td>115</td>
    </tr>
    <tr>
      <td>5.4.1</td>
      <td>Long lived α-decays</td>
      <td>116</td>
    </tr>
    <tr>
      <td>5.4.2</td>
      <td>Long lived β-decays</td>
      <td>118</td>
    </tr>
    <tr>
      <td>5.4.3</td>
      <td>Conclusions</td>
      <td>120</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>Quasar absorption spectra</td>
      <td>121</td>
    </tr>
    <tr>
      <td>5.5.1</td>
      <td>Elements for a α<sub>EM</sub> measurement</td>
      <td>121</td>
    </tr>
    <tr>
      <td>5.5.2</td>
      <td>Evolution of the debate</td>
      <td>125</td>
    </tr>
    <tr>
      <td>5.5.3</td>
      <td>Different methods to track for the variation of constants</td>
      <td>128</td>
    </tr>
    <tr>
      <td>5.5.4</td>
      <td>Single ion differential measurement (SIDAM)</td>
      <td>134</td>
    </tr>
    <tr>
      <td>5.5.5</td>
      <td>Summary of the observable constraints on α<sub>EM</sub></td>
      <td>136</td>
    </tr>
    <tr>
      <td>5.6</td>
      <td>Quasar absorption spectra: constraints on combination of α<sub>EM</sub>, μ and g<sub>p</sub></td>
      <td>137</td>
    </tr>
    <tr>
      <td>5.6.1</td>
      <td>H t-21 cm vs. UV: xg<sub>p</sub>α<sub>EM</sub>μ</td>
      <td>137</td>
    </tr>
    <tr>
      <td>5.6.2</td>
      <td>H t vs. molecular transitions: y ≡ g<sub>p</sub>α<sub>EM</sub><sup>2</sup></td>
      <td>140</td>
    </tr>
    <tr>
      <td>5.6.3</td>
      <td>OH - 18 cm: Fg<sub>p</sub>(α<sub>EM</sub><sup>2</sup>μ<sup>1.57</sup>)</td>
      <td>141</td>
    </tr>
    <tr>
      <td>5.6.4</td>
      <td>Far infrared fine-structure lines: F'α<sub>EM</sub><sup>2</sup>μ</td>
      <td>143</td>
    </tr>
    <tr>
      <td>5.6.5</td>
      <td>“Conjugate” satellite OH lines: Gg<sub>p</sub>(α<sub>EM</sub><sup>2</sup>μ<sup>1.85</sup>)</td>
      <td>144</td>
    </tr>
    <tr>
      <td>5.6.6</td>
      <td>Summary of the constraints on xyFF' and G</td>
      <td>145</td>
    </tr>
    <tr>
      <td>5.7</td>
      <td>Molecular absorption spectra and the electron-to-proton mass ratio</td>
      <td>145</td>
    </tr>
    <tr>
      <td>5.7.1</td>
      <td>Introduction</td>
      <td>145</td>
    </tr>
    <tr>
      <td>5.7.2</td>
      <td>Constraints with H<sub>2</sub></td>
      <td>147</td>
    </tr>
    <tr>
      <td>5.7.3</td>
      <td>Amonia observations</td>
      <td>151</td>
    </tr>
    <tr>
      <td>5.7.4</td>
      <td>Methanol observations</td>
      <td>152</td>
    </tr>
    <tr>
      <td>5.7.5</td>
      <td>CO observations</td>
      <td>154</td>
    </tr>
    <tr>
      <td>5.8</td>
      <td>Emission spectra</td>
      <td>155</td>
    </tr>
    <tr>
      <td>5.9</td>
      <td>Stellar constraints</td>
      <td>159</td>
    </tr>
    <tr>
      <td>5.10</td>
      <td>Cosmic microwave background</td>
      <td>163</td>
    </tr>
    <tr>
      <td>5.11</td>
      <td>21 cm</td>
      <td>172</td>
    </tr>
  </tbody>
</table>
<table><tbody><tr><td>5.12 Galaxy clusters</td><td>176</td></tr><tr><td>5.13 Fast radio bursts</td><td>177</td></tr><tr><td>5.14 Big Bang nucleosynthesis</td><td>178</td></tr><tr><td>5.14.1 Overview</td><td>178</td></tr><tr><td>5.14.2 Status of standard BBN</td><td>180</td></tr><tr><td>5.14.3 Constants everywhere</td><td>181</td></tr><tr><td>5.14.4 From BBN parameters to fundamental constants</td><td>185</td></tr><tr><td>5.14.5 Constants and the lithium problem</td><td>187</td></tr><tr><td>5.14.6 Conclusion</td><td>188</td></tr><tr><td>6 The gravitational sector and G</td><td>189</td></tr><tr><td>6.1 Solar system constraints</td><td>190</td></tr><tr><td>6.2 Pulsar timing</td><td>192</td></tr><tr><td>6.3 Stellar constraints</td><td>194</td></tr><tr><td>6.3.1 Ages of globular clusters</td><td>195</td></tr><tr><td>6.3.2 Solar and stellar seismology</td><td>195</td></tr><tr><td>6.3.3 White dwarfs pulsation</td><td>197</td></tr><tr><td>6.3.4 Late stages of stellar evolution and supernovae</td><td>197</td></tr><tr><td>6.3.5 New developments from gravitational-wave astronomy</td><td>199</td></tr><tr><td>6.4 Cosmological constraints</td><td>200</td></tr><tr><td>6.4.1 Cosmic microwave background</td><td>200</td></tr><tr><td>6.4.2 Big Bang nucleosynthesis</td><td>202</td></tr><tr><td>6.4.3 New proposals: strong lensing</td><td>204</td></tr><tr><td>6.4.4 New proposals: gravitational waves</td><td>205</td></tr><tr><td>7 Models with varying constants</td><td>206</td></tr><tr><td>7.1 String dilaton and runaway dilaton models</td><td>206</td></tr><tr><td>7.2 Damour–Donoghue interaction light dilaton model</td><td>210</td></tr><tr><td>7.3 Bekenstein and related models</td><td>216</td></tr><tr><td>7.4 Light scalar quintessence</td><td>220</td></tr><tr><td>7.5 Other ideas</td><td>225</td></tr><tr><td>8 Spatial variations</td><td>226</td></tr><tr><td>8.1 Generalities</td><td>227</td></tr><tr><td>8.2 Local scales</td><td>227</td></tr><tr><td>8.2.1 Solar system scales</td><td>227</td></tr><tr><td>8.2.2 Atomic clock constraints</td><td>230</td></tr><tr><td>8.2.3 Milky Way</td><td>237</td></tr><tr><td>8.2.4 Towards strong field environments</td><td>239</td></tr><tr><td>8.3 Cosmological scales</td><td>242</td></tr><tr><td>8.3.1 Primordial fluctuations</td><td>242</td></tr><tr><td>8.3.2 Dipolar modulation</td><td>244</td></tr><tr><td>8.3.3 Wall of constants</td><td>246</td></tr><tr><td>8.3.4 Non-Copernican field configuration</td><td>249</td></tr><tr><td>8.4 Universality of free fall</td><td>249</td></tr><tr><td>9 Why are the constants just so?</td><td>250</td></tr><tr><td>9.1 Universe and multiverse approaches</td><td>250</td></tr><tr><td>9.2 Fine-tunings and determination of the anthropic range</td><td>251</td></tr><tr><td>9.3 Anthropic predictions</td><td>255</td></tr><tr><td>10 Conclusions</td><td>257</td></tr><tr><td>11 Epilogue: Towards the collective creation of a community data base</td><td>259</td></tr><tr><td>Appendix A. System of units and the international system of units (SI)</td><td>260</td></tr><tr><td>A.1. System of units</td><td>260</td></tr><tr><td>A.2. SI systems and the number of basic units</td><td>261</td></tr><tr><td>A.3. Natural units</td><td>262</td></tr><tr><td>A.4. The new (2018) SI system</td><td>263</td></tr><tr><td>A.5. Do we actually need 3 natural units?</td><td>267</td></tr><tr><td>Appendix B. Notations</td><td>269</td></tr></tbody></table>

<figure><img src="image_4.png" /></figure>
<table><tr><td>B.1 Constants</td><td>269</td></tr><tr><td>B.2 Sensitivity coefficients</td><td>269</td></tr><tr><td>Appendix C. Background cosmological spacetime</td><td>272</td></tr><tr><td>Appendix D. Equation of motion for a charged particle</td><td>273</td></tr><tr><td>References</td><td>274</td></tr></table>

# 1 Introduction

Fundamental constants appear everywhere in the mathematical formulation of the laws we use to describe the phenomena of nature. They seem to contain some truth about the properties of the physical world while their real nature seem to evade us.

While initiated about one century ago, the interest of fundamental constants attracted little attention until the surprising claim by Webb et al. (1999) that quasar absorption spectra indicated that the fine structure constant was smaller in the past. This relaunched the interest on fundamental constants on two different but complementary fronts: (1) *theoretical*, as a way to constrain deviations from General Relativity, the validity of which on astrophysical scales is questioned due to the need of *dark energy* and (2) *experimental and observational*, motivated by both the necessity to confirm or infirm the claim by Webb et al. (1999) and by the developments of many complementary methods to test the constancy of fundamental constants on a large band of redshifts (Uzan 2003b). This is now recognized as a key issue in views of the current anomalies of our cosmological models (Peebles 2022). To finish, the new definition of the International System of units adopted in 2018 has radically changed the way we shall conceive the relations between constants and units, which give a third motivation for a better understanding of their nature.

The question of the constancy of the constants of physics was probably first addressed by Dirac (1937, 1938) who expressed, in his “*Large Numbers hypothesis*”, the opinion that very large (or small) dimensionless universal constants cannot be pure mathematical numbers and must not occur in the basic laws of physics. He suggested, on the basis of this purely numerical principle, that these large numbers should rather be considered as variable parameters characterizing the state of the universe. Dirac formed five dimensionless ratios among which<fnref n="1" /> $\delta \equiv H_0\hbar/m_p c^2 \sim 2h_0 \times 10^{-42}$ and $\epsilon \equiv G\rho_0/H_0^2 \sim 5h_0^{-2} \times 10^{-4}$ and asked the question of which of these ratios is constant as the universe evolves. Usually, $\delta$ varies as the inverse of the cosmic time while $\epsilon$ varies also with time if the universe is not described by an Einstein-de Sitter solution—i.e., when a cosmological constant, curvature or radiation are included in the cosmological model; see Eq. (453). Dirac then noticed<fnref n="2" /> that $\alpha_G/\mu\alpha_{EM}$, representing the relative magnitude of electrostatic and gravitational forces between a proton and an electron,

<footnote n="1">¹ $H_0$ is the Hubble constant today, $\rho_0$ the mean matter energy density today. $h_0$ is the reduced Hubble parameter, a dimensionless number defined in Eq. (456). The other constants are defined in Tables 1 and 2 and the definitions of the cosmological quantities are summarized in Appendix C.</footnote>

<footnote n="2">² See Eq. (2) below for the definitions of $\alpha_{EM}$, $\alpha_G$ and $\mu$.</footnote>
was of the same order as $H_0e^2/m_ec^2 = \delta\alpha_{EM}\mu$ representing the age of the universe in atomic units. Hence, this five numbers can be "harmonized" if one assumes that $\alpha_G$ vary as the inverse of the cosmic time, as $\delta$ does.

This numerological argument by Dirac is indeed not a physical theory but it opened many doors in the investigation on physical constants, both on questioning whether they are actually constant and on trying to understand the numerical values we measure. Both questions have initiated two fields of research, the first on the use of constants to test General Relativity on cosmological scales and, the second, more speculative, on ways to account for fine tunings, and in particular the one for life.

First, the implementation of Dirac's phenomenological idea into a field-theory framework was proposed by Jordan (1937), who realized that the constants have to become dynamical fields. He proposed a theory in which both the gravitational and fine-structure constants can vary—Unzicker (2009) and Section II of Uzan (2003b) provides a summary of some earlier attempts to quantify the cosmological implications of Dirac's argument and to formulate alternatives. Fierz (1956) then realized that in such a case, atomic spectra shall become spacetime-dependent so that these theories can be observationally tested. The sub-case in which only $G$ can vary led to the definition of scalar-tensor theories of gravity, further explored by Brans and Dicke (1961). These theories were generalized to obtain various functional dependencies for $G$ in the formalization of universal scalar-tensor theories (see, e.g., Damour and Esposito-Farèse 1992). Today many extensions of General Relativity are, as we shall see, accompanied by the variation of some constants.

Second, Dicke (1961) pointed out that in fact the density of the universe is determined by its age, the later being related to the time needed to form galaxies, stars, heavy nuclei.... This led him to formulate that the presence of an observer in the universe places constraints on the physical laws that can be observed. In fact, what is meant by observer is the existence of (highly?) organized systems and this principle can be seen as a rephrasing of the question "why is the universe the way it is?" (see e.g., Hogan 2000). Carter (1974, 1983), who actually coined the term "anthropic principle", showed that the numerological coincidences found by Dirac can be derived from physical models of stars and the competition between the weakness of gravity with respect to nuclear fusion. Carr and Rees (1979) then showed how one can scale up from atomic to cosmological scales only by using combinations of $\alpha_{EM}$, $\alpha_G$ and $m_e/m_p$.

To summarize, Dirac's insight was to question whether some numerical coincidences between very large numbers, that cannot be themselves explained by the theory in which they appear, was a mere coincidence or whether it can reveal the existence of some new underlying physical laws. This paved the way to three main roads of investigation:

* how do we construct theories in which what were thought to be constants are in fact dynamical fields?

* how can we constrain, experimentally or observationally, the spacetime dependencies of the constants that appear in our physical laws?
* how can we explain the values of the fundamental constants and the fine-tuning that seems to exist between their numerical values? This latter issue led to the notion of *naturalness*, that is the not-so-well-defined idea that a model of physics is expected to work without requiring ad-hoc coincidences or “fine-tunings” of its parameters and that instead of such “conspiracies” one needs to find a cause.<fnref n="3" />

While “varying constants” may seem, at first glance, to be a cheap oxymoron, it has to be considered merely as jargon to be understood as “revealing new degrees of freedom and their coupling to the known fields of our theory”, that is a powerful method to test, or eventually reveal, the existence of new fields beyond those of the standard model of particle physics and General Relativity. It has the advantage to be applicable on astrophysical scales and to complement the tests of General Relativity that rely on the large scale structure (see Uzan 2003a, 2007 for an early discussion of this complementarity). As such, the tests on the constancy of the fundamental constants are indeed very important tests of fundamental physics and of the laws of nature we are currently using. Detecting any such variation will indicate the need for new physical degrees of freedom in our theories, that is *new physics*.

Besides, given the rising number of anomalies in cosmology and the growing importance of the dark sector, constraints on the stability of fundamental constants have an important role to play in particular in their possibility to reveal a new long range composition dependent force and the need for an extension of General Relativity. This was discussed and highlighted for the first time in Uzan (2003a); see also Uzan (2007, 2010c). This point was recently raised by Peebles (2022) who argued for “a well-supported Fine-Structure Survey”, which is indeed what the *constantology* (Uzan 2003b; Uzan and Leclercq 2008) and *α-cosmography* (Martins et al. 2022a) are supposed to be. We hope that the expansion of this field and the connection with fundamental physics and cosmology and the power of constants to reveal scalar fields will be fully recognized, as well as their power to bridge local, astrophysical and cosmological scales.

In order to use fundamental constants to set constraints on theories beyond the standard model, one needs to go through a series of complementary steps, which are all very technical, namely

1. it is necessary to understand and to model the physical systems used to set the constraints. In particular one needs to relate the effective parameters that can be

<footnote n="3">³ The concept has been widely used in particle physics in the context of renormalisation. According to Hooft (1980), 'it is unlikely that the microscopic equations contain various free parameters that are carefully adjusted by Nature to give cancelling effects such that the macroscopic systems have some special properties. This is a philosophy which we would like to apply to the unified gauge theories: the effective interactions at [...] a low energy scale $\mu_1$ should follow from the properties at a much higher energy scale $\mu_2$ without the requirement that various different parameters at the energy scale $\mu_2$ match with an accuracy of the order of $\mu_1/\mu_2$. That would be unnatural." or, according to Gell-Mann (1983) "The situation of having numerous arbitrary dimensional parameters is even more humiliating when some of them are very small. [...] First of all, we'd like to interpret small or nearly symmetric quantities as coming from a slightly broken symmetry; otherwise they don't make sense to us. [...] When we can't avoid dialing some renormalized quantity to a small value [...] that situation has recently been described as a problem of 'naturalness'." We refer to Nelson (1985), Giudice (2008), Craig (2023) for further discussions on this concept, as well as Sect. 9.</footnote>
observationally constrained to a set of fundamental constants. One also needs to list and understand all systematics a bad modelization of which could mimic the variation of a constant;

2. it is necessary to relate and compare different constraints that are obtained at different spacetime positions. This requires a spacetime dynamics and thus to specify a model as well as a cosmology. It follows that these constraints combining several systems at different redshifts are usually more model-dependent;

3. it is necessary to relate the variations of different fundamental constants in order to discuss degeneracies or to highlight amplifications.

Therefore, we shall start in Sect. 2.1 by recalling the links between the constants of physics and the theories in which they appear, as well as with metrology. This will

<table><caption>**Table 1** List of the 22 fundamental constants of our standard model</caption><thead><tr><th>Constant</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>Speed of light</td><td>c</td><td>299 792 458 m s<sup>-1</sup></td></tr><tr><td>Planck constant</td><td>h</td><td>6.626 070 15 &times; 10<sup>-34</sup> J s</td></tr><tr><td>Newton constant</td><td>G</td><td>6.674 30(15) &times; 10<sup>-11</sup> m<sup>2</sup> kg<sup>-1</sup> s<sup>-2</sup></td></tr><tr><td>Weak coupling constant (at m<sub>Z</sub>)</td><td>g<sub>2</sub>(m<sub>Z</sub>)</td><td>0.6520 &plusmn; 0.0001</td></tr><tr><td>Strong coupling constant (at m<sub>Z</sub>)</td><td>g<sub>3</sub>(m<sub>Z</sub>)</td><td>1.221 &plusmn; 0.022</td></tr><tr><td>Weinberg angle</td><td>sin<sup>2</sup> &theta;<sub>W</sub> (91.2 GeV)<sub>&overline{MS}</sub></td><td>0.231 21 &plusmn; 0.00004</td></tr><tr><td>Electron Yukawa coupling</td><td>h<sub>e</sub></td><td>2.94 &times; 10<sup>-6</sup></td></tr><tr><td>Muon Yukawa coupling</td><td>h<sub>&mu;</sub></td><td>0.000607</td></tr><tr><td>Tauon Yukawa coupling</td><td>h<sub>&tau;</sub></td><td>0.0102156</td></tr><tr><td>Up Yukawa coupling</td><td>h<sub>u</sub></td><td>0.000016 &plusmn; 0.000007</td></tr><tr><td>Down Yukawa coupling</td><td>h<sub>d</sub></td><td>0.00003 &plusmn; 0.00002</td></tr><tr><td>Charm Yukawa coupling</td><td>h<sub>c</sub></td><td>0.0072 &plusmn; 0.0006</td></tr><tr><td>Strange Yukawa coupling</td><td>h<sub>s</sub></td><td>0.0006 &plusmn; 0.0002</td></tr><tr><td>Top Yukawa coupling</td><td>h<sub>t</sub></td><td>1.002 &plusmn; 0.029</td></tr><tr><td>Bottom Yukawa coupling</td><td>h<sub>b</sub></td><td>0.026 &plusmn; 0.003</td></tr><tr><td>Quark CKM matrix angle</td><td>sin &theta;<sub>12</sub></td><td>0.22650 &plusmn; 0.00048</td></tr><tr><td></td><td>sin &theta;<sub>23</sub></td><td>0.04053<sup>+0.00083</sup><sub>-0.00061</sub></td></tr><tr><td></td><td>sin &theta;<sub>13</sub></td><td>0.00361<sup>+0.00011</sup><sub>-0.00009</sub></td></tr><tr><td>Quark CKM matrix phase</td><td>&delta;<sub>CKM</sub></td><td>1.196<sup>+0.045</sup><sub>-0.043</sub></td></tr><tr><td>Higgs potential quadratic coefficient</td><td>&mu;&#773;</td><td>&asymp; 88.4 GeV</td></tr><tr><td>Higgs potential quartic coefficient</td><td>&lambda;</td><td>&asymp; 0.13</td></tr><tr><td>QCD vacuum phase</td><td>&theta;<sub>QCD</sub></td><td>&lt; 10<sup>-10</sup></td></tr></tbody></table>

