# Exploration of new experimental strategies for the detection of ultralight dark matter : laboratory searches on ground and in space

## Soutenue par

Jordan Gué

Le 01 Octobre 2024

## École doctorale n°127

Ecole Doctorale
Astronomie et Astrophysique
d'Ile de France (AAIF)

## Spécialité

Astronomie et Astrophysique

## Composition du jury :

**Franck Pereira Dos Santos**
CNRS Research director
SYRTE, Observatoire de Paris

Président

**Diego Blas**
Professor
IFAE, Universitat Autònoma de Barcelona

Rapporteur

**Hartmut Grote**
Professor
Cardiff University

Rapporteur

**Clare Burrage**
Professor
University of Nottingham

Examinatrice

**Antoine Petiteau**
CEA Researcher
IRFU, CEA Saclay

Examinateur

**Marianna Safronova**
Professor
University of Delaware

Examinatrice

**Peter Wolf**
CNRS Research director
SYRTE, Observatoire de Paris

Directeur de thèse

**Aurélien Hees**
CNRS Researcher
SYRTE, Observatoire de Paris

Co-encadrant de thèse

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<figure><img src="image_2.png" /></figure>
# Contents

I General context and framework 4

1 Special relativity 6

2 Quantum mechanics 8

3 General relativity 10

3.1 Equivalence principle 10

3.2 Einstein field equations 11

3.3 Experimental successes 11

3.4 Limits 12

4 The Standard Model of particle physics 14

4.1 Symmetries 14

4.2 Gauge groups 14

4.3 Higgs boson 16

4.4 Matter content 16

4.5 Lagrangian 17

4.6 Experimental successes and limits 18

5 Dark matter 19

5.1 Some smoking guns of existence of dark matter 19

5.2 Dark matter candidates 21

5.3 Ultralight dark matter intrinsic characteristics 23

6 Alternatives to dark matter 27

II Phenomenology of some ultralight dark matter candidates 32

7 Cosmological evolution 33

7.1 Action 33

7.2 Klein Gordon equation in an expanding Universe 34

7.3 Oscillating field as cold dark matter 35

7.4 Expressing the dark matter field in various reference frames 39

8 Couplings to Standard Model fields 41

8.1 Scalar field 41

8.2 Pseudo-scalar field 46

8.3 Vector field 56
III Electromagnetic cavities probes 64

9 Search for axions with an optical cavity and an optical fiber 65

9.1 Birefringence and oscillation of the length of the cavity 65

9.2 Phase shift at the output of the cavity 67

9.3 Phase shift along an optical fiber 67

10 Search for dark photons in a microwave cavity with Rydberg atoms 69

10.1 Contributions from dark photons and applied electric field inside the cavity 70

10.2 Total electric field squared 73

10.3 Measurement with atoms through the quadratic Stark effect 75

IV Dish antennas detectors 79

11 Search for dark photons using dish antennas 80

11.1 Electromagnetic emission from the dish 80

11.2 Propagation of the field from the dish to an antenna 82

11.3 Detection of the electric field with a horn antenna 85

V Search through tests of the equivalence principle 92

12 Classical tests of the equivalence principle 93

12.1 Experiments testing the equivalence principle 93

12.2 Acceleration difference induced by oscillating rest mass 94

13 Atom interferometry as a quantum test of the equivalence principle 96

13.1 Two-photon transitions : $\pi/2 - \pi - \pi/2$ setups 96

13.2 Are the calculations doable in the laboratory frame ? 108

13.3 Gradiometers 110

13.4 Single Photon Isotope Differential interferometer 112

VI Effects on gravitational waves detectors 118

14 Gravitational waves and LISA 119

14.1 Gravitational waves physics basics 119

14.2 LISA, the first european space-based gravitational waves detector 121

14.3 LISA's expected performance 123

15 Search for oscillations of rest mass in LISA 125

15.1 Signatures of dark matter and gravitational waves on LISA arms 125

15.2 Bayesian inference to discriminate between dark matter and gravitational waves 132

15.3 Realistic limit on the sensitivity of LISA to ultralight dark matter couplings 146

15.4 Conclusion 151
<table><tr><td>16</td><td>Search for vacuum birefringence and dichroism in LISA</td><td>152</td></tr><tr><td>16.1</td><td>Birefringence</td><td>154</td></tr><tr><td>16.2</td><td>Dichroism</td><td>160</td></tr><tr><td>16.3</td><td>Conclusion</td><td>160</td></tr></table>

## VII Sensitivity estimates 164

<table><tr><td>17</td><td>Experimental considerations</td><td>165</td></tr><tr><td>17.1</td><td>DAMNED and optical fibers</td><td>165</td></tr><tr><td>17.2</td><td>Rydberg atoms in a microwave cavity</td><td>166</td></tr><tr><td>17.3</td><td>SHUKET</td><td>172</td></tr><tr><td>17.4</td><td>MICROSCOPE</td><td>173</td></tr><tr><td>17.5</td><td>Atom interferometers</td><td>174</td></tr><tr><td>17.6</td><td>LISA</td><td>177</td></tr></table>

<table><tr><td>18</td><td>Sensitivity results on ultralight dark matter couplings</td><td>179</td></tr><tr><td>18.1</td><td>Dilaton-SM couplings $d'_s$</td><td>179</td></tr><tr><td>18.2</td><td>Axion-gluon coupling $f_a^{-1}$</td><td>184</td></tr><tr><td>18.3</td><td>Axion-photon coupling $g_{ay}$</td><td>187</td></tr><tr><td>18.4</td><td>Dark photon-photon coupling $\chi$</td><td>189</td></tr><tr><td>18.5</td><td>Discussion</td><td>199</td></tr></table>

## VIII Conclusion 207

## IX Appendix 212

<table><tr><td>A</td><td>Electric fields inside a cavity</td><td>213</td></tr><tr><td>A.1</td><td>Phase shift induced by vacuum birefringence</td><td>213</td></tr><tr><td>A.2</td><td>Amplitude of the square of the total electric field inside a cavity</td><td>215</td></tr><tr><td>A.3</td><td>Amplitude fluctuation inside a cavity</td><td>215</td></tr></table>

## B Motion of atomic wavepackets inside an atom interferometer 218

<table><tr><td>C</td><td>Projection of galactic velocity onto sensitive axes of various experiments</td><td>220</td></tr><tr><td>C.1</td><td>Atom interferometers</td><td>220</td></tr><tr><td>C.2</td><td>MICROSCOPE</td><td>221</td></tr></table>

## D Fast likelihood for scalar ultralight dark matter 223

## E Propagation of the electric field from the dish to the fictional plane in SHUKET 226
# Part I

## General context and framework
At the end of the nineteenth century, classical physics, which includes among others, electromagnetism, statistical mechanics, thermodynamics and classical mechanics, was able to describe accurately many different physical systems, through, for example, Maxwell's equations and Newton's classical laws of motion.

However, classical physics was lacking explanations for various phenomena, such as Mercury's perihelion, black-body radiation, internal structure of atoms and people were having trouble finding the so-called aether, a postulated medium for the propagation of electromagnetic waves.

In the twentieth century, a complete change of paradigm appeared, as new fundamental theories emerged, which are the basis of what is known today as modern physics: relativity and quantum mechanics. These theories were able to address many of the unanswered questions from classical physics.

Based on that, two major fundamental physics theories arose, which respectively describe the infinitely big and the infinitely small: general relativity (GR) and the Standard Model of particle physics (SM). They are considered as the most rigorously and extensively confirmed theories of all time through experiments. As imperfect physics theories, it still remains open questions, one of which is the inability to reconcile the two theories into a single *theory of everything*. Another major issue arising from these theories, and relevant for this thesis, is the nature of dark matter, a hypothetical form of matter, abundantly present in the universe, which we can only notice through its gravitational effect. In this opening part, we will first introduce the theoretical framework, necessary for the understanding of the dark matter puzzle. After reviewing the key concepts related to dark matter relevant for this thesis, we will focus on one specific class of solutions as for the microscopic nature of dark matter, ultralight dark matter. Afterwards, we will discuss the main aspects that characterize ultralight dark matter, which will work as a basis of the next chapters of this thesis.
# Chapter 1

## Special relativity

Special relativity is a pillar of modern physics as it is the first theory of space and time, as two parameters of a deeper mathematical object : a four-dimensional spacetime. The theory, developed by Einstein in 1905 [1], addressed the inability to reconcile Maxwell's electromagnetism with classical mechanics. It is based on two postulates : the principle of relativity and the invariance of the speed of light. The former states that the laws of physics are the same in every inertial reference frame, i.e with zero net acceleration. The latter states that the speed of light in vacuum is invariant and always equals $c$ in all global inertial reference frames. From these two postulates, one can show that the speed of light $c$ is the maximum velocity allowed by special relativity, from the principle of causality.

The main idea behind this theory is that time is no longer an absolute parameter but depends on the velocity of the reference frame. Before the development of special relativity, if one wanted to transform time and space coordinate from one reference frame $R$ to another frame $R'$ where the latter travels with constant velocity $v$ in the $x$ direction, compared to the former, one used the non-relativistic Galilean transformations, i.e

$$ ct' = ct, \quad x' = x - vt, \quad y' = y, \quad z' = z. \qquad (1.1) $$

Instead, Lorentz transformations transform the time coordinate as a linear combination of initial time and space coordinates, i.e (for the same situation as before)

$$ ct' = \gamma \left( ct - \frac{vx}{c} \right), \quad x' = \gamma(x - vt), \qquad (1.2) $$

where the $y$ and $z$ coordinates are unchanged, and where $\gamma = 1/\sqrt{1 - (v/c)^2}$ is the Lorentz factor. The first equation transforms the quantity $ct$, instead of time $t$ alone, to make sure that all components have the same unit of length. One can see that Lorentz transformations deviate notably from Galilean ones when $v$ approaches the speed of light $c$, in which case effects of time dilation (moving clock ticks slower than resting one) and length contraction (moving observer measures contracted length of objects in the direction of movement) appear. The symmetry group of Lorentz transformations is the Lorentz group which includes space rotations and boosts.

The comprehension of space and time as two sides of the same coin is the basis of the existence of four-vectors $X^\mu = (X^0, \vec{X})$, which is a four-dimensional generalization of usual vectors in 3-space $\vec{X}$. The components of four-vectors transform in a specific way under
Lorentz transformations. In a situation similar than previously, the component of a four-vector $X^\nu$ in a given inertial frame transforms in $X'^\mu$ in another inertial frame, which moves in the x-direction with velocity $v$ with respect to the first one, as

$$X'^{\mu} = \Lambda_{\nu}^{\mu} X^{\nu} \qquad (1.3a)$$

where we introduce the $\Lambda$ matrix of Lorentz group (the group of all Lorentz transformations)

$$\Lambda_{\nu}^{\mu} = \begin{pmatrix} \gamma & -\frac{\gamma v}{c} & 0 & 0 \\ -\frac{\gamma v}{c} & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \qquad (1.3b)$$

as the transformation matrix of Eq. (1.2). There exist four-vectors describe many properties of objects, in particular the four-position $x^\mu = (ct, \vec{x})$, the four-momentum $p^\mu = (E/c, \vec{p})$, where $E$ is the energy, or the electromagnetic four-potential $A^\mu = (\phi/c, \vec{A})$, where $\phi, \vec{A}$ are respectively the scalar and vector potentials.

In a (flat and cartesian) four dimensional spacetime (i.e with no energy or gravity involved), distances are computed using the line element

$$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2 \equiv \eta_{\mu\nu} dx^\mu dx^\nu, \qquad (1.4)$$

where $\eta_{\mu\nu}$ is the Minkowski metric. This rank-2 tensor is central in special relativity and general relativity (where it will be generalized to $g_{\mu\nu}$ which in general differs from $\eta_{\mu\nu}$ to account for energy content of spacetime) as it is used to describe e.g. time, distances and curvature.

The theory of special relativity has been confirmed experimentally many times in various systems. In particular, it allowed to understand why one were able to detect cosmic muons on Earth ground. Muons are unstable particles which can be produced in cosmic showers when highly energetic particles enter the Earth atmosphere and recoil with atoms. They are relativistic particles, i.e they travel at a speed close to $c$, but they are unstable and have a lifetime of around 2.2 $\mu$s, after which they decay to lighter particles. Considering their lifetime and altitude at which they are produced, it is only through time dilation that their detection on ground can be explained.

Special relativity is very important in various fields of physics, and in particular it is a pillar of the two cornerstones of fundamental physics that will be discussed in the next chapters: general relativity and the Standard Model of particle physics.
# Chapter 2

## Quantum mechanics

Quantum mechanics is the theory that describes phenomena at very small scales (typically at the level of atoms or small molecules, i.e 10⁻¹⁰ m, and below).

As its name suggests, it is based on the quantization of physical quantities. This has mainly two consequences. The first one is the discovery of particles, as quanta of energies, such as photons. This allows to explain many experimental results, like the photoelectric effect or the Compton effect [2]. This is at the origin of the wave-particle duality, i.e the fact that some systems, such as light or atoms, can be described by a wave or by a particle, depending on the specific experiment considered. The second consequence is that the energy, the momentum, the angular momentum or spin of quantum systems can only take discrete values, while classical mechanics allow a continuum spectrum of numerical values for such quantities.

When shining light through a material, some electrons of the material are ejected, because the energy of light is larger than the binding energy of the electron in the atom. This is the photoelectric effect. If one measures the energy of the electrons *E* as function of the frequency of the incident light *ω*, one finds a linear dependence, whose slope is the so-called Planck constant *ħ*, i.e *E* = *ħ*ω. The Planck constant, also known as the quanta of action, is a fundamental quantity of quantum mechanics, as it is at the basis of quantization. Similarly to the speed of light *c* in special relativity, quantum mechanical effects start to be relevant when the action of the system is of order *ħ*. This is the reason why, macroscopically, when the system's action is much greater than *ħ*, quantum mechanics is negligible and classical mechanics works well.

As a probabilistic theory, it is fundamentally different from classical mechanics which is deterministic. Indeed, in the latter, if the initial position and velocity of an object is known, then it is possible to compute the motion of such object at any point in time. Quantum mechanics, however, describe the state of a particle with a wavefunction, usually denoted *ψ*, a complex function of the state. By the Born rule, the square modulus of the wavefunction |*ψ*(*s*)|² gives the probability of finding the particle in a given eigenstate *s*.

Quantities that can be measured experimentally are known as observables, and are described by hermitian operators, i.e matrices with real eigenvalues. As the eigenvectors of a given observable *A* form an orthonormal basis, a physical state *ψ* can be expressed as the superposition of eigenstates of *A*. This is what is known as the superposition principle [2].

Assuming we want to measure two quantities of a quantum state *ψ* that are represented by two observables that are canonically conjugate variables, such as the position *X* and
the momentum P, the dispersion of both operators around their expectation values obey $\Delta P\Delta X \ge \hbar/2$ [2], which is known as the Heisenberg uncertainty principle. This means that there exists a fundamental limit on the knowledge of both conjugate variables of a system when measuring them simultaneously. This uncertainty exists between any pairs of conjugate variables, for example time and energy, which are canonically related through Schrödinger equation (see below).

The fundamental equation of quantum mechanics is the Schrödinger equation [2]

$$i\hbar\frac{d\psi}{dt} = H\psi, \qquad (2.1)$$

which describes the evolution of a quantum state $\psi$ using the Hamiltonian operator $H$, which describes the total energy of the system (kinetic and potential). In a sense, it is the quantum generalization of Newton's second law of motion.
# Chapter 3

## General relativity

10 years after its famous annus mirabilis, Einstein published in 1915 its first paper [3] on general relativity (GR) which extends the laws of special relativity to gravitational fields. GR is a theory of gravitation which revolutionized our view of gravity. At that time, Newton's theory of gravitation was the leading framework, and despite explaining various phenomena, lacked explanation of several effects, such as Mercury perihelion precession. In Newton's theory, gravity is a force that acts instantaneously from the emitter to the receiver, which breaks causality and therefore special relativity. Instead, Einstein introduces the same four-dimensional spacetime as special relativity, but which curves under the energy-matter distribution of the manifold. In this sense, gravity is simply the curvature of spacetime, affecting the motion of freely falling massive objects through the curvature of the geodesics they follow. The effects of GR are visible when velocities are close to the speed of light and/or when gravitational field becomes strong (when deviations from flat spacetime are non negligible). When these conditions are not fulfilled, GR is well approximated by Newtonian mechanics.

This theory is one of the most successful theory ever created, as it was tested experimentally in various situations for more than 100 years now, and has never been disproved.

### 3.1 Equivalence principle

In addition to special relativity, the second pillar of GR is known as the equivalence principle (EP) between gravitation and acceleration. EP includes three different forms : the weak EP (WEP), the Einstein EP (EEP) and the strong EP. Here, we will be interested in the EEP which includes three different facets [4]. The first one is the WEP also known as the universality of free fall (UFF) which states that all bodies fall with the same acceleration in the same gravitational potential, no matter their composition. In other words, gravity is universal. It is satisfied if the gravitational theory is metric (which is the case for GR), which means that all matter fields are universally coupled to gravity. UFF implies that the gravitational mass (which appears in Newton's gravitational law) is equivalent to the inertial mass (which appears in Newton's second law). The second hypothesis is the Local Lorentz Invariance (LLI) whose principle is that the results of any local experiment, where gravity effects are excluded, do not depend on the position and velocity of the reference frame. The third principle is the Local Position Invariance (LPI), stating that results of any experiment do not depend on the spacetime position of the laboratory. This principle is closely related to the
steadiness of fundamental constants of nature [4].

## 3.2 Einstein field equations

The central GR equations are 6 independent metric $g_{\mu\nu}$ field equations, which are known as the Einstein field equations (EFE) [3]

$$G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} - \Lambda g_{\mu\nu} \qquad (3.1)$$

The left-hand side contains only the Einstein tensor $G_{\mu\nu}$ which contains second order derivatives of the metric. It involves in particular the Ricci scalar, which describes the curvature of spacetime. The right-hand side represents the matter-energy content. The tensor $T_{\mu\nu}$ is the stress-energy tensor of all matter and energy of spacetime. The constant $8\pi G/c^4$, which in the following will be denoted as $\kappa$ is important on dimensional level (to relate $T_{\mu\nu}$ components in $J/m^3$ units with $G_{\mu\nu}$ components in units of $m^{-2}$), but it also indicates the entity of the phenomena involves, i.e in that case gravitational physics (through $G$) with relativistic effects (through $c$). In addition to these terms whose links were theoretically demonstrated by Einstein in his original papers from GR's first principles, $\Lambda g_{\mu\nu}$, where $\Lambda$ stands for the cosmological constant, was included to take into account the acceleration of the expansion of the Universe. This constant can be interpreted as a perfect fluid with negative pressure whose nature is not understood, and which is usually denoted as *dark energy* (see below).

John Archibald Wheeler said "Space tells matter how to move, matter tells space how to curve" [5], which is a very simple statement to describe EFE and to show their non linearity. The non linearity of Einstein field equations make them very complicated to solve, and only partial solutions or solutions in very easy systems (such as the Schwartzschild solution describing the vacuum solution of EFE outside a spherically symmetric body with no charge or angular momentum) have been found. A whole branch of GR studies aims at solving EFE in peculiar systems using numerical methods, known as numerical relativity.

Despite being non linear, EFE are very similar in their form to Maxwell's equations of electromagnetism (EM) where distribution of matter (charges and currents in EM) sources curvature of spacetime (electromagnetic fields in EM).

## 3.3 Experimental successes

Overall, GR is a very successful theory as it predicted a large number of experimental results or explained various misunderstood phenomena.

Most solar system gravitational physics can be accurately described by Newtonian physics because the gravitational field is weak and the velocities are small (compared to speed of light $c$). However, the precession of the orbit of Mercury does not exactly follow Newton's predictions [6]. Thanks to GR corrections, which are necessary since the Sun's gravitational potential on Mercury starts to be large, this discrepancy was solved.

Other effects from GR at Solar system scale were experimentally verified. Some of them are the gravitational redshift [7] and Shapiro time delay [8], where both effects describe how light frequency and geodesics, despite being massless particles, are also affected by massive objects. The former states that two clocks located at different gravitational potential will
not tick at the same rate, the one located in a weaker gravitational potential ticking faster. Even though the effect is small, this correction is necessary to ensure the Global Navigation Satellite System (GNSS) to work as expected. The latter, predicted by Irwin Shapiro [9], describes how the propagation time for a light ray depends on the gravitational potential encountered by photons, more precisely how a time delay appears in signal when light travels close to a massive object. It was experimentally confirmed by the emission towards bodies close to the Sun (Mercury or Venus) and measuring the light round trip time, affected by the Sun's gravitational potential [10].

At galactic scale, GR predicts gravitational lensing effects, which describe how light geodesics are curved around galactic objects, such that the galaxy acts as a lens. As a consequence, several effects can be observed such as the Einstein ring, where the same astrophysical object is observed at all locations around the lens galaxy if source, lens and observer are perfectly aligned, making a light ring around the galaxy.

A very important field of research resulting from GR is the standard model of cosmology. This model aims at describing the evolution of the universe. In its simplest form, the universe is assumed homogeneous and isotropic on large scales, such that the spacetime metric is the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric $g_{\mu\nu} = \det(-1, a^2(t), a^2(t), a^2(t))$ in our sign convention, where $a(t)$ is the dimensionless scale factor accounting for the expansion of the Universe. This metric essentially describes a flat expanding universe. The standard model of cosmology [11] provides good theoretical account for the Cosmic Microwave Background (CMB), the large scale structures and the accelerating expansion of the Universe. Based on measurements from e.g. [12], the energy content of the Universe is 5% of ordinary baryonic matter, around 25% of non-baryonic matter, more commonly known as cold dark matter (CDM) (see Section 5.1) and about 70% of an unknown dark energy ($\Lambda$), responsible for the expansion of the Universe. Following these observations, the standard model of cosmology is referred to as $\Lambda$CDM-model.

Finally, the recent measurement of gravitational waves (GW) by the LIGO/VIRGO collaboration [13] is an important discovery, as it validates even more GR but it also provides a new way of observing the Universe. GW are currently visible as emitted by extremely massive objects such as black holes or neutron stars, therefore their detection allows to test GR in strong gravitational field regimes. Moreover, the Universe being transparent to GW (there exists no structure that can absorb them or reflect them) which is not the case for EM waves, GW can be used to discover new regions of the Universe.

## 3.4 Limits

As described earlier, EFE Eq. (3.1) contains an heuristical term, related to the acceleration of the expansion of the Universe, by some unknown energy. The simplest form of energy is the cosmological constant, with constant energy density over whole spacetime, but other models assumed e.g scalar fields as the cause of the acceleration of expansion. Some efforts were made to explain dark energy by the vacuum energy of the Universe, but it leads to a discrepancy between theory and observations of more than 120 orders of magnitude [14]. This leads to what is known as the cosmological constant problem.