See Tiesinga et al. (2021) for further details on the measurements. It is important to stress that, since the detection of the Higgs boson, all free parameters of the standard model have been measured, which makes it a fully predictive theory. Note that this list does not include the charge of the electron $e = 1.602176634 \times 10^{-19}$ C that is related to $g_2$ and $\theta_W$ thanks to Eq. (1)
<table><caption>**Table 2** List of some related constants that appear in our discussions</caption><thead><tr><th>Constant</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>Electromagnetic coupling constant</td><td>g<sub>EM</sub> = e = g<sub>2</sub> sin θ<sub>W</sub></td><td>0.313429 ± 0.000022</td></tr><tr><td>Higgs mass</td><td>m<sub>H</sub></td><td>(125.20 ± 0.11) GeV</td></tr><tr><td>Higgs vev</td><td>v</td><td>(246.7 ± 0.2) GeV</td></tr><tr><td>Fermi constant</td><td>G<sub>F</sub> = 1/√2ν<sup>2</sup></td><td>1.166 378 8(6) × 10<sup>-5</sup> GeV<sup>-2</sup></td></tr><tr><td>Mass of the W<sup>±</sup></td><td>m<sub>W</sub></td><td>80.369 2(133) GeV</td></tr><tr><td>Mass of the Z<sup>0</sup></td><td>m<sub>Z</sub></td><td>91.188 0(20) GeV</td></tr><tr><td>Fine structure constant</td><td>α<sub>EM</sub></td><td>1/137.035 999 084(21)</td></tr><tr><td>Fine structure constant at m<sub>Z</sub></td><td>α<sub>EM</sub>(m<sub>Z</sub>)</td><td>1/(1127.930 ± 0.008)</td></tr><tr><td>Weak structure constant at m<sub>Z</sub></td><td>α<sub>W</sub>(m<sub>Z</sub>)</td><td>0.03383 ± 0.00001</td></tr><tr><td>Strong structure constant at m<sub>Z</sub></td><td>α<sub>S</sub>(m<sub>Z</sub>)</td><td>0.118 0(9)</td></tr><tr><td>Gravitational structure constant</td><td>α<sub>G</sub> = Gm<sub>p</sub><sup>2</sup>/ħc</td><td>~ 5.905 × 10<sup>-39</sup></td></tr><tr><td>Electron mass</td><td>m<sub>e</sub> = h<sub>e</sub>ν/√2</td><td>510.998 950 00(15) keV<br>9.1093837015(28) × 10<sup>-31</sup> kg</td></tr><tr><td>Mu mass</td><td>m<sub>μ</sub> = h<sub>μ</sub>ν/√2</td><td>(105.658 375 5±0.000 002 3) MeV</td></tr><tr><td>Tau mass</td><td>m<sub>τ</sub> = h<sub>τ</sub>ν/√2</td><td>1776.93 ± 0.09 MeV</td></tr><tr><td>Up quark mass</td><td>m<sub>u</sub> = h<sub>u</sub>ν/√2</td><td>(2.16 ± 0.07) MeV</td></tr><tr><td>Down quark mass</td><td>m<sub>d</sub> = h<sub>d</sub>ν/√2</td><td>(4.70 ± 0.07) MeV</td></tr><tr><td>Strange quark mass</td><td>m<sub>s</sub> = h<sub>s</sub>ν/√2</td><td>(93.5 ± 0.8) MeV</td></tr><tr><td>Charm quark mass</td><td>m<sub>c</sub> = h<sub>c</sub>ν/√2</td><td>(1.2730 ± 0.0046) GeV</td></tr><tr><td>Bottom quark mass</td><td>m<sub>b</sub> = h<sub>b</sub>ν/√2</td><td>(4.183 ± 0.007) GeV</td></tr><tr><td>Top quark mass</td><td>m<sub>t</sub> = h<sub>t</sub>ν/√2</td><td>(172.57 ± 0.29) GeV</td></tr><tr><td>QCD energy scale</td><td>Λ<sub>QCD</sub></td><td>(190–240) MeV</td></tr><tr><td>Mass of the proton</td><td>m<sub>p</sub></td><td>938.272 088 16(29) MeV<br>1.672 621 923 69(51) × 10<sup>-27</sup> kg</td></tr><tr><td>Mass of the neutron</td><td>m<sub>n</sub></td><td>939.565 420 52(54) MeV</td></tr><tr><td>proton-neutron mass difference</td><td>Q<sub>np</sub></td><td>(1.293 332 4 ± 0.000 000 5) MeV</td></tr><tr><td>Proton-to-electron mass ratio</td><td>μ = m<sub>p</sub>/m<sub>e</sub></td><td>1836.152 673 43(11)</td></tr><tr><td>Electron-to-proton mass ratio</td><td>μ̄ = m<sub>e</sub>/m<sub>p</sub></td><td>1/1836.152 673 43(11)</td></tr><tr><td>d-u quark mean mass</td><td>m<sub>q</sub> = (m<sub>u</sub> + m<sub>d</sub>)/2</td><td>(2.5–5.0) MeV</td></tr><tr><td>d-u quark mass difference</td><td>δm<sub>q</sub> = m<sub>d</sub> – m<sub>u</sub></td><td>(0.2–4.5) MeV</td></tr><tr><td>Proton gyromagnetic factor</td><td>g<sub>p</sub></td><td>5.586</td></tr><tr><td>Neutron gyromagnetic factor</td><td>g<sub>n</sub></td><td>–3.826</td></tr><tr><td>Rydberg constant</td><td>R<sub>∞</sub></td><td>10 973 731.568 527(73) m<sup>-1</sup></td></tr><tr><td>Photon mass</td><td>m<sub>γ</sub></td><td>&lt; 10<sup>-18</sup> eV</td></tr><tr><td>Photon charge</td><td>q<sub>γ</sub></td><td>&lt; 10<sup>-35</sup> e</td></tr><tr><td>Graviton mass</td><td>m<sub>g</sub></td><td>&lt; 1.76 × 10<sup>-23</sup> eV</td></tr></tbody></table>

See Navas et al. (2024)
offer a clear *definition of fundamental constants* and of their splitting as either *fundamental units* or *fundamental parameters*—see Sect. 2.1.3 for detailed definitions—, each playing a deep but different role in the laws of physics. From a theoretical point of view, the constancy of the fundamental constants is deeply linked with the equivalence principle and General Relativity. In Sect. 2.2 we recall this relation and in particular the link with the universality of free fall and then highlight the connection with cosmology and the dark sector in Sect. 2.3. We then consider two theoretical topics. First, Sect. 3 describes the *dynamical aspects*, that is the way to make constants dynamical, hence introducing gravity theories beyond General Relativity. Then, Sect. 4 addresses the *structural issue* of the connection between the variations of several constants as the consequence of unification. It must also describe the computation of the effective parameters required to interpret experiments and observations in terms of fundamental constants, focusing on nuclear and atomic physics. Then, we describe and summarize the various constraints that exist on the variation of non-gravitational constants and of the gravitational constant respectively in Sects. 5 and 6. Section 7 describes the most popular models used to analyze the different constraints in a consistent way. We finish by a discussion on their spatial variations in Sect. 8 and then on the possibility to understand their numerical values in Sect. 9.

Various reviews have been written on this topic over the past years. I will refer to the first review by Uzan (2003b) as FVC03 and mention the following later reviews by Uzan (2004), Barrow (2005b), Bronnikov and Kononogov (2006), Karshenboim (2005a, 2006), García-Berro et al. (2007), Olive (2009b), Damour (2009), Uzan (2009, 2010b, 2015), Berengut and Flambaum (2010), Martins (2017) and I refer to Tiesinga et al. (2021) for the numerical values of the constants adopted in this review.

The present review builds on its previous version (Uzan 2011) of which it is an update. First, it contains a detailed description of the role of the fundamental constant in the new SI unit system. While it keeps mostly the same organisation and notations, it updates all the constraints of the former version and includes new techniques that have been proposed since then. In particular, the debate on the results of quasar absorption spectra have witnessed a series of dedicated works and a deeper understanding. We shall mention huge progresses on the constraints from atomic clocks and their possible applications in space and the improvements on the constraints of the violation of the universality of free fall. On the cosmological side, the cosmic microwave background and primordial nucleosynthesis predictions and observations have been improved as well as the constraints on spatial variations. New methods include molecular spectroscopy, the Sunyaev–Zel'dovich effect, and the use of gravitational waves. On the theory sides, many works have tried to clarify the relations between constants.

<figure><img src="image_5.png" /></figure>
# 2 Constants and fundamental physics

## 2.1 Toward a definitions of fundamental constants

Any physical theories introduce various mathematical structures to describe the phenomena of nature. They involve various fields, symmetries and constants. These structures are postulated in order to construct a mathematically-consistent description of the known physical phenomena in the most unified and simple way. As such they reflect empirical choices guided by some principles and intuition. Hence, they need to be well-defined and confirmed by experiment.

We define the fundamental constants of a physical theory as the minimal set of *any parameters that cannot be explained by this theory*.

Indeed, we are often dealing with other constants that in principle can be expressed in terms of these fundamental constants. The existence of these two sets of constants is important and arises from two different considerations. From a theoretical point of view we would like to extract the minimal set of fundamental constants, but often these constants are not directly measurable. From a more practical point of view, we need to measure constants, or combinations of them, which allow us to reach the highest accuracy.

Therefore, these fundamental constants are *contingent quantities* that can only be measured. Such parameters have to be assumed constant in this theoretical framework for two reasons:

* from a *theoretical point of view*, the postulated theoretical framework does not— and cannot—provide any way to compute these parameters, i.e., it does not have any equation of evolution for them since otherwise it would be considered as a dynamical field,

* from an *experimental point of view*, these parameters can only be measured. If the theories in which they appear have been validated experimentally, it means that, at the precisions of these experiments, these parameters have indeed been checked to be constant, as required by the necessity of the reproducibility of the experimental results.

This means that testing for the constancy of these parameters is a test of the theories in which they appear and allows us to extend our knowledge of their domain of validity. This also resonates with the definition by Weinberg (1983b) who stated that they cannot be calculated in terms of other constants "...not just because the calculation is too complicated (as for the viscosity of water) but because we do not know of anything more fundamental". We could say that the more constants a theory involves the higher its contingency, which may link with the discussion we shall have on fine tuning in Sect. 9.

This has a series of implications. *First*, the list of fundamental constants to consider depends on our theories of physics and, thus, on time. Indeed, when introducing new, more unified or more fundamental, theories the number of constants may change so that this list reflects both our knowledge of physics and, more important, our ignorance. *Second*, it also implies that some of these
fundamental constants can become dynamical quantities in a more general
theoretical framework so that the tests of the constancy of the fundamental
constants are tests of fundamental physics, which can reveal that what was thought
to be a fundamental constant is actually a field whose dynamics cannot be neglected.
If such fundamental constants are actually dynamical fields it also means that the
equations we are using are only approximations of other and more fundamental
equations, in a kind of adiabatic limit, and that an equation for the evolution of this
new field has to be obtained.

The reflections on the nature of the constants and their role in physics are
numerous. We refer to the books by Barrow (2002), Uzan and Lehoucq (2005),
Fritzsch (2009a), Uzan and Leclercq (2008, 2020) and to Weinberg (1983b), Okun
(1996), Bjorken (2001), Duff et al. (2002), Wilczek (2007), Fritzsch (2009b),
Volovik (2009), Langacker (2017) for various discussions of this issue that we
cannot develop at length here. This paragraph summarizes some of the properties of
the fundamental constants that need to be kept in mind.

2.1.1 Characterizing the fundamental constants

Physical constants seem to play a central role in our physical theories since, in
particular, they determine the magnitudes of the physical processes. Let us sketch
briefly some of their main properties.

*How many fundamental constants shall we consider?*

The set of constants, which are conventionally considered as fundamental
(Flowers and Petley 2001), consists of the electron charge *e*, the electron mass *m*<sub>e</sub>,
the proton mass *m*<sub>p</sub>, the reduced Planck constant *ħ*, the velocity of light in vacuum *c*,
the Avogadro constant *N*<sub>A</sub>, the Boltzmann constant *k*<sub>B</sub>, the Newton constant *G*, the
permeability and permittivity of vacuum, *ε*<sub>0</sub> and *μ*<sub>0</sub>. The latter had a fixed exact
value in the SI system of unit before 2018 (*μ*<sub>0</sub> = 4π × 10<sup>-7</sup> H m<sup>-1</sup>), which was
implicit in the definition of the Ampere; *ε*<sub>0</sub> is then fixed by the relation *ε*<sub>0</sub>*μ*<sub>0</sub> = c<sup>-2</sup>
(See Sect. 2.2 and Appendix A for a discussion on the *new SI*).

However, it is clear that this cannot correspond to the list of the fundamental
constants, as defined earlier as the free parameters of the theoretical framework at
hand. To identify such a list we must specify this framework since it is clear that one
cannot discuss fundamental constants without first defining the theoretical
framework assumed to give the most fundamental description of nature. As
discussed in the previous section, fundamental constants are the unknown constant
parameters of this fundamental theory so that this list has evolved with our
understanding of physics (see Uzan and Lehoucq 2005 for a documented history of
the evolution of the fundamental constants).

Today, gravitation is described by General Relativity, and the three other
interactions and the matter fields are described by the standard model of particle
physics. It follows that one has to consider 22 unknown constants (i.e., 19 unknown
dimensionless parameters): the Newton constant *G*, 6 Yukawa couplings for the
quarks (*h*<sub>u</sub>, *h*<sub>d</sub>, *h*<sub>c</sub>, *h*<sub>s</sub>, *h*<sub>t</sub>, *h*<sub>b</sub>) and 3 for the leptons (*h*<sub>e</sub>, *h*<sub>μ</sub>, *h*<sub>τ</sub>), 2 parameters of the
Higgs field potential (μ̂, λ̂), 4 parameters for the Cabibbo–Kobayashi–Maskawa
matrix (3 angles $\theta_{ij}$ and a phase $\delta_{\text{CKM}}$), 3 coupling constants for the gauge groups $SU(3)_c \times SU(2)_L \times U(1)_Y$ ($g_1, g_2, g_3$ or equivalently $g_2, g_3$ and the Weinberg angle $\theta_W$), and a phase for the QCD vacuum ($\theta_{\text{QCD}}$), to which one must add the speed of light $c$ and the Planck constant $h$. See Table 1 for a summary and their numerical values.

Again, this list relies on what we accept as a fundamental theory. Today we have many hints that the standard model of particle physics has to be extended, in particular to include the existence of massive neutrinos. Such an extension comes with at least seven new constants (3 Yukawa couplings and 4 Maki–Nakagawa–Sakata (MNS) parameters, similar to the CKM parameters). On the other hand, the number of constants can decrease if some unifications between various interaction exist (see Sect. 4.1 for more details) since the various coupling constants may be related to a unique coupling constant $\alpha_U$ and an energy scale of unification $m_u$ through

$$ \alpha_i^{-1}(E) = \alpha_U^{-1} + \frac{b_i}{2\pi} \ln \frac{m_u}{E}, $$

where the $b_i$ are numbers, that depend on the explicit model of unification. Note that this would also imply that the variations, if any, of various constants shall be correlated.

### Relation to other usual constants

The free parameters of the standard model are related to various constants that appear in this review (see Table 2) and actually to all the other constants of physics but also of other fields.

First, the quartic and quadratic coefficients of the Higgs field potential are related to the Higgs mass and vev,

$$ m_H = \sqrt{-\mu^2/2} \quad \text{and} \quad v = \sqrt{-\mu^2/\lambda}. $$

The latter is related to the Fermi constant is derived from the $\mu$-lifetime as (see Sect. 10 of Navas et al. 2024)

$$ G_F = (v^2\sqrt{2})^{-1} = 1.166\,378\,8(6) \times 10^{-5} \text{ GeV}^{-2}, $$

which imposes that $v = (246.7 \pm 0.2) \text{ GeV}$. Since its discovery, the Higgs mass has been measured with increasing precision to reach, thanks to the ATLAS experiment at CERN, the unprecedented precision of 0.09%, $m_H = (125.20 \pm 0.11) \text{ GeV}$ (ATLAS 2023).

Then, the masses of the quarks and leptons are related to their Yukawa couplings and the Higgs vev by $m = h\nu/\sqrt{2}$. Note however that, due to the long-distance confining property of the strong QCD interaction, free quarks have never been observed. All quarks but the top hadronize, i.e., become part of a meson or baryon, on a timescale of order $1/\Lambda_{\text{QCD}}$, the top decaying before it can hadronize. This implies that the question of what is meant by the quark mass is a complex one, which requires one to adopt a specific prescription; see Section 9 of Navas et al.
(2024). Perturbatively, one can use the prescription of the pole mass $m_q$ of the divergence of the quark propagator as a definition which, while close to the physical picture of mass, however suffers from ambiguities of order $\Lambda_{QCD}$ when relating it to observable quantities. This anticipates the difficult discussion on the relations between the free parameters of the standard model and the observables and nuclear properties.

The values of the gauge couplings depend on energy via the renormalization group so that they are given at a chosen energy scale, here the mass of the Z-boson, $m_Z$. $g_1$ and $g_2$ are related by the Weinberg angle as

$$g_1 = g_2 \tan \theta_W.$$

The electromagnetic coupling constant is not $g_1$ since $SU(2)_L \times U(1)_Y$ is broken to $U(1)_{\text{elec}}$ so that

$$g_{\text{EM}}(m_Z) = e = g_2(m_Z) \sin \theta_W. \qquad (1)$$

Defining the fine-structure constant as $\alpha_{\text{EM}} = g_{\text{EM}}^2/\hbar c$, the (usual) zero energy electromagnetic fine structure constant can be extracted from the anomalous magnetic moment of the electron, $a_e = (1159652180.59 \pm 0.13) \times 10^{-12}$ applying QED corrections up to five loops allows one to extract to get $\alpha_{\text{EM}} = 1/137.035999166(15)$ (Fan et al. 2023). The combination of the measurements of the Rydberg constant and atomic masses with interferometry of atomic recoil kinematics of $^{87}$Rb and $^{133}$Cs leads to $\alpha_{\text{EM}} = 1/137.035999206(11)$ (Morel et al. 2020) and $\alpha_{\text{EM}} = 1/137.035999046(27)$ (Parker et al. 2018) respectively. It is to be noted that these two latter values differ by $5.5\sigma!$ Combining all the latest data yields the world average $\alpha_{\text{EM}} = 1/137.035999178(8)$, which differs from the CODATA recommended value at $Q^2 = 0$ $\alpha_{\text{EM}} = 1/137.035999084(21)$, which is the value we report in Table 2, even though it does not use the latest experimental inputs. We refer to Section 10 of Navas et al. (2024) for a discussion of these values and their differences.

From electroweak renormalization schemes, one can characterize the running of $\alpha_{\text{EM}}$ on the energy scale of the process with, given the discussion of the previous paragraph, $\alpha_{\text{EM}} \simeq 1/137.036$ at very low energy, i.e., close to the Thomson limit. The running has been observed in small and large angle Bhabha scattering (Abbiendi et al. 2006; Achard et al. 2005). At energies larger than a few MeV, the hadronic contribution to vacuum polarization introduces a theoretical uncertainty in the determination of $\alpha_{\text{EM}}$. Using the modified minimal subtraction (MS) scheme and $\alpha_s(m_Z) = 0.1187 \pm 0.0017$, one gets

$$\alpha_{\text{EM}}(m_Z) = 1/(1127.930 \pm 0.008),$$

that corresponds to the quark contribution (without the top) to the conventional (on-shell) QED coupling

$$\alpha_{\text{EM}}(m_Z) = \frac{\alpha_{\text{EM}}}{1 - \Delta\alpha_{\text{EM}}(m_Z)}$$

with $\Delta\alpha_{\text{EM}}(m_Z) = 0.02783 \pm 0.00006$. We refer to Table 10.1 of Navas et al.
(2024) for the various evaluations of $\Delta z_{EM}(m_Z)$. This already illustrates the model-dependence that will enter the relation between the QCD parameters and the low energy quantities, even when one sticks to the standard model.

We define the QCD energy scale, $\Lambda_{QCD}$, as the energy at which the strong coupling constant diverges. Note that it implies that $\Lambda_{QCD}$ also depends on the Higgs and fermion masses through threshold effects.

More familiar constants, such as the masses of the proton and the neutron are, as we shall discuss in more details below (see Sect. 4.2), more difficult to relate to the fundamental parameters because they depend not only on the masses of the quarks but also on the electromagnetic and strong binding energies.

*Are some constants more fundamental than others?*

As pointed-out by Lévy-Leblond (1979), all constants of physics do not play the same role, and some have a much deeper role than others. Following his analysis, we can define three classes of fundamental constants,

* class A being the class of the constants characteristic of a particular system,

* class B being the class of constants characteristic of a class of physical phenomena,

* class C being the class of universal constants.

Indeed, the status of a constant can change with time and our understanding of the laws of nature; see Fig. 1. For instance, the velocity of light was initially a class A constant (describing a property of light alone), which then became a class B constant when it was realized within Maxwell theory that light electromagnetic phenomena so that its velocity was a general property of all electromagnetic waves and as such related to the vacuum permeability and permittivity. To finish, it ended as a type C constant: it enters special relativity and is related to the notion of causality, whatever the physical phenomena at stakes. It has even become a much more fundamental constant: since 1983 it enters in the definition of the meter (Petley 1983); see Uzan and Lehoucq (2005) for a more detailed discussion. We thus define

* class D being the class of the dimensionful constants that enter the definition of units.

This classification has to be contrasted with the proposition of Wilczek (2007) to distinguish the standard model free parameters as the gauge and gravitational couplings (which are associated to internal and spacetime curvatures) and the other parameters entering the accommodation of inertia in the Higgs sector.

*Relation with physical laws*

As explained in the previous paragraph, Lévy-Leblond (1979) ranked the constants in terms of their universality and he proposed that only three constants be considered to be of class C, namely $G$, $\hbar$ and $c$. He pointed out two important roles of these constants in the laws of physics.
<figure><img src="image_6.png" /><figcaption>**Fig. 1** (Over-simplified) timeline of the evolution of the status of the three fundamental constants *c*, *G* and *h* that are used to define the Planck units. Only the gravitational constant does not enter the definition of the International System of units. The changes of status of these three constants is related to unification between different physical theories and the emergence of new concepts such as spacetime and wave function that unified previous non-commensurable concepts (space and times, wave and particle) thanks to one constant. From Uzan and Lehoucq (2005)</figcaption></figure>

First, they act as *concept synthesizer* during the process of our understanding of the laws of nature: contradictions between existing theories have often been resolved by introducing new concepts that are more general or more synthetic than older ones. Constants build bridges between quantities that were thought to be incommensurable and thus allow for new concepts to emerge. For example the speed of light *c* underpins the synthesis of space and time while the Planck constant *h* allowed one to relate the concept of energy and frequency and the gravitational constant *G* creates a link between matter and spacetime.

Second, it follows that these constants are related to the *domains of validity* of these theories. For instance, as soon as velocity approaches *c*, relativistic effects become important and cannot be neglected. On the other hand, for speed much below *c*, Galilean kinematics is sufficient. Similarly the Planck constant signals when quantum theory has to be used since if the action *S* of a system greatly exceeds the value *ħ*, classical mechanics will be appropriate to describe this system.

Third, these two constants are also *related to two universal principles*, i.e., that are independent on the exact form of the laws of nature: *c* is related to causality and *ħ* to the quantum uncertainty principle. These two principles lie above the specific formulation of a precise field theory. No such principle associated to *G* has been exhibited even though, within General Relativity, one can identify it to a measure of the universal elasticity of spacetime to any matter form. Figure 1 summarizes the
(simplified) evolution of the status of these 3 constants in regards of the evolution of theoretical physics theories.

Hence, while the place of *c* (related to the notion of causality) and *ħ* (related to the quantum) in this list are well argued, the place of *G* remains debated since it is thought that it will have to be replaced by some energy scale of a quantum theory of gravity.

### Evolutions

There are many ways the list of constants can change with our understanding of physics.

First, *new constants may appear* when new systems or new physical laws are discovered; this is, for instance, the case of the charge of the electron or more recently the gauge couplings of the nuclear interactions.

A constant can also *move from one class to a more universal class*. An example is that of the electric charge, initially of class A (characteristic of the electron), which then became class B when it was understood that it characterizes the strength of the electromagnetic interaction.

A constant can also *disappear* from the list, because it is either replaced by more fundamental constants (e.g., the Earth acceleration due to gravity and the proportionality constant entering Kepler law both disappeared because they were “explained” in terms of the Newton constant and the mass of the Earth or the Sun) or because it can happen that a better understanding of physics teaches us that two hitherto distinct quantities have to be considered as a single phenomenon (e.g., the understanding by Joule that heat and work were two forms of energy led to the fact that the Joule constant, expressing the proportionality between work and heat, lost any physical meaning and became a simple conversion factor between units used in the measurement of heat (calories) and work (Joule). Nowadays the calorie has fallen in disuse.

Indeed demonstrating that a constant is varying will have direct implications on our list of constants since it would mean that it has to be replaced by a field, the dynamics of which will introduce new constant parameters that would need (1) to be measurable and measured for the theory to be predictive and (2) related to the former “constants” it replaces is some limiting regime. This also means that the definitions of most dictionaries shall be modified since a constant cannot be defined by the property that “it does not change with time”, that is by an empirical statement that shall be tested and as a consequence cannot be elevated to a category.

### Conclusion

The evolution of the number, status of the constants teaches us a lot about the evolution of the ideas and theories in physics since they reflect the birth of new concepts, their evolution and unification with other ones. Fundamental constants are both a door on new physics and a pedagogical way to illustrate the evolution of ideas in physics. The discussion of this paragraph highlights that they cannot be defined without a set of reference theories of which they are the free parameters that, indeed, cannot be determined by these theories and hence must be measurable and
measured for it to be fully predictive. This discussion also illustrates that an inadequate definition may lead to the conclusion that the question “Are the constants varying?” is a dull oxymoron while it is actually a deep question on the domain of validity of the theories that are used to describe nature. Constants stand at the crossroad between theoretical physics and history of science, offering a guideline to read the status and evolution of our physical theories. As we shall see, being among the most fundamental concepts of a theory, they have also modified the way we think physical units.

## 2.1.2 Constants and metrology

Since we cannot compute them in the theoretical framework in which they appear, it is a crucial property of the fundamental constants (but in fact of all the constants) that their value can be measured. The relation between constants and metrology is a huge subject to which we just draw the attention on some selected aspects, specially in view of the redefinition of the International System of units that was adopted on November 16th 2018, placing constants at the heart of metrology. For more discussions, see Birge (1929) and Karshenboim (2005a, 2005c).

The introduction of constants in physical laws is closely related to the existence of systems of units. For instance, Newton’s law states that the gravitational force between two masses is proportional to each mass and inversely proportional to the square of their separation. To transform the proportionality to an equality one requires the use of a quantity with dimension of $m^3 kg^{-1} s^{-2}$ independent of the separation between the two bodies, of their masses, of their composition (equivalence principle) and on the position (local position invariance). With an other system of units the numerical value of this constant could have simply been anything. Indeed, the numerical value of any constant crucially depends on the definition of the system of units.

### The need for constants

For centuries, physical laws were expressed only with dimensional numbers. For instant, considering falling bodies from different heights, Galileo established a scaling relation between the height $h$ and duration $t$ of the fall,

$$ \frac{h_1}{h_2} = \left(\frac{t_1}{t_2}\right)^2, $$

which implies no constant at all. This is a pure comparison between the evolution of two physical systems. Hence no need for units and constants. The drawback is that the comparison has to be performed locally. Hence, one can think of the introduction of a units system as the invention of some abstract reference physical system to which one can compare local experimental measurements. It follows that we would rewrite the previous law of falling bodies as
$$ \frac{h}{1 \text{ m}} \propto \left( \frac{t}{1 \text{ s}} \right)^2, $$

which is only a proportionality because in general a body does not fall from 1 m in 1 s, and then as

$$ h = \frac{1}{2} g t^2 $$

with g a constant of proportionality which shall be independent of h, t and the mass, with dimension of an acceleration. It is clear that the units used to report the value of g and to measure h and t have to be consistent so that the numerical value of g depends in the choice of the abstract reference physical system (i.e., the units system) to which the experiment is compared.

Hence, the introduction of constants and the one of units are related and they allow for a globalisation of the comparison of experiments performed at different epoch and different laboratories: one shifted from a “local equality” to a “global proportionality” using arbitrary units and then a “global equality” requiring a proportionality constant. Indeed the system of units is completely arbitrary, and thus the numerical values of the dimensionful constants. We shall now describe in more details the various connections between constants and units.

## *Measuring constants*

The determination of the laboratory value of constants relies mainly on the measurements of lengths, frequencies, times, ... (see Petley 1985 for a treatise on the measurement of constants, and Flowers and Petley (2001) for a review). Hence, any question on the variation of constants is linked to the definition of the system of units and to the theory of measurement. The behavior of atomic matter is determined by the value of many constants. As a consequence, if, e.g., the fine-structure constant is spacetime dependent, the comparison between several devices such as clocks and rulers will also be spacetime dependent. This dependence will differ from one clock to another so that metrology becomes both *device and spacetime dependent*, a property that will actually be used to construct tests of the constancy of the constants since it questions the reproductibility of experiments.

Indeed a measurement is always a comparison between two physical systems of the same dimensions. This is thus a relative measurement, which gives as result a pure number. This trivial statement is oversimplifying since in order to compare two similar quantities measured separately, one actually needs to perform a number of comparisons. In order to reduce their number—and in particular to avoid creating every time a chain of comparisons—, a certain set of them has been included in the definitions of units. Each unit can then be seen as an abstract physical system, which has to be realized effectively in the laboratory, and to which another physical system is compared. A measurement in terms of these units is usually called an absolute measurement. Most fundamental constants are related to microscopic physics and their numerical values can be obtained either from a pure microscopic comparison (as is, e.g., the case for $m_e/m_p$) or from a comparison between microscopic and macroscopic values (for instance to deduce the value of the mass of the electron in
kilogram). This shows that the choice of the units has an impact on the accuracy of the measurement since the pure microscopic comparisons are in general more accurate than those involving macroscopic physics. This implies that only the variation of dimensionless constants can be measured and in case such a variation is detected, it is impossible to determine, which dimensional constant is varying (Ellis and Uzan 2005).

It is also important to stress that in order to deduce the value of constants from an experiment, one usually needs to use theories and models. An example (Karshenboim 2005a) is provided by the Rydberg constant. It can easily be expressed in terms of some fundamental constants as $R_\infty = \alpha_{EM}^2 m_e c / 2h$. It can be measured from, e.g., the triplet $1s-2s$ transition in hydrogen, the frequency of which is related to the Rydberg constant and other constants by assuming QED so that the accuracy of $R_\infty$ is much lower than that of the measurement of the transition. This could be solved by defining $R_\infty$ as $4v_H(1s-2s)/3c$ but then the relation with more fundamental constants would be much more complicated and actually not exactly known. This illustrates the relation between a practical and a fundamental approach and the limitation arising from the fact that we often cannot both exactly calculate and directly measure some quantity. Note also that some theoretical properties are plugged in the determination of the constants.

The laboratory determination of the fine structure constant $\alpha_{EM}$ arises from the comparison of theory and experiment for anomaly $a_e = (|g_e| - 2)/2$ of the electron magnetic-moment $\mu_e = g_e e s / 2m_e$, $s$ being its spin. It inherits contributions from the pure QED, predominantly electroweak, and predominantly hadronic sectors (Mohr et al. 2016). The most accurate measurement of $a_e$ was obtained from a single electron that was suspended for months at a time in a cylindrical Penning trap (Hanneke et al. 2008) leading to $\alpha_{EM}^{-1} = 137.035999084(51)$. It can also be obtained from the expression $\alpha_{EM}^2 = 2R_\infty(M/m_e)(h/M)/c$ (Wicht et al. 2002). Using Bloch oscillations of $^{87}$Rb atoms in an optical lattice, Bouchendira et al. (2013) determined $h/M_{Rb}$ and deduced $\alpha_{EM}^{-1} = 137.035999037(91)$. The agreement of these determinations of the fine structure constant is a strong validation of QED and of the standard model.

As a conclusion, let us recall that (1) in general, the values of the constants are not determined by a direct measurement but by a chain involving both theoretical and experimental steps, (2) they depend on our theoretical understanding, (3) the determination of a self-consistent set of values of the fundamental constants results from an adjustment to achieve the best match between theory and a defined set of experiments (which is important because we actually know that the theories are only good approximation and have a domain of validity) (4) that the system of units plays a crucial role in the measurement chain, since for instance in atomic units, the mass of the electron could have been obtained directly from a mass ratio measurement (even more precise!) and (5) fortunately the test of the variability of the constants does not require a priori to have a high-precision value of the considered constants.
## System of units

Thus, one needs to define a coherent system of units. This has a long, complex and fascinating history that was driven by simplicity and universality but also by increasing stability and accuracy (Barrow 2002; Uzan and Leclercq 2008; LNE 2019; Boudjaaba et al. 2019; Tzalenchuk et al. 2022).

The international system of units (SI) has evolved to slowly replace physical artefacts by fundamental constants. This important story is summarized in Appendix A. First, Sect. A.1 highlights the process in which it was understood that one shall look for the most fundamental physical objects to define units, opening a debate on the number of basic units; see Sect. A.2. It follows that the definitions were driven by the will to use more stable and more fundamental quantities so that they closely follow the progress of physics and eventually fundamental constants. While the role of fundamental constants was clear in the construction of natural units described in Sect. A.3, it increased to lead to the definition of the new (2018) SI system that is described in Sect. A.4.

The important point is that a, in principle arbitrary, choice of at least 3 units—see however the discussion of Sect. A.5—needs to be made in order to allow ease of comparison of global experimental results. But that choice has no impact on the more important question at the heart of this review that is on whether dimensionless constants de keep the same value over space and time.

### 2.1.3 Definitions: fundamental parameters and fundamental units

Once a set of three independent constants has been chosen as natural units, then all other constants are dimensionless quantities. The values of these combinations of constants does not depend on the way they are measured (Cook 1957; Duff 2002; Rich 2003; Ellis and Uzan 2005), on the definition of the units etc. It follows that any variation of constants that will leave these numbers unaffected is actually just a redefinition of units.

Hence, we shall split the constants into two sets respectively defined as (see Fig. 2)

* *fundamental units*, the arbitrary set of dimensionful constants used to define a system of units. Generally, there are 3 fundamental units.

* *fundamental parameters*, the set of dimensionless constants that remains. Changing their value will change the physics; the actual list depends on the laws chosen to describe nature and they can only be measured.

These latter dimensionless numbers represent, e.g., the mass ratio, relative magnitude of strength etc..... Changing their values will indeed have an impact on the intensity of various physical phenomena, so that they encode some properties of our world. They have specific values (e.g., $\alpha_{EM} \sim 1/137$, $m_p/m_e \sim 1836$, etc.) that
<figure><img src="image_7.png" /><figcaption>Fig. 2 Once a set of theories is assumed to describe nature, one can identify its free parameters to get a list of fundamental constants that can then be split in dimensionful fundamental units, freely chosen to define a system of units, that can indeed be mapped to the SI units, and a list of dimensionless fundamental parameters the value of which are independent of any units system. The latter define the physical properties of the world. One can question their being fine-tuned and their constancy. Translated from Uzan and Lehoucq (2005)</figcaption></figure>

we may hope to understand. Are all these numbers completely contingent, or are some (why not all?) of them related by relations arising from some yet unknown and more fundamental theories. In such theories, some of these parameters may actually be dynamical quantities and, thus, vary in space and time. These are our potential varying constants.

In this review several fundamental parameters will be considered, namely the fine structure constant $\alpha_{EM}$ and the gravitational structure constant $\alpha_G$, the electron-to proton mass ratio,<fnref n="4" />

$$ \alpha_{EM} \equiv \frac{e^2}{4\pi\epsilon_0\hbar x}, \quad \alpha_G \equiv \frac{Gm_p^2}{\hbar c}, \quad \bar{\mu} \equiv \frac{m_e}{m_p}, \quad \mu \equiv \frac{m_p}{m_e}, \qquad (2) $$

the elementary masses to QCD scale ratio,

$$ X_q \equiv \frac{m_q}{\Lambda_{\text{QCD}}}, \qquad (3) $$

for the light quark q of mass $m_q = (m_u + m_d)/2$, the quark s and the electron.

<footnote n="4">⁴ The introduction of $\mu$ and $\bar{\mu}$, while redundant, is due to the use of two conventions in the literature. In so doing, we stay close to the published results without having to change signs in the constraints.</footnote>
## 2.2 The constancy of constants as a test of general relativity

The previous paragraphs have emphasize why testing for the consistency of the constants is a test of fundamental physics since it can reveal the need for new physical degrees of freedom in our theory. We now need to stress the relation with other tests of General Relativity and with cosmology.

### 2.2.1 General relativity

The tests of the constancy of fundamental constants take all their importance in the realm of the tests of the equivalence principle (Will 1993; Damour 2012). Einstein General Relativity is based on two independent hypotheses, which can conveniently be described by decomposing the action of the theory as $S = S_{grav} + S_{matter}$. We shall detail both aspects and highlight the connection between variation of the fundamental constants and the existence of a fifth force.

#### *Equivalence principle*

The equivalence principle has strong implication for the functional form of $S_{matter}$. This principle includes three hypotheses:

* the universality of free fall (UFF), that is the fact that the trajectory of any freely falling test-body in an external gravitational field does not depend on its internal structure, mass, and chemical composition. In Newtonian gravity, it leads to the requirement that the weight of a body is proportional to its inertial mass, i.e., that the inertial and gravitational masses are equal;

* the local position invariance (LPI), that is the statement that any non-gravitational experiment is independent when and where it is performed;

* the local Lorentz invariance (LLI). that is the statement that any non-gravitational experiment in a freely falling laboratory is independent of the velocity of its rest frame.

In its weak form (that is for all interactions but gravity), it is satisfied by any metric theory of gravity and General Relativity is conjectured to satisfy it in its strong form (that is for all interactions gravity included). We refer to Will (1993) for a detailed description of these principles. The weak equivalence principle (WEP) can be mathematically implemented by assuming that all matter fields are minimally coupled to a single metric tensor $g_{\mu\nu}$. This metric defines the length and times measured by laboratory clocks and rods so that it can be called the *physical metric*. This implies that the action for any matter field, $\psi$ say, can be written as

$$S_{\text{matter}}(\psi; g_{\mu\nu}). \qquad (4)$$

This universal *metric coupling* ensures in particular the validity of the universality of free-fall. Since locally, in the neighborhood of the worldline, there always exists a change of coordinates so that the metric takes a Minkowskian form at lowest order, the gravitational field can be locally“effaced” (up to tidal effects). If we identify this neighborhood to a small lab, this means that any physical properties
that can be measured in this lab must be independent of where and when the experiments are carried out. This is indeed the assumption of *local position invariance*, which implies that the constants must take the same value independent of the spacetime point where they are measured.

Thus, testing the constancy of fundamental constants is a direct test of this principle and therefore of the metric coupling. Interestingly, the tests we are discussing in this review allow one to extend them much further than Solar system scales and even in the early universe, an important information to check the validity of relativity in cosmology.

### Test-body in free fall

The action of a point-body subjected to no other force than gravity reads

$$S_{\text{matter}} = - \int mc\sqrt{-g_{\mu\nu}(x^\alpha)\nu^\mu\nu^\nu} dt, \quad (5)$$

with $\nu^\mu \equiv dx^\mu/dt$. The equation of motion that one derives from this action is the usual geodesic equation

$$a^\mu \equiv u^\nu \nabla_\nu u^\mu = 0, \quad (6)$$

where $u^\mu = dx^\mu/cd\tau$, $\tau$ being the proper time; $\nabla_\mu$ is the covariant derivative associated with the metric $g_{\mu\nu}$ and $a^\nu$ is the 4-acceleration. Hence, any test particle in free fall follows a geodesic independent of its mass and chemical composition, as anticipated from the property (4) that translates the UFF. Again, this property relies only on the universal minimal coupling in the matter action, independently of the field equations for the spacetime metric.

Any metric theory of gravity enjoys such a matter Lagrangian and the worldline of any test particle shall be a geodesic of the spacetime with metric $g_{\mu\nu}$, as long as there is no other long range force acting on it (see Esposito-Farèse 2011 for a detailed review of motion in alternative theories of gravity). In the weak field limit $g_{00} = -1 - 2U/c^2$ where $U$ is the gravitational potential. Note that $U$ may differ from the Newtonian potential $\Phi_N$. It follows from Eq. (6) that, in the slow velocity limit, the geodesic equation reduces to

$$\dot{\mathbf{v}} = \mathbf{a} = -\nabla U \equiv \mathbf{g}, \quad (7)$$

hence defining the acceleration $\mathbf{g}$. If the metric theory reduces to General Relativity then $U = \Phi_N$ and $g_N = -\nabla\Phi_N$. Recall that the proper time of a clock is related to the coordinate time by $d\tau = \sqrt{-g_{00}}dt$. Thus, if one exchanges electromagnetic signals between two identical clocks in a stationary situation, the apparent difference between their frequencies is

$$\frac{v_1}{v_2} = 1 + \frac{U(2) - U(1)}{c^2}, \quad (8)$$

at lowest order. This is the *universality* of gravitational redshift, showing a structural relation between the redshift and the free-fall motion (7) of massive particles.
## Tests of the metric coupling

The assumption of a metric coupling is actually well tested in the Solar system through 4 types of experiments; see Will (1993, 2006) for a detailed discussion and Peik et al. (2021), Herrmann and Rätzel (2023) for recent developments on quantum tests of relativity.

* First, the local position invariance implies that all non-gravitational constants are spacetime independent, which have been tested to a very high accuracy in many physical systems and for various fundamental constants; this is the subject of the present review.

* Second, the isotropy of space has been tested from the constraint on the possible quadrupolar shift of nuclear energy levels (Chupp et al. 1989; Lamoreaux et al. 1986; Prestage et al. 1985) proving that different matter fields couple to a unique metric tensor at the $10^{-27}$ level.

* Third, the Einstein effect (or gravitational redshift) given by Eq. (8) shall be compatible with the potential $U$ given by the equation of motion (7). This has been tested at the $2 \times 10^{-4}$ level (Vessot and Levine 1979). It is standard to express the redshift $z_{12}$ between two spacetime positions as

$$z_{12} = (1 + \gamma)\Delta_{12}U \qquad (9)$$

where, following Will (1993), $U$ is the potential the gradient of which determines in the Newtonian limit the acceleration, i.e., $\mathbf{a} = -\nabla U$, of a free particle. Testing the consistency of $\gamma$ with 0 is a test of LPI. Indeed in General Relativity $U = \Phi_N$ and $\gamma = 0$; see Eq. (8). In alternative theories, one shall

<figure><img src="image_8.png" /><figcaption>Fig. 3 The universality of free fall has been a red thread in the understanding of gravitation since Galileo. This explains that it has been tested with higher and higher accuracy with different methods, starting from the study of the universality of free fall, comparison of pendulums with different chemical compositions, torsion balances, Lunar laser ranging and space-designed experiments of free fall in Earth orbit. Adapted from Uzan and Lehoucq (2005)</figcaption></figure>
identify $U$ and the redshift independently and how they can be measured. As an example, scalar-tensor theories discussed in Sect. 3.1.1 enjoy $\gamma = 0$. The parameter is constrained at the $10^{-5}$ level (Delva et al. 2018; Herrmann et al. 2018).

* Fourth, the universality of free fall can be tested by comparing the accelerations of two test bodies in an external gravitational field. We shall review its status in the next paragraph.

This set of experiments lead us to conclude that the hypothesis of metric coupling is extremely well-tested in the Solar system.

### Tests of the universality of free fall

The test of the universality as a long history, briefly summarized on Fig. 3. The deviations from UFF are characterized by the Eötvös parameter $\eta_{12}$ defined as

$$ \eta_{12} \equiv 2 \frac{|\mathbf{a}_1 - \mathbf{a}_2|}{|\mathbf{a}_1 + \mathbf{a}_2|}, \qquad (10) $$

that shall be compatible with 0 as long as UFF holds.

Indeed, in the Newtonian limit, the Einstein equations and the equation of free fall reduce to the Newton second law and the expression for the weight with the same mass. Nevertheless, contrary to General Relativity, in which UFF is hardwired, one can introduce two notions of masses, the inertial mass $m_I$ and the gravitational mass$^5 m_G$ so that the free fall in an external gravitational field is dictated by

$$ m_I \mathbf{a} = \mathbf{P}, \qquad \mathbf{P} = m_G g. $$

Clearly if the UFF holds, then $(m_I/m_G)_1 = (m_I/m_G)_2$ so that the inertial and gravitational masses can be chosen equal, from which it follows trivially that

$$ \eta_{12} = 2 \frac{|m_G^{(1)}/m_I^{(1)} - m_G^{(2)}/m_I^{(2)}|}{|m_G^{(1)}/m_I^{(1)} + m_G^{(2)}/m_I^{(2)}|}. $$

It is an empirical fact, established by Galileo (in his experiments on inclined planes rather than the probably apocryphal experiment in the Leaning Tower of Pisa), that in the absence of friction, all objects, no matter what their inertial mass, or the nature of their constituents, or the internal energy or cohesive forces of their constituents, fall in the same way in an external gravitational field (in contrast to, for example, the behavior in an electric field of two individual charges of opposite sign and of their neutral ensemble). This principle was stated in the introduction of its *Principia* by Newton and was then one of the cornerstone of General Relativity.

<footnote>$^5$ We recall that actually one shall distinguish between the *passive gravitational masse* that characterizes the response of a body to the gravitational action of external objects, i.e., defining its weight, that is, the gravitational force $m_G g$ acting on the body and the *active gravitational mass* that characterizes the object which creates a gravitational field, i.e., its 'gravitational charge'. Thanks to Newton's third law of equal action and reaction, one can always choose these two gravitational masses to be equal.</footnote>
Newton manages to test it using pendulums, using the fact that the equation of motion actually becomes

$$\ddot{\theta} + \omega^2\theta = 0 \quad \text{with} \quad \omega = \omega_0\sqrt{\frac{m_G}{m_1}} \quad \text{and} \quad \omega_0 = \sqrt{\frac{g}{L}}$$

so that comparing the oscillations of two different bodies suspended by cables of the same length, one gets $\eta_{12} = 2|\omega_1 - \omega_2|/\omega_0$. This method was invented by Galileo, who estimated that $\eta_{12} < 1\%$. Newton repeated the experiment taking into account air resistance to reach an accuracy of $10^{-3}$. In 1827 Bessel reached an accuracy of $10^{-5}$. Then, the tests used torsion balances, as initiated in 1922 by Eötvös to get to a constraint of $10^{-8}$. Finally, Lunar laser ranging, orbitography and dedicated free fall experiments in orbit manage to reach the higher constraints described below.