As shown in EFE through the constant $\kappa = 8\pi G/c^4$, GR is a theory combining gravitation and special relativity. However, it does not include quantum mechanical effects (which are relevant when the action is close to $\hbar$). While GR describes accurately physics of the
macroscopic scale, another theory is needed to predict physics of the microscopic scale, namely quantum field theory, whose principles will be detailed in the next section. Running backward in time in the cosmological evolution of the Universe, one should arrive at the conclusion that there was a time in the history of the Universe, where its full energy content was compressed into a very small spacetime region such that temperature, energy density, etc.. were extremely high. In such a case, our current theories (GR and the Standard Model of particle physics (SM), see next section) are not valid anymore because both gravitational and quantum mechanical effects must be taken into account at the same time, which GR and SM are individually not able to do. A new high energy theoretical framework, sometimes referred to as quantum gravity, including both theories is therefore needed in order to understand the physics of the early universe. In this sense, even though the current standard model of cosmology assumes an initial Big Bang singularity as "birth" of the Universe and then an inflationary epoch that stretched spacetime at an exponential rate, one does not have any complete proof of their existence.

As stated in Section 3.1, the equivalence principle is very important in GR and is the basis of metric theories, i.e an universal coupling between gravity and all types of matter. However, this principle is only heuristical, i.e it is only supported by observations of test masses which fall at the same rate. In particular, it is not based on an underlying symmetry of the universe [15], in contrast of e.g gauge principle, as we shall see in the next section. It is also very intriguing that gravitation does not rely on any internal charge, compared to other well known fundamental interactions, such as electromagnetism. Therefore, in some theoretical scenarios, EP is expected to be broken at some scale, see e.g. [16–18], which would naturally invalidate GR, as a metric theory.

Last but not least, the problem of dark matter, which is the most relevant limit for this thesis. Together with the Standard Model of particle physics (see Section 4), GR does not predict the existence of dark matter, a new form of matter, different from baryonic matter that we are able to describe microscopically (see Section 4), and which, in our current understanding, only interact gravitationnally with visible matter. As we shall see in Section 5, its introduction allows one to explain various cosmological phenomena.
# Chapter 4

# The Standard Model of particle physics

The second cornerstone of fundamental physics, describing the interactions between elementary particles at the microscopic scale is the Standard Model of particle physics (SM). As its name suggests, it is based on quantum mechanics. Many scientists consider SM as the most successful theory ever created.

In the fourth century before J.-C., Democritus argued that atoms were the fundamental bricks of matter, and therefore could not be cut. It is only in the nineteenth and twentieth centuries that scientists such as Joseph Thomson and Ernest Rutherford discovered the electron and the proton.

Later in the twentieth century and after the work of Werner Heisenberg, Max Born and others on the creation of quantum mechanics on one side and Albert Einstein with its relativistic theory on the other side, the first attempts of quantizing the electromagnetic field were made by Paul Dirac. This is the birth of quantum field theory (QFT) which states that particles are not the most fundamental bricks of the universe, but are simply quantum excitation of something more fundamental : quantum fields. This is the theoretical framework at the basis of the SM.

## 4.1 Symmetries

As a relativistic quantum theory, QFT enjoys the symmetries of quantum mechanics, i.e space rotations and translations, and time translations, and of special relativity, i.e Lorentz symmetry. Space translations extend Lorentz symmetry to Poincaré symmetry. Another important symmetry of particle interactions is what is known as gauge symmetry. The various quantum fields introduced to describe the interactions between particles are not observables, i.e they cannot be measured. Therefore, the observables do not depend on the way we define those fields. The degree of freedom to shift or rotate the quantum fields is gauge symmetry.

## 4.2 Gauge groups

The current version of SM is based on three different gauge groups, which form the three fundamental interactions between particles that SM is able to describe accurately : electromagnetism described by quantum electrodynamics (QED), the strong interaction, described
by the quantum chromodynamics (QCD) and the weak interaction.

4.2.1 QED

Quantum electrodynamics [19–21] is a quantum generalization of the classical Maxwell’s equations for electromagnetism. It is based on a $U(1)$ gauge symmetry. The gauge field associated with this symmetry is the electromagnetic four-potential $A^\mu$ and the gauge particle (force carrier) is the photon. The gauge symmetry implies that the group transformation is $e^{iq}$, with $q$ the electric charge. This means that QED is an abelian theory in the sense that the group operations are commutative. The consequence of that is the gauge boson of the symmetry, the photon, must be neutral under the symmetry transformation. Therefore, photons cannot self interact. These are the reason why the photon is electrically neutral and that Maxwell’s equations (or their relativistic version) are linear. In addition, as gauge bosons, photons must be massless.

4.2.2 Weak interaction

Weak interaction [22] is based on a $SU(2)$ symmetry, which stands for special unitary matrices of dimension 2. There are three gauge fields associated with this symmetry: they are known as the $W^\pm$ bosons, which are electrically charged and the $Z^0$ boson, which is neutral. All fermions are affected by the weak interaction whose particular effect is to change the flavor of such fermions, i.e their fundamental nature. Weak interaction is the only charge (C) and charge-parity (CP) symmetry breaking interactions (for example, a broken CP symmetry means that a left handed fermion and right handed antifermion do not interact in the same way under the weak interaction). In addition, while they are gauge bosons, $W^\pm, Z^0$, are not massless (their mass is ~ 100 GeV/c² [23]), and this implies that weak interaction is a very short-range interaction. The generation of mass for $W^\pm$ and $Z^0$ bosons is known as the Higgs mechanism and will be discussed in Section 4.3.

4.2.3 QCD

The strong force (also known as the quantum chromodynamics (QCD)) [22] describes the interactions between gluons and quarks, which are the particles components of baryons, such as protons or neutrons. The associated conserved charge of the symmetry is an additional quantum number that only quarks and gluons possess, called the color charge, hence the name of the theory. There exists six different color charges: red, blue, green, anti-red, anti-blue and anti-green. The idea behind this formulation of color is that some combinations of colorful states can be colorless, e.g a mixture of red - green - blue state or red - anti-red does not carry color. Each quark carries one color, while gluons, the QCD force carriers carry two different colors. There exists eight independent gluon color states, which represent the generators of the $SU(3)$ gauge symmetry.

These bring two important consequences for the theory. The first one is that, contrary to QED, gluons carry the conserved charge of the symmetry, therefore they can interact with themselves, i.e QCD is a non-abelian gauge theory. The second one is the color confinement. There exists no bound states which is colorful, i.e quarks cannot be isolated, and therefore cannot be observed alone. Only colorless bound states composed of two (mesons, like pions) or three (hadrons, like protons and neutrons) quarks can be observed.
## 4.3 Higgs boson

At energies larger than what is known as the weak scale ($E = 246$ GeV), weak interaction and electromagnetism are unified into a single interaction, known as the electroweak interaction [22]. Gauge symmetry is required for any quantum theory to be valid, since it is related to the fact that the definition of the field (up to shift and rotation) should not impact the physical observables. Therefore, at this energy scale, in order for the electroweak theory to be gauged, fermions and gauge bosons, including $W^{\pm}$, must be massless.

However, we know from various experimental results that fermions are not massless, which means that one needs a mechanism to arise between this highly energetic state and the lower energy state of the current experiments and of the world we are living in.

In order to address this issue, one can introduce a massive scalar field $\phi$ with a new $U(1)$ gauge symmetry and a quartic potential. When the energy of the system gets below the electroweak scale, the potential of the field changes and gets the form of a Mexican hat, where the vacuum state is not aligned with the zero-particle state $\langle\phi\rangle = 0$. The vacuum states are therefore degenerate, and in order to minimize the energy, the system must choose one of its vacua with a given state $\langle\phi_{min}\rangle \neq 0$, which does not correspond to the zero-particle state. One says that the $U(1)$ symmetry has been spontaneously broken. The difference between vacuum state and zero-particle state is known as the vacuum expectation value (vev) of the field, therefore, by spontaneously breaking the symmetry, the field acquires a non-zero vev. Since the vacuum state $\phi_{min}$ does not correspond to the zero-particle state, it contains a non-zero number of particles. If we now assume that the SM fermionic and bosonic fields, initially massless, interact with the field $\phi$ in its vacuum state $\phi_{min}$, they effectively get "weighed down" by those interactions, i.e they behave as massive particles. In the SM, $\phi$ breaks three degrees of freedom (out of four) of the original electroweak symmetry group $SU(2) \times U(1)$, and therefore only three gauge bosons (out of four), $W^{\pm}$, $Z_0$, gain mass after symmetry breaking. This is the reason why the photon remains massless.

This mechanism, i.e fields acquiring mass by interacting with another field that spontaneously break symmetries, is known as the Higgs mechanism. We call $\phi$ the Higgs field, and its associated quantum is the Higgs boson. This idea was developed by three independent groups [24–26], and Peter Higgs and François Englert received the Nobel Prize in 2013 for this discovery.

## 4.4 Matter content

In addition to the Higgs boson and the interaction carriers described in the previous sections, i.e the photon $\gamma$, the two $W^{\pm}$ bosons, the $Z^0$ boson and the eight gluons $g$, one needs to describe the matter content of the universe, i.e the fermions [22].

There exist three generations of fermions, : the first, second and third generations. From one generation to another, fermions are quite similar, the only relevant parameter to distinguish them is the rest mass increase (or equivalently the lifetime decrease because decay to lighter particles is energetically favored) from generation $N$ to generation $N + 1$. All fermions are also distinguished by the interactions they are sensitive to, i.e if there are charged over the gauge group associated to that interaction. First, we have the leptons, which carry electric charge and weak charges (weak hypercharge, weak isospin), therefore
<figure><img src="image_3.png" /><figcaption>Figure 4.1: All fundamental particles of the Standard Model (with the hypothetical graviton), with their electric charge, color charge, mass and spin (Credit: CERN)</figcaption></figure>

they can be involved in electromagnetic and weak interactions. Among the leptons, we find the electron $e^-$ and the electron neutrino $\nu_e$ (first generation), the muon $\mu^-$ and the muon neutrino $\nu_\mu$ (second generation), and the tau $\tau^-$ and the tau neutrino $\nu_\tau$ (third generation). Second, we have the quarks which, in addition to electric and weak charges, carry color charge, which allows them to be sensitive to the third fundamental interaction, the strong force. There exist in total six different quarks. The first generation contains the up $u$ and down $d$ quarks, the second generation contains the strange $s$ and charm $c$ quarks, and the third generation contain the top $t$ and the bottom $b$ quarks.

Note that for each fermion, there exists an antifermion which has the same mass, but carries opposite charges, implying that a given antifermion is sensitive to the same interactions as its associated fermion.

The whole zoo of fundamental particles is summarized in Fig. 4.1, where electric charge, color charge, spin and mass of each particle are provided.

## 4.5 Lagrangian

The action of the theory is a number and can be calculated by integration of the Lagrangian $\mathcal{L}$ over the four dimensional spacetime measure $d^4x$ on a given manifold. This implies that the Lagrangian must be a scalar under Lorentz transformation, i.e a Lorentz scalar.

The full Lagrangian of SM being extremely lengthy, a reduced version can be used for our practical purposes [27]

$$ \mathcal{L} = -\frac{1}{4} F^{\mu\nu} F_{\mu\nu} + i\bar{\Psi} D\!\!\!/\Psi + \Psi_i y_{ij} \Psi_j \phi + \text{h.c.} + |D_{\mu}\phi|^2 - V(\phi) \quad (4.1) $$

The first term, represented by the contraction of the interaction carriers fields strength tensors $F^{\mu\nu}$ with themselves (to construct a Lorentz scalar), gathers the kinetic energies of all interaction carriers. Depending on the particle, this term allows self interaction (e.g for gluons) or simply kinetic energy (e.g for photons).
The second term includes the interaction between force carriers and the fermionic matter
fields (leptons and quarks), represented by the spinor $\Psi$, which is a mathematical object
used to describe half-integer spin particles, i.e fermions. The $\mathcal{D}$ is the mathematical operator
of gauged covariant derivative, which is a generalization of the partial derivative $\partial$ to which
we incorporate the gauge fields to account for fermion-boson interactions.

The third (respectively fourth h.c. for hermitian conjugate) term account for the interaction
between fermionic (respectively antifermionic) matter fields and the Higgs field $\phi$, which
gives rise to the mass of the fermions. $y_{ij}$ represents the various components of the Yukawa
matrix and describes the coupling constant of a given fermion to the Higgs field.

The fifth term describes the interaction of the weak force bosons with the Higgs field, such that they acquire mass.

Finally, the sixth term represents the potential energy of the Higgs field, which has the
form of the so-called "Mexican-hat". As mentioned before, this implies an infinite number
of different potential minima, leading to spontaneous symmetry breaking when the field
chooses a particular one. This idea is at the basis of the Higgs mechanism for the generation
of mass for many of the gauge and fermions fields.

4.6 Experimental successes and limits

After the theoretical completion of the SM, several particle accelerators were built, in par-
ticular at CERN, in Geneva, Switzerland, to test experimentally this theory. It consists of
beams of particles with large kinetic energy which collide in order to produce large mass
particles, to observe them through decay products. More than 40 years ago, the first particle
accelerator built at CERN was the Super-Proton-Antiproton-Synchrotron (SpπS), then came
the Large Electron-Positron Collider (LEP) and finally the Large Hadron Collider (LHC); and
the main difference between these accelerators (except the nature of particles colliding) is
the kinetic energies reached by the colliding particles, and therefore one is able to produce
particles with higher and higher masses.

Since their launch, they allowed scientists to observe the W and Z bosons, gluons, the $\tau$
lepton and the top and bottom quarks. More recently, in 2012, the Higgs boson was found
[28] at LHC, which implied that all particles theoretically predicted by the SM have been
experimentally detected.

SM has still issues explaining various physical phenomena. First, as explained in the
previous chapter about GR, there is still no unification between QFT and GR, i.e no one has
shown how to quantize gravity, i.e explain gravitation interaction completely from quantum
fields and particles.

Additionally, some fine-tuned problems arise in SM, such as the strong CP problem.
In short, QCD is theoretically allowed to break the Charge-Parity symmetry, which would
be visible experimentally by measuring a non-zero electric dipole moment of the neutron.
However, this measurement is consistent with zero with high accuracy [29], leading us to
think that QCD preserve CP symmetry. This problem will be more deeply studied in Chapter
8.2 on axions. Finally, one of the major issues in fundamental physics is the microscopic
nature of dark matter, which is still unknown, and which is the main topic of this thesis.
# Chapter 5

## Dark matter

As quickly stated in Section 3, a non baryonic matter, more commonly known as dark matter (DM) was first introduced to explain astrophysical observations at galactic scale, as we shall see in the following. Then, it was added into the cosmological equations in order to match various other observations, at the cosmological scale. In the first part of this section, we review the various observations that led to the conjecture of DM, based on the gravitational interaction between DM and the rest of the content of the Universe, and which essentially implies that DM, as its name suggests, is massive. Then, in the second part, we review the various possibilities as for the microscopic nature of DM, and in particular we introduce ultralight dark matter (ULDM) candidates, which will as serve as a foundation for the following chapters.

### 5.1 Some smoking guns of existence of dark matter

In this section, we will discuss some astrophysical and cosmological hints for the existence of DM.

#### 5.1.1 Galaxy rotation curves

The first historical hint of DM was made in the 1930s by Fritz Zwicky. He measured the velocity $v$ of stars of the Coma galaxy clusters as function of their distance to the galactic center $R$ and by using the virial theorem, he deduced the total mass $M$ of the galaxy cluster [30]

$$Mv^2 \propto \frac{GM^2}{R} \Rightarrow M \propto \frac{v^2 R}{G}. \qquad (5.1)$$

However, the total mass (~ 3 × 10¹⁴ solar masses), deduced by the dynamics of the cluster is much larger than the total luminous mass (~ 8 × 10¹¹ solar masses), deduced by the number of galaxies inside the cluster. DM was first introduced to account for this missing, non luminous, mass.

Afterwards, several observations of different galaxy rotation curves, i.e the measurement of the velocity of stars as function of their distance to the galactic center, indicate that non luminous mass has to be added. As it can be seen from Fig. 5.1, the discrepancy between the measurement of the mass from luminous objects and the measurement of mass through
<figure><img src="image_4.png" /></figure>

Figure 5.1: Velocity profile of the NGC 6503 galaxy (from [31]). The measured velocity are shown with dots and corresponding error bars. The velocity curve that fits the amount of luminous matter and gas do not fit the observations, and one needs to add a "halo" component, which corresponds to dark matter.

stellar dynamics becomes significant at large distance to the galactic center, which implies
that DM would be spherically distributed, encompassing the whole galaxy.

**5.1.2 Gravitational lensing**

Gravitational lensing, another observation at galactic scale, reveals the presence of an invis-
ible mass. As detailed in Section 3, GR allows massive objects to curve the geodesics of
light such that they act as a lens. By detecting the deviated photons on Earth, one is able
to reconstruct the mass distribution of the massive object that bent their trajectories. Such
measurement was done using the bullet cluster as lens [32] in addition to the measurement
of the visible mass distribution inside the cluster. As it is shown in Fig. 5.2, the measurement
of the cluster mass reconstructed from gravitational lensing is shown in blue, while the X-ray
measurement, reveals in pink the location of the "visible" mass, which is mostly made of gas.
One can clearly see that both distributions do not coincide, which leads us to the conclusion
that most of the mass in the cluster is invisible, i.e is DM.

**5.1.3 Cosmic microwave background**

At cosmic scale, the measurement of the Cosmic Microwave Background (CMB) suggests the existence of DM. In the Standard Model of Cosmology, about 300 000 years after the presupposed Big-Bang, while the temperature of the universe fell down to ~3000K, electrons and baryons were able to recombine and form atoms. This epoch, known as recombination, also marks the time of decoupling between such neutral atoms and photons, such that the latter were able to freely propagate through space, thus making the universe transparent. At that moment, the whole universe was immersed in a 3000 K photon bath. With the expansion of the universe, the wavelength of those photons were stretched such that their temperature is roughly 2.7 K today, which corresponds to a microwave wavelength. This phenomena is known as the CMB, and was first measured by Penzias and Wilson in 1964, which valued
<figure><img src="image_5.png" /><figcaption>Figure 5.2: Composite image of the bullet cluster, where the blue area shows the gravita-
tional lensing measurement of the mass and the pink area is the X-ray measurement of the
visible mass in the cluster. X-ray: NASA/CXC/CfA/ M.Markevitch et al.; Lensing Map:
NASA/STScI; ESO WFI; Magellan/U.Arizona/ D.Clowe et al. Optical image: NASA/STScI;
Magellan/U.Arizona/D.Clowe et al.</figcaption></figure>

them the Nobel prize in 1978. The Planck satellite measured with great accuracy the CMB temperature map [33], and the collaboration was able to measure its power spectrum which is with exceptional agreement with the presence of non baryonic matter, i.e DM, which accounts for ~ 26.5% of the total energy density of the universe [33]. In short, temperature anisotropies of the CMB are linked to the density anisotropies at the time of recombination, which depends on the densities of baryons, DM and photons and their interactions. As DM mainly interact gravitationally with the rest, it collapsed and formed dense regions before decoupling. Through gravitational redshift, photons imprinted those different gravitational potentials and this is why we see temperature anisotropies of those photons in the CMB.

**5.2 Dark matter candidates**

Despite the current gravitational evidence of the presence of DM in the universe, through
the various observations, described in the previous section, we still have no clue on the
microscopic nature of DM, i.e the fundamental particle behind it, its intrinsic properties
and its interactions with SM fields. Our current understanding of the microscopic world, in
particular from QFT, requires the introduction of an underlying field, which will be denoted
in the following by the DM field.

In the Standard Model of Cosmology, the DM field decouples quickly from the rest of the energy, i.e baryons and photons, during the early times of the Universe. Before recombination, from the observation of the CMB, one can conclude that the energy density is not exactly the same everywhere in space, i.e there are some underdense and overdense regions. Overdense regions, containing more DM, baryonic matter and photons, attract more mass, therefore the interaction between baryonic matter and photons increase locally, creating pressure force. This creates sound waves propagating outwards from the overdensities. At the time of recombination, baryons and photons decoupled, which led the photons to propagate freely and which relieved the pressure away. At this point, the left behind DM and baryons stopped propagating but they still formed an overdensed regions of matter, which many believe is the seed of galaxies that we see today. The distance travelled by the
<figure><img src="image_6.png" /><figcaption>Figure 5.3: Mass scale of DM candidates, from massive primordial black holes to the light fields, from [36].</figcaption></figure>

sound wave is known as the sound horizon and by measuring the two-point correlation function of the distance between galaxies, we expect to see a bump in the distribution, corresponding to the sound horizon (magnified by the expansion of the universe), what is known as the Baryonic Acoustic Oscillations (BAO) peak.

In this process, DM and baryonic matter attract each other gravitationally to form the galaxies we see today. We define the de Broglie wavelength of the field, as the typical wavelength at which one can describe DM as a matter wave (see Section 5.2.3)

$$ \lambda_{\text{DM}}^{\text{dB}} = \frac{2\pi\hbar}{m_{\text{DM}}v_{\text{DM}}} \sim 10^{-3} \left( \frac{\text{eV}}{m_{\text{DM}}c^2} \right) \text{m.} \qquad (5.2) $$

This wavelength is associated to the momentum of the DM wave $m_{\text{DM}}v_{\text{DM}} = \hbar k_{\text{DM}}$, where $k_{\text{DM}}$ is its wavenumber. Since DM allowed the formation of large scale structures, in particular galaxies, we require the DM de Broglie wavelength to be at most of the typical size of dwarf galaxies [34], corresponding to the primordial galaxies to form and which are roughly 1000 light years diameter long. This makes a lower bound on the DM mass to be $m_{\text{DM}}c^2 \gtrsim 10^{-22}$ eV [34].

There is no observational evidence for a higher bound on the DM mass candidate. However, out of all the possible candidates, the most massive objects which could explain DM would be primordial black holes (PBH) with maximum mass of about $10^2$ solar masses [35] (which is equivalent to $\sim 10^{35}$ g). This corresponds roughly to a mass of $10^{68}$ eV/c$^2$, such that, overall, the bounds on the mass of the DM candidates are

$$ 10^{-22} \text{ eV} \le m_{\text{DM}}c^2 < 10^{68} \text{ eV}. \qquad (5.3) $$

This means that the range of possible mass for the DM candidate covers 90 orders of magnitude, implying that its search constitutes an experimental challenge.

Multiple DM candidates exist, depending on the mass, and are summarized in Fig. 5.3.

### 5.2.1 Macroscopic mass scale

As stated previously, DM with very large mass, typically larger than $10^{16}$ g [35] (which corresponds to $\sim 10^{50}$ eV/c$^2$) would be constituted of PBH. This kind of black holes would theoretically be generated by inflation [35].
### 5.2.2 Particles mass scale

At lower mass, typically from $1 \text{ eV/c}^2$ to $10^{19} \text{ GeV/c}^2$ (which corresponds to the Planck mass), DM would be made of fermionic particles. As shown in Fig. 5.3, several theoretical models exist, such as sterile neutrinos, but the most studied ones are weakly interacting massive particles (WIMPs), which arise in beyond the Standard Model scenarios. As their name suggests, they are weakly interacting with SM particles and their mass is large, in the GeV-TeV scale, therefore they can be detected in particle accelerators, such as at LHC. Historically, WIMPs were thought to be the most promising DM candidates but lacking a detection at LHC, other models arose, in particular with particles with lower mass.