As seen from Fig. 3, the universality of free fall has been confirmed by all experiments so far, with increasing accuracy. Still it remains an empirical principle that needs to be better understood. As we shall see later, many theories beyond General Relativity describing varying constant lead to the existence of a fifth force and thus to a violation of the UFF. Note also that if UFF did not hold, one would have to take extra-care in the measurement of masses and definition of the kilogram.

First, the UFF has been constrained experimentally, e.g., in the laboratory by comparing the acceleration of a beryllium and a copper masses in the Earth gravitational field (Adelberger 2001) to get $\eta_{Be,Cu} = (-1.9 \pm 2.5) \times 10^{-12}$. Similarly the comparison of Earth-core-like and Moon-mantle-like bodies gave (Baeßler et al. 1999) $\eta_{Earth,Moon} = (0.1 \pm 2.7 \pm 1.7) \times 10^{-13}$, and experiments with torsion balance using beryllium-aluminum and beryllium-titanium test body pairs set the constraints (Schlamminger et al. 2008; Wagner et al. 2012a) $\eta_{Be,Ti} = (0.3 \pm 1.8) \times 10^{-13}$ and $\eta_{Be,Al} = (-0.7 \pm 1.3) \times 10^{-13}$.

While free fall experiments can be performed on Earth, they are limited by the duration of the fall and are limited by errors in initial conditions at release coupling to the gravity gradient of the Earth, they allow one to reach a precision of order $10^{-10}$. High drop towers such as the ZARM tower of the University of Bremen allows for free-fall times up to 9.3 s. Indeed, space allows for much longer free falls with a similar acceleration in low-Earth orbits. This led to the MICROSCOPE experiment (Touboul et al. 2017, 2019, 2022) that, after two-years mission from 2016 to 2018, provided a sharp constraint on the Eötvös parameter between one proof mass made of titanium and another made of platinum,

$$\eta_{Ti,Pt} = (-1.5 \pm 2.3 \pm 1.5_{\text{syst}}) \times 10^{-15}, \quad (11)$$

where the statistical error is $1\sigma$. Besides, the reference instrument provided a null result, $\eta_{Pt,Pt} = [0.0 \pm 1.1 \pm 2.3_{\text{syst}}] \times 10^{-15}$ showing no sign of unaccounted systematics. We refer to Bergé (2023) for an up-to-date review of the MICROSCOPE experiment, its results and implications for theories of gravity beyond General Relativity.

The Lunar Laser ranging experiment (Williams et al. 2004), compares the relative acceleration of the Earth and Moon in the gravitational field of the Sun. It
<table><caption>**Table 3** Summary of the constraints on the violation of the universality of free fall</caption><thead><tr><th>Constraint</th><th>Body 1</th><th>Body 2</th><th>Ref</th></tr></thead><tbody><tr><td>(-0.2 ± 2.8) × 10<sup>-12</sup></td><td>Be</td><td>Al</td><td>Su et al. (1994)</td></tr><tr><td>(-1.9 ± 2.5) × 10<sup>-12</sup></td><td>Be</td><td>Cu</td><td>Su et al. (1994)</td></tr><tr><td>(5.1 ± 6.7) × 10<sup>-12</sup></td><td>Si/Al</td><td>Cu</td><td>Su et al. (1994)</td></tr><tr><td>(0.1 ± 2.7 ± 1.7) × 10<sup>-13</sup></td><td>Earth-like rock</td><td>Moon-like rock</td><td>Baeßler et al. (1999)</td></tr><tr><td>(1.1 ± 3.0) × 10<sup>-9</sup></td><td>Cu-Pt</td><td>U-238</td><td>Smith et al. (2000)</td></tr><tr><td>(-1.9 ± 2.5) × 10<sup>-12</sup></td><td>Be</td><td>Cu</td><td>Adelberger (2001)</td></tr><tr><td>(-1.0 ± 1.4) × 10<sup>-13</sup> [☉]</td><td>Earth</td><td>Moon</td><td>Williams et al. (2004)</td></tr><tr><td>(0.3 ± 1.8) × 10<sup>-13</sup></td><td>Be</td><td>Ti</td><td>Schlamminger et al. (2008)</td></tr><tr><td>(-3.1 ± 4.7) × 10<sup>-13</sup> [☉]</td><td>Be</td><td>Ti</td><td>Schlamminger et al. (2008)</td></tr><tr><td>(-0.7 ± 1.3) × 10<sup>-13</sup></td><td>Be</td><td>Al</td><td>Wagner et al. (2012a)</td></tr><tr><td>(-5.3 ± 4.0) × 10<sup>-13</sup> [☉]</td><td>Be</td><td>Al</td><td>Wagner et al. (2012a)</td></tr><tr><td>(-0.8 ± 1.3) × 10<sup>-13</sup> [☉]</td><td>Earth</td><td>Moon</td><td>Williams et al. (2012)</td></tr><tr><td>(-1.5 ± 2.3 ± 1.5<sub>syst</sub>) × 10<sup>-15</sup></td><td>Ti</td><td>Pt</td><td>Touboul et al. (2022)</td></tr><tr><td>(0.4 ± 1.2) × 10<sup>-7</sup></td><td>Rb-85 (F=2)</td><td>Rb-85 (F=3)</td><td>Fray et al. (2004)</td></tr><tr><td>(0.3 ± 5.4) × 10<sup>-7</sup></td><td>Rb-87</td><td>K-39</td><td>Schlippert et al. (2014)</td></tr><tr><td>(0.2 ± 1.2) × 10<sup>-7</sup></td><td>Rb-85 (m<sub>F</sub> = +1)</td><td>Rb-85 (m<sub>F</sub> = -1)</td><td>Duan et al. (2016)</td></tr><tr><td>(1.9 ± 3.2) × 10<sup>-7</sup></td><td>Rb-87</td><td>K-39</td><td>Albers et al. (2020)</td></tr><tr><td>(1.6 ± 1.8<sub>stat</sub> ± 3.4<sub>syst</sub>) × 10<sup>-12</sup></td><td>Rb-87</td><td>Rb-85</td><td>Asenbaum et al. (2020)</td></tr><tr><td>(0.8 ± 1.4) × 10<sup>-10</sup></td><td>Rb-87</td><td>Rb-85</td><td>Zhou et al. (2021)</td></tr><tr><td>(0.9 ± 2.9) × 10<sup>-7</sup></td><td>Rb-85 (F=1)</td><td>Rb-85 (F=2)</td><td>Xu et al. (2022)</td></tr></tbody></table>

The bounds on the Eötvös parameter are given with a $1\sigma$ uncertainty. When no mention is added, the two test has been performed in Earth gravitational field unless [☉] is indicated to specify that it refers to the Sun gravitational field. The top-half of the table concerns macroscopic bodies while the bottom-half summarizes the main tests with atom interferometry

sets the constraint $\eta_{Earth,Moon} = (-1.0 \pm 1.4) \times 10^{-13}$, slightly reevaluated to $\eta_{Earth,Moon} = (-0.8 \pm 1.3) \times 10^{-13}$ (Williams et al. 2012). Note that since the core represents only 1/3 of the mass of the Earth, and since the Earth's mantle has the same composition as that of the Moon (and thus shall fall in the same way), one loses a factor of three, so that this constraint is actually similar to the one obtained in the lab. Further constraints are summarized in Table 3. The latter constraint also contains some contribution from the gravitational binding energy and thus includes the strong equivalence principle. When the laboratory result of Baeßler et al. (1999) is combined with the LLR results of Williams et al. (1996) and Müller et al. (1991), one gets a constraints on the strong equivalence principle parameter, respectively $\eta_{SEP} = (3 \pm 6) \times 10^{-13}$ and $\eta_{SEP} = (-4 \pm 5) \times 10^{-13}$—see Merkowitz (2010) for a review. While similar analysis with satellite laser ranging have reached $\eta \sim 2 \times 10^{-9}$ for LAGEOS and LAGEOS II, they remain not competitive with LLR (Nobili et al. 2008).
Thanks to the advent of atom interferometry (Cronin et al. 2009), one can now use atoms, neutrons or charged particles as test masses, allowing one for a better control of systematic effects in free-fall experiments. Atoms have well-known and reproducible properties and it is possible to make very small atomic probes, the position of which can be very precisely controlled, and the possibility to use different atomic states and isotopes allow for the rejection of systematic errors. Using cold rubidium atoms in an atomic fountain interferometer, Fray et al. (2004) compared the free fall of Rb-85 and Rb-87 to reach $\eta = (1.2 \pm 1.7) \times 10^{-7}$ while the comparison between the two different hyperfine ground states $F = 2, m_F = 0$ and $F = 3, m_F = 0$ of Rb-85 concluded that $\eta_{85F=2,F=3} = (0.4 \pm 1.2) \times 10^{-7}$ while the comparison of $m_F = +1$ and $m_F = -1$ of Rb-87 (Duan et al. 2016) gave $\eta = (0.2 \pm 1.2) \times 10^{-7}$. Bonnin et al. (2013) reached $\eta = (1.2 \pm 3.2) \times 10^{-7}$ for Rb-87 vs Rb-88 while the comparison of Rb-87 to K-39 concluded that $\eta = (0.3 \pm 5.4) \times 10^{-7}$ (Schlippert et al. 2014) and $\eta = (0.2 \pm 1.6) \times 10^{-7}$ for the comparison of Sr-88 (no spin) and Sr-87 (half integer spin) (Tarallo et al. 2014). Rb-85/87 reached the level $10^{-9}$ (Bonnin et al. 2015) and then Asenbaum et al. (2020) obtained

$$ \eta = [1.6 \pm 1.8_{\text{stat}} \pm 3.4_{\text{syst}}] \times 10^{-12}. \qquad (12) $$

We refer to Yuan et al. (2023) for a dedicated review on the quantum tests of the WEP with cold atom interferometry. These experiments are expected to reach higher precision in space and could test the WEP with an-matter (Charlton et al. 2020; Rousselle et al. 2022).

These constraints are summarized in Table 3. While they show no sign of violation of the UFF, they require physical models to be interpreted and related to the variation of the fundamental constants, as we shall describe.

## Dynamics

The second building block of General Relativity concerns the dynamics of the gravitational sector, assumed to be described by the Einstein-Hilbert action

$$ S_{\text{grav}} = \frac{c^3}{16\pi G} \int \sqrt{-g} R_* d^4 x. \qquad (13) $$

This defines the dynamics of a massless spin-2 field $g_{\mu\nu}^*$, called the Einstein metric. General Relativity then assumes that both metrics coincide,

$$ g_{\mu\nu} = g_{\mu\nu}^* \qquad (14) $$

(which is related to the strong equivalence principle), but it is possible to design theories in which this is indeed not the case (see the example of scalar-tensor theories below; Sect. 3.1.1) so that General Relativity is one out of a large family of metric theories.

The variation of the total action with respect to the metric yields the Einstein field equations
$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}, \qquad (15)$$

where $T^{\mu\nu} = (2/\sqrt{-g})\delta S_{\text{matter}}/\delta g_{\mu\nu}$ is the matter stress-energy tensor. The coefficient

$$\kappa = \frac{8\pi G}{c^4} \qquad (16)$$

is determined by the weak-field limit of the theory that should reproduce the Newtonian predictions. In this regime, the Einstein equations reduce to the Poisson equation

$$\Delta\Phi_N = 4\pi G\rho$$

defining the Newtonian potential.

To grasp the lowest order relativistic effects in the weak field limit, and in order to test the dynamics of General Relativity in the Solar system, one can rely on the parameterized post-Newtonian formalism (PPN). Its is a general formalism that introduces 10 phenomenological parameters to describe any possible deviation from General Relativity at the first post-Newtonian order (Will 1993, 2006); see also Blanchet 2006 for a review on higher orders. The formalism assumes that gravity is described by a metric and that it does not involve any characteristic scale. In the simplest case and very special case of a single gravitating body, it reduces to the two Eddington parameters introduced in the Schwarzschild metric in isotropic coordinates as

$$g_{00} = -1 + 2\frac{Gm}{rc^2} - 2\beta^{\text{PPN}}\left(\frac{Gm}{rc^2}\right)^2, \qquad g_{ij} = \left(1 + 2\gamma^{\text{PPN}}\frac{Gm}{rc^2}\right)\delta_{ij}.$$

Indeed, General Relativity predicts $\beta^{\text{PPN}} = \gamma^{\text{PPN}} = 1$. These two phenomenological parameters are constrained (1) by the shift of the Mercury perihelion (Shapiro 1990), which implies that $|2\gamma^{\text{PPN}} - \beta^{\text{PPN}} - 1| < 3 \times 10^{-3}$, (2) the Lunar laser ranging experiments (Williams et al. 2004), which implies that $|4\beta^{\text{PPN}} - \gamma^{\text{PPN}} - 3| = (4.4 \pm 4.5) \times 10^{-4}$ and (3) by the deflection of electromagnetic signals, which are all controlled by $\gamma^{\text{PPN}}$. For instance the very long baseline interferometry (Shapiro et al. 2004) implies that $|\gamma^{\text{PPN}} - 1| = 4 \times 10^{-4}$, while the measurement of the time delay variation to the Cassini spacecraft (Bertotti et al. 2003) sets $\gamma^{\text{PPN}} - 1 = (2.1 \pm 2.3) \times 10^{-5}$.

### Phenomenological modifications of Newton gravity

The PPN formalism does not allow one to test finite range effects that could be caused, e.g., by a massive degree of freedom. In that case one expects a Yukawa-type deviation from the Newton potential. The total gravitational potential created by a point-mass of mass $m$ at distance $r$ is parameterized as
$$V = -\frac{Gm}{r} \left(1 + \alpha e^{-r/\lambda}\right), \qquad (17)$$

a form that will be physically justified later, where $\alpha$ is the strength of the deviation compared to Newtonian gravity (which may depend on the composition of the interacting masses) and $\lambda$ is the range of the corresponding. For macroscopic extended bodies, the potential is modified by a shape factor arising from the integration of the point-particle potential. As we shall see later, such a potential corresponds to the static limit of an interaction mediated by virtual bosons of mass $m_\phi = \hbar/\lambda c$.

The constraints on $\eta$ from Table 3 can be translated on constraints in the $(\lambda, \alpha)$-plane. For each experiment, one first needs to compute the shape factor to take into account that all element of the body do not contribute similarly to the macroscopic Yukawa interaction. For instance, a homogenous sphere of radius $R$ has a shape factor

$$\Phi(x) \equiv 3(x \cosh x - \sinh x)/x^3 \qquad (18)$$

with $x = R/\lambda$, which indeed tends to 1 when $\lambda \to +\infty$ since in this limit the Gauss theorem is recovered. Then, the new interaction may be composition dependent so that one shall replace $\alpha$ in Eq. (17) by $\alpha_{ij}$ that depends on the dimensionless Yukawa charge $q$ characteristic of each material

$$\alpha_{ij} = \alpha \left(\frac{q}{\mu}\right)_i \left(\frac{q}{\mu}\right)_j \qquad (19)$$

with $\mu$ the atomic mass in atomic units (e.g., $\mu = 12$ for carbon-12, or $\mu = 47.948$ for titanium). The definition of $q$ depends on the microphysics of the coupling of the new massive degree of freedom to standard matter fields; see Sect. 2.2.2 below for explicit examples. Taking into account the electromagnetic and nuclear binding energies, the charge usually reduces to the baryon or lepton numbers ($B$ and $L$) of the test mass materials; see e.g., Fayet (1990) and Sect. 2.2.2 below. While purely phenomenological, this description is useful since it describes the macroscopic fifth force induced by a massive scalar field in the Newtonian regime. It follows that the Eötvös parameter due to a Yukawa potential between 2 bodies in free-fall around the Earth is

$$\eta = \alpha \left[ \left( \frac{q}{\mu} \right)_i - \left( \frac{q}{\mu} \right)_j \right] \left( \frac{q}{\mu} \right)_E \Phi \left( \frac{R_E}{\lambda} \right) \left( 1 + \frac{r}{\lambda} \right) e^{-\frac{r}{\lambda}} \qquad (20)$$

($\Phi = 1$ for the test masses since their sizes are much smaller than $\lambda$) with $r = R_E + h$, $h$ being the altitude of the satellite.

Figure 4 summarizes the constraints obtained thanks to the MICROSCOPE experiments (Bergé et al. 2018a). The constraints on $(\lambda, \alpha)$ are summarized in Hoyle et al. (2004); Bergé (2023), which typically shows that $\alpha < 10^{-2}$ on scales ranging from the millimeter to the Solar system size. One shall however be careful with these constraints since geodesy experiments shall also take into account the shape of
<figure><img src="image_9.png" /><figcaption>**Fig. 4** Constraints on the Yukawa potential parameters ($\alpha, \lambda$) assuming $q = B$ (left) or $q = B - L$ (right) obtained by MICROSCOPE and compared to previous constraints (dotted: Smith et al. (2000), Dashed: Schlamminger et al. (2008), Dot-dashed: Talmadge et al. (1988)). Images reproduced from Bergé et al. (2018a), copyright by APS</figcaption></figure>

the Earth while many analysis assume a spherical Earth; see Bergé et al. (2018b) for a discussion of the impact of the shape of the Earth when constraining gravity and the geoid simultaneously.

### Conclusions

General Relativity is our reference theory to describe gravitation, that is the long range non-screened interaction. As we have seen the UFF is extremely well-tested in the Solar system and complemented by the test of the LPI described in this review. Besides, it is also tested with pulsars (Damour and Esposito-Farèse 1998; Esposito-Farèse 2005) in the strong field regime (Psaltis 2008) and, more recently thanks to the detection of gravitational waves. For more details we refer to Damour and Lilley (2008), Turyshev (2008), Will (1993, 2006). Needless to say that any extension of General Relativity has to pass these constraints. However, deviations from General Relativity can be larger in the past, as we shall see, which makes cosmology an interesting physical system to extend these constraints in spacetime but also to other matter component such as the dark matter that cannot be used, for now, in laboratory experiments.

### 2.2.2 Varying constants and free fall: the link between LPI and UFF

As the previous discussion shows, the constancy of the fundamental constants and the universality are two pillars of the equivalence principle. Dicke (1964) realized that they are actually not independent and that if the coupling constants are spatially dependent then this will induce a violation of the universality of free fall.

#### Fifth force

The connection lies in the fact that the mass of any composite body, starting, e.g., from neutrons and protons and then nuclei, depends on the mass of the elementary particles that constitute it (this means that it will depend on the various Yukawa couplings and on the Higgs sector parameters) but also a contribution, $E_{\text{binding}}/c^2$,
arising from the binding energies of the different interactions (i.e., strong, weak and electromagnetic) but also gravitational for massive bodies. Thus, the mass of any body is a function of all the constants, $m[\alpha_i]$.

It follows that the action for a point particle is no more given by Eq. (5) but by

$$S_{\text{matter}} = - \int m_A[\alpha_j] c \sqrt{-g_{\mu\nu}(x^\alpha) v^\mu v^\nu} dt, \quad (21)$$

where $\alpha_j$ stands for the list of constants on which $m_A$ depends. It includes $\alpha_{EM}$ but also many others constants. The index $A$ in $m_A$ recalls that the dependence in these constants is a priori different for bodies of different chemical composition. The variation of this action gives the equation of motion

$$u^\nu \nabla_\nu u^\mu = - \left( \sum_i \frac{\partial \ln m_A}{\partial \alpha_i} \frac{\partial \alpha_i}{\partial x^\beta} \right) (g^{\beta\mu} + u^\beta u^\mu). \quad (22)$$

It departs from a pure geodesic motion due to the appearance of a fifth force

$$F_5^\mu = - \left( \sum_i \frac{\partial \ln m_A}{\partial \alpha_i} \frac{\partial \alpha_i}{\partial x^\beta} \right) (g^{\beta\mu} + u^\beta u^\mu) \quad (23)$$

that is, thanks to the projection operator $(g^{\beta\mu} + u^\beta u^\mu)$, purely spatial, i.e., $F_5^\mu u_\mu = 0$. The first parenthesis contains the sensitivities of the mass on a variation of the various constants $\alpha_i$,

$$f_{A,i} \equiv \frac{\partial \ln m_A}{\partial \alpha_i}. \quad (24)$$

To finish, the force is non-vanishing only if the constants enjoy a 4-gradient, $\partial\alpha_i/\partial x^\beta \neq 0$, that is if at least one of them varies in space and/or time.

The expression (22) provides the link between variation of fundamental constants and the universality of free fall. This simple argument shows that if the constants depend on time then there must exist an anomalous acceleration that will depend on the chemical composition of the body $A$. It implies that, given a microscopic physical model to compute the sensitivities $f_{A,i}$, any constraint on the UFF can be translated on a constraint on $\partial\alpha_i/\partial x^\beta$. Similarly, any model with varying constants e.g., on cosmological scales, will inevitably induce a violation of UFF in the Solar system, which creates a strong link between cosmological and local scales. Concrete examples of such fifth force will be discussed in Sect. 3.

### Newtonian limit

In the Newtonian limit $g_{00} = -1 + 2\Phi_N/c^2$, and at first order in $v/c$, the equation of motion of a test particle reduces to

$$\mathbf{a} = \mathbf{g}_N + \delta\mathbf{a}_A, \quad \delta\mathbf{a}_A = -c^2 \sum_i f_{A,i} \left( \nabla\alpha_i + \dot{\alpha}_i \frac{\mathbf{v}_A}{c^2} \right) \quad (25)$$

so that in the slow velocity (and slow variation) limit it reduces to
$$ \delta \mathbf{a}_A = -c^2 \sum_i f_{A,i} \nabla \alpha_i. $$

This allows one to get the Eötvös parameter as

$$ \eta_{AB} = \frac{c^2}{g_N} \sum_i |(f_{A,i} - f_{B,i}) \nabla \alpha_i|. \qquad (26) $$

so that the constraints summarized in Table 3 can be translated on a constraint on the gradient of the fundamental constants on Earth length scale.

The anomalous acceleration (25) is generated by the change in the (electromagnetic, gravitational, ...) binding energies (Dicke 1964; Haugan and Will 1976; Nordtvedt 1990) but also in the Yukawa couplings and in the Higgs sector parameters so that the $\alpha_i$-dependencies are a priori composition-dependent. As a consequence, any variation of a fundamental constant will entail a violation of the universality of free fall: the total mass of the body being space dependent, an anomalous force appears if energy is to be conserved. The variation of the constants, deviation from General Relativity and violation of the weak equivalence principle are in general expected together, with quantitative relationships depending on the particular theory and model used; see e.g., Sect. 7 for concrete examples.