### 5.2.3 Classical fields mass scale

In the $10^{-22} \text{ eV/c}^2$ to $1 \text{ eV/c}^2$ mass range, we find DM candidates denoted as ultralight dark matter (ULDM) candidates. At such low masses, by computing the number density, i.e the average number of particles occupying a box with phase space volume $(\lambda_{\text{DM}}^{\text{dB}})^3$ as $\langle N \rangle = (\lambda_{\text{dB}}/\ell)^3$ where $\ell$ is the distance between particles, we find that

$$ \langle N \rangle = \frac{\rho_{\text{DM}}(2\pi\hbar c^2)^3}{(m_{\text{DM}}c^2)^4 v_{\text{DM}}^3} \approx 10^6 \left( \frac{1\text{eV}}{m_{\text{DM}}c^2} \right)^4, \qquad (5.4) $$

where we used the de Broglie wavelength of the field Eq. (5.2) and inserting DM parameters, especially DM energy density in the Milky Way $\rho_{\text{DM}} = 0.4 \text{ GeV/cm}^3$ [37] and galactic velocity $v_{\text{DM}} = 10^{-3} \text{ c. Eq. (5.4)}$ means that for DM particles with masses lower than roughly 10 eV, the average occupation number is much larger than 1, implying that the underlying particle is necessarily a boson, due to the Pauli exclusion principle. For sub-eV masses, this occupation number is so large, there is no need to study particles individually, and a standard classical field theory is enough to describe the field.

An important requirement for the field to be a suitable DM candidate is that it behaves as cold dark matter (CDM) at cosmological scales [38], i.e as pressureless matter ($P \sim 0$) and that its energy density redshiftes as $a^{-3}$, with $a$ the cosmological scale factor. As we shall see in Chapter 7, all ULDM models (i.e with mass between $10^{-22}$ to 1 eV.) fulfill this requirement since the angular frequency of the field exceeds the Hubble constant today.

ULDM includes a large number of models, which depend on the nature of the field, scalar, pseudo-scalar, vector and tensor, and each model has different couplings to SM fields, leading to different observable phenomenology. As we shall see in Chapter 8, a various number of ULDM models are studied in this thesis, namely scalar, pseudo-scalar and vector fields, and their respective phenomenology is derived.

## 5.3 Ultralight dark matter intrinsic characteristics

In this section, we discuss some of the ULDM characteristics that are central for its detection. In Table 5.1, we summarize the experimental values for those ULDM parameters.
<table><caption>Table 5.1: ULDM intrinsic parameters</caption><thead><tr><th>Parameters</th><th>Symbol</th><th>Numerical value</th><th>Unit</th></tr></thead><tbody><tr><td>Local energy density</td><td>ρ<sub>DM</sub></td><td>0.4 [37]</td><td>GeV/cm<sup>3</sup></td></tr><tr><td>Rest mass</td><td>m<sub>DM</sub></td><td>[10<sup>-22</sup>; 1]</td><td>eV/c<sup>2</sup></td></tr><tr><td>Coherence time</td><td>τ(f<sub>DM</sub>)</td><td>10<sup>6</sup> f<sub>DM</sub><sup>-1</sup></td><td>s</td></tr><tr><td>DM mean velocity in heliocentric frame</td><td>v<sub>DM</sub></td><td>3 × 10<sup>5</sup> [39, 40]</td><td>m/s</td></tr><tr><td>DM velocity dispersion</td><td>σ<sub>v</sub></td><td>1.5 × 10<sup>5</sup> [39]</td><td>m/s</td></tr></tbody></table>

### 5.3.1 Rest mass

As it was mentioned in the last section, the mass of ULDM candidates is contained in

$$10^{-22} \text{ eV} \le m_{\text{DM}}c^2 \lesssim 1 \text{ eV}. \qquad (5.5)$$

### 5.3.2 Local energy density

Several methods exist to estimate the dark matter energy density, and this, at different scales (e.g. galactic or local) [40]. Here, we are only interested in the DM energy density in the vicinity of the Sun, as it is what experiments on Earth will be sensitive too. Such density can be estimated by a parametric fit of the entire rotation curves of the galaxy (global method) $\rho_{\text{DM}}(r)$ as function of the distance $r$ to the galactic center. Recent determinations of $\rho_{\text{DM}}(r = r_{\odot}) \equiv \rho_{\text{DM}}$ points towards [40]

$$\rho_{\text{DM}} = 0.4 \text{ GeV/cm}^3. \qquad (5.6)$$

Eq. (5.6) means that the total DM energy contained in a sphere of radius ~ 30 astronomical units (AU) (which corresponds approximately to the Sun-Neptune distance) is ~ $10^{-13} M_{\odot}$. This shows how DM is locally subdominant, and therefore, how its gravitational impact on the orbit of the surrounding bodies is completely negligible [40].

### 5.3.3 Velocity distribution and coherence time

Galactic DM models assume that DM follows a spherical distribution around galaxies, making the so-called DM halo, as explained in Section 5.1. During its formation, the DM halo virialized, and therefore, acquired a non-zero velocity dispersion $\sigma_v$. More precisely, in the heliocentric reference frame, we assume the galactic DM follows a Maxwellian velocity distribution $\mathcal{F}(\vec{v})$ [41]

$$\mathcal{F}(\vec{v}) = \frac{1}{(2\pi\sigma_v^2)^{3/2}} e^{-\frac{(\vec{v}-\vec{v}_\odot)^2}{2\sigma_v^2}} \qquad (5.7a)$$

where $\vec{v}_\odot \equiv \vec{v}_{\text{DM}}$ is the mean velocity of the Solar system in the galactic frame and $\sigma_v$ is the dispersion (virial) velocity. Eq. (5.7a) can be integrated over a full sphere to become a distribution over the magnitude of the velocity, i.e [42]

$$\mathcal{F}(v) = \sqrt{\frac{2}{\pi}} \frac{v}{\sigma_v v_{\text{DM}}} e^{-\frac{v^2+v_{\text{DM}}^2}{2\sigma_v^2}} \sinh\left(\frac{vv_{\text{DM}}}{\sigma_v^2}\right) \qquad (5.7b)$$
Since the DM frame moves compared to any laboratory frame on Earth with a velocity following the distribution Eq. (5.7b), the DM field itself acquires a kinetic energy $E_k = m_{DM}v^2/2$ in addition to its rest energy $E = m_{DM}c^2$ such that the measured DM frequency $f_{DM}^{\text{meas.}}$ is in reality

$$f_{\text{DM}}^{\text{meas.}}(v) = f_{\text{DM}} \left( 1 + \frac{v^2}{2c^2} \right), \qquad (5.8a)$$

where

$$f_{\text{DM}} = \frac{m_{\text{DM}}c^2}{2\pi\hbar}, \qquad (5.8b)$$

is the intrinsic DM Compton frequency. In the following of this manuscript, when deriving sensitivities of experiments to ULDM fields at a given frequency $f_{DM}$, the kinetic energy correction in Eq. (5.8a) will be neglected at leading order, such that we will consider the observed DM frequency $f_{DM}^{\text{meas.}}$ to be equal to the intrinsic DM frequency $f_{DM}$.

Nonetheless, using Eqs. (5.7b) and (5.8a), we can now construct a frequency distribution which reads [43]

$$\tilde{\mathcal{F}}(f_{\text{DM}}^{\text{meas.}}) = \sqrt{\frac{2}{\pi}} \frac{c^2}{f_{\text{DM}}v_{\text{DM}}\sigma_v} e^{\frac{-v_{\text{DM}}^2+2c^2\left(1-\frac{f_{\text{DM}}^{\text{meas.}}}{f_{\text{DM}}}\right)}{2\sigma_v^2}} \sinh\left(\frac{v_{\text{DM}}c}{\sigma_v^2}\sqrt{2\left(\frac{f_{\text{DM}}^{\text{meas.}}}{f_{\text{DM}}} - 1\right)}\right). \qquad (5.9)$$

This equation essentially means that, a more rigorous modeling of galactic DM implies that we cannot model the field as monochromatic: it is a stochastic sum of $N$ different fields oscillating at different frequencies, which all follow the distribution Eq. (5.9) [44].

The frequency broadening Eq. (5.9) induces a characteristic coherence time of the field $\tau(\omega_{\text{DM}})$. For periods shorter than the coherence time, the field behaves as monochromatic (i.e with only one amplitude and phase, but which are still stochastic), while for times longer than the coherence time, the field is a superposition of plane waves with different frequencies. One can compute the first (mean $\mu_{f_{\text{DM}}^{\text{meas.}}}$) and second (standard deviation $\delta f_{\text{DM}}^{\text{meas.}}$) moments of the frequency distribution by integration over its domain of definition ($[f_{\text{DM}}, +\infty[$). They are given by

$$\mu_{f_{\text{DM}}^{\text{meas.}}} = f_{\text{DM}} \left( 1 + \frac{v_{\text{DM}}^2 + 3\sigma_v^2}{2c^2} \right) \approx f_{\text{DM}} \qquad (5.10a)$$

$$\delta f_{\text{DM}}^{\text{meas.}} = \frac{f_{\text{DM}}}{2c^2} \sqrt{v_{\text{DM}}^4 + 10v_{\text{DM}}^2 \sigma_v^2 + 15\sigma_v^4} \qquad (5.10b)$$

such that we can define the coherence time of the field as

$$\tau(f_{\text{DM}}) = \frac{1}{\delta f_{\text{DM}}^{\text{meas.}}} = \frac{2c^2}{f_{\text{DM}}\sqrt{v_{\text{DM}}^4 + 10v_{\text{DM}}^2\sigma_v^2 + 15\sigma_v^4}} \approx \frac{10^6}{f_{\text{DM}}}, \qquad (5.11)$$

where we used $v_{\text{DM}} \sim 2.33 \times 10^5 \text{ m/s} \approx 10^{-3} c$ [39] and $\sigma_v \sim v_{\text{DM}}/\sqrt{2} \approx 5 \times 10^{-4} c$ [39]. Note that for these calculations, we neglected the truncation of the velocity distribution Eq. (5.7a)
at the galactic escape velocity $v_{esc} \sim 5.5 \times 10^5$ m/s as it was done in [39]. However, it can be shown analytically that it would lead to a slight change in mean and standard deviation of the frequency distribution (of less than 10%), i.e it will not alter significantly our estimates.

While, in the following of this manuscript, we will model DM as a single monochromatic field, and not as a superposition of fields oscillating at different frequencies following Eq. (5.9), we will still take into account the coherence time of the field when deriving the various sensitivities of experiments to DM, and we show here why. Let us assume a given (linear) coupling $\zeta$ between DM and any SM sector (electromagnetic, fermionic, etc..) which induces a given signal $s(t)$ in our apparatus. Very generically, the signal searched for $s(t)$ is

$$s(t) = [X_s] \zeta \cos (2\pi f_{DM}t + \Phi), \qquad (5.12)$$

where we factorize the coupling $\zeta$ from the rest of the amplitude of the signal $[X_s]$. If the total time of integration of the experiment $T_{obs}$ is much shorter than the coherence time, i.e $T_{obs} \ll \tau(f_{DM})$, Eq. (5.12) is fully valid; i.e the signal is monochromatic, and the experiment sensitivity on the coupling at frequency $f_{DM}$ is simply

$$\zeta(f_{DM}) = \frac{1}{[X_s]} \sqrt{\frac{S_s(f_{DM})}{T_{obs}}} = \frac{\sqrt{SNR}}{[X_s]} \sqrt{\frac{S_n(f_{DM})}{T_{obs}}}, \qquad (5.13a)$$

where $S_s, S_n$ are respectively the signal and noise power spectral densities (PSD) of the experiment and the SNR (signal-to-noise ratio) is defined as the ratio of signal to noise PSD, i.e $S_s/S_n$. The SNR is used an an estimator of the signal strength, and it is useful in data analysis because it fixes a detection threshold in frequency domain, for possible discovery of new physics, compared to statistical anomaly.

Note that in this regime, where $T_{obs} \ll \tau(f_{DM})$, a correction factor to the sensitivity arises due to the stochastic nature of the amplitude of the field [45]. In our case where we will always consider a 68% detection threshold (i.e. SNR = 1), this correction factor induces a loss in signal of $\sim 1.5^1$.

On the other hand, when the integration time is much longer than the coherence time of the field, i.e $T_{obs} \gg \tau(f_{DM})$, this means that the signal searched for and parameterized by Eq. (5.12) is no longer coherent, i.e. it should be modeled as a sum of several stochastic harmonics, as we discussed it before. Another method to analyse the data is to cut the dataset in fragments with duration smaller than $\tau(f_{DM})$ and search for a coherent signal in each of these blocks of data. In such a case, the experimental sensitivity to the coupling is reduced and becomes [46]

$$\zeta(f_{DM}) = \frac{\sqrt{SNR}}{[X_s]} \sqrt{\frac{S_n(f_{DM})}{\sqrt{T_{obs}}\tau(f_{DM})}}. \qquad (5.13b)$$

<footnote n="1">¹As it is pointed out in [45], this correction factor depends on the nature of the signal, i.e if the apparatus couples to the field itself or e.g. to its gradient. However, we will neglect this subtlety, because for gradient coupling, the exact value of this correction factor highly depends on how the sensitive axis of the experiment evolves with time, and therefore it must be calculated case by case [45].</footnote>
# Chapter 6

## Alternatives to dark matter

As we saw in Section 5.1, the various indicators of DM arise by using GR as our theory of gravitation and Newtonian mechanics as its small velocity and weak field limit. Another possibility to overcome the DM problem would be somehow to modify those theories of gravitation, to match again theory with observations. The most famous alternative theory to Newton's second law is MOND (MOdified Newtonian Dynamics) introduced by Milgrom in 1983 [47]. The main idea of MOND is to explain the galaxy rotation curves by introducing an universal acceleration $a_0 \sim 10^{-10} \text{ m/s}^2$ [47] in order to avoid smaller acceleration for stars very far away from center of galaxies. For a body of mass m, Newton's second law then becomes

$$F = m\mu\left(\frac{g}{a_0}\right)g, \qquad (6.1a)$$

where g is the gravitational field and $\mu(x)$ is an asymptotic function that tends to 1 for $x \gg 1$ and to $x$ when $x \ll 1$. Therefore, in extreme weak acceleration environment such that $g \ll a_0$, Eq. (6.1a) becomes

$$F = m\frac{g^2}{a_0}. \qquad (6.1b)$$

A relativistic version of the MOND paradigm called TeVeS for Tensor-Vector-Scalar gravity was developed in 2004 by Bekenstein [48].

While the MOND theory and its extensions suffered for a long time to explain various phenomena, in particular in cluster of galaxies [49], and therefore would still require DM, recent studies (see e.g. [50–52]) were able to solve these problems. Nonetheless, in this thesis, we will be interested in solutions for dark matter involving new particles, and not through modified gravity.
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# Part II

## Phenomenology of some ultralight dark matter candidates
# Chapter 7

# Cosmological evolution

## 7.1 Action

In this section, we will show how a light bosonic field can act as CDM at cosmological scales. In the following of this manuscript, we will be interested in several types of light fields : a scalar field $\phi$, a pseudo-scalar field $a$ and a vector field $\phi^\mu$.

The dynamics of such fields are encoded in their action which we define as

$$S_{\phi} = \frac{1}{c} \int d^4 x \sqrt{-g} \left[ \frac{R}{2\kappa} - \frac{1}{2\kappa} g^{\mu\nu} \partial_{\mu} \phi \partial_{\nu} \phi - \frac{V(\phi)}{2\kappa} + \mathcal{L}_{\text{SM}}[\Psi_i] + \mathcal{L}_{\text{int}}[\Psi_i, \phi] \right] \quad (7.1a)$$

$$S_a = \frac{1}{c} \int d^4 x \sqrt{-g} \left[ \frac{R}{2\kappa} - \frac{1}{2\kappa} g^{\mu\nu} \partial_{\mu} a \partial_{\nu} a - \frac{V(a)}{2\kappa} + \mathcal{L}_{\text{SM}}[\Psi_i] + \mathcal{L}_{\text{int}}[\Psi_i, a] \right] \quad (7.1b)$$

$$S_{\phi^{\mu}} = \frac{1}{c} \int d^4 x \sqrt{-g} \left[ \frac{R}{2\kappa} - \frac{1}{4\mu_0} g^{\mu\nu} g^{\alpha\beta} \phi_{\mu\alpha} \phi_{\nu\beta} - \frac{V(\phi^{\mu})}{2\mu_0} + \mathcal{L}_{\text{SM}}[\Psi_i] + \mathcal{L}_{\text{int}}[\Psi_i, \phi^{\mu}] \right] \quad (7.1c)$$

for the scalar, pseudo-scalar and vector fields, where $\kappa = 8\pi G/c^4$ is the Einstein gravitational constant, $\mu_0$ is the vacuum permeability, $g = \det(g_{\mu\nu})$ and $\phi_{\mu\nu} = \partial_{\mu}\phi_{\nu} - \partial_{\nu}\phi_{\mu}$ is the $\phi^\mu$ field strength tensor. We choose a convention where the scalar $\phi$ and pseudo scalar $a$ fields are dimensionless, while the vector field $\phi^\mu$ has units of V.s/m, as the usual EM potential. In each expression, the first three terms represent the gravitational action which contains GR (through the Ricci scalar $R$) in addition to the kinetic and potential energy of the new field, while the last term represents the coupling of the new field with existing SM fields. In this section, we will focus on the former while the latter will be considered further away as it is responsible for the possible observed phenomenology for direct DM detection.

In Eq. (7.1), we choose a convention on the scalar field normalization factor in front of the kinetic and potential terms, such that it matches the 1/4 normalization of the vector kinetic term<fnref n="1" />. Other conventions exist with a different normalization factor, e.g in [1], which can be recovered by a simple redefinition of the field $\phi \to \sqrt{2}\phi$.

In general, the potential for the various fields is a polynomial expansion of the field.

<footnote n="1">¹For those respective normalization factors, one can show that integrating by parts the vector field Lagrangian leads to a term $\partial_{\mu}\phi^{\nu}\partial^{\mu}\phi_{\nu}/2$, which is similar to the scalar field kinetic term.</footnote>
Without loss of generality, considering a scalar field $\phi$, it has the form

$$V(\phi) = \frac{m_{\phi}^2 c^2}{\hbar^2} \phi^2 + \mu \phi^3 + \lambda \phi^4 + \dots \qquad (7.2)$$

where the first term is the mass term with $m_\phi$ the mass of the field. The second and third terms represent respectively cubic and quartic self interaction of the field with strength $\mu, \lambda$, in units of $[L^{-2}]$. Higher order self interactions are not represented.

In all models of interest in this manuscript, self interactions will not be considered, such that the potential of the field reduces to the mass term, with mass $m_\phi, m_a, m_U$ respectively for the scalar, pseudo-scalar and vector fields. Therefore, the GR action is invariant under $\{\phi, a, \phi^\mu\} \to \{-\phi, -a, -\phi^\mu\}$ parity transformation, such that both scalar and pseudo-scalar fields have the exact same dynamics at the GR level (i.e neglecting their respective interactions with SM fields). In addition, each vector field spatial component $\phi^i$ (which are the only relevant degrees of freedom<fnref n="2" />) can be described by a scalar field $\phi_U$, which obeys the following action

$$S_{\phi U}^{\text{Grav.}} = \frac{1}{c} \int d^4 x \sqrt{-g} \left[ -\frac{1}{2\mu_0} g^{\mu\nu} \partial_{\mu} \phi U \partial_{\nu} \phi U - \frac{m_U^2 c^2}{2\mu_0 \hbar^2} \phi_U^2 \right] \qquad (7.3)$$

in the Coulomb gauge, up to a total derivative, and neglecting GR for practical purposes. Eq. (7.3) has a very similar form of the action of a pure scalar field $\phi$, described in Eq. (7.1a) (up to a constant due to the difference in units between scalar and vector fields). Therefore, we will only derive the cosmological evolution of a pure scalar field, whose result will be easily extended to pseudo-scalar and vector fields.

## 7.2 Klein Gordon equation in an expanding Universe

Let us first discuss scalar fields. As noted in the end of the last section, neglecting their respective interactions with SM fields, both scalar and pseudo-scalar fields have the same dynamics, such that we will only make the calculations in the case of a pure scalar field $\phi$. From Eq. (7.1), the Lagrangian describing the kinetic and potential energies of the new scalar field is

$$\mathcal{L}_{\phi} = -\frac{\sqrt{-g}}{2\kappa} \left( g^{\mu\nu} \partial_{\mu} \phi \partial_{\nu} \phi + \frac{m_{\phi}^2 c^2}{\hbar^2} \phi^2 \right) \qquad (7.4)$$

We represent the universe as flat, homogeneous and isotropic and described using the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric defined as

$$g_{\mu\nu} = \text{diag}(-1, a^2(t), a^2(t), a^2(t)) \qquad (7.5)$$

<footnote n="2">²Solving the equations of motion for the vector field, one can show that there is no $\ddot{\phi}^0$ term, therefore the temporal component of the field is non dynamical. This naturally yields that its three degrees of freedom (the mass and the two polarization states) can be described by its spatial components only.</footnote>
with a the scale factor, such that $\sqrt{-g} = a^3(t)$. Using the Euler-Lagrange equation, we obtain the equation of motion of the field $\phi$

$$ \partial_{\mu} \left( -a^3(t) \partial^{\mu} \phi \right) + \frac{m_{\phi}^2 c^2}{\hbar^2} a^3(t) \phi = 0 \qquad (7.6a) $$

$$ \Rightarrow \Box \phi - \frac{3H(t)}{c^2} \dot{\phi} - \frac{m_{\phi}^2 c^2}{\hbar^2} \phi = 0 \qquad (7.6b) $$

where $\Box = \eta^{\mu\nu} \partial_{\mu} \partial_{\nu}$ is the d'Alembertian operator, with $\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$, the Minkowski metric, the dot represents derivative with respect to time and where we defined $H(t) = \dot{a}(t)/a(t)$, the Hubble constant. As it can be seen from Eq. (7.6b), the unit of the field does not impact the equation of motion, since it is a linear differential equation. Therefore, the vector field will follow the exact same Klein-Gordon equation, despite its different unit.

Eq. (7.6b) describes a damped harmonic oscillator. At early times of the universe, when $H \gg m_{\phi}c^2/\hbar$, the solution of this equation is a constant $\phi \to \phi_0$, i.e the field is frozen at $\phi_0$ and behaves effectively as massless with a flat potential. As soon as $H \sim m_{\phi}c^2/\hbar$, the potential of the field gets curved by the mass, and as long as $\phi_0$ does not exactly correspond to the local minimum of the potential, the field rolls down the potential and gets massive. When $H \ll m_{\phi}c^2/\hbar$, such that we can neglect the field friction, the solution of the Klein-Gordon equation is an oscillating solution of the form

$$ \phi(t, \vec{x}) = \phi_0 \cos(\omega_{\phi} t - \vec{k}_{\phi} \cdot \vec{x} + \Phi) \qquad (7.7a) $$

where $\phi_0, \Phi$ are respectively the amplitude of oscillation and a phase and where

$$ |\vec{k}_{\phi}| = \sqrt{\omega_{\phi}^2/c^2 - m_{\phi}^2 c^2/\hbar^2} \qquad (7.7b) $$

is the wavevector of the field. A vector field will oscillate in the exact same manner, but with a vectorial amplitude, i.e

$$ \phi^{\mu}(t, \vec{x}) = \gamma^{\mu} \cos(\omega_{U} t - \vec{k}_{U} \cdot \vec{x} + \Phi) \qquad (7.8) $$

This generation of field oscillation, with non-zero amplitude, which as we shall see in the next section leads to a non-zero energy density for the field, is more commonly known as the misalignment mechanism. Indeed, if by any chance, $\phi_0$ corresponded to the exact potential minimum, the field would not oscillate, and as a consequence would still be effectively massless.

In the following of this manuscript, we require the field to oscillate, such that the necessary condition $H \ll m_{\phi}c^2/\hbar$ equates to

$$ m_{\phi} \gg 10^{-33} \, \text{eV}/c^2 \qquad (7.9) $$

with the current value of the Hubble constant $H_0 \sim 10^{-18} \, \text{s}^{-1}$. Note that this condition is looser than the bound of DM mass from observations Eq. (5.3).

## 7.3 Oscillating field as cold dark matter

We now focus on the energy density and pressure of such fields, and we show that they behave at cosmological scale as cold dark matter, i.e pressureless matter with equation of state $\omega = P/\rho \sim 0$.
### 7.3.1 Scalar field

Still neglecting interaction, the stress-energy tensor of the scalar field $T^{\mu\nu}$ can be defined from the field Lagrangian Eq. (7.4), as

$$T^{\mu\nu} = \frac{2 \delta(\sqrt{-g} \mathcal{L}_{\phi})}{\sqrt{-g} \delta g_{\mu\nu}} \qquad (7.10a)$$

$$= \frac{1}{\sqrt{-g} \kappa \delta g_{\mu\nu}} \left( \delta(\sqrt{-g}) \left( -g^{\mu\nu} \partial_{\mu} \phi \partial_{\nu} \phi - \frac{m_{\phi}^2 c^2}{\hbar^2} \phi^2 \right) + \sqrt{-g} (-\delta g^{\mu\nu} \partial_{\mu} \phi \partial_{\nu} \phi) \right) \qquad (7.10b)$$

where we only vary the metric parameters. Using Jacobi's formula, we have

$$\delta\sqrt{-g} = \frac{1}{2}\sqrt{-g}g^{\mu\nu}\delta g_{\mu\nu} \qquad (7.11a)$$

$$\delta g^{\mu\nu} = -g^{\mu\alpha} g^{\nu\beta} \delta g_{\alpha\beta} \qquad (7.11b)$$

which simplifies the stress-energy tensor expression to

$$T^{\mu\nu} = \frac{-g^{\mu\nu} \left( \frac{1}{2} \partial_{\alpha} \phi \partial^{\alpha} \phi + \frac{m_{\phi}^2 c^2}{2\hbar^2} \phi^2 \right) + \partial^{\mu} \phi \partial^{\nu} \phi}{\kappa} \qquad (7.12)$$

For simplicity, we assume the field behaves as a perfect fluid at cosmological scales, such that its stress-energy tensor has the form

$$T^{\mu\nu} = \left( \rho + \frac{P}{c^2} \right) u^{\mu} u^{\nu} + P g^{\mu\nu} \qquad (7.13)$$

where $P, \rho$ are respectively the pressure and energy density of the fluid and $u^{\mu}$ its 4-velocity. By identification with Eq. (7.12), we find the average pressure over several field oscillations to be

$$\langle P \rangle = -\frac{1}{\kappa} \left\langle \frac{1}{2} \partial_{\alpha} \phi \partial^{\alpha} \phi + \frac{m_{\phi}^2 c^2}{2\hbar^2} \phi^2 \right\rangle \equiv 0 \qquad (7.14a)$$

while the average energy density is defined as the 00 component of the stress-energy tensor, i.e

$$\langle \rho \rangle = \langle T_{00} \rangle = \frac{1}{\kappa} \left\langle \frac{1}{2} \partial_{\alpha} \phi \partial^{\alpha} \phi + \frac{m_{\phi}^2 c^2}{2\hbar^2} \phi^2 + \left( \frac{1}{c} \frac{\partial \phi}{\partial t} \right)^2 \right\rangle = \frac{\omega_{\phi}^2 \phi_0^2}{2\kappa c^2} \qquad (7.14b)$$

where we used Eqs. (7.7a), (7.7b), and $\langle A + B \cos(2\omega t + \Phi) \rangle = A$. The observed equation of state of the field is defined as

$$\frac{\langle P \rangle}{\langle \rho \rangle} = 0, \qquad (7.15)$$
Therefore, the scalar field behaves cosmologically as a pressureless fluid, i.e cold dark matter with energy density

$$ \langle \rho \rangle = \frac{\omega_{\phi}^2 \phi_0^2}{2\kappa c^2} \equiv \rho_{\text{DM}} \qquad (7.16) $$

where $\rho_{\text{DM}}$ is the local DM energy density, such that we define the amplitude of oscillation of all scalar ULDM candidates to be

$$ \phi_0 = \frac{\sqrt{16\pi G\rho_{\text{DM}}}}{\omega_{\phi}c} \qquad (7.17) $$

In conventions where the scalar field Lagrangian Eq. (7.4) is multiplied by 2 compared to ours (e.g in [1], with the redefinition of the field $\phi \to \sqrt{2}\phi$), the associated amplitude of the scalar field is $\phi_0 = \sqrt{8\pi G\rho_{\text{DM}}}/\omega_{\phi}c$. We shall see in Chapter 8 that the observables of interest remain unchanged, as these fields redefinitions are unphysical, as expected.

### 7.3.2 Vector field

From Eq. (7.1c), we can define the vector field Lagrangian as

$$ \mathcal{L}_{\phi^{\mu}} = -\sqrt{-g} \left( \frac{1}{4\mu_0} g^{\mu\nu} g^{\alpha\beta} \phi_{\mu\alpha} \phi_{\nu\beta} - \frac{m_U^2 c^2}{2\mu_0 \hbar^2} g^{\mu\nu} \phi_{\mu} \phi_{\nu} \right), \qquad (7.18) $$

The vector field equation reads

$$ \partial_{\mu} \phi^{\mu\nu} - \frac{m_U^2 c^2}{\hbar^2} \phi^{\nu} = 0, \qquad (7.19a) $$

where we assumed an expanding Universe, with $H \ll m_U c^2/\hbar$, such that the Hubble constant is neglected. Taking the divergence of Eq. (7.19a) and using the antisymmetric property of $\phi^{\mu\nu}$, we find the continuity equation of $\phi^\mu$ i.e

$$ \partial_{\mu} \phi^{\mu} = 0. \qquad (7.19b) $$

While a similar equation can be found in electromagnetism by choosing the Lorenz gauge, Eq. (7.19b) is genuinely a field equation, which means that we are still free to choose a gauge. As in the scalar field case, the stress-energy tensor of the vector field is defined as

$$ T^{\mu\nu} = \frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g} \mathcal{L}_{\phi^{\mu}})}{\delta g_{\mu\nu}} = \frac{-g^{\mu\nu} \left( \frac{1}{4} \phi_{\alpha\beta} \phi^{\alpha\beta} + \frac{m_U^2 c^2}{2\hbar^2} \phi_{\alpha} \phi^{\alpha} \right) + \phi^{\alpha\mu} \phi_{\alpha}^{\nu} + \frac{m_U^2 c^2}{\hbar^2} \phi^{\mu} \phi^{\nu}}{\mu_0}, \qquad (7.20) $$

where we used again the antisymmetricity property of $\phi^{\mu\nu}$. We now assume an equivalent form of the plane wave solution Eq. (7.8), $\phi^\mu = \Upsilon^\mu \mathcal{R}[e^{ik_\mu x^\mu}]$ (forgetting about the irrelevant
phase), such that each field term of Eq. (7.20) can be written as

$$ \phi_{\alpha\beta}\phi^{\alpha\beta} = 2 \left[ (ik_{\alpha}\gamma_{\beta}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}]) (ik^{\alpha}\gamma^{\beta}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}]) - (ik_{\beta}\gamma_{\alpha}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}]) (ik^{\alpha}\gamma^{\beta}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}]) \right] \quad (7.21a) $$

$$ \phi_{\alpha}\phi^{\alpha} = (\gamma_{\alpha}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}]) (\gamma^{\alpha}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}]) \quad (7.21b) $$

$$ \phi^{\alpha\mu}\phi^{\nu}_{\alpha} = (ik^{\alpha}\gamma^{\mu}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}] - ik^{\mu}\gamma^{\alpha}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}]) (ik_{\alpha}\gamma^{\nu}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}] - ik^{\nu}\gamma_{\alpha}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}]) \quad (7.21c) $$

$$ \phi^{\mu}\phi^{\nu} = (\gamma^{\mu}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}]) (\gamma^{\nu}\mathcal{R}[e^{ik_{\alpha}x^{\alpha}}]). \quad (7.21d) $$

Considering

$$ \langle \mathcal{R}[a^{\mu}e^{ik_{\alpha}x^{\alpha}}]\mathcal{R}[b^{\nu}e^{ik_{\alpha}x^{\alpha}}] \rangle = \frac{1}{2} \mathcal{R}[a^{\mu}b^{\nu}] \quad (7.22) $$

where the star denotes the complex conjugate, the average over several oscillations of the stress-energy tensor of the vector field can be written as

$$ \langle T^{\mu\nu} \rangle = \frac{-g^{\mu\nu} \left( \frac{1}{4} k_{\alpha} k^{\alpha} \gamma_{\beta} \gamma^{\beta} + \frac{m_{U}^{2} c^{2}}{4\hbar^{2}} \gamma_{\alpha} \gamma^{\alpha} \right) + \frac{1}{2} k_{\alpha} k^{\alpha} \gamma^{\mu} \gamma^{\nu} + \frac{1}{2} k^{\mu} k^{\nu} \gamma_{\alpha} \gamma^{\alpha} + \frac{m_{U}^{2} c^{2}}{2\hbar^{2}} \gamma^{\mu} \gamma^{\nu}}{\mu_{0}} \quad (7.23) $$

where we used $k_{\mu}\gamma^{\mu} = 0$ from Eq. (7.19b). Using the same identification with perfect fluid as before, we find that the average pressure is

$$ \langle P \rangle = -\frac{1}{\mu_0} \left\langle \frac{1}{4} k_{\alpha} k^{\alpha} \gamma_{\beta} \gamma^{\beta} + \frac{m_{U}^{2} c^{2}}{4\hbar^{2}} \gamma_{\alpha} \gamma^{\alpha} \right\rangle = 0 \quad (7.24a) $$

since $k^{\mu}k_{\mu} = -(\omega_{U}/c)^{2} + |\vec{k}_{U}|^{2}$ and using Eq. (7.7b). The average energy density is

$$ \langle \rho \rangle = \frac{1}{\mu_0} \left\langle \frac{1}{4} k_{\alpha} k^{\alpha} \gamma_{\beta} \gamma^{\beta} + \frac{m_{U}^{2} c^{2}}{4\hbar^{2}} \gamma_{\alpha} \gamma^{\alpha} + \frac{1}{2} k_{\alpha} k^{\alpha} (Y_{0})^{2} + \frac{1}{2} \left(\frac{\omega_{U}}{c}\right)^{2} \gamma_{\alpha} \gamma^{\alpha} + \frac{m_{U}^{2} c^{2}}{2\hbar^{2}} (Y_{0})^{2} \right\rangle \\ \equiv \frac{\omega_{U}^{2}}{2\mu_{0}c^{2}} \gamma_{\alpha} \gamma^{\alpha}. \quad (7.24b) $$

Then, the amplitude of the total vector field is simply

$$ Y_{\alpha} Y^{\alpha} = \frac{2\mu_{0}\rho_{\text{DM}}c^{2}}{\omega_{U}^{2}}. \quad (7.25a) $$

Due to the small galactic velocity in the DM halo, the temporal contribution of the dot product in the previous equation can be neglected. Indeed, parametrizing $\gamma^{\mu} = (\phi/c, \vec{Y})$, we have

$$ Y_{\alpha} Y^{\alpha} = -\left(\frac{\phi}{c}\right)^{2} + |\vec{Y}|^{2} = -\left(\frac{c}{\omega_{U}}\right)^{2} (\vec{k}_{U} \cdot \vec{Y})^{2} + |\vec{Y}|^{2} = \frac{v_{\text{DM}}^{2}}{c^{2}} (\hat{e}_{v} \cdot \vec{Y})^{2} + |\vec{Y}|^{2} \approx |\vec{Y}|^{2}, \quad (7.25b) $$
where we used the continuity equation $k_\mu Y^\mu = 0$ to set $\phi/c = (\vec{k}_U \cdot \vec{Y})c/\omega_U$ and $\vec{k}_U = -\omega_U \vec{v}_{DM}/c^2 = -\omega_U v_{DM} \hat{e}_v/c^2$. This implies

$$|\vec{Y}| = \frac{\sqrt{2\mu_0\rho_{DM}c}}{\omega_U}. \qquad (7.26)$$

Eq. (7.26) is a good approximation for experiments where the propagation of the field is negligible, since we assume $\vec{k}_U = 0$, such that $Y^0 = 0$ (from $k_\mu Y^\mu = 0$). Note that Eqs. (7.17) and (7.26) are related by the change $\mu_0 \to \kappa$, as expected from the respective Lagrangian densities.

## 7.4 Expressing the dark matter field in various reference frames

### 7.4.1 Galactic frame

In the galactic frame, which is assumed to be the local DM rest frame<fnref n="3" />, the expression of the (scalar) field Eq. (7.7a) reduces to

$$\phi = \phi_0 \cos(\omega_{\phi} t + \Phi) \qquad (7.27)$$

with $\hbar\omega_\phi = m_\phi c^2$, since $|\vec{k}_\phi| = 0$ on average in this frame. However, for experiments lasting longer than the typical coherence time of the field Eq. (5.11), one must take into account the velocity distribution introduced in Eq. (5.7a). In this case, the mean velocity of the distribution is $v_\odot \to v_{DM}^{\text{RF}} = 0$, while the frequency dispersion is given by Eq. (5.10b) (with $v_{DM} = 0$) $\delta f_\phi^{\text{RF}} \sim 5 \times 10^{-7} f_\phi$.

The form of the field Eq. (7.27) will be used in our study of atom interferometry probes of ULDM in Chapter 13. As mentioned previously, we will neglect the frequency broadening due to the velocity distribution Eq. (5.7a), such that we will consider DM as a pure monochromatic field.

### 7.4.2 Laboratory frame

Eq. (7.7a) is the exact solution of the ULDM field in any other reference frame, in particular laboratory frames on Earth. In such cases, the field is propagating with velocity $|\vec{v}_{DM}| \sim 10^{-3}c$ (see Table 5.1, and neglecting the Earth velocity around the Sun, $|\vec{v}_{\text{Earth}}| \sim 3 \times 10^4 \text{ m/s}$, which averages to 0 over the year.).

We would like to know if it is still possible to neglect the propagation term $\vec{k}_\phi \cdot \vec{x}$ for the sensitivity estimates of the various experiments under consideration in the following of this thesis. To answer this question, the first point is to know whether the experiment couples to the field itself or to its gradient. In the latter case, it is clear that the full form of the field Eq. (7.7a) must be used, otherwise its gradient becomes 0.

In the former case, in order to neglect the propagation, and therefore considering the field as a pure standing wave, the main requirement is that the de Broglie wavelength of the

<footnote n="3">³Note that we can find a rest frame of DM because the field is massive, as opposed to the electromagnetic field.</footnote>
field $\lambda_{\text{DM}}^{\text{dB}}$ must be much larger than the length scale of the experiment $l_{\text{exp}}$, i.e, in terms of DM mass

$$l_{\text{exp}} \ll 10^{-3} \left( \frac{1 \text{ eV}}{m_{\phi}c^2} \right) \text{ m.} \qquad (7.28)$$

As we shall see in the corresponding chapters, this requirement is fulfilled for the experiments detailed in Chapters 9, 10, 11 and 12, therefore we will assume the field to be homogeneous over the full size of experiment with the form

$$\phi = \phi_0 \cos(\omega_{\phi} t + \Phi) \qquad (7.29a)$$

with

$$\omega_{\phi} = \sqrt{|\vec{k}_{\phi}|^2 c^2 + \frac{m_{\phi}^2 c^4}{\hbar^2}} = \frac{m_{\phi} c^2}{\hbar} \sqrt{1 + \left(\frac{v_{\text{DM}}}{c}\right)^2} \approx \frac{m_{\phi} c^2}{\hbar} \qquad (7.29b)$$

where we used $|\vec{k}_{\phi}| = 2\pi/\lambda_{\text{DM}}^{\text{dB}}$ and where we neglected the $(v_{\text{DM}}/c)^2 \sim 10^{-6}$ correction. One can easily notice that this form is equivalent (up to $(v_{\text{DM}}/c)^2$ corrections) to the one in the galactic frame Eq. (7.27).

In Chapter 15, we will be interested in the direct detection of ULDM fields by a space-based experiment, LISA, which couples to the gradient of the field. and therefore, we will consider the exact form of the field Eq. (7.7a).
# Chapter 8

## Couplings to Standard Model fields

### 8.1 Scalar field

#### 8.1.1 Theoretical motivations

A scalar particle is the simplest model of ULDM. Light scalar fields appear in various theoretical models, such as dilaton and moduli fields in string theory [2], or the relaxion field to understand the electroweak hierarchy problem, i.e the large energy difference between the electroweak scale (set by the Higgs field vacuum expectation value $v \sim 246$ GeV) and the gravity scale (set by the Planck mass $M_P c^2 \sim 10^{19}$ GeV) [3]. As we shall see in the following, a coupling between such scalar field with SM fields naturally leads to violation of the equivalence principle, which is expected to happen at some scale, as discussed in Section 3.4.

#### 8.1.2 Rest mass and transition frequency oscillations

We define the linear interaction Lagrangian of a dimensionless scalar field $\phi$ appearing in Eq. (7.1a) as [1]

$$ \mathcal{L}_{\text{int}} = \frac{\phi}{\sqrt{2}} \left( \frac{d_e}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{d_g\beta_3}{2g_3} G_{\mu\nu}^a G^{a\mu\nu} - \sum_{i=e,u,d} (d_{m_i} + \gamma_{m_i} d_g) m_i \bar{\psi}_i \psi_i \right), \quad (8.1) $$

where as stated previously, our convention in the scalar field normalization implies an additional $1/\sqrt{2}$ factor compared to [1, 4]. $F^{\mu\nu}$, $G^{a\mu\nu}$ represent the electromagnetic and gluonic strength tensors respectively, $\mu_0$ is the vacuum magnetic permeability, $g_3$, $\beta_3$ are the dimensionless QCD coupling constant and QCD beta function for the running of $g_3$, $m_i$ the mass of the fermions fields in units of energy, with spinors $\psi_i$ and $\gamma_{m_i}$ the anomalous dimension giving the energy running of the masses of the QCD coupled fermions. The $d_i$ represent the dimensionless coupling constants between the scalar dilaton and the different matter fields.

Since the energy scale we are interested in is very low (compared to second generation leptons' masses), this low energy effective Lagrangian involves only the low mass fermions, i.e the electron, up and down quarks, as can be noticed in the last term in Eq. (8.1).