## Sensitivity parameters

The composition dependence of $\delta \mathbf{a}_A$ and thus of $\eta_{AB}$ can be used to optimize the choice of materials for the experiments testing the equivalence principle (Damour 1996; Damour and Donoghue 2008, 2010b) but also to distinguish between several models if data from the universality of free fall and atomic clocks are combined (Dent 2007).

From a theoretical point of view, the computation of $\eta_{AB}$ requires the determination of the coefficients $f_{Ai}$. This can be achieved in two steps by first relating the new degrees of freedom of the theory to the variation of the fundamental constants and then relating them to the variation of the masses. As we shall detail in Sect. 3, the first issue is very model dependent while the second is especially difficult (see Sect. 4.2), particularly when one wants to understand the effect of the quark masses, since it is related to the intricate structure of QCD and its role in low energy nuclear reactions.

As an example, the mass of a nuclei of charge Z and atomic number A can be expressed as

$$ m(A, Z) = Zm_p + (A - Z)m_n + Zm_e + E_S + E_{EM}, \qquad (27) $$

where $E_S$ and $E_{EM}$ are respectively the strong and electromagnetic contributions to the binding energy. The Bethe–Weizäcker formula allows one to estimate the latter as

$$ E_{EM} = 98.25 \frac{Z(Z-1)}{A^{1/3}} \alpha_{EM} \text{ MeV}. \qquad (28) $$

If we decompose the proton and neutron masses as (Gasser and Leutwyler 1982)
$$m_{(p,n)} = u_3 + b_{(u,d)}m_u + b_{(d,u)}m_d + B_{(p,n)}\alpha_{EM} \quad (29)$$

where $u_3$ is the pure QCD approximation of the nucleon mass ($b_u$, $b_d$ and $B_{(n,p)}/u_3$ being pure numbers), it reduces to

$$m(A, Z) = (Au_3 + E_S) + (Zb_u + Nb_d)m_u + (Zb_d + Nb_u)m_d + \left(ZB_p + NB_n + 98.25 \frac{Z(Z-1)}{A^{1/3}} \text{MeV}\right)\alpha_{EM}, \quad (30)$$

with $N = A - Z$, the neutron number. For an atom, one would have to add the contribution of the electrons, $Zm_e$. This expression clearly depends on strong, weak and electromagnetic quantities. The numerical coefficients $B_{(n,p)}$ are given explicitly by (Gasser and Leutwyler 1982)

$$B_p\alpha_{EM} = 0.63 \text{ MeV} \qquad B_n\alpha_{EM} = -0.13 \text{ MeV}. \quad (31)$$

Such estimations were used in the first analysis of the relation between variation of the constant and the universality of free fall (Damour and Polyakov 1994b; Dvali and Zaldarriaga 2002; Uzan 2003b) but the dependence on the quark mass is still not well understood and we refer to Damour and Donoghue (2008, 2010b), Dmitriev and Flambaum (2003), Donoghue (2006), Flambaum and Shuryak (2003) for some attempts to refine this description. It is also important to stress that a microscopic model with only a varying fine structure constant will generate varying proton and neutron masses due from their binding energies, an effect that cannot be neglected.

For macroscopic bodies, the total mass also inherits a negative contribution,

$$\Delta m(G) = -\frac{G}{2c^2} \int \frac{\rho(\mathbf{r})\rho(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} d^3\mathbf{r}d^3\mathbf{r}', \quad (32)$$

from the gravitational binding energy. As a conclusion, from (30) and (32), we expect the masses to depend on all the coupling constants. We shall discuss this issue in more detail in Sect. 4.2.

To finish, note that varying coupling constants can also be associated with violations of local Lorentz invariance and CPT symmetry (Kostelecký et al. 2003; Bertolami et al. 2004; Gurzadyan et al. 2010).

## 2.3 Relations with cosmology

Most constraints on the time variation of the fundamental constants will not be local and related to physical systems at various epochs of the evolution of the universe. It follows that the comparison of different constraints requires a full cosmological model. Besides, many varying fundamental constant models have been related to the study of dark energy models involving a light field. Thus, we need to recall the basics of the current cosmological model.
### 2.3.1 ΛCDM standard cosmological model

Our current cosmological model is known as the ΛCDM (see Peter and Uzan 2009; Uzan 2021 for a detailed description, Appendix C for the main definitions and Table 30 for the typical value of the cosmological parameters).

It is important to recall that its construction relies on 4 main hypotheses, as discussed in Uzan (2007),

(H1) a theory of gravity;

(H2) a description of the matter components contained in the Universe and their non-gravitational interactions;

(H3) symmetry hypothesis; and

(H4) a hypothesis on the global structure, i.e., the topology, of the Universe.

These hypotheses are not on the same footing since H1 and H2 refer to the physical theories. However, these hypotheses are not sufficient to solve the field equations and we must make an assumption on the symmetries (H3) of the solutions describing our Universe on large scales while H4 is an assumption on some global properties of these cosmological solutions, with same local geometry. But the last two hypothesis are unavoidable because the knowledge of the fundamental theories is not sufficient to construct a cosmological model (Uzan 2010a).

The ΛCDM model assumes that gravity is described by General Relativity (H1), that the Universe contains the fields of the standard model of particle physics plus some dark matter and a cosmological constant, the latter two having no physical explanation at the moment. It also deeply involves the Copernican principle as a symmetry hypothesis (H3), without which the Einstein equations usually cannot be solved, and assumes most often that the spatial sections are simply connected (H4). H2 and H3 imply that the description of the standard matter reduces to a mixture of a pressureless and a radiation perfect fluids. This model is compatible with all astronomical data despite the fact that anomalies have been showing up in the past years (Peebles 2022). It roughly indicates that $\Omega_{A0} \simeq 0.73$, $\Omega_{\text{mat0}} \simeq 0.27$, and $\Omega_{K0} \simeq 0$. Thus, cosmology roughly imposes that $|\Lambda_0| \le H_0^2$, that is $\ell_{\Lambda} \le H_0^{-1} \sim 10^{26} \text{ m} \sim 10^{41} \text{ GeV}^{-1}$.

### 2.3.2 Dark energy and its implications

The analysis of the cosmological dynamics of the universe and of its large scale structures requires the introduction of a new constant, the cosmological constant, associated with a recent acceleration of the cosmic expansion, that can be added to the Einstein-Hilbert action as

$$S_{\text{grav}} = \frac{c^3}{16\pi G} \int \sqrt{-g} (R - 2\Lambda) d^4 x.$$

This constant can equivalently be introduced in the matter action. Note, however, that it is disproportionately small compared to the natural scale fixed by the Planck length
$$ \rho_{A_0} \sim 10^{-120} M_{Pl}^4 \sim 10^{-47} \text{ GeV}^4. \qquad (33) $$

Classically, this value is no problem but it was pointed out that at the quantum level, the vacuum energy should scale as $M^4$, where M is some energy scale of high-energy physics. In such a case, there is a discrepancy of 60–120 orders of magnitude between the cosmological conclusions and the theoretical expectation. This is the cosmological constant problem (Weinberg 1989).

Two approaches to solve this problem have been considered. Either one accepts such a constant and such a fine-tuning and tries to explain it on anthropic ground. Or, in the same spirit as Dirac, one interprets it as an indication that our set of cosmological hypotheses have to be extended, by either abandoning the Copernican principle (Uzan et al. 2008b) or by modifying the local physical laws (either gravity or the matter sector). The way to introduce such new physical degrees of freedom were classified in Uzan (2007). In that latter approach, the tests of the constancy of the fundamental constants are central, since they can reveal the coupling of this new degree of freedom to the standard matter fields. Note, however, that the cosmological data still favor a pure cosmological constant.

### 2.3.3 Light fields phenomenology

Among all the proposals, quintessence (Ratra and Peebles 1988) involves a scalar field rolling down a runaway potential hence acting as a fluid with an effective equation of state in the range $-1 \le w \le 1$ if the field is minimally coupled. It was proposed that the quintessence field is also the dilaton (Riazuelo and Uzan 2000; Gasperini et al. 2002). The same scalar field then drives the time variation of the cosmological constant and of the gravitational constant and it has the property to also enjoy tracking solutions (Uzan 1999). Such models do not solve the cosmological constant problem but only relieve the coincidence problem. One of the underlying motivation to replace the cosmological constant by a scalar field comes from superstring models in which any dimensionful parameter is expressed in terms of the string mass scale and the vacuum expectation value of a scalar field. However, the requirement of slow roll (mandatory to have a negative pressure) and the fact that the quintessence field dominates today imply, if the minimum of the potential is zero, that it is very light, roughly of order $m \sim 10^{-33}$ eV (Carroll 1998).

Such a light field will inevitably lead to violations of the universality of free fall and variation of the fundamental constants if it is non-universally coupled to the matter fields. Carroll (1998) considered the effect of the coupling of this very light quintessence field to ordinary matter via a coupling to the electromagnetic field as $\phi F^{\mu\nu} \tilde{F}_{\mu\nu}$. Chiba and Khori (2002) also argued that an ultra-light quintessence field induces a time variation of the coupling constant if it is coupled to ordinary matter and studied a coupling of the form $d_e \phi F^{\mu\nu} F_{\mu\nu} / M_P$, as, e.g., expected from Kaluza-Klein theories (see Sect. 3.4 below). Such a coupling with $d_e \ll 1$ generically arise if quantum gravity effects weakly break an underlying global symmetry of $\phi$ near the Planck scale (Kallosh et al. 1995).
As will be described in Sect. 7, many models link the acceleration of the cosmic expansion and the variation of fundamental constants. While a deviation from $w = -1$ is difficult to detect, quintessence may reveal itself through the variation of fundamental constants.

### 2.3.4 Ultra-light dark matter models

As we shall see in more details below, in a Friedmann-Lemaître spacetime, a massive scalar field with linear coupling enjoys a Klein-Gordon equation of the form $\ddot{\phi} + 3H\phi + m^2\phi = 4\pi G\sigma$ where $H = \dot{a}/a$ is the Hubble parameter. The second term sources a Hubble damping that can be neglected as long as $m_\phi \gg H_0 \sim 1.5 \times 10^{-33}$ eV. $\sigma$ is a source term that arise from the coupling of $\phi$ to the standard model field, see Eq. (61) with $\alpha(\phi) = \text{cst}$ below. Being related to the baryon density, it will evolve with a characteristic time scale of $H^{-1}$ so that for periods much shorter, it can be considered constant. In this regime, the solution of the Klein-Gordon as a superposition of a term sourced by the standard matter and an oscillating component,

$$ \phi(t) = \frac{4\pi G\sigma}{m_{\phi}^2} + \phi_0 \cos(m_{\phi}t + \delta). \quad (34) $$

Averaging its energy and pressure, $4\pi G\rho_\phi = \dot{\phi}^2/2 + V$ and $4\pi G P_\phi = \dot{\phi}^2/2 - V$— using the notation and normalisation of Sect. 3.1.1 and Eq. (48)— over a period much smaller than $H^{-1}$, the scalar field behaves as a pressureless component that can be identified with dark matter (DM) with density

$$ 4\pi G\rho_{\phi} = \frac{1}{2}m_{\phi}^{2}\phi_{0}^{2}. \quad (35) $$

Note however that light scalar fields do not behave exactly as perfect cold dark matter on short length scales, where their density perturbations have a nonzero sound speed. For $m_\phi < 10^{-22}$ eV, they would have inhibited cosmological structure growth (Marsh and Silk 2014) in conflict with observations. In this mass range, the field would behave classically. Hence, a light dilaton with mass in the range

$$ m_{\phi} \in (10^{-22} - 1) \text{ eV} \quad (36) $$

can act as a dark matter component, such an ultralight bosonic field as dark matter has large occupation numbers per mode and behaves as a classical wave with a frequency proportional to $m_\phi$.

It was realized, as already discussed in the previous section on quintessence, that the light field shall couple to standard model fields so that these dark matter models can be detected in laboratory experiments, in particular through the variation of fundamental constants (Arvanitaki et al. 2015; Stadnik and Flambaum 2015c; Van Tilburg et al. 2015; Stadnik and Flambaum 2015a; Derevianko 2016). Stadnik and Flambaum (2016a) proposed to search for this dark component with laser interferometer measurements, such as large-scale gravitational-wave detectors (e.g.,
LIGO, Virgo, GEO600,...) and investigate the design of a strontium optical lattice clock—silicon single-crystal cavity interferometer as a novel small-scale platform. Savalle et al. (2019b) investigate the d-detection of light DM thanks to spacetime separated clocks that are shown to probe different parameter combinations than the “usual” co-located clock experiments. We refer to Stadnik and Flambaum (2015b) for an overview of recent developments in the detection of light bosonic dark matter, including axion, pseudoscalar axion-like and scalar dark matter, to Safronova et al. (2018a), Safronova (2019) for extensive reviews on the search for light dark matter, and to Budker et al. (2025) for a study of spacecraft missions to detect such halos with instruments such as quantum clocks, atomic and molecular spectrometers designed to search for fast (tens of hertz to gigahertz) oscillations of fundamental constants. Gué et al. (2024) presented the theoretical investigation of the expected experimental signals produced by freely falling atoms with time oscillating mass and transition frequency that may be induced by the coupling of scalar DM to standard matter showing.

From a theoretical point of view, most analysis rely of the light dilaton model described in details in Sect. 7.2 with linear or quadratic couplings. Note that Gan and Liu (2023) proposed to consider a dark photon coupled to electromagnetism with a $\phi$ dependent coupling, which allows one to open the early time parameter space to produce dark matter while being compatible with actual laboratory constraints, making the dark photon a still viable candidate. Note the claim by Bauer and Chakraborti (2024) that the quadratic interactions of axion dark matter lead to a divergence of the axion field value for decay constant smaller than $10^{15}$ GeV so that the field value for the axion close to a massive body like Earth is divergent in a similar way as the spontaneous scalarization (Damour and Esposito-Farèse 1993, 1996) arising in scalar-tensor theories for light scalars in the presence of massive bodies.

### 2.3.5 Dark matter topological defects

Note also that such a scalar field can also form topological defects (Vilenkin and Shellard 1994) that can result in the existence of domains with different values of the fundamental constants (Olive et al. 2011), as described in Sect. 8.3.3. Besides, the dark matter density would be concentrated into many distinct, compact spatial regions. Essig et al. (2013) proposed a classification of the way the fields forming the defect interact with the standard matter, using the so-called “portals”, the collection of gauge-invariant operators of the standard matter coupled with the operators from the dark sector. Derevianko and Pospelov (2014) described many field set-ups that could lead to monopole, strings or wall assuming linear and quadratic coupling. Besides the linear, $\phi m_\psi \bar{\psi} \psi$, and quadratic, $\phi^2 m_\psi \bar{\psi} \psi \psi$, couplings they considered an axionic portal $\partial_\mu \phi \bar{\psi} \gamma_\mu \psi$ and a current-current portal $\phi^* \partial_\mu \phi \bar{\psi} \gamma_\mu \psi$. They emphasized that networks of correlated atomic clocks, such as the Global Positioning System, are a powerful tool to search for this topological defect dark matter, hence providing a new fundamental physics application to the ever-improving accuracy of atomic clocks. Indeed, during the encounter with an
extended dark matter object, as it sweeps through the network, initially synchro-
nized clocks will become desynchronized. Time discrepancies between spatially-
separated clocks are expected to exhibit a distinct signature, encoding defect's space
structure and its interaction strength with atoms. Stadnik and Flambaum (2015c)
detailed the detection of topological defect dark matter through transient-in-time
effects and for a relic, coherently oscillating condensate. Roberts et al. (2017) found
no evidence for DM in the form of domain walls from the mining 16 years of
archival GPS data. A Global Network of Optical Magnetometers for Exotic physics
searches (GNOME) was proposed to reach higher sensitivities (Afach et al. 2021).

2.3.6 Varying constants and cosmology

We have three means of investigation and types of relations between varying
constants and the cosmological model within the scalar field extension of the
standard ΛCDM.

Test field

The field driving the time variation of the fundamental constants is not taylored
for explaining the acceleration of the universe—and it never dominates the matter
content today or its equation of state is not negative enough. In such a case, the
variation of the constants is disconnected from the dark energy problem. Cosmology
then allows us to determine the dynamics of this field during the whole history of
the universe and thus to compare local and cosmological constraints and interpret all
the data consistently but the new field does not modify the cosmic history. An
example is provided by scalar-tensor theories (see Sect. 3.1.1) for which one can
compare, e.g., primordial nucleosynthesis to local constraints (Damour and Pichon
1999).

Note however that in such a situation one should take into account the effect of
the variation of the constants on the astrophysical observations since it can affect
local physical processes and bias, e.g., the luminosity of supernovae and indirectly
modify the distance luminosity-redshift relation derived from these observations
(Barrow and Magueijo 2000; Riazuelo and Uzan 2002).

Dark sector candidate

The field driving the time variation of the fundamental constants is also
responsible for the “dark sector”. It can account for dark matter or cause the late
time acceleration of the cosmic expansion (see Sect. 7.4 for concrete examples and
the discussion of Sect. 2.3.3 above).

It follows that the local value of the scalar field and time derivatives are related
either to the dark matter abundance or to the equation of state of dark energy.
Hence, the dynamics of the universe, the level of variation of the constants and the
other deviations from general relativity are connected (Damour and Polyakov
1994b; Damour et al. 2002b; Uzan 2004; Martin et al. 2006; Uzan 2007; Damour
and Lilley 2008) so that the study of the variation of the constants can improve,
within the model, the reconstruction of the equation state of the dark energy
(Avelino et al. 2006; Doran 2005; Nunes et al. 2009; Parkinson et al. 2004).
## Dark-visible matter equivalence principle

Note that the question of the validity of the equivalence principle in the dark sector is also crucial since it cannot be tested in the laboratory or in Solar system experiments. To that purpose, Coc et al. (2009c) derived the constraints from BBN on a scalar-tensor theory with non-universal coupling with dark sector so that no violation of the EEP can be witnessed in the visible sector, following earlier proposals by Damour et al. (1990), Fuzfa and Alimi (2007). Mohapi et al. (2016) constrain such a coupling from galactic observations of strong lensing and of velocity dispersion, showing that data favor violations through coupling strengths that are of opposite signs for ordinary and dark matter even though no deviation from GR.

From a theoretical perspective, this opens the question of whether one can modify the dark matter gravity sector independently of the standard matter sector, in particular for since such models have become candidates to a solution of the Hubble tension (Pitrou and Uzan 2024; Uzan and Pitrou 2024). In some models, a scalar fifth force may potentially lead to a violation of the WEP that could be probed (Carroll et al. 2010; Mantry 2010), in particular if dark and standard matter fields interact. Such models are however strongly constrained (Carroll et al. 2009). It has also been suggested that the anomalies in the positron/electron spectra may arise from a long-range dark force mediating the dark matter annihilation and that it could be detected at the LHC (Bai and Han 2009). Then, one shall investigate the effects of the scalar force in astrophysical environments and how it modifies DM haloes, see e.g., Frieman and Gradwohl (1991), Gradwohl and Frieman (1992), Nusser et al. (2005), Kesden and Kamionkowski (2006), Bean et al. (2008), Keselman et al. (2010), Mohapi et al. (2016) for tests of the WEP between the visible and DM sectors. Peebles (2012) suggested that a DM fifth force seems to have beneficial effects for DM distribution on small scales and that an extra evanescent component of matter with evolving mass and a fifth force large enough may be needed to reach a better understanding of early assembly of more nearly isolated protogalaxies.

## The Hubble tension

A new connection has appeared with the *Hubble tension*. In a simplified way, it arises from the fact that the value of the Hubble constant $H_0$ derived from the analysis of CMB+BAO data, which point toward $H_0 = (67.4 \pm 0.5)$ km/s/Mpc, differs from the one obtained at low redshifts from the Hubble diagram, estimated to be $H_0 = (73.04 \pm 1.04)$ km/s/Mpc; hence a $5\sigma$ discrepancy (Schöneberg et al. 2022).

The problem is usually formulated as a low/high-redshift tension since in first approximation, the key physical parameters at the background level are the comoving sound horizon

$$r_s = \frac{1}{H_0} \int_{z_s}^{\infty} \frac{c_s dz}{H(z)/H_0} \qquad (37)$$

with $z_s \sim 1088$ the redshift at recombination, and the comoving angular diameter distance
$$R_{\text{ang}}(z) = \frac{1}{H_0} f_{\kappa} \left[ \int_{0}^{z} \frac{dz}{H(z)/H_0} \right], \qquad (38)$$

related to the angular (or luminosity) distance, $D_A(z) = R_{\text{ang}}(z)/(1+z)$ or $D_L = (1+z)R_{\text{ang}}(z)$; see Appendix C for definitions. Their ratio fixes the physical angular scales of the acoustic peaks of the CMB.

The precision of CMB observations set a strong constraint on this quantity from which one can get the early time estimations of $H_0$ given the knowledge of the baryon and dark matter (DM) energy densities. It follows that most of the arguments on the $H_0$ tension circle around the sound horizon with two main categories of models (Di Valentino et al. 2021; Schöneberg et al. 2022; Abdalla et al. 2022): (1) “Late time models” modify the expansion history after recombination, increasing $H_0$ but keeping $r_s$ unchanged; (2) “Early time models” modify the expansion history before recombination, e.g., through energy injection around the recombination, changing both $H_0$ and $r_s$ so as to have a lower sound horizon to allow for a larger $H_0$ to which one can add (3) “earlier recombination models” (Jedamzik et al. 2021) and (4) “modified DM gravity models” (Pitrou and Uzan 2024; Uzan and Pitrou 2024) in which the gravitation in the DM sector is assumed to enjoy a scalar-tensor gravity while the visible sector remains unaffected. Many of these models assume a form of dark energy or an interaction of dark matter with a new matter component and in particular a scalar field. The dynamics of these 4 categories have been compared on Fig. 6 of Uzan and Pitrou (2024).

It was realized that the Einstein-Boltzmann equations used to predict CMB+BAO is invariant (Rich 2015; Cyr-Racine et al. 2022; Ge et al. 2023) under the rescaling transformations<fnref n="6" />

$$H \rightarrow \lambda H \qquad G\rho \rightarrow \lambda^2 G\rho \qquad \sigma_T n_e \rightarrow \lambda \sigma_T n_e, \qquad A_s \rightarrow A_s \lambda^{1-n_s}$$

of the Hubble parameter, gravitational constant, Thomson scattering and amplitude of the cosmological perturbations, the latter involving a dependence on the spectral index $n_s$ of the initial power spectrum. Hence, it led to the idea that the Hubble tension may be related to the variation of constants. A varying gravitational constant was proposed by Bégué et al. (2019) and implemented in different ways (Knox and Millea 2020; Di Valentino et al. 2021; Abdalla et al. 2022) as a generalisation of old extended quintessence models (Uzan 1999). Most of these models face difficulties with either primordial nucleosynthesis and/or local constraints on deviations from General Relativity. Cyr-Racine et al. (2022) implemented this scaling by invoking a mirror world but the model requires primordial helium abundance in contradiction with BBN results. Zhang and Frieman (2023) extended this to a model of a massive field with linear and quadratic coupling able to resolve the tension while keeping the same helium-4 abundance from BBN provided $\Delta\alpha_{\text{EM}}/\alpha_{\text{EM}} \simeq -2 \times 10^{-5}$ at recombination.

<footnote n="6">⁶ Rich (2015) derives the dependence of the CMB temperature angular power spectrum to the 4 dimensionless quantities where $m_\chi$ is the mass of the dark matter particle, demonstrating in particular that $\Delta[m_p/m_\chi]/(m_p/m_\chi) = 0.09 \pm 0.15$ while $m_\chi$ is not known.</footnote>
Indeed, in order for the sound horizon, and thus CMB predictions, to remain unchanged, one needs to also modify the Thomson cross-section $\sigma_T$, i.e., the non-gravitational physics which is highly constrained. The idea to “compensate” the variation of the gravitational constant by a variation of the fine structure constant and/or electron mass and thus the possibility to solve or alleviate the Hubble tension thanks to a variation of the constant has first been noticed by Hart and Chluba (2020). It has attracted a huge attention since then; see the report of the latest constraints in Sect. 5.10. Hart and Chluba (2022) then showed that varying $(\alpha_{EM}, m_e)$ can alleviate the Hubble tension despite the CMB (Ade et al. 2015) while being compatible with other constraints on the variation of constants, as discussed in this review, that shall indeed be included together with the cosmological observations.

From the study of the scaling of the parameters, Sekiguchi and Takahashi (2021) demonstrated that if the Hubble constant and electron mass are scaled $\Delta h/h \sim 3.23 \Delta m_e / m_e$ then the CMB is barely unchanged. Hence, one could resolve the Hubble tension at the expanse of a higher $m_e$. However this would also require that $\Delta \Omega_m / \Omega_m = -5.46 \Delta m_e / m_e$ resulting in a lower $\Omega_m$ (Lee et al. 2023b), which is potentially a critical issue since this would be in contraction with SNIa and BAO data. As understood by Sekiguchi and Takahashi (2021), this could be resolved by allowing for a non-vanishing space curvature, which actually turned to be the best-fit model in the extensive model comparisons by Schöneberg et al. (2022) or by including early dark energy (Hart and Chluba 2023) highlighting clearly the connection between $m_e$ and $H_0$. Note that a non-vanishing space curvature needs to face model-building issues in inflation and in tension with the analysis by Di Dio et al. (2016).

Most of the studies (Hart and Chluba 2018, 2020; Schöneberg et al. 2022; Khalife et al. 2024; Baryakhtar et al. 2024; Seto and Toda 2024; Toda et al. 2024; Schöneberg and Vacher 2025) considered a *constant shift* of the electron mass between CMB and today; see Sect. 5.10 and Table 17. Following Martins et al. (2015b), some models assume a phenomenological evolution of the constants parameterized as $[(1+z)/1000]^p$ for $\alpha_{EM}$ and or $m_e$ and include $p$ in the fit; see e.g., Hart and Chluba (2020). While this accounts for a cosmic time evolution such a parameterisation is actually a choice of a “solution” to a non-specified underlying theory. While this could be sufficient to analyse background data, it shall be taken with a grain of salt for CMB analysis since perturbations cannot be treated consistently. As illustrated by Uzan and Pitrou (2024) on a scalar-tensor theory, neglecting the treatment of the perturbations can lead to an error larger than a percent for the CMB predictions compared to the full treatment, with same background evolution. Indeed, many dynamical models can and have been constructed on the basis of a light scalar field; see e.g., Barrow and Magueijo (2005), Avelino (2008), Hoshiya and Toda (2023) as well as all the models discussed in Sect. 7. As Eq. (358) shows, all these models induce a time variation of the constant in the Solar system that is sharply constrained. On the example of $B(\varphi)F^2$ models discussed in Sect. 7.4 Vacher and Schöneberg (2024) demonstrated that this is a critical hurdle to construct a viable model of varying constant that
would resolve the Hubble tension since the field would have to relax rapidly enough to avoid the local constraints. In particular they demonstrate that in single-field models one has to extremely fine-tune the shape of the potential and/or the initial conditions. Indeed, for single field models in a potential that is not fine-tuned we can put a generic bound at recombination of $\Lambda\alpha_{EM}/\alpha_{EM} < 5 \times 10^{-4}$ 95 C.L.. To finish, let us also mention that Schöneberg and Vacher (2025) cooked up a model in which $m_e \propto m_p$ to evade the local constraints on the variation of $\bar{\mu}$ while allowing $m_e$ to vary between recombination and today.

## 2.4 Conclusion

The cosmological model requires a new constant, the cosmological constant $\Lambda$, that needs to be included in our list of fundamental constants. Its value questions naturalness, hence motivating many lines of research.

Indeed, it may be replaced, as for any other constants, by a new degree of freedom accounting for a new component of dark energy. Note that since the cosmological constant is not connected with the standard model of particle physics, this does not imply a violation of the universality of free fall. The cosmological dynamics can be hope to constrain the dark energy equation of state in order to conclude whether it is varying or not. Cosmology also provides a link between the microphysics and macrophysics description of our universe, as foreseen by Dirac. The tests of fundamental constants can discriminate between various explanations of the acceleration of the universe and eventually indicate if the dark sector can be explained by a modification of General Relativity. Once a specific model is specified, cosmology also allows one to set stronger constraints since it relates observables that cannot be compared otherwise; we give several examples in Sect. 7. The cosmological model is also needed to get the look-back time/redshift relation as depicted on Fig. 6 for the standard $\Lambda$CDM model.

To finish, and as summarized on Fig. 5, the study of the stability of fundamental constants in term of a drift, modulation, oscillation, transient or fluctuation links to various phenomena of cosmological interest such as dark matter, dark energy, topological defects. This creates a strong bond between local experiments and the global description of the universe, making the fundamental constants a key system to unveil the dark sector.

As we shall discuss dimensionless constants, we refer to Narimani et al. (2012) and Rich (2013) for a rewriting of the cosmological observables in terms of dimensionless parameters, which is a useful insight.

## 3 Theories with varying constants

Since the variation of one constant would result on a violation of the LPI and of the UFF, theories with varying constants are strongly related to the study of gravity theories beyond General Relativity. This Section describes this link by first recalling the standard universal scalar-tensor theories in Sect. 3.1.1 mostly to clarify the connection between dynamical fields and constants as well as the role of the choice
<figure><img src="image_10.png" /><figcaption>**Fig. 5** The local studies of the variation of fundamental constants through either atomic clock experiments or bounds on the universality of free fall offer different connections with cosmology. Constraints on a *linear drift* open to dark energy models (see Sects. 7.1, 7.2 or 7.4 for concrete examples), while *transients* allow one to study topological defects (see Sects. 2.3.5 and 8.3.3 for a concrete example). The study of *oscillations* give access to light dark matter models (see Sects. 2.3.4 and 7.2 for concrete examples) while annual or diurnal *modulations* set bounds on the coupling to the gravitational potential, as described in Sect. 8.2. Finally, the so-far experimentally unexplored *fluctuations* link to cosmological fluctuations; see e.g., Sect. 8.3</figcaption></figure>

of a frame. This will be extended in Sect. 3.1.2 to discuss the simplest varying fine structure constant theory and discuss general properties of non universal coupling while giving a concrete example of the connection between the amplitude of the variation of $\alpha_{EM}$ and the Eötvös parameter. Since both are constrained to a high level in the Solar system, a new dynamical field shall be hidden which has led to the formulation of screening mechanisms described in Sect. 3.2. Section 3.3 will then highlight the role of the constants in the general search for any new physical degrees of freedom beyond the standard model, a standard prediction of many high-energy physics theories, summarized in Sect. 3.4, that are generically predicting dynamical constants and hence set to the challenge of explaining why the latter are actually observed to vary so little, i.e., they have to provide an efficient stabilisation mechanism to be viable.
## Early insights

As explained in the introduction, Dirac postulated that $G$ varies as the inverse of the cosmic time. Such an hypothesis is indeed not a theory since the evolution of $G$ with time is postulated and not derived from an equation of evolution<fnref n="7" /> consistent with the other field equations, that indeed shall take into account that $G$ is no more a constant. In particular, in a Lagrangian formulation one needs to take into account that $G$ is no more constant and thus to be considered as a field when the action is varied. This will indeed have consequences in all the field equations and provide a new equation of motion.

The first implementation of Dirac’s phenomenological idea into a field-theory framework (i.e., modifying Einstein’s gravity and incorporating non-gravitational forces and matter) was proposed by Jordan (1937). He realized that the constants have to become dynamical fields and proposed the action

$$S = \int \sqrt{-g} d^4 x \phi^\eta \left[ R - \xi \left( \frac{\nabla \phi}{\phi} \right)^2 - \frac{\phi}{2} F^2 \right], \quad (39)$$

$\eta$ and $\xi$ being two parameters. It follows that both $G$ and the fine-structure constant have been promoted to the status of dynamical fields.

Fierz (1956) realized that with such a Lagrangian, atomic spectra will be spacetime-dependent, and he proposed to fix $\eta$ to the value $-1$ to prevent such a spacetime dependence. This led to the definition of a one-parameter ($\xi$) class of scalar-tensor theories in which only $G$ is assumed to be a dynamical field. This was then further explored by Brans and Dicke (1961) (with the change of notation $\xi \to \omega_{BD}$). In this Jordan-Fierz-Brans-Dicke theory the gravitational constant is replaced by a scalar field, which can vary both in space and time. It follows that, for cosmological solutions, $G \propto t^{-n}$ with $n^{-1} = 2 + 3\omega_{BD}/2$. Thus, Einstein's gravity is recovered when $\omega_{BD} \to \infty$. This kind of theory was further generalized to obtain various functional dependencies for $G$ in the formalisation of scalar-tensor theories of gravitation (see, e.g., Damour and Esposito-Farèse 1992 or Will 1993).

The number of theoretical frameworks describing varying constants and gravity beyond General Relativity is too large to be exhaustively reviewed. Hence, we first present generalities on scalar-tensor theories and their extensions to include non universal couplings. This will bridge with the presentation of Sect. 2.2. Then, the connection with high-energy physics will be considered.

<footnote n="7">⁷ Note that the Dirac hypothesis can also be achieved by assuming that $e$ varies as $t^{1/2}$, as pointed out by Gamow (1967). Indeed this reflects a choice of units, either atomic or Planck units. However, there is a difference: assuming that only $G$ varies violates the strong equivalence principle while assuming a varying $e$ results in a theory violating the weak equivalence principle. It does not mean that we are detecting the variation of a dimensionful constant but simply that either $e^2/\hbar c$ or $Gm_e^2/\hbar c$ is varying. This shows that many implementations of the same idea are a priori possible.</footnote>
## 3.1 Introducing new fields: generalities

### 3.1.1 The example of scalar-tensor theories

Let us start to remind how the standard general relativistic framework can be extended to make $G$ dynamical on the example of scalar-tensor theories, in which gravity is mediated not only by a massless spin-2 graviton but also by a spin-0 scalar field that couples universally to matter fields (this ensures the universality of free fall).

#### General action

In the Jordan frame, using the notations by Esposito-Farèse and Polarski (2001), the action of the theory takes the form

$$S = \int \frac{d^4 x}{16\pi G_*} \sqrt{-g} [F(\varphi)R - g^{\mu\nu}Z(\varphi)\varphi_{,\mu}\varphi_{,\nu} - 2U(\varphi)] + S_{\text{matter}}[\psi; g_{\mu\nu}], \quad (40)$$

where $G_*$ is the bare gravitational constant. This action involves three arbitrary functions ($F$, $Z$ and $U$) but only two are physical since there is still the possibility to redefine the scalar field. The function $F$ needs to be positive to ensure that the graviton carries positive energy. $S_{\text{matter}}$ is the action of the matter fields that are minimally coupled to the metric $g_{\mu\nu}$. It follows that the lengths and times as measured by laboratory apparatus are defined in this frame. Besides, the UFF and the WEP hold but not the SEP.

Note that the normalisation of the scalar fields $\varphi$, as well as $\varphi_*$ in the Einstein frame formulation below, has been chosen to be dimensionless and we set the mass scale

$$M_* = \frac{1}{\sqrt{8\pi G_*}}$$

which would reduce to the Planck mass in General Relativity. We set

$$\phi = M_* \varphi, \quad \phi_* = M_* \varphi_* \qquad (41)$$

Among all the scalar-tensor theories, the Brans–Dicke theory is defined by the single free dimensionless parameter $\omega_{\text{BD}}$ as

$$F = \varphi, \quad Z = \frac{\omega_{\text{BD}}}{\varphi}, \quad U = 0. \qquad (42)$$

#### Field equations in the Jordan frame

The variation of the action (40) gives the following field equations
$$F(\varphi) \left( R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \right) = 8\pi G_* T_{\mu\nu} + Z(\varphi) \left[ \partial_{\mu} \varphi \partial_{\nu} \varphi - \frac{1}{2} g_{\mu\nu} (\partial_{\alpha} \varphi)^2 \right] \\ + \nabla_{\mu} \partial_{\nu} F(\varphi) - g_{\mu\nu} \Box F(\varphi) - g_{\mu\nu} U(\varphi) \quad (43)$$

$$2Z(\varphi) \Box \varphi = - \frac{dF}{d\varphi} R - \frac{dZ}{d\varphi} (\partial_{\alpha} \varphi)^2 + 2 \frac{dU}{d\varphi} \quad (44)$$

$$\nabla_{\mu} T_{\nu}^{\mu} = 0 \qquad (45)$$

where $T \equiv T_{\mu}^{\mu}$ is the trace of the matter energy-momentum tensor $T^{\mu\nu} \equiv (2/\sqrt{-g}) \times \delta S_{m}/\delta g_{\mu\nu}$. As expected (Ellis and Uzan 2005), we have one equation, (43), which reduces to the standard Einstein equation when $\varphi$ is constant and a new equation, (44), to describe the dynamics of the new degree of freedom while the conservation equation (45) of the matter fields is unchanged, as expected from the weak equivalence principle so that any test-particle will follow geodesic of the metric $g_{\mu\nu}$.

### Field equations in Einstein frame

It is useful to define an Einstein frame action through a conformal transformation of the metric

$$g_{\mu\nu}^{*} = F(\varphi)g_{\mu\nu} \iff g_{\mu\nu} = A^{2}(\varphi_{*})g_{\mu\nu}^{*}. \qquad (46)$$

In the following all quantities labeled by a star (*) refer to Einstein frame. Defining the field $\varphi_{*}$ and the two functions $A(\varphi_{*})$ and $V(\varphi_{*})$ (see, e.g., Esposito-Farèse and Polarski 2001) by

$$\begin{aligned} \left(\frac{d\varphi_{*}}{d\varphi}\right)^{2} &= \frac{3}{4}\left(\frac{d\ln F(\varphi)}{d\varphi}\right)^{2} + \frac{1}{2F(\varphi)}, \\ A(\varphi_{*}) &= F^{-1/2}(\varphi), \\ 2V(\varphi_{*}) &= U(\varphi)F^{-2}(\varphi), \end{aligned}$$

the action (40) takes the form

$$S = \frac{1}{16\pi G_*} \int d^4 x \sqrt{-g_*} [R_* - 2g_*^{\mu\nu} \partial_\mu \varphi_* \partial_\nu \varphi_* - 4V] + S_{\text{matter}}[A^2 g_*^{\mu\nu}; \psi]. \quad (47)$$

The kinetic terms have been diagonalized so that the spin-2 and spin-0 degrees of freedom of the theory are perturbations of $g_{\mu\nu}^{*}$ and $\varphi_{*}$ respectively. In this frame the field equations are

$$\begin{aligned} R_{\mu\nu}^{*} - \frac{1}{2} R^{*} g_{\mu\nu}^{*} = 8\pi G_{*} T_{\mu\nu}^{*} &+ 2\partial_{\mu}\varphi_{*}\partial_{\nu}\varphi_{*} - g_{\mu\nu}^{*}(g_{*}^{\alpha\beta}\partial_{\alpha}\varphi_{*}\partial_{\beta}\varphi_{*}) \\ &- 2V(\varphi)g_{\mu\nu}^{*}, \end{aligned} \quad (48)$$
$$ \Box_{*} \varphi_{*} = -4\pi G_{*} \alpha(\varphi_{*}) T_{*} + dV(\varphi)/d\varphi_{*}, \qquad (49) $$

$$ \nabla_{\mu}^{*} T_{*\nu}^{\mu} = \alpha(\varphi_{*}) T_{*} \hat{\alpha}_{\nu} \varphi_{*}, \qquad (50) $$

with

$$ \alpha \equiv d \ln A/d\varphi_{*} \quad \text{and} \quad \beta \equiv d\alpha/d\varphi_{*}. \qquad (51) $$

In this version, the Einstein equations are not modified. But, since the theory can now be seen as the theory in which all the mass are varying in the same way, there is a source term to the conservation equation. It follows that the motion of a test-body will enjoy a fifth force and will not be a geodesic of the metric $g_{\mu\nu}^*$, except for photons.

This shows that the same theory can be interpreted as a varying $G$ theory or a universally varying mass theory, but remember that whatever its form the important parameter is the dimensionless quantity $Gm^2/\hbar c$. Still, all other dimensionless parameters will remain constant, and in particular mass ratios and the fine structure constant, simply as a consequence of the universal coupling.

### Gravitational constant

The action (40) defines an effective gravitational constant $G_{\text{eff}} = G_{*}/F = G_{*}A^2$. This constant does not correspond to the gravitational constant effectively measured in a Cavendish experiment,

$$ G_{\text{cav}} = G_{*} A_{0}^{2} (1 + \alpha_{0}^{2}) = \frac{G_{*}}{F} \left( 1 + \frac{F_{\phi}^{2}}{2ZF + 3F_{\phi}^{2}} \right) \qquad (52) $$

where the first term, $G_{*}A_{0}^{2}$, corresponds to the exchange of a graviton while the second term $G_{*}A_{0}^{2}\alpha_{0}^{2}$, is related to the long range scalar force, a subscript 0 referring to the quantity evaluated today. The gravitational constant depends on the scalar field and is thus dynamical. In the case of the Brans-Dicke parameterisation (42) reduces to

$$ G_{\text{cav}} = \frac{G_{*} 2\omega_{\text{BD}} + 4}{\varphi (2\omega_{\text{BD}} + 3)}. \qquad (53) $$

### Local constraints

This illustrates the main features common to all models: (i) new dynamical fields appear (here a scalar field), (ii) some constant will depend on the value of this scalar field (here $G$ is a function of the scalar field). It follows that the Einstein equations will be modified and that there will exist a new equation dictating the propagation of the new degree of freedom.

In this particular example, the coupling of the scalar field is universal so that no violation of the universality of free fall is expected. The deviation from General Relativity can be quantified in terms of the post-Newtonian parameters, which can be expressed in terms of the values of $\alpha$ and $\beta$ today as
<figure><img src="image_11.png" /><figcaption>Fig. 6 Once a cosmological model is specified, one can deduce the look-back time-redshift relation, here for the standard ΛCDM model with the cosmological parameters of Table 30. We have also indicated some of the systems that will be studied in Sect. 5</figcaption></figure>

$$ \gamma^{\text{PPN}} - 1 = -\frac{2\alpha_0^2}{1 + \alpha_0^2}, \qquad \beta^{\text{PPN}} - 1 = \frac{1}{2} \frac{\beta_0 \alpha_0^2}{(1 + \alpha_0^2)^2}. \qquad (54) $$

These expressions are valid only if the field is light on the Solar system scales. If this is not the case then these conclusions may be changed (Khoury and Weltman 2004a). The Solar system constraints imply $\alpha_0$ to be very small, typically $\alpha_0^2 < 10^{-5}$ while $\beta_0$ can still be large. Binary pulsar observations (Damour and Esposito-Farèse 1998; Esposito-Farèse 2005) impose that $\beta_0 > -4.5$. The time variation of G is then related to $\alpha_0$, $\beta_0$ and the time variation of the scalar field today as

$$ \frac{\dot{G}_{\text{cav}}}{G_{\text{cav}}} = 2\alpha_0 \left( 1 + \frac{\beta_0}{1 + \alpha_0^2} \right) \dot{\varphi}_{*0}. \qquad (55) $$

This example shows that the variation of the constant and the deviation from General Relativity quantified in terms of the PPN parameters are of the same magnitude, because they are all driven by the same new scalar field.

The example of scalar-tensor theories is also very illustrative to show how deviation from General Relativity can be fairly large in the early universe while still being compatible with Solar system constraints. It relies on the attraction mechanism toward General Relativity (Damour and Nordtvedt 1993a, b).
**Fig. 7** Dynamics of a non-minimally scalar field arising from the Klein–Gordon equation (56). Since it couples to the trace of the stress-energy tensor, it generates an effective potential during the matter dominated era while it freezes to a constant value during the radiation era. The attraction toward the minimum of the coupling function $a(\varphi) = \ln A(\varphi) = \frac{1}{2}\beta\varphi^2$ is more efficient for large $\beta$. The oscillations during the radiation era are caused by the electron-antielectron, muon-antimuon etc. annihilations. Images reproduced with permission from Coc et al. (2006), copyright by APS

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

Cosmological evolution

Consider the simplest model of a massless dilaton with quadratic coupling ($\ln A = \frac{1}{2}\beta\varphi_*^2$). Note that the linear case corresponds to a Brans–Dicke theory with a fixed deviation from General Relativity. It follows that $\alpha_0 = \beta\varphi_{*0}$ and $\beta_0 = \beta$. As
long as $V = 0$, the Klein–Gordon equation can be rewritten in terms of the variable $p = \ln a$ as

$$ \frac{2}{3 - \varphi_{*}^{\prime 2}} \varphi_{*}^{\prime\prime} + (1 - w)\varphi_{*}^{\prime} = -\alpha(\varphi_{*})(1 - 3w), \qquad (56) $$

with $w$ the equation of state of the fluid dominating the cosmological dynamics. As emphasized by Damour and Nordtvedt (1993a), this is the equation of motion of a point particle with a velocity dependent inertial mass, $m(\varphi_*) = 2/(3 - \varphi_*^{\prime 2})$ evolving in a potential $\alpha(\varphi_*)(1 - 3w)$ and subject to a damping force, $-(1 - w)\varphi_*^{\prime}$. During the cosmological evolution the field is driven toward the minimum of the coupling function. If $\beta > 0$, it drives $\varphi_*$ toward 0, that is $\alpha \to 0$, so that the scalar-tensor theory becomes closer and closer to General Relativity. When $\beta < 0$, the theory is driven away from General Relativity and is likely to be incompatible with local tests unless $\varphi_*$ was initially unrealistically close from 0.