These interactions between the SM particles and the dilaton lead to variation of several fundamental constants of Nature in SM, namely the QCD energy scale $\Lambda_{\text{QCD}}$, the EM fine
structure constant $\alpha$, the electron mass $m_e$ and the light quark masses $m_i$, with $i = \{u, d\}$. Specifically, assuming the usual EM kinetic term in addition to the coupling of the dilaton to the EM sector (which is the first term of Eq. (8.1)), one has

$$ \mathcal{L} \supset -\frac{1}{4\mu_0} \left(1 - \frac{d_e \phi}{\sqrt{2}}\right) F_{\mu\nu} F^{\mu\nu} \approx -\frac{1}{4\mu_0 \left(1 + \frac{d_e \phi}{\sqrt{2}}\right)} F_{\mu\nu} F^{\mu\nu}, \qquad (8.2) $$

where we used the Taylor expansion of the term in brackets as $d_e \phi \ll 1$. Since $\mu_0 \propto \alpha$, this Lagrangian leads to a small dependency of the fine structure constant on the scalar field as

$$ \alpha(\phi) = \left(1 + \frac{d_e \phi}{\sqrt{2}}\right) \alpha, \qquad (8.3a) $$

where $\alpha$ is the bare value of the fine structure constant, i.e without the dilaton-EM coupling. The other constants of Nature evolve as [4]

$$ \Lambda_3(\phi) = \left(1 + \frac{d_g \phi}{\sqrt{2}}\right) \Lambda_3, \qquad (8.3b) $$

$$ m_e(\phi) = \left(1 + \frac{d_{m_e} \phi}{\sqrt{2}}\right) m_e, \qquad (8.3c) $$

$$ m_i(\Lambda_{QCD})(\phi) = \left(1 + \frac{d_{m_i} \phi}{\sqrt{2}}\right) m_i(\Lambda_{QCD}). \qquad (8.3d) $$

If one of these couplings is non-zero, the corresponding variable would not be constant at two different spacetime positions where the field $\phi$ takes different values, hence the violation of the local position invariance, or more generally of the equivalence principle. Following the work of [4], we introduce the mean light quark mass $\hat{m} = (m_u + m_d)/2$ and the difference of light quark masses $\delta m = m_d - m_u$. Following Eq. (8.3), the numerical value of these new parameters will change as

$$ \hat{m}(\phi) = \left(1 + \frac{d_{\hat{m}} \phi}{\sqrt{2}}\right) \hat{m}, \qquad (8.3e) $$

$$ \delta m(\phi) = \left(1 + \frac{d_{\delta m} \phi}{\sqrt{2}}\right) \delta m, \qquad (8.3f) $$

with

$$ d_{\hat{m}} = \frac{m_u d_{m_u} + m_d d_{m_d}}{m_u + m_d}, \qquad (8.3g) $$

$$ d_{\delta m} = \frac{m_d d_{m_d} - m_u d_{m_u}}{m_d - m_u}. \qquad (8.3h) $$

Following [4], we can define dimensionless dilatonic mass and frequency charges for the atom A respectively given by

$$ [Q_M^A]_d = \frac{\partial \ln m_A(\phi)}{\partial \phi} \left(\equiv \alpha_A / \sqrt{2}\right), \qquad (8.4a) $$

$$ [Q_\omega^A]_d = \frac{\partial \ln \omega_A(\phi)}{\partial \phi}, \qquad (8.4b) $$
<table><caption>Table 8.1: Dilatonic charges for some species of atoms. The transition is hyperfine for <sup>87</sup>Rb and optical for the rest. The charges are derived from Eqs. (8.4a), (8.8) and (8.9). The value of Q<sub>ω,m<sub>e</sub></sub> is universal for all atomic transitions, see Eqs. (8.8) and (8.9), so it is not provided. As we shall see in Chapter 13 and Chapter 15, we will not consider the atomic transition of <sup>195</sup>Pt, <sup>48</sup>Ti, <sup>87</sup>Rb, Au-Pt and SiO<sub>2</sub>, reason why their frequency charges are not provided.</caption><thead><tr><th>Atomic</th><th>Q<sub>M,e</sub><br>[×10<sup>-3</sup>]</th><th>Q<sub>M,m<sub>e</sub></sub><br>[×10<sup>-3</sup>]</th><th>Q<sub>M,ṁ</sub><br>[×10<sup>-3</sup>]</th><th>Q<sub>M,δm</sub><br>[×10<sup>-3</sup>]</th><th>Q<sub>ω,e</sub></th><th>Q<sub>ω,ṁ</sub><br>[×10<sup>-3</sup>]</th><th>Q<sub>ω,δm</sub><br>[×10<sup>-3</sup>]</th></tr></thead><tbody><tr><td><sup>195</sup>Pt [5]</td><td>4.278</td><td>0.220</td><td>85.25</td><td>0.340</td><td>—</td><td>—</td><td>—</td></tr><tr><td><sup>48</sup>Ti [5]</td><td>2.282</td><td>0.253</td><td>82.58</td><td>0.138</td><td>—</td><td>—</td><td>—</td></tr><tr><td><sup>87</sup>Rb</td><td>2.869</td><td>0.234</td><td>83.95</td><td>0.254</td><td>2.34 [6]</td><td>-67 [6]</td><td>-17.3 [6]</td></tr><tr><td><sup>85</sup>Rb</td><td>2.961</td><td>0.239</td><td>83.98</td><td>0.220</td><td>—</td><td>—</td><td>—</td></tr><tr><td><sup>40</sup>Ca</td><td>2.409</td><td>0.275</td><td>82.08</td><td>0</td><td>2.02 [7]</td><td>0.0007</td><td>0</td></tr><tr><td><sup>44</sup>Ca</td><td>2.116</td><td>0.250</td><td>82.29</td><td>0.155</td><td>2.02</td><td>0.00065</td><td>O(10<sup>-7</sup>)</td></tr><tr><td><sup>86</sup>Sr</td><td>3.074</td><td>0.243</td><td>84.06</td><td>0.198</td><td>2.06</td><td>0.00030</td><td>O(10<sup>-7</sup>)</td></tr><tr><td><sup>87</sup>Sr</td><td>3.027</td><td>0.240</td><td>84.05</td><td>0.215</td><td>2.06 [8]</td><td>0.00030</td><td>O(10<sup>-7</sup>)</td></tr><tr><td><sup>88</sup>Sr</td><td>2.980</td><td>0.238</td><td>84.03</td><td>0.232</td><td>2.06</td><td>0.00030</td><td>O(10<sup>-7</sup>)</td></tr><tr><td><sup>171</sup>Yb</td><td>4.114</td><td>0.225</td><td>85.14</td><td>0.308</td><td>2.31 [9]</td><td>0.00015</td><td>O(10<sup>-7</sup>)</td></tr><tr><td><sup>176</sup>Yb</td><td>3.957</td><td>0.219</td><td>85.05</td><td>0.348</td><td>2.31</td><td>0.00020</td><td>O(10<sup>-7</sup>)</td></tr><tr><td><sup>196</sup>Hg</td><td>4.469</td><td>0.224</td><td>85.35</td><td>0.312</td><td>2.81 [7]</td><td>0.00013</td><td>O(10<sup>-7</sup>)</td></tr><tr><td><sup>202</sup>Hg</td><td>4.291</td><td>0.218</td><td>85.25</td><td>0.353</td><td>2.81</td><td>0.00010</td><td>O(10<sup>-7</sup>)</td></tr><tr><td>SiO<sub>2</sub></td><td>1.607</td><td>0.275</td><td>79.62</td><td>0.003</td><td>—</td><td>—</td><td>—</td></tr><tr><td>73%Au-27%Pt</td><td>2.204</td><td>0.432</td><td>85.27</td><td>0.337</td><td>—</td><td>—</td><td>—</td></tr></tbody></table>

which encode the spacetime variation of the mass and of the transition frequency of the atom A. In Eq. (8.4a), we show in parenthesis the relation between our definition of the mass charge with the previously defined coupling function in [1, 4]. In a general frame (i.e where the gradient of the DM field is not neglected), using Eqs. (7.7a) and (8.4), the rest mass and transition frequency of an atom A oscillate as

$$m_A(t, \vec{x}) = m_A^0 \left( 1 + \frac{\sqrt{16\pi G \rho_{\text{DM}} [Q_M^A]_d}}{\omega_\phi c} \cos(\omega_\phi t - \vec{k}_\phi \cdot \vec{x} + \Phi) \right), \quad (8.5a)$$

$$\omega_A(t, \vec{x}) = \omega_A^0 \left( 1 + \frac{\sqrt{16\pi G \rho_{\text{DM}} [Q_\omega^A]_d}}{\omega_\phi c} \cos(\omega_\phi t - \vec{k}_\phi \cdot \vec{x} + \Phi) \right), \quad (8.5b)$$

where $m_A^0, \omega_A^0$ are respectively the unperturbed rest mass and transition frequency of A.

The rest mass depending differently on all the varying constants of Nature, we can define partial dilatonic mass charges Q<sub>M,i</sub> such that [4]

$$[Q_M^{\text{atom}}]_d = \frac{1}{\sqrt{2}} \left( Q_{M,m_e}^{\text{atom}}(d_{m_e} - d_g) + Q_{M,e}^{\text{atom}}d_e + Q_{M,\hat{m}}^{\text{atom}}(d_{\hat{m}} - d_g) + Q_{M,\delta m}^{\text{atom}}(d_{\delta m} - d_g) \right), \quad (8.6a)$$
with [4]

$$Q_{M,\hat{m}} = 0.093 - \frac{0.036}{A^{1/3}} - 0.02 \frac{(A - 2Z)^2}{A^2} - 1.4 \times 10^{-4} \frac{Z(Z - 1)}{A^{4/3}}, \quad (8.6b)$$

$$Q_{M,\delta m} = 0.0017 \frac{A - 2Z}{A}, \qquad (8.6c)$$

$$Q_{M,m_e} = 5.5 \times 10^{-4} \frac{Z}{A}, \qquad (8.6d)$$

$$Q_{M,e} = \left( -1.4 + 8.2 \frac{Z}{A} + 7.7 \frac{Z(Z-1)}{A^{4/3}} \right) \times 10^{-4}, \qquad (8.6e)$$

where A and Z are respectively the mass and charge numbers of the atom and where the couplings $d_i$ are the ones defined in Eq. (8.3).

The frequency charge depends on dilatonic charges $Q_{\omega,X}$ and on the dilaton/matter coupling coefficients $d_X$ (see e.g. [1])

$$[Q_{\omega}^{\text{atom}}]_{d} = \frac{1}{\sqrt{2}} \left( Q_{\omega,m_e}^{\text{atom}} (d_{m_e} - d_g) + Q_{\omega,e}^{\text{atom}} d_e + Q_{\omega,\hat{m}}^{\text{atom}} (d_{\hat{m}} - d_g) + Q_{\omega,\delta m}^{\text{atom}} (d_{\delta m} - d_g) \right). \quad (8.7)$$

In case of hyperfine transitions, we have $f_{\text{atom}}^{\text{hyp}} \propto \alpha^{k_{\alpha}}(m_e/m_p)(m_q/\Lambda_{\text{QCD}})^{k_q}$ [10], where $k_{\alpha}, k_q$ represent respectively the sensitivity coefficients of the hyperfine transition to the fine structure constant and to the ratio of the light quark masses $m_q$ to the QCD mass scale $\Lambda_{\text{QCD}}$ ratio. Subsequently, the corresponding dilatonic frequency charges are

$$Q_{\omega,m_e}^{\text{hyp}} = 1, \qquad (8.8a)$$

$$Q_{\omega,e}^{\text{hyp}} = k_{\alpha}, \qquad (8.8b)$$

$$Q_{\omega,\hat{m}}^{\text{hyp}} = -0.048 + k_q, \qquad (8.8c)$$

$$Q_{\omega,\delta m}^{\text{hyp}} = 0.0017 + k_q, \qquad (8.8d)$$

where we used the dependency of the proton mass to the light quark masses $\partial \ln m_p / \partial \ln \hat{m} = 0.048$, $\partial \ln m_p / \partial \ln \delta m = -0.0017$ [4].

The optical transition frequencies depend mainly on the electron mass and on the fine structure constant $f_{\text{atom}}^{\text{opt}} \propto (m_e m_N) \alpha^{2+\epsilon_{\text{atom}}}/(m_e + m_N)$ [11, 12] (where $m_N$ is the total nucleus mass and $\epsilon_{\text{atom}}$ accounts for the relativistic correction that determines the frequency dependence on $\alpha$). Therefore, the frequency dilatonic charges write

$$Q_{\omega,m_e}^{\text{opt}} = 1, \qquad (8.9a)$$

$$Q_{\omega,e}^{\text{opt}} = 2 + \epsilon_{\text{atom}}, \qquad (8.9b)$$

$$Q_{\omega,\hat{m}}^{\text{opt}} \approx \frac{2.6 \times 10^{-5}}{A}, \qquad (8.9c)$$

$$Q_{\omega,\delta m}^{\text{opt}} \approx 9.0 \times 10^{-7} \frac{A - 2Z}{A^2}, \qquad (8.9d)$$

where for the third charge $Q_{\omega,\hat{m}}^{\text{opt}}$, we assumed $m_n = m_p$. In Table 8.1, we show the dilatonic mass and frequency charges, for some atoms and corresponding atomic transitions (in particular, for each optical transition, we consider the intercombination line (ICL), i.e the transition $^1S_0 \rightarrow {}^3P_1$, see Table 8.2).
<table><caption>Table 8.2: Some optical transition frequencies of interest.</caption><thead><tr><th>Species</th><th>Transition</th><th>Frequency (rad/s)</th></tr></thead><tbody><tr><td><sup>88</sup>Sr</td><td>5s<sup>2</sup> <sup>1</sup>S<sub>0</sub> → 5s5p <sup>3</sup>P<sub>1</sub></td><td>2.73 × 10<sup>15</sup></td></tr><tr><td><sup>40</sup>Ca</td><td>4s4s <sup>1</sup>S<sub>0</sub> → 4s4p <sup>3</sup>P<sub>1</sub></td><td>2.87 × 10<sup>15</sup></td></tr><tr><td><sup>171</sup>Yb</td><td>6s<sup>2</sup> <sup>1</sup>S<sub>0</sub> → 6s6p <sup>3</sup>P<sub>1</sub></td><td>3.39 × 10<sup>15</sup></td></tr><tr><td><sup>196</sup>Hg</td><td>6s<sup>2</sup> <sup>1</sup>S<sub>0</sub> → 6s6p <sup>3</sup>P<sub>1</sub></td><td>7.45 × 10<sup>15</sup></td></tr></tbody></table>

### 8.1.3 UFF violating acceleration of a test mass

The point mass action [4]

$$S_{\text{mat}}[\Psi_i] = - \sum_A \int_A d\tau (m_A(t, \vec{x})c^2 + E_A^{\text{int}}(t, \vec{x})), \quad (8.10)$$

describes the motion of an ensemble of particles with rest mass energy $m_A c^2$ and internal energy $E^{\text{int}} = \hbar\omega_A$, with $\omega_A$ the transition frequency of the body under consideration<fnref n="1" />. $\Psi_i$ are the different SM fields, $d\tau$ is the proper time interval defined as $c^2 d\tau^2 = -g_{\alpha\beta} dx^\alpha dx^\beta$, where $g_{\mu\nu}$ is the spacetime metric. The body A has its own composition therefore its own coupling with the dilaton field. The equivalent Lagrangian description reads

$$\mathcal{L}_A = - \left( m_A(t, \vec{x})c + \frac{\hbar\omega_A(t, \vec{x})}{c} \right) \sqrt{-g_{\mu\nu} \frac{dx^\mu}{dt} \frac{dx^\nu}{dt}}, \quad (8.11a)$$

which, for non relativistic velocities and considering flat spacetime, i.e $\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$, can be approximated by

$$\mathcal{L}_A = - \left( m_A(t, \vec{x})c^2 + \hbar\omega_A(t, \vec{x}) \right) \left( 1 - \frac{v_A^2}{2c^2} \right), \quad (8.11b)$$

to first order in $(v_A/c)^2$ (where $v_A$ is the coordinate velocity of the atom). Using Eq. (8.5), a simple Euler-Lagrange derivation of Eq. (8.11b) gives

$$\vec{a}_A(t, \vec{x}) = \left[ \omega_{\phi} \vec{v}_A - \vec{k}_{\phi} c^2 \right] \frac{\sqrt{16\pi G \rho_{\text{DM}}}}{\omega_{\phi} c} \left( [Q_M^A]_d + [Q_{\omega}^A]_d \frac{\hbar \omega_A^0}{m_A^0 c^2} \right) \sin(\omega_{\phi} t - \vec{k}_{\phi} \cdot \vec{x} + \Phi), \quad (8.12)$$

at lowest order in $v_A/c$, where the superscript 0 indicates the bare value of the quantity, at zeroth order in the field $\phi$. In addition, we considered the lowest order expansion in factors of $\hbar\omega_A^0/m_A^0 c^2$, as in general $m_A^0 c^2 \gg \hbar\omega_A^0$. As we have seen previously, the various $[Q_M^A]_d, [Q_{\omega}^A]_d$ charges are atom-dependent, implying that the acceleration Eq. (8.12) leads to a violation of the UFF.

<footnote n="1">¹This contribution will only be relevant for atoms, see Chapter 13.</footnote>
## 8.2 Pseudo-scalar field

### 8.2.1 Theoretical motivations

The QCD Lagrangian in SM reads

$$ \mathcal{L}_{\text{QCD}} = -\frac{1}{4} G_{\mu\nu}^a G^{a,\mu\nu} + \bar{\Psi}_q (i\not D - m_q e^{i\theta_q}) \Psi_q, \qquad (8.13) $$

where the first term represents the kinetic energy of the non abelian gluon gauge field, through the strength tensors $G_{\mu\nu}^a$ with $\sqrt{J/m^3}$ units, the second term represents the interaction between gluons and quarks $\Psi_q$ ($\not D = \gamma^\mu D_\mu$, where $\gamma^\mu$ are the dimensionless gamma matrices<fnref n="2" />) and the third term is the quarks mass term. One could add to this Lagrangian a total derivative term (which would not change the fields equations of motion) of the form

$$ \mathcal{L}_{\theta} = \theta \frac{g_3^2}{32\pi^2} G_{\mu\nu}^a \tilde{G}^{a,\mu\nu}, \qquad (8.14) $$

where $\tilde{G}^{a,\mu\nu} = \epsilon^{\mu\nu\rho\sigma} G_{\rho\sigma}^a/2$ is the dual gluon field strength tensor, with superscript a running over the 8 gluon fields. Despite leaving the equations of motion invariant, the Lagrangian Eq. (8.14) breaks CP (Charge-Parity) symmetry if the $\theta$ parameter, being an angle, is different from 0 or $\pi$ [13]. To test CP symmetry of QCD, one can measure the neutron electric dipole moment $d_n$ whose best upper limit constraint is $|d_n| < 3 \times 10^{-26}$ e.cm (90% confidence level) [14], which translates into an upper limit of the $\theta$ parameter to be $\theta < 10^{-10}$ [13]. This extremely small value is puzzling because we should expect $\theta \sim O(1)^3$, and makes the so-called strong CP problem. In the 1970s, Peccei and Quinn [15] proposed a solution to solve this problem: promote the previously fixed $\theta$ parameter to a dynamical massive field with a potential with a zero minimum, such that $\theta$ relaxes naturally to 0 [16–18]. More precisely, a new $U(1)_{PQ}$ symmetry is introduced which is spontaneously broken at a given energy scale $f_a$. The QCD axion is the Goldstone boson of this broken symmetry, which acquires a mass $m_a$ via its interaction with pions. See [19] for a complete review of the Peccei-Quinn mechanism. One can show that the QCD axion mass is inversely proportional to $f_a$, and that it also couples to the photon, nucleon, electron fields and nucleon electric dipole moment with respective couplings $g_{a\gamma}, g_{aN}/m_N c^2, g_{ae}/m_e c^2, g_d$ (with $m_N, m_e$ the nucleon and electron mass). All these couplings are proportional to $1/f_a$ such that the first three have units of 1/energy and the last coupling $g_d$ has units of $C.m/J$ (because it is also proportional to $d_n$) [20, 21]. Therefore, there is only one free parameter of the theory, which is the QCD axion mass. One can define a more generic pseudo-scalar field, the Axion-Like-Particle (ALP) a which couples to the same fields and has the same mass as the QCD axion, but where the relation of proportionality between all the parameters disappears, i.e all couplings and mass are free parameters of the theory. Therefore, ALP does not solve the strong CP problem, but is still a

<footnote n="2">²The gamma matrices are a set of 4 × 4 matrices which allow one to construct Lorentz scalar with fermionic spinors. These are the 4-dimensional generalization of Pauli matrices.</footnote>

<footnote n="3">³In reality, $\theta = \theta_{QCD} + \theta_q$ where $\theta_{QCD}$ is a contribution from QCD and where $\theta_q$ is a contribution from electroweak sector (coming from the quark masses), which are two very different sectors, therefore one should not expect those two contributions to cancel each other exactly.</footnote>
DM candidate, as shown in Section 7.3. The dimensionless ALP field is defined as

$$ \theta = \sqrt{\frac{\hbar c^5}{8\pi G}} \frac{a}{f_a}, \qquad (8.15) $$

where $\sqrt{\hbar c^5/8\pi G} \sim 2 \times 10^{18}$ GeV $\equiv E_P$ is the reduced Planck energy (which is usually defined in terms of the inverse of the $\kappa$ parameter, reason for the $8\pi$ factor [4]) and its full interaction Lagrangian can be written as [20]

$$ \mathcal{L}_{\text{int}} = E_P \left( \frac{g_3^2}{32\pi^2} \frac{a}{f_a} G_{\mu\nu}^a \tilde{G}^{a,\mu\nu} - \frac{g_{a\gamma}}{4\mu_0} a F_{\mu\nu} \tilde{F}^{\mu\nu} + \frac{g_{aN}\hbar c}{2m_N c^2} \partial_\mu a \bar{\Psi}_N \gamma^\mu \gamma_5 \Psi_N + \frac{g_{ae}\hbar c}{2m_e c^2} \partial_\mu a \bar{\Psi}_e \gamma^\mu \gamma_5 \Psi_e + \frac{c}{2i} g_d a \bar{\Psi}_N \sigma_{\mu\nu} \gamma_5 \Psi_N F^{\mu\nu} \right), \qquad (8.16) $$

where $\Psi_N, \Psi_e$ are respectively the nucleon and electron fermion fields, where $\gamma_5 = i\gamma^0\gamma^1\gamma^2\gamma^3$ the fifth gamma matrix and $\sigma^{\mu\nu} = [\gamma^\mu, \gamma^\nu]/2$.

### 8.2.2 Axion electrodynamics

Following Eqs. (7.1b) and (8.16) in flat spacetime ($\sqrt{-g} = 1$) and using space positive metric $g_{\mu\nu} = \det(-1, 1, 1, 1)$, we write the Lagrangian describing the interaction between the classical dimensionless axion pseudo scalar field $a$ with mass $m_a$ and EM through the coupling $g_{a\gamma}$ [13]

$$ \mathcal{L} = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} + j_\mu A^\mu - \frac{1}{2\kappa} \partial_\mu a \partial^\mu a - \frac{1}{2\kappa} \frac{m_a^2 c^2}{\hbar^2} a^2 - E_P \frac{g_{a\gamma}}{4\mu_0} a F_{\mu\nu} \tilde{F}^{\mu\nu}, \qquad (8.17) $$

where the EM gauge field $A^\alpha$ has usual units of V.s/m and where $\tilde{F}^{\mu\nu}$ is the dual electro-magnetic strength tensor defined as

$$ \tilde{F}^{\mu\nu} = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma} F_{\rho\sigma} \equiv \epsilon^{\mu\nu\rho\sigma} \partial_{\rho} A_{\sigma}, \qquad (8.18a) $$

with $\epsilon^{\mu\nu\rho\sigma}$ the antisymmetric 4-dimensional Levi-Civita tensor, which has the following properties

$$ \epsilon^{\mu\nu\rho\sigma} = g^{\mu\alpha} g^{\nu\beta} g^{\rho\delta} g^{\sigma\delta} \epsilon_{\alpha\beta\delta\gamma}, \qquad (8.18b) $$

$$ \epsilon_{0ijk} = \epsilon_{ijk}, \qquad (8.18c) $$

where $\epsilon_{ijk}$ is the usual 3-dimensional Levi-Civita tensor. Using these properties, one can show that

$$ F_{\mu\nu} \tilde{F}^{\mu\nu} = 4 \frac{\vec{E} \cdot \vec{B}}{c}, \qquad (8.19) $$

where we defined $\partial_\mu = (\partial_t/c, \vec{\nabla})$, $j^\mu = (\rho c, \vec{j})$ and $A^\mu = (\phi/c, \vec{A})$ with $\phi, \vec{A}$ respectively the scalar and vector potentials and introduce the electric and magnetic fields defined as

$$ \vec{E} = -\vec{\nabla}\phi - \partial_t \vec{A}, \qquad (8.20a) $$

$$ \vec{B} = \vec{\nabla} \times \vec{A}. \qquad (8.20b) $$
The equation of motion of the axion and photon fields are derived from the usual Euler-Lagrange equations and reads [22, 23]

$$ \left( \square - \frac{m_a^2 c^2}{\hbar^2} \right) a = \frac{\kappa E_p}{\mu_0 c} g_{a\gamma} \vec{E} \cdot \vec{B}, \qquad (8.21a) $$

for the axion field and

$$ \square A^\nu - \partial^\nu \partial_\mu A^\mu + E_p g_{a\gamma} (\partial_\mu a) \epsilon^{\mu\nu\rho\sigma} \partial_\rho A_\sigma = -\mu_0 j^\nu, \qquad (8.21b) $$

for the photon field. We can treat separately the temporal and spatial part of Eq. (8.21b), i.e

$$ \square \phi + \partial_t \partial_\mu A^\mu + c E_p g_{a\gamma} \epsilon_{ijk} (\partial_i a) \partial_j A_k = -\mu_0 \rho c^2, \qquad (8.22a) $$

$$ \square A^i - \partial^i \partial_\mu A^\mu + E_p g_{a\gamma} \epsilon^{\mu i \rho \sigma} (\partial_\mu a) \partial_\rho A_\sigma = -\mu_0 j^i. \qquad (8.22b) $$

Using the temporal equation Eq. (8.22a), the definition of the electric and magnetic fields Eq. (8.20) and the zero divergence of the magnetic field, we obtain

$$ \vec{\nabla} \cdot (\vec{E} - cE_p g_{a\gamma} a \vec{B}) = \frac{\rho}{\epsilon_0}, \qquad (8.23) $$

which makes the modified Gauss's law. From the spatial equation of motion, Eq. (8.22b), we obtain

$$ \vec{\nabla} \times \vec{B} = \mu_0 \left( \vec{j} + \epsilon_0 \frac{\partial \vec{E}}{\partial t} \right) - \frac{E_p}{c} g_{a\gamma} (\dot{a}\vec{B} + \vec{\nabla} a \times \vec{E}), \qquad (8.24) $$

where the dot represents derivative with respect to time and which corresponds to the modified Ampère's law.

Eqs.(8.23) and (8.24) together with the two unchanged Maxwell's equations form what is more commonly known as axion electrodynamics [22, 23]

$$ \vec{\nabla} \cdot (\vec{E} - cE_p g_{a\gamma} a \vec{B}) = \frac{\rho}{\epsilon_0}, \qquad (8.25a) $$

$$ \vec{\nabla} \times \vec{E} + \frac{\partial \vec{B}}{\partial t} = 0, \qquad (8.25b) $$

$$ \mu_0 \left( \dot{j} + \epsilon_0 \frac{\partial \vec{E}}{\partial t} \right) - \frac{E_p}{c} g_{a\gamma} (\dot{a}\vec{B} + \vec{\nabla} a \times \vec{E}) = \vec{\nabla} \times \vec{B}, \qquad (8.25c) $$

$$ \vec{\nabla} \cdot \vec{B} = 0. \qquad (8.25d) $$

The various new terms of Eq. (8.25) break the electric-magnetic duality which is present in the usual Maxwell's equations : in vacuum, these equations are not invariant under the change $(\vec{E}, \vec{B}) \rightarrow (c\vec{B}, -\vec{E}/c)$.

### 8.2.3 Vacuum birefringence and dichroism

We now focus on the spatial equation of motion of the EM field Eq. (8.22b). We assume propagation in vacuum $\dot{j} = 0$, both temporal and Coulomb gauges for the EM field $A_0 = 0$ and $\vec{\nabla} \cdot \vec{A} = 0^4$ and the axion field is spatially homogeneous at the scale of the experiment<fnref n="5" />

<footnote n="4">⁴Out of the four degrees of freedom encoded in the 4-potential $A^\mu$, only 2, representing the photon polarization states, are physical, reason why we can fix two gauges.</footnote>

<footnote n="5">⁵As shown in Eq. (7.28), the constraint on the axion mass depends on the size of experiment considered.</footnote>
such that the form Eq. (7.29) is valid. Then, Eq. (8.22b) becomes [24]

$$ \ddot{A}_i - c^2 \nabla^2 A_i + c E_P g_{a\gamma} \dot{a} \epsilon_{ijk} \partial_j A_k = 0. \qquad (8.26) $$

We can decompose $A_i$ into the left and right circular polarization modes of the photon with wave number $\vec{k}$ as

$$ A_i(t, x) = \int \frac{d^3 k}{(2\pi)^3} (A_k^+(t) e_i^+ + A_k^-(t) e_i^-) e^{i\vec{k} \cdot \vec{x}}. \qquad (8.27) $$

Without loss of generality, if we assume the propagation direction to be along the $\hat{z}$ direction, $\vec{k} = (0, 0, k)$, the normalized right/left helicities of the photon can be expressed as $e^+ = e^R = (1, -i, 0)/\sqrt{2}$ and $e^- = e^L = (1, i, 0)/\sqrt{2}$ such that

$$ \epsilon_{ijk} k_j e_i^\pm = \pm i k e_i^\pm. \qquad (8.28) $$

Then, we can write two separate equations of motion for both polarization modes [24]

$$ \ddot{A}_i^+ + k^2 c^2 \left( 1 + \frac{E_P}{c^2} \frac{\sqrt{16\pi G \rho_{\text{DM}} g_{a\gamma}}}{k} \sin(\omega_a t + \Phi) \right) A_i^+ = 0 \qquad (8.29\text{a}) $$

$$ \ddot{A}_i^- + k^2 c^2 \left( 1 - \frac{E_P}{c^2} \frac{\sqrt{16\pi G \rho_{\text{DM}} g_{a\gamma}}}{k} \sin(\omega_a t + \Phi) \right) A_i^- = 0, \qquad (8.29\text{b}) $$

where we used Eq. (7.17). Considering the plane wave ansatz $A_i^\pm(t) = \zeta_i^\pm \exp(-i(\omega_\pm t - \vec{k} \cdot \vec{x}))$, Eq. (8.29) leads to

$$ \omega_{\pm} = kc\sqrt{1 \pm \frac{E_P \sqrt{16\pi G \rho_{\text{DM}} g_{a\gamma}}}{c^2 k} \sin(\omega_a t + \Phi)}, \qquad (8.30) $$

or in other words, the dispersion relation of the right/left circular polarizations is modified such that they travel with different phase velocities $c_\pm = \omega_\pm/k$, respectively

$$ c_\pm = c \sqrt{1 \pm \frac{E_P \sqrt{16\pi G \rho_{\text{DM}} g_{a\gamma}}}{c^2 k} \sin(\omega_a t + \Phi)}. \qquad (8.31) $$

Note that Eq. (8.30) leads also to a modification of the group velocity of the left and right polarization of light

$$ v_g = \frac{\partial \omega_{\pm}}{\partial k} \approx c \left( 1 + \frac{16\pi G \rho_{\text{DM}} E_P^2 g_{a\gamma}^2}{8k^2 c^4} \sin^2(\omega_a t + \Phi) \right) + O\left( \left( \frac{E_P}{c^2} \frac{\sqrt{16\pi G \rho_{\text{DM}} g_{a\gamma}}}{k} \right)^3 \right). \qquad (8.32) $$

This modification of the group velocity does not break special relativity since we now assume that light propagates inside a medium full of axions, which is not pure vacuum.

In other words, vacuum becomes birefringent in presence of an axion background. Experiments such as DANCE [25], LIDA [26] or ADBC [27] attempt to detect axion-photon coupling through this effect.
We now study dichroism effects from axion-photon coupling. To do so, we focus specifically on the axion-photon interaction Lagrangian, written in terms of electric and magnetic fields i.e

$$ \mathcal{L}_{a\gamma} = -E_P \frac{g_{a\gamma}}{\mu_0 c} a \vec{E} \cdot \vec{B}. \qquad (8.33) $$

In a medium of permittivity $\epsilon$ and permeability $\mu$ and where external electric and magnetic fields $\vec{E}_0$ and $\vec{B}_0$ are applied, the modified Maxwell's equations Eqs. (8.23) and (8.24) show that the axion field is a source of electric charge $\rho_a$ and current density $\vec{j}_a$ [13]

$$ \rho_a = \epsilon c E_P g_{a\gamma} \vec{\nabla} a \cdot \vec{B}_0 \qquad (8.34a) $$

$$ \vec{j}_a = \frac{\mu E_P}{c} g_{a\gamma} (\dot{a}\vec{B}_0 + \vec{\nabla} a \times \vec{E}_0), \qquad (8.34b) $$

implying a conversion between axions and photons.

As derived in [13], if we assume light propagating in the z direction whose polarization lies in the x-y plane and a static and homogeneous magnetic field polarized along the x axis $\vec{B}_0 \equiv B_0 \hat{e}_x$, with no electric field, the photon-axion conversion will happen in the same direction as the B polarization direction. This means that the polarization component of light along the x-direction is depleted, implying a rotation of the plane of polarization. This is what is called dichroism. The conversion probability is given by [13]

$$ P(\gamma \rightarrow a, L) = \sqrt{\frac{\mu}{\epsilon}} \sqrt{\frac{\epsilon_0}{\mu_0}} \frac{\hbar}{\mu_0} \frac{g_{a\gamma}^2 B_0^2}{\beta_a q^2} \sin^2\left(\frac{qL}{2}\right), \qquad (8.35a) $$

where L is the travelling distance, q is the difference in wavevector between axions and photons, i.e $q = |\vec{k}_a - \vec{k}_\gamma|$ and $\beta_a = |\vec{k}_a|/\omega_a \equiv v_a/c^2$ with $v_a$ the speed of incident axions. Note that, by energy conservation, $\omega_a = \omega_\gamma$, since the static magnetic field does not transfer energy.

In addition, the relative phase between the polarizations parallel and perpendicular to the magnetic field changes, and light acquires ellipticity [13]. If we assume plane wave solutions for axions and photons, one can show that the $\gamma \rightarrow a$ conversion leads to a modification of the dispersion relation of photons of x polarization, which depends on the amplitude of the magnetic field in this direction. This is equivalent to a relative phase added to the x-polarization compared to the y-one that reads [13]

$$ \phi(L) = \frac{\hbar c g_{a\gamma}^2 B_0^2}{\mu_0 4q^2} (qL - \sin(qL)). \qquad (8.35b) $$

### 8.2.4 UFF violation through the coupling with gluons

We now consider the coupling between the ALP and gluons which reads, from Eq. (8.16),

$$ \mathcal{L}_{\text{int}} = E_P \frac{g_3^2}{32\pi^2} \frac{a}{f_a} G_{\mu\nu}^a \tilde{G}^{a,\mu\nu} = \frac{g_3^2}{32\pi^2} \theta G_{\mu\nu}^a \tilde{G}^{a,\mu\nu}. \qquad (8.36) $$
In [12], it is shown that the interaction Lagrangian from Eq. (8.36) induces a dependency of the mass of pions to the axion field, which implies a dependency of the mass of nucleons and atomic binding energy on the axion field. More precisely,

$$ \frac{\partial \ln m_N}{\partial (\theta^2)} = \frac{\partial \ln m_N}{\partial \ln (m_\pi^2)} \frac{\partial \ln (m_\pi^2)}{\partial (\theta^2)}, \qquad (8.37) $$

where $m_N$ is the rest mass of the nucleon N. As a consequence, the mass of any atom will also depend on the axion field and its coupling strength with gluons. In this sense, we can define the dimensionless axionic mass and frequency charges of an atom A as

$$ [Q_M^{\text{atom}}]_a = \frac{\partial \ln m_{\text{atom}}}{\partial (\theta^2)} \qquad (8.38a) $$

$$ [Q_{\omega}^{\text{atom}}]_a = \frac{\partial \ln \omega_{\text{atom}}}{\partial (\theta^2)}, \qquad (8.38b) $$

in a similar way as in Eq. (8.4). Therefore, a similar oscillation on both rest mass and transition frequency of an atom A arise

$$ m_A(t, \vec{x}) = m_A^0 \left( 1 + \frac{8\pi G \rho_{\text{DM}} E_P^2 [Q_M^A]_a}{f_a^2 \omega_a^2 c^2} \cos(2(\omega_a t - \vec{k}_a \cdot \vec{x} + \Phi)) \right), \qquad (8.39a) $$

$$ \omega_A(t, \vec{x}) = \omega_A^0 \left( 1 + \frac{8\pi G \rho_{\text{DM}} E_P^2 [Q_\omega^A]_a}{f_a^2 \omega_a^2 c^2} \cos(2(\omega_a t - \vec{k}_a \cdot \vec{x} + \Phi)) \right). \qquad (8.39b) $$

We made a redefinition of the unperturbed mass as $m_A^0 \rightarrow m_A^0(1 + 8\pi G \rho_{\text{DM}} E_P^2 [Q_\omega^A]_a / f_a^2 \omega_a^2 c^2)$, and similarly for the frequency. Then, the axion-gluon coupling would induce a UFF violating acceleration on the atom A of the form

$$ \vec{a}_A(t, \vec{x}) = \left[ \omega_a \vec{v}_A - \vec{k}_a c^2 \right] \frac{16\pi G \rho_{\text{DM}} E_P^2}{f_a^2 \omega_a^2 c^2} \left( [Q_M^A]_a + [Q_\omega^A]_a \frac{\hbar \omega_A^0}{m_A^0 c^2} \right) \sin \left( 2\omega_a t - 2\vec{k}_a \cdot \vec{x} + 2\Phi \right), \qquad (8.40) $$

where we did not include contributions from the axion spin-induced fifth force (see e.g. [28]), because we are only interested in the oscillating solution. Now, we wish to get an analytical expression for both axionic charges.

The rest mass of an atom, with charge number Z and neutron number N, can be parameterized as

$$ m_{\text{atom}}c^2 = m_{\text{const.}}c^2 + E_{\text{bind}} \equiv Z(m_p + m_e)c^2 + Nm_n c^2 + E_{\text{bind}}, \qquad (8.41) $$

where $m_p, m_e, m_n$ are respectively the rest masses of the proton, the electron and the neutron, and where $m_{\text{const.}}$, $E_{\text{bind}}$ represent respectively the rest mass of the particle constituents of the atom (proton, neutron, electron) and the nuclear binding energy.
## Nucleons rest mass

In [12], it is shown the pion mass is $\theta$ dependent and influence the nucleon mass $m_N$ through

$$ \frac{\partial \ln m_N}{\partial \ln m_{\pi}^2} \approx 0.06, \qquad (8.42a) $$

$$ \frac{\partial \ln m_{\pi}^2}{\partial (\theta^2)} = -\frac{m_u m_d}{2(m_u + m_d)^2} = -0.109 \qquad (8.42b) $$

$$ \Rightarrow \frac{\partial \ln m_N}{\partial (\theta^2)} \approx -0.065, \qquad (8.42c) $$

such that, for an atom made of (N+Z) nucleons, the contribution of the nucleons rest mass to the atom rest mass to the axionic charge is given by

$$ [Q_M^{\text{atom}}]_a \Big|_{\text{const.}} = \frac{\partial \ln m_N}{\partial (\theta^2)} \approx -0.065, \qquad (8.43) $$

meaning that it is independent of the number of nucleons inside the atom, or in other words of the atomic species.

## Binding energy of the nuclei

Let us now focus on the contribution of the binding energy of the nuclei to the axionic mass charge and show that it is composition dependent.

As computed in [4], the binding energy of the nuclei depends to first order on the mass of the pions and therefore, following Eq. (8.42), on the ALP. We will now use the results from Section IV of [4] to infer the analytical expression of the dependency of the binding energy to the ALP and to the mass number A and the charge number Z.

Four different interactions contribute to the binding energy [4]: the central force $E_{\text{central}}$ coming from the isospin symmetric central nuclear force, the asymmetry energy $E_{\text{asym}}$, i.e the residual energy from the asymmetry between neutrons and protons inside the nucleus, the Coulomb force $E_{\text{Coulomb}}$ depending on how tightly the nucleons are packed together and the pairing energy $E_{\text{pairing}}$ leading to

$$ E_{\text{bind}} = E_{\text{central}} + E_{\text{asym}} \frac{(A - 2Z)^2}{A} + E_{\text{Coulomb}} \frac{Z(Z - 1)}{A^{1/3}} - \delta \frac{E_{\text{pairing}}}{A^{1/2}}, \qquad (8.44) $$

where $\delta = \pm 1$ respectively for even-even and odd-odd number of protons-neutrons in the nuclei, and $\delta = 0$ otherwise.

**Central force** This interaction comes from the isospin symmetric central nuclear force, which is the dominant contribution in the binding of heavy nuclei [4]. The dominant interactions are an attractive scalar $\eta_S$ and a repulsive vector $\eta_V$, and [4] shows that the former is more sensitive to the pion mass, implying in our case

$$ \frac{\partial E_{\text{Central}}}{\partial (\theta^2)} \approx \frac{\partial E_{\text{Central}}}{\partial \eta_S} \frac{\partial \eta_S}{\partial \ln m_{\pi}^2} \frac{\partial \ln m_{\pi}^2}{\partial (\theta^2)}, \qquad (8.45a) $$
with [4, 29]

$$E_{\text{Central}} \approx -(120A - 97A^{2/3})\eta_S, \qquad (8.45\text{b})$$

$$\frac{\partial \eta_S}{\partial \ln m_{\pi}^2} = -0.35 \text{ MeV}. \qquad (8.45\text{c})$$

**Asymmetry energy** The residual energy from the asymmetry between neutrons and protons inside the nucleus contains two components : 1) from the Pauli exclusion principle, requiring that when there are more neutrons than protons, the extra neutrons are in the higher energy states than the protons; and 2) from the nuclear force, which is more attractive for a neutron and a proton than with a pair of neutrons or a pair of protons. The asymmetry energy depends mainly on the scalar coupling strength between the nucleons $G_S$ [4] implying

$$\frac{\partial E_{\text{Asym}}}{\partial(\theta^2)} = \frac{\partial E_{\text{Asym}}}{\partial G_S} \frac{\partial G_S}{\partial \ln m_{\pi}^2} \frac{\partial \ln m_{\pi}^2}{\partial(\theta^2)}, \qquad (8.46\text{a})$$

with [4, 29]

$$\frac{\partial E_{\text{Asym}}}{\partial G_S} \frac{\partial G_S}{\partial \ln m_{\pi}^2} = -19 \text{ MeV}. \qquad (8.46\text{b})$$

**Coulomb force** The Coulomb energy has a dependency on the strong interaction coupling terms since it depends on how tightly the nucleons are packed together. It can be shown that this contribution depends on the same scalar coupling as the asymmetry energy, hence

$$\frac{\partial E_{\text{Coulomb}}}{\partial(\theta^2)} = \frac{\partial E_{\text{Coulomb}}}{\partial G_S} \frac{\partial G_S}{\partial \ln m_{\pi}^2} \frac{\partial \ln m_{\pi}^2}{\partial(\theta^2)}, \qquad (8.47\text{a})$$

with [4, 29]

$$\frac{\partial E_{\text{Coulomb}}}{\partial G_S} \frac{\partial G_S}{\partial \ln m_{\pi}^2} = -0.13 \text{ MeV}. \qquad (8.47\text{b})$$

**Pairing energy** The pairing interaction contributes to binding energy and its numerical value is [29]

$$E_{\text{Pairing}} = 12 \text{ MeV}. \qquad (8.48)$$

[4] shows that this contribution is subdominant for all atoms, due to its dependency to mass number A, compared to the other interactions, hence we will not consider the pairing energy in the calculation of the axionic charges of atoms.

Adding all contributions together, we get the analytic expression for the axionic mass
charge

$$
\begin{align}
[Q_M^{\text{atom}}]_a &= \frac{1}{m_{\text{atom}}} \frac{\partial m_{\text{atom}}}{\partial(\theta^2)} \nonumber \\
&= \frac{1}{m_{\text{atom}} c^2} \left( \frac{\partial(m_{\text{rest mass}} c^2)}{\partial(\theta^2)} + \frac{\partial E_{\text{bind}}}{\partial(\theta^2)} \right) \nonumber \\
&\approx -0.065 \frac{m_{\text{rest mass}}}{m_{\text{atom}}} + \frac{1 \text{ MeV}}{m_{\text{atom}} c^2} \left( -4.578A + 3.701A^{2/3} + 2.071 \frac{(A-2Z)^2}{A} + 0.014 \frac{Z(Z-1)}{A^{1/3}} \right) \nonumber \\
&\approx -0.065 + F_A \left( -4.92 + \frac{3.98}{A^{1/3}} + 2.22 \frac{(A-2Z)^2}{A^2} + 1.50 \frac{Z(Z-1)}{A^{4/3}} \times 10^{-2} \right) \times 10^{-3}. \tag{8.49}
\end{align}
$$

At the last line, we considered $m_{\text{rest mass}}/m_{\text{atom}} \approx 1+\langle E_{\text{bind}} \rangle/m_{\text{amu}}c^2$ with $\langle E_{\text{bind}} \rangle \sim 8$ MeV, the average binding energy per nucleon and $m_{\text{amu}}$ the atomic mass unit [4]. Also, we factorized $F_A = A m_{\text{amu}}/m_{\text{atom}}$ which is of order unity at first order for all species of atoms (the relative error is $O(10^{-3})$)[4].

Now, we focus on the axionic frequency charge. We will first consider hyperfine atomic transitions ($\omega_{\text{atom}}^{\text{hyp}}$) which are impacted by the axion-gluon coupling from Eq. (8.36) as [12]

$$
\frac{\partial \ln \omega_{\text{atom}}^{\text{hyp}}}{\partial(\theta)^2} = \left( \frac{\partial \ln g}{\partial \ln m_{\pi}^2} - \frac{\partial \ln m_p}{\partial \ln m_{\pi}^2} \right) \frac{\partial \ln m_{\pi}^2}{\partial(\theta)^2}, \quad (8.50)
$$

where g is the nucleon g-factor whose dependence on pion mass is given by [12]

$$
\frac{\partial \ln g}{\partial \ln m_{\pi}^2} = K_n \frac{\partial \ln g_n}{\partial \ln m_{\pi}^2} + K_p \frac{\partial \ln g_p}{\partial \ln m_{\pi}^2} - 0.17 K_b, \quad (8.51a)
$$

with

$$
\frac{\partial \ln g_n}{\partial \ln m_{\pi}^2} \approx -0.25, \qquad (8.51b)
$$

$$
\frac{\partial \ln g_p}{\partial \ln m_{\pi}^2} \approx -0.17, \qquad (8.51c)
$$

where $g_n$ ($g_p$) respectively the neutron (proton) gyromagnetic factors and $K_n$, $K_p$ and $K_b$, co-
efficients computed from chiral perturbation theory (nuclear shell model). These coefficients
are not measurable, but they are related to observable parameters, namely, the sensitivity
coefficient of the nuclear magnetic moment to light quarks masses $\kappa_q$, to strange quarks
masses $\kappa_s$ and to light quarks masses over QCD energy scale $\kappa$ through [10]

$$
\kappa_q = -0.118 K_n - 0.087 K_p, \qquad (8.52a)
$$

$$
\kappa_s = 0.0013 K_n - 0.013 K_p, \qquad (8.52b)
$$

$$
\kappa = -0.12 K_n - 0.10 K_p - 0.11 K_b . \qquad (8.52c)
$$

Using Eqs. (8.42), (8.50), (8.51) and (8.52), we define the dimensionless axionic frequency charge of the atom as

$$
[Q_{\omega}^{\text{atom}}]_a \approx (-16.8\kappa - 5.69\kappa_q + 25.1\kappa_s + 0.65) \times 10^{-2}. \quad (8.53)
$$
In particular, Eq. (8.53) is relevant for $^{87}$Rb hyperfine transition, whose associated $\kappa$ parameters are $\kappa = -0.016$, $\kappa_q = -0.046$, $\kappa_s = -0.010$ [10].