It follows that the deviation from General Relativity remains constant during the radiation era—up to threshold effects in the early universe (Damour and Pichon 1999; Coc et al. 2006) and quantum effects (Cembranos et al. 2009)—and the theory is then attracted toward General Relativity during the matter era. Note that it implies that postulating a linear or inverse variation of $G$ with cosmic time is actually not realistic in this class of models. Since the theory is fully defined, one can easily compute various cosmological observables: late time dynamics (Martin et al. 2006), CMB anisotropy (Riazuelo and Uzan 2002), weak lensing (Schind et al. 2005), BBN (Damour and Pichon 1999; Coc et al. 2006, 2009b) in a consistent way to confront them with data.

### Further WEP preserving theories

Scalar-tensor theories are a simple example of gravity theories beyond General Relativity. Many such theories have been formulated today and are of interest in particular to study the dark energy problem in cosmology (see e.g., Clifton et al. 2012 for a review). The Lovelock theorem (Lovelock 1971) states that the theory of gravity derived from an action principle. Provided (1) the theory is diffeomorphism-invariant, (2) the spacetime geometry is described by a Lorentzian $D$-dimensional manifold and the connection compatible with the metric is torsion and metricity free, i.e., it reduces to the Levi-Civita connection, (3) the action depends solely on the metric manifold, and (4) the field equations are of second order in derivatives of the metric then the action functional is of the form

$$ S[g] = \int d^D x \sqrt{-g} \sum_{i=0}^{(d-1)/2} \beta_i \mathcal{R}^{(i)} $$

with $\beta_i$ coupling constants and $\mathcal{R}^{(i)}$ the curvature invariants of order $i$ of the Riemann tensor,
$$ \mathcal{R}^{(i)} = \frac{k!}{2^k} \delta^{[\mu_1 \nu_1 \dots \mu_k \nu_k]} \prod_{j=1}^{i} R^{\alpha_j \beta_j} \mu_j \nu_j. $$

In $D=4$ dimensions, it reduces to General Relativity plus a cosmological constant together with the Gauss-Bonnet topological term, that does not modify the equations of motion,

$$ S[g] = \int d^D x \sqrt{-g} [\beta_0 + \beta_1 R + \beta_2 (R^2 - 4R_{\mu\nu} R^{\mu\nu} + R_{\mu\nu\alpha\beta} R^{\mu\nu\alpha\beta})] $$

This tells us that there are 5 ways in which GR can be modified, by relaxing these assumptions, respectively,

1. Add new field involved in mediating the gravitational force, i.e., new fields coupled to the metric tensor in the Einstein-Hilbert action; this is the situation we have considered here.

2. Work in more than $D=4$ for dimensions, such as, e.g., string theory, Kaluza-Klein theory, or the inclusion of a Gauss-Bonnet term in the action which only becomes relevant in higher dimensions; see Sect. 3.4 below.

3. Build a higher-order theory whose field equations contain greater than second-order derivatives, see e.g., the class of dhost theories (Langlois and Noui 2016).

4. Give up locality, add torsion or break diffeomorphism invariance.

5. Give up on the action principle; which we do not consider in this review.

While this provides a convenient guideline to classify gravity theories beyond General Relativity, this remains an idealized picture since indeed, one can relax several hypothesis simultaneously and that the same theory can be “read” differently given the way it is written mathematically. Figure 8 summarizes the different avenues to extend General Relativity and their relations. Scalar-tensor theories can be considered for many reasons as the simplest extensions of General Relativity. They are part of the larger class of Horndeski theories (Horndeski 1974), which are the most general 4-dimensional scalar-tensor theory whose Lagrangian leads to second-order equations of motion. To finish, note that a similar theorem relying on the Hamiltonian formulation was derived by Hojman et al. (1976). It suggests another route for extensions beyond General Relativity by considering Dirac’s hypersurface deformation algebra, leading to the idea of emergent spacetime equipped with a deformed general covariance so that modifications of gravity appear even if one does not break the gauge symmetry but instead deform it.

### 3.1.2 Making non-gravitational constants dynamical

#### Simplistic varying-$\alpha_{EM}$ model

Given the previous example, it seems a priori simple to cook up a theory that describes a varying fine-structure constant by coupling a scalar field to the electromagnetic Faraday tensor as
<figure><img src="image_15.png" /><figcaption>**Fig. 8** DHOST theories satisfies the WEP at the Lagrangian level (left). They include as subcases the Horndeski theories (among which Galileon and K-essence) and scalar-tensor theories with many models, depending on the choice of the potential and coupling function. They are also a subcase of all non-universal theories (right). While the DHOST theories satisfies the WEP at the microscopic level, note that they may violate if at the macroscopic level, as described in Sect. 3.2</figcaption></figure>

$$S = \frac{1}{16\pi G} \int \left[ R - 2(\partial_{\mu}\varphi)^2 - \frac{1}{4}B(\varphi)F_{\mu\nu}^2 \right] \sqrt{-g} d^4 x + S_{\text{matter}}[\psi; g_{\mu\nu}] \quad (57)$$

so that the fine-structure is given by $\alpha_{EM} = B^{-1}$. However, such a simple implementation may have dramatic implications on the universality of free fall because the contribution of the electromagnetic binding energy to the mass of any nucleus implies that it become $\varphi$-dependent. One can easily reproduce the derivation of the fifth force in Eqs. (22) to get

$$u^{\nu} \nabla_{\nu} u^{\mu} = F_{5}^{\mu} \equiv - \frac{\partial \ln m_{A}}{\partial \varphi} (g^{\beta\mu} + u^{\beta} u^{\mu}) \partial_{\beta} \varphi. \quad (58)$$

Hence the sensitivity to $\varphi$ is

$$\alpha_A(\varphi) \equiv \frac{\partial \ln m_A}{\partial \varphi} = \frac{\partial \ln m_A}{\partial \alpha_i} \frac{\partial \alpha_i}{\partial \varphi} = \sum_i \alpha_i f_{A,i} s_i(\varphi) \quad (59)$$

where the different sensitivity coefficients are $f_{A,i}$ and $s_i$ are defined in Eqs. (443) and (446) in Appendix B.2. Indeed with the choice of a scalar field with dimension of mass, one would have got

$$\alpha_A(\phi) \equiv M_* \frac{\partial \ln m_A}{\partial \phi}$$

making use of the relation (41).
Let us note before proceeding that in $D = 4$ dimensions such a simple coupling cannot be eliminated by a conformal rescaling $g_{\mu\nu} = A^2(\varphi)g_{\mu\nu}^*$ since

$$ \int B(\varphi) g^{\mu\rho} g^{\nu\rho} F_{\nu\sigma} F_{\rho\sigma} \sqrt{-g} d^4 x \rightarrow \int B(\varphi) A^{D-4}(\varphi) g_{*}^{\mu\rho} g_{*}^{\nu\rho} F_{\nu\sigma} F_{\rho\sigma} \sqrt{-g} d^4 x $$

so that the action is invariant in $D = 4$ dimensions, i.e., if included in a general scalar-tensor theories as discussed in Sect. 3.1.1 it will be present both in Einstein and Jordan frames so that the violation of UFF appears in both frames.

### Amplitude of the violation of the UFF

In the case at hand, the $\varphi$-dependent part of the mass arises from the electromagnetic binding energy that we can estimate thanks to the Bethe-Weizäcker formula (28). Using the expression (27) and assuming that the protons and neutrons all have the same mass $m_N$ and neglecting the $\varphi$-dependent contributions of the binding energies of the proton and neutron, it follows that

$$ \begin{aligned} \alpha_A = \frac{\partial \ln m_A}{\partial \varphi} &\approx -98.25 \alpha_{EM} \frac{Z(Z-1)}{A^{1/3}} \times \frac{1 \text{ MeV}}{Am_N} \times (\partial_{\varphi} \ln \alpha_{EM})_0 \\ &\sim 10^{-3} \frac{Z(Z-1)}{A^{4/3}} (\partial_{\varphi} \ln B)_0 \end{aligned} $$

since $\alpha_{EM} = 1/B$. Then, the level of the violation of the universality of free fall is expected to be of the level of $\eta_{12} \sim \alpha_{\oplus} |\alpha_1 - \alpha_2| \sim 10^{-6} X(A_1, Z_1; A_2, Z_2) \times (\partial_{\varphi} \ln B)_0^2$. Since the factor $X(A_1, Z_1; A_2, Z_2)$ typically ranges as $O(0.1 - 10)$, we deduce that $(\partial_{\varphi} \ln B)_0$ has to be very small then $O(10^{-5} - 10^{-4})$ for the Solar system constraints to be satisfied—see Fig. 26 for a concrete detailed example. It follows that today the scalar field has to be very close to the minimum of the coupling function $\ln B$. This led to the idea of the least coupling mechanism (Damour and Polyakov 1994a, b) discussed in Sect. 7.1. The constraints on such models are described in Sect. 7.4.

### 3.1.3 Summary

This example is indeed very simplistic because it only takes into account the effect of the electromagnetic binding energy (see Sect. 8.4) but it clearly illustrates that the amplitude of variation of the constants and of the violation of the universality free fall are linked and both determined by the $\varphi$-dependence of the mass, $m_A(\varphi)$ of the extended objects starting from the nuclei. It explicitly shows that one cannot couple a light field blindly to, e.g., the Faraday tensor, to make the fine-structure constant dynamics and that some mechanism for reconciling this variation with local constraints, and in particular the university of free fall, will be needed. Hence, the use of both local and cosmological constraints will be an asset to constrain all physical models.

#### Generic action

To finish, let us note that this hints that generic varying-constants theories will take the form
$$ S = \frac{1}{16\pi G} \int \left[ R - 2(\partial_{\mu}\varphi)^2 - 4V(\varphi) \right] \sqrt{-g} d^4 x \qquad (60) $$
$$ \sum_{\text{matterfields}} [\psi_i; A_i^2(\varphi)g_{\mu\nu}], $$

that would depend on a set of coupling functions $A_i(\varphi)$ that will determine the strength of the fifth force acting on each fundamental fields and from which one can compute the fifth force on a test-body once its chemical composition and mass are specified. It is naturally written in Einstein frame and contrary to the universal scalar-tensor theories, one cannot define a unique and universal Jordan frame. Hence, one shall be careful when referring to densities etc. since one would need to precise in which frame it is defined and how it is measured. Picking-up a Jordan frame for a component $i$, then all the other components $j \neq i$ will have coupling $A_i/A_j$ and then experience a fifth force. In particular models, see e.g., Sect. 7, one can construct an approximate “hadronic” frame to disentangle between composition-independent and composition-dependent effect.

## Generic weak and cosmological solutions of the Klein–Gordon equation

Before we turn to specific models, the action (60) has some generic properties, in particular concerning the profile et evolution of the scalar field since the Klein–Gordon equation (49) generalises to

$$ \Box \varphi = dV(\varphi)/d\varphi + \sum_{i} 4\pi G \alpha_{i}(\varphi)(\rho_{i} - 3P_{i}) \qquad (61) $$

using that $T_i = 3P_i - \rho_i$ so that one identifies the effective potential

$$ V_{\text{eff}} = V + 4\pi G \sum_{i} (\rho_{i} - 3P_{i}) \ln A_{i}(\varphi) \qquad (62) $$

The sum may be on different species that the sum in Eq. (60) since the latter concerns fundamental fields while the former is on macroscopic components, obtained as averaging, hence it could be baryonic matter, dark matter, radiation in cosmology, gas of different chemical compositions, etc.

First, let us consider this equation in a Minkowski spacetime. Assuming that $V$ has a minimum so that locally it reduces to a mass term, and that the couplings $\ln A_i$ are linear, it reduces to

$$ -\ddot{\varphi} + \Delta\varphi = \frac{dV_{\text{eff}}}{d\varphi} = m_{\varphi}^2\varphi + 4\pi G \sum_{i} \alpha_{i}(\rho_{i} - 3P_{i}). $$

This equation can be solved by Fourier decomposing $\varphi$ to get a solution of the homogeneous equation and then use the Yukawa Green function to determine a particular solution, hence
$$ \varphi = \int \frac{d^3\mathbf{k}}{(2\pi)^{3/2}} \varphi_{\mathbf{k}} e^{i(\mathbf{k}\cdot\mathbf{x} - \omega_{\mathbf{k}}\mathbf{t})} - \int d^3\mathbf{x}' G\alpha_i[\rho_i(\mathbf{x}') - 3P_i(\mathbf{x}')] \frac{e^{-m_{\varphi}|\mathbf{x}-\mathbf{x}'|}}{|\mathbf{x}-\mathbf{x}'|} $$

with the condition $\omega_k^2 = \mathbf{k}^2 + m_\varphi^2$ and $\varphi_{\mathbf{k}}^* = \varphi_{-\mathbf{k}}$ for a real-valued scalar field. In the case of a point mass, $\rho = M_A\delta(\mathbf{x})$, the static solution is indeed the usual Yukawa potential for a point particle

$$ \varphi_{\text{pp}}(\mathbf{x}) = -\frac{G\alpha_A M_A}{r} e^{-m_{\varphi}r} $$

while for a homogeneous sphere of density $\rho_A$ and radius $R_A$,

$$ \varphi_{\text{sphere}}(\mathbf{x}) = \varphi_{\text{pp}}(\mathbf{x})\Phi(m_{\varphi}R_A) $$

where the shape factor $\Phi$ is given in Eq. (18). This confirms the standard result of Eq. (20). The case of quadratic coupling was demonstrated by Hees et al. (2018) to “naturally” lead to screening or scalarization, which makes their study in the sSlar system particularly interesting. Such a solution will be useful to study the spatial variation of the constants in the Solar system.

Then, on cosmological scales, as long as the spacetime can be described by a Friedmann–Lemaître spacetime with scale factor $a$—see Appendix C for definitions—, the conservation equation (50) that holds for each component $i$ independently reduces to

$$ \dot{\rho}_i + 3H(\rho_i + P_i) = (\rho_i - 3P_i)\alpha_i(\varphi)\dot{\varphi}. $$

Assuming a constant equation of state $w_i = P_i/\rho_i$ and rewriting the r.h.s. as $\rho_i(1 - 3w_i)(\ln A_i)^*$, it and can be integrated as

$$ \rho_i(a) = \rho_{i0} a^{-3(1+w_i)} \left[ \frac{A_i(\varphi)}{A_i(0)} \right]^{4-3(1+w_i)}. $$

This shows that the cosmic dilution will be modified in a composition-dependent way. While radiation is not affected, baryonic matter and eventually dark matter will not redshift as $a^{-3}$, which could be interpreted effectively as a non-vanishing equation of state.

## 3.2 Hiding light degrees of freedom

While the introduction of a new light degree of freedom is an easy recipe to extend General Relativity and to make some constants dynamical, they face the problem to generate a too large violation of the UFF or of the tests of General Relativity in the Solar system, in particular when the new field is assumed to be light in order to be a dark energy candidate, typically $m \sim 10^{-33}$ eV for a scalar field (Ratra and Peebles 1988). Hence, the scalar field shall decouple from the matter sector, at least on Solar system scales. Several mechanisms have been considered to implement such a screening; see e.g., Khoury (2010).
### 3.2.1 Decoupling mechanisms

Two mechanisms fall in this category in which the coupling of the new degree of freedom becomes small in the Solar system.

#### Least coupling principle

First, as we have already foreseen in the previous paragraph 3.1.2, if the field potential and coupling function enjoy the same minima, then the scalar field is attracted toward its minimum during the cosmological evolution. Hence its coupling to standard matter becomes weak. This idea of the least coupling mechanism was first proposed by Damour and Nordtvedt (1993a, 1993b) for scalar-tensor theories and adapted by Gasperini et al. (2002) to quintessence models and then developed by Damour and Polyakov (1994a, 1994b) for non-universal couplings. It derives from the attraction of the scalar-tensor theory toward General Relativity during the cosmological evolution since the Klein-Gordon equation (49) is dictated by the effective potential

$$V_{\text{eff}}(\varphi_*) = V(\varphi_*) - 4\pi G_* T_* \ln A(\varphi_*). \quad (63)$$

As later detailed in Sect. 7.1 can be implemented by

$$A = \exp\left(\frac{1}{2}\beta\varphi^2\right) \qquad V = \frac{1}{2}m_{\varphi}\varphi^2 \quad (64)$$

as originally considered by Damour and Nordtvedt (1993a, 1993b) and then for the light-dilaton model by Damour and Polyakov (1994a, 1994b), or as

$$A = \exp(-\lambda\varphi) \qquad V \propto \exp(-\varphi) \quad (65)$$

as for the runaway dilaton (Gasperini et al. 2002; Damour et al. 2002b).

#### Symmetron mechanism

The symmetron mechanism (Hinterbichler and Khoury 2010; Hinterbichler et al. 2011) is a mechanism that makes the strength of the coupling environmentally dependent so that it becomes sufficiently small in regions of high density, or equivalently in the Newtonian limit of high Newtonian potential. It relies on the properties that the effective potential vacuum has an expectation value of the scalar field is nonzero in low-density environment and the fact that the $Z_2$-symmetry (i.e., $\varphi_* \leftrightarrow -\varphi_*$) is restored in high-density regions, so that the field have a zero vev in such regions and does not couple to matter. It can be implemented with the choice

$$A(\varphi_*) = 1 + \frac{\varphi_*^2}{2M^2} \qquad V(\varphi_*) = -\frac{1}{2}\mu^2\varphi_*^2 + \frac{1}{4}\lambda\varphi_*^4. \quad (66)$$

While the two mechanisms lead to a suppression of the fifth force in the Solar system today, they differ in the sense that the least coupling principle depends on the actual cosmological value of the scalar field, $\varphi_0$ and thus on the strength of the attraction toward General Relativity and its initial value at the end of inflation while
the symmetron mechanism depends on the local matter density. It follows that they have different phenomenologies for cosmology and the variation of the constants. Let us also stress that for non-universal couplings, the effective potential is given by Eq. (62) and the mechanism requires the potential and the coupling functions have the same minima.

### 3.2.2 Chameleon mechanism

#### Mechanism

Since the “local” mass of the scalar field is related to the range of the fifth force it mediates, another possibility to screen its effect is to make it heavy in high-density environment and light in low-density regions. The chameleon mechanism is an interesting model of this kind (Khoury and Weltman 2004a, b). It is defined as a scalar-tensor theory as described in Sect. 3.1.1 with the special feature that the potential V and the coupling function A do not share the same minimum. Hence, the scalar field acquires a density-dependent mass that controls the range of the associated fifth force, from small-range in high-density regions to long-range in low-density regions. This makes it not only a promising candidate to explain the accelerated expansion of the Universe, but also a potentially easily testable model. It is defined by the action (47) in the Einstein frame with a coupling constant associated to the matter fields,

$$A_i(\phi_*) = e^{\beta_i \phi_*/M_*} \qquad (67)$$

and a potential monotonically decreasing, tending to 0 at infinity with a null derivative. A prototypical potential is given by the Ratra-Peebles inverse power-law potential (Ratra and Peebles 1988) of energy scale A and exponent n

$$V(\phi_*) = A^4 \left( 1 + \frac{\Lambda^n}{\phi_*^n} \right), \qquad (68)$$

in the case of a universal coupling $\beta$. Otherwise the minimum will depend on both the density and chemical composition. Clearly, the minimum of the effective potential depends on the local density of matter as

$$\phi_{\min}(\rho) = \left( \frac{n M_* \Lambda^{n+4}}{\beta \rho} \right)^{1/n+1}.$$

In a medium of constant density, the field is expected to relax exponentially to this minimum on a typical distance of the order of its local Compton wavelength $\lambda_c(\rho) = 1/\sqrt{V_{eff}''([\phi_{min}(\rho)]).}$

#### Macroscopic violation of the UFF in a theory satisfying the WEP

To characterize the effect of the screening on the scalar fifth force, consider a spherical body with density $\rho$, radius $R$ and mass $M$ in a outside medium of density $\rho_{out}$. The scalar field will relax to the minimum of its potential $\phi_{in}$ inside if the
sphere only if the object is dense enough for the field's Compton wavelength to be much smaller than $R$, and to $\phi_{out}$ outside. One can define the skin thickness $\Delta R$ for $\phi$ to relax to $\phi_{in}$ and the thin-shell parameter

$$Q = 3 \frac{\Delta R}{R} \qquad (69)$$

One shall then consider two effects. First, outside the spherical body, the environmental screening induces a Yukawa suppression on a typical length $\lambda_c(\rho_{out})$ so that the fifth force will cancel for any test mass placed further than this distance. Hence the interaction between two bodies is Yukawa-suppressed in a high-density environment but may still be significant in low-density regions. Then if $Q \ll 1$, the field will settle to $\phi_{in}$ so that $\nabla\phi$ will vanish but on a thin portion of the body. It follows that only a thin shell contributes to the fifth force once integrated on the whole body so that it becomes marginal compared to that of the Newtonian gravity and is effectively hidden to experiments and the source is screened. This is in particular the reason for which planetary orbits do not show any deviation from pure Newtonian gravity: even if the chameleon's Compton wavelength in space becomes

<figure><img src="image_16.png" /><figcaption>**Fig. 9** Chameleon's parameter space for $n = 1$ adapted from Burrage and Sakstein (2018). It includes a purple line indicating the regimes in which the MICROSCOPE satellite is screened/unscreened (Pernot-Borràs et al. 2019) as well as the constraints derived from the stiffness measurement sessions (Pernot-Borràs et al. 2021)—the region excluded at $2\sigma$ is above the four lines described in the legend. The horizontal dotted line denotes the energy scale of dark energy. The experiments include atomic interferometry (Hamilton et al. 2015; Jaffe et al. 2017), Eötv-Wash group's torsion balance experiments (Upadhye 2012; Kapner et al. 2007), Casimir effect measurements (Brax et al. 2007; Decca et al. 2007), astrophysics tests (Jain et al. 2013a; Cabré et al. 2012; Vikram et al. 2018) and lensing (Wilcox et al. 2016), precision atomic tests (Brax and Burrage 2011; Jaeckel and Roy 2010) while the dashed-blue line corresponds to microsphere experiments (Rider et al. 2016) and the blue and red points to neutron interferometry (Lemmel et al. 2015; Li et al. 2016)</figcaption></figure>
larger than typical distances between planets, the chameleon is hidden as soon as planets are screened. On the opposite, if $Q \sim 1$, the chameleon field does not reach its minimum within objects that just appears as a perturbation for $\phi$ and then subject to a chameleon-induced fifth force. As a consequence, the amplitude of the fifth force between two spherical masses can be shown to be

$$F_{12}(r) = 2\beta^2 Q_1 Q_2 \frac{GM_1 M_2}{r^2}.$$

Hence two macroscopic bodies with different thin-shell parameters undergo different fifth forces. Even if the fifth force is universal for point particles, it will depend on the density and shapes of macroscopic bodies. This gives an example of violation of the UFF at the macroscopic level while it still holds at the microscopic level. The screening analysis of the MICROSCOPE satellite was a key issue to demonstrate that, contrary to what was expected, it was not designed to test chameleon models, even if it were in space; see Fig. 9.