<table><caption>Table 8.3: Axionic charges for some species of atoms. The transition is hyperfine for $^{87}$Rb and optical for the rest (as in Table 8.1, all of the optical transition are the $^1S_0 \rightarrow {}^3P_1$ ICL transition, see Table 8.2). The charges are derived from Eqs. (8.49), (8.53), (8.54) and (8.55). As we shall see in Chapters 13 and 15, we will not consider the atomic transition of $^{195}$Pt, $^{48}$Ti, SiO$_2$ and Au-Pt, reason why their frequency charge is not provided.</caption><thead><tr><th>Species</th><th>$Q_M$ [$\times 10^{-3}$]</th><th>$Q_\omega$ [$\times 10^{-3}$]</th></tr></thead><tbody><tr><td><sup>195</sup>Pt [5]</td><td>-69.065</td><td>—</td></tr><tr><td><sup>48</sup>Ti [5]</td><td>-68.770</td><td>—</td></tr><tr><td><sup>87</sup>Rb</td><td>-68.920</td><td>9.30 [10]</td></tr><tr><td><sup>85</sup>Rb</td><td>-68.924</td><td>—</td></tr><tr><td><sup>40</sup>Ca</td><td>-68.715</td><td>-0.00188</td></tr><tr><td><sup>44</sup>Ca</td><td>-68.738</td><td>-0.182</td></tr><tr><td><sup>86</sup>Sr</td><td>-68.933</td><td>-0.554</td></tr><tr><td><sup>87</sup>Sr</td><td>-68.932</td><td>-0.552</td></tr><tr><td><sup>88</sup>Sr</td><td>-68.932</td><td>-0.00550</td></tr><tr><td><sup>171</sup>Yb</td><td>-69.054</td><td>-0.0121</td></tr><tr><td><sup>176</sup>Yb</td><td>-69.043</td><td>-0.0119</td></tr><tr><td><sup>196</sup>Hg</td><td>-69.077</td><td>-0.00684</td></tr><tr><td><sup>202</sup>Hg</td><td>-69.066</td><td>-0.00677</td></tr><tr><td>SiO<sub>2</sub></td><td>-68.442</td><td>—</td></tr><tr><td>73%Au-27%Pt</td><td>-69.067</td><td>—</td></tr></tbody></table>

The second contribution, arising from the coupling of the ALP to the nuclear charge radius has the form [31–33]

$$[Q_{\omega}^{A}]_{a} = \frac{\partial \ln \omega_{A}}{\partial \ln \langle r_{N}^{2} \rangle} \frac{\partial \ln \langle r_{N}^{2} \rangle}{\partial \ln m_{\pi}^{2}} \frac{\partial \ln m_{\pi}^{2}}{\partial(\theta)^{2}} = \frac{\beta F}{\omega_{A}} \frac{\partial \ln m_{\pi}^{2}}{\partial(\theta)^{2}} \left( \langle r_{N}^{2} \rangle + \frac{C_{2}}{C_{1}} \langle r_{N}^{4} \rangle \right), \quad (8.55a)$$

where $\langle r_N^2 \rangle$, $\langle r_N^4 \rangle$ are respectively the mean squared and quartic charge nuclear radius, $F$ is the difference in field shift factor between excited and ground state, $\beta = \partial \ln \langle r_N^2 \rangle / \partial \ln m_\pi^2 \equiv -0.2$ [31] and $C_1, C_2$ are Seltzer coefficients, to account for higher order radial moments in the computation of the field shift (which are relevant for heavy atoms) [33]. For $^{171}$Yb optical transition, $F = -2\pi \times 10.955 \text{ GHz/fm}^2$ [34], $r_N = 5.2906 \text{ fm}$ [35], $C_1 = 6.23 \times 10^{-4}$ and $C_2 = -0.579$ [32], which leads to

$$[Q_{\omega}^{Yb}]_{a} \sim -1.21 \times 10^{-5}. \qquad (8.55b)$$

Let us now consider optical transitions whose frequency does depend neither on the nucleon g-factor nor on the proton mass at lowest order. This implies that the axionic frequency charge at lowest order is 0. Higher order contributions would lead to non-zero axionic frequency charge, in particular if one considers its dependence on the fine structure constant $\alpha$ which arises at loop level [30], or to the nuclear charge radius [31] and are therefore highly suppressed. We now compute the axionic optical frequency charge of an atom A, considering both of these effects for some atomic optical transition. Using the dependence of optical transition on the fine structure constant $\alpha$, the gluon-photon coupling at loop level leads to a charge

$$[Q_{\omega}^{A}]_{a} = c_{F2} \frac{(2 + \epsilon_{A})\alpha}{4\pi^{2}}, \quad (8.54a)$$

where $c_{F2}$ is the parameter encoding the explicit symmetry-breaking, generating the one-loop coupling. While for QCD axion, this parameter is $O(1)$, it is much smaller in the case of ALP (due to the introduction of an additional dark sector, e.g a dark photon, see [30]). It is nevertheless possible to have $c_{F2} \sim 10^{-2}$ [30]. Therefore, the corresponding axionic charge is, e.g. for $^{171}$Yb,

$$[Q_{\omega}^{Yb}]_{a} \sim 1.2 \times 10^{-8}. \qquad (8.54b)$$
For other atomic transition, we will use the fact that the field shift $F\langle r_N^2 \rangle$ approximately scales as $Z^2/A^{1/3}$ [33]. Note that, as expected, those two contributions are at least two orders of magnitude smaller than the axionic charges of hyperfine transitions (see Table 8.3).

## 8.3 Vector field

### 8.3.1 Theoretical motivations

In addition to the current SM gauge group $SU(3) \times SU(2) \times U(1)$, many theories predict an additional gauge group which would resolve various particle physics and/or cosmology problems, e.g. the vMSM model solves DM, neutrino masses and baryogenesis by incorporating three right-handed $SU(2)_R \times U(1)$ neutrinos [36]. The minimal gauged extension would be an additional $U(1)$ symmetry, carried by a new vector field, coupled to a local gauge symmetry current, in particular a linear combination of the baryon number $B$, the lepton number $L$ and the hypercharge $Y$ [37, 38]. Supersymmetry predicts a coupling to the $B-L$ current [37, 38]. This new vector field, sometimes referred as U-boson, heavy photon or dark photon (DP), would induce a new force, proportional to the current to which it couples.

### 8.3.2 Oscillating electric field through the coupling with photons

We now are interested with the interaction of the DP vector field $\phi^\mu$ of mass $m_U$ with EM. Using Eq. (7.1c), the Lagrangian describing such interaction is given by [39]

$$ \mathcal{L} = -\frac{1}{4\mu_0} F^{\mu\nu} F_{\mu\nu} + j^\mu A_\mu - \frac{1}{4\mu_0} \phi^{\mu\nu} \phi_{\mu\nu} - \frac{m_U^2 c^2}{2\mu_0 \hbar^2} \phi^\mu \phi_\mu - \frac{\chi}{2\mu_0} F_{\mu\nu} \phi^{\mu\nu}, \quad (8.56) $$

where $\chi$ is the dimensionless kinetic mixing coupling parameter which characterizes the coupling between the DP and the EM field.

In order to ease the calculations, we make a non-unitary transformation of the fields, i.e [40]

$$ \bar{A}^{\mu} = A^{\mu} + \chi \phi^{\mu} \quad (8.57) $$

while $\phi^\mu$ remains unchanged at first order in $\chi$. This allows us to redefine the fields in terms of mass eigenstates called massless and massive photons, which couples to the standard EM current. Using this change of variable, Eq. (8.56) becomes, at first order in $\chi$,

$$ \mathcal{L} = -\frac{1}{4\mu_0} \bar{F}^{\mu\nu} \bar{F}_{\mu\nu} + j^\mu (\bar{A}_\mu - \chi \phi_\mu) - \frac{1}{4\mu_0} \phi^{\mu\nu} \phi_{\mu\nu} - \frac{m_U^2 c^2}{2\mu_0 \hbar^2} \phi^\mu \phi_\mu, \quad (8.58) $$

with $F_{\mu\nu} = \bar{F}_{\mu\nu} - \chi \phi_{\mu\nu}$. The field equations read

$$ \partial_{\alpha} \bar{F}^{\beta\alpha} = \mu_0 j^{\beta}, \quad (8.59a) $$

$$ \partial_{\alpha} \phi^{\beta\alpha} = -\chi \mu_0 j^{\beta} - \frac{m_U^2 c^2}{\hbar^2} \phi^{\beta}. \quad (8.59b) $$
The antisymmetry of both strength tensors leads to the conservation of the electromagnetic 4-current $\partial_{\mu}j^{\mu} = 0$, and the continuity equation for the DP field $\partial_{\mu}\phi^{\mu} = 0$, as shown in Section 7.3.2. Using the Lorenz gauge for EM, $\partial^{\mu}\bar{A}_{\mu} = 0$, Eqs. (8.59) become

$$ \Box \bar{A}^{\beta} = -\mu_0 j^{\beta}, \qquad (8.60a) $$

$$ \Box \phi^{\beta} = \frac{m_U^2 c^2}{\hbar^2} \phi^{\beta} + \chi \mu_0 j^{\beta}, \qquad (8.60b) $$

where $\Box = \eta^{\mu\nu}\partial_{\mu}\partial_{\nu} = -\frac{1}{c^2}\partial_t^2 + \nabla^2$. These equations admit two classes of solutions in vacuum (where the current vanishes): a massless vector field (standard EM) and a massive one, the latter being characterized by solutions:

$$ \phi^{\beta} = \gamma^{\beta} e^{ik_{\mu}x^{\mu} + \Phi}, \qquad (8.61a) $$

$$ \bar{A}^{\alpha} = 0, \qquad (8.61b) $$

where $k_{\mu}k^{\mu} = -(\omega_U/c)^2 + |\vec{k}_U|^2 = -m_U^2c^2/\hbar^2$, as derived in Eq. (7.7b), and where Eq. (8.61a) is equivalent to Eq. (7.8). As can be noticed from Eq. (8.57), this solution induces an ordinary electromagnetic field in a vacuum (see also [39])

$$ A^{\beta} = -\chi \gamma^{\beta} e^{ik_{\mu}x^{\mu} + \Phi}. \qquad (8.62) $$

Due to the coupling between the DP and the EM field (coupling characterized by the mixing parameter $\chi$), the DP field will induce a small electromagnetic field, whose strength is proportional to $\chi$ and to the DP field amplitude, see Eq. (8.62). Then the induced electromagnetic field consists in an oscillating electric and magnetic fields of the form [39]

$$ E_{\text{DM}}^{j} = -\frac{\partial A^{0}}{\partial x_{j}} - \frac{\partial A^{j}}{\partial t} = \chi \left( k_{U}^{j} \gamma^{0} - \omega_{U} \gamma^{j} \right) \sin \left( \omega_{U} t - \vec{k}_{U} \cdot \vec{x} + \Phi \right) \qquad (8.63a) $$

$$ B_{\text{DM}}^{j} = \epsilon_{ijk} \partial^{j} A^{k} = -\chi \epsilon_{ijk} k_{U}^{j} \gamma^{k} \sin \left( \omega_{U} t - \vec{k}_{U} \cdot \vec{x} + \Phi \right), \qquad (8.63b) $$

The magnetic component is suppressed by a factor $v_{\text{DM}}/c \sim 10^{-3}$, therefore only the electric field component will be considered in Chapters 10 and 11. In addition, if one considers that $\vec{k}_U = 0$, we have $\gamma^0 = 0$ using the continuity equation $k_{\mu}\gamma^{\mu} = 0$, and the electric field does not propagate, i.e

$$ E_{\text{DM}}^{j} = -\chi \omega_{U} \gamma^{j} \sin (\omega_{U} t + \Phi) \qquad (8.63c) $$

where its amplitude is directly related to the local DM density (from Eq. (7.26))

$$ |\vec{E}_{\text{DM}}| = \chi c \sqrt{2\mu_0 \rho_{\text{DM}}}. \qquad (8.63d) $$

The idea of several experiments searching for DP is to focus this small electromagnetic field in order to enhance it and hopefully make it detectable [41–46].
### 8.3.3 UFF violation through the coupling with the B-L current

We now study the interaction between the vector field and the B-L current $j_{B-L}^{\mu}$ through the coupling $\epsilon$, which reads [47, 48]

$$ \mathcal{L} = -\frac{1}{4\mu_0} \phi^{\mu\nu} \phi_{\mu\nu} - \frac{m_U^2 c^2}{2\mu_0 \hbar^2} \phi^{\mu} \phi_{\mu} + \epsilon e j_{B-L}^{\mu} \phi_{\mu}, \qquad (8.64) $$

where the coupling is parameterized with respect to the usual EM charge $e$. Varying Eq. (8.64) with respect to $\phi^{\mu}$ leads to its field equation

$$ \partial_{\nu} \phi^{\mu\nu} = -\frac{m_U^2 c^2}{\hbar^2} \phi^{\mu} + \mu_0 \epsilon e j_{B-L}^{\mu}. \qquad (8.65) $$

The conservation of the B-L current leads to the continuity equation of the DP field $\partial_{\mu} \phi^{\mu} = 0$. Therefore, one can simplify the field equation to

$$ \Box \phi^{\mu} = \frac{m_U^2 c^2}{\hbar^2} \phi^{\mu} - \mu_0 \epsilon e j_{B-L}^{\mu}. \qquad (8.66) $$

Using Eq. (8.11b), one finds the equation of motion for a test particle A in vacuum of mass $m_A$, velocity $\vec{v}_A$ and acceleration $\vec{a}_A$

$$ m_A \vec{a}_A = \epsilon e [Q_{B-L}^A] (\vec{E}_{DP} + \vec{v}_A \times \vec{B}_{DP}), \qquad (8.67a) $$

where we have defined the B-L current associated to A as $j_{B-L}^{\mu} = (c, \vec{v}_A)[Q_{B-L}^A] \delta^{(3)}(\vec{x} - \vec{x}_A)$, with $[Q_{B-L}^A]$, the B-L charge of A, and where

$$ \vec{E}_{DP} = -\frac{\partial \vec{\phi}}{\partial t} - \vec{\nabla} \phi^0 = (-\vec{k}_U Y^0 + \omega_U \vec{Y}) \sin (\omega_U t - \vec{k}_U \cdot \vec{x} + \Phi) \qquad (8.67b) $$

$$ \vec{B}_{DP} = \vec{\nabla} \times \vec{\phi} = (\vec{k}_U \times \vec{Y}) \sin (\omega_U t - \vec{k}_U \cdot \vec{x} + \Phi), \qquad (8.67c) $$

using Eqs. (8.61a) and (8.63a) $(\vec{E}_{DP}, \vec{B}_{DP} = \vec{E}_{DM}/\chi, \vec{B}_{DM}/\chi)$, as we are considering the vacuum solution of Eq. (8.66). Note that a static solution of such equation exists and is associated with a Yukawa-type force sourced by a massive body.

The small galactic velocity implies that 1) $|\vec{k}_U|c \ll \omega_U$, and together with the continuity equation, 2) $\vec{k}_U Y^0 \ll \omega_U \vec{Y}$, such that Eq. (8.67a) simplifies to [25, 49–51]

$$ \vec{a}_A(t, \vec{x}) = \epsilon e \omega_U \vec{Y} \frac{[Q_{B-L}^A]}{m_A} \sin(\omega_U t - \vec{k}_U \cdot \vec{x}_A + \Phi), \qquad (8.68) $$

which, in the same way as Eq. (8.12), violates UFF.
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# Part III

## Electromagnetic cavities probes
# Chapter 9

## Search for axions with an optical cavity and an optical fiber

Usually, cavities can be used to search for axion-photon coupling $g_{ay}$ through the axion-photon conversion under a strong magnetic field. This is the case for ADMX [1–5] or CAPP [6–8] experiments. In practice, these experiments aim at detecting photons sourced by this strong magnetic field and the DM axion field through the inverse Primakoff effect. Then, the photons would resonate inside the cavity in order to increase drastically the signal. In this case, the frequency of the produced photons correspond to the axion mass, and therefore these experiments are sensitive to a very narrow mass interval (which is essentially the width of the resonant frequency of the cavity).

Cavities can also search for this $g_{ay}$ coupling through the effect derived in Section 8.2.3, i.e the phase velocity difference between the left and right polarization of light. In the following, we will consider an optical cavity to compute the phase observable. Then, in Chapter 18, we will make some estimation on the sensitivity of the optical cavity used in the DAMNED experiment [9]. We will show that unfortunately, such experiments are not competitive for the search of the axion-photon coupling.

### 9.1 Birefringence and oscillation of the length of the cavity

We consider a setup where two optical cavities are used : one in which a linear polarized light is used, i.e the phase velocity is exactly the speed of light $c$, and another where we send a right polarized light. This could be done, e.g, by inserting a quarter wave plate in front of one of the cavities which would change a linear polarization to a circular one. Then, we measure the phase difference between the output of the two cavities.

From Eq. (8.31), the left and right circular polarized light respectively evolve with phase velocity

$$c \left( 1 - \frac{E_P \sqrt{4\pi G \rho_{\text{DM}} g_{ay}}}{c^2} \sin(\omega_a t + \Phi) \right) \equiv c - \delta c \sin(\omega_a t + \Phi) \quad (9.1a)$$

$$c \left( 1 + \frac{E_P \sqrt{4\pi G \rho_{\text{DM}} g_{ay}}}{c^2} \sin(\omega_a t + \Phi) \right) \equiv c + \delta c \sin(\omega_a t + \Phi), \quad (9.1b)$$
with $\omega_a$ the oscillation frequency of the axion field, $E_P$ the reduced Planck energy and

$$ \delta c = \frac{\sqrt{4\pi G\rho_{\text{DM}}E_P g_{a\gamma}}}{\omega_0}, \qquad (9.1c) $$

with $\omega_0 = kc$, the unperturbed light angular frequency. In this model, the cavity length $l$ is unchanged, but is directly related to the light travel time $\tau_{\leftrightarrow}(t)$ inside the cavity as

$$ 2l = \int_{0}^{\tau_{\leftrightarrow}(t)} c(t')dt', \qquad (9.2) $$

where $c(t)$ is the time dependent phase velocity of light whose exact form depends on the light polarization Eq. (8.31). Assuming a right handed photon starting its propagation at the left mirror at time $t - \tau_{\leftrightarrow}(t)$, it will come back to the same spatial point at time $t$, where the travel time is given by

$$ \tau_{\leftrightarrow}(t) = \frac{2l}{c} - \int_{t-\tau_{\leftrightarrow}}^{t} \frac{\delta c_{\leftrightarrow}(t')}{c} dt' - \int_{t-\tau_{\leftrightarrow}-\tau_{\leftrightarrow}}^{t-\tau_{\leftrightarrow}} \frac{\delta c_{\rightarrow}(t')}{c} dt', \qquad (9.3a) $$

where $\tau_{\leftrightarrow}$ and $\tau_{\leftarrow}$ are the time intervals the photon takes on the trip there and back respectively and

$$ \delta c_{\rightarrow}(t) = +\delta c \sin(\omega_a t + \Phi) \qquad (9.3b) $$

$$ \delta c_{\leftarrow}(t) = -\delta c \sin(\omega_a t + \Phi), \qquad (9.3c) $$

respectively for right and left handed polarizations, since the photon has opposite circular polarization on the way there and back due to the mirror reflection. Indeed, the mirror reverses one of the linear polarizations while keeping the other unchanged, which implies that a circular polarization changes other (up to an irrelevant phase) We can then interpret this oscillation of phase velocity of light as an oscillation of the cavity length itself, i.e we can write

$$ \tau_{\leftrightarrow}(t) = \frac{2(l + \delta l(t))}{c}, \qquad (9.4a) $$

where

$$ \delta l(t) = \delta c f(t), \qquad (9.4b) $$

with

$$ f(t) = \frac{1}{2} \left[ \int_{t-\tau_{\leftarrow}}^{t} \sin(\omega_a t' + \Phi) dt' - \int_{t-\tau_{\leftarrow}-\tau_{\rightarrow}}^{t-\tau_{\rightarrow}} \sin(\omega_a t' + \Phi) dt' \right]. \qquad (9.4c) $$

As we are interested in the first order solution of the phase shift in the perturbation $g_{a\gamma}$, and since $\delta c = O(g_{a\gamma})$, we can solve explicitly $f(t)$ at zeroth order in the perturbation, i.e using $\tau_{\leftarrow} = \tau_{\rightarrow} = \tau_0 \equiv l/c$ in the integral bounds

$$ f(t) = \frac{2}{\omega_a} \sin^2\left(\frac{\omega_a l}{2c}\right) \cos\left(\omega_a \left(t - \frac{l}{c}\right) + \Phi\right). \qquad (9.5) $$

such that the variation of length of the cavity becomes

$$ \delta l(t) = \frac{\sqrt{16\pi G\rho_{\text{DM}}E_P g_{a\gamma}}}{\omega_0\omega_a} \sin^2\left(\frac{\omega_a l}{2c}\right) \cos\left(\omega_a \left(t - \frac{l}{c}\right) + \Phi\right) \qquad (9.6) $$
## 9.2 Phase shift at the output of the cavity

In order to obtain the phase shift of the electric field resulting from the length oscillation Eq. (9.6), we will sum over a large number $N$ of field contributions that are reflected back and forth in the cavity (i.e $N \to \infty$).

The calculation is explicitly presented in Appendix A.1. Here, we immediately show the amplitude of the phase shift (oscillating at the axion frequency $\omega_a$) between the transmitted field and the input field as

$$|\Delta\phi(t)| \approx \frac{2\sqrt{16\pi G\rho_{\text{DM}}E_P g_{ay}}}{\omega_a c\sqrt{1 - 2r^2 \cos\left(\frac{2\omega_a l}{c}\right) + r^4}} \sin^2\left(\frac{\omega_a l}{2c}\right), \quad (9.7)$$

where $r$ is the mirror reflectivity of the cavity. This means the phase observable is independent of the laser frequency. When the axion frequency corresponds to an even mode of the cavity, $\omega_a l/c = (2n + 1)\pi$, and considering that $r = 1 - \epsilon$, $\epsilon \ll 1$, the signal scales in $\epsilon^{-1}$, and therefore increases drastically. This phase shift expression has not been derived in the literature explicitly, in the case of the axion-photon coupling. In Chapter 18, we will use it to derive the sensitivity of DAMNED to $g_{ay}$.

## 9.3 Phase shift along an optical fiber

We now consider a very simple experiment where we use two fibers of same length L, in which we send a left polarized light A in the first one and a right polarized light B in the other one. At the output, we measure the phase shift between both light signals. Mathematically, this situation is equivalent to having only one single fiber and sending both signals in the two opposite directions of the fiber. Both signals have the same angular frequency $\omega_0$ (at zeroth order in the perturbation), and after having traveled through the fiber, the electric fields have the form

$$E_A(t) = E_A^0 e^{-i\omega_0(t-t_{\leftarrow})} \quad (9.8a)$$

$$E_B(t) = E_B^0 e^{-i\omega_0(t-t_{\rightarrow})} \quad (9.8b)$$

with $t_{\leftarrow}$ and $t_{\rightarrow}$ being respectively the time for the left and right polarized signals to cross the full fiber. At the output of the fiber, the phase shift between the two electric fields is just given by

$$\Delta\phi(t) = \omega_0(t_{\leftarrow} - t_{\rightarrow}) \quad (9.9a)$$

where

$$t_{\leftarrow} = \frac{L}{c} - \int_{0}^{L/c} \frac{\delta c_{\leftarrow}}{c} \sin(\omega_a t + \Phi) \quad (9.9b)$$

$$t_{\rightarrow} = \frac{L}{c} - \int_{0}^{L/c} \frac{\delta c_{\rightarrow}}{c} \sin(\omega_a t + \Phi), \quad (9.9c)$$
where the amplitudes $\delta c_{\to} = -\delta c_{\leftarrow}$ are given in Eq. (9.1c). Then, it is straightforward to find the amplitude of the phase shift as

$$|\Delta\phi(t)| = \frac{2\sqrt{16\pi G\rho_{DM}E_P g_{a\gamma}}}{\omega_a c} \left|\sin\left(\frac{\omega_a L}{2c}\right)\right|. \qquad (9.10)$$

Similarly as in the previous section, we will use this result to derive the sensitivity of an optical fiber to $g_{a\gamma}$ in Chapter 18.
# Chapter 10

## Search for dark photons in a microwave cavity with Rydberg atoms

As mentioned in Section 8.3.2, an oscillating DP coupled to the standard electromagnetic field will induce a small oscillating electric field in a vacuum. One very peculiar feature of this electric field is that it does not propagate, i.e. its wave-vector vanishes $\vec{k}_U = 0$ (to first order in $v_{DM}/c$). This is due to the fact that this electric field is induced by a massive vector field and therefore its dispersion relation is given by Eq. (7.7b). In this chapter, we will show how an electromagnetic cavity can be used to search for the electric field induced by a DP. In addition, we will show how to use atoms as a tool to detect this electric field through the Stark effect, i.e. the displacement of the energy levels of an atom under a perturbation by a static electric field (or by an electric field whose frequency is much lower than the transition frequency of the atom). This experimental proposal to search for DP using atoms inside a microwave cavity is the object of a published article [10].

There are mainly two reasons to consider a cavity as an experiment to search for DP. First of all, as for other DP experiments using resonators, the mirrors of the cavity will enhance the electric field induced by the DP. Indeed, the electric field parallel to the surface of a perfect conductor has to vanish. Therefore, because of the presence of the oscillating DP-induced electric field, the mirror will generate a standard electromagnetic field that will propagate perpendicularly to the mirror and whose amplitude is such that it will cancel the DP-induced field parallel to the surface. Physically, the DP-induced electric field will induce an oscillation of the electrons within the mirror which will create a standard electromagnetic field. Since a cavity consists in two mirrors, this boundary condition can, under some conditions, produce resonances that will significantly enhance the small DP-induced electric field.

The second reason to consider a cavity is related to the use of atoms to measure the electric field inside the cavity through the Stark effect, which is sensitive to the square of the electric field. If one applies a standard electromagnetic wave inside the cavity (whose electric field will be denoted by $\vec{E}_A$) the DP contribution to the square of the electric field inside the cavity is $\sim \vec{E}_{DM} \cdot \vec{E}_A$ (to first order in $\chi$, the kinetic mixing between EM and DP), which can also be enhanced by a resonant $\vec{E}_A$. In addition to enhancing the amplitude of the signal to be measured, applying an external field is also important to produce a signal at low, but non-zero, frequency, where the Stark effect can be realistically measured. More precisely, for a cavity whose length is of the order of a few cm, we will be interested in searching for DP oscillating at a frequency $f_U = \omega_U/2\pi$ of the order of a few GHz. The difficulty is that
such a rapid oscillation of the atomic transition frequency will be very hard to measure as the interrogation cycle of the atoms is much longer. But, if one applies en external field at a angular frequency $\omega_A$ which is close to $\omega_U$, then the cross term between the DP electric field and the applied electric field will have a component oscillating at the low angular frequency $|\omega_U - \omega_A|$, which can be measured by the atoms.

As in the previous chapter, we will compute the expected signal, leaving the experimental parameters free. Afterwards, in Chapter 17, we will discuss the sources of noise of such experiment, and in Chapter 18, we will derive the expected sensitivity of the experiment to the $\chi$ coupling, by considering specific experimental parameters.

## 10.1 Contributions from dark photons and applied electric field inside the cavity

In this section, we derive the electric field induced by both the DP and the applied field at the center of the cavity. We summarize the methodology, the derivation and discuss the main results.

The finesse $\mathcal{F}$ of a cavity is a measure of the narrowness of resonances $\delta f$ compared to their frequency. More precisely, it is defined as the ratio between $\delta f$ and the frequency separation of resonances $\Delta f$

$$ \mathcal{F} = \frac{\Delta f}{\delta f}, \qquad (10.1) $$

which is frequency independent. Considering a cavity consisting of two flat mirrors of reflectivity $r$ separated by a distance $L^1$, we can relate the reflectivity and the finesse of a cavity through ($\mathcal{F} \gg 1$)[11]

$$ r \approx 1 - \frac{\pi}{2\mathcal{F}}. \qquad (10.2) $$

While the finesse is usually associated to an optical cavity, it is the quality factor $Q$ that is a more common parameter for microwave cavities. The quality factor is usually defined as the ratio between the energy stored inside the cavity with the energy dissipated. It then becomes obvious that finesse and quality factor are two quantities profoundly connected. The quality factor can also be defined as [11]

$$ Q = \frac{f}{\delta f}, \qquad (10.3) $$

Although being frequency dependent, one can notice that the quality factor of the first resonance of the cavity corresponds exactly to the finesse. In the following of this chapter, we will assume only the first resonances of our microwave cavity, such that $\mathcal{F} \sim Q$.

The principle of calculation is similar for both the DP and the applied electric fields and has already been used in Chapter 9.2, inspired from [12]. The idea is to propagate the electric field an infinite number of round trips inside the cavity and to sum these infinite number

<footnote n="1">¹For simplicity, we assume that the transverse size of the mirrors is $\gg L, \lambda$, where $\lambda$ is the wavelength of the fields of interest.</footnote>
of contributions at a given location. To perform cavity round trips, the field is propagated along one direction and when it reaches a mirror, its amplitude is multiplied by $-r$ and its wave-vector is flipped. For $r < 1$, the infinite sum converges and can be calculated explicitly.

<figure><img src="image_7.png" /><figcaption>Figure 10.1: Experimental scheme that we propose for the search of the DP-photons coupling: an external field (in blue) is applied at the cavity edge. The standing DP electric field (in yellow) generates a propagating electric field inside the cavity (in orange). At the center of the cavity, the transition frequency of Rydberg atoms is impacted by $|E|^2$ through the Stark effect (see text.).</figcaption></figure>

First, let us apply this procedure to the applied external field. We assume that the external electric field is applied on the left edge of the cavity, see Fig. 10.1. Ideally, the applied field can be parameterized as $\vec{E}_A = \vec{X}_A \mathfrak{R} [e^{-i(\omega_A t - k_A(x + \frac{L}{2}) + \phi_A)}]$, with its amplitude $X_A$, angular frequency $\omega_A$ and phase $\phi_A$. Still assuming a transmission coefficient of the mirror $\sqrt{1-r^2}$, the first contribution at the center of the cavity reads

$$ \vec{E}_A^0(x = 0, t) = \mathfrak{R} \left[ \sqrt{1 - r^2} \vec{X}_A e^{-i(\omega_A t - k_A \frac{L}{2} + \phi_A)} \right]. \quad (10.4) $$

This contribution propagates until the other cavity boundaries, gets reflected once with coefficient $-r$ such that boundary conditions are respected, then comes back to the center, implying that the second contribution reads

$$ \vec{E}_A^1(x = 0, t) = -r \vec{E}_A^0 \left( x = 0, t + \frac{L}{c} \right) = \mathfrak{R} \left[ -r \sqrt{1-r^2} e^{ik_A L} \vec{X}_A e^{-i(\omega_A t - k_A \frac{L}{2} + \phi_A)} \right], \quad (10.5) $$

the additional phase $e^{ik_A L}$ shows the time delay of $E^1$ compared to $E^0$ after half a round-trip. This occurs several times and after an infinite number of round trips $N$, the full contribution
of the external applied field inside the cavity is

$$ \vec{E}_A^{\text{tot}}(x=0,t) = \sum_{n=0}^{N=+\infty} \vec{E}_A^n(x=0,t) \qquad (10.6a) $$

$$ = \sqrt{1 - r^2} \vec{X}_A \mathfrak{R} \left[ e^{-i(\omega_A t + \phi_A)} \frac{e^{i \frac{k_A L}{2}}}{1 + r e^{ik_A L}} \right] \qquad (10.6b) $$

$$ \equiv \vec{A}(\omega_A) \cos(\omega_A t + \phi_A) + \vec{B}(\omega_A) \sin(\omega_A t + \phi_A), \qquad (10.6c) $$

where we assumed $r < 1$ such that $r^N \to 0$ and with

$$ \vec{A}(\omega_A) = \frac{\vec{X}_A (1+r) \sqrt{1-r^2} \cos\left(\frac{\omega_A L}{2c}\right)}{1 + 2r \cos\left(\frac{\omega_A L}{c}\right) + r^2}, \qquad (10.7a) $$

$$ \vec{B}(\omega_A) = \frac{\vec{X}_A (1 - r) \sqrt{1 - r^2} \sin\left(\frac{\omega_A L}{2c}\right)}{1 + 2r \cos\left(\frac{\omega_A L}{c}\right) + r^2}. \qquad (10.7b) $$

One can notice a resonance at the center of the cavity for even modes of the cavity, as expected.

Let us now focus on the contribution from the DP field. From Eqs. (8.63c), one can, without loss of generality, write the expression of the electric field related to the DP as $\vec{E}_{DM} = \vec{X}_{DM} \mathfrak{R} [e^{-i\omega_U t}] = X_{DM} \hat{e}_{DM} \mathfrak{R} [e^{-i\omega_U t}]$ where $\hat{e}_{DM}$ is a unit vector characterizing the polarization of the DP field and $X_{DM} = \chi c \sqrt{2\mu_0 \rho_{DM}}$ (from Eq. (8.63d)), and where the phase of the field $\Phi$ disappeared because it can be reabsorbed in $\phi_A$ when computing the interference between the DP and applied fields. Because of this electric field, both mirrors will generate a propagating standard electromagnetic field such that the total component of the electric field parallel of the mirrors' surface vanishes. The same procedure as above can be realized to know the DM electric field amplitude at the center of the cavity. The subtleties of this calculation are that: 1) the field is emitted by the mirrors towards the center of the cavity, therefore the transmission coefficient $\sqrt{1-r^2}$ factor is not present; 2) only the DM polarization transverse to the mirror is re-emitted, noted $\vec{X}_{DM,\parallel}$; and 3) there are two different contributions, in phase, each being emitted from one of the edges of the cavity. The total DM contribution at the center is then

$$ \vec{E}_{DM}^{\text{tot}}(x=0,t) = \mathfrak{R} \left[ \vec{X}_{DM} e^{-i\omega_U t} + 2\vec{X}_{DM,\parallel} e^{-i(\omega_U t - \frac{k_L L}{2})} \frac{1}{1+r e^{ik_L}} \right] \qquad (10.8a) $$

$$ \equiv \vec{C}(\omega_U) \cos(\omega_U t) + \vec{D}(\omega_U) \sin(\omega_U t), \qquad (10.8b) $$

with

$$ \vec{C}(\omega_U) = \vec{X}_{DM} + \frac{2\vec{X}_{DM,\parallel}(1+r)\cos\left(\frac{\omega_U L}{2c}\right)}{1 + 2r\cos\left(\frac{\omega_U L}{c}\right) + r^2}, \qquad (10.9a) $$

$$ \vec{D}(\omega_U) = \frac{2\vec{X}_{DM,\parallel}(1-r)\sin\left(\frac{\omega_U L}{2c}\right)}{1 + 2r\cos\left(\frac{\omega_U L}{c}\right) + r^2}. \qquad (10.9b) $$
The first term of Eq. (10.8a) corresponds to the background oscillating DM field at the center, which is always present, even without the cavity. The second term is the DM contribution from the cavity, which is almost equivalent, in its form, to the total contribution of the applied field, with an additional factor two, due to the emission of a field from both edges of the cavity (instead of only one for the applied field).

The calculations presented in this section have been carried out to leading order in $O(\chi)$. In particular, at each interaction between the EM waves and the mirrors a small quantity of EM energy will be transformed into DP. The amplitude of such a process is proportional to $\chi$ and therefore neglected as it contributes terms of order $O(\chi^2)$. Furthermore the corresponding energy loss is much smaller than the one coming from the finite reflection coefficient $r$.

## 10.2 Total electric field squared

As mentioned in the beginning of this chapter, the main idea of the experiment proposed is to detect the hypothetical electric field induced by the DP field by using atoms to measure it through the quadratic Stark effect. The Stark effect consists in a shift in the energy levels of an atom under the perturbation of a static (or slowly evolving<fnref n="2" />) electric field, and is given by [13]

$$ \Delta\nu = -\frac{\Delta\alpha}{2h} |\vec{E}|^2, \qquad (10.10) $$

at quadratic order in the electric field, thus the quadratic Stark effect, where $h$ is the Planck constant, $\Delta\nu$ is the frequency shift induced by the slowly evolving electric field $\vec{E}$ and $\Delta\alpha$ is the differential polarizability of the atomic transition considered. Taking into account both contributions from the applied electric field and the DP field computed previously, the total electric field power at the center of the cavity is

$$ |\vec{E}(\omega_U, \omega_A)|^2 = |\vec{E}_A^{\text{tot}} + \vec{E}_{\text{DM}}^{\text{tot}}|^2 = (\vec{A}(\omega_A) \cdot \vec{C}(\omega_U) + \vec{B}(\omega_A) \cdot \vec{D}(\omega_U)) \cos(\Delta\omega t + \phi_A) + (\vec{B}(\omega_A) \cdot \vec{C}(\omega_U) - \vec{A}(\omega_A) \cdot \vec{D}(\omega_U)) \sin(\Delta\omega t + \phi_A) + \text{constant and fast oscillating terms}, \quad (10.11) $$

with $\Delta\omega = \omega_A - \omega_U$. In the following, we will not consider the constant terms. Indeed, in the experimental scheme proposed here, we will be interested in the oscillatory behavior of the atomic frequencies. Moreover, we discarded the fast oscillating terms whose angular frequencies are $2\omega_A$, $2\omega_U$ or $\omega_A + \omega_U$, with periods ($O(10^{-9})$ s) much shorter than the atom interrogation time, such that on average, their impact vanishes.

From Eq. (10.11), the signal amplitude can be written as (see Appendix A.2 for more details)

$$ \frac{\sqrt{1 - r^2} X_A X_{\text{DM}} \beta}{\sqrt{1 + 2r \cos(\frac{\omega_A L}{c}) + r^2}} \sqrt{1 + 4 \frac{1 + (1 + r) \cos(\frac{\omega_U L}{2c})}{1 + 2r \cos(\frac{\omega_U L}{c}) + r^2}} \equiv \chi S(\omega_U, \omega_A; \rho_{\text{DM}}, X_A; L, r), \quad (10.12a) $$

<footnote n="2">²As long as the angular frequency of oscillation is much smaller than the atomic transition angular frequency from state k to i, $\Delta\omega \ll \omega_{ik}$.</footnote>
where

$$ \beta = \hat{e}_{\text{DM}} \cdot \frac{\vec{X}_A}{X_A}, \qquad (10.12b) $$

and $X_{\text{DM}} \propto \chi$, following Eq. (8.63d).

If the polarization is fixed and does not change each coherence time, $\beta = \cos \theta$, with $\theta$ the angle between $\vec{X}_A$ and $\vec{X}_{\text{DM}}$. If the DP field is isotropically distributed $\beta = 1/\sqrt{3}$. To avoid any orthogonality between the two polarizations, i.e $\beta = 0$, one must run the experiment for a significant time, i.e at least several days (see discussion in [14]). In Fig. 10.2 are shown 3D

<figure><img src="image_8.png" /><figcaption>Figure 10.2: Signal contribution Eq. (10.12a) (arb. units) as function of $\omega_U$, $\omega_A$, both presented in units of $c/L$. The resonance peaks appear clearly when both frequencies correspond to an odd mode of the cavity.</figcaption></figure>

plots of the signal contribution Eq. (10.12a) as function of the angular frequencies of the DM field $\omega_U$ and the applied field $\omega_A$. One can notice the various resonance peaks when both frequencies correspond to an odd mode of the cavity. In the experiment scheme presented here, we require the two frequencies to be close, hence we are only interested in regions where $\omega_U \sim \omega_A$. This corresponds to the diagonal of the plot seen from the top. However, in other types of experiment where both frequencies could be separated by some hundreds of MHz or more, other regions of this plot could be studied.

The amplitude of the signal is linear in $\chi$ and depends non-linearly on the local DM density, on the amplitude and frequency of the applied external field and on the mass ($m_U c^2 = \hbar \omega_U$) of the DP field. It depends also on some geometric factor of order unity that characterizes how the detector couples to the DP field polarization.

For DM and applied frequencies close to odd resonances of the cavity, i.e $\omega_U L/c \approx \omega_A L/c \approx \pi + 2n\pi$, with $n \in \mathbb{N}$, the amplitude Eq. (10.12a) becomes proportional to the quality factor $Q$, as expected. Additionally, as it is also proportional to the injected amplitude $X_A$, we directly see the interesting feature of looking at the field squared amplitude, the applied field acting as an amplification for the DM field.

Finally, the signal we are interested in consists in atomic frequency oscillations with respect to an unperturbed reference

$$ v(t) = v_0 + \Delta v \cos (\Delta \omega t + \phi), \qquad (10.13a) $$
whose amplitude is given by

$$ \Delta\nu = -\chi \frac{\Delta\alpha S(\omega_U, \omega_A; \rho_{DM}, X_A; L, r)}{2h}, \qquad (10.13b) $$

which depends on both the applied external field and the DP field.

## 10.3 Measurement with atoms through the quadratic Stark effect

### 10.3.1 Rydberg atoms

The transition frequency measurement can be performed using a regular atomic clock, which allows very good uncertainty on the frequency measurement, but is not very sensitive to the Stark effect e.g. for the $5s^2 \text{ } ^1S_0 \rightarrow 5s5p \text{ } ^3P_0$ clock transition in Sr the differential polarizability $\Delta\alpha/2h \approx 3.1 \times 10^{-6}$ Hz/(V/m)$^2$ [15], thus requiring a strong applied field.

To overcome this lack of sensitivity to $E^2$, “regular” atoms can be replaced by Rydberg atoms, which are in a quantum state with high principal quantum number $n$ [16]. The electrons are much further from the nucleus thus the atom has much higher polarizability. For large $n$ the corresponding polarizabilities in Sr can reach $\Delta\alpha/2h \approx 10^5$ Hz/(V/m)$^2$ ($n > 60$) [17].

Typically, Sr Rydberg measurements use laser cooled Sr atoms that are excited to Rydberg states using two photon transitions [17, 18] and direct spectroscopy of e.g. the $5s5p \text{ } ^3P_1 \rightarrow 5sns \text{ } ^3S_1$ transition is performed, with $n$ up to 81 [17, 18]. We give a more explicit summary of the measurement process done in [18]. The atoms are initially in a given atomic state $|i\rangle$ and they transition to a Rydberg state $|r\rangle$ using a UV beam whose frequency (of the order of some hundreds of THz) is continuously controlled to match the transition frequency, that can be perturbed by, e.g., environmental noise. More explicitly, this UV beam is locked on a reference cavity, whose length is modified using a piezo-tuning. This spectroscopic measurement allows one to determine the Rydberg transition frequency with or without the effect of the DP-EM coupling. Now, let us add the effect of such coupling. The quadratic electric field induces a Stark shift on the Rydberg state, implying that at a given time $t$, the transition frequency corresponds to $v(t)$ from Eq. (10.13a). Therefore, in order to maximize the transition probability, the UV light will continuously follow $v(t)$, and therefore it will also slowly oscillate at $\Delta\omega$. Afterwards, when the electrons have transitioned to the Rydberg state, another laser pulse is used to ionize them and a small electric field is used to drift the ions to a metallic plate and count them. By counting the ions, one can infer the number of atoms which made the transition to the Rydberg state and therefore measure their transition frequency from the UV beam.

### 10.3.2 Bandwidth of detection

In order to see the oscillations at $\Delta\omega$ in the total field power Eq. (10.11), we require the DM and the applied fields to have different frequencies. We also require the Nyquist frequency of the apparatus to be higher than the angular frequency of the signal $\Delta\omega < \pi f_s$ to be able to detect any oscillatory behavior in the transition frequency of the atoms.
For a given applied angular frequency $\omega_A$, sampling frequency $f_s$, Rydberg atoms perform the measurements of the electric field squared during $T_{obs}$ at an angular frequency $\Delta\omega$ (more precisely, $T_{obs} \times f_s$ measurements will be taken for each $\omega_A$). The time $T_{obs}$ is arbitrary; if it is longer than the measurement process comprising excitation and ionisation, one has to prepare again the atoms to their Rydberg state after de-excitation accordingly. As detailed above, the experiment is sensitive to any $\Delta\omega$ such that $2\pi/T_{obs} \le |\Delta\omega| \le \pi f_s$, which, in terms of DM angular frequency, is equivalent to

$$ \omega_U \in [\omega_A - \pi f_s; \omega_A + \frac{2\pi}{T_{obs}}] \cup [\omega_A + \frac{2\pi}{T_{obs}}; \omega_A + \pi f_s]. \quad (10.14) $$

This consists in one measurement sequence which will probe the range of DP masses corresponding to Eq. (10.14). The logic of the experiment is then to shift the frequency of the applied electric field to scan another part of the mass range. By applying this procedure for various well chosen $\omega_A$, one can scan a large part of the DP mass range.

This scheme can be repeated N times, as much as time allows. At the end of this loop, the total experimental time is simply $T_{tot} = N T_{obs}$. The corresponding total DM frequencies band scanned is $f_{tot} = N f_s = T_{tot} f_s / T_{obs}$. The larger the total experimental time, the larger the band of scanned DM frequencies. Additionally, the blind spots at exact $\omega_A$ can be avoided, and sensitivity can be optimised (see below) by shifting $\omega_A$ by a little less than $2\pi f_s$, at the expense of increasing the overall experimental duration.

For DM frequencies in the GHz range, the coherence time of the DM field Eq. (5.11) $\tau(\omega_U/2\pi) \sim 10^{-3}$ s, i.e it will be considered much smaller than $T_{obs}$, of the order of several seconds.

Here, we have presented the experimental scheme of an experiment involving Rydberg atoms inside a microwave cavity for the detection of the EM-DP coupling $\chi$. In Chapter 17, we will discuss the sources of noises of the apparatus, which will allow us to derive an expected sensitivity of the experiment in Chapter 18.
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