## Constraints

We refer to Joyce et al. (2015), Burrage and Sakstein (2018), Brax et al. (2018) for reviews on the experimental tests of the chameleon, see Fig. 9. While most of them rely on fifth force effects, screened scalar-tensor models can be constrained by means of gravitational redshift measurements performed with atomic clocks (Brax et al. 2023c; Lévy and Uzan 2025) and unconstrained regions of the chameleon parameter space could be ruled out by future redshift experiments with some technical issues to be resolved such as the influence of the nucleus on the chameleon field at the electron cloud. The analysis of the chameleon in space experiments such as MICROSCOPE requires to propagate the field inside the instrument to take into account the screening. Two numerical tools (Briddon et al. 2021; Lévy et al. 2022) are now available to compute the chameleon field profile and the fifth force it generates for realistic geometries. All this limits the constraints that can be obtained (Pernot-Borràs et al. 2020). Note that at low orbits, one has to model the shape of the Earth (Burrage et al. 2015). Still, it was demonstrated that a constraint can be derived from the modelization of the electrostatic stiffness (Pernot-Borràs et al. 2021; Bergé et al. 2022).

## Dependence of the constants on the mass density of the environment

Indeed, the chameleon mechanism can easily be implemented with non-universal couplings and for any of the model described in Sect. 7. They will induce a dependence of the fundamental constant on the energy density $\rho$ of the environment due to the shift of the field (Olive and Pospelov 2008). Brax (2014) studied the effect of screening to the detectability of the variation of the constants in particular in astrophysical environment showing that if the dipolar variation of $\alpha_{EM}$—see Sect. 8.3.2 below,—were to be confirmed this would rule out screened modified gravity models. The cosmological variation of $\alpha_{EM}$ in chameleon models was investigated in Brax et al. (2004, 2010b). Models based on the Lagrangian (364) and exhibiting the chameleon mechanism were investigated by Olive and Pospelov
(2008). Chameleon models coupled to $F^2$—see Sect. 7.4—were discussed by Brax et al. (2011) and constrained with Solar physics (O’Shea et al. 2024). The axion-homeopathy model was extended to an axio-ahameleon mechanism by Brax et al. (2024) involving two light scalar fields, an axion and a Brans-Dicke dilaton.

In particular, interstellar absorbing clouds have typical densities $10^{-10}$ smaller than in terrestrial environments (Levshakov et al. 2010b) so that the scalar field driving the dynamics of the fundamental constant will be much lighter in space will screened for Earth measurements. Besides, for all models enjoying screening, two absorbing regions at the same redshift would show different values of the fundamental constants if they lie in screened and unscreened regions respectively. This suggests new ways of analyzing data where a tomographic description of the Universe mapping screened and unscreened regions would be correlated to the measured variations of constants (Cabre et al. 2012).

The study of molecular CO, CH, NH$_3$ and CH$_3$OH absorption spectra, discussed in details in Sect. 5.7, have actually allowed to study fundamental constants in the different regions of the Milky Way, opening the way to set limits on matter-density dependence of fundamental constants in different regions of our Galaxy with different densities. The possible shift in the value of $\mu$ in the Milky Way described in Sect. 8.2.3 was related by Levshakov et al. (2008a, 2010a, 2010b) to the model of Olive and Pospelov (2008) to conclude that such a shift was compatible with a chameleon-like mechanism. Similarly, Truppe et al. (2013) constrained the variation of $\alpha_{EM}$ and $\mu$ between high and low-density environments of the Earth and the interstellar medium thanks to CH and OH, in the Milky way.

A proper and in depth analysis of the dependence of the constant on the local energy density still remains to be performed.

### 3.2.3 Environment-dependent dilaton model

Following Brax et al. (2018), the quadratic dilaton model was combined (Brax et al. 2010a) with a runaway potential,

$$A(\varphi) = e^{\frac{1}{2}\beta\varphi^2} \sim 1 + \frac{1}{2}\beta\varphi^2 \quad V(\varphi) = V_0e^{-\lambda\varphi}, \qquad (70)$$

to make the dilaton dependent on the trace of the matter stress-energy tensor, i.e., on the local energy density $\rho$ for non-relativistic matter. The effective potential (62), $V_{\text{eff}} = V_0e^{-\lambda\varphi} + 2\pi G\rho\beta\varphi^2$ has a minimum explicitly given by

$$\varphi_{\min} = \lambda^{-1} W \left( \frac{\lambda^2 V_0}{2\pi G \beta \rho} \right)$$

where $W$ stands for the Lambert function, i.e., the inverse function of $xe^x$. As for the chameleon, the field is screened in high-density environments. The cosmological dynamics was studied in (Brax et al. 2022) and the effect on Lunar laser ranging and laboratory experiments detailed in Fischer et al. (2024a, 2024b). The mechanism is similar to the chameleon and can be combined with composition-dependent couplings, even though it has not been investigated in the literature so far.
<table><caption>**Table 4** Main decoupling and screening mechanisms in scalar-tensor gravity</caption><thead><tr><th>Model</th><th>V</th><th>A</th><th>Environment dependence</th></tr></thead><tbody><tr><td>Light dilaton</td><td><sup>1</sup>&frasl;<sub>2</sub>m&phi;<sup>2</sup></td><td><sup>1</sup>&frasl;<sub>2</sub>&beta;&phi;<sup>2</sup></td><td>No</td></tr><tr><td>Runaway dilaton</td><td>e<sup>-&phi;</sup></td><td>e<sup>-&lambda;&phi;</sup></td><td>No</td></tr><tr><td>Environment-dep. dilaton</td><td>e<sup>-&lambda;&phi;</sup></td><td>1 + <sup>1</sup>&frasl;<sub>2</sub>&beta;&phi;<sup>2</sup></td><td>Yes</td></tr><tr><td>Chameleon</td><td>e<sup>&beta;&phi;</sup></td><td>A<sup>4</sup>(1 + <sup>&lambda;<sup>n</sup></sup>&frasl;<sub>&phi;<sup>n</sup></sub>)</td><td>Yes</td></tr><tr><td>Symmetron</td><td>-<sup>1</sup>&frasl;<sub>2</sub>&mu;<sup>2</sup>&phi;<sup>2</sup> + <sup>1</sup>&frasl;<sub>4</sub>&lambda;&phi;<sup>4</sup></td><td>1 + <sup>1</sup>&frasl;<sub>2</sub>&beta;&phi;<sup>2</sup></td><td>Yes</td></tr></tbody></table>

### 3.2.4 Conclusion

The interaction between light scalar fields and matter are generically highly constrained by Solar system tests unless, as discussed their couplings to ordinary matter are much suppressed relative to gravitational strength. This is a major hindrance to construct realistic models of light dilatons coupled to matter. We have described four mechanism of decoupling or screening that allow these models to evade local constraints. They are summarized in Table 4. Even though they have been discussed within composition-independent scalar-tensor theories, they are easily adapted for all the models described in Sect. 7.

## 3.3 Searching for new degrees of freedom

This section has illustrated the variety of the extensions of standard framework. As soon as a new degree of freedom is introduced it modifies the equation of motion and consistently a new equation of motion for this degree of freedom follows. Hence the set of equations remains consistent and let no freedom to specify an ad hoc law of variation, i.e., an arbitrary function of time or redshift. This also means that the constants have to be promoted to a dynamical field at the level of the action and indeed not in field equations derived assuming these parameters were constants.

The new degree of freedom can act as *matter* if it is not coupled to the standard matter fields. As such, it only modifies the solution of the Einstein field equations through its stress-energy tensor. This implies that the relative abundance of this new species is arbitrary. When coupled to standard matter, it mediates a *fifth force* that can be long-range if the field is light enough. Among those theories, a large class of models involve universal coupling so that the WEP still holds at the microscopic level, but not necessarily for macroscopic objects; see e.g., chameleon models. When the field is non-universally coupled, the phenomenology exhibits a violation of the UFF and a variation of the fundamental parameters. Both can be tested and are well-constrained, as we shall see, in the Solar system. For this reason, viable theories usually incorporate a screening mechanism. The new degree of freedom can remain a negligible part of the total energy budget but act through the force it generates.
The existence of these new degrees of freedom are strongly motivated by the hypothesis of the existence of a dark matter and dark energy components required by the cosmological model. It is important to remind that their interpretation as *matter* or *new force* depends on the fact that they modify the Friedmann equations through their energy, which is then non-negligible, or by modifying the gravitational interaction, in which case their self-energy may remain negligible. The question is then to prove their existence, characterize them and their interactions, see e.g., Fig. 10. This is a large research program that bridges particle physics, gravity and cosmology.

Many tools can be conjointly used: the study of the expansion of the universe and the growth of its large-scale structure to constrain the dark energy equation of state and the properties of dark matter, the use of the large-scale structure as first proposed in Uzan and Bernardeau (2001) to test the Einstein field equation, in particular thanks to weak lensing, the test of the distance duality relation and many laboratory to constrain a dark matter component. Fundamental constants play a key role in this program and complement the long list of tests of General Relativity in the Solar system (Will 2014) and on astrophysical scales (Uzan 2003a, 2010c).

<figure><img src="image_17.png" /><figcaption>**Fig. 10** The discussion on physics the standard model can be addressed by trying to unveil the existence of a new dynamical degrees of freedom, their nature and couplings to the standard model fields. While it can be considered as a matter component if it remains universally coupled, it mediates an interaction otherwise. This is in particular the case for non-universal coupling that can be revealed by the variation of the fundamental constants. After Uzan et al. (2004)</figcaption></figure>
## 3.4 High-energy theories and varying constants

Many high-energy theories involve scalar fields that couple non-universally do the standard model. It seems a generic feature of higher-dimensional gravity theories and of string theories to lead to low-energy effective theories with dynamical constants. Hence, such theories have to face the question: “why are the low-energy constants so stable?” which sets a challenge for them to exhibit a stabilisation mechanism.

### 3.4.1 Higher-dimension and Kaluza-Klein theory

Extra-scalar field with non-minimal couplings naturally appear when compactifying a higher-dimensional theory. As an example, let us consider the 5-dimensional Einstein-Hilbert action (Peter and Uzan 2009, Chapt. 13)

$$S = \frac{1}{12\pi^2 G_5} \int \bar{R} \sqrt{-\bar{g}} d^5 x.$$

Decomposing the 5-dimensional metric $\bar{g}_{AB}$ as

$$\bar{g}_{AB} = \begin{pmatrix} g_{\mu\nu} + \frac{A_{\mu}A_{\nu}}{M^2} \phi^2 & \frac{A_{\mu}}{M} \phi^2 \\ \frac{A_{\nu}}{M} \phi^2 & \phi^2 \end{pmatrix},$$

where $M$ is a mass scale, we obtain

$$S = \frac{1}{16\pi G_*} \int \left( R - \frac{\phi^2}{4M^2} F^2 \right) \phi \sqrt{-g} d^4 x, \qquad (71)$$

where the 4-dimensional gravitational constant is $G_* = 3\pi G_5/4 \int dy$. The scalar field couples explicitly to the kinetic term of the vector field and cannot be eliminated by a redefinition of the metric: again, this is the well-known conformal invariance of electromagnetism in four dimensions discussed in Sect. 3.1.2 in which we explicitly showed that such a term induces a variation of the fine-structure constant as well as a violation of the universality of free-fall. Such dependencies of the masses and couplings are generic for higher-dimensional theories and in particular string theory. It is actually one of the definitive predictions for string theory that there exists a dilaton, that couples directly to matter (Taylor and Veneziano 1988) and whose vacuum expectation value determines the string coupling constants (Witten 1984).

In the models by Kaluza (1921) and Klein (1926) the 5-dimensional spacetime was compactified assuming a single spatial extra-dimension with topology $S^1$ and radius $R_{KK}$. It follows that any field $\chi(x^\mu, y)$ can be Fourier transformed along the compact dimension (with coordinate $y$), so that, from a 4-dimensional point of view, it gives rise to a tower of of fields $\chi^{(n)}(x^\mu)$ of mass $m_n = nR_{KK}$. At energies small compared to $R_{KK}^{-1}$ only the $y$-independent part of the field remains and the physics looks 4-dimensional.
Assuming that the action (71) corresponds to the Jordan frame action, as the coupling $\phi R$ may suggest, the gravitational constant and the Yang–Mills coupling associated with the vector field $A^\mu$ must scale as

$$G \propto \phi^{-1}, \quad g_{YM}^{-2} \propto \phi^2/G \propto \phi^3. \qquad (72)$$

As explained in Eq. (52), $G$ is not the gravitational constant that would be measured in a Cavendish experiment that is given by $G_{cav} \propto G_* \phi^{-1} (1 + \frac{1}{2\phi^3})$. This was generalized to the case of $D$ extra-dimensions (Cremmer and Scherk 1977) to get

$$G \propto \phi^{-D}, \quad \alpha_i(m_{KK}) = K_i(D)G\phi^{-2} \qquad (73)$$

where the constants $K_i$ depends only on the dimension and topology of the compact space (Weinberg 1983a) so that the only fundamental constant of the theory is the mass scale $M_{4+D}$ entering the $4+D$-dimensional theory. A theory on $M_4 \times M_D$ where $M_D$ is a $D$-dimensional compact space generates a low-energy quantum field theory of the Yang–Mills type related to the isometries of $M_D$. For instance Witten (1981) showed that for $D=7$, it can accommodate the Yang–Mills group $SU(3) \times SU(2) \times U(1)$. The two main problems of these theories are that (1) one cannot construct chiral fermions in four dimensions by compactification on a smooth manifold with such a procedure and (2) that gauge theories in five dimensions or more are not renormalizable.

In such a framework, the variation of the gauge couplings and of the gravitational constant arises from the variation of the size of the extra dimensions so that one can derives stronger constraints that by assuming independent variation, but at the expense of being more model-dependent. Let us mention the works by Marciano (1984) and Wu and Wang (1986) in which the structure constants at lower energy are obtained by the renormalization group, and the work by Veneziano (2002) for a toy model in $D \ge 4$ dimensions, endowed with an invariant UV cut-off $\Lambda$, and containing a large number $N$ of non-self-interacting matter species.

Kolb et al. (1986) used the scalings (73) to constrain the time variation of the radius of the extra dimensions during primordial nucleosynthesis to conclude that $|\Delta R_{KK}/R_{KK}| < 1\%$. Barrow (1987) took the effects of the variation of $\alpha_S \propto R_{KK}^{-2}$ and deduced from the helium-4 abundance that $|\Delta R_{KK}/R_{KK}| < 0.7\%$ and $|\Delta R_{KK}/R_{KK}| < 1.1\%$ respectively for $D=2$ and $D=7$ Kaluza–Klein theory and that $|\Delta R_{KK}/R_{KK}| < 3.4 \times 10^{-10}$ from the Oklo data. An analysis of most cosmological data (BBN, CMB, quasar etc.) assuming that the extra dimension scales as $R_0(1 + \Delta t^{-3/4})$ and $R_0[1 + \Delta](1 - \cos \omega(t - t_0))$ concluded that $\Lambda$ has to be smaller than $10^{-16}$ and $10^{-8}$ respectively (Landau and Vucetich 2002), while Li and Chu (2006a) assumed that gauge fields and matter fields can propagate in the bulk, that is in the extra dimensions. Lorén-Aguilar et al. (2003) evaluated the effect of such a couple variation of $G$ and the structures constants on distant supernova data, concluding that a variation similar to the one reported by Webb et al. (2001) would make the distant supernovae brighter, hence having an effect opposite to the one of the cosmological constant. We shall stress that Tahara et al. (2020) proposed a
model to freeze the extra-dimensions thanks to hight-order curvature terms in the Lovelock theory.

### 3.4.2 String theory

There exist five anomaly-free, supersymmetric perturbative string theories respectively known as type I, type IIA, type IIB, SO(32) heterotic and $E_8 \times E_8$ heterotic theories; see, e.g., Polchinski (1998).

One of the definitive predictions of these theories is the existence of a scalar field, the dilaton, that couples directly to matter (Taylor and Veneziano 1988) and whose vacuum expectation value determines the string coupling constant (Witten 1984). There are two other excitations that are common to all perturbative string theories, a rank two symmetric tensor (the graviton) $g_{\mu\nu}$ and a rank two antisymmetric tensor $B_{\mu\nu}$. The field content then differs from one theory to another.

It follows that the 4-dimensional couplings are determined in terms of a string scale and various dynamical fields (dilaton, volume of compact space, ...). When the dilaton is massless, we expect three effects:

1. a scalar admixture of a scalar component inducing deviations from General Relativity in gravitational effects,

2. a variation of the couplings, and

3. a violation of the weak equivalence principle.

Our purpose is to show how the 4-dimensional couplings are related to the string mass scale, to the dilaton and the structure of the extra-dimensions mainly on the example of heterotic theories.

To be more specific, let us consider an example. The two heterotic theories originate from the fact that left- and right-moving modes of a closed string are independent. This reduces the number of supersymmetry to $N=1$ and the quantization of the left-moving modes imposes that the gauge group is either SO(32) or $E_8 \times E_8$ depending on the fermionic boundary conditions. The effective tree-level action is (see, e.g., Gross and Sloan 1987)

$$S_H = \int d^{10}\mathbf{x} \sqrt{-g_{10}} e^{-2\Phi} \left[ M_H^8 \left\{ R_{10} + 4\Box\Phi - 4(\nabla\Phi)^2 \right\} - \frac{M_H^6}{4} F_{AB}F^{AB} + \dots \right]. \quad (74)$$

When compactified on a 6-dimensional Calabi-Yau space, the effective 4-dimensional action takes the form

$$S_H = \int d^4 \mathbf{x} \sqrt{-g_4} \phi \left[ M_H^8 \left\{ R_4 + \left(\frac{\nabla \phi}{\phi}\right)^2 - \frac{1}{6} \left(\frac{\nabla V_6}{V_6}\right)^2 \right\} - \frac{M_H^6}{4} F^2 \right] + \dots \quad (75)$$

where $\phi \equiv V_6 e^{-2\Phi}$ couples identically to the Einstein and Yang-Mills terms. It follows that
$$M_4^2 = M_H^8 \phi, \quad g_{YM}^{-2} = M_H^6 \phi \qquad (76)$$

at tree-level. Note that to reach this conclusion, one has to assume that the matter fields—in the ‘dots’ of Eq. (75)—are minimally coupled to $g_4$; see, e.g., Maeda (1988).

The strongly coupled SO(32) heterotic string theory is equivalent to the weakly coupled type I string theory. Type I superstring admits open strings, the boundary conditions of which divide the number of supersymmetries by two. It follows that the tree-level effective bosonic action is $N = 1$, $D = 10$ supergravity, which takes the form, in the string frame,

$$S_I = \int d^{10}\mathbf{x} \sqrt{-g_{10}} M_I^6 e^{-\phi} \left[ e^{-\phi} M_I^2 R_{10} - \frac{F^2}{4} + \dots \right] \qquad (77)$$

where the dots contains terms describing the dynamics of the dilaton, fermions and other form fields. At variance with (74), the field $\Phi$ couples differently to the gravitational and Yang–Mills terms because the graviton and Yang–Mills fields are respectively excitation of close and open strings. It follows that $M_I$ can be lowered even to the weak scale by simply having $\exp \Phi$ small enough. Type I theories require D9-branes for consistency. When $V_6$ is small, one can use T-duality (to render $V_6$ large, which allows one to use a quantum field theory approach) and turn the D9-brane into a D3-brane so that

$$S_I = \int d^{10}\mathbf{x} \sqrt{-g_{10}} e^{-2\Phi} M_I^8 R_{10} - \int d^4\mathbf{x} \sqrt{-g_4} e^{-\Phi} \frac{1}{4} F^2 + \dots \qquad (78)$$

where the second term describes the Yang–Mills fields localized on the D3-brane. It follows that

$$M_4^2 = e^{-2\Phi} V_6 M_I^8, \quad g_{YM}^{-2} = e^{-\Phi} \qquad (79)$$

at tree-level. If one compactifies the D9-brane on a 6-dimensional orbifold instead of a 6-torus, and if the brane is localized at an orbifold fixed point, then gauge fields couple to fields $M_i$ living only at these orbifold fixed points with a (calculable) tree-level coupling $c_i$ so that

$$M_4^2 = e^{-2\Phi} V_6 M_I^8, \quad g_{YM}^{-2} = e^{-\Phi} + c_i M_i. \qquad (80)$$

The coupling to the field $c_i$ is a priori non universal. At strong coupling, the 10-dimensional $E_8 \times E_8$ heterotic theory becomes M-theory on $R^{10} \times S^1/Z_2$ (Hořava and Witten 1996). The gravitational field propagates in the 11-dimensional space while the gauge fields are localized on two 10-dimensional branes. At one-loop, one can derive the couplings by including Kaluza–Klein excitations to get (Dudas 2000)

$$g_{YM}^{-2} = M_H^6 \phi - \frac{b_d}{2} (RM_H)^2 + \dots \qquad (81)$$

when the volume is large compared to the mass scale and in that case the coupling is no more universal. Otherwise, one would get a more complicated function